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CQDC-Warp: GPU-Accelerated Differentiable Quasi-Dynamic Simulation for Contact-Rich Manipulation

Yanchen Ding, Jiayi Chen, Jianfeng Xu, Wei Zhao, Li Yi, He Wang

arXiv:2610.21349v1cs.RO

TL;DR

Contact-rich trajectory optimization must contend with coupled motion and contact decisions, while existing differentiable simulators face throughput and evaluation-scope limitations. CQDC-Warp combines GPU-batched differentiable quasi-dynamic simulation with contact-guided MPC; on the Allegro benchmark it improves success and generation time over CFree and IMPACT.

  • Problem

    Efficient contact-rich trajectory generation matters for downstream robot learning, but dexterous data collection is costly and existing differentiable simulators have throughput and task-coverage limitations.

    Source: §I · p. 1

  • Method

    CQDC-Warp uses GPU-batched differentiable quasi-dynamic simulation for MPC, with intended contacts guiding planning while simulated dynamics determine their realization.

    Source: §I · p. 2

  • Results

    Across 17,408 Allegro trials, CQDC-Warp achieves 79.8% mean success versus 73.7% for CFree and 65.2% for IMPACT, with mean generation time of 0.794 s versus 21.422 s and 39.844 s.

    Source: A. In-Hand Rotation · p. 6

  • Takeaways & Limitations

    The framework is demonstrated across contact-rich tasks and robot platforms, including real-world trajectory replay.

    Source: §I · p. 2, §VII · p. 7

    The planner relies on suitable initializations and intended contacts specified before planning.

    Source: §VII · p. 7

Abstract

from arXiv · show

Efficient synthesis of contact-rich manipulation trajectories is essential for downstream robot learning. Gradient-based planning offers a promising approach, but jointly optimizing motion and contact decisions remains challenging. Meanwhile, many existing planners run on CPUs with limited throughput. We present CQDC-Warp, a GPU-accelerated differentiable quasi-dynamic simulator for batched model predictive control (MPC). We use it to study contact-guided trajectory optimization, in which intended robot--object contacts are specified before planning and encouraged through soft costs, while the simulated dynamics determine contact geometry and frictional response without a prescribed contact-mode schedule. Across $17{,}408$ trials on the $17$-object Allegro in-hand rotation benchmark, our framework achieves $79.8\%$ success, compared with $73.7\%$ for CFree and $65.2\%$ for IMPACT, and reduces mean trajectory-generation time by factors of $27.0$ and $50.2$, respectively. On a single NVIDIA RTX 4090 with $2{,}048$ parallel worlds, forward and backward throughput reach $2.06\times10^5$ and $2.25\times10^5$ world-steps per second, respectively. Further experiments cover manipulation with LEAP and Shadow hands, environmental contact exploitation, bimanual manipulation, whole-body control, and real-world trajectory replay on a Sharpa hand.

I. INTRODUCTION

Contact-rich dexterous trajectory generation could support robot learning, but data collection is costly and existing differentiable planners face throughput and task-diversity limitations. CQDC-Warp combines GPU-batched differentiable simulation with contact-guided planning, and the paper evaluates it across varied robots and tasks.

I. INTRODUCTION

  • Dexterous demonstration collection is costly, while model-free RL typically requires extensive task-specific trial and error and relies on sampled experience rather than dynamics derivatives.Learned policies can remain sensitive to their training distributions, limiting reuse across contact-rich tasks and scenes.Source: §I · p. 1
  • Existing differentiable contact simulators may smooth contact forces or neglect constraint coupling, while many run on CPUs or optimize one trajectory at a time.Published GPU-based evaluations also typically cover only one or two robot embodiments and a limited task set.Source: §I · pp. 1–2
  • CQDC-Warp extends the Convex Quasi-Dynamic Contact model with compliant frictional contact, revised inertial coupling, implicit dynamics derivatives, sampled contact-geometry derivatives, and GPU-batched MPC.Evaluations include in-hand manipulation, environmental contact exploitation, bimanual manipulation, whole-body control, and real-world trajectory replay.Source: §I · p. 2
  • Intended contacts are specified before planning, but the simulated dynamics determine their exact geometry and frictional response without a prescribed contact-mode schedule.The intended contact identifies which robot link should touch the object.Source: §I · pp. 1–2

II. RELATED WORK

Related work spans rigid-body contact models, differentiable dynamics and collision geometry, and trajectory-optimization formulations. CQDC-Warp builds on convex quasi-dynamic contact and sampling-based surrogate geometry gradients, while using GPU-batched single-shooting MPC.

