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Spline Trajectory Tracking and Obstacle Avoidance for Mobile Agents via Convex Optimization

Akua Dickson, Christos G. Cassandras, Roberto Tron

TL;DR

This work proposes a motion planning technique based on output feedback that enable agents to converge to a specified polynomial trajectory while avoiding collisions within a polygonal environment and synthesizes a controller for each convex cell via a semi-definite programming problem that includes the derived constraints.

Abstract

We propose an output feedback control-based motion planning technique for agents to enable them to converge to a specified polynomial trajectory while imposing a set of safety constraints on our controller to avoid collisions within the free configuration space (polygonal environment). To achieve this, we 1) decompose our polygonal environment into different overlapping cells 2) write out our polynomial trajectories as the output of a reference dynamical system with given initial conditions 3) formulate convergence and safety constraints as Linear Matrix Inequalities (LMIs) on our controller using Control Lyapunov Functions (CLFs) and Control Barrier Functions (CBFs) and 4) solve a semi-definite programming (SDP) problem with convergence and safety constraints imposed to synthesize a controller for each convex cell. Extensive simulations are included to test our motion planning method under different initial conditions and different reference trajectories. The synthesized controller is robust to changes in initial conditions and is always safe relative to the boundaries of the polygonal environment.

Spline Trajectory Tracking and Obstacle Avoidance for Mobile Agents via Convex Optimization

TL;DR

This work proposes a motion planning technique based on output feedback that enable agents to converge to a specified polynomial trajectory while avoiding collisions within a polygonal environment and synthesizes a controller for each convex cell via a semi-definite programming problem that includes the derived constraints.

Abstract

We propose an output feedback control-based motion planning technique for agents to enable them to converge to a specified polynomial trajectory while imposing a set of safety constraints on our controller to avoid collisions within the free configuration space (polygonal environment). To achieve this, we 1) decompose our polygonal environment into different overlapping cells 2) write out our polynomial trajectories as the output of a reference dynamical system with given initial conditions 3) formulate convergence and safety constraints as Linear Matrix Inequalities (LMIs) on our controller using Control Lyapunov Functions (CLFs) and Control Barrier Functions (CBFs) and 4) solve a semi-definite programming (SDP) problem with convergence and safety constraints imposed to synthesize a controller for each convex cell. Extensive simulations are included to test our motion planning method under different initial conditions and different reference trajectories. The synthesized controller is robust to changes in initial conditions and is always safe relative to the boundaries of the polygonal environment.
Paper Structure (19 sections, 5 theorems, 53 equations, 2 figures, 1 table)

This paper contains 19 sections, 5 theorems, 53 equations, 2 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Dynamical system
  4. Polynomial trajectories
  5. Bounding Bernstein polynomials and their derivatives
  6. Polygonal Environment Decomposition
  7. Problem statement
  8. PROPOSED SOLUTION
  9. Polynomial trajectories
  10. Combined Dynamics
  11. Convergence Conditions via Linear Matrix Inequalities
  12. Safety Constraints by Control Barrier Functions
  13. Safety Constraints on $K$
  14. Control Synthesis
  15. Initialization and switching controllers
  16. ...and 4 more sections

Key Result

Lemma 1

Polynomial representations with both matrix $P$ and $\mathcal{A}$ can be written as Given the standard polynomial basis ($\mathcal{A}$), $D$ is an invertible transformation matrix that derives the Bernstein polynomial basis ($P$) from the standard polynomial basis such that: Then $\mathcal{A}=PD$.

Figures (2)

  • Figure 1: Simulation results without noise. This figure presents multiple initializations of an agent that tracks the predefined polynomial trajectory. Each polygonal cell with colored walls represents a convex cell. Convex cells may overlap. In the middle portion of the environment, several convex cells overlap together. The reference polynomial trajectories corresponding to the convex cells are shown as grey colored lines. The control points defining the segment for each cell have the same colors as the corresponding cell (except for the final point, which has the color of the following cell). We randomly place an initial position for the agent system; the agent's trajectories are given by the colored lines (although, thanks to our controller, they quickly converge to the gray reference trajectory). The agent trajectory colors correspond to the colored walls of the region in which they are initialized.
  • Figure 2: Simulation results with noise. We present results for one random initialization of an agent in the presence of Gaussian noise. The various colored trajectories represent the trajectory of the agent within the polygonal environment. Each colored trajectory corresponds to a convex cell with the same colored walls. Each color represents the segment of the polynomial trajectory under a corresponding controller.

Theorems & Definitions (12)

  • Lemma 1
  • Lemma 2
  • proof
  • Corollary 1
  • Definition 1
  • Proposition 1
  • proof
  • Lemma 3
  • proof
  • Example 1
  • ...and 2 more