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PRISM: Projection-Integrated Sampling-Based MPC with Bayesian Cost Tuning for Bimanual Manipulation

Alinjar Dan, Iryna Hurova, Karl Kruusamäe, Arun Kumar Singh

arXiv:2608.25666v1cs.RO

TL;DR

Bimanual manipulation in cluttered, contact-rich settings requires coordinated, constraint-respecting planning that can adapt to changing geometry and interactions. PRISM combines projection-integrated sampling MPC with GPU-accelerated physics rollouts and offline Bayesian cost tuning. Across four bimanual tasks, it reports improved success and consistent performance, operates at 10–16 Hz, and transfers successfully to dual UR5e hardware.

  • Problem

    Bimanual manipulation remains difficult because learned policies can be sensitive to distribution shifts, while sampling-based controls may violate smoothness and kinematic limits.

  • Method

    PRISM samples joint-velocity trajectories, projects them onto position, velocity, acceleration, jerk, and boundary constraints, evaluates feasible rollouts in a GPU-accelerated MuJoCo model, and tunes cost weights offline with Bayesian optimization.

  • Results

    Across tray moving, ball lifting and transport, cube handover, and box lifting, PRISM improves success and consistency against sampling-based baselines, runs at 10–16 Hz, and transfers successfully to UR5e manipulators.

  • Takeaways & Limitations

    PRISM provides an online model-based alternative that adapts to novel geometric configurations without retraining while supporting contact-rich bimanual planning.

  • Takeaways & Limitations

    The cost-function structure remains manually designed, and MuJoCo-based modeling may leave sim-to-real gaps for deformable objects, fluids, or highly dynamic contacts.

Abstract

from arXiv · show

Bimanual manipulation in cluttered, contact-rich environments remains challenging because it requires coordinated motion generation, interaction-aware planning, and reliable execution under tight kinematic constraints. We present PRISM, a projection-integrated sampling-based Model Predictive Control (MPC) framework that uses a GPU-accelerated physics simulator as an online world model for complex dual-arm manipulation. The main algorithmic contribution is a QP-guided control sampling strategy that decouples trajectory exploration from kinematic feasibility. At each MPC step, sampled joint-velocity trajectories are projected onto the set of motions satisfying joint position, velocity, acceleration, and jerk bounds, together with an initial-velocity boundary condition, before rollout evaluation. This enables broad yet feasible exploration of coordinated bimanual behaviors. To support efficient online execution, we derive a custom ADMM/Bregman-splitting QP solver that exploits joint-wise separability and reusable matrix factorizations. We further use Bayesian optimization to tune task-cost weights offline, reducing manual parameter selection. We evaluate PRISM on challenging variants of PerAct$^{2}$ tasks, including obstacle-constrained ball transport, tray transport, cube handover, and box lifting. Experiments show improved robustness and task success relative to representative sampling-based baselines, while maintaining real-time or near-real-time execution. We also demonstrate successful sim-to-real transfer on dual UR5e manipulators, highlighting the practical potential of physics-based online planning for contact-rich bimanual manipulation. Project details, including code and supplementary videos, are available at \href{https://sites.google.com/view/prismbimanual}{\texttt{https://sites.google.com/view/prismbimanual}}.

I. INTRODUCTION

PRISM addresses distribution-shift and smoothness challenges in bimanual manipulation through online, physics-based sampling MPC with QP projection and offline Bayesian cost tuning. It is evaluated on contact-rich tasks where coordinated adaptation to novel geometries is important.

  • Introduction: PRISM uses GPU-accelerated physics rollouts to evaluate large batches of candidate bimanual motion sequences against the current scene and contact state.The approach targets previously unseen obstacle layouts, object poses, and contact interactions without relying on a monolithic learned policy.
  • Introduction: QP projection separates exploration from kinematic feasibility by making sampled trajectories satisfy position, velocity, acceleration, jerk, and boundary-continuity constraints before rollout evaluation.This permits aggressive exploration covariances while retaining smooth, executable candidates.
  • Introduction: PRISM is tested on obstacle-constrained ball transport, tray transport, cube handover, and box lifting, with reported gains in robustness, task success, and execution relative to representative sampling-based alternatives.The tasks involve cluttered, contact-rich bimanual interactions and include geometric distribution shifts challenging for learned policies.
  • Introduction: Compared with post-hoc filtering and derivative-space lifting, QP projection enforces strict kinematic constraints during optimization rather than only smoothing selected controls afterward.The projection-based strategy is described as enabling substantially larger effective exploration covariances than vanilla sampling and frequency-domain alternatives.
  • Introduction: Bayesian optimization tunes planner cost weights offline over randomized full-task episodes, improving success rates across all four evaluated tasks without online computational overhead.The procedure reduces reliance on labor-intensive, task-specific manual tuning.

