ConstructiveControlRobotsAbridged

Nicholas Kottenstette
© Copyright 2012 All rights reserved.
Constructive Networked Control
Constructive Approaches for Design of
Networked Control Systems
November 18th 2015
Dr. Nicholas Kottenstette PhD
Principal Robotic Control Systems Engineer
Corindus Vascular Robotics
Nicholas.E.Kottenstette@ieee.org
nicholas.kottenstette@corindus.com
https://www.linkedin.com/in/nicholas-kottenstette-phd-b1b6aa14

OVERVIEW
• Multirate Networked Control Framework for Robotic Systems
– Interior Conic Systems – Properties & Feedback Control Laws
– Wireless Networked Control Architecture for Conic Systems
– Application I: Networked Control of a Robotic System
– Application II: Telemanipulation of Haptic Paddles
– Application III: Networked Coordination of Robotic Arms

Continuous Time (CT) Interior Conic Systems
0 0 0
( ) ( ) ( ( ) ( ) ( ) () ) 0
T T T
T T T
p p p p p py t y t dt a b y t u t dt ab u t u t dt     
2 2
2 2) ,|| ( ) || ( || ( ) || 0T p p p p p p T
T
p py a b uy bu a  
2
passive systems are inside the sector [0, ],
strictly input passive systems are inside the sector [ , ,
strictly output passive systems are in
] 0
]
-stable systems are in
side the sect
sid
or [0,
e thm
a
b
L
b
a

 
 
e sector [a,b] , .a b  
Interior conic systems are inside the sector
0
[ , ]
b
a b
a   

Discrete Time (DT) Interior Conic Systems
1
0
1 1
0 0
( ) ( ) ( ( ) ( ) ( ) () ) 0
N N
T T T
p p p p p p
i
N
ii
y i y i a b y i u i ab u i u i
 
 
    
2 2
2 2) ,|| ( ) || ( || ( ) || 0N p p p p p p N
N
p py a b uy bu a  
2
passive systems are inside the sector [0, ],
strictly input passive systems are inside the sector [ , ,
strictly output passive systems are in
] 0
]
-stable systems are in
side the sect
sid
or [0,
e thm
a
b
l
b
a

 
 
e sector [a,b] , .a b  
Interior conic systems are inside the sector
0
[ , ]
b
a b
a   

Properties: Interior Conic Systems – Sum Rule
1 1 1:H u y

1u
2 2 2:H u y
1y
2u
u

1 2y y y 
2y
1 1inside [ , ]a b
2 2inside [ , ]a b
 1 2 1 2: inside the sector ,H u y a a b b  

Properties: Interior Conic Systems – Scaling Rule
:H ku y
inside [ , ]a b
ku y
 
 
: inside the sector , , if 0
: inside the sector , , if 0
H u y ka kb k
H u y kb ka k
 
 

Properties: Interior Conic Systems – Concatenation Rule
   
 
 
   
: inside the sector , in which 1,...,
max , min
max 1
max max 4min .
2 2
l l
l l
l l
l l l l
l l l l
l l
H u y a b l m
a bab
a b a b
a b a b
a b a b
a a b a b
a b
  
 
     
  
   
         
1
m
u
u
u
 
   
  
1
m
y
y
y
 
   
  

Properties: Interior Conic Systems – Rotation Rule
 
 
T
T
If the matrix is an orthogonal matrix ( ) then
: is inside the sector , iff
: is inside the sector , .R
R R R I
H R u y a b
H u Ry a b




Sufficient Condition for Feedback Stability of
Interior Conic Systems


Dkd


D
D
D cl
If : is inside the sector [ , ] ([0, ]),
, 0 and the feedback law is
( ) ( ( ) ( )) in which the gain satisfies:
1 1
when 0;
1
when 0 then : is stable.
d
d
H a b
a b b
t k t t
k a
b a
k a H
b
 
  
 
 
   
 

   
     
‘D-Term’

Feedback Stability Result for Cascades of
Interior Conic Systems
The following control structure, typically exploited in robotic
control applications is stable under the following set of conditions.
d
cl : dH  Pkd


cl cl
cl
P
P P
If : is inside the sector [0, ] ([0,1]),
0 and : is inside the sector [0, ] ([0, ]),
0 and the feedback law is ( ) ( ( ) ( ))
in which the gain satisfies 0 ,then the syst
d
d d
H b
b H b
b t k t t
k k
 
 
  

    
    
   em is stable.



