3D magnetic steering wheel angle and suspension travel detection
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This document describes a thesis submitted by Bruno Paes Spricigo to the Federal University of Santa Catarina in Brazil. The thesis proposes using a 3D magnetic sensor to simultaneously measure steering wheel angle and suspension travel in vehicles. This could increase safety by providing more data inputs for systems like traction control and stability control. The document outlines the objectives, structure, background topics on suspensions, steering, and magnetic position sensing, and the proposed approach of using magnetic fields around a sensor to calculate linear and angular displacement. Details are provided on the materials used, including software, microcontrollers, sensors and magnets. The approach aims to decrease sensor overhead while improving measurement precision for smart suspension applications.
![LIST OF FIGURES
Figure 1 – Unibody and chassi vehicle. . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 2 – Dampened Oscillation Movement . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 3 – Vehicle Perception by Passenger . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 4 – Rack-pinion steering box . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 5 – A front-wheel-steering vehicle and the Ackerman condition . . . . . . . . . 30
Figure 6 – Lorentz Force and Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 7 – Angular measurement with magnet sensor . . . . . . . . . . . . . . . . . . 37
Figure 8 – Linear measurement with magnet sensor . . . . . . . . . . . . . . . . . . . 38
Figure 9 – Linear movement into angular . . . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 10 – Linear and angular measurements combined . . . . . . . . . . . . . . . . . 39
Figure 11 – Epson E2C351S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Figure 12 – Arduino Leonardo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Figure 13 – Sensor TLV493D-A1B6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Figure 14 – Sensor axes of measurement . . . . . . . . . . . . . . . . . . . . . . . . . 44
Figure 15 – ProJet 3510 HDPlus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 16 – Printing Materials Properties . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 17 – Two Circles Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Figure 18 – Mechanism calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Figure 19 – Linear fitting of C(β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 20 – 3rd degree polynomial fitting of C(β) . . . . . . . . . . . . . . . . . . . . 51
Figure 21 – 5th degree polynomial fitting of C(β) . . . . . . . . . . . . . . . . . . . . . 52
Figure 22 – Time test for polynomial approximation . . . . . . . . . . . . . . . . . . . 52
Figure 23 – PCB holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Figure 24 – System cut and partial assembly . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 25 – Final assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Figure 26 – Linear position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Figure 27 – Angular position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Figure 28 – Combination of movements . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Figure 29 – Magnetic field simulation in sensor plane . . . . . . . . . . . . . . . . . . . 57
Figure 30 – Magnetic field simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Figure 31 – Linear position simulation at ∆ = 2 mm . . . . . . . . . . . . . . . . . . . 58
Figure 32 – Linear position simulation at ∆ = 5 mm . . . . . . . . . . . . . . . . . . . 59
Figure 33 – Linear position simulation with noise . . . . . . . . . . . . . . . . . . . . . 60
Figure 34 – Contour Plots for dout and βout . . . . . . . . . . . . . . . . . . . . . . . . 61
Figure 35 – Contour Plots for [dout]2
+ [βout]2
. . . . . . . . . . . . . . . . . . . . . . . 61
Figure 36 – Robot Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-13-320.jpg)
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Figure 2 – Dampened Oscillation Movement
0 1 2 3 4 5 6
Time [s]
−0.020
−0.015
−0.010
−0.005
0.000
0.005
0.010
0.015
0.020
Amplitude[m]
Simple Harmonic Motion
Under-damping ζ = 0.3
Critical-damping ζ = 1.0
Over-damping ζ = 2.5
Source: Author, 2016
predicted. Only vertical dynamic behavior can be studied with quarter vehicle model, so horizon-
tal (braking and accelerating dynamics) and lateral studies (handling and dynamics of a vehicle
under cornering) are neglected. Nevertheless, the quarter vehicle "[. .. ]contains the most basic
features of the real problem and includes a proper representation of the problem of controlling
wheel and wheel-body load variations." (JAZAR, 2009). It is also the initial model that is used
to begin estimating the spring stiffness and damping coefficient needed to achieve the expected
dynamic behavior. This behavior must consider that certain frequencies e.g., the range from 1 to
80 Hz, are more critical because they affect the human body, causing some kind of discomfort
(REIMPELL; STOLL; BETZLER, 2001).
A vehicle is a very complex dynamic system that only exhibits vibration in consequence
to excitation inputs (GILLESPIE, 1992), thus, its response properties are responsible by the
vibration’s magnitude and direction affecting the passengers’ perception of ride/vehicle as seen
in Figure 3.
Figure 3 – Vehicle Perception by Passenger
Excitation
Sources
Road Roughness
Tire/Wheel
Driveline
Engine
Vehicle
Dynamic
Response
Vibrations
Ride
Perception
Source: Adapted from Gillespie (1992).](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-27-320.jpg)
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r1 can be defined. h is then calculated by:
h = r2
1 − l2 (7)
The x and y coordinates of B are then calculated:
x =
l
d
(x2 − x1) ±
h
d
(y2 − y1) + x1 (8)
y =
l
d
(y2 − y1)
h
d
(x2 − x1) + y1 (9)
Two coordinates of x and y are presented in Equation 8 and in Equation 9, because
there are two points of intersection between two circles. In this case though, only one set of
coordinates is important and the lowest point is the chosen to be the connection between the two
rods. With the coordinates of point A and B it is possible to calculate the β angle of sensor as
seen in Equation 10.
β = arctan
xB
yB
(10)
The graphical interface shown in Figure 18 is of a calculator developed using Python to
verify the range of angular output when varying the position of point C, the length of r1 and r2
and the steer movement. It allows a preciser design of the parameters and a better knowledge of
the expected output. A larger version of this figure is shown in Appendix A.
Figure 18 – Mechanism calculator
−250 −200 −150 −100 −50 0 50 100
[mm]
−150
−100
−50
0
50
[mm]
Sensor arm movement
Mechanism
−80 −60 −40 −20 0 20 40 60 80
Steering linear movement [mm]
−60
−40
−20
0
20
40
60
β:sensorangle[◦
]
Linear vs rotatory movement
Sensor Arm [mm]
100.00
Connection Rod [mm]
175.00
Steer [mm]
-30.00
Cx [mm]
-174.71
Cy [mm]
-90.00
Reset Values
Source: Author, 2016
Through the calculator some parameter values were defined as shown:
• Sensor arm: 100 mm
• Connection rod: 175 mm
• Point C coordinates: [174.71, -90] mm](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-51-320.jpg)
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The value of Cx is more precise because it was adjusted so when the steering wheel is at
0◦
, the sensor is also at 0◦
. Assuming a range of 150 mm of rack linear travel, the output of the
sensor is a range of 97.6◦
varying from -47.6◦
to 50.0◦
, when the steering wheel is completely to
the left and right side, respectively. This asymmetry is not a problem because the output angle
can be translated to rack linear movement and, with the steering ratio, the steering angle is easily
found. So finding the range of β is important to ensure that there is enough range of sensor
rotation to map the whole steering wheel positions range. But the mechanism’s main function is
to find the steering angle using the sensor angle β as input, and it can be done by first calculating
point B using Equation 11:
B = [−r1 sin β, −r1 cos β] (11)
Then, the value of linear rack travel can be calculated if point B and the coordinate y of
point C are known, through:
r2
1 = (xB − xC)2
+ (yB − yC)2
(12)
Since r1, yB, yC and xB are known and invariable, it is easier to create a constant k to
group all of this values:
−x2
C − 2xCxB = −x2
B + r2
1 − (yB − yC)2
(13)
And then:
k = −x2
B + r2
1 − (yB − yC)2
(14)
So, applying Equation 14 in Equation 13 and multiplying it by -1 gives:
x2
C + 2xCxB − k = 0 (15)
That can be easily solved to find two solutions to xC, where the smallest is the actual
value of the instant rack linear movement, this method needs a previous calculation of point
B and an input of the coordinate y of C though. An approach of curve fitting can make this
an one-step calculation, therefore reducing the time to translate β into rack movement, while
keeping the error below a threshold. Both methods still give the position of the rack as output,
which needs to be multiplied by the steering ratio to get steering wheel angle.
C(β) = 1.6095β − 172.60 (16)
A first linear approximation is shown in Figure 19 using Equation 16, presenting the
comparison between the curve of C as a function of β using the previous approach and a linear
fit. The red curve presents the error between the two curves, with a peak of more than two percent
when the angle is greater than -40◦
. The R2
value is 0.9968, which is good for some applications,](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-52-320.jpg)
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but in this experiment (together with some over 2% peaks or error) is not acceptable because this
is a systematic error that influences every measurement and result.
Figure 19 – Linear fitting of C(β)
−40 −30 −20 −10 0 10 20 30 40
Beta Angle [◦
]
−250
−230
−210
−190
−170
−150
−130
−110
−90
RackTravel[mm]
−2
−1
0
1
2
Error[%]
C vs β
C vs β - 1st degree Approximation
Error
Source: Author, 2016
C(β) = −8e−5
β3
+ 2.1e−3
β2
+ 1.736β − 170.67 (17)
A third degree polynomial equation can be fitted to the original curve to decrease
the error. The Equation 17 is shown in Figure 20 together with the comparison curve and the
maximum error, that in this case is slightly smaller than two percent. Even though the maximum
error is still big, the average error is not, fluctuating in between -0.3% and 0.3% for most of the
values. The R2
value in this case is 0.9999531.
Figure 20 – 3rd degree polynomial fitting of C(β)
−40 −30 −20 −10 0 10 20 30 40
Beta Angle [◦
]
−250
−230
−210
−190
−170
−150
−130
−110
−90
RackTravel[mm]
−2.0
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Error[%]
C vs β
C vs β - 3rd degree Approximation
Error
Source: Author, 2016
C(β) = 6e−10
β5
+ 7e−7
β4
− 9e−5
β3
− 3.5e−3
β2
+ 1.7449β − 170.37 (18)](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-53-320.jpg)
![52
Using a fifth degree polynomial fit naturally produces less error values. The Equation 18
is shown in Figure 21 and both curves fit so well that the two lines look one. The R2
value in this
case is greater than 0.9999999. The maximum error is around 0.25% but almost the whole range
of angles has almost zero error.
Figure 21 – 5th degree polynomial fitting of C(β)
−40 −30 −20 −10 0 10 20 30 40
Beta Angle [◦
]
−250
−230
−210
−190
−170
−150
−130
−110
−90
RackTravel[mm]
−2.0
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
Error[%]
C vs β
C vs β - 5th degree Approximation
Error
Source: Author, 2016
Increasing the degree of the polynomial fit would continuously decrease the error, but
would also increase the processing time. A small test was performed with an algorithm in Python
to test the time needed to process different amount of points using the direct method and the
equations fitting. Figure 22 shows the amount of time needed for each one of these equations. As
expected the behavior when increasing the number of points is linear, the results in Figure 22 are
plotted in a log-log plot though, because the number of points to process also increased by the
power of 10. Results smaller than 10−3
s or 1 ms were not caught by the algorithm, but the plot
still shows clearly the ratios between each method.
Figure 22 – Time test for polynomial approximation
101
102
103
104
105
106
107
Number of points
10−4
10−3
10−2
10−1
100
101
102
103
Processingtime[s]
Direct method
Linear fit
3rd degree polynomial
5th degree polynomial
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-54-320.jpg)
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Figure 30 – Magnetic field simulation
−30 −20 −10 0 10 20 30
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx(0.0)
By(0.0)
Bz(0.0)
Btan
Bamp
(a) Magnetic field simulation
−30 −20 −10 0 10 20 30
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx(45.0)
By(45.0)
Bz(45.0)
Btan
Bamp
(b) Magnetic field simulation (45◦)
Source: Author, 2016.
Figure 30a shows the components of magnetic field together with the calculated values
of Btan and Bamp. This scenario occurs when the sensor and magnet are aligned and β = 0◦
and d = 0. In Figure 30a all the components are symmetric because this simulation shows the
ideal behavior of the system. The Bx component is constant and null because β = 0◦
, which is
expected since an input of Bx = 0 in Equation 20 results in β = 0◦
. Figure 30b shows the same
magnetic setup of Figure 30a, but with β = 45◦
. In this case, Bx and By components have the
same value as expected. In both cases, Bz, Btan and Bamp are the same because these values are
not affected by β.
Several cases of building misalignments were simulated to help the calibration process
of the system. To predict the behavior or to compare the experimental results, some simulations
were performed introducing noise or displacements, this allows a better understanding of the
physical system and which components and tolerances are more important.
Figure 31 – Linear position simulation at ∆ = 2 mm
−30 −20 −10 0 10 20 30
Travel [mm]
−40
−20
0
20
40
MagneticField[mT]
Bx
By
Bz
−30 −20 −10 0 10 20 30
Travel [mm]
−3
−2
−1
0
1
2
3
CalculatedTravel[mm]
−15
−10
−5
0
5
10
15
Error[%]
Calculated travel
Linearization of slope
Error
Source: Author, 2016
Results from the simulation show that the linear position measurement through Equa-
tion 19 works better if the air gap ∆ increases. When ∆ is too small, i.e., ∆ < 2 mm, the K](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-60-320.jpg)
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factor of the Equation 19 has less effect and the result is less linear for every case, as shown in
Figure 31. The Figure 31 shows the linear position calculation along with a linearization of the
output. The maximum error is 12.28 % and the R value is R = 0.99919. The error formula used
in this approach divides the absolute deviation Di by the value of d, as shown in Equation 25.
Di = |xi − m(X)|;
Error = Di
xi
· 100
(25)
Where Di is the absolute deviation, xi is the calculated travel value d, m(X) is the
central tendency (the linearization) and the Error value is the error in [%].
Figure 32 – Linear position simulation at ∆ = 5 mm
−30 −20 −10 0 10 20 30
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−30 −20 −10 0 10 20 30
Travel [mm]
−3
−2
−1
0
1
2
3
CalculatedTravel[mm]
−4
−3
−2
−1
0
1
2
3
4
Error[%]
Calculated travel
Linearization of slope
Error
Source: Author, 2016
The opposite case, where ∆ ≥ 2mm, is easily linearized with the K value, in Figure 32
a simulation with ∆ = 5 mm is shown. The maximum error is −3.84 % and the R value is
R = 0.99994.
It is also visible the decrease of the maximum value of magnetic field from Figure 31 to
Figure 32.
Both results in Figure 31 and Figure 32 should be multiplied by a factor to translate the
real values of displacement.
The introduction of small noises in the simulation also influences the result making it
less stable. The same simulation with ∆ = 5 mm with noise inserted is shown in Figure 33. The
maximum error peaks over 100 % and the R value is 0.99941.
The angular position detection through Equation 20 works properly if the system is
aligned, but simulations revealed that some parameters influence the precision of the angular
position calculation. Misalignments between sensor and magnet can be harmless if they occur
in y and z components, but in the sensor’s x axis direction any small displacement makes the
system really inaccurate, therefore the calculation of the beta angle would only be possible to
values of d close to 0.
To improve the precision of the angular measurement two small modifications were
proposed in Equation 20: application of a constant value Krot to adjust the curve and make](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-61-320.jpg)
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Figure 33 – Linear position simulation with noise
−30 −20 −10 0 10 20 30
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−30 −20 −10 0 10 20 30
Travel [mm]
−3
−2
−1
0
1
2
3
CalculatedTravel[mm]
−100
−50
0
50
100
Error[%]
Calculated value
Linearization of slope
Error
Source: Author, 2016
it more linear or using the linear position result f(Spos) as an input. The results of the first
approach were still inaccurate. The best solution was to use the linear position result as input in
a polynomial equation inside Equation 20 as shown in Equation 26.
Sang = arctan
Bx
By
+ f(Spos) (26)
Where f(Spos) is a polynomial equation in the form f(Spos) = p0S3
pos + p1Spos. The
coefficients p0 and p1 are calculated through an algorithm that tests values to find the ones
delivering the smallest mean squared error. This approach delivers good results, but is too
dependent of the system’s calibration. Any movement or vibration requires a new run of the
algorithm to optimize the coefficients.
5.2.3 Lookup Table Method
Another method to calculate the values β and d is to use a lookup table. This approach
consists of running several sweeps to gather all the values of magnetic field through all the range
of linear displacement and angular motion. With these values of magnetic field, the positions d
and β are calculated using Equation 19 and Equation 26 for the whole range of d and β. Two
two-dimensional arrays were created to store these calculated values of d and β, respectively.
This is important because of the measurement’s systematic error, for example when measuring
d = −10mm and β = 10◦
, the results might be slightly different (some close values but not
exactly d = −10mm and β = 10◦
) because of the misalignments and small noises inserted to
simulate the real experiment.
The goal is to create two functions dout(d, β) and βout(d, β) using d and β as input, so
two interpolation arrays were created using a 3rd
degree spline. Since the spectrum of values is
mapped inside dout and βout, it is possible to get the measurement of a point, use Equation 19
and Equation 20 to find dcalc and βcalc and compare these values to the lookup table to find the](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-62-320.jpg)
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closest values of dout and βout. Also, dout and βout are functions of d and β, so the real values of
d and β can be found. The two contour plots are shown in Figure 34.
Figure 34 – Contour Plots for dout and βout
−10 −5 0 5 10
Displacement d [mm]
−40
−20
0
20
40
Angleofsensorβ[◦]
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0 −10 −5 0 5 10
Displacement d [mm]
−40
−20
0
20
40
Angleofsensorβ[◦]
-45.00 -45.00
-30.00
-15.00
0.00
15.00
30.00
45.00
Source: Author, 2016
To find the real value of d and β, the dout and βout arrays are subtracted by dcalc and βcalc,
respectively, so the new arrays dout and βout are null for dout(dcalc, βcalc) and βout(dcalc, βcalc).
Then, another array is created by adding [dout]2
+ [βout]2
. This array has only one null point,
exactly for dcalc and βcalc.
Figure 35 – Contour Plots for [dout]2
+ [βout]2
−10 −5 0 5 10
Displacement d [mm]
−40
−20
0
20
40
Angleofsensorβ[◦]
10.0
100.0
300.0
300.0
1000.0
1000.0
2000.0
Source: Author, 2016
The black dot comes from a random measurement that returns dcalc = 4 mm and
βcalc = −15◦
, for example. Graphically is possible to get the real values of d and β by getting
the x and y value on Figure 35, but for an automated process an algorithm that finds the roots of
the array was implemented.
