Guaranteed Trajectory Tracking under Learned Dynamics with Contraction Metrics and Disturbance Estimation

TL;DR

This work addresses safe trajectory tracking for nonlinear systems with matched uncertainties during the learning phase. It introduces a disturbance-estimation-based contraction control (DE-CCM) framework that uses a learned dynamics $\hat{d}(x)$, computable error bounds $\bar{\delta}_{de}(t)$, and a robust Riemann energy condition to ensure universal exponential stability of the true system despite imperfect learned dynamics. Learning enhances robustness and enables better trajectory planning, as demonstrated on a planar quadrotor. The method remains valid even when the learned model is imperfect, provided error bounds hold, and includes a low-pass filtering mechanism to mitigate high-frequency estimation signals.

Abstract

This paper presents an approach to trajectory-centric learning control based on contraction metrics and disturbance estimation for nonlinear systems subject to matched uncertainties. The approach uses deep neural networks to learn uncertain dynamics while still providing guarantees of transient tracking performance throughout the learning phase. Within the proposed approach, a disturbance estimation law is adopted to estimate the pointwise value of the uncertainty, with pre-computable estimation error bounds (EEBs). The learned dynamics, the estimated disturbances, and the EEBs are then incorporated in a robust Riemann energy condition to compute the control law that guarantees exponential convergence of actual trajectories to desired ones throughout the learning phase, even when the learned model is poor. On the other hand, with improved accuracy, the learned model can help improve the robustness of the tracking controller, e.g., against input delays, and can be incorporated to plan better trajectories with improved performance, e.g., lower energy consumption and shorter travel time.The proposed framework is validated on a planar quadrotor example.

Figures (8)

• Figure 1: Proposed control architecture incorporating learned dynamics
• Figure 2: Top: Planned and executed trajectories under our proposed DE-CCM for Tasks 1$\sim$3 under different learning scenarios. Bottom: Tracking performance of DE-CCM, Ad-CCM lopez2020adaptive-ccm, RCCM (which can be seen as a special case of DE-CCM with uncertainty estimate set to $0$), and CCM for Task 1 under different learning scenarios Start points of planned and actual trajectories are intentionally set to be different to reveal the transient response under different scenarios.
• Figure 3: Safe exploration of the state space for learning uncertain dynamics under DE-CCM
• Figure 4: Trajectories of true, learned, and estimated disturbances in the presence of no and good learning for Task 1. The notations $d_1$, $\hat{d}_1$ and $\check d_1$ denote the first element of $d$, $\hat{d}$ and $\check d$, respectively.
• Figure 5: Trajectories of Riemann energy $E(x^\star,x)$ (left) and the first element of control input (right), i.e., $u_1$, in the presence of no, poor and good learning for Task 1

Theorems & Definitions (26)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Lemma 1
• Lemma 2
• Lemma 3
• Remark 6
• Remark 7