A. Sampling-Based Optimization

The sampling optimizer iteratively samples joint-velocity sequences, projects them into a kinematically feasible set, evaluates parallel physics rollouts, and updates its Gaussian search distribution using cost-weighted statistics.

  • Sampling and initialization: The optimizer initializes a Gaussian distribution over horizon-length joint-velocity sequences and samples a batch of candidate controls at each iteration.Its mean and covariance define the current search space for candidate trajectories.
  • Distribution update: Temperature-scaled exponential weights assign greater influence to lower-cost samples, replacing elite-only selection with importance-weighted updates.The mean follows an MPPI-style update, while the covariance uses a regularized weighted diagonal approximation inspired by CEM.
  • QP projection: Each raw sample is projected by a quadratic program onto trajectories satisfying joint position, velocity, acceleration, jerk, and boundary constraints.Discrete integration and finite differencing connect joint velocities to positions and higher-order derivatives, while the QP runs in parallel across samples on the GPU.
  • Rollout evaluation: Projected sequences are rolled out in parallel through the MuJoCo world model, which predicts manipulator and environment trajectories for task-cost evaluation.The algorithm aggregates rollout cost components into a total cost for each candidate.
  • Output: After iterative sampling, projection, simulation, evaluation, and updating, the optimizer returns the lowest-cost joint-velocity sequence and final distribution parameters.The returned parameters support warm-starting in the subsequent receding-horizon MPC cycle.

B. MPC Planner

PRISM embeds the sampling optimizer inside a receding-horizon MPC loop that repeatedly incorporates current robot and environment observations. QP projection supports broad feasible exploration while improving trajectory continuity and warm-start behavior.

  • Receding-horizon execution: Replanning limits sensitivity to simulator-to-reality mismatch, unmodeled contact effects, object motion, and obstacle displacement.The full open-loop sequence is not executed because changing physical conditions can invalidate its predictions.
  • Receding-horizon execution: At each control step, the MPC loop observes robot configuration, joint velocity, and environment state before optimizing a finite-horizon feasible joint-velocity sequence.Only the first few controls are applied, after which the system is observed again and replanning continues.
  • Warm-starting: Previous-cycle distribution parameters are time-shifted and recycled to initialize the next optimization, preserving useful trajectory information across MPC updates.This warm-starting mechanism is intended to improve sample efficiency and reduce the iterations required at the next step.
  • Decoupling exploration and smoothness: QP projection decouples exploration from motion smoothness, allowing large sampling covariance while guaranteeing evaluated trajectories obey strict kinematic bounds.The projection finds the closest feasible trajectory in the ℓ2 sense, preserving as much of each exploratory sample as possible.
  • System architecture: The architecture combines an outer feedback-and-warm-start MPC loop with an inner QP-constrained sampler and parallel MJX rollout evaluator.This division assigns feedback and distribution reuse to the outer loop while the inner optimizer handles feasible exploration and task-cost evaluation.

2) Stabilizing Warm-Start Across MPC Iterations:

QP projection stabilizes MPC warm-starts by enforcing smooth, state-consistent trajectories while preserving broad feasible exploration. Compared with baseline and LPF sampling, it substantially expands coverage without violating kinematic limits.

  • Warm-start stability: QP derivative bounds prevent the sharp temporal variations that make successive unconstrained MPC solutions diverge.Bounded velocity, acceleration, and jerk produce smoother trajectories and improve the prior used for the next replanning step.
  • Comparison with LPF: Low-pass filtering attenuates high-frequency perturbations but cannot guarantee strict bounds on individual control magnitudes or derivatives.Large-amplitude filtered samples may still violate derivative limits.
  • Boundary consistency: Every QP-projected sample shares the current initial velocity, preventing discontinuities at the MPC replanning boundary.The initial-velocity equality constraint has no analogue in a stateless perturbation filter.
  • Experimental scope: The single-joint experiment is a simplified diagnostic rather than a manipulation benchmark.It isolates feasible-sample distributions independently of task costs, contact