5.3 Robot Tests
The robot adopted to perform the tests has four axes of movement: x, y, z and u, where
the first three are translational movement and u is a rotation axis. Although only z is used to](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-63-320.jpg)
![63
Figure 37 – Robot Experiment Sketch
x
y
z Steering
Sensor
Suspesion
x
z
y
u
Robot axes
Sensor axes
Source: Author, 2016
A sketch of the magnet holder is shown in Figure 37, as well as a magnet representation
with its north and south pole. The figure shows the two Cartesian coordinates of the system: the
robot’s and the sensor’s one. The simulated suspension and steering movement are presented
right next to the respective moving elements.
After the calibration of the system, a sweep test delivers a result very close to the
simulated data. The methodology for the following tests is to generate random points of position
d and angle β, where −10 ≤ d ≤ 10 [mm] and −45 ≤ β ≤ 45 [◦
]. The displacement d and
angle β are then calculated through the two methods and compared to the reference values.](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-65-320.jpg)
![65
6 RESULTS
This chapter presents the results from the robot experiments with the magnetic field
readings, the linear and angular positions calculation and the comparison with selected random
points. These first experiments were performed with a sweep moving the magnet on the z axis
of the robot from −15 mm to 15 mm. The air gap value is of ∆ = 3 mm. The results of the
magnetic fields Bx, By and Bz are shown in Figure 38.
Figure 38 – Initial results of magnetic fields Bx, By and Bz
−15 −10 −5 0 5 10 15
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
Source: Author, 2016
The results of the magnetic fields Bx, By and Bz are shown in Figure 39, along with
the calculation of the linear position of the magnet through Equation 19, where the value of K
used was K = 1.06.
Figure 39 – Magnetic fields Bx, By, Bz and the linear position calculation
−15 −10 −5 0 5 10 15
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−15 −10 −5 0 5 10 15
Travel [mm]
−15
−10
−5
0
5
10
15
CalculatedTravel[mm]
−15
−10
−5
0
5
10
15
Error[%]
Calculated travel
Correct value
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-67-320.jpg)
![66
It is also shown in Figure 39 a line from (x1, y1) = (−15, −15) to (x2, y2) = (15, 15),
which represents the correct values. There is also the error line, with a peak when x is close to 0.
The peak happens because, as seen in Equation 25, the absolute deviation Di is divided by d, so
the closer to d = 0, higher the error in [%].
The angular position detection, using the Equation 20, is shown in Figure 40 in a test
performed with β = 0◦
, along with a linearization of the angular calculation result from −7 mm
to 7 mm.
Figure 40 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 0◦
−15 −10 −5 0 5 10 15
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−15 −10 −5 0 5 10 15
Travel [mm]
−4
−2
0
2
4
CalculatedAngle[◦]
−100
−80
−60
−40
−20
0
20
40
Error[%]
Calculated Angle
Linearization of slope
Error
Source: Author, 2016
Because of the setup’s misalignments and construction tolerances, the Bx component is
not null for the whole displacement d, as expected. So, as seen in Equation 20, if Bx = 0, the β
angle will, naturally, not be zero. The angular position results shown in Figure 40 also have two
peaks, located where d is close to the end of the magnet. This happens because at the end of the
magnet, By = 0, so Bx
By
→ ±∞. From this point, the results are shown for d from −10 mm to
10 mm, to avoid showing the areas where the angle calculation is inherently inaccurate.
Figure 41 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 30◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
0
10
20
30
40
50
60
CalculatedAngle[◦]
−4
−3
−2
−1
0
1
2
3
4
Error[%]
Calculated Angle
Linearization of slope
Error
Source: Author, 2016
The angular position detection, using the Equation 20, is shown in Figure 41 in a test
performed with β = 30◦
. As seen in Figure 41, the results for angular position calculation when](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-68-320.jpg)
![67
β = 30◦
are significantly better if compared to the ones shown in Figure 40. The percentage
error is, naturally, smaller than in the results for β = 0◦
because the absolute deviation Di is now
divided by a greater value. The error value is always under 4%.
Figure 42 – Magnetic fields Bx, By, Bz and the angular position calculation for β = −30◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−60
−50
−40
−30
−20
−10
0
CalculatedAngle[◦]
−4
−3
−2
−1
0
1
2
3
4
Error[%]
Calculated Angle
Linearization of slope
Error
Source: Author, 2016
The results for angular position calculation for β = −30◦
are presented in Figure 42.
As in Figure 41, the percentage error is small and the general result is acceptable for this first
approach. The error value is always under 3%.
More results of linear and angular position calculation using the equations Equation 19
and Equation 20 are presented on Appendix D.
6.1 Lookup Table Method Application
This section shows the lookup table application to the experiment data. Using the results
in Appendix D, two contour plots were made to represent the values of d and β for the spectrum
of motion of the experiment, i.e., −10 ≤ d ≤ 10 mm and −45 ≤ β ≤ 45◦
. Figure 43 shows
these two contour plots.
Figure 43 – Lookup contour plots for experiment data
−10 −5 0 5 10
Displacement d [mm]
−40
−20
0
20
40
Angleofsensorβ[◦]
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
−10 −5 0 5 10
Displacement d [mm]
−40
−20
0
20
40
Angleofsensorβ[◦]
-45.00 -45.00
-30.00
-15.00
0.00
15.00
30.00
45.00
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-69-320.jpg)
![68
These contour plots show the result for d = dout(dcalc, βcalc) and β = βout(dcalc, βcalc).
With the spectrum mapped into the lookup table and into these two contour plots, when a new
measurement is done, the calculated outputs of dcalc and βcalc are compared with the values of
dout and βout in Figure 43 to find the real values of d and β.
6.2 Random points analysis
This section presents the random points analysis, where a set of random displacement d
points is generated in order to create a road profile. The robot moves the magnet as it was the car
suspension going trough this road profile, and the magnetic data is collected. The calculation of
dcalc through Equation 19 returns a value that is plotted in Figure 44, along with the reference
values of the road profile and the percentage error between the real value and the achieved dcalc.
Figure 44 – Set of random points for d and β
0 200 400 600 800 1000
Distance [m]
−15
−10
−5
0
5
10
15
Height[mm]
Road Profile
Calculated Position
0.0 0.2 0.4 0.6 0.8 1.0
−20
−15
−10
−5
0
5
10
15
20
Error[%]
Error
Source: Author, 2016
These experiments are presented for a single steering angle at a time. Figure 44 presents
the output for β = −45◦
and Figure 45 presents the output for β = 30◦
.
Figure 45 – Set of random points for d and β
0 200 400 600 800 1000
Distance [m]
−15
−10
−5
0
5
10
15
Height[mm]
Road Profile
Calculated Position
0.0 0.2 0.4 0.6 0.8 1.0
−20
−15
−10
−5
0
5
10
15
20
Error[%]
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-70-320.jpg)
![69
The other approach, using the lookup table method, is presented in Figure 46. In this
case, a parametric curve was created, to vary d and β at the same time, which is closer to reality.
So a bi-dimensional array with random values of d and β is created forming a single curve. The
experiment was performed for these values and their dcalc and βcalc values are calculated.
Figure 46 – Set of random points for d and β
−10 −5 0 5 10
Suspension travel [mm]
−40
−20
0
20
40
Steeringangle[◦]
Road Profile
Calculated Position
0.0 0.2 0.4 0.6 0.8 1.0
−15
−10
−5
0
5
10
15
Error[%]
Linear error
Angular error
Source: Author, 2016
The parametric curve and the calculated points of dcalc and βcalc are presented in
Figure 46, along with the error in the linear and angular positions calculated data.
6.3 Discussion
A system perfectly aligned delivers the best result with no systematic error, but that
is not feasible. Improving the building tolerances of this system is possible because this work
developed an initial small prototype, therefore, there are big opportunities of improvement
regarding design and manufacturing. Other configurations could be proposed to make the system
tighter or the mechanism holder could be designed to attach directly to the robot arm, so there
would be no need for calibration of the x and y axes of the robot. The PCB was manually welded,
so small misalignments are expected. Although the errors caused by this were minimal, this
could be improved by a better method of welding.
The application of this system in a car is the ultimate goal of this project. It would
require small changes in the mechanism configuration to allow it to connect to the suspension
linkages, but since the system does not suffer any stress while in movement, the parts could also
be 3D printed. Another important task of this project goal would be to retrieve the real values
of suspension displacement and steering angle. Either because the vehicle does not have these
sensors or this data is not easily accessible. In any case, the correct values of displacement and
angle are needed to compare with the proposed system and, to calculate the error.
The magnetic map could be improved to get better results from the system. The an-
gular detection is the main issue, becoming highly instable in the region where the tangential](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-71-320.jpg)
![75
BIBLIOGRAPHY
3D Systems. ProJet 3510 SD Professional 3D Printer. 2016. Available at:
<http://www.3dsystems.com/3d-printers/professional/projet-3500-sd>.
Arduino LLC. Arduino Board Leonardo. 2012. Available at: <https://www.arduino.cc/en/Main/
ArduinoBoardLeonardo>.
ChenYang Technologies GmbH. ChenYang Hard Ferrite (Ceramic) Magnets. 2006. Available at:
<http://www.cy-magnetics.com/CY-Mag-HFerrite.pdf>.
Delphi Corporation. Delphi MagneRide. 2005. Available at: <http://www.motor-talk.de/forum/
aktion/Attachment.html?attachmentId=488981>.
GENTA, G.; MORELLO, L. The Automotive Chassis. Vol. 1: Components design. 1. ed. Berlin:
Springer, 2009. (Mechanical Engineering Series).
GILLESPIE, T. D. Fundamentals of vehicle dynamics. 1. ed. Warrendale: Society of Automotive
Engineers, 1992.
HEREMANS, J. P. Magnetic field sensors for magnetic position sensing in automotive
applications. Materials Research Society Proceedings, Cambridge University Press, New York,
USA, v. 475, 1997.
Infineon Technologies Austria AG. Low Power 3D Magnetic Sensor with I2C Interface:
TLV493D-A1B6 3D Magnetic Sensor. 2016. 23 p.
JAZAR, R. N. Vehicle dynamics: theory and applications. 3. ed. New York, NY: Springer, 2009.
KNOWLES, D. Automotive Suspension and Steering Systems. 5. ed. New York: Delmar,
Cengage learning, 2011. 595 p.
LONG range magnetic field measurement with magnetic sensors. In: IEEE Industrial
Automation, Information and Communications Technology (IAICT 2014). Bali, Indonesia:
[s.n.].
Mercedes-Benz. Magic Body Control. 2013. Available at: <https://www.mercedes-benz.com/en/
mercedes-benz/innovation/magic-body-control/>.
MILANO, S. Product Information, Allegro Hall-Effect Sensor ICs. 2013. Avail-
able at: <http://www.allegromicro.com/en/Design-Center/Technical-Documents/
Hall-Effect-Sensor-IC-Publications/Allegro-Hall-Effect-Sensor-ICs.aspx>.
REIMPELL, J.; STOLL, H.; BETZLER, J. W. The automotive chassis: engineering principles.
2. ed. Oxford: Butterworth-Heinemann, 2001.
RIBEIRO, M.; ORTNER, M. Magnetic Field Shaping for Improved 1-D Linear Position
Measurement. In: IEEE 9th International Conference on Sensing Technology (ICST 2015).
Auckland, New Zealand: IEEE, 2015.
Seiko Epson Corporation. Scara Roboter Manipulator Handbuch: E2C Serie. 2003. 69 p.](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-77-320.jpg)
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Figure A.1 – Mechanism calculator
−250−200−150−100−50050100
[mm]
−150
−100
−50
0
50
[mm]
Sensorarmmovement
Mechanism
−80−60−40−20020406080
Steeringlinearmovement[mm]
−60
−40
−20
0
20
40
60
β:sensorangle[◦
]
Linearvsrotatorymovement
SensorArm[mm]
100.00
ConnectionRod[mm]
175.00
Steer[mm]
-30.00
Cx[mm]
-174.71
Cy[mm]
-90.00
ResetValues
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-82-320.jpg)
![81
APPENDIX B – PYTHON CODES
This appendix presents some Python codes used in this work. Each section presents a
small explanation and the code used.
B.1 Mechanism
This code calculates the B coordinates from A and C coordinates and the sensor arm
and connection rod length. It also calculates the β angle for the same setup.
Code B.1 – Python example
1 import numpy as np
2
3 class mechCalculator:
4
5 def __init__(self, a, c, sensorarm, connectionrod):
6 self.A = a
7 self.C = c
8 self.r1 = sensorarm
9 self.r2 = connectionrod
10
11 def calculateB(self):
12 d = np.sqrt((self.A[0] - self.C[0]) ** 2 + (self.A[1] - self.C[1]) ** 2)
13 l = (self.r1 ** 2 - self.r2 ** 2 + d ** 2) / (2 * d)
14 h = np.sqrt(self.r1 ** 2 - l ** 2)
15 self.x1 = (l / d) * (self.C[0] - self.A[0]) + (h / d) * (self.C[1] - self.A[1]) + self.A
[0]
16 self.x2 = (l / d) * (self.C[0] - self.A[0]) - (h / d) * (self.C[1] - self.A[1]) + self.A
[0]
17 self.y1 = (l / d) * (self.C[1] - self.A[1]) - (h / d) * (self.C[0] - self.A[0]) + self.A
[1]
18 self.y2 = (l / d) * (self.C[1] - self.A[1]) + (h / d) * (self.C[0] - self.A[0]) + self.A
[1]
19
20 return [self.x2, self.y2]
21
22 def calculateBeta(self):
23 B = self.calculateB()
24 if self.x2 >= 0:
25 return np.degrees(np.arctan(abs(self.x2 - self.A[0]) / abs(self.y2 - self.A[1])))
26 else:
27
28 return -np.degrees(np.arctan(abs(self.x2 - self.A[0]) / abs(self.y2 - self.A[1])))
29
30 def calculateB2(self):
31 newCalc = self.calculateB()
32 c1 = self.x2**2 - self.r2**2 + (self.y2 + 90.0)**2
33 pos = -(-2*self.x2 + np.sqrt(4*self.x2**2 - 4*c1)) / 2](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-83-320.jpg)
![82
34
35 return pos
B.2 Other Python Code
This code calculates the C position when the β angle is given.
Code B.2 – Python example 2
1 import numpy as np
2
3 class cposcalc:
4
5 def __init__(self, beta):
6 self.beta = beta
7
8 def cpos(self):
9 if self.beta >= 0:
10 B = [100*np.sin(np.radians(self.beta)), -100*np.cos(np.radians(self.beta))]
11 else:
12 B = [-100*np.sin(np.radians(-self.beta)), -100*np.cos(np.radians(-self.beta))]
13 c1 = B[0]**2 - 175**2 + (B[1] + 60.0)**2
14 pos = -(-2*B[0] + np.sqrt(4*B[0]**2 - 4*c1)) / 2
15
16 return pos
B.3 Robot experiment output read and plot
This code is used to read the files with results from the robot experiment. The text files
are read and put into variables and plotted. The figure containing the plot is saved as EPS.
Code B.3 – Python example
1 # ---------- Import ----------
2 from plotConfig_v15BrunoAx1 import *
3
4 # ---------- Data Read ----------
5 dat1 = genfromtxt(’filename.txt’,delimiter=’,’,skip_header=0,skip_footer=0)
6 dat1 = dat1[:,5:8]
7 dat1 = array([average(dat1[100*i:100*(i+1)],axis=0) for i in range(len(dat1)/100)])
8
9 Bx = dat1[:,0]
10 By = dat1[:,1]
11 Bz = dat1[:,2]
12
13 # ---------- Plot ----------
14 xs = linspace(-15,15,len(Bx))
15
16 ax1.plot(xs,Bx,color=ORANGE,label=r’$B_x$’)
17 ax1.plot(xs,By,color=TEAL,label=r’$B_y$’)
18 ax1.plot(xs,Bz,color=PURPLE,label=r’$B_z$’)
19 ax1.grid(True)
20 ax1.set_xlabel(r’Travel [$mm$]’)
21 ax1.set_ylabel(r’Magnetic Field [$mT$]’)
22 ax1.legend(loc=4,prop={’size’:14})](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-84-320.jpg)
![87
APPENDIX D – RESULTS OF LINEAR AND ANGULAR POSITION
CALCULATIONS
This appendix presents all the results for the magnetic field and the linear and angular
positions calculation. As stated in chapter 6, the d range in this results is from −10 mm to 10 mm,
to avoid showing the range where the calculation is inaccurate. The results will be presented in
two sections, where the first shows the linear position calculation results in seven scenarios, i.e.,
β = [−45, −30, −15, 0, 15, 30, 45]◦
and the second section shows the angular position detection
for the same scenarios.
