Properties of Expected Residual Minimization Model for a Class of Stochastic Complementarity Problems

Luo, Meiju; Lu, Yuan

doi:10.1155/2013/497586

Cited by 5 publications

(5 citation statements)

References 12 publications

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“…The preceding formulation assumes perfect knowledge of the contact parameters. If any of the terms in (1c)-(1e) are uncertain or random, then resolving the contact forces becomes a stochastic complementarity problem (SCP) [29]:…”

Section: B Stochastic Complementarity Constraintsmentioning

confidence: 99%

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Robust Trajectory Optimization Over Uncertain Terrain With Stochastic Complementarity

Drnach

Zhao

2021

IEEE Robot. Autom. Lett.

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Trajectory optimization with contact-rich behaviors has recently gained attention for generating diverse locomotion behaviors without pre-specified ground contact sequences. However, these approaches rely on precise models of robot dynamics and the terrain and are susceptible to uncertainty. Recent works have attempted to handle uncertainties in the system model, but few have investigated uncertainty in contact dynamics. In this study, we model uncertainty stemming from the terrain and design corresponding risk-sensitive objectives under the framework of contact-implicit trajectory optimization. In particular, we parameterize uncertainties from the terrain contact distance and friction coefficients using probability distributions and propose a corresponding expected residual minimization cost design approach. We evaluate our method in three simple robotic examples, including a legged hopping robot, and we benchmark one of our examples in simulation against a robust worst-case solution. We show that our risksensitive method produces contact-averse trajectories that are robust to terrain perturbations. Moreover, we demonstrate that the resulting trajectories converge to those generated by a traditional, non-robust method as the terrain model becomes more certain. Our study marks an important step towards a fully robust, contact-implicit approach suitable for deploying robots on real-world terrain.

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Section: B Stochastic Complementarity Constraintsmentioning

confidence: 99%

“…Theoretical works have studied robust solutions to Eq. ( 2) in both the case when F is affine [24], [25], [30] and in the case when F is nonlinear [29]. In these works, it is common to define a residual function ψ such that the residual is zero when the complementarity conditions are satisfied:…”

Section: Expected Residual Minimizationmentioning

confidence: 99%

Robust Trajectory Optimization Over Uncertain Terrain With Stochastic Complementarity

Drnach

Zhao

2021

IEEE Robot. Autom. Lett.

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“…The complementarity constraints in (1) assume that perfect information about the contact model is available. However, if any of the model parameters are uncertain, the problem has stochastic complementarity constraints (SCP) ( Luo and Lu, 2013 ) 0 ≤ z ⊥ F ( z , ω ) ≥ 0, ω ∈ Ω where z is a deterministic variable, and F (⋅) is a vector-valued stochastic function, and ω represents a random variable on probability space

, with sample space Ω, event space

, and probability distribution

.…”

Section: Problem Formulationmentioning

confidence: 99%

“…Prior works on SCPs ( Chen et al, 2009 ; Tassa and Todorov, 2010 ; Luo and Lu, 2013 ) commonly replace the complementarity constraint with a residual function ψ that attains its roots when the complementarity constraints are satisfied: ψ ( z , F ) = 0⇔ z ≥ 0, F ≥ 0, zF = 0. Although this formulation is for scalars z and F , it generalizes to the case when z and F are vectors by applying the complementarity constraints and/or the residual function elementwise.…”

Section: Problem Formulationmentioning

confidence: 99%

Mediating Between Contact Feasibility and Robustness of Trajectory Optimization Through Chance Complementarity Constraints

2022

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As robots move from the laboratory into the real world, motion planning will need to account for model uncertainty and risk. For robot motions involving intermittent contact, planning for uncertainty in contact is especially important, as failure to successfully make and maintain contact can be catastrophic. Here, we model uncertainty in terrain geometry and friction characteristics, and combine a risk-sensitive objective with chance constraints to provide a trade-off between robustness to uncertainty and constraint satisfaction with an arbitrarily high feasibility guarantee. We evaluate our approach in two simple examples: a push-block system for benchmarking and a single-legged hopper. We demonstrate that chance constraints alone produce trajectories similar to those produced using strict complementarity constraints; however, when equipped with a robust objective, we show the chance constraints can mediate a trade-off between robustness to uncertainty and strict constraint satisfaction. Thus, our study may represent an important step towards reasoning about contact uncertainty in motion planning.

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“…see [32, equation (3.8)]. In [36], Luo and Lin minimize the expected residual while convergence analysis of the expected residual minimization (ERM) technique has been carried out in the context of stochastic Nash games [37] and stochastic variational inequality problems [35]. In more recent work, Chen, Wets, and Zhang [7] revisit this problem and present an alternate ERM formulation, with the intent of developing a smoothed sample average approximation scheme.…”

Section: The Expected-residual Minimization (Erm) Formulationmentioning

confidence: 99%

On the existence of solutions to stochastic quasi-variational inequality and complementarity problems

Ravat

Shanbhag

2017

Math. Program.

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Variational inequality problems allow for capturing an expansive class of problems, including convex optimization problems, convex Nash games and economic equilibrium problems, amongst others. Yet in most practical settings, such problems are complicated by uncertainty, motivating the examination of a stochastic generalization of the variational inequality problem and its extensions in which the components of the mapping contain expectations. When the associated sets are unbounded, ascertaining existence requires having access to analytical forms of the expectations. Naturally, in practical settings, such expressions are often difficult to derive, severely limiting the applicability of such an approach. Consequently, our goal lies in developing techniques that obviate the need for integration and our emphasis lies in developing tractable and verifiable sufficiency conditions for claiming existence. We begin by recapping almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued maps provided in our prior work [42,43] and provide extensions to multi-valued mappings. Next, we extend these statements to quasi-variational regimes where maps can be either single or set-valued. Finally, we refine the obtained results to accommodate stochastic complementarity problems where the maps are either general or co-coercive. The applicability of our results is demonstrated on practically occuring instances of stochastic quasi-variational inequality problems and stochastic complementarity problems, arising as nonsmooth generalized Nash-Cournot games and power markets, respectively.

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Properties of Expected Residual Minimization Model for a Class of Stochastic Complementarity Problems

Cited by 5 publications

References 12 publications

Robust Trajectory Optimization Over Uncertain Terrain With Stochastic Complementarity

Robust Trajectory Optimization Over Uncertain Terrain With Stochastic Complementarity

Mediating Between Contact Feasibility and Robustness of Trajectory Optimization Through Chance Complementarity Constraints

On the existence of solutions to stochastic quasi-variational inequality and complementarity problems

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