Agents, Modes, and Dynamics Clause Samples

Agents, Modes, and Dynamics. Consider a control scenario with an ego agent (which we control) and m N other interacting agents (which we do not control, but we can predict their motion). We assume an upper bound on m is known a priori. We consider multi-modal interaction dynamics, where each mode y ∈ Y = {1, 2, · · · , ny} is a unique homotopy s.t. pose(Ry(t)) ∩ posi(Ry(t)) = ∅ ∀ t, y, i (3b)‌ pose(Ry(t)) ∩ Xobs = ∅ ∀ t, y (3c) ua(τ ) = ub(τ ) ∀ τ ∈ [0, tc], a, b ∈ Y (3d) where J is an arbitrary cost function, t ∈ T , y ∈ Y, and i ∈ { T → U} {1, · · · , m}. Program (3) seeks a set of |Y| control signals uy : y∈Y for which the ego agent does not collide with any other agent (Constraint (3b)) or any obstacles (Constraint (3c)). Per (2), the reachable set constraints (3b) and (3c) implicitly require the multi-agent system to start from the initial condition set (0) and obey the interaction dynamics f . Finally, the control signals must agree up to a consensus horizon tc ∈ [∆t, tf] (Constraint (3d)); this means know as an H-representation, and represented by a matrix A and a vector b such that x ∈ Z ⇐⇒ min(Ax − b) ≥ 0: Proposition 2 ([43, Theorem 2.1]). Let Z = Z(c, G) ⊂ R2, that the ego agent must apply the same control at least up to time tc for all modes. We adopt this strategy because we do not necessarily know the actual mode when solving (3), so we must be able to simultaneously plan a control strategy for each mode. This idea, Contingency MPC [42], has been with nG generators. Assume that G has no generators of length 0. Let ℓG ∈ Rm be a vector of the lengths of each generator: ℓG[i] = G[:, i] 2. Let applied successfully in multi-agent settings [6]. The key challenge is that one can never perfectly represent A[:, i] = ℓG[i] · " # G[1,i] G[2,i] ∈ R2×(2nG), and (5a) R the reachable sets i (t). Furthermore, to enable a numerical solution, we require an efficient, differentiable representation of collision detection per (3b) and (3c).