D.1 Linear position calculation
Figure D.1 – Magnetic fields Bx, By, Bz and the linear position calculation for β = −45◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−10
−5
0
5
10
CalculatedTravel[mm]
−120
−100
−80
−60
−40
−20
0
20
Error[%]
Calculated travel
Correct value
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-89-320.jpg)
![88
Figure D.2 – Magnetic fields Bx, By, Bz and the linear position calculation for β = −30◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−10
−5
0
5
10
CalculatedTravel[mm]
−100
−80
−60
−40
−20
0
20
Error[%]
Calculated travel
Correct value
Error
Source: Author, 2016
Figure D.3 – Magnetic fields Bx, By, Bz and the linear position calculation for β = −15◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−10
−5
0
5
10
CalculatedTravel[mm]
−80
−60
−40
−20
0
20
Error[%]
Calculated travel
Correct value
Error
Source: Author, 2016
Figure D.4 – Magnetic fields Bx, By, Bz and the linear position calculation for β = 0◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−10
−5
0
5
10
CalculatedTravel[mm]
−60
−50
−40
−30
−20
−10
0
10
20
Error[%]
Calculated travel
Correct value
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-90-320.jpg)
![89
Figure D.5 – Magnetic fields Bx, By, Bz and the linear position calculation for β = 15◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT] Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−10
−5
0
5
10
CalculatedTravel[mm]
−20
−15
−10
−5
0
5
10
15
Error[%]
Calculated travel
Correct value
Error
Source: Author, 2016
Figure D.6 – Magnetic fields Bx, By, Bz and the linear position calculation for β = 30◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−10
−5
0
5
10
CalculatedTravel[mm]
−60
−50
−40
−30
−20
−10
0
10
20
Error[%]
Calculated travel
Correct value
Error
Source: Author, 2016
Figure D.7 – Magnetic fields Bx, By, Bz and the linear position calculation for β = 45◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−10
−5
0
5
10
CalculatedTravel[mm]
−40
−30
−20
−10
0
10
20
Error[%]
Calculated travel
Correct value
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-91-320.jpg)
![90
D.2 Angular position calculation
Figure D.8 – Magnetic fields Bx, By, Bz and the angular position calculation for β = −45◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−60
−50
−40
−30
−20
−10
0
CalculatedAngle[◦] −4
−3
−2
−1
0
1
2
3
4
Error[%]
Calculated Angle
Linearization of slope
Error
Source: Author, 2016
Figure D.9 – Magnetic fields Bx, By, Bz and the angular position calculation for β = −30◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−60
−50
−40
−30
−20
−10
0
CalculatedAngle[◦]
−4
−3
−2
−1
0
1
2
3
4
Error[%]
Calculated Angle
Linearization of slope
Error
Source: Author, 2016
Figure D.10 – Magnetic fields Bx, By, Bz and the angular position calculation for β = −15◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−30
−25
−20
−15
−10
−5
0
CalculatedAngle[◦]
−4
−3
−2
−1
0
1
2
3
4
Error[%]
Calculated Angle
Linearization of slope
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-92-320.jpg)
![91
Figure D.11 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 0◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT] Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
−10
−5
0
5
10
CalculatedAngle[◦]
−150
−100
−50
0
50
100
150
200
250
Error[%]
Calculated Angle
Linearization of slope
Error
Source: Author, 2016
Figure D.12 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 15◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
0
5
10
15
20
25
30
CalculatedAngle[◦]
−8
−6
−4
−2
0
2
4
6
8
Error[%]
Calculated Angle
Linearization of slope
Error
Source: Author, 2016
Figure D.13 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 30◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
0
10
20
30
40
50
60
CalculatedAngle[◦]
−4
−3
−2
−1
0
1
2
3
4
Error[%]
Calculated Angle
Linearization of slope
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-93-320.jpg)
![92
Figure D.14 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 45◦
−10 −5 0 5 10
Travel [mm]
−20
−15
−10
−5
0
5
10
15
20
MagneticField[mT]
Bx
By
Bz
−10 −5 0 5 10
Travel [mm]
0
10
20
30
40
50
60
CalculatedAngle[◦]
−3
−2
−1
0
1
2
3
Error[%]
Calculated Angle
Linearization of slope
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-94-320.jpg)
![93
APPENDIX E – RESULTS OF THE ROAD PROFILE CALCULATION
This appendix presents the results for the linear position calculation in compari-
son to a randomly generated road profile. As stated in chapter 6, the d range in this results
is from −10 mm to 10 mm. The results will be presented in two sections, where the first
shows the linear position calculation results using the Equation 19 in seven scenarios, i.e.,
β = [−45, −30, −15, 0, 15, 30, 45]◦
and the second section shows the linear position detection
using the lookup table method.
E.1 Linear position calculation
Figure E.1 – Linear position calculation for β = −45◦
0 200 400 600 800 1000
Distance [m]
−15
−10
−5
0
5
10
15
Height[mm]
Road Profile
Calculated Position
0.0 0.2 0.4 0.6 0.8 1.0
−20
−15
−10
−5
0
5
10
15
20
Error[%]
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-95-320.jpg)
![94
Figure E.2 – Linear position calculation for β = −30◦
0 200 400 600 800 1000
Distance [m]
−15
−10
−5
0
5
10
15
Height[mm]
Road Profile
Calculated Position
0.0 0.2 0.4 0.6 0.8 1.0
−20
−15
−10
−5
0
5
10
15
20
Error[%]
Error
Source: Author, 2016
Figure E.3 – Linear position calculation for β = −15◦
0 200 400 600 800 1000
Distance [m]
−15
−10
−5
0
5
10
15
Height[mm]
Road Profile
Calculated Position
0.0 0.2 0.4 0.6 0.8 1.0
−20
−15
−10
−5
0
5
10
15
20
Error[%]
Error
Source: Author, 2016
Figure E.4 – Linear position calculation for β = 0◦
0 200 400 600 800 1000
Distance [m]
−15
−10
−5
0
5
10
15
Height[mm]
Road Profile
Calculated Position
0.0 0.2 0.4 0.6 0.8 1.0
−20
−15
−10
−5
0
5
10
15
20
Error[%]
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-96-320.jpg)
![95
Figure E.5 – Linear position calculation for β = 15◦
0 200 400 600 800 1000
Distance [m]
−15
−10
−5
0
5
10
15
Height[mm] Road Profile
Calculated Position
0.0 0.2 0.4 0.6 0.8 1.0
−20
−15
−10
−5
0
5
10
15
20
Error[%]
Error
Source: Author, 2016
Figure E.6 – Linear position calculation for β = 30◦
0 200 400 600 800 1000
Distance [m]
−15
−10
−5
0
5
10
15
Height[mm]
Road Profile
Calculated Position
0.0 0.2 0.4 0.6 0.8 1.0
−20
−15
−10
−5
0
5
10
15
20
Error[%]
Error
Source: Author, 2016
Figure E.7 – Linear position calculation for β = 45◦
0 200 400 600 800 1000
Distance [m]
−15
−10
−5
0
5
10
15
Height[mm]
Road Profile
Calculated Position
0.0 0.2 0.4 0.6 0.8 1.0
−20
−15
−10
−5
0
5
10
15
20
Error[%]
Error
Source: Author, 2016](https://image.slidesharecdn.com/57b9c33c-4d50-4faf-8322-6a8e996b9fd9-161220205147/85/3D-magnetic-steering-wheel-angle-and-suspension-travel-detection-97-320.jpg)
3D magnetic steering wheel angle and suspension travel detection
- 1. UNIVERSIDADE FEDERAL DE SANTA CATARINA CENTRO TECNOLÓGICO DE JOINVILLE CURSO DE ENGENHARIA AUTOMOTIVA BRUNO PAES SPRICIGO 3D MAGNETIC STEERING WHEEL ANGLE AND SUSPENSION TRAVEL DETECTION A novel application of 3D magnetic sensing techniques to increase vehicle safety decreasing sensor overhead Joinville, Brazil 2016
- 3. BRUNO PAES SPRICIGO 3D MAGNETIC STEERING WHEEL ANGLE AND SUSPENSION TRAVEL DETECTION A novel application of 3D magnetic sensing techniques to increase vehicle safety decreasing sensor overhead Thesis submitted as partial fulfillment of the requirements for the degree of Bachelor in Automotive Engineering of the Federal University of Santa Catarina, Technologic Center of Joinville, Brazil. Supervisor: Prof. Thiago Antônio Fiorentin, Dr. Eng. Joinville, Brazil 2016
- 5. Bruno Paes Spricigo 3D Magnetic Steering Wheel Angle and Suspension Travel Detection A novel application of 3D magnetic sensing techniques to increase vehicle safety decreasing sensor overhead Thesis submitted as partial fulfillment of the re- quirements for the degree of Bachelor in Auto- motive Engineering of the Federal University of Santa Catarina, Technologic Center of Joinville, Brazil. Work Approved. Joinville, November 28th 2016: Prof. Thiago Antônio Fiorentin, Dr. Eng. Supervisor Prof. Marcos Alves Rabelo, Dr. Eng. Guest 1 Prof. Antônio Otaviano Dourado, Dr. Eng. Guest 2 Joinville, Brazil 2016
- 7. ACKNOWLEDGMENTS Here, I would like to thank all the people that made this work possible. First of all, my parents, because without their endless support, I would not be able to get anywhere. To my love and best friend, the one who always encouraged and inspired me to be a better person. To all my friends that make life easier and so much fun. To my great professors that inspired me so much and made me eager to learn. To the Formula CEM team and all of its fun and hard moments that showed me the value of working hard. To BSc. Marcelo Ribeiro and Dr. Michael Ortner for the endless support during this work. To CTR and its research sponsors for giving me the opportunity to make this work abroad. Thank you!
- 9. ABSTRACT Electronics and control are continuously growing subjects in the automotive industry. The de- velopment of new technologies to reduce consumption, increase comfort and handling is the number one priority of many manufactures. Various systems that make nowadays vehicles more secure like TCS (Traction Control System) and ESP (Electronic Stability Program) rely on sensing several variables like individual wheel speed and suspension displacement to compare it to an analytical model and decide if action is needed or not. The main target of this work is to propose a smart use of a new 3D magnetic sensor to improve the quality and precision of the suspension displacement measurement and, because of the greater capabilities of the sensor, detect the steering wheel angle at the same time. The mechanical and magnetic implementations are discussed in detail. Key-words: smart suspension. magnetic sensor. suspension displacement measurement.
- 11. RESUMO Eletrônica e controle são crescentes tópicos na indústria automotiva. O desenvolvimento de novas tecnologias para reduzir consumo, aumentar conforto e dirigibilidade é uma grande prioridade para muitas empresas. Muitos dos sistemas que fazem os veículos atuais mais seguros como TCS (sistema de controle de tração) e ESP (sistema de controle de estabilidade) dependem da detecção de diversas variáveis, como velocidade individual das rodas e curso da suspensão, para que seus respectivos valores sejam comparados a um modelo analítico para tomar a decisão se há necessidade de ação ou não. O principal objetivo desse trabalho é propor o uso de um novo sensor magnético 3D para aumentar a qualidade e precisão da medição do curso da suspensão e, devido às grandes capacidades do sensor, detectar também o ângulo de direção simultaneamente. As implementações dos sistemas mecânicos e magnéticos serão discutidas em detalhe. Palavras-chaves: suspensão inteligente. sensor magnético. detecção do curso da suspensão.
- 13. LIST OF FIGURES Figure 1 – Unibody and chassi vehicle. . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Figure 2 – Dampened Oscillation Movement . . . . . . . . . . . . . . . . . . . . . . . 25 Figure 3 – Vehicle Perception by Passenger . . . . . . . . . . . . . . . . . . . . . . . 25 Figure 4 – Rack-pinion steering box . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Figure 5 – A front-wheel-steering vehicle and the Ackerman condition . . . . . . . . . 30 Figure 6 – Lorentz Force and Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . 32 Figure 7 – Angular measurement with magnet sensor . . . . . . . . . . . . . . . . . . 37 Figure 8 – Linear measurement with magnet sensor . . . . . . . . . . . . . . . . . . . 38 Figure 9 – Linear movement into angular . . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 10 – Linear and angular measurements combined . . . . . . . . . . . . . . . . . 39 Figure 11 – Epson E2C351S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Figure 12 – Arduino Leonardo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Figure 13 – Sensor TLV493D-A1B6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Figure 14 – Sensor axes of measurement . . . . . . . . . . . . . . . . . . . . . . . . . 44 Figure 15 – ProJet 3510 HDPlus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Figure 16 – Printing Materials Properties . . . . . . . . . . . . . . . . . . . . . . . . . 45 Figure 17 – Two Circles Intersection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Figure 18 – Mechanism calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Figure 19 – Linear fitting of C(β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Figure 20 – 3rd degree polynomial fitting of C(β) . . . . . . . . . . . . . . . . . . . . 51 Figure 21 – 5th degree polynomial fitting of C(β) . . . . . . . . . . . . . . . . . . . . . 52 Figure 22 – Time test for polynomial approximation . . . . . . . . . . . . . . . . . . . 52 Figure 23 – PCB holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Figure 24 – System cut and partial assembly . . . . . . . . . . . . . . . . . . . . . . . 54 Figure 25 – Final assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Figure 26 – Linear position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Figure 27 – Angular position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Figure 28 – Combination of movements . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Figure 29 – Magnetic field simulation in sensor plane . . . . . . . . . . . . . . . . . . . 57 Figure 30 – Magnetic field simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Figure 31 – Linear position simulation at ∆ = 2 mm . . . . . . . . . . . . . . . . . . . 58 Figure 32 – Linear position simulation at ∆ = 5 mm . . . . . . . . . . . . . . . . . . . 59 Figure 33 – Linear position simulation with noise . . . . . . . . . . . . . . . . . . . . . 60 Figure 34 – Contour Plots for dout and βout . . . . . . . . . . . . . . . . . . . . . . . . 61 Figure 35 – Contour Plots for [dout]2 + [βout]2 . . . . . . . . . . . . . . . . . . . . . . . 61 Figure 36 – Robot Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
- 14. Figure 37 – Robot Experiment Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Figure 38 – Initial results of magnetic fields Bx, By and Bz . . . . . . . . . . . . . . . . 65 Figure 39 – Magnetic fields Bx, By, Bz and the linear position calculation . . . . . . . . 65 Figure 40 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 0◦ . 66 Figure 41 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 30◦ 66 Figure 42 – Magnetic fields Bx, By, Bz and the angular position calculation for β = −30◦ 67 Figure 43 – Lookup contour plots for experiment data . . . . . . . . . . . . . . . . . . 67 Figure 44 – Set of random points for d and β . . . . . . . . . . . . . . . . . . . . . . . 68 Figure 45 – Set of random points for d and β . . . . . . . . . . . . . . . . . . . . . . . 68 Figure 46 – Set of random points for d and β . . . . . . . . . . . . . . . . . . . . . . . 69 Figure 47 – Angular Range Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
- 15. LIST OF TABLES Table 1 – Semi-active suspension / Mechanical vs. Sensor technology . . . . . . . . . 36 Table 2 – Active suspension / Mechanical vs. Sensor technology . . . . . . . . . . . . 37
- 17. LIST OF SYMBOLS −→ B Magnetic field; B⊥ Perpendicular magnetic field; Bamp Amplitude of magnetic field; Btan Tangential component of magnetic field; Bvert Vertical component of magnetic field; Bx Magnetic field in x; By Magnetic field in y; Bz Magnetic field in z; Br Remanence magnetic field; d Displacement of magnet; Di Absolute deviation; −→ F Force; I Current; K Constant to improve linear results; l Wheelbase; O Turning center; q Electrical charge of the electron; r Radius; Sang Calculated value of sensor’s angle; Spos Calculated value of magnet’s displacement; t Thickness; T Temperature; −→v Velocity; Vcc IC power supply voltage; VH Hall voltage; Vin Input voltage; Voutput Output voltage; w Track of front axis; β Sensor angle; δ Average value of steering angle;
- 18. δi Steering angle of inner wheel; δo Steering angle of outer wheel; ∆ Air gap; ρn Number of carriers per volume;
- 19. CONTENTS 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2 BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1 Suspension Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Smart Suspension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.3 Steering System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Magnetic Position Detection . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.1 Hall Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.2 Position Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 CONTEXT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1 Suspension Displacement Measurement . . . . . . . . . . . . . . . . . . 35 3.2 Suspension Measurement with Magnetic Sensor . . . . . . . . . . . . . . 37 3.2.1 Steering Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4 MATERIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.1 Softwares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Microcontroller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.4 Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.5 Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.6 Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5 DEVELOPMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1.2 Components Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 Magnetic Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2.1 Mathematical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2.2 Magnetic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2.3 Lookup Table Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3 Robot Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.1 Lookup Table Method Application . . . . . . . . . . . . . . . . . . . . . 67 6.2 Random points analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.1 Future Work Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
- 20. BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 APPENDIX 77 APPENDIX A – MECHANISM CALCULATOR . . . . . . . . . . . . 79 APPENDIX B – PYTHON CODES . . . . . . . . . . . . . . . . . . . . 81 B.1 Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 B.2 Other Python Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 B.3 Robot experiment output read and plot . . . . . . . . . . . . . . . . . . . 82 APPENDIX C – PCB LAYOUT AND ELECTRONIC COMPONENTS 85 APPENDIX D – RESULTS OF LINEAR AND ANGULAR POSITION CALCULATIONS . . . . . . . . . . . . . . . . . . . . 87 D.1 Linear position calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 87 D.2 Angular position calculation . . . . . . . . . . . . . . . . . . . . . . . . . 90 APPENDIX E – RESULTS OF THE ROAD PROFILE CALCULATION 93 E.1 Linear position calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 93 ANNEX 97 ANNEX A – ROBOT SPECIFICATION TABLE . . . . . . . . . . . . . 99 ANNEX B – MAGNET SPECIFICATION TABLE . . . . . . . . . . . 101
- 21. 19 1 INTRODUCTION The automotive industry invests lots of resources in embedded electronics and control, developing new technologies aiming for fuel consumption reduction, increase of combustion efficiency and improvement of comfort and handling with chassis systems. The safety level of nowadays vehicles is mainly a consequence of the always growing amount of sensors, actuators and controllers inside these vehicles. It is known that increasing driver’s assistance with electron- ics has lead to more safety. The increasing development of autonomous driving vehicles by some manufacturers is a great evidence that using computer to make some decisions and adjustments in a vehicle is a safety increaser feature. Chassis systems like TCS (Traction Control System) and ESP (Electronic Stability Program) rely on precise sensing to perform efficiently. These assistance systems measure several variables like individual wheel speed, individual suspension displacement, acceleration of vehicle body and steering wheel angle. A comparison between these values and a mathematical model output is made to identify if the driver is loosing control of the vehicle or if some unwanted wheel-spin or wheel-lock occurs. The actuation of these systems is very important, but it also depends on good sensing and robust control algorithms. Although better and more precise sensors are developed everyday, the automotive industry is very concerned about costs, therefore, a sensor considered the best and most precise would be ideal to make a vehicle more secure and comfortable, but the costs of implementation might not be feasible on regular vehicles, limiting the application to high-end luxury models or more expensive vehicles like transportation trucks or tractors (agricultural machinery in general). Magnetic sensors are already a reality in the automotive industry with several applications like assessing pedals, crankshaft and camshaft positions. Magnetic sensors are an alternative for current measurement systems because of their numerous advantages like small packaging sizes, low production costs, contactless measurement and an excellent robustness against vibrations, temperature, moisture and dirt (LONG..., ). Most of the applications of magnetic sensors require a 1D or 2D sensor, but with the introduction of the 3D magnetic Hall sensors, a whole new range of applications can be explored and improvements of the existing systems and applications are also possible. The suspension is responsible for the comfort and to maintain the tire-ground contact (GENTA; MORELLO, 2009) and the steering system is responsible for changing the vehicle’s direction (GENTA; MORELLO, 2009), so it is fair to say that they influence directly the dynamic behavior of a vehicle. Precise measurement of these variables is mandatory to get good results
- 22. 20 of the assistance systems. Potentiometers and accelerometers are among the technologies used to measure suspension displacement. As listed by Genta and Morello (2009), there are at least two approaches to use accelerometers: using two of them, one moving with the vehicle body and the other moving with the wheel (the difference between signals is integrated to get the displacement) or using one accelerometer to estimate the acceleration from the body. Steering angle measurement can be made with optical sensors, potentiometers and magnetic systems. The main motivation of this work is to improve the quality of these measurements and, to suggest a more reliable and efficient system that requires only one sensor and one magnet to measure both system’s variables simultaneously. A 3D sensor has the advantage of the third component of magnetic field, making it possible to map more complex movements. To use its full potential of measurement, this work presents a solution to measure two distinct movements keeping the setup simple with only one magnet and sensor. The working principle is: the suspension movement is attached to the magnet, moving it linearly, while the steering movement is connected to the sensor, rotating it. Simulations were performed to find a precise and robust magnetic setup that fulfills both linear and angular movements range requirements. To transform the magnetic field into position measurement, a magnetic map was developed with two equations taking the three components of the magnetic field as input and outputting two positions. These equations were an initial calculation and needed some constant adjustments to get more precise results. A third method to calculate the positions is presented, where the whole spectrum of angles and linear position is mapped and a interpolation is used to create contours of the positions. The interpolation method is more precise than the others because all the systematic errors are included in the data used to map the spectrum. 1.1 Objectives Main Objective • Develop a system that, with one 3D sensor and one magnet, is able to read suspension displacement and steering angle simultaneously. Specific Objectives • Suggest an automotive application involving the use of a 3D Hall sensor; • Use magnetic field to determine relative points; • Improve performance of actual systems by innovating the measurement method.
