Online Time-Informed Kinodynamic Motion Planning of Nonlinear Systems

TL;DR

This work tackles online time-optimal kinodynamic motion planning for nonlinear systems by introducing a Time-Informed Set (TIS) framework enhanced with deep learning and operator theory. A Deep Invertible Koopman Operator with Control U (DIKU) enables long-horizon, bidirectional prediction in a lifted space, while Random Set theory via ASKU provides online, adversarially inflated TIS approximations to guide sampling. An Online Time-Informed SKMP framework directly samples within the TIS, pruning and expanding the search based on forward/backward reachability analyses. Experiments across six nonlinear systems show near real-time TIS estimation and improved planning efficiency compared with baselines, demonstrating practical impact for fast, kinodynamic planning in high-dimensional, nonlinear domains.

Abstract

Sampling-based kinodynamic motion planners (SKMPs) are powerful in finding collision-free trajectories for high-dimensional systems under differential constraints. Time-informed set (TIS) can provide the heuristic search domain to accelerate their convergence to the time-optimal solution. However, existing TIS approximation methods suffer from the curse of dimensionality, computational burden, and limited system applicable scope, e.g., linear and polynomial nonlinear systems. To overcome these problems, we propose a method by leveraging deep learning technology, Koopman operator theory, and random set theory. Specifically, we propose a Deep Invertible Koopman operator with control U model named DIKU to predict states forward and backward over a long horizon by modifying the auxiliary network with an invertible neural network. A sampling-based approach, ASKU, performing reachability analysis for the DIKU is developed to approximate the TIS of nonlinear control systems online. Furthermore, we design an online time-informed SKMP using a direct sampling technique to draw uniform random samples in the TIS. Simulation experiment results demonstrate that our method outperforms other existing works, approximating TIS in near real-time and achieving superior planning performance in several time-optimal kinodynamic motion planning problems.

Figures (6)

• Figure 1: An illustration of online time-informed kinodynamic motion planning of nonlinear control systems. (a) A Deep Invertible Koopman operator with control U model, DIKU, is trained offline for the nonlinear systems to obtain equivalent linear systems that enable forward and backward dynamics prediction in the lifted space. (b) Our algorithm randomly samples states in the start and goal sets. It then uses the ASKU to bidirectionally propagate the learned linear dynamics and then recover, generating the forward reachable set $\mathcal{X}_k^f$ (independent blue dashed convex hull) and backward reachable tube $\mathcal{X}^b_{-cost:0}$ (serials of green dashed convex hulls) in near real-time. Their intersections constitute the time-informed set (TIS) $\Omega(cost)$ (black convex polygons). Time-informed tree growth is achieved for the off-the-shelf SKMPs by directly sampling in the TIS. The TIS updates according to the cost of the search tree returned after sufficient iterations to help refine the solution (red line).
• Figure 2: An overview of Deep Invertible Koopman operator with control U (DIKU) neural network model for long-horizon forward and backward dynamics prediction
• Figure 3: Comparison of forward and backward dynamics prediction by our DIKU and the DKU with consistency loss. The early predictions are more accurate than the late ones, which can be observed from the lines with plus markers of forward prediction from left to right and from those lines with dot markers of backward from right to left. The lower blue lines indicate smaller bidirectional dynamics evolution errors by DIKU, especially for nonlinear systems.
• Figure 4: Comparison results of the example 2D system tang2022reachability (a) forward reachable sets and tubes and (b) the backward counterparts of TIS computed by the level set toolbox mitchell2005time (Ground truth, black), ellipsoidal toolbox kurzhanski2000ellipsoidal (ET, cyan), relaxed HJB equation method tang2022reachability (RHJB, purple), our basic sampling-based convex approximation (Ours, red), and our adversarial sampling for over-approximation, ASKU, (Ours (AS), green). Our method provides inner-approximated and tight over-approximated TISs online.
• Figure 5: Volume convergence (green line) and speed (blue line) on DIKU with a varying number of samples $M$ for our basic BRT approximation. The volume accuracy is computed by being divided by the volume of ground truth. A high degree of accuracy can be obtained at the cost of computation speed.

Theorems & Definitions (1)

• Lemma III.1