Path planning using positive invariant sets

Abstract

Abstract-We present an algorithm for steering the output of a linear system from a feasible initial condition to a desired target position, while satisfying input constraints and nonconvex output constraints. The system input is generated by a collection of local linear state-feedback controllers. The pathplanning algorithm selects the appropriate local controller using a graph search, where the nodes of the graph are the local controllers and the edges of the graph indicate when it is possible to transition from one local controller to another without violating input or output constraints. We present two methods for computing the local controllers. The first uses a fixed-gain controller and scales its positive invariant set to satisfy the input and output constraints. We provide a linear program for determining the scale-factor and a condition for when the linear program has a closed-form solution. The second method designs the local controllers using a semi-definite program that maximizes the volume of the positive invariant set that satisfies state and input constraints. We demonstrate our path-planning algorithm on docking of a spacecraft. The semi-definite programming based control design has better performance but requires more computation.

Enhancing motion safety by identifying safety-critical passageways

“…Invariant sets are sets of states which allow a system to remain within this set for an infinite time horizon. In [18]- [22], invariant sets are applied to motion planning of autonomous systems. Invariant sets are also used for safety verification.…”

Necessary and sufficient conditions for constraint satisfaction in switched systems using switch‐robust control invariant sets

“…Figure 3 shows the lateral position of the vehicle in closed-loop with a switched LQR and switched MPC controllers, where the mode was selected by a high-level path-planner, in this case, an invariant-set path-planner. [29][30][31] While the LQR provides smooth tracking of the reference, it does not satisfy the constraints specified by the path-planner, which may result in a collision. Similarly, if the switch-invariant terminal constraints (6d) are omitted, the MPC problem (6) could become infeasible after a mode switch.…”

“…The only restriction that the controller imposes on the path‐planner is the minimum amount of time between lane change requests, ie, the minimum dwell‐time. Thus, switch‐RCI sets could be used to simplify graph‐based path‐planners exploiting invariant sets …”

“…Thus, switch-RCI sets could be used to simplify graph-based path-planners exploiting invariant sets. [29][30][31] The theoretical results presented in this paper make no assumptions about the state and input constraint sets. However, for the specific vehicle lane changing case study, we consider polytopic sets, 32,33 rather than ellipsoidal sets, 34,35 for several reasons.…”

A Time-optimal Trajectory Generation Approach with Non-uniform B-splines

“…Moreover, solving a MILP requires high computational effort. An alternative algorithm is proposed, in which state and input constraints are handled by designing a graph of local controllers using positive invariant sets [16]. Afterwards, a graph search algorithm is applied to find a sequence of the local controllers.…”