- 23. 21 1.2 Thesis Structure This work is structured in seven chapters, being the first one the introduction and the last one the conclusion. The second chapter brings the theoretical background to this work, presenting suspension and steering systems of vehicles, as well as magnetic sensors. It contains explanations about suspension systems, its mathematical models, smart suspensions, steering systems and magnetic sensors. The third chapter presents the context of this work with the state of the art in suspension displacement measurement and the main ideas presented of use for magnetic sensor in suspensions and steering systems. The fourth chapter presents the materials used to complete this thesis, like the softwares used, the hardware involved and the technology used to manufacturer the developed mechanism. The fifth chapter shows the development of the calculations, starting from how to calculate the mechanism to convert linear movement into rotation, the development of the parts, the magnetic map needed to translate three components of magnetic field into two distinct movements, and the robot setup. The chapters three to five are the methodology part of this work. The sixth chapter presents the results of the experiment using a robot and their respective errors. Random points were created to simulate a road profile with different suspension displacement and steering angles. The results of positioning of the experiment were compared with the known points and the errors of the experiment are presented. The final chapter has the conclusion and discussion of the results.
- 25. 23 2 BACKGROUND This chapter presents explanations about suspension systems (including smart suspen- sions) and magnetic sensors. The approach is to answer what is a suspension system, why is it important, how does it work and how it is classified. After understanding the compromises of passive suspensions, it will be easier to understand the advantages of smart suspension systems and how suspension displacement measurement is important. 2.1 Suspension Systems A vehicle’s suspension is the system that connects vehicle’s wheels with its body, in case of a unibody1 vehicle (Figure 1a). If the vehicle is a body-on-frame2 one (Figure 1b), the suspension system makes the connection between wheels and chassis. In both cases though, the system is responsible for allowing relative movement between ground and, vehicle increasing comfort for passengers or improving vehicle’s stability and safety. Figure 1 – Unibody and chassi vehicle. (a) Unibody vehicle. (b) Chassis frame. Source: Knowles (2011). Comfort improvement is achievable because the suspension absorbs and smooths out shocks from road irregularities that would be transfered to the wheel and, furthermore, transmitted to the body as vibrations (GENTA; MORELLO, 2009). Vehicle’s stability and safety are obtained by keeping body roll3 in a level that doesn’t compromise the dynamics of movement and maintains a high grip between road and tire under all conditions. 1 Type of body/frame construction in which the body of the vehicle, its floor plan and chassis form a single structure. 2 Automotive construction method that mounts the body of vehicle to a rigid frame called chassis. 3 Rotational movement of vehicle body towards the outside of a turn.
- 26. 24 According to Genta and Morello (2009, p. 133), a suspension system fulfills its require- ments if it: • Allows a distribution of forces, exchanged by the wheels with ground, complying with design specifications in every load condition; • Determines the vehicle trim4 under the action of static and quasi-static forces. A suspension is mainly composed by two components: an elastic component or spring, and a damper or shock absorber. The spring’s main target is to allow relative movement between road and vehicle. The damper’s main task is to dissipate this extra energy transfered to the system from road surface variation. The geometry of the linkages5 is the most used classification mean for suspension systems. These linkages highly affect the efficiency and characteristics of the system because they can either be simple, cheap and deliver average results or extremely complex and expensive, allowing a great number of configuration to improve overall ride quality under many scenarios. Genta and Morello (2009, p. 134) say that "in theory, tires alone could isolate the vehicle body from forces coming from the road, but their elastic and damping properties are not sufficient to achieve suitable handling and comfort targets, unless at very low speed and on smooth roads.", making suspension systems essential to achieve the adequate amount of handling and comfort. Springs and dampers have a major role in achieving the goals of a vehicle’s project because their parameters almost completely define its dynamic behavior. By transforming relative movement in potential energy, the spring element leaves the static deformation6 to a different position. It is known that this change of position will result in an oscillatory movement (simple harmonic motion) of the mass connected to the spring, i.e., the vehicle body. Theoretically, in an ideal scenario, this movement would not stop because no energy is lost by the system7 . Employing a damper or shock absorber is, therefore, mandatory to be able to control this oscillation movement. Essentially, a damper’s main task is to take energy out of the system, usually accomplishing it by transforming it in heat, obtaining then a damped oscillatory movement where the amplitude decreases with time. Simple harmonic and damped motion are very common and, usually, simple to model. In Figure 2, four types of movement are shown in order to illustrate how differently the amplitudes decay with time when varying the damping’s coefficient only. Using a mathematical model named quarter vehicle model (JAZAR, 2009), it is possible to estimate the frequency response of the vehicle, and the suspension behavior can be roughly 4 Trim control is a quasi-static control aiming constant vertical static displacement of the rear axle or of both axles, at any vehicle load (GENTA; MORELLO, 2009, p. 341). 5 Links connecting the wheels with vehicle’s body or frame. 6 Spring’s length at rest when supporting the weight (or part of) of the vehicle. 7 Considering an ideal system with no friction losses, where all potential energy is repeatedly transformed into kinetic energy and the other way around.
- 27. 25 Figure 2 – Dampened Oscillation Movement 0 1 2 3 4 5 6 Time [s] −0.020 −0.015 −0.010 −0.005 0.000 0.005 0.010 0.015 0.020 Amplitude[m] Simple Harmonic Motion Under-damping ζ = 0.3 Critical-damping ζ = 1.0 Over-damping ζ = 2.5 Source: Author, 2016 predicted. Only vertical dynamic behavior can be studied with quarter vehicle model, so horizon- tal (braking and accelerating dynamics) and lateral studies (handling and dynamics of a vehicle under cornering) are neglected. Nevertheless, the quarter vehicle "[. .. ]contains the most basic features of the real problem and includes a proper representation of the problem of controlling wheel and wheel-body load variations." (JAZAR, 2009). It is also the initial model that is used to begin estimating the spring stiffness and damping coefficient needed to achieve the expected dynamic behavior. This behavior must consider that certain frequencies e.g., the range from 1 to 80 Hz, are more critical because they affect the human body, causing some kind of discomfort (REIMPELL; STOLL; BETZLER, 2001). A vehicle is a very complex dynamic system that only exhibits vibration in consequence to excitation inputs (GILLESPIE, 1992), thus, its response properties are responsible by the vibration’s magnitude and direction affecting the passengers’ perception of ride/vehicle as seen in Figure 3. Figure 3 – Vehicle Perception by Passenger Excitation Sources Road Roughness Tire/Wheel Driveline Engine Vehicle Dynamic Response Vibrations Ride Perception Source: Adapted from Gillespie (1992).
- 28. 26 Comfort and active safety are conflicting objectives that suspension systems must fulfill (GENTA; MORELLO, 2009). Suspensions with a high level of comfort should be soft, while to guarantee a constant contact of wheels with the ground they should be rigid (GENTA; MORELLO, 2009). Both these characteristics involve damping coefficient settings, the first requiring small values, while the latter is reachable with higher damping coefficients. 2.2 Smart Suspension Passive suspension systems8 can only react to forces coming from the road, because of their nature of only dissipating energy of the system (GENTA; MORELLO, 2009). And because they have fixed values9 of spring stiffness and damping coefficient, choosing between ride comfort and handling/safety is needed. "The limit of these passive suspensions can be easily explained by the impossi- bility of managing two independent parameters - body vertical accelerations (related to comfort) and vertical force variations (related to active safety) - with a single parameter, the suspension damping coefficient. The two objectives are independent and their optimum values are obtained with different damping coefficients." (GENTA; MORELLO, 2009, p. 340). Manufactures must then design this parameters according to the target of each project. Sportive cars should be more stable under high cornering or other high demanding situations, therefore, their suspensions should be stiffer, meaning that ride comfort is compromised. Normal cars, in the other hand, should be comfortable and pleasant, but over-improving these character- istics would definitely decrease safety so a balance point is best solution. Sport Utility Vehicles (SUV) and trucks face the same problem, mainly because the wide range of weight they carry requires a wide range of suspension parameters for optimal dynamic behavior under all loads. Adapting these mechanic parameters is ideal to improve dynamic behavior, but to be able to optimize these parameters adding microelectronics and controllers is recommended. The electronics are responsible to gather and interpret data to know how this parameters should be changed. These are called adaptive, controlled or active suspensions depending on the level of contribution they make (GENTA; MORELLO, 2009). The most common classification of these systems is: • Semi-active: Usually is able to adapt its damping coefficient (therefore, adapting to different situations and loads to keep the level of comfort and handling), by measuring suspension displacement, g-force and/or body roll and calculating an according value. There is no addition of energy into the movement (the system is not able to actively change the displacement of suspension), instead it controls how the system will dissipate the extra energy. 8 Systems made only with mechanical parts like springs, dampers and linkages with little or no electronics. 9 Even though some more advanced systems might have ranges of values instead of a single value, i.e., a gradient behavior, they are optimized for just certain conditions, they cannot adapt.
- 29. 27 • Active: The main difference is that adding energy to the movement is possible, therefore, the system can change displacement of the suspension. The suspension sensors measures the kinematic parameters and a controller interprets the data to decide if actuation is needed, if so, it might actuate changing parameters (passive approach), introducing movement to balance displacement (active approach) or both. In both cases, the system relies on sensors to measure kinematic parameters, a control unit and some actuators. The task of the actuators of a semi-active suspension is to change damper’s behavior, what can be done mechanically10 or electronically11 . In an active suspension, the actuators are generally more robust because they add energy to the movement using pneumatic, hydraulic or, in same cases, electromagnetic-based systems. Control units might be a specific component for suspension systems or be part of a central unit as the Electronic Control Unit (ECU). Sensing might be simple as a single accelerometer to measure g-force or complex with several sensors monitoring from suspension’s displacement to acceleration in different axes of vehicle body. According to Genta and Morello (2009), the goals of a smart suspension system are trim, roll, damping and/or full active control. Trim control means that the suspension system is able to detect differences in static deformation between the front and rear axes caused by different payloads, maintaining the same ride height at any vehicle load. This allows the vehicle to detect load variation that would completely affect the ride quality and correct it before the movement starts. Roll control is responsible for controlling vehicle roll and roll speed dynamically. Anti-roll bars with static values are used to prevent excessive roll of vehicle body in passive systems. Smart suspensions use sensors to detect movement and interpret the intensity of body roll and decide if action is required, if so, actuators impose an anti-roll bar preload to adapt its performance. Damping control changes damping coefficient of the shock absorber to adapt to various situations, i.e., changing between a sport setup to comfort and vice-versa, dynamically. These controlled dampers are classified as Adaptive or Semi-active: • Adaptive: The damping coefficient is set to higher levels while in very low or very high (lower than 20 km/h and greater than 120 km/h, respectively) speeds, the first to avoid car bounce while maneuvering and the latter to improve vehicle stability. In between these two situations the value of the damping coefficient is kept low to increase comfort. Even though this is far more range than passive systems, it is not completely adjustable, i.e., hitting a bump while in high speed (high values of damping coefficient) would mean a very uncomfortable ride, the same goes for stability requiring situations while in medium speeds (low damping coefficient); 10 Internal valves inside the damper with variable diameter change fluid’s motion. 11 A damper containing magnetorheological fluid that changes its viscosity according to a magnetic field.
- 30. 28 • Semi-active: Based on the Skyhook theory12 , it’s more complex but enables more control and improved performance. This system relies on sensing accurately the kinematic param- eters to be able to adapt itself to various situations. It calculates the damping force needed to balance the vertical force on suspension using mathematical models with body and wheel speed as input. The system will detect the need of changing damping coefficient and will perform so dynamically, in certain time intervals. It typically includes accelerometers in vehicle body (measuring three axes), a lateral accelerometer on the front of vehicle (or a steering angle sensor), a braking circuit pressure sensor and a car speed sensor. Full active control fulfill all above objectives in any dynamic situation. It means that for any load or road variation the suspension system can control trim, roll and damping coefficient to keep higher levels of ride comfort and safety. The energy requested by the control system is significant in the third system (damping control) and maximum in the fourth (full active control) (GENTA; MORELLO, 2009). Smart Suspension systems are strongly dependent on good sensing, because external parameters and resultant movement of vehicle body must be precisely measured in order to provide good actions. Although acting (whether changing damping coefficient or displacement movement) is an important part of the system, precisely sensing and interpreting data is crucial to decide how and when to act. 2.3 Steering System Depending on how a vehicle’s path is controlled, vehicles are classified in two distinct categories, guided and piloted. The first is controlled by a set of kinematic constraints while the later has a guidance system controlled by a human or some kind of device exerting forces that change vehicle’s trajectory (GENTA; MORELLO, 2009). Piloted system is usually composed by a steering mechanism13 , a steering box14 and a steering column (connecting steering wheel and steering box). The change of direction begins with a rotatory movement of steering wheel that acts on the rack-pinion (or other type of mechanism), shown in Figure 4, generating a linear movement of the steering rack and, therefore, the steering tie rods. The rack moves linearly, pushing or pulling the back (or front) part of the wheel hub producing a rotation around the king-pin axis. Being responsible for maneuvering and changing vehicle’s direction, as well as affecting greatly the dynamic behavior are the main reasons why steering system is among the most 12 The theory says that it would be ideal to have the vehicle body connected to an inertial reference system, such as the sky, by a shock absorber (GENTA; MORELLO, 2009). It is not feasible, but presents the mathematical model to a setup, in which, the vehicle body would be completely isolated from road variation, body roll and other phenomenon that affect handling and comfort. 13 The system of linkages steering the front wheels in a particular way around the king-pin axis, connecting steering arms moving with the suspension stroke to the steering box (GENTA; MORELLO, 2009). 14 Transforms steering wheel rotation into a displacement of the steering tie rods or rack (GENTA; MORELLO, 2009).
- 31. 29 Figure 4 – Rack-pinion steering box Source: Genta and Morello (2009). important systems in any vehicle. For modern chassis systems like traction control system (TCS), electronic stability program (ESP) or electronic stability control (ESC), the input of the driver at steering wheel is essential because they mainly focus on predicting some dynamic variables such as speed, intended direction, body roll and others. Theses reference values are extracted from input of various sensors throughout the vehicle and compared to actual values of wheel spin and lateral force to analyze if corrections are needed. A big difference between steering and suspension systems is that assisting systems keep focus on sensing and don’t act on steering, with exception of the latest collision avoidance systems that actually change vehicle’s trajectory to dodge obstacles. Low speed or kinematic steering is the definition of a motion of vehicle where the velocity is small enough so the slip of tires are really small and, therefore, there is almost no capability of exerting corner force (GENTA; MORELLO, 2009). Ackerman geometry is defined as the setup of wheel angles where in small speeds the tire slip is zero (JAZAR, 2009). Figure 5 shows the Ackerman geometry where δi is the steering angle of the inner wheel while δo is the angle of the outer wheel, being δi always greater than δo because of the difference of radius. This condition is achieved when the projected lines from the rotation axis of all wheels meet in a single point O called turning center. The distance between steering axes is called track and is represented by w. The distance between rear and front axles is called wheelbase and is represented by l. The Ackerman condition is achieved when: cot δo − cot δi = w l (1) Ackerman is one of the many solutions when designing a steering system geometry, it is ideal for slow speeds and no suspension movement, which, usually, is not the average use of a
- 32. 30 Figure 5 – A front-wheel-steering vehicle and the Ackerman condition Source: Jazar (2009). vehicle. There are other geometries with their own advantages and disadvantages, optimized for different situations like race cars employing reverse Ackerman or simpler vehicles with parallel δi and δo angles. According to Jazar (2009) "there is no four-bar linkage steering mechanism that can provide the Ackerman condition perfectly. However, we may design a multi-bar linkages to work close to the condition and be exact at a few angles." Also, because suspension movement interferes with steering system’s angles like camber, caster and toe, the iteration of these two systems must be precisely designed. The initial development of steering system is often made using the bicycle model, in which a 4 wheel based vehicle is transformed into a 2 wheel based one. This model defines δ as the average value of inner and outer steer angles to simplify the further steps of development. cot δ = cot δo + cot δi 2 (2) High speed cornering results in high lateral acceleration which demands high levels of slip angles and Ackerman geometry might not be the ideal solution anymore. Body roll is a major consequence of high speed cornering, it changes normal load in inner and outer wheels and, therefore, their capabilities of exerting corner force. Suspension movement resulted from body roll also interferes with steering parameters making the vehicle handle better or worse depending on the project. Even though steering parameters (angles like caster and camber) may vary because of vehicle movement, they are not adaptive and can not be changed on-the-fly. Meaning that different situations require them to be set and optimized for an intended use while considering all the suspension geometry and movement.
