Learning-based Robust Motion Planning With Guaranteed Stability: A Contraction Theory Approach

Abstract

This paper presents Learning-based Autonomous Guidance with RObustness and Stability guarantees (LAG-ROS), which provides machine learning-based nonlinear motion planners with formal robustness and stability guarantees, by designing a differential Lyapunov function using contraction theory. LAG-ROS utilizes a neural network to model a robust tracking controller independently of a target trajectory, for which we show that the Euclidean distance between the target and controlled trajectories is exponentially bounded linearly in the learning error, even under the existence of bounded external disturbances. We also present a convex optimization approach that minimizes the steady-state bound of the tracking error to construct the robust control law for neural network training. In numerical simulations, it is demonstrated that the proposed method indeed possesses superior properties of robustness and nonlinear stability resulting from contraction theory, whilst retaining the computational efficiency of existing learning-based motion planners.

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

“…Although computationally tractable, it still has some limitations in that the problem size grows exponentially with the number of variables and basis functions [100]. Learning-based and data-driven control using contraction theory [14,15,101,102] has been developed to refine these ideas, using the high representational power of DNNs [103][104][105] and their scalable training realized by stochastic gradient descent [47,106].…”

“…The major advantage of using contraction theory for learning-based and data-driven control is that, by regarding its internal learning error as an external disturbance, we can ensure the distance between the target and learned trajectories to be bounded exponentially with time as in the CV-STEM results [14,15,101,102], with its steadystate upper bound proportional to the learning error. Such robustness and incremental stability guarantees are useful for formally evaluating the performance of machine learning techniques such as reinforcement learning [107][108][109][110], imitation learning [111][112][113][114][115], or neural networks [103][104][105].…”

“…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].…”

“…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].…”

“…Deep Neural Networks (DNNs) have been shown to generalize well to the set of unseen events that are from almost the same distribution as their training set [60,[152][153][154][155], and consequently, obtaining a tighter and more general upper bound for the learning error as in (120) has been an active field of research [150,151,156,157]. Thus, the condition (115) has become a common assumption in analyzing the performance of learning-based and data-driven control techniques [60,101,102,116,[158][159][160][161][162].…”