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Contact-Implicit Stein Projected ADMM for Discovery of Diverse Contact-Rich Manipulation Strategies
Hrishikesh Sathyanarayan, Christian Hughes, Ian Abraham
TL;DR
Contact-implicit optimization can settle on one of many valid contact strategies, and its outcomes are sensitive to initialization. This paper combines Stein variational inference with ADMM, applying repulsion to the split variable rather than the primal variable. The resulting solver discovers diverse contact-rich strategies across manipulation tasks without sacrificing feasibility.
Problem
Contact-implicit optimization has many feasible contact modes but can collapse onto a single local optimum, while existing diversification approaches rely on restarts or generative models with noted limitations.
Source: §I · p. 1, A. Contact-Implicit Trajectory… · p. 2Method
The method integrates Stein repulsion into ADMM through a desired projection for the split variable, while the primal step maintains feasibility.
Source: B. Stein Variational Inference · p. 2, §I · p. 2Results
The solver achieves better diversity than naive repulsion and existing methods without sacrificing feasibility across motivating problems and four contact-rich manipulation tasks.
Source: §VI · p. 7Takeaways & Limitations
The solver is simpler in form and facilitates discovery of diverse contact modes at lower computational cost.
Source: §I · p. 2
Abstract
from arXiv · showhide
Contact-implicit trajectory optimization formulates contact-rich manipulation as a single constrained program; however, that single program run collapses onto one local optimum out of many equally valid contact modes, grasps, or push directions. As a consequence, the resulting manipulation strategy is reluctant to change and sensitive to initialization. In order to promote robust manipulation, this paper investigates how contact-implicit solvers can discover diverse contact-rich strategies. Our approach derives a variation of Consensus Alternating Direction Method of Multipliers (ADMM) combined with Stein variational inference methods to output a set of distinct contact-rich solutions. We find that applying the Stein repulsive force to ADMM's split variable (rather than its primal form) allows for effective coverage over the set of feasible contact strategies without prematurely stalling the solver. We demonstrate the effectiveness of our approach on a variety of contact-rich manipulation tasks, including pushing, grasping, and multi-robot handover. Last, we find the proposed solver is simpler in form and capable of discovering unique contact modes when compared with existing solvers. Videos and code with examples are found in https://anon-website-submission.github.io/stein-admm-website/.
I. INTRODUCTION
Contact-implicit optimization can represent many valid contact strategies, but individual solver runs tend to settle on one, making results sensitive to initialization. The paper combines Stein-based particle diversity with ADMM to discover multiple contact strategies in a single run.
I. INTRODUCTION
- Stein repulsion is applied to ADMM’s split variable, while the primal step tracks the objective and the split variable promotes diversity.The approach is presented as a simpler solver that facilitates discovery of diverse contact modes at lower computational cost.Source: §I · pp. 1–2
A. Contact-Implicit Trajectory Optimization
- Contact-implicit optimization lifts contact forces into the decision variables and enforces contact through complementarity constraints, allowing plans without a pre-specified contact sequence.Its feasible set contains distinct, lower-dimensional contact-mode manifolds; the paper notes that restarts and generative models have limitations, while complementarity-free reformulations forgo the unified treatment of contact modes.Source: A. Contact-Implicit Trajectory… · p. 2
B. Stein Variational Inference
- SVGD uses a particle population and kernel repulsion to promote diversity, but unconstrained SVGD and prior constrained variants have limitations for nonlinear inequalities and non-convex feasible sets.The paper routes the repulsive gradient through ADMM as a desired projection for the split variable while the primal step maintains feasibility.Source: B. Stein Variational Inference · p. 2
III. PRELIMINARIES
The preliminaries describe contact dynamics through complementarity constraints and formulate contact-implicit trajectory optimization over a trajectory of states, controls, and contact forces. They also introduce a split-variable construction in which Stein repulsion is projected onto the feasible set.
A. Contact Dynamics via Complementarity
- Contact impulses comprise normal and friction forces subject to non-penetration complementarity and a Coulomb friction cone.The contact Jacobian’s normal row is the gap-function gradient, and its tangential rows map velocity to relative sliding velocity.Source: A. Contact Dynamics via Complementarity · p. 2
B. Contact-Implicit Trajectory Optimization
- The projected variable-splitting construction measures Stein repulsion on split variables, projects its target onto the feasible set, and uses the x-update to track the objective.The split variable pulls the primal variable toward feasibility.Source: Figure 2 · p. 3
- Contact-implicit trajectory optimization stacks the dynamics over a state, contact-force, and control trajectory, with boundary conditions and additional equality or inequality constraints.The contact forces are decision variables subject to the LCP, so the optimization reasons jointly about forces and contact mode.Source: B. Contact-Implicit Trajectory… · pp. 2–3
C. Stein Variational Gradient Descent
SVGD moves a particle population toward a target distribution using score and repulsion terms, but its updates do not account for constraints. Consensus ADMM splits the constrained problem into objective and feasible-set variables, alternating updates until they reach consensus.