- 33. 31 Steering can be a purely mechanic system but in most cases it includes some sort of assistance to decrease the driver’s effort and improve comfort. Power steering might be hydraulic or electrical with some configurations having no actual mechanical connection between steering wheel and steering box, the system relies on sensing the driver’s input and acting accordingly. Steering ratio is the angle of rotation of the steering wheel compared to the angle of rotation of wheels (JAZAR, 2009). The average magnitude for street vehicles is around 10:1 (varying widely between models and brands), while other vehicles for different applications like race cars or heavy duty vehicles will, naturally, have different values. Also, some applications use a non-constant value configuration, where vehicles feature a variable ratio steering box, whether applying a variable mechanical rack-and-pinion system or power steering. Steering ratio might vary to increase comfort, safety or aggressiveness for different steering wheel angles, speeds or drive modes. 2.4 Magnetic Position Detection 2.4.1 Hall Sensor A current-carrying conductive plate crossed by a magnetic field perpendicular to the plane of the plate develops a crossing potential voltage, and this is called Hall Effect. The Lorentz force is the main physical principle of this effect. A moving electron traveling in a magnetic field generates a force shown in Equation 3. −→ F = q−→v × −→ B (3) Where −→ F is the resulting force, q is the electrical charge of the electron, −→v is the velocity of motion and −→ B is the magnetic field. The trajectory of the electron changes because of the resulting force, developing a potential voltage across the plate shown in Equation 4. VH = IB⊥ ρnqt (4) Where VH is the Hall Voltage, I is the current passing through the plate, B⊥ is the perpendicular magnetic field, ρn is the number of carriers per volume, q is the charge and t is the thickness of the plate. This setup is shown in Figure 6. The magnetic Hall sensor uses a Hall element to generate the Hall voltage VH and some electronics to amplify the signal. Some have ratiometric output with Vcc 2 when there is no magnetic field applied, Vcc when a south pole is detected and GND when a north pole is detected (MILANO, 2013). 2.4.2 Position Detection The detection of a device’s position can be done via a magnetic system. A magnet moving attached to the device to be measured with relative motion to a magnetic sensor results in
- 34. 32 Figure 6 – Lorentz Force and Hall Effect Source: Milano (2013). a modulation of the magnetic field that is detected by the sensor and translated into mechanical position, orientation or both. A new vehicle can hold up to 80 applications of magnetic sensors like: wheel speed detection, pedals’ positions, steering wheel angle, crankshaft and camshaft position, valve position, transmission speed/gear position/actuator, gear stick, oil pump, control elements, window lifter, among others. (TREUTLER, 2001), (HEREMANS, 1997), (RIBEIRO; ORTNER, 2015). Some advantages of a magnetic position detection system compared to others is: • Robustness: – Contactless measurement and, thus, wear free; – Long lifetimes, because modern magnets last up to decades; – Perfect for machinery because oil, water, grease and dirt do not influence the magnetic field, so there is no need for airtight seals or other environmental contamination control; – Temperature and mechanical pressure are compensated on chip. • The source (magnet) is mounted on the moving part and does not require cable access; • High precision, low power requirements, potential for miniaturization; • Inexpensive to manufacture since the advent of Hall sensors. One and two dimensional Hall sensors are very common and most of the current applications require only 1D and 2D field detection. The recent introduction of a 3D magnetic
- 35. 33 Hall sensor opens the questions for new applications with more complex movements and position detection, for improved old applications and for additional information that can be obtained from the third component of the field. Modern Hall type sensors are highly linear, which means that the sensor output signals can be directly interpreted as the magnetic field components themselves. These sensors have several embedded digital circuitry for signal processing, most of the time aiming for temperature or pressure compensation or signal amplification. There is also the possibility for on-chip signal processing to give position output directly.
- 37. 35 3 CONTEXT This chapter presents the most common methods used to measure suspension displace- ment such as potentiometers and accelerometers and ideas to replace these system with magnetic devices. Other types of sensor to measure displacement such as optical devices, capacitive and inductive sensors are not commonly used to measure suspension displacement even though they are well known in industry. 3.1 Suspension Displacement Measurement There are two main approaches to use accelerometers, the first uses two sensors, one in vehicle body and the other in the wheel. This configuration allows the system to measure the difference in acceleration between unsprung15 and sprung16 masses, and by integrating two times the response, the system gets the displacement of the suspension. Another approach involves only one accelerometer in vehicle body and the acquisition of the body acceleration. The acceleration along with the parameters of the suspension components allow the system to estimate the movement of suspension. Potentiometers can be easily used to measure displacement because of their simplicity and direct correlation between output signal and proportion of movement. There are some linear models, but the most widely used ones have rotational movement. Simple potentiometers have three connectors, a Vin, a ground pin (GND) and the Voutput, where the value of Voutput varies between 0 V and Vin linearly proportional to the range of movement. Some simple mechanical system is then attached to the sensor to convert linear movement from suspension to rotational movement of sensor. The value of Voutput is translated directly into displacement, therefore this method is simple and reliable. In semi-active suspension there are two types of action, using a damper made out of ferromagnetic fluid with capability of changing its viscosity when in the presence of magnetic fields and with an electromechanical valves that vary their diameter to change fluid’s flow velocity. In active suspension the types of action are usually pneumatic system. The Bose system uses electromagnetic valves with the same type of action of loudspeakers to control suspension movement. Other types of active suspensions with hydraulic systems or other technologies are 15 The mass of the wheels system and part of the mass of the suspension system, this is the mass directly in contact with the ground. 16 The mass of the vehicle that is supported by the suspension system such as vehicle body and occupants.
- 38. 36 not widely used. A list of the different makes and their sensing technology is presented in Table 1 and Table 2. The sensing systems approached by Table 1 and Table 2 are: • Two accelerometers; • Potentiometers; • Magnetic sensors; • One accelerometer with estimation of displacement. Table 1 – Semi-active suspension / Mechanical vs. Sensor technology Semi Active Suspension Types of action Ferro-fluid / MagneRide Electro-Mechanical valves Types of sensor Accelerometer No make found. Ducati; BMW. Potentiometer Audi; Ferrari; Lam- borghini; Vauxhall; Cadillac; Buick; Chevrolet; GMC. BMW; Audi; Ford; Mercedes- Benz; Volkswagen ; Volvo. Magnetic Sensor No make found. No make found. G-Force based 1 No make found. Öhlins-Kawasaki. 1 Does not actually measure the displacement, gets data from acceleration of vehicle among with other information such as throttle and brake pedals position and steering angle. Source: Author, 2016. All companies using ferromagnetic fluid as a mean to vary damping coefficient apply rotational movement potentiometer with some mechanism to transform the linear displacement range into angular measurement. They rely in a ratiometric analog output signal along with a stand-alone ECU that might be integrated with other chassis systems (Delphi Corporation, 2005). Contrarily to magneride, electro-mechanical valves are developed by several companies, therefore automakers can do partnerships or design themselves a specific system to each model or brand as seen in Table 1. Full active suspensions are not really common in large scale, Mercedes-Benz have their Magic Body Control that uses potentiometers to measure the displacement of each suspension together with a stereo camera that scans the road up to 15 m ahead of the vehicle with accuracy of 3 mm in height measurement (Mercedes-Benz, 2013). Other systems of full active suspension either have no data about their sensors or were not implemented in production vehicles, such as the Bose System.
- 39. 37 Table 2 – Active suspension / Mechanical vs. Sensor technology Active Suspension Types of action Full Active Suspension Full Active Bose Suspen- sion Types of sensor Accelerometer No make found. No make found. Potentiometer Mercedes-Benz. No make found. Magnetic Sensor No make found. No make found. G-Force based No make found. No make found. Source: Author, 2016. 3.2 Suspension Measurement with Magnetic Sensor A trivial solution to measure suspension displacement with magnetic sensor is shown in Figure 7. A magnet and a magnetic sensor replace the potentiometer, also using the same mechanism to transform linear displacement into angular movement. The angle of the magnet can be measured easily from this approach and, therefore the suspension displacement. This setup also allows any range of displacement because it depends solely on the mechanism. Figure 7 – Angular measurement with magnet sensor Plug Angle Sensor Source: Author, 2016. Another simple solution consists in a linear position measurement of a magnet that moves along with the suspension by a magnetic sensor. This setup is shown in Figure 8 and in this case, the displacement range depends on the magnetic system so it has to be located where it measures a rate of the suspension movement inside its limits. Because of the arc nature of
- 40. 38 the movement, the closer the magnetic system is to vehicle body (attachment point between suspension linkages and vehicle body), the smaller is the amplitude of magnet’s movement. Figure 8 – Linear measurement with magnet sensor Hub Suspension Position Sensor Vehicle Body A-Arms A-Arms Source: Author, 2016. Both systems are simple, reliable and would meet the standards to replace other systems in smart suspensions although they are not achieving the full potential of nowadays magnetic sensors, since new sensors have three components of magnetic field and a simple system only uses two of them. 3.2.1 Steering Measurement Including the measurement of steering angle movement using the same sensor and magnet is an improvement possible because of 3D sensors. These extra components allow a magnet-mechanic coupling that measure more complex movements than strictly linear or angular ones. An angular input in the steering wheel gives a linear output in the rack, this movement is then again transformed into angular movement of the sensor by a mechanism shown in Figure 9. The suspension movement is acting directly over the magnet, so the magnetic system has to be precisely placed where the full range of the suspension arms is within the limits of measurement of the sensor. A full sketch of the proposed solution to measure both suspension and steering movements with one sensor and one magnet are shown in Figure 10. Two simple movements generate a complex movement into the central point where the sensor is located. This mechanical setup allows completely isolation from one movement to the other, and even though the magnetic fields are affected by both movements, some magnetic mapping can extract accurately the two(linear and angular) positions out of the three magnetic field components measured by the sensor.
- 41. 39 Figure 9 – Linear movement into angular Steering wheel angle (Rack) Rotary Movement Source: Author, 2016. Figure 10 – Linear and angular measurements combined A-Arms Vehicle Body Steering wheel angle (Rack) Suspension Linear Rotary Source: Author, 2016.
- 43. 41 4 MATERIALS This chapter presents all the materials used to develop this work. It covers the majority of the used software, hardware, sensors, magnets, manufacturing methods, etc. 4.1 Softwares The section of used software consists of all the softwares used to create algorithms, to design the parts, to read the sensor values, and others. All the algorithms of this work were developed using Python 2.7.11 Anaconda distribu- tion, using the PyCharm Community Edition 2016.1.3 as integrated development environment (IDE). All magnetic simulations were performed using a Python module created by Mr. Dr. Michael Ortner - Researcher from CTR AG that supervised this work. Also, a plotting configura- tion module was used to improve the graphical quality of the plots in this thesis. The parts were designed completely using SolidWorks 2016. The whole system was assembled inside SolidWorks and all the movements were simulated generating some videos to illustrate the motion of the proposal. All the printed parts were saved in stereo-lithography (STL) as well as the SolidWorks standard part format. To organize the bibliography and the several links, blog posts and documents, the software Zotero was used. It creates a bibtex file to be used as the bibliography source for this work. The illustrations and sketches were drawn using Inkscape 0.91. The motivation to use this software was mainly its open-source nature and the good capabilities of vectorial drawing. Some files were exported as eps or pgf file format to keep its vectorial nature, allowing text selection in some graphs and no resolution loss when zooming. Regarding the communication softwares, between robot and computer and between micro controller and computer, all of them were developed by BSc. Marcelo Ribeiro - Researcher from CTR AG that supervised this works. A computer is used as a master in a master-slave communication between the robot controller and a computer. The master computer controls the robot position sending commands to the slave computer that sends information back if the movement was done correctly or not. Velocity and power of the robot may be configurable through the master computer’s software. The software is able to get the raw data from the sensor and process it to get the magnetic field. Inside the communication software, a plot with the three components of magnetic field is shown.
- 44. 42 4.2 Robot A robot arm is used to calibrate the system and to move the magnet. The robot is manu- factured by Seiko Epson Corporation and the model is the E2C351S and is shown in Figure 11. It has four axes of movement, three with rotational movement and one with translational movement, that allow the robot arm to move in four directions, i.e., x, y, z and u. Figure 11 – Epson E2C351S Source: (Seiko Epson Corporation, 2003). The Annex A shows a table with the specifications of the robot. Among them, the maximum velocity of the axes are: 3600 mm/s for the first axis, 3600 mm/s for the second axis, 1100 mm/s for the third axis and 2600 deg/s for the fourth axis. The maximum precision is: ±0, 010 mm for the first axis, ±0, 010 mm for the second axis, ±0, 010 mm for the third axis and ±0, 015 deg for the fourth axis. The maximum amplitude is: ±110 deg for the first axis, ±145 deg for the second axis, ±120 mm for the third axis and ±360 deg for the fourth axis. The power consumption is 150 W for all servomotors that move the four axes. 4.3 Microcontroller The microcontroller and the protocol used to read the sensor signal were one Arduino Leonardo board and I2 c bus, respectively. The Arduino Leonardo is a microcontroller board based on the ATmega32u4 processor. It has 20 digital input/output pins, a 16 MHz processor, a micro USB connection, a power jack and a reset button (Arduino LLC, 2012). From the 20 i/o pins of the board, the sensor connects to only 4 of them: Ground (GND), Vcc of 3.3 V , pin 2 (SDA) and pin 3 (SCL). The software developed for the microcontroller reads the sensor 12-bit raw values and can deliver as output the raw values of the sensor or the processed values of magnetic field.
- 45. 43 Figure 12 – Arduino Leonardo Source: (Arduino LLC, 2012). 4.4 Sensor The 3D magnetic sensor TLV493D-A1B6 detects the magnetic flux density in three directions; x, y and z as shown in Figure 13. Figure 13 – Sensor TLV493D-A1B6 Source: (Infineon Technologies Austria AG, 2016). Among its features are: • 3D Magnetic sensing; • Very low power consumption, i.e., 10 µA during operations; • Digital output via 2-wire based standard I2 C interface up to 1 MBit/s; • 12-bit data resolution for each measurement direction; • Bx, By and Bz linear field measurement up to ±130 mT; • Supply voltage range from 2.8 V to 3.5 V ; • Working temperature range Tj from −40◦ C to 125◦ C; • Temperature measurement; The directions of the axes of measurement are shown in Figure 14.
- 46. 44 Figure 14 – Sensor axes of measurement Source: (Infineon Technologies Austria AG, 2016). 4.5 Magnet Several magnets were considered during the process of this work and after some simu- lation and test, the chosen magnet to perform the experiments is the 7x7x25mm Y30BH Ferrite Magnet. It is 25 mm long with a square section with 7 mm side. The remanence magnetization field is Br = 380 ∼ 400 mT, the coercivity is bHc = 230 ∼ 275 kA/m and the maximum operation temperature is Tmax = 250◦ . The ferrite magnet is, usually, produced by powder metallurgical method with chemical composition of BaO : 6Fe2O3. The ferrite magnets are relatively brittle and hard and special machining techniques should be used in case machining is needed. (ChenYang Technologies GmbH, 2006). Some other information and physical properties are presented next: • Good resistance to demagnetization; • Excellent corrosion resistance; • Good temperature stability; • Curie temperature: 450◦ ; • Hardness: 480 ∼ 580 Hv; • Temperature coefficient of Br: −0.2 % ◦C ; • Temperature coefficient of iHc: 0.3 % ◦C ; • Tensile strength: < 100 N/mm; • Transverse rupture strength: 300 N/mm. More information on the magnet can be found on the Annex B.
- 47. 45 4.6 Manufacturing The 3D printer used to produce the parts for this work is the ProJet 3510 HDPlus, manufactured by 3D Systems and presented in Figure 15. It has three printing resolutions, i. e., High definition (HD), Ultra high definition (UHD) and Extreme high definition (XHD). (3D Systems, 2016). The HD resolution has 375 by 375 by 790 dots per inch (dpi) in the x, y and z axis, respectively, and the result is a 32 µm high layer of print. The UHD resolution has 750 by 750 by 890 dpi and 29 µm high layer. The finer resolution, XHD, has 750 by 750 by 1600 dpi and a 16 µm high layer. The accuracy is from 0.025 to 0.05 mm per 25.4 mm in all resolutions. Figure 15 – ProJet 3510 HDPlus Source: (3D Systems, 2016). The materials used to print the parts are the VisiJet S300 for the support material and the VisiJet M3 Crystal as the basic material. The properties of the materials are presented in Figure 16. Figure 16 – Printing Materials Properties Source: (3D Systems, 2016).
- 48. 46 The software used for printing is the 3D Modeler Client Manager also from 3D Systems. The software is used to send the STL files to the printer.
