Simple and Efficient Algorithms for Computing Smooth, Collision-free Feedback Laws Over Given Cell Decompositions

Abstract

Abstract-This paper presents a novel approach to computing feedback laws in the presence of obstacles. Instead of computing a trajectory between a pair of initial and goal states, our algorithms compute a vector field over the entire state space; all trajectories obtained from following this vector field are guaranteed to asymptotically reach the goal state. As a result, the vector field globally solves the navigation problem and provides robustness to disturbances in sensing and control. The vector field's integral curves (system trajectories) are guaranteed to avoid obstacles and are C ∞ smooth. We construct a vector field with these properties by partitioning the space into simple cells, defining local vector fields for each cell, and smoothly interpolating between them to obtain a global vector field. We present an algorithm that computes these feedback controls for a kinematic point robot in an arbitrary dimensional space with piecewise linear boundary; the algorithm requires minimal preprocessing of the environment and is extremely fast during execution. For many practical applications in two-dimensional environments, full computation can be done in milliseconds. We also present an algorithm for computing feedback laws over cylindrical algebraic decompositions, thereby solving a smooth feedback version of the generalized piano movers' problem.

Decoupled Formal Synthesis for Almost Separable Systems with Temporal Logic Specifications

“…The experiment setting considered is illustrated in Figure 1. The underlying dynamics can be driven among cells using gradient methods [11], so to simplify the presentation we model the robots as entirely discrete transition systems.…”

Efficient navigation for anyshape holonomic mobile robots in dynamic environments

“…Even though numerous methods for motion planning and trajectory optimization can be found in the literature, the trade-off between the optimality of the solution and the computational effort is still crucial. Stabilizing receding horizon controllers (RHC) are motion planners that avoid local minima [10,13]. For practical reasons, many methods assume that the state space is collision-free.…”

“…However, in real environments this assumption does not hold and as soon as the configuration space is non-convex, RHC cannot be applied [12]. This problem can be solved by splitting the configuration space into convex parts and by optimizing them separately [10]. The convergent dynamic window approach (CDW) [13] uses an interpolated continuous version of the navigation function [7] as a control Lyapunov function.…”

Global vector field computation for feedback motion planning

“…In order to overcome these problems, feedback motion planning approaches take into account feedback concerns during collision-free path computation. Rather than planning a single collision-free path between the initial and goal configurations, these approaches compute a feedback plan over the entire free space of the robot that can converge towards the goal [4], [5]. A feedback plan is often represented as a vector field over the free space.…”

“…Most of the prior work on computing global vector fields for feedback motion planning is based on sequential composition [4], [5], [6], [7]. In these approaches, the robot's free space is decomposed into cells.…”