‘P-Term’

Inner Product Equivalent Sample and Hold
(IPESH) Transform
   2
2
1
( )
p
p
s
H sz
H z
T z s
   
  
  
 
 
( ) is inside the sector , iff
( ) is inside the sector , .
p
p
H s a b
H z a b
 z-1
sT
 2
z-transform of sampled
time series of ( )pH s s

Further Reading
• Kottenstette, N., LeBlanc, H., Eyisi, E., Koutsoukos, X. “Multi-rate
networked control of conic (dissipative) systems,” 2011 ACC pp.
274-280 (TR: ISIS-09-108).
• Kottenstette, N., McCourt, M.J., Xia, M., Gupta, V., Antsaklis, P.J.,
“On relationships among passivity, positive realness, and
dissipativity in linear systems”, Automatica 50 (4), pp. 1003-1016.
• Kottenstette, N., Antsaklis, P.J., "Relationships between positive
real, passive dissipative, & positive systems," 2010 ACC, pp.409-
416.
• Kottenstette, N., Porter, J., “Digital passive attitude and altitude
control schemes for quadrotor aircraft,” 2009 ICCA, pp.1761-1768
(TR: ISIS-08-911).
• G. Zames, “On the input-output stability of time-varying nonlinear
feedback systems. i. conditions derived using concepts of loop gain,
conicity and positivity,” TAC, vol. AC-11, no. 2, pp. 228 – 238, 1966.
• J. C. Willems, The Analysis of Feedback Systems. Cambridge, MA,
USA: MIT Press, 1971.

Wireless Networked Control Architecture for Conic Systems
• Render robotic system dynamics (Hp) To remain inside the
sector [a,b].
• Architecture allows for accommodation of time varying
channel capacity (due to communication faults/uncertainty)
• If a >=0 and b is finite, then use discrete time lag
compensator (Hc) to control 𝑦𝑝(t).

Wave Variables – Plant Interface
• Wave variables allow for a continuous time system to be
interconnected to a discrete time controller while allowing for time
varying communication delays.
pe
         
 
 
 
 
dc p
pdc
The wave variable transformation relates the continuous time plant "outputs"
, to the corresponding "inputs" , as follows:
2
2
p p
pp
u t y t v t y t
v tu t I I
y ty t I I

 
     
     
     

Multirate Passive Hold (PH:MTs) allows for adjustment
to time varying channel capacity
      
      
1
, ,..., 1 1
1
, , 1
p c
p p s s
s
v i v j i Mj M j
M
v t v i t iT i T
T
   
  
   
2 2
s
p c NMNT
v v 
   
 
   
 
 
1 1 11 12 2 22 2
1 0 1 0
1 1s
l l
s
s
i T M jm MN m N
p p p c c NMNT MN
l i l j i MjsiT
v v i dt v v j v
T M
   
    
      

Multirate Passive Sampler (PS:MTs) allows for
adjustment to time varying channel capacity
• Filters can be classic Butterworth antialiasing filters which allows for
improved noise rejection and performance.
   
 
 
LPc LP
pLP
1 1
1 if 1 if
,
1, otherwise1, otherwise
1
( )
j
s
jM
c
i j M
MTH j H e M
u j u i
M

  


  
   
  
  
 
   
22
s
c pN MNT
u u 

Multirate Passive Sampler (PS:MTs): Proof
       
 
 
2 2 2
2
2 2 2
1 1 1 1 1 1 1 1
2
12
pLP
0 1 1
Use Cauchy–Schwarz Inequality (CSI): ,
From CSI we have: ( )1 ( ) 1 ( )
1
( )l
jM jM jM jM
i j M i j M i j M i j M
jMN
c N
j i j M
y u y u
y i y i M y i
u u i
M
           

   

    
         
    
  
      
   
 
 
   
 
   
 
1 2 2
2
pLP
1 1 0 1 1
2
1 1 22
2
1 0 1 01 1
( )
1
( ) ( )
l
s s
pLPc
s
s s
jMm m N
pLP pMN MN
l l j i j M
iT iTm MN m MN
s
p pLPc pMN MNT
l i l i si T i Ts
u i u u
T
u u t dt u t dt u
TT

     
 
    
  
          
    
     
        
   
    
( )pLPcu t

Wave Variables – Controller Interface
• Digital controllers can be implemented while explicitly allowing for
continuous time stability.
         