C. Stein Variational Gradient Descent
- SVGD moves particles toward a target density using a score term and kernel repulsion, but its updates do not account for constraints.The repulsive term pushes particles away from nearby particles; the empirical direction approximates a direction that decreases KL divergence between the particle distribution and the target.Source: C. Stein Variational Gradient Descent · p. 3
D. Consensus ADMM
- Consensus ADMM duplicates the constraint output as a variable, then alternates objective and feasible-set updates with a dual ascent step.The split variables reach consensus when v(x) = z; the z-update projects onto the feasible set C.Source: D. Consensus ADMM · p. 3
- The ADMM split can represent constraint equations or the identity map when C contains the feasible set of x.In practice, the indicator function can be replaced by an equivalent smooth approximation.Source: D. Consensus ADMM · p. 3
IV. STEIN PROJECTED ADMM
The method applies Stein repulsion through the projection of ADMM’s split variable rather than directly to the primal variable.
IV. STEIN PROJECTED ADMM
- Stein repulsion is routed through the split-variable projection, using the Stein gradient as a reference within the feasible set while the dual variable pulls the primal variable toward it.This design aims to preserve repulsion in the constraint manifold.Source: §IV · p. 4
A. Stein Projected Variable Splitting
The split-variable construction applies the Stein gradient in ADMM’s constrained update while leaving its objective update unchanged.
A. Stein Projected Variable Splitting
- The algorithm instantiates the split-variable updates for each particle and creates copies of z and u for each one.It takes initial particles, constraints, objective, kernel, repulsion step size, tolerances, and a maximum iteration count as inputs, and outputs x and z particle sets.Source: A. Stein Projected Variable Splitting · p. 4
- The Stein gradient is computed with respect to the split variable z rather than the primal variable x.The method injects it as a reference target in the z-update because applying it directly to z or x may fail to preserve feasible-set diversity.Source: A. Stein Projected Variable Splitting · p. 4
- ADMM’s x-update remains unchanged, tracking the objective while being pulled toward distinct z values within the feasible set.The dual update remains standard and drives x and z toward consensus as the algorithm converges.Source: A. Stein Projected Variable Splitting · p. 4
V. RESULTS
The results section tests Stein Projected ADMM on motivational problems and contact-rich manipulation tasks.
V. RESULTS
- The section evaluates Stein Projected ADMM on motivational problems and contact-rich manipulation tasks.The supplied passage does not report specific outcomes; it says findings are summarized in bold paragraph headers.Source: §V · p. 4
A. Motivating Examples: Annulus and Complementarity
The projected variable-splitting construction applies Stein repulsion to a split variable and projects it onto the feasible set, promoting spread without sacrificing feasibility. At larger particle counts it spreads particles more evenly than the baselines, while reaching the complementarity tolerance about twice as fast as SteinSQP.
A. Motivating Examples: Annulus and Complementarity
- At N = 512, the proposed approach spreads particles evenly within the feasible set, while baselines overemphasize regions at large particle sizes.At low particle counts, baseline performance is comparable.Source: Figure 3 · p. 4, A. Motivating Examples: Annulus and… · p. 5
- Applying Stein repulsion to the split variable z and projecting it onto the feasible set encourages diversity there while the ADMM dual variable pulls solutions toward feasibility.Naive Stein updates to the x- or z-update emphasize the objective, clumping particles at the feasible-set boundary and leaving many infeasible.Source: A. Motivating Examples: Annulus and… · p. 5
- On the complementarity problem, Stein Projected ADMM reaches constraint tolerance 10^-4 approximately 2× faster than SteinSQP, with less variability.The comparison reports wall-clock time over particle counts and iterations to convergence; the experiments used an Apple M4 Pro chip.Source: A. Motivating Examples: Annulus and… · p. 5, Figure 4 · p. 5
- ADMM variable splitting smooths the competing pulls of the objective, constraints, and Stein repulsion, yielding more consistent iterations to convergence.The comparison is shown for the complementarity problem.Source: A. Motivating Examples: Annulus and… · p. 5, Figure 4 · p. 5
B. Box Pivoting with Two Fingers
Across box pivoting and cube pushing, Stein Projected ADMM produces diverse contact strategies while maintaining competitive constraint satisfaction. In pushing, it outperforms the modified SteinSQP comparison in diversity, while kernel choice yields only modest differences in trajectory diversity.
