Trajectory Optimization under Contact Timing Uncertainties

TL;DR

This work presents a novel optimal control formulation to find robust control policies under contact timing uncertainties that ensures robustness criterion satisfaction of candidate pre-contact states and optimizes for contact-relevant objectives.

Abstract

Most interesting problems in robotics (e.g., locomotion and manipulation) are realized through intermittent contact with the environment. Due to the perception and modeling errors, assuming an exact time for establishing contact with the environment is unrealistic. On the other hand, handling uncertainties in contact timing is notoriously difficult as it gives rise to either handling uncertain complementarity systems or solving combinatorial optimization problems at run-time. This work presents a novel optimal control formulation to find robust control policies under contact timing uncertainties. Our main novelty lies in casting the stochastic problem to a deterministic optimization over the uncertainty set that ensures robustness criterion satisfaction of candidate pre-contact states and optimizes for contact-relevant objectives. This way, we only need to solve a manageable standard nonlinear programming problem without complementarity constraints or combinatorial explosion. Our simulation results on multiple simplified locomotion and manipulation tasks demonstrate the robustness of our uncertainty-aware formulation compared to the nominal optimal control formulation.

Figures (10)

• Figure 1: Illustration of Uncertain Hybrid Systems
• Figure 2: Illustration of the difference between (a) the nominal optimal control and (b) the proposed approach. The proposed method does not switch the mode of the states within the robust phase, but generates a trajectory of feasible pre-switch states over the uncertainty set.
• Figure 3: Illustration of the planar two-link point-footed robot. (a) The robot has two joints (hip and knee) and a 2-DoF base. The black dot denotes the CoM. (b) When landing, the ground position is uncertain.
• Figure 4: Simulation data during the robust phase. The weights for maximizing the uncertainty in (a),(c) are respectively 1000x that in (b),(d). 'mi' denotes the impact minimization over the given uncertainty [-0.05, 0.05]m. 'mu_(x)' denotes uncertainty maximization for known impact limits x (unit: N), where the flat region denotes the uncertainty. 'nom' denotes the nominal optimal control data. Flat parts of 'mu_(x)' denote the optimized uncertainty region where the impact limits are satisfied.
• Figure 5: Illustration of how the manipulator catches a free-falling object. The manipulator (a) lifts its end-effector (EF) to a high position and then (b) lowers its EF to reduce the velocity w.r.t. the object.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3