Chance-Constrained Trajectory Optimization for Safe Exploration and Learning of Nonlinear Systems

Abstract

Learning-based control algorithms require data collection with abundant supervision for training. Safe exploration algorithms ensure the safety of this data collection process even when only partial knowledge is available. We present a new approach for optimal motion planning with safe exploration that integrates chance-constrained stochastic optimal control with dynamics learning and feedback control. We derive an iterative convex optimization algorithm that solves an Information-cost Stochastic Nonlinear Optimal Control problem (Info-SNOC). The optimization objective encodes control cost for performance and exploration cost for learning, and the safety is incorporated as distributionally robust chance constraints. The dynamics are predicted from a robust regression model that is learned from data. The Info-SNOC algorithm is used to compute a sub-optimal pool of safe motion plans that aid in exploration for learning unknown residual dynamics under safety constraints. A stable feedback controller is used to execute the motion plan and collect data for model learning. We prove the safety of rollout from our exploration method and reduction in uncertainty over epochs, thereby guaranteeing the consistency of our learning method. We validate the effectiveness of Info-SNOC by designing and implementing a pool of safe trajectories for a planar robot. We demonstrate that our approach has higher success rate in ensuring safety when compared to a deterministic trajectory optimization approach.

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

“…Policy to be learned State tracking error x − x d Computational load (a) Learning-based motion planner [5]- [11] (x, o ,t) → u d Increases exponentially (Lemma 1) One neural net evaluation (b) Robust tube-based motion planner [13]- [26] (…”

“…One neural net evaluation demonstrations for training, so that the learned policy will not violate given state constraints even with the learning error and external disturbances. In this phase, LAG-ROS learns the contracting control law independently of a target trajectory, making it implementable without solving any motion planning problems online unlike [13]- [26]. The performance of LAG-ROS is evaluated in cart-pole balancing [28] and nonlinear motion planning of multiple robotic agents [29] in a cluttered environment, demonstrating that LAG-ROS indeed satisfies the formal exponential bound as in [13]- [21] with its computational load as low as that of existing learning-based motion planners [5]- [11].…”

“…sup x,t d(x,t) =d, o g ∈ R g is a vector containing global environment information such as initial and terminal states, states of obstacles and other agents, etc., and x d : R g × R ≥0 → R n and u d : R n × R g × R ≥0 → R n are the target trajectories given by existing motion planning algorithms, e.g., [13]- [26].…”

“…, modeled by a neural network, where o : R n × R g → R with ≤ g is local environment information extracted from o g ∈ R g [5]. (b) Robust tube-based motion planner [13] or [14]- [26]:…”

“…Control theoretical approaches to circumvent such difficulties include robust tube-based motion planning [13]- [26] equipped with a tracking control law for robustness and stability certificates. Among these are contraction theorybased robust control [13]- [21], which tracks a target trajectory computed externally by existing motion planners, and thereby robustly keeps the system trajectories in a control invariant tube that satisfies given state constraints.…”