Minimum sensitivity control for planning with parametric and hybrid uncertainty

Abstract

This paper introduces a method to minimize norms on nonlinear trajectory sensitivities during open-loop trajectory optimization. Specifically, we derive new parametric sensitivity terms that measure the variation in nonlinear (continuous-time) trajectories due to variations in model parameters, and hybrid sensitivities, which account for variations in trajectory caused by sudden transitions from nominal dynamics to alternative dynamic modes. We adapt continuous trajectory optimization to minimize these sensitivities while only minimally changing a nominal trajectory. We provide appended states, cost, and linearizations, required so that existing open-loop optimization methods can generate minimally sensitive feedforward trajectories. Although there are several applications for sensitivity optimization, this paper focuses on robot motion planning, where popular sample-based planners rely on local trajectory generation to expand tree/graph structures. While such planners often use stochastic uncertainty propagation to model and reduce uncertainty, this paper shows that trajectory uncertainty can be reduced by minimizing first-order sensitivities. Simulated vehicle examples show parametric sensitivity optimization generates trajectories optimally insensitive to parametric model uncertainty. Similarly, minimizing hybrid sensitivities reduces uncertainty in crossing mobility hazards (e.g. rough terrain, sand, ice). Examples demonstrate the process yields a planner that uses approximate hazard models to automatically and optimally choose when to avoid hazardous terrain and when controls can be adjusted to traverse hazards with reduced uncertainty. Sensitivity optimization offers a simple alternative to stochastic simulation and complicated uncertainty modeling for nonlinear systems.