- 49. 47 5 DEVELOPMENT This chapter presents the development of this work. It explains the mathematical analysis, design and production of the mechanism responsible to convert linear movement of steering rack into rotational movement of the sensor, the magnetic system and magnetic mapping used to translate three components of magnetic field into two independent movements and the robot experiments setup. 5.1 Mechanism Designing the mechanism with feasible dimensions (as if it is going to be applied in a car) was the main target of development. The relation between linear movement and rotational output has to be well known and designed to allow enough angular range of the sensor from the rack displacement. The mathematical analysis and the values considered in the development process are shown in subsection 5.1.1. 5.1.1 Mathematical Analysis Considering the sensor a fixed point with only rotational movement possible, a Cartesian coordinates system zero can be set on sensor’s position, called point A (the final mechanism can be seen in ??). This crank mechanism is composed of two connection rods, one connected directly to the sensor, or point A, and the other directly to the rack of steering system, or point C, the connection point between these two rods is called point B. The crank mechanism usually converts continuous rotational movement into reciprocating linear movement (or vice-versa), but it can also travel just a part of both movements, like in this application. Point A has coordinates (0, 0) in this system, point C can be assumed to only move in one axis to ease the calculation of point B. Because points A, C and the linear movement of C are not collinear, the approach used to calculate point B coordinates is to find the intersection points of two circles with radius r1 and r2, where r1 is the length of the connection rod between rack and point B and r2 is the length of the sensor arm connecting the sensor with point B. Assuming points A and C as the center of the two circles, depending on the position of these two points, there might be zero, one or two points of intersection betweens the two circles. Zero or one points are not useful in this setup because it would mean no actual connection between the two rods or a connection forming a straight line impossible to move because the
- 50. 48 force direction and the two rods would be collinear, resulting in an attempt of translational movement instead of rotational. The two intersection points configuration is shown in Figure 17 with one of the points highlighted as point B. Assuming that C only moves in x axis, its y coordinate, as well as A coordinates are invariable. The yellow line shows the movement of C, generating point C , a new circle that is shown dashed in Figure 17 and, the point B for the new intersection point. The angle between the sensor arm and y axis of this coordinate system is defined as β and when its value is zero, the sensor axes are completely aligned with the coordinate system. Positive values of β are assumed when point B is in the third quadrant and negative values when B is in the fourth quadrant. Figure 17 – Two Circles Intersection Source: Author, 2016 For this configuration the following can be defined: d = (x1 − x2)2 + (y1 − y2)2 (5) Where d is the distance between points A and C, x1 and y1 are the x and y coordinates of point A, respectively and x2 and y2 are x and y coordinates of point C. l = r2 1 − r2 2 + d2 2d (6) Where l is the distance from C to the line joining the two points of intersection. They are perpendicular to each other so a small right-angled triangle with legs l and h and hypotenuse
- 51. 49 r1 can be defined. h is then calculated by: h = r2 1 − l2 (7) The x and y coordinates of B are then calculated: x = l d (x2 − x1) ± h d (y2 − y1) + x1 (8) y = l d (y2 − y1) h d (x2 − x1) + y1 (9) Two coordinates of x and y are presented in Equation 8 and in Equation 9, because there are two points of intersection between two circles. In this case though, only one set of coordinates is important and the lowest point is the chosen to be the connection between the two rods. With the coordinates of point A and B it is possible to calculate the β angle of sensor as seen in Equation 10. β = arctan xB yB (10) The graphical interface shown in Figure 18 is of a calculator developed using Python to verify the range of angular output when varying the position of point C, the length of r1 and r2 and the steer movement. It allows a preciser design of the parameters and a better knowledge of the expected output. A larger version of this figure is shown in Appendix A. Figure 18 – Mechanism calculator −250 −200 −150 −100 −50 0 50 100 [mm] −150 −100 −50 0 50 [mm] Sensor arm movement Mechanism −80 −60 −40 −20 0 20 40 60 80 Steering linear movement [mm] −60 −40 −20 0 20 40 60 β:sensorangle[◦ ] Linear vs rotatory movement Sensor Arm [mm] 100.00 Connection Rod [mm] 175.00 Steer [mm] -30.00 Cx [mm] -174.71 Cy [mm] -90.00 Reset Values Source: Author, 2016 Through the calculator some parameter values were defined as shown: • Sensor arm: 100 mm • Connection rod: 175 mm • Point C coordinates: [174.71, -90] mm
- 52. 50 The value of Cx is more precise because it was adjusted so when the steering wheel is at 0◦ , the sensor is also at 0◦ . Assuming a range of 150 mm of rack linear travel, the output of the sensor is a range of 97.6◦ varying from -47.6◦ to 50.0◦ , when the steering wheel is completely to the left and right side, respectively. This asymmetry is not a problem because the output angle can be translated to rack linear movement and, with the steering ratio, the steering angle is easily found. So finding the range of β is important to ensure that there is enough range of sensor rotation to map the whole steering wheel positions range. But the mechanism’s main function is to find the steering angle using the sensor angle β as input, and it can be done by first calculating point B using Equation 11: B = [−r1 sin β, −r1 cos β] (11) Then, the value of linear rack travel can be calculated if point B and the coordinate y of point C are known, through: r2 1 = (xB − xC)2 + (yB − yC)2 (12) Since r1, yB, yC and xB are known and invariable, it is easier to create a constant k to group all of this values: −x2 C − 2xCxB = −x2 B + r2 1 − (yB − yC)2 (13) And then: k = −x2 B + r2 1 − (yB − yC)2 (14) So, applying Equation 14 in Equation 13 and multiplying it by -1 gives: x2 C + 2xCxB − k = 0 (15) That can be easily solved to find two solutions to xC, where the smallest is the actual value of the instant rack linear movement, this method needs a previous calculation of point B and an input of the coordinate y of C though. An approach of curve fitting can make this an one-step calculation, therefore reducing the time to translate β into rack movement, while keeping the error below a threshold. Both methods still give the position of the rack as output, which needs to be multiplied by the steering ratio to get steering wheel angle. C(β) = 1.6095β − 172.60 (16) A first linear approximation is shown in Figure 19 using Equation 16, presenting the comparison between the curve of C as a function of β using the previous approach and a linear fit. The red curve presents the error between the two curves, with a peak of more than two percent when the angle is greater than -40◦ . The R2 value is 0.9968, which is good for some applications,
- 53. 51 but in this experiment (together with some over 2% peaks or error) is not acceptable because this is a systematic error that influences every measurement and result. Figure 19 – Linear fitting of C(β) −40 −30 −20 −10 0 10 20 30 40 Beta Angle [◦ ] −250 −230 −210 −190 −170 −150 −130 −110 −90 RackTravel[mm] −2 −1 0 1 2 Error[%] C vs β C vs β - 1st degree Approximation Error Source: Author, 2016 C(β) = −8e−5 β3 + 2.1e−3 β2 + 1.736β − 170.67 (17) A third degree polynomial equation can be fitted to the original curve to decrease the error. The Equation 17 is shown in Figure 20 together with the comparison curve and the maximum error, that in this case is slightly smaller than two percent. Even though the maximum error is still big, the average error is not, fluctuating in between -0.3% and 0.3% for most of the values. The R2 value in this case is 0.9999531. Figure 20 – 3rd degree polynomial fitting of C(β) −40 −30 −20 −10 0 10 20 30 40 Beta Angle [◦ ] −250 −230 −210 −190 −170 −150 −130 −110 −90 RackTravel[mm] −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Error[%] C vs β C vs β - 3rd degree Approximation Error Source: Author, 2016 C(β) = 6e−10 β5 + 7e−7 β4 − 9e−5 β3 − 3.5e−3 β2 + 1.7449β − 170.37 (18)
- 54. 52 Using a fifth degree polynomial fit naturally produces less error values. The Equation 18 is shown in Figure 21 and both curves fit so well that the two lines look one. The R2 value in this case is greater than 0.9999999. The maximum error is around 0.25% but almost the whole range of angles has almost zero error. Figure 21 – 5th degree polynomial fitting of C(β) −40 −30 −20 −10 0 10 20 30 40 Beta Angle [◦ ] −250 −230 −210 −190 −170 −150 −130 −110 −90 RackTravel[mm] −2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 Error[%] C vs β C vs β - 5th degree Approximation Error Source: Author, 2016 Increasing the degree of the polynomial fit would continuously decrease the error, but would also increase the processing time. A small test was performed with an algorithm in Python to test the time needed to process different amount of points using the direct method and the equations fitting. Figure 22 shows the amount of time needed for each one of these equations. As expected the behavior when increasing the number of points is linear, the results in Figure 22 are plotted in a log-log plot though, because the number of points to process also increased by the power of 10. Results smaller than 10−3 s or 1 ms were not caught by the algorithm, but the plot still shows clearly the ratios between each method. Figure 22 – Time test for polynomial approximation 101 102 103 104 105 106 107 Number of points 10−4 10−3 10−2 10−1 100 101 102 103 Processingtime[s] Direct method Linear fit 3rd degree polynomial 5th degree polynomial Source: Author, 2016
- 55. 53 The ratio between the direct method and the 5th order polynomial (the slowest of the fittings) equation is around 100. This means that some processing time can be saved in this task so it can be used in other parts of the system. But assuming this was the only task to be executed by the system, because of the order of maximum calculating frequencies of theses equations the time of execution is not an issue. Even the direct method has a frequency of 30 kHz in this test, which is a lot bigger than the expected 5.7 kHz of the update rate of the sensor in Fast Mode (Infineon Technologies Austria AG, 2016). 5.1.2 Components Development The components were designed to be as compact as possible but robust enough to handle all the movements and forces. The first prototype was designed for the robot tests, but using real geometry that could be applied to a car. The mechanism system design consisted of three central parts that hold the Printed Circuit Board (PCB), a magnet holder that is fixed to the robot, a stem connected to the sensor and the system holder. The PCB layout was defined as a circle of 16 mm of diameter. Two resistors, one capacitor and the sensor are the components of the PCB. It has four connections: VDD (power supply), GND, Serial clock (SCL) and Serial data signal (SDA). Appendix C shows the PCB layout made with the software Eagle and electronic circuit. The upper part of the magnet holder has a fixed geometry because it is fastened to the robot while the lower part is designed according to the magnet size. It needs to fit the magnet tightly so no relative movement occurs while the robot moves simulating the suspension displacement. A round part is designed to fix the PCB, it has an internal duct for cables and its shape needs to be circular because this is the main rotation part of the system. Figure 23 shows the PCB holder part with its cable channel. Figure 23 – PCB holder Source: Author, 2016 Two other parts are developed to involve the PCB holder preventing it to make trans- lational movement while it is loose enough to keep friction low so rotational movement is performed easily. One of these parts contains a duct where the magnet holder slides through and
- 56. 54 the other part contains the fasteners that keep the system together. Figure 24 shows a cut of the three parts and their final assembly. Figure 24 – System cut and partial assembly Source: Author, 2016 The sensor arm is a stem with 100 mm between the two connection’s centers. It connects with the PCB holder through a hexagonal shaped hole to ensure that every motion of the sensor arm is transfered to the sensor. The final assembly of this part of the mechanism is shown in Figure 25. All components were exported as stereo-lithography (STL) files and printed in order to keep the system light, precise and robust enough. Figure 25 – Final assembly Source: Author, 2016 5.2 Magnetic Map This section presents the development of the magnetic map with the mathematical analysis to detect linear and angular positions, as well as a combination of these two positions using the magnetic field components. 5.2.1 Mathematical Analysis State of the art linear position detection requires two components of magnetic field, the tangential component in the direction of the movement and the vertical. Figure 26 shows
- 57. 55 the magnetic setup and the simulated x and z components of magnetic field in a linear position measurement. Figure 26 – Linear position Source: Author, 2016 Because of the dipole field geometry one component will be an even function (x component) of the displacement and the other will be odd (z component). Magnetic position sensing is based on properly choosing the system parameters so the field components are approximately harmonic in a (small) range of d around the origin. The error of this approximation is called intrinsic and depends on the specific setup, i.e. the geometry of the magnet, the stroke S and the air gap ∆. To achieve a linear map between the position d and the sensor output Spos(d) the chip combines the two signals using Equation 19. Spos = arctan KBvert Btan − π 2 sign(Btan) (19) The output of the Equation 19 is typically monotone but not completely linear. The tune of the factor K makes the output more linear inside a range of d slightly smaller than the length of the magnet. An optimal K factor depends on the choice of the magnet, the air gap ∆ as well as the length S of the desired stroke. Using the ratio between magnetic fields outperforms systems with only one component because: • The total amplitude of the field shifts between the two components along the stroke, the combination via arctan provides a much larger possible measurement range; • The system is more stable against variations of ∆ because, while the individual components of magnetic field strongly varies with ∆, the ratio between the different components does not. A simple angular detection magnetic system features a 2D sensor and rotating diamet- rically magnetized magnet. The rotation center of the magnet and the center of the sensor are
- 58. 56 aligned, therefore, the magnetic field is always anti-parallel to the direction of the magnetization and the two transversal components of the sensor can easily determine the angular position using Equation 20. Figure 27 – Angular position Source: Author, 2016 Sang = arctan Bx By (20) The magnetic setup of an angular position detection system is shown in Figure 27, together with the x and y components of the magnetic field when the magnet rotates from −180◦ to 180◦ . All the 360◦ of rotation are covered by this magnetic map and in case of a small misalignment, a calibration process is needed. The output signal’s precision is dependent only of the signal to noise ratio. Figure 28 – Combination of movements Source: Author, 2016 Both position detection systems are shown in Figure 28 in the same magnetic setup. The magnet moves in d and its position is calculated using Equation 19. The sensor rotates in β and its value is calculated with Equation 20. The values used by Equation 19 and Equation 20 are calculated by: Bvert = Bz (21)
- 59. 57 Btan = B2 x + B2 y (22) The angle β is the instant angular position of the sensor and is found by: β = arccos Bx Btan = arcsin By Btan = arctan By Bx (23) A combination of all components is calculated by Equation 24 and is named Bamp, as shown in Equation 24. Bamp = B2 x + B2 y + B2 z (24) 5.2.2 Magnetic System Simulations of the magnetic field were performed to evaluate the magnetic setup. The main variables of the setup are: the magnet dimensions, the air gap ∆ and the magnet’s magnetic remanence Br. A sweep movement - varying linearly the position from a point to another - of the magnet is simulated with a constant value of sensor angle β. The direction of the magnet’s movement is the sensor z axis, moving from −d to d, where d is a value adjusted to deliver the best results of linear and angular position. The results of the simulations are the three components of magnetic field, which are used to calculate Bvert, Btan and Bamp are then calculated. Figure 29 – Magnetic field simulation in sensor plane −30 −20 −10 0 10 20 30 −20 −15 −10 −5 0 5 10 15 20 Source: Author, 2016 Figure 29 shows a plane with the vectors of the magnetic field that are ∆ distant from the surface of the magnet, this is a two dimension representation of the three dimension vectors of the field. The black rectangle is representing the sensor with β = 0◦ , and the red and blue rectangle represents the magnet, where the red side is the north pole and the blue side is the south pole, when d = 0. In Figure 29, both parts are perfectly aligned so the magnetic field is symmetric and behaves as expected. But because of building tolerances and construction imperfections, misalignments occur and the magnetic system is not completely symmetric anymore. The three components of magnetic field are shaped accordingly and follow this non-symmetric behavior.
- 60. 58 Figure 30 – Magnetic field simulation −30 −20 −10 0 10 20 30 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx(0.0) By(0.0) Bz(0.0) Btan Bamp (a) Magnetic field simulation −30 −20 −10 0 10 20 30 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx(45.0) By(45.0) Bz(45.0) Btan Bamp (b) Magnetic field simulation (45◦) Source: Author, 2016. Figure 30a shows the components of magnetic field together with the calculated values of Btan and Bamp. This scenario occurs when the sensor and magnet are aligned and β = 0◦ and d = 0. In Figure 30a all the components are symmetric because this simulation shows the ideal behavior of the system. The Bx component is constant and null because β = 0◦ , which is expected since an input of Bx = 0 in Equation 20 results in β = 0◦ . Figure 30b shows the same magnetic setup of Figure 30a, but with β = 45◦ . In this case, Bx and By components have the same value as expected. In both cases, Bz, Btan and Bamp are the same because these values are not affected by β. Several cases of building misalignments were simulated to help the calibration process of the system. To predict the behavior or to compare the experimental results, some simulations were performed introducing noise or displacements, this allows a better understanding of the physical system and which components and tolerances are more important. Figure 31 – Linear position simulation at ∆ = 2 mm −30 −20 −10 0 10 20 30 Travel [mm] −40 −20 0 20 40 MagneticField[mT] Bx By Bz −30 −20 −10 0 10 20 30 Travel [mm] −3 −2 −1 0 1 2 3 CalculatedTravel[mm] −15 −10 −5 0 5 10 15 Error[%] Calculated travel Linearization of slope Error Source: Author, 2016 Results from the simulation show that the linear position measurement through Equa- tion 19 works better if the air gap ∆ increases. When ∆ is too small, i.e., ∆ < 2 mm, the K
- 61. 59 factor of the Equation 19 has less effect and the result is less linear for every case, as shown in Figure 31. The Figure 31 shows the linear position calculation along with a linearization of the output. The maximum error is 12.28 % and the R value is R = 0.99919. The error formula used in this approach divides the absolute deviation Di by the value of d, as shown in Equation 25. Di = |xi − m(X)|; Error = Di xi · 100 (25) Where Di is the absolute deviation, xi is the calculated travel value d, m(X) is the central tendency (the linearization) and the Error value is the error in [%]. Figure 32 – Linear position simulation at ∆ = 5 mm −30 −20 −10 0 10 20 30 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −30 −20 −10 0 10 20 30 Travel [mm] −3 −2 −1 0 1 2 3 CalculatedTravel[mm] −4 −3 −2 −1 0 1 2 3 4 Error[%] Calculated travel Linearization of slope Error Source: Author, 2016 The opposite case, where ∆ ≥ 2mm, is easily linearized with the K value, in Figure 32 a simulation with ∆ = 5 mm is shown. The maximum error is −3.84 % and the R value is R = 0.99994. It is also visible the decrease of the maximum value of magnetic field from Figure 31 to Figure 32. Both results in Figure 31 and Figure 32 should be multiplied by a factor to translate the real values of displacement. The introduction of small noises in the simulation also influences the result making it less stable. The same simulation with ∆ = 5 mm with noise inserted is shown in Figure 33. The maximum error peaks over 100 % and the R value is 0.99941. The angular position detection through Equation 20 works properly if the system is aligned, but simulations revealed that some parameters influence the precision of the angular position calculation. Misalignments between sensor and magnet can be harmless if they occur in y and z components, but in the sensor’s x axis direction any small displacement makes the system really inaccurate, therefore the calculation of the beta angle would only be possible to values of d close to 0. To improve the precision of the angular measurement two small modifications were proposed in Equation 20: application of a constant value Krot to adjust the curve and make
- 62. 60 Figure 33 – Linear position simulation with noise −30 −20 −10 0 10 20 30 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −30 −20 −10 0 10 20 30 Travel [mm] −3 −2 −1 0 1 2 3 CalculatedTravel[mm] −100 −50 0 50 100 Error[%] Calculated value Linearization of slope Error Source: Author, 2016 it more linear or using the linear position result f(Spos) as an input. The results of the first approach were still inaccurate. The best solution was to use the linear position result as input in a polynomial equation inside Equation 20 as shown in Equation 26. Sang = arctan Bx By + f(Spos) (26) Where f(Spos) is a polynomial equation in the form f(Spos) = p0S3 pos + p1Spos. The coefficients p0 and p1 are calculated through an algorithm that tests values to find the ones delivering the smallest mean squared error. This approach delivers good results, but is too dependent of the system’s calibration. Any movement or vibration requires a new run of the algorithm to optimize the coefficients. 5.2.3 Lookup Table Method Another method to calculate the values β and d is to use a lookup table. This approach consists of running several sweeps to gather all the values of magnetic field through all the range of linear displacement and angular motion. With these values of magnetic field, the positions d and β are calculated using Equation 19 and Equation 26 for the whole range of d and β. Two two-dimensional arrays were created to store these calculated values of d and β, respectively. This is important because of the measurement’s systematic error, for example when measuring d = −10mm and β = 10◦ , the results might be slightly different (some close values but not exactly d = −10mm and β = 10◦ ) because of the misalignments and small noises inserted to simulate the real experiment. The goal is to create two functions dout(d, β) and βout(d, β) using d and β as input, so two interpolation arrays were created using a 3rd degree spline. Since the spectrum of values is mapped inside dout and βout, it is possible to get the measurement of a point, use Equation 19 and Equation 20 to find dcalc and βcalc and compare these values to the lookup table to find the
- 63. 61 closest values of dout and βout. Also, dout and βout are functions of d and β, so the real values of d and β can be found. The two contour plots are shown in Figure 34. Figure 34 – Contour Plots for dout and βout −10 −5 0 5 10 Displacement d [mm] −40 −20 0 20 40 Angleofsensorβ[◦] -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 −10 −5 0 5 10 Displacement d [mm] −40 −20 0 20 40 Angleofsensorβ[◦] -45.00 -45.00 -30.00 -15.00 0.00 15.00 30.00 45.00 Source: Author, 2016 To find the real value of d and β, the dout and βout arrays are subtracted by dcalc and βcalc, respectively, so the new arrays dout and βout are null for dout(dcalc, βcalc) and βout(dcalc, βcalc). Then, another array is created by adding [dout]2 + [βout]2 . This array has only one null point, exactly for dcalc and βcalc. Figure 35 – Contour Plots for [dout]2 + [βout]2 −10 −5 0 5 10 Displacement d [mm] −40 −20 0 20 40 Angleofsensorβ[◦] 10.0 100.0 300.0 300.0 1000.0 1000.0 2000.0 Source: Author, 2016 The black dot comes from a random measurement that returns dcalc = 4 mm and βcalc = −15◦ , for example. Graphically is possible to get the real values of d and β by getting the x and y value on Figure 35, but for an automated process an algorithm that finds the roots of the array was implemented. 5.3 Robot Tests The robot adopted to perform the tests has four axes of movement: x, y, z and u, where the first three are translational movement and u is a rotation axis. Although only z is used to
- 64. 62 perform the tests, the other axes are needed to calibrate the system. A sketch of the experiment’s setup is shown in Figure 36a, and a picture of the final assembly is shown in Figure 36b. Figure 36 – Robot Experiment Connection Rod SensorArm Rack Travel Sensor (a) Robot Experiment Sketch (b) Robot experiment picture Source: Author, 2016. Calibrating the system is the first part of the experiment, it is needed to position all four axes in a configuration where the central point of the magnet has no relative x and y positions regarding the central point of the sensor and the surface of the magnet is parallel with the xy plane of the sensor. To accomplish this, several sweeps varying the z position of the robot are performed. Doing an initial sweep, the first coordinate is calibrated. A graphical analysis of the three components of magnetic field reveals the z absolute position of the robot where the value of Bz is zero, therefore, this is the new relative zero z position. The y coordinate of the robot only affects the distance between magnet surface and sensor, so it can be easily adjusted to the desired value without performing any measurement. The x and u coordinates of the robot affect greatly the Bx and By components of magnetic field. This coordinates’ calibration involves a trial and error approach. When the sensor arm, and therefore the sensor, is aligned the Bx component of magnetic field, ideally, is a straight line in the x axis of the plot. Any misalignment of robot’s x and u will generate a non-constant and, sometimes, non-symmetric curve of the expected Bx component for β = 0. The robot precision of 0.010 mm or 10 µm allows a very precise calibration of all axes, but other important factor to ensure a very tight alignment of the system is the sensor’s position. But as precise as the weld to the PCB might be, there will always be some minimal misalignments, meaning that the whole calibration process will not be perfect at the end, either by not giving the expected result when β = 0 or when β varies. Because the methods to calculate the displacement d or the angle β involve ratios of the different components, the system can handle minor misalignments and errors.
