Computation of Feedback Control Laws Based on Switched Tracking of Demonstrations

TL;DR

This paper presents an algorithm that uses a demonstrator (typically given by a trajectory optimizer) to automatically synthesize feedback controllers for steering a system described by ordinary differential equations into a goal set.

Abstract

A common approach in robotics is to learn tasks by generalizing from special cases given by a so-called demonstrator. In this paper, we apply this paradigm and present an algorithm that uses a demonstrator (typically given by a trajectory optimizer) to automatically synthesize feedback controllers for steering a system described by ordinary differential equations into a goal set. The resulting feedback control law switches between the demonstrations that it uses as reference trajectories. In comparison to the direct use of trajectory optimization as a control law, for example, in the form of model predictive control, this allows for a much simpler and more efficient implementation of the controller. The synthesis algorithm comes with rigorous convergence and optimality results, and computational experiments confirm its efficiency.

Figures (5)

• Figure 1: Switching control input and the resulting switching trajectory (blue) and the corresponding parts of target demonstrations (red) with time intervals between switches (gray). The simulation of the quadcopter example (see description in Section 6.1., only one state and one control input) with the LQR switching controller produced by our implementation of Algorithm 2.
• Figure 2: System simulations in Algorithm \ref{['alg:unknown']}: goal set (green ellipse), demonstrations (magenta), compatible reachability certificate candidate (contour plot) on the set of initial states $I$ (rectangle). Simulations that meet the sufficient decrease condition are blue, others are red.
• Figure 3: Assumption ( III ): Assignment rule $\phi$ must always assign a sufficiently close demonstration (i.e. from a neighborhood $\alpha$) if at least one demonstration exists in a neighborhood $\beta$.
• Figure 4: Several iterations of LQR switching algorithm for the pendulum example: demonstrations (magenta), reachability certificate candidate (contour plot) on the set of initial states $I$ (inner rectangle) compatible with these demonstrations. System simulations in Algorithm 2 (blue) are terminated, when the sufficient decrease condition is met.
• Figure 5: Solution produced by LQR switching algorithm: demonstrations (magenta) and switching trajectories (gray).

Theorems & Definitions (23)

• Definition 1
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• Proposition 2