 
 
 
 
dp c
dp
The wave variable transformation relates the discrete time controller "outputs"
, to the corresponding "inputs" , as follows:
2
2 1
c c
c c
c
v j y j u j y j
I I
v j u j
y j y j
I I

 
 
    
    
    
 
 

Resulting Lag Compensator From IPESH Transform
 
 
   
2
2
Desired Analog Lag Compensator:
Application of the IPESH Transform Results In
2
1 2
1
2 1
I
c P I
I s
c I s I s
c P I
s
s
H s K K
s
T
z
z H s T T
H z K K
T z s z


 
 
   
 


    
      
  
Z
   
   
 
 
 
Denote and as the respective
z-transforms of and such that
.
c c
c c
c
c
c
Y z E z
y j e j
Y z
H z
E z


Assumptions for Sampling Rate Independent Stability
 
i) : is a continous conic system inside the sector ,
ii) : is a discrete conic dissipative system inside the sector ,
iii) a) if 0 then 0< <
1 1
b) if 0 then 0< <
2
p p p p p
c c c c c
p
p
p
H e y a b
H e y a b
a
a
a


   

 
  
1
iv) the networked control system is well posed (no instantaneous loop gains 1).
pb
 
  
 


Sampling Rate Independent Stability Lemma
Lemma 1: For our aforementioned digital control network in which
the plant, : and controller : satisfy the conditions of our
aforementioned assumptions. If the plant and controller satis
p p p c c cH e y H e y 
 
   
fy any one of the
following listed set of conditions:
I. is inside the sector 0, and is inside the sector 0, ;
II. is inside the sector 0, and is inside the sector , such that
p p c c
p c c c
H b H b
H H a b
  

  2
TT T T
0 < (or vice-versa);
III. is inside the sector , in which 0 and is inside the
1
sector , such that < , < ;
then the digital control network : , ,
c c
p p p c
c c p c c
p
p c p
a b
H a a H
a b a a b
a
H r r y y

  
   
 
   
TT
2is -stable.m
c L  

Additional Assumption for Delay Independent Stability
           
   
Assume that delays are introduced at the wave-variable interface
between the controller and multi rate passive sampler and hold
devices (PS:MT / PH:MT ) such that:
, , , , 0,1,...
s s
cd c
j p j c j j p j j c j
u j u j p j
  
     
 
   
   
   
2 2
and v ,
Then stability is assured as long as for all , , 0,1,... :
i) and
ii) holds.
Condition i) ensures that always holds.
Condition ii) ensures that
cd c
cd cN N
j v j c j
j l l j
j p j l p l
j c j l c l
u u
      
 
  
  

   
2 2
always holds.cd cN N
v v

Additional Reading for Proof of Lemma 1
• Kottenstette, N., Hall, J.F., Koutsoukos, X., Sztipanovits, J.,
Antsaklis, P., “Design of Networked Control Systems Using
Passivity”, IEEE Transactions on Control Systems Technology
Volume: 21 Issue: 3 2012 pp. 649 - 665 – Lemma 1-I.
• Kottenstette, N., LeBlanc, H., Eyisi, E., Koutsoukos, X. “Multi-Rate
Networked Control of Conic (Dissipative) Systems,” 2011 ACC pp.
274-280 (TR: ISIS-09-108) – Lemma 1-II, 1-III.

