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Bezier Reachable Polytopes: Efficient Certificates for Robust Motion Planning with Layered Architectures

Noel Csomay-Shanklin, Aaron D. Ames

TL;DR

The notion of Bézier Reachable Polytopes is introduced – certificates of reachable points in the space of Bézier polynomial reference trajectories that serve as a constructive tool for layered architectures, enabling long-horizon tasks to be reasoned about in a computationally tractable manner.

Abstract

Control architectures are often implemented in a layered fashion, combining independently designed blocks to achieve complex tasks. Providing guarantees for such hierarchical frameworks requires considering the capabilities and limitations of each layer and their interconnections at design time. To address this holistic design challenge, we introduce the notion of Bezier Reachable Polytopes -- certificates of reachable points in the space of Bezier polynomial reference trajectories. This approach captures the set of trajectories that can be tracked by a low-level controller while satisfying state and input constraints, and leverages the geometric properties of Bezier polynomials to maintain an efficient polytopic representation. As a result, these certificates serve as a constructive tool for layered architectures, enabling long-horizon tasks to be reasoned about in a computationally tractable manner.

Bezier Reachable Polytopes: Efficient Certificates for Robust Motion Planning with Layered Architectures

TL;DR

The notion of Bézier Reachable Polytopes is introduced – certificates of reachable points in the space of Bézier polynomial reference trajectories that serve as a constructive tool for layered architectures, enabling long-horizon tasks to be reasoned about in a computationally tractable manner.

Abstract

Control architectures are often implemented in a layered fashion, combining independently designed blocks to achieve complex tasks. Providing guarantees for such hierarchical frameworks requires considering the capabilities and limitations of each layer and their interconnections at design time. To address this holistic design challenge, we introduce the notion of Bezier Reachable Polytopes -- certificates of reachable points in the space of Bezier polynomial reference trajectories. This approach captures the set of trajectories that can be tracked by a low-level controller while satisfying state and input constraints, and leverages the geometric properties of Bezier polynomials to maintain an efficient polytopic representation. As a result, these certificates serve as a constructive tool for layered architectures, enabling long-horizon tasks to be reasoned about in a computationally tractable manner.

Paper Structure

This paper contains 11 sections, 5 theorems, 54 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Bézier Curves
  4. State and Input Constraint Satisfaction
  5. Bézier Reachable Polytopes
  6. Reducing Conservatism
  7. Results
  8. Simulation Results
  9. Hardware Results
  10. Conclusion
  11. Acknowledgements

Key Result

Theorem 1

Let system $\Sigma_d$ be a planning model for system $\Sigma$ with tracking certificate $\mathcal{E}$. There exist matrices $\mathbf{F}$ and $\mathbf{G}$ such that any Bézier curve $\mathbf{B} :I\to \mathcal{X}_d$ with control points $\mathbf{p}$ satisfying: when tracked results in the closed loop system satisfying $\mathbf{\Pi} ( \mathbf{x} _{\rm cl}) \in \mathcal{C}_\mathcal{X}$ and $\mathbf{k

Figures (6)

  • Figure 2: A depiction of the layered architectures investigated in this work, where the reachable set of the combined planning and tracking layers can be represented via a linear inequality in the space of Bézier polynomials.
  • Figure 3: A visual guide to the properties of Bézier curves.
  • Figure 4: A selection of Bézier curves and forward reachable sets. The top row depicts curves with exact tracking, the middle row with a fixed size tracking certificate, and the bottom row with a tracking certificate whose upper bound scales linearly with the planning input $\mathbf{u} _d$.
  • Figure 5: A depiction of the forward reachable sets as a function of system parameters. (Top Row) 1-step reachable sets, as in Theorem \ref{['thm:BezPoly']}. (Bottom Row) 20-step reachable sets, as in Corollary \ref{['coll:ref']}. (Left Column) Varying the input constraint $\mathbf{ u }_{\text{max}}$ (Middle Column) Varying the time horizon T (Right Column) Varying the initial condition.
  • Figure 6: The proposed method applied to the pendulum swingup problem. As the input bounds are tightened from 5 Nm (bottom) to 0.5 Nm (top), the resulting graph search trajectory increases in complexity and length.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Example 1
  • Definition 3: Dynamically Feasible Trajectory
  • proof
  • proof
  • Remark 1
  • Definition 4
  • Theorem 1
  • Lemma 1
  • ...and 8 more