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VBOC: Learning the Viability Boundary of a Robot Manipulator using Optimal Control

Asia La Rocca, Matteo Saveriano, Andrea Del Prete

TL;DR

This letter presents a new approach for numerically approximating the viability kernel of robot manipulators that solves optimal control problems to compute states that are guaranteed to be on the boundary of the set, therefore learning in a smaller dimensional space.

Abstract

Safety is often the most important requirement in robotics applications. Nonetheless, control techniques that can provide safety guarantees are still extremely rare for nonlinear systems, such as robot manipulators. A well-known tool to ensure safety is the Viability kernel, which is the largest set of states from which safety can be ensured. Unfortunately, computing such a set for a nonlinear system is extremely challenging in general. Several numerical algorithms for approximating it have been proposed in the literature, but they suffer from the curse of dimensionality. This paper presents a new approach for numerically approximating the viability kernel of robot manipulators. Our approach solves optimal control problems to compute states that are guaranteed to be on the boundary of the set. This allows us to learn directly the set boundary, therefore learning in a smaller dimensional space. Compared to the state of the art on systems up to dimension 6, our algorithm resulted to be more than 2 times as accurate for the same computation time, or 6 times as fast to reach the same accuracy.

VBOC: Learning the Viability Boundary of a Robot Manipulator using Optimal Control

TL;DR

This letter presents a new approach for numerically approximating the viability kernel of robot manipulators that solves optimal control problems to compute states that are guaranteed to be on the boundary of the set, therefore learning in a smaller dimensional space.

Abstract

Safety is often the most important requirement in robotics applications. Nonetheless, control techniques that can provide safety guarantees are still extremely rare for nonlinear systems, such as robot manipulators. A well-known tool to ensure safety is the Viability kernel, which is the largest set of states from which safety can be ensured. Unfortunately, computing such a set for a nonlinear system is extremely challenging in general. Several numerical algorithms for approximating it have been proposed in the literature, but they suffer from the curse of dimensionality. This paper presents a new approach for numerically approximating the viability kernel of robot manipulators. Our approach solves optimal control problems to compute states that are guaranteed to be on the boundary of the set. This allows us to learn directly the set boundary, therefore learning in a smaller dimensional space. Compared to the state of the art on systems up to dimension 6, our algorithm resulted to be more than 2 times as accurate for the same computation time, or 6 times as fast to reach the same accuracy.
Paper Structure (16 sections, 5 theorems, 19 equations, 5 figures, 3 tables, 2 algorithms)

This paper contains 16 sections, 5 theorems, 19 equations, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Notation
  4. Problem statement
  5. Backward Reachability vs Viability
  6. Data-Driven Learning
  7. Viability-Boundary Optimal Control (VBOC)
  8. Viability for robot manipulators
  9. Star-convex Viability Set Representation
  10. Uniform Data Distribution
  11. Algorithmic Implementation
  12. Results
  13. Tests on a 2D system
  14. Tests on a 4D system
  15. Tests on a 6D system
  16. ...and 1 more sections

Key Result

Lemma 1

Let us consider a locally-optimal state trajectory $\{x_i^* \}_{0}^N$ computed by solving eq:viab_boundary_ocp. Let us assume that $P_S a \neq 0$, where $P_S \triangleq (I - S^\dagger S)$ is a null-space projector of $S$. Then, if $N$ is sufficiently large to allow reaching $\mathcal{S}$ from any vi Moreover, for any sufficiently small value $\epsilon > 0$:

Figures (5)

  • Figure 1: A star-convex set, with two possible examples of choices of $a$. The orange parts of the set boundary cannot be discovered by any choice of $a$.
  • Figure 2: Viability kernel for the single pendulum. The background color represents the set learned using VBOC. The black and blue dots represent the training data from the two generated trajectories. The axis limits correspond to the joint position and velocity limits.
  • Figure 3: Comparison between the RMSE evolution for the 4D system.
  • Figure 4: Cumulative error distribution for the 4D system at then end of the training. This plot shows how many test samples (y axis) obtained a prediction error (x axis) below a certain value.
  • Figure 5: Comparison between the RMSE evolution for the 6D system.

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Theorem 1
  • Lemma 3