Application I: Networked Control of a Robotic System - Robot
 
Gravity Compensated Robotic systems from the torque
to joint velocity output are inside the sector 0, .
Notation & Assumptions:
( ) : joint angles, (t): joint torques,
( ( )) : joint torques due to grav
t
g t




 
     
 
T
T T
ity and joint position
( ( )) : mass matrix
C ( ) : matrix of centrifugal and Coriolis effects
- 2 2 2 0.
Dynamics:
( ) , 0
.
u p
u p
M t M M
t C
M C M C M C
g M C g
M C
   
 
  
 
       
           
     
pH
 ( )py t
p I

Application I: Networked Control of a Robotic System - Robot
 
T
T T
T T
T T T T T
0
Use Lyapunov Function:
1
( ( )) 0, 0.
2
1
( ( ))
2
1
( ( ))
2
1
( ( )) 2
2
( ( )) ( ( )) ( (0))
s
u p
u p u p
NT
s
V t M
V t M M
V t C M
V t M C
V t dt V NT V
 
   
      
      
           
              
       
 
T T
0
T T
0 0
1 1 1
(0) inside sector 0,
s
s s
NT
u p
NT NT
u
p p p
dt
dt dt V
 

  
   
 
        
  

 
p I

Application I: Networked Control of a Robotic System - Controller
 
T T
1 1
Synthesize Passive MIMO Lag Type Control Law Using
inner product equivalent sample and hold (IPESH).
Passive CT Control Law ( 0, 0):
( ) ( ), ( ) ( ) ( )
Notation: [ ]
p p d d
uc p d
d
K K K K
x t e t t K x t K e t
e j j

   
  
        
 
 
1
1
( 1)
1 1
1
1
; [ ] [ ] [ ]
Synthesized Passive DT Control Law [ ] :
1
( ) [ ] ,( 1) , [ ] ( )
[ 1] [ ] [ ]
[ ] [ ] [ ]
2
s
s
sr c uc
pc
j MT
s s uc uc
s jMT
s
s
uc p p d
j e j e j j
G e j
e t e j t jMT j MT j t dt
T
x j x j MT e j
MT
j K x j K K e j
 
 


 
    
  
  
    
  

cH
 j  j
 j
 j
 j
 j
  cy j
  ce j

 
 
   1 1
1
1 1
1
If >0 then : [ ] is insided the sector 0, .
Denote: ;
2
Resulting Strictly Output Passive DT Control Law:
[ 1] [ ] [ ]
[ ] [ ]
c c uc
c
s
p p d c p
c s p s c s p
uc p
H e i i
T
D K K G I D
x i I T G K x i T I T G D e i
i G K x i G
 


 

 
 
 
  
 
   
    
  1[ ]pD e i cH
 j  j
 j
 j
 j
 j
  cy j
  ce j

   
   
1
1 1
1
1 1 1 1
1 1
Accounting for wave variables and
Denote: ;
2 2
1
, ,
,
2
Resulting DT Control
c c
s s
c p d sp p d
sp sp c s p sp s c s p
s
fe p fe p d
u j v j
T T
G I K K D G K K
G I D I T G K T I T G D
T
C G G K D G G K K

 


 
   
    
        
    
       
 
   
 
 
     
Law:
1 1 2
[ 1] [ ] [ ]
2 2
[ ] [ ]
sp sp fe sp fe c sr
c c fe fe c sr
x j C x j I D u j j
v j u j C x j D u j j
  
 
    
                  
  
       
   

Application I: Networked Control of a Robotic System - Setup
• Pioneer 3 (P3) arm simulated on PC Host 1 - Plant.
• Digital Control Law implemented on PC Host 2 – Controller
• Wave variables communicated using TCP/IP over wireless network
subject to disturbance node (D1, D2, D3, D4) flood attacks.
 cu j
 cdv j  cv j
 cdu j

Application I: Networked Control of a Robotic System - Parameters
Nominal Position Tracking Response
(no steady-state tracking error)
 
 
 
Control Tunning Rule:
1
1. Approx. Plant , in which
is the inertia around joint 1.
2. Approx. Controller G
3. Desire and
phase margin of degrees.
pm
p d
c
c
s
G s
Js
J
k k s
s
s
N MT







 
 
2
45
Therefore: i) ,
5
ii) , iii)
1
c
c
p d p
c
J
k k k








 

2
80, 2, .293 kg-m , .1 s,
1.0 6, 0.5, 8.02, 4.1
1, 1.
s
c p p d
N J T
e k k
M

 

   
    
 

Application I: Networked Control of a Robotic System - Results
Network Flood Attack 2 Disturbance Nodes Active

Application I: Networked Control of a Robotic System - Results
Network Flood Attack 4 Disturbance Nodes Active

Application I: Networked Control of a Robotic System - Summary
• Exploited passive relationships between the control torque and
angular velocity of the robots joints.
• Synthesized a DT passive lag compensator which is effective in
controlling the angular joint position of the robotic arm which
requires only the exchange of angular velocity of the joints.
• The use of wave variables ensures system stability while allowing
for system performance to degrade gradually as the round trip
delays increases.
• Stability is assured due to Lemma 1-I.
• See also: Kottenstette, N., Hall, J.F., Koutsoukos, X., Sztipanovits,
J., Antsaklis, P., “Design of Networked Control Systems Using
Passivity”, IEEE Transactions on Control Systems Technology
Volume: 21 Issue: 3 2012 pp. 649 – 665.