B. Box Pivoting with Two Fingers
- Final box configurations and finger placements compare Stein Projected ADMM with repulsion-off ADMM and SteinSQP.The figure shows overlaid configurations for the box-pivoting task.Source: Figure 6 · p. 5, B. Box Pivoting with Two Fingers · p. 6
- In two-finger box pivoting, Stein Projected ADMM achieves competitive constraint violation, greater diversity than repulsion-off ADMM, and diversity competitive with SteinSQP.The task pivots a box from an initial pose to a target pose while maintaining contact and respecting friction constraints.Source: B. Box Pivoting with Two Fingers · p. 5
C. Non-prehensile Pushing
- For cube pushing, RBF produces subtly greater trajectory diversity than Cauchy and Laplace, while constraint satisfaction varies across kernels and can be adjusted with penalty ρ.The particles reach the same goal pose using different contact sides, points, and forces.Source: C. Non-prehensile Pushing · p. 6, Figure 8 · p. 6, Figure 7 · p. 6
- In cube pushing, Stein Projected ADMM achieves better diversity while maintaining feasibility than modified SteinSQP, which has lower constraint violation but reduced diversity.The SteinSQP comparison optimizes contact as an explicit variable, unlike its original implementation.Source: C. Non-prehensile Pushing · p. 6
D. Quasi-static Grasping and Humanoid Hand-over
Stein Projected ADMM is reported to improve diversity without sacrificing feasibility across contact-rich manipulation tasks. In quasi-static grasping, it finds different finger placements, orientations, and forces for the same equilibrium, while the section also describes hardware grasps and humanoid hand-over.
D. Quasi-static Grasping and Humanoid Hand-over
- The quasi-static grasping and humanoid hand-over tasks optimize contact points and forces for a target object pose under equilibrium, friction, and contact constraints.The optimization also accounts for differential kinematics, manipulability, and collision to ensure feasibility within lower-level kinematic coordination.Source: D. Quasi-static Grasping and Humanoid… · p. 6
- For the same quasi-static equilibrium, grasp solutions spread across different contact regions, finger placements, orientations, and forces.The ADMM formulation incorporates multi-body kinematics through constraint functions or penalties, with projection performed by a differential kinematic solver respecting joint limits and contact constraints.Source: D. Quasi-static Grasping and Humanoid… · pp. 6–7
- Allegro-hand experiments evaluate real-time four-finger quasi-static grasps of a non-convex object, selecting the executed grasp by an optimistic best-cost heuristic.The candidate grasps are shown in Fig. 9.Source: Figure 9 · p. 7
VI. CONCLUSION
- Across four contact-rich manipulation tasks, Stein Projected ADMM achieves better diversity than naive repulsion and existing methods without sacrificing feasibility, and scales favorably with particle number.It is also competitive with existing methods in wall-clock time to convergence and solution diversity.Source: §VI · p. 7
APPENDIX
The appendix describes contact and friction formulations, diversity measurement, illustrative constrained problems, and task-specific setups for pushing, pivoting, grasping, and hand-over.
Friction formulation
- Friction forces are approximated as non-negative combinations of fixed tangent directions, with complementarity conditions enforcing maximum dissipation and friction-cone constraints.A slip-rate variable enforces that friction opposes relative sliding at the cone boundary.Source: APPENDIX · p. 7
Measuring diversity
- Particle diversity is measured by the log-determinant of the kernel Gram matrix; values closer to zero indicate greater diversity, with sensitivity depending on kernel and bandwidth.The kernel and bandwidth are chosen to reflect the problem's scale.Source: APPENDIX · p. 7
Annulus problem
- The annulus example pulls particles toward a point outside a non-convex feasible band defined by inner and outer radius inequalities.Its objective is centered at μ = (5, 0)ᵀ with σ = 2.Source: APPENDIX · p. 7
Complementarity problem
- The complementarity example uses a quadratic objective centered at an infeasible point, with exact complementarity placing minima evenly between the two axis-aligned solutions.The infeasible center is (λ⋆, ϕ⋆) = (1, 1).Source: APPENDIX · p. 7
Box pivoting
- The box-pivoting setup uses two finger contacts and allows terminal orientations within a two-sided band around π/4, with complementarity and friction constraints on finger and ground forces.The stated band is |α_T − π/4| ≤ 1°; the setup also specifies a 25-step horizon with Δt = 0.05 s.Source: APPENDIX · pp. 7–8
Non-prehensile pushing
- The pushing setup uses one pusher contacting the box at a body-frame point; its kernel study compares median-bandwidth RBF, Cauchy, and Laplace kernels.SteinSQP optimizes over box pose, contact point, and normal and tangential forces.Source: APPENDIX · p. 8
Quasi-static grasping and humanoid hand-over
- Quasi-static grasping generates diversity through Stein repulsion within the force-closure manifold, while hand-over uses four hand contacts subject to equilibrium, friction, complementarity, and arm-reach constraints.Both setups set the objective to zero; the hand-over example specifies a 3 kg cube with 0.18 m halfwidth and friction coefficient 0.5.Source: APPENDIX · p. 8