- 65. 63 Figure 37 – Robot Experiment Sketch x y z Steering Sensor Suspesion x z y u Robot axes Sensor axes Source: Author, 2016 A sketch of the magnet holder is shown in Figure 37, as well as a magnet representation with its north and south pole. The figure shows the two Cartesian coordinates of the system: the robot’s and the sensor’s one. The simulated suspension and steering movement are presented right next to the respective moving elements. After the calibration of the system, a sweep test delivers a result very close to the simulated data. The methodology for the following tests is to generate random points of position d and angle β, where −10 ≤ d ≤ 10 [mm] and −45 ≤ β ≤ 45 [◦ ]. The displacement d and angle β are then calculated through the two methods and compared to the reference values.
- 67. 65 6 RESULTS This chapter presents the results from the robot experiments with the magnetic field readings, the linear and angular positions calculation and the comparison with selected random points. These first experiments were performed with a sweep moving the magnet on the z axis of the robot from −15 mm to 15 mm. The air gap value is of ∆ = 3 mm. The results of the magnetic fields Bx, By and Bz are shown in Figure 38. Figure 38 – Initial results of magnetic fields Bx, By and Bz −15 −10 −5 0 5 10 15 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz Source: Author, 2016 The results of the magnetic fields Bx, By and Bz are shown in Figure 39, along with the calculation of the linear position of the magnet through Equation 19, where the value of K used was K = 1.06. Figure 39 – Magnetic fields Bx, By, Bz and the linear position calculation −15 −10 −5 0 5 10 15 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −15 −10 −5 0 5 10 15 Travel [mm] −15 −10 −5 0 5 10 15 CalculatedTravel[mm] −15 −10 −5 0 5 10 15 Error[%] Calculated travel Correct value Error Source: Author, 2016
- 68. 66 It is also shown in Figure 39 a line from (x1, y1) = (−15, −15) to (x2, y2) = (15, 15), which represents the correct values. There is also the error line, with a peak when x is close to 0. The peak happens because, as seen in Equation 25, the absolute deviation Di is divided by d, so the closer to d = 0, higher the error in [%]. The angular position detection, using the Equation 20, is shown in Figure 40 in a test performed with β = 0◦ , along with a linearization of the angular calculation result from −7 mm to 7 mm. Figure 40 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 0◦ −15 −10 −5 0 5 10 15 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −15 −10 −5 0 5 10 15 Travel [mm] −4 −2 0 2 4 CalculatedAngle[◦] −100 −80 −60 −40 −20 0 20 40 Error[%] Calculated Angle Linearization of slope Error Source: Author, 2016 Because of the setup’s misalignments and construction tolerances, the Bx component is not null for the whole displacement d, as expected. So, as seen in Equation 20, if Bx = 0, the β angle will, naturally, not be zero. The angular position results shown in Figure 40 also have two peaks, located where d is close to the end of the magnet. This happens because at the end of the magnet, By = 0, so Bx By → ±∞. From this point, the results are shown for d from −10 mm to 10 mm, to avoid showing the areas where the angle calculation is inherently inaccurate. Figure 41 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 30◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] 0 10 20 30 40 50 60 CalculatedAngle[◦] −4 −3 −2 −1 0 1 2 3 4 Error[%] Calculated Angle Linearization of slope Error Source: Author, 2016 The angular position detection, using the Equation 20, is shown in Figure 41 in a test performed with β = 30◦ . As seen in Figure 41, the results for angular position calculation when
- 69. 67 β = 30◦ are significantly better if compared to the ones shown in Figure 40. The percentage error is, naturally, smaller than in the results for β = 0◦ because the absolute deviation Di is now divided by a greater value. The error value is always under 4%. Figure 42 – Magnetic fields Bx, By, Bz and the angular position calculation for β = −30◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −60 −50 −40 −30 −20 −10 0 CalculatedAngle[◦] −4 −3 −2 −1 0 1 2 3 4 Error[%] Calculated Angle Linearization of slope Error Source: Author, 2016 The results for angular position calculation for β = −30◦ are presented in Figure 42. As in Figure 41, the percentage error is small and the general result is acceptable for this first approach. The error value is always under 3%. More results of linear and angular position calculation using the equations Equation 19 and Equation 20 are presented on Appendix D. 6.1 Lookup Table Method Application This section shows the lookup table application to the experiment data. Using the results in Appendix D, two contour plots were made to represent the values of d and β for the spectrum of motion of the experiment, i.e., −10 ≤ d ≤ 10 mm and −45 ≤ β ≤ 45◦ . Figure 43 shows these two contour plots. Figure 43 – Lookup contour plots for experiment data −10 −5 0 5 10 Displacement d [mm] −40 −20 0 20 40 Angleofsensorβ[◦] -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 −10 −5 0 5 10 Displacement d [mm] −40 −20 0 20 40 Angleofsensorβ[◦] -45.00 -45.00 -30.00 -15.00 0.00 15.00 30.00 45.00 Source: Author, 2016
- 70. 68 These contour plots show the result for d = dout(dcalc, βcalc) and β = βout(dcalc, βcalc). With the spectrum mapped into the lookup table and into these two contour plots, when a new measurement is done, the calculated outputs of dcalc and βcalc are compared with the values of dout and βout in Figure 43 to find the real values of d and β. 6.2 Random points analysis This section presents the random points analysis, where a set of random displacement d points is generated in order to create a road profile. The robot moves the magnet as it was the car suspension going trough this road profile, and the magnetic data is collected. The calculation of dcalc through Equation 19 returns a value that is plotted in Figure 44, along with the reference values of the road profile and the percentage error between the real value and the achieved dcalc. Figure 44 – Set of random points for d and β 0 200 400 600 800 1000 Distance [m] −15 −10 −5 0 5 10 15 Height[mm] Road Profile Calculated Position 0.0 0.2 0.4 0.6 0.8 1.0 −20 −15 −10 −5 0 5 10 15 20 Error[%] Error Source: Author, 2016 These experiments are presented for a single steering angle at a time. Figure 44 presents the output for β = −45◦ and Figure 45 presents the output for β = 30◦ . Figure 45 – Set of random points for d and β 0 200 400 600 800 1000 Distance [m] −15 −10 −5 0 5 10 15 Height[mm] Road Profile Calculated Position 0.0 0.2 0.4 0.6 0.8 1.0 −20 −15 −10 −5 0 5 10 15 20 Error[%] Error Source: Author, 2016
- 71. 69 The other approach, using the lookup table method, is presented in Figure 46. In this case, a parametric curve was created, to vary d and β at the same time, which is closer to reality. So a bi-dimensional array with random values of d and β is created forming a single curve. The experiment was performed for these values and their dcalc and βcalc values are calculated. Figure 46 – Set of random points for d and β −10 −5 0 5 10 Suspension travel [mm] −40 −20 0 20 40 Steeringangle[◦] Road Profile Calculated Position 0.0 0.2 0.4 0.6 0.8 1.0 −15 −10 −5 0 5 10 15 Error[%] Linear error Angular error Source: Author, 2016 The parametric curve and the calculated points of dcalc and βcalc are presented in Figure 46, along with the error in the linear and angular positions calculated data. 6.3 Discussion A system perfectly aligned delivers the best result with no systematic error, but that is not feasible. Improving the building tolerances of this system is possible because this work developed an initial small prototype, therefore, there are big opportunities of improvement regarding design and manufacturing. Other configurations could be proposed to make the system tighter or the mechanism holder could be designed to attach directly to the robot arm, so there would be no need for calibration of the x and y axes of the robot. The PCB was manually welded, so small misalignments are expected. Although the errors caused by this were minimal, this could be improved by a better method of welding. The application of this system in a car is the ultimate goal of this project. It would require small changes in the mechanism configuration to allow it to connect to the suspension linkages, but since the system does not suffer any stress while in movement, the parts could also be 3D printed. Another important task of this project goal would be to retrieve the real values of suspension displacement and steering angle. Either because the vehicle does not have these sensors or this data is not easily accessible. In any case, the correct values of displacement and angle are needed to compare with the proposed system and, to calculate the error. The magnetic map could be improved to get better results from the system. The an- gular detection is the main issue, becoming highly instable in the region where the tangential
- 72. 70 component shows a root. Other important limitation is that the angular range with the current approaches is always smaller than the magnet length. The angular detection has a high (linear) error behavior for the whole range of movement when a misalignment δt is present, and because building tolerances have a feasibility limit, this type of imperfection is always expected. Transversal displacement δt and the position of the magnet d are the only variables that affect the error, the β angle has no influence. So it is correct to affirm that = (δt, d). For small values of δt, the error is always of a similar analytical form: (δt, d) ≈ δt · ¯(d). It means that if ¯(d) is known, the error can be easily corrected. Given that Btan(x) and Bvert(x) ≡ Bz(x) and Spos(x) ≡ x, three approaches are suggested: • Scheme 1 - Linear correction: Use the position x to correct the output of Sang; Scorr ang = arctan Bx By + P · x (27) • Scheme 2 - Tangent correction: Use the ratio Bvert Btan to correct the error because both have similar shapes; Scorr ang = arctan Bx By + P · Bvert Btan (28) • Scheme 3 - Bz = By correction: The error of the y and z components are very similar for small displacements δt; Scorr ang = arctan Bx + P · Bz By (29) In all cases, P is a coefficient that must be optimized to each magnetic setup. Figure 47 shows the implementation of the three schemes to calculate the angle. In this simulation the angle β = 0◦ , so the expected result is a constant of value 0. The closer the schemes’ simulation results are to a 0 valued constant line, smaller the error. This simulation shows very good results regarding the application of theses schemes, specially the third scheme which has a maximum error of 1◦ for the whole range of movement. Applying these schemes is easier than the lookup table because it does not need a map with the whole spectrum of values, the new schemes have an output with smaller errors than the old ones and comparable to the lookup table and the third scheme makes it possible to calculate the angle even further than the tangential components roots, improving greatly the possibilities of different configurations for this system. Possible further improvements regarding magnetic mapping are: • Use two parallel magnets to improve transversal stability; • Develop other schemes that deliver smaller errors and improve stability of the system.
- 73. 71 Figure 47 – Angular Range Study Source: Author, 2016
- 75. 73 7 CONCLUSIONS The main objective of this work was accomplished, the developed system uses only one 3D magnetic sensor and one magnet and is able to detect simulated suspension and steer- ing movements successfully. The specific objectives were also accomplished because a new application involving the 3D Hall sensor is presented, the system was able to use magnetic field to determine relative points and the presented setup uses an innovative method to measure the variables in comparison with actual systems. Chassis systems and electronic assistance are growing features of vehicles, so is the need of better and more robust sensing systems. The proposal to use only one sensor to measure the combination of suspension displacement and steering angle is feasible. Both systems were simulated in the experiments: the robot moving linearly the magnet to simulate the suspension displacement and the mechanism rotating the sensor to measure steering angle. The initial results with the magnetic formulas demonstrated that the system is capable of identifying the two distinct movements from three magnetic field components. The precision of the position detection using magnetic formulas is proportional to the building tolerances, as expected. The alignment of the system is crucial to get good results, specially when it comes to the x axis influencing the angular detection. The first approach with the magnetic formulas was successful in detecting the movements, but with no correction of the systematic error the system becomes inaccurate. With the application of constants to adjust the result curves, the error goes down but it is still too high to use in a real vehicle application. The third method of calculation used a spline interpolation with values that included the systematic error, therefore, this approach gives the smallest errors. The principle of action works, it is possible to combine the measurement of suspension displacement and steering angle in a single sensor. With three components of magnetic field and a proper magnet map, the system can detect easily the positions of this complex movement. Applying the correct method of calculation, that has the smallest possible error, it would be possible to apply the system in a real vehicle and use the data as input for the chassis systems. 7.1 Future Work Proposal The suggestions for future work can be resumed in three topics: • Improve mechanical tolerances; • Change magnetic map approach;
- 76. 74 • Application of the system in a vehicle.
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- 79. Appendix
- 81. 79 APPENDIX A – MECHANISM CALCULATOR This appendix presents an image of the Python calculator developed to design the mechanism of this work.