Application II: Telemanipulation of Haptic Paddles - Overview
• Two delta robots (Novint
Falcons) can be considered as
individual entities in a
diagonalized point mass plant
model.
• A velocity feedback loop applied
to each robot renders each robot
to be constrained to be interior
conic with position as the output.
• A passivity preserving orthogonal
rotation is applied at the
controller side in order to allow
each robot to track each other.

Application II: Telemanipulation of Haptic Paddles – Lemma 1-III
  2
2
For our digital control network if:
III. is inside the sector , in which 0 and
is inside the sector , such that and
1
then the digital control network is stable.
p p p
c c c p c
m
c
p
H a a
H a b a a
b L
a

   
 
  

Application II: Telemanipulation of Haptic Paddles - Plant Model
 
T TT T 6 T T 6
1 2 1 2: , , , ,
,
: is inside the sector , .
p p p p p p p p p
ps
pl
p p p
H e y e e e y y y
MMT
K
H e y

 

          
 
  
 1,2l 

Application II: Telemanipulation of Haptic Paddles - Controller Model
 
T
0
is inside the21
, ,
2 sector ,2 0
c c
c
c c c c
c c
a b
I
I I
R R R I K
a b a bI I
I
a b
 
  
       
  
 1,2l 

Application II: Telemanipulation of Haptic Paddles - Results
2
Parameter Summary:
1, 0.164 kg
=1, , K
1
,
p
ps
p
c c
M M
MMT
a b
 
 
 

 
 
 

Application II: Telemanipulation of Haptic Paddles - Proof
 
 
 
1
1
2
2
0
2 2
is inside the sector , . Since
2 2
0
we can consider the concatenation result : is inside the sector , :
max ,
c c c c
c c c
c c c c
c c c c
l l
a b a b
I u
y
K a b
a b a by
I u
a b a b
H u y a b
a b a b
    
    
     
    
       

    
   
 
1 1 2 2
2 2
2
2
min , 1,2 , ,
2
44
Since 0
4
min , , since
4 4 4
l l c c c c
l l c c
c c c c c cc c
c c c c
c c c c c c
c c c cc c c c c c c c
c c c c c
a b a b a bab
l a b a b
a b a b a b
a b a b a ba b
a b a b a b
a b a b a b
a b a ba b a b a b a bab
a b a b a b a
  
      
   
  
        
  
   
   
      
 
 
2
0
4
then .
c c
c c c
c c
c c
a b
b a b
a bab
a b a b

 


 

Application II: Telemanipulation of Haptic Paddles - Summary
• Demonstrated how direct position information can be exchanged
over a network in order to create leader-follower haptic systems.
• Using an orthogonal rotation matrix we derived a low complexity
control law which works within our network control architecture to
allow each paddle to track each other.
• The use of wave variables ensures system stability while allowing
for system performance to degrade gradually as the round trip
delays increases.
• Stability is assured due to Lemma 1-III.
• Kottenstette, N., LeBlanc, H., Eyisi, E., Koutsoukos, X. “Multi-rate
networked control of conic (dissipative) systems,” 2011 ACC pp.
274-280 (TR: ISIS-09-108).

Application III: Networked Coordination of Robotic Arms - Overview
• A wave variable networking interface known as a power junction
(PJ) is used to interconnect a single lag compensator to multiple (2)
robotic arms.
• The PJ and lag compensator ensure that each robotic arm will track
each other
• A classic PD controller is applied to each robotic arm, and a high
pass filtered output is augmented to the position output of each arm
in order to assure passivity while allowing position tracking at
steady state operation.
• Stability is assured through the use of wave variables and
constraining the controllers and robotic arms interfaced to the
network to possess a strictly output passive mapping.

