Uncertainty-aware Safe Exploratory Planning using Gaussian Process and Neural Control Contraction Metric

Abstract

In this paper, we consider the problem of using a robot to explore an environment with an unknown, state-dependent disturbance function while avoiding some forbidden areas. The goal of the robot is to safely collect observations of the disturbance and construct an accurate estimate of the underlying disturbance function. We use Gaussian Process (GP) to get an estimate of the disturbance from data with a high-confidence bound on the regression error. Furthermore, we use neural Contraction Metrics to derive a tracking controller and the corresponding high-confidence uncertainty tube around the nominal trajectory planned for the robot, based on the estimate of the disturbance. From the robustness of the Contraction Metric, error bound can be pre-computed and used by the motion planner such that the actual trajectory is guaranteed to be safe. As the robot collects more and more observations along its trajectory, the estimate of the disturbance becomes more and more accurate, which in turn improves the performance of the tracking controller and enlarges the free space that the robot can safely explore. We evaluate the proposed method using a carefully designed environment with a ground vehicle. Results show that with the proposed method the robot can thoroughly explore the environment safely and quickly.

Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview

“…Proof. As proven in Theorems 3.1 and 4.1, mI M mI of (26), the contraction constraint (69), and the objective (88) reduce to (89) with c 1 = 0 and c 2 = 0. Since the resultant constraints are convex and the objective is affine in terms of the decision variables ν, χ, and W , the problem (89) is indeed convex.…”

“…In this context, we could utilize Control Contraction Metrics (CCMs) [33,[77][78][79][80][81] for extending contraction theory to the systematic design of differential feedback control δu = k(x, δx, u, t) via convex optimization, achieving greater generality at the expense of computational efficiency in obtaining u. Applications of the CCM to estimation, adaptive control, and motion planning are discussed in [82], [83][84][85], and [78,[86][87][88][89], respectively, using geodesic distances between trajectories [49]. It is also worth noting that the objective function of CV-STEM has the condition number of a positive definite matrix that defines a contraction metric as one of its arguments, rendering it applicable and effective even to machine learning-based automatic control frameworks as shall be seen in Sec.…”

“…It can be proven that the NCM and NSCM still yield robustness and optimality associated with the CV-STEM framework despite having non-zero modeling errors [116]. These metrics could also be synthesized and learned simultaneously with their feedback control laws directly by DNNs [89,116,117] at the expense of the convex property in the CV-STEM formulation.…”

“…7) In [78,118], contraction theory is leveraged to develop a tracking feedback controller with an optimized control invariant tube, solving the problem of robust motion planning under bounded external disturbances. This problem is also considered for systems with changing operating conditions [86] and parametric uncertainty [87][88][89] for its broader use in practice. As these methods still require online computation of a target trajectory, Learningbased Autonomous Guidance with RObustness and Stability (LAG-ROS) [102] is developed to model such robust control laws including the CV-STEM by a DNN, without explicitly requiring the target or desired trajectory as its input.…”

“…While this considers motion planning algorithms only implicitly to avoid solving them in real-time, it is shown that contraction theory still allows us to assure a property of robustness against deterministic and stochastic disturbances following the same argument as in the NCM and NSCM work [102]. Note that LAG-ROS using contraction theory is not intended to derive new learningbased motion planning, but rather to augment any existing motion planner with a real-time method of guaranteeing formal incremental robustness and stability, and thus still applicable to other methods such as tube-based robust Model Predictive Control (MPC) [78,[118][119][120][121][122], its dynamic and adaptive counterparts [86][87][88][89], and CCMbased learning certified control [117].…”