- 82. 80 Figure A.1 – Mechanism calculator −250−200−150−100−50050100 [mm] −150 −100 −50 0 50 [mm] Sensorarmmovement Mechanism −80−60−40−20020406080 Steeringlinearmovement[mm] −60 −40 −20 0 20 40 60 β:sensorangle[◦ ] Linearvsrotatorymovement SensorArm[mm] 100.00 ConnectionRod[mm] 175.00 Steer[mm] -30.00 Cx[mm] -174.71 Cy[mm] -90.00 ResetValues Source: Author, 2016
- 83. 81 APPENDIX B – PYTHON CODES This appendix presents some Python codes used in this work. Each section presents a small explanation and the code used. B.1 Mechanism This code calculates the B coordinates from A and C coordinates and the sensor arm and connection rod length. It also calculates the β angle for the same setup. Code B.1 – Python example 1 import numpy as np 2 3 class mechCalculator: 4 5 def __init__(self, a, c, sensorarm, connectionrod): 6 self.A = a 7 self.C = c 8 self.r1 = sensorarm 9 self.r2 = connectionrod 10 11 def calculateB(self): 12 d = np.sqrt((self.A[0] - self.C[0]) ** 2 + (self.A[1] - self.C[1]) ** 2) 13 l = (self.r1 ** 2 - self.r2 ** 2 + d ** 2) / (2 * d) 14 h = np.sqrt(self.r1 ** 2 - l ** 2) 15 self.x1 = (l / d) * (self.C[0] - self.A[0]) + (h / d) * (self.C[1] - self.A[1]) + self.A [0] 16 self.x2 = (l / d) * (self.C[0] - self.A[0]) - (h / d) * (self.C[1] - self.A[1]) + self.A [0] 17 self.y1 = (l / d) * (self.C[1] - self.A[1]) - (h / d) * (self.C[0] - self.A[0]) + self.A [1] 18 self.y2 = (l / d) * (self.C[1] - self.A[1]) + (h / d) * (self.C[0] - self.A[0]) + self.A [1] 19 20 return [self.x2, self.y2] 21 22 def calculateBeta(self): 23 B = self.calculateB() 24 if self.x2 >= 0: 25 return np.degrees(np.arctan(abs(self.x2 - self.A[0]) / abs(self.y2 - self.A[1]))) 26 else: 27 28 return -np.degrees(np.arctan(abs(self.x2 - self.A[0]) / abs(self.y2 - self.A[1]))) 29 30 def calculateB2(self): 31 newCalc = self.calculateB() 32 c1 = self.x2**2 - self.r2**2 + (self.y2 + 90.0)**2 33 pos = -(-2*self.x2 + np.sqrt(4*self.x2**2 - 4*c1)) / 2
- 84. 82 34 35 return pos B.2 Other Python Code This code calculates the C position when the β angle is given. Code B.2 – Python example 2 1 import numpy as np 2 3 class cposcalc: 4 5 def __init__(self, beta): 6 self.beta = beta 7 8 def cpos(self): 9 if self.beta >= 0: 10 B = [100*np.sin(np.radians(self.beta)), -100*np.cos(np.radians(self.beta))] 11 else: 12 B = [-100*np.sin(np.radians(-self.beta)), -100*np.cos(np.radians(-self.beta))] 13 c1 = B[0]**2 - 175**2 + (B[1] + 60.0)**2 14 pos = -(-2*B[0] + np.sqrt(4*B[0]**2 - 4*c1)) / 2 15 16 return pos B.3 Robot experiment output read and plot This code is used to read the files with results from the robot experiment. The text files are read and put into variables and plotted. The figure containing the plot is saved as EPS. Code B.3 – Python example 1 # ---------- Import ---------- 2 from plotConfig_v15BrunoAx1 import * 3 4 # ---------- Data Read ---------- 5 dat1 = genfromtxt(’filename.txt’,delimiter=’,’,skip_header=0,skip_footer=0) 6 dat1 = dat1[:,5:8] 7 dat1 = array([average(dat1[100*i:100*(i+1)],axis=0) for i in range(len(dat1)/100)]) 8 9 Bx = dat1[:,0] 10 By = dat1[:,1] 11 Bz = dat1[:,2] 12 13 # ---------- Plot ---------- 14 xs = linspace(-15,15,len(Bx)) 15 16 ax1.plot(xs,Bx,color=ORANGE,label=r’$B_x$’) 17 ax1.plot(xs,By,color=TEAL,label=r’$B_y$’) 18 ax1.plot(xs,Bz,color=PURPLE,label=r’$B_z$’) 19 ax1.grid(True) 20 ax1.set_xlabel(r’Travel [$mm$]’) 21 ax1.set_ylabel(r’Magnetic Field [$mT$]’) 22 ax1.legend(loc=4,prop={’size’:14})
- 85. 83 23 24 # ---------- Save Plot ---------- 25 plt.savefig(’/Directory/filename.eps’, format=’eps’, dpi=100) 26 plotstyle() 27 plt.show()
- 87. 85 APPENDIX C – PCB LAYOUT AND ELECTRONIC COMPONENTS This appendix presents the physical PCB layout made with the software EAGLE. Figure C.1 shows the physical connections and Figure C.2 presents the sensor circuit with the obligatory components. Figure C.1 – PCB Layout Source: Author, 2016 Figure C.2 – Sensor Circuit Source: Author, 2016
- 89. 87 APPENDIX D – RESULTS OF LINEAR AND ANGULAR POSITION CALCULATIONS This appendix presents all the results for the magnetic field and the linear and angular positions calculation. As stated in chapter 6, the d range in this results is from −10 mm to 10 mm, to avoid showing the range where the calculation is inaccurate. The results will be presented in two sections, where the first shows the linear position calculation results in seven scenarios, i.e., β = [−45, −30, −15, 0, 15, 30, 45]◦ and the second section shows the angular position detection for the same scenarios. D.1 Linear position calculation Figure D.1 – Magnetic fields Bx, By, Bz and the linear position calculation for β = −45◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −10 −5 0 5 10 CalculatedTravel[mm] −120 −100 −80 −60 −40 −20 0 20 Error[%] Calculated travel Correct value Error Source: Author, 2016
- 90. 88 Figure D.2 – Magnetic fields Bx, By, Bz and the linear position calculation for β = −30◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −10 −5 0 5 10 CalculatedTravel[mm] −100 −80 −60 −40 −20 0 20 Error[%] Calculated travel Correct value Error Source: Author, 2016 Figure D.3 – Magnetic fields Bx, By, Bz and the linear position calculation for β = −15◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −10 −5 0 5 10 CalculatedTravel[mm] −80 −60 −40 −20 0 20 Error[%] Calculated travel Correct value Error Source: Author, 2016 Figure D.4 – Magnetic fields Bx, By, Bz and the linear position calculation for β = 0◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −10 −5 0 5 10 CalculatedTravel[mm] −60 −50 −40 −30 −20 −10 0 10 20 Error[%] Calculated travel Correct value Error Source: Author, 2016
- 91. 89 Figure D.5 – Magnetic fields Bx, By, Bz and the linear position calculation for β = 15◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −10 −5 0 5 10 CalculatedTravel[mm] −20 −15 −10 −5 0 5 10 15 Error[%] Calculated travel Correct value Error Source: Author, 2016 Figure D.6 – Magnetic fields Bx, By, Bz and the linear position calculation for β = 30◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −10 −5 0 5 10 CalculatedTravel[mm] −60 −50 −40 −30 −20 −10 0 10 20 Error[%] Calculated travel Correct value Error Source: Author, 2016 Figure D.7 – Magnetic fields Bx, By, Bz and the linear position calculation for β = 45◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −10 −5 0 5 10 CalculatedTravel[mm] −40 −30 −20 −10 0 10 20 Error[%] Calculated travel Correct value Error Source: Author, 2016
- 92. 90 D.2 Angular position calculation Figure D.8 – Magnetic fields Bx, By, Bz and the angular position calculation for β = −45◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −60 −50 −40 −30 −20 −10 0 CalculatedAngle[◦] −4 −3 −2 −1 0 1 2 3 4 Error[%] Calculated Angle Linearization of slope Error Source: Author, 2016 Figure D.9 – Magnetic fields Bx, By, Bz and the angular position calculation for β = −30◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −60 −50 −40 −30 −20 −10 0 CalculatedAngle[◦] −4 −3 −2 −1 0 1 2 3 4 Error[%] Calculated Angle Linearization of slope Error Source: Author, 2016 Figure D.10 – Magnetic fields Bx, By, Bz and the angular position calculation for β = −15◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −30 −25 −20 −15 −10 −5 0 CalculatedAngle[◦] −4 −3 −2 −1 0 1 2 3 4 Error[%] Calculated Angle Linearization of slope Error Source: Author, 2016
- 93. 91 Figure D.11 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 0◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] −10 −5 0 5 10 CalculatedAngle[◦] −150 −100 −50 0 50 100 150 200 250 Error[%] Calculated Angle Linearization of slope Error Source: Author, 2016 Figure D.12 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 15◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] 0 5 10 15 20 25 30 CalculatedAngle[◦] −8 −6 −4 −2 0 2 4 6 8 Error[%] Calculated Angle Linearization of slope Error Source: Author, 2016 Figure D.13 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 30◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] 0 10 20 30 40 50 60 CalculatedAngle[◦] −4 −3 −2 −1 0 1 2 3 4 Error[%] Calculated Angle Linearization of slope Error Source: Author, 2016
- 94. 92 Figure D.14 – Magnetic fields Bx, By, Bz and the angular position calculation for β = 45◦ −10 −5 0 5 10 Travel [mm] −20 −15 −10 −5 0 5 10 15 20 MagneticField[mT] Bx By Bz −10 −5 0 5 10 Travel [mm] 0 10 20 30 40 50 60 CalculatedAngle[◦] −3 −2 −1 0 1 2 3 Error[%] Calculated Angle Linearization of slope Error Source: Author, 2016
- 95. 93 APPENDIX E – RESULTS OF THE ROAD PROFILE CALCULATION This appendix presents the results for the linear position calculation in compari- son to a randomly generated road profile. As stated in chapter 6, the d range in this results is from −10 mm to 10 mm. The results will be presented in two sections, where the first shows the linear position calculation results using the Equation 19 in seven scenarios, i.e., β = [−45, −30, −15, 0, 15, 30, 45]◦ and the second section shows the linear position detection using the lookup table method. E.1 Linear position calculation Figure E.1 – Linear position calculation for β = −45◦ 0 200 400 600 800 1000 Distance [m] −15 −10 −5 0 5 10 15 Height[mm] Road Profile Calculated Position 0.0 0.2 0.4 0.6 0.8 1.0 −20 −15 −10 −5 0 5 10 15 20 Error[%] Error Source: Author, 2016
- 96. 94 Figure E.2 – Linear position calculation for β = −30◦ 0 200 400 600 800 1000 Distance [m] −15 −10 −5 0 5 10 15 Height[mm] Road Profile Calculated Position 0.0 0.2 0.4 0.6 0.8 1.0 −20 −15 −10 −5 0 5 10 15 20 Error[%] Error Source: Author, 2016 Figure E.3 – Linear position calculation for β = −15◦ 0 200 400 600 800 1000 Distance [m] −15 −10 −5 0 5 10 15 Height[mm] Road Profile Calculated Position 0.0 0.2 0.4 0.6 0.8 1.0 −20 −15 −10 −5 0 5 10 15 20 Error[%] Error Source: Author, 2016 Figure E.4 – Linear position calculation for β = 0◦ 0 200 400 600 800 1000 Distance [m] −15 −10 −5 0 5 10 15 Height[mm] Road Profile Calculated Position 0.0 0.2 0.4 0.6 0.8 1.0 −20 −15 −10 −5 0 5 10 15 20 Error[%] Error Source: Author, 2016
- 97. 95 Figure E.5 – Linear position calculation for β = 15◦ 0 200 400 600 800 1000 Distance [m] −15 −10 −5 0 5 10 15 Height[mm] Road Profile Calculated Position 0.0 0.2 0.4 0.6 0.8 1.0 −20 −15 −10 −5 0 5 10 15 20 Error[%] Error Source: Author, 2016 Figure E.6 – Linear position calculation for β = 30◦ 0 200 400 600 800 1000 Distance [m] −15 −10 −5 0 5 10 15 Height[mm] Road Profile Calculated Position 0.0 0.2 0.4 0.6 0.8 1.0 −20 −15 −10 −5 0 5 10 15 20 Error[%] Error Source: Author, 2016 Figure E.7 – Linear position calculation for β = 45◦ 0 200 400 600 800 1000 Distance [m] −15 −10 −5 0 5 10 15 Height[mm] Road Profile Calculated Position 0.0 0.2 0.4 0.6 0.8 1.0 −20 −15 −10 −5 0 5 10 15 20 Error[%] Error Source: Author, 2016
- 99. Annex
- 101. 99 ANNEX A – ROBOT SPECIFICATION TABLE This annex presents the specification table of the robot used on the experiments. (3D Systems, 2016).
- 102. Einrichten und Betrieb 2. Spezifikationen 18 E2C Rev.1 2.4 Spezifikationen Element E2C251* E2C351* Armlänge 1. & 2. Arm 125 mm / 125 mm 225 mm / 125 mm Gewicht (das Gewicht der Kabel ausgenommen) E2C**1S, C: 14 kg / E2C**1SM, CM: 16 kg Antriebsmethode Alle Achsen AC Servomotor 1. & 2. Achse 2900 mm/s 3600 mm/s Max. Geschwindigkeit *1 3. Achse 1100 mm/s 4. Achse 2600 Grad/s 1. & 2. Achse ± 0.008 mm ± 0.010 mm Wiedergenauigkeit 3. Achse ± 0.010 mm 4. Achse ± 0.015 Grad 1. Achse ± 90 Grad ± 110 Grad 2. Achse ± 135 Grad ± 145 Grad Max. Arbeitsbereich *2 3. Achse E2C**1S, SM 120 mm / E2C**1C, CM: 100 mm 4. Achse ± 360 Grad 1. Achse 0 bis +163840 -18205 bis +182045 2. Achse ± 76800 ± 82489 Max. Pulsebereich 3. Achse E2C**1S, SM: -36864 bis 0 / E2C**1C, CM: -30720 bis 0 4. Achse ± 46695 1. Achse 0.0010986 Grad/Pulse 2. Achse 0.0017578 Grad/Pulse Auflösung 3. Achse 0.0032552 mm/Pulse 4. Achse 0.0077097 Grad/Pulse 1. Achse 150 W 2. Achse 150 W Motorstromverbrauch 3. Achse 150 W 4. Achse 150 W Nutzlast Nominal / max. 1 kg / 3 kg 4. Achse erlaubtes Trägheitsmoment *3 Nominal / max. 0.005 kg⋅m 2 / 0.050 kg⋅m 2 Schaftdurchmesser / Durchgangsbohrung φ φ16 (h7) mm / φ 11 mm 3. Achse Abwärts-Kraft 150 N Anwenderverkabelung Elektrisch 15 Adrig (15-Pin D-Sub Anschluss) Anwenderverkabelung Pneumatik 2 Pneumatikschläuche (φ 6 mm), 1 Pneumatikschlauch (φ 4 mm) Zulässiger Druck: 0.59 MPa (6 kgf/cm 2 ) Umgebungstemperatur 5 bis 40°C (mit minimaler Temperaturschwankung)Umgebungsbed ingungen Relative Luftfeuchtigkeit 10 bis 80% (keine Kondensation) Equivalenter ununterbrochener A-gewichteter Schalldruckpegel *4gel) LAeq = 66.3 dB (A) oder darunter Verwendbare Steuerungen RC520 / RC420 SPEED 5 ACCEL 10,10 SPEEDS 50 ACCELS 200 FINE 10,10,10,10 Vorgabewerte WEIGHT 1,125 Reinlichheits-Grad Reinheitsklasse: 10 oder entsprechend Staubmenge: 10 Partikel oder weniger (0.13µm Durchmesser oder größer) (In 28317cm 3 (1cft) Probeluft um den Mittelpunkt des Arbeitsbereiches herum) Abluftschlauch Polyurethan-Schlauch Außendurchmesser φ 8 mm Reinraum- Modell Abluftsyste m *5 Empfohlene Abluft- Durchflussmeng e Etwa 1000 cm 3 /s (Normal) 100
- 103. 101 ANNEX B – MAGNET SPECIFICATION TABLE This annex presents the specification table of the magnet used on the experiments. (ChenYang Technologies GmbH, 2006).
- 104. 3 Magnetic Properties of Hard Ferrite (Ceramic) Magnets Material Remanence Coercivity Intrinsic Coercivity Max. Energy Product Br (mT) Br (kGs) bHc (kA/m) bHc (kOe) iHc (kA/m) iHc (kOe) (BH)max (KJ/m 3 ) (BH)max (MGOe) Y10 200-235 2.00-2.35 125-160 1.57-2.01 210-280 2.64-3.52 6.5-9.5 0.8-1.2 Y10T >200 >2.00 128-160 1.60-2.00 128-160 1.60-2.00 6.4-9.6 0.8-1.2 Y20 320-380 3.20-3.80 135-190 1.70-2.38 140-195 1.76-2.45 18.0-22.0 2.3-2.8 Y22H 310-360 3.10-3.60 220-250 2.77-3.14 280-320 3.52-4.02 20.0-24.0 2.5-3.0 Y23 320-370 3.20-3.70 170-190 2.14-2.38 190-230 2.39-2.89 20.0-25.5 2.5-3.2 Y25 360-400 3.60-4.00 135-170 1.70-2.14 140-200 1.76-2.51 22.5-28.0 2.8-3.5 Y25BH 360-390 3.60-3.90 176-216 2.20-2.70 215-231 2.70-2.90 23.9-27.1 3.0-3.4 Y26H 360-390 3.60-3.90 220-250 2.77-3.14 225-255 2.83-3.21 23.0-28.0 2.9-3.5 Y27H 370-400 3.70-4.00 205-250 2.58-3.14 210-255 2.64-3.21 25.0-29.0 3.1-3.7 Y28 370-400 3.70-4.00 175-210 2.20-2.64 180-220 2.26-2.77 26.0-30.0 3.3-3.8 Y30 385-405 3.85-4.05 176-224 2.20-2.80 184-226 2.30-2.84 27.5-30.5 3.45-3.95 Y30BH 380-400 3.80-4.00 230-275 2.89-3.46 235-290 2.95-3.65 27.0-32.5 3.4-4.1 Y32 400-420 4.00-4.20 160-190 2.01-2.38 165-195 2.07-2.45 30.0-33.5 3.8-4.2 Y33 410-430 4.10-4.30 220-250 2.77-3.14 225-255 2.83-3.21 31.5-35.0 4.0-4.4 Y35 400-420 4.00-4.20 160-190 2.01-2.38 165-195 2.07-2.45 30.0-33.5 3.8-4.2 Y35H1 395-415 3.95-4.15 251-259 3.15-3.25 255-271 3.20-3.40 29.6-32.8 3.7-4.1 Y35H2 390-410 3.90-4.10 236-295 3.30-3.70 275-299 3.45-3.75 28.8-32.0 3.6-4.04 Y35H3 405-425 4.05-4.25 223-247 2.80-3.10 231-255 2.90-3.20 30.2-35.4 3.8-4.4 Y35H-4H 370-390 3.70-3.90 270-302 3.40-3.80 326-358 4.10-4.50 25.6-28.8 3.2-3.6 Y38B 410-430 4.10-4.30 251-275 3.15-3.45 255-279 3.20-3.50 31.8-35.0 4.0-4.4 Y38H 395-415 3.95-4.15 287-309 3.60-3.90 311-333 3.90-4.20 29.5-32.7 3.7-4.1 Y40E 370-390 3.70-3.90 279-301 3.50-3.80 382-414 4.80-5.20 25.6-29.4 3.2-3.6 Y40B 410-430 4.10-4.30 290-324 3.65-3.95 307-329 3.85-4.15 32.6-34.4 4.0-4.4 Y45E 420-440 4.20-4.40 318-342 4.00-4.30 386-410 4.85-5.15 33.5-36.5 4.2-4.6 Y45B 430-450 4.30-4.50 247-271 3.10-3.40 251-275 3.15-3.45 35.1-38.3 4.4-4.8 102
- 105. 4 Typical Demagnetization Curves of Ceramic Magnets (Hard Ferrite) Y10 Isotropic Ferrite Grade Y25 Anisotropic Ferrite Grade Y30 Anisotropic Ferrite Grade Y33 Anisotropic Ferrite Grade Y30BH Anisotropic Ferrite Grade Y35 Anisotropic Ferrite Grade 103