Application III: Networked Coordination of Robotic Arms - Overview
• Novint Falcon paddle used by human operator to provide reference to
Crust Crawler robotic arms.
• Three Host Computers are used: Robot 1; Robot 2; & Network
Controller which is interfaced to haptic paddle.
• Communication between devices through wave variables over TCP/IP.

Application III: Networked Coordination of Robotic Arms - Architecture
Lag
Compensator
Robot 2
(rendered SOP) Power Junction
Robot 1
(rendered SOP)


1


Application III: Networked Coordination of Robotic Arms - Robots
 
 
 
 
 
 
Derivative Filter:
IPESHˆ
1
1 1ˆ ,
exp .
High Pass Filter:
IPESH
1 1
,
exp .
s
s
s
s
s
H s
s
p z
H z
T z p
T
p
s
H s
s
p z
H z
T z p
p T











 
 









 


 
   
 


 


 
Robot

Application III: Networked Coordination of Robotic Arms - PJ
     
   
   
 
1 1 2 2 3 3Wave variable pairs: , , , , ,
power-output pairs: , , 1,..., , 1
power-input pairs: , , 1,..., , 3
, , , , 1,...,s
j j
k k
m
j k j k s
u v u v u v
u v j m m
u v k m n n
u u v v l m
 
  
 
 
2
1 11
1
1
The outgoing waves are computed from the incoming waves v as follows:
sf , sf sgn , 2,3 .
3 1
l l l
l l
l
k j
m m
mj jj j
v k v jm
jjj
v
v v v
v v k
n mv
 


 
    
  
 


2 3
31 21
1 2
1 2
1
The outgoing waves are computed from the incoming waves as follows:
sf , sf sgn sf sgn .
l
l
l l l l l
l
j k
n n
nkk m kk m
u j u k u k kn k
k m kkk m
u u
u u
u u u u u
mu
   

  
 
   
      
   
 
  


Application III: Networked Coordination of Robotic Arms – Lag Comp.
1
1
( ) 1 .
2 1
In order to account for scaling effects of PJ, upsamplers and downsamplers
( )
select .
s
c
s
T z
H z k
z
M n m
k
m

 
  
 



Application III: Networked Coordination of Robotic Arms – Experimental

Application III: Networked Coordination of Robotic Arms – Experimental
• Open-Loop Approach Involves Transmitting only a desired setpoint
to each robot which originated from the haptic paddles setpoint
interface.
• It is clear that the lag compensator and PJ network allow for
improved tracking between each robotic arm (similar results for
Joints 1 and 2).

Application III: Networked Coordination of Robotic Arms – Summary
• A PJ and single lag compensator makes it possible to coordinate
motion of (n-m) robotic arms in spite of network time delay.
• Demonstrated that a high pass filter can be augmented to the
position output of the robot in order to derive a strictly output
passive robotic interface which allows for steady state reference
tracking.
See Also:
1. Kottenstette, N., Hall, J.F., Koutsoukos, X., Antsaklis, P., “Digital Control of Multiple
Discrete Passive Plants Over Networks,” International Journal of Systems, Control
and Communications, Vol. 3, No. 2, pp. 194 – 228 April 2011.
2. Koutsoukos, X., Kottenstette, N., Hall, J., Eyisi, E., Leblanc, H., Porter, J.,
Sztipanovits, J., “A Passivity Approach for Model-Based Compositional Design of
Networked Control Systems,”, ACM Transactions on Embedded Computing
Systems, Special Issue on the Synthesis of Cyber-Physical Systems Volume 11
Issue 4, December 2012 Article No. 75.
3. Kottenstette, N., Hall, J.F., Koutsoukos, X., Sztipanovits, J., Antsaklis, P., “Design
of Networked Control Systems Using Passivity”, IEEE Transactions on Control
Systems Technology Volume: 21 Issue: 3 2012 pp. 649 – 665.

Conclusions
• Interior conic system formulation allows numerous
systems to be modeled and considered for low
complexity control laws including robotic and quadrotor
systems.
• Multirate digital control architectures for these systems
can be realized with wave variables and novel signal
processing abstractions such as the PS, PH, and PJ.
• Questions?

ConstructiveControlRobotsAbridged

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