A. Rigid-Body Contact Models

  • Classical penalty models trade lower penetration at higher stiffness against worse numerical conditioning, while complementarity models include rigid frictional contact and CQDC uses a convex cone-complementarity formulation.Polyhedral friction approximations yield LCPs; cone complementarity retains circular friction cones but can permit gliding artifacts.Source: A. Rigid-Body Contact Models · p. 2

B. Differentiable Contact Simulation

  • Implicit differentiation computes sensitivities at a converged contact solution without differentiating through solver iterations, while geometry-gradient approaches trade SDF overhead or coupled-system size against local surface information.CQDC-Warp follows implicit differentiation and adopts sampling-based surrogate gradients that aggregate information across neighboring surface features.Source: B. Differentiable Contact Simulation · p. 2

C. Contact-Rich Trajectory Optimization

  • Single shooting optimizes controls through complete rollouts, whereas multiple shooting and direct collocation expose temporal sparsity and allow state-trajectory initialization at the cost of more variables and constraints.CQDC-Warp uses GPU-batched differentiable rollouts for single-shooting MPC; its contact-guided setting lies between predefined contact modes and contact-implicit optimization.Source: C. Contact-Rich Trajectory Optimization · p. 3

A. CQDC: Convex Quasi-Dynamic Contact Dynamics

CQDC formulates quasi-dynamic contact through an optimization problem whose optimality conditions encode the contact dynamics. For collision detection, sampled surface points provide a smooth surrogate for differentiating witness-point selection.

A. CQDC: Convex Quasi-Dynamic Contact Dynamics

  • The CQDC model's generalized-inequality KKT conditions identify its contact dynamics as the optimality conditions of a convex optimization problem.The formulation includes actuated and unactuated configurations, actuator targets, external forces, and stacked per-contact Jacobians and forces.Source: A. CQDC: Convex Quasi-Dynamic Contact… · p. 3
  • The quasi-dynamic approximation neglects transient dynamics and focuses on transitions between steady states, supporting larger planning steps and fewer steps to look further ahead.It is described as useful for quasi-static manipulation dominated by friction and damping.Source: A. CQDC: Convex Quasi-Dynamic Contact… · p. 3

B. Differentiable Collision Detection

  • DiffCollision makes sampled witness-point selection differentiable with a distance-based backward surrogate, using a softmax-weighted average of surface samples.Equivalent gradient transport transfers a fixed object's gradient to the optimized object when only one pose is adjustable.Source: B. Differentiable Collision Detection · p. 3

IV. CQDC-WARP

CQDC-Warp replaces classical hard contact with compliant frictional contact and revises the inertial model. The compliant formulation penalizes deformation and supports a batched GPU solve.

A. Compliant Contact

  • CQDC-Warp replaces hard contact with compliant frictional contact that retains the friction-cone constraint while penalizing contact deformation.Source: A. Compliant Contact · p. 3
  • The compliance matrix R = ρI sets contact softness, while η scales the reference gap and controls penetration recovery.In the stiction branch, the normal response acts as a unilateral linear spring with stiffness 1/ρ and zero-force reference gap (1 − η)g_i.Source: A. Compliant Contact · pp. 3–4

IV. CQDC-WARP

  • The revised inertial metric reproduces MuJoCo-Warp’s trajectory, unlike original CQDC, which misses joint–base coupling.Source: Figure 2 · p. 4

A. Compliant Contact

  • Cone projection eliminates the contact-space slack variables, leaving a strictly convex problem solved by a batched GPU Newton method.Source: A. Compliant Contact · p. 4

B. Revised Inertial Model

CQDC-Warp retains inertial coupling between actuated and unactuated coordinates and uses shared witness-pair weights to estimate contact geometry. Its preliminary witness-point benchmark matches DiffCollision without EGT.

B. Revised Inertial Model

  • The revised model retains joint–base inertial coupling through the full generalized mass matrix, which classical CQDC’s block-diagonal metric omits.The omission can contribute to sim-to-sim and sim-to-real discrepancies.Source: B. Revised Inertial Model · p. 4

C. Contact-Geometry Surrogate Gradients

D. Contact-Solve Implicit Gradients

CQDC-Warp computes contact-solve gradients by implicitly differentiating the converged solution. The method uses the stationarity equation’s displacement Jacobian to solve for downstream sensitivities.

D. Contact-Solve Implicit Gradients

E. Implementation

CQDC-Warp combines GPU-parallel simulation and differentiation with batched gradient-based MPC. Contact intent enters through soft costs, while each planning cycle executes the best sequence’s first command and replans.

E. Implementation

  • CQDC-Warp uses NVIDIA Warp and adapts MuJoCo-Warp’s batched GPU pipeline, adding adjoints for kinematics, surrogate gradients for collision geometry, and implicit gradients for contact solves.Source: E. Implementation · p. 4

V. CQDC-WARP MODEL PREDICTIVE CONTROL

  • Intended robot–object contacts enter the MPC objective as soft costs, but collision detection and the contact solve determine realized geometry, forces, and frictional response.Contact intent does not prescribe a complete contact-mode schedule.Source: §V · p. 5

A. Batched Receding-Horizon Optimization

B. Trajectory Costs

CQDC-Warp defines trajectory costs over residuals, using pose errors to track goals and regularization terms to constrain motion and commands.

B. Trajectory Costs

1) Positional Costs:

  • Pose residuals track desired translation and rotation, with additional goal-error-rate penalties to address insufficient positional gradients near the goal.The rotation error uses the relative rotation’s axis–angle representation.Source: 1) Positional Costs · p. 5
  • Detaching the previous endpoint’s error preserves the gradient at the new endpoint and prevents reducing the rate by moving the previous endpoint.Source: 1) Positional Costs · p. 5

2) Regularization Costs:

  • Velocity residuals activate above configured limits, while command velocity and acceleration residuals regularize command magnitude and temporal variation.Source: 2) Regularization Costs · p. 5

3) Contact Costs:

Contact costs encourage designated robot links to realize intended contacts while leaving witness-point placement and frictional mode to the dynamics.

3) Contact Costs:

  • For each task, designated robot links define intended contacts; distance residuals measure signed-gap deviation from a target, and optional position residuals anchor witness points along selected axes.Source: 3) Contact Costs · p. 5
  • Distance costs encourage intended contact and optional position costs bias its location, without fixing the witness point or frictional mode in the dynamics.Source: 3) Contact Costs · p. 5

VI. EXPERIMENT

  • Planning uses one NVIDIA RTX 4090, 20 gradient iterations, and horizon H = 5; success normally requires position and rotation errors below 2 cm and 5° for 20 consecutive steps.Source: §VI · p. 5
  • Table II reports success and mean trajectory-generation time over 1,024 trials per object.Source: Table II · p. 6
  • Figure 3 groups representative trajectories into in-hand rotation, environmental contact, bimanual manipulation, and whole-body control; light-green shapes mark goal poses.Source: Figure 3 · p. 6

A. In-Hand Rotation

The section reports Allegro in-hand rotation results and extends evaluation to other hands, environmental and bimanual tasks, whole-body control, and physical replay.

A. In-Hand Rotation

  • 79.8% mean success over 17,408 Allegro trials, versus 73.7% for CFree and 65.2% for IMPACT; mean trajectory-generation time is 0.794 s versus 21.422 s and 39.844 s.The reported speedups are 27.0× and 50.2× under these execution settings.Source: A. In-Hand Rotation · p. 6
  • Success rates were 79.88% for LEAP, 65.75% for Shadow, and 54% for box flip, with targets perturbing initial orientations by 15◦–30◦ around uniformly sampled axes, supporting transfer across hand embodiments.Source: A. In-Hand Rotation · p. 6

B. Additional Robots and Tasks

  • CQDC-Warp succeeds in 938/1024 bottle-tilting trials and 902/1024 wall-supported object-raising trials using environmental contacts as passive support.Source: B. Additional Robots and Tasks · p. 6
  • Two Shadow hands achieve 1024/1024 successes opening a bottle and 1007/1024 successes lifting a truncated cone by 0.20 m while preserving its orientation.Source: B. Additional Robots and Tasks · p. 6
  • Whole-body evaluations include A1 rising to a three-legged stance, GO2-D1 lifting a bottle while maintaining its base pose, and G1 posture and bottle-lifting tasks.Source: B. Additional Robots and Tasks · p. 6
  • A physical Sharpa hand replays the bottle–table trajectory at 10 Hz, supporting its physical plausibility and potential use for robot learning.The bottle model was obtained by scanning the target bottle and reconstructing its digital twin.Source: B. Additional Robots and Tasks · pp. 6–7

C. Time Analysis

CQDC-Warp reaches high batched simulation throughput, while solver time and iteration count decrease as contact becomes softer. Ablations show that contact-geometry derivatives and planning lookahead improve Allegro benchmark success; the planner nevertheless depends on suitable initializations and intended contacts specified in advance.

C. Time Analysis

  • At 2,048 worlds, CQDC-Warp reaches 2.06 × 10^5 forward and 2.25 × 10^5 backward world-steps/s.Forward throughput is comparable to MuJoCo-Warp's 2.24 × 10^5 world-steps/s at that batch size.Source: C. Time Analysis · p. 7
  • At ρ = 10^-4, CQDC-Warp's combined forward and backward time is 9.07 ms, versus 97.52 s for Clarabel and 11.54 s for MOREAU.As ρ increases, forward time falls from 47.97 to 4.60 ms and mean Newton iterations from 14.83 to 3.41.Source: C. Time Analysis · p. 7

D. Ablation Study

  • 79.81% success falls to 77.37% without contact-geometry derivatives, a 2.44-percentage-point drop.The ablation fixes witness points in each object's local frame.Source: D. Ablation Study · p. 7
  • H = 5 achieves 79.77% success; success falls to 69.39% at H = 3 and zero at H = 1.Longer horizons increase generation time without improving success.Source: D. Ablation Study · p. 7

VII. CONCLUSION AND LIMITATIONS

  • The planner requires suitable initializations and intended contacts specified before planning, and its objective weights require tuning across tasks and embodiments.Evaluation on long-horizon tasks with frequent contact transitions, and the impact of generated data on downstream policy learning, remain to be assessed.Source: §VII · p. 7
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