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.

Trajectory Generation for Minimum Closed-Loop State Sensitivity

“…The quantity Π(t) cannot be, in general, computed in closed-form, but its dynamics can be given a closed-form expression. This has, for instance, been detailed in [15] for an open-loop version of the problem considered in this paper, in which the (indirect) dependency of the control inputs u(·) on the system parameters is not taken into account. In this case, since u(·) is considered independent from p, the dynamics of Π takes the expressioṅ…”

“…However, estimating the parameters online can, again, introduce undesired transients and coupled estimation/system dynamics that can be hard to analyze. Other interesting recent works have instead focused on the generation of feedforward trajectories meant to minimize the state sensitivity of a system, see, e.g., [13]- [15] and references therein. However, the main limitation of these approaches is in their 'open-loop' nature, as they do not fully take into account the effects of parametric uncertainty on the coupled system/controller pair (i.e., the 'closed-loop' system).…”

“…One can then, for instance, choose and fine tune a control law for a nominal model of the system, and then generate optimized trajectories that fulfill some given task and whose realization by the chosen controller results as insensitive as possible to variations in the nominal parameters. This is achieved by extending the 'open-loop' optimization proposed in [15] to the 'closed-loop' case, in which the coupling system/controller, and the dependency of the controller (in both its feedforward and feedback terms) on the (possibly uncertain) parameters, is fully taken into account.…”

Toward Aerial Physical Locomotion: The Contact-Fly-Contact Problem

“…Reasonable choices for the cost functions J 1 (x), J 2 (x) and J 3 (x) could be the execution time or control effort/energy for generating minimum-time or minimumeffort/energy trajectories from x 0 to x f . Motivated by the recent [21], in this work we however choose to also consider optimality of the state sensitivity w.r.t. variations in the system parameters (e.g., mass, inertia, CoM location, propeller characteristics, and so on).…”

Optimal Active Sensing with Process and Measurement Noise

“…One exception is represented by the POMPD-based methods such as [11], [12] where the authors propose a motion planning algorithm that minimises the motion and the sensing uncertainties in environments with static obstacles. Another approach is the one in [13], where an optimal control algorithm selects trajectories that optimises a cost function composed by the state tracking goals, the control effort, and the sensitivity of planned motion to variations in model parameters, including actuation noise. However, these approaches are more devoted to an offline implementation (also because of their computational complexity), while the goal of this paper is to propose an online solution to the active sensing problem.…”