Table 2 Summary of notation used in this paper
Symbol | Description |
|---|---|
Problem formulation (Section 3.1) | |
k | Number of agents |
T | planning horizon |
\(\mathcal {X}\) | general continuous state space |
\(\mathcal {X}_I\) | set of initial states |
\(\mathcal {X}_g\) | set of final states |
\(x^i_t\) | state of agent i at time t |
\({\bar{x}}^i_t\) | observable state of agent i to other agents |
\(\Omega ^i\) | observation set of agent i |
\(o^i_t\) | observation of agent i at time t |
\(\mathcal {O}^i\) | observation function (\(\mathcal {O}^i: \mathcal {X}\rightarrow \Omega ^i\)) |
\(\mathcal {N}^i\left( x^i_t\right) \) | set of robots detected by i |
\(\Gamma ^i\) | agent i’s trajectory |
\({\widetilde{\Gamma }}^i\) | set of preferred trajectories |
\(\mathcal {T}\) | transition dynamics (Eq. 1) |
\(\mathcal {U}^i\) | action space for agent i |
\(\mathcal {J}^i\) | running cost for agent i (\(\mathcal {J}^i_t:\mathcal {X} \times \mathcal {U}^i \rightarrow \mathbb {R}\)) |
\(\mathcal {J}^i_f\) | running cost at time T |
\(\mathcal {C}^i\left( x^i_t \right) \in \mathcal {X}\) | convex hull of agent i |
\(\pi ^i \in \mathcal {K}\) | controller belonging to set of controllers |
Control Barrier Function (Section 4) | |
\(h^i:\mathcal {X} \longrightarrow \mathbb {R}\) | control barrier function |
\(\mathscr {C}^i\) | safe set |
\(\partial \mathscr {C}^i\) | boundary of \(\mathscr {C}^i\) |
\( L_f h^i\left( x^i_t\right) , L_g h^i\left( x^i_t\right) \) | lie derivatives of \(h^i\left( x^i_t\right) \) w.r.t f and g |
Deadlock Prevention (Section 5) | |
[1; N] | set of integers \(\{1,\,2,\dots ,\,N\}\) |
\(\mathscr {C}_\ell (t)\) | liveness set |
\(v_t^i\) | linear velocity of agent i |
\(v_t\) | joint velocity of all agents |
\({\widetilde{v}}_t\) | perturbed joint velocity |
\(h_v\left( x_t\right) \) | CBF of \(\mathscr {C}_\ell (t)\) |
\(p_t^i\) | position of agent i |
\(\theta _t^i\) | angle of agent i |
\(\ell _j\left( p_t^i,v_t^i\right) \) | liveness function for agent i |
\(\sigma \) | priority ordering |
\(\sigma _{\textsc {opt}}\) | optimal ordering |
\(\alpha _q\) | time-based reward for receiving an order position q |
\(b^i\) | bid made by agent i |
\(\left( r^i, p^i\right) \) | auction specified by allocation and payment rule |
\(\zeta ^i\) | private priority incentive parameter |