Robotic Manipulation Under Stochastic Contact Constraints

The patent introduces a barrier function as an intermediary mathematical tool to handle the stochastic complementarity constraints. The barrier function transforms the infeasible stochastic complementarity constraints into a feasible optimization problem by penalizing constraint violations in a differentiable manner, allowing the use of standard optimization algorithms while maintaining the physical meaning of contact constraints.

The patent transforms the stochastic complementarity constraints by changing the parameter representation from direct complementarity conditions to expected value constraints with variance penalties. This parameter transformation converts the infeasible stochastic problem into a feasible deterministic equivalent that can be solved using standard optimization techniques while accounting for uncertainty.

The patent transforms the static hard constraints into dynamic soft constraints that adapt during optimization. By using expected residual minimization with penalty functions, the constraints become flexible during the optimization process, allowing the solver to navigate the feasible region more easily while still converging to solutions that satisfy the original hard constraints with high probability.

The patent applies beforehand cushioning by introducing penalty terms that anticipate constraint violations. The expected residual minimization approach pre-penalizes potential constraint breaches in the objective function, cushioning against infeasibility before it occurs and guiding the optimization toward feasible solutions without requiring complex constraint handling during the solving process.

The patent incorporates feedback by using the expected residual information from the soft constraints to guide the optimization toward more robust solutions. The penalty terms provide feedback about constraint satisfaction levels, allowing the optimizer to adjust control inputs to maintain feasibility under uncertainty, thereby improving robustness while keeping the problem computationally tractable.

The patent substitutes the mechanical hard constraint enforcement with a statistical mechanics approach using expected value constraints and penalty functions. This substitution replaces the rigid feasibility enforcement with a probabilistic framework that achieves similar robustness goals while enabling the use of efficient gradient-based optimization methods.