cs.ROSep 23, 2026

Contact-Implicit Stein Projected ADMM for Discovery of Diverse Contact-Rich Manipulation Strategies

Authors: Hrishikesh Sathyanarayan, Christian Hughes, Ian Abraham

Organizations: Department of Mechanical Engineering, Yale University, New Haven, CT, USA. · Department of Electrical Engineering, University of Sydney, NSW, Australia.

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Abstract

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/.

Figures & tables

Fig. 2: Overview of projected variable-splitting construction on a motivating complementarity problem with constraints 0  ≤  λ  ⊥  ϕ  ≥  00\;\leq\;\lambda\;\perp\;\phi\;\geq\;0 , and objective f=12∥[λ,ϕ]⊤−[λ⋆,ϕ⋆]⊤∥22f=\frac{1}{2}\left\lVert[\lambda,\phi]^{\top}-[\lambda^{\star},\phi^{\star}]^{\top}\right\rVert_{2}^{2} . The initial particles xx are first split into duplicate zz variables. The Stein repulsion is measured according to zz to produce z~\tilde{z} as a target reference which is projected onto the feasible set CC . The xx -update then tracks the objective while being pulled toward the feasible set CC through the split variable.
Fig. 3: Illustrative example of N=512N=512 particles, on the annulus problem (top) and the complementarity problem (bottom). At large samples, the proposed approach evenly spreads particles within the feasible set within tolerance. Baseline methods have comparable performance at low-particle numbers but overemphasize certain regions with larger particle sizes.
MethodNNIters.FeasibleMMD 2
Stein ADMM xx -repulsion8840082.4%1.64×10−31.64\times 10^{-3}
Stein ADMM zz -repulsion8849577.3%3.77×10−43.77\times 10^{-4}
SteinSQP [ 16 ]88238100%2.52×10−32.52\times 10^{-3}
Stein Projected ADMM88235100.0%1.29×10−41.29\times 10^{-4}
TABLE I: Annulus problem, median over 1010 seeds.
Fig. 4: Numerical comparison on complementarity problem. (Left) Wall-clock time to first reach constraint tol=10−4\text{tol}=10^{-4} vs. particle count NN (over 99 seeds). (Right) iterations to convergence for Stein Projected ADMM (ours) and SteinSQP [ 16 ] . Experiments were conducted on an Apple M4 Pro chip.
MethodFeasibleMMD 2Convergence (Iters.).Term. Viol.ms/iterMode ϕ\phiMode λ\lambda
Stein ADMM xx -repulsion0.0%1.97×10−21.97\times 10^{-2}200 (never)6.74×10−36.74\times 10^{-3}0.51±0.030.51\pm 0.0336.0±3.736.0\pm 3.730.0±3.730.0\pm 3.7
Stein ADMM zz -repulsion88.6%6.72×10−46.72\times 10^{-4}369.39×10−59.39\times 10^{-5}0.60±0.010.60\pm 0.0132.5±1.132.5\pm 1.133.5±1.133.5\pm 1.1
SteinSQP [ 16 ]100.0%3.21×10−33.21\times 10^{-3}62.99×10−52.99\times 10^{-5}1.00±0.041.00\pm 0.0430.0±3.830.0\pm 3.836.0±3.836.0\pm 3.8
Stein Projected ADMM100.0%3.41×10−33.41\times 10^{-3}36.03×10−56.03\times 10^{-5}0.54±0.010.54\pm 0.0133.0±0.733.0\pm 0.733.0±0.733.0\pm 0.7
TABLE II: Complementarity problem ( 0≤ϕ⊥λ≥00\leq\phi\perp\lambda\geq 0 ), median over 1010 seeds.
Fig. 5: Convergence curves of the box pivoting problem. (a) Maximum constraint violation for each particle solution. (b) Diversity measured by ( 17 ) and objective value per trajectory. Faded colors are the individual constraint curves, solid colors are the mean of the particle set, mean objective (left axis, solid) and log-determinant diversity (right axis, dashed).
Fig. 6: Final overlaid configurations and finger placements in the box pivoting problem for Stein Projected ADMM (ours), the repulsion-off ADMM baseline, and SteinSQP [ 16 ] .
Fig. 7: Diverse cube-pushing trajectories generated from Stein Projected ADMM. Each trajectory particle reaches the same goal pose via different contact strategies. (Left) Rendered particle trajectories with contact forces and points. (Right) Overlaid trajectory distribution.
Fig. 8: Diverse cube-pushing trajectories on the same initialization under three SVGD kernels (RBF, Cauchy, Laplace) and [ 16 ] ’s SteinSQP with contact as an explicit variable (differing from the original implementation).
MethodMax ∣eq∣|eq|Max ineqDiversityMean obj.
RBF6.62×10−36.62\times 10^{-3}3.22×10−33.22\times 10^{-3}−12.37-12.370.0370.037
Cauchy2.33×10−22.33\times 10^{-2}7.67×10−47.67\times 10^{-4}−14.35-14.350.0300.030
Laplace1.12×10−41.12\times 10^{-4}1.58×10−41.58\times 10^{-4}−13.53-13.530.0360.036
SteinSQP (RBF) [ 16 ]1.50×10−71.50\times 10^{-7}1.10×10−51.10\times 10^{-5}−25.25-25.250.2320.232
TABLE III: Kernel/solver comparison on the cube-pushing trajectory problem ( N=8N{=}8 , 300 iterations).
Fig. 9: Diverse grasps on Allegro hand hardware solved with Stein-projected ADMM. We evaluate four-fingered quasi-static grasps of a non-convex object that runs in real-time (we refer the reader to the multimedia material). The executed grasp is chosen via an optimistic (best cost) heuristic from the solved candidate grasps.
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