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Learning Barrier-Certified Polynomial Dynamical Systems for Obstacle Avoidance with Robots

Martin Schonger, Hugo T. M. Kussaba, Lingyun Chen, Luis Figueredo, Abdalla Swikir, Aude Billard, Sami Haddadin

TL;DR

This work proposes to use polynomial representations for DSs, which yields an optimization problem that can be tackled by sum-of-squares techniques and can handle obstacle shapes that fall outside the scope of assumptions typically found in the literature concerning obstacle avoidance within the DS learning framework.

Abstract

Established techniques that enable robots to learn from demonstrations are based on learning a stable dynamical system (DS). To increase the robots' resilience to perturbations during tasks that involve static obstacle avoidance, we propose incorporating barrier certificates into an optimization problem to learn a stable and barrier-certified DS. Such optimization problem can be very complex or extremely conservative when the traditional linear parameter-varying formulation is used. Thus, different from previous approaches in the literature, we propose to use polynomial representations for DSs, which yields an optimization problem that can be tackled by sum-of-squares techniques. Finally, our approach can handle obstacle shapes that fall outside the scope of assumptions typically found in the literature concerning obstacle avoidance within the DS learning framework. Supplementary material can be found at the project webpage: https://martinschonger.github.io/abc-ds

Learning Barrier-Certified Polynomial Dynamical Systems for Obstacle Avoidance with Robots

TL;DR

This work proposes to use polynomial representations for DSs, which yields an optimization problem that can be tackled by sum-of-squares techniques and can handle obstacle shapes that fall outside the scope of assumptions typically found in the literature concerning obstacle avoidance within the DS learning framework.

Abstract

Established techniques that enable robots to learn from demonstrations are based on learning a stable dynamical system (DS). To increase the robots' resilience to perturbations during tasks that involve static obstacle avoidance, we propose incorporating barrier certificates into an optimization problem to learn a stable and barrier-certified DS. Such optimization problem can be very complex or extremely conservative when the traditional linear parameter-varying formulation is used. Thus, different from previous approaches in the literature, we propose to use polynomial representations for DSs, which yields an optimization problem that can be tackled by sum-of-squares techniques. Finally, our approach can handle obstacle shapes that fall outside the scope of assumptions typically found in the literature concerning obstacle avoidance within the DS learning framework. Supplementary material can be found at the project webpage: https://martinschonger.github.io/abc-ds
Paper Structure (10 sections, 2 theorems, 8 equations, 8 figures, 1 algorithm)

This paper contains 10 sections, 2 theorems, 8 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Learning stable dynamical systems
  4. Barrier certificates
  5. Proposed method
  6. Simulations
  7. Comparison of polynomial DSs with LPV-DS
  8. Encoding demonstrations using ABC-DS
  9. Experiments
  10. Conclusion

Key Result

Theorem 1

Let ${\mathcal{X}_0, \mathcal{X}_u \subset \mathbb{R}^n}$, and ${\varepsilon_1 > 0}$. If there exists a continuously differentiable function ${B\colon \mathbb{R}^n \to \mathbb{R}}$ such that then for all trajectories $\bm{\xi}(t)$ of the system ${\dot{\bm{\xi}}(t) = f(\bm{\xi}(t))}$ such that ${\bm{\xi}(0)\in\mathcal{X}_0}$, one has that ${\bm{\xi}(t)\not\in\mathcal{X}_u}$ for all ${t \ge 0}$.

Figures (8)

  • Figure 1: In a pick-and-place task a robot can be taught how to move from $\bm{\xi}_0$ to $\bm{\xi}^*$ by encoding this trajectory in a dynamical system (DS). While the reference trajectory avoids the two obstacles, enclosed by the unsafe set $\mathcal{X}_u$, a disturbance (blue) can push the robot to a state $\tilde{\bm{\xi}}$ where there is no reference velocity available. Nonetheless, it is crucial that the trajectory starting in $\tilde{\bm{\xi}}$ also does not collide with any obstacle, i.e. does not enter $\mathcal{X}_u$. For example, the left dashed partial trajectory is unsafe, whereas the right one is safe (up to where it is shown). It is desired from the DS to generate safe trajectories for regions of the state space that go beyond the reference trajectories.
  • Figure 2: A dynamical system $f$ is safe if none of its trajectories starting from a state in the initial set $\mathcal{X}_0$ reach a state in the unsafe set $\mathcal{X}_u$. In the particular case of obstacle avoidance, $\mathcal{X}_u$ can be specified to enclose any obstacles. The certified safe set $\mathcal{X}_s$ of $f$ with respect to a barrier certificate $B$ is a superset of $\mathcal{X}_0$ and amounts to all states $\bm{\xi}$ for which ${B(\bm{\xi}) \le 0}$. When starting at any state in $\mathcal{X}_s$, the trajectory generated by $f$ is guaranteed to not enter $\mathcal{X}_u$. The specific DS shown in the figure was generated by our proposed approach, with reference data obtained from a robot. The trajectory generated by the DS when starting in the center of $\mathcal{X}_0$ is plotted in pink. The trajectory executed by the robot when controlled by this DS is shown in blue.
  • Figure 3: Polynomial DS generated by our proposed method (without barrier) on a representative subset of the LASA handwriting dataset. Black stream lines depict the vector field with a globally asymptotically stable equilibrium (black dot). The yellow contours depict the values of the Lyapunov function, where lighter areas indicate lower values. The reference trajectories are shown in blue, and a trajectory starting from the references' mean initial point and simulated by the DS is plotted in pink.
  • Figure 4: Mean and standard deviation of the MSE of our polynomial approach (without barrier) versus LPV-DS on a subset of the LASA handwriting dataset. We evaluate each one 10 times with different random seeds for the random number generation, which influences the solver. The reported values are for normalized reference data.
  • Figure 5: Performance of ABC-DS on different data from the LASA dataset in the presence of elliptical unsafe sets. The obstacles are safely avoided with non-conservative barrier certificates and the trajectories simulated with the computed DS (black stream lines) are still close to the reference trajectories.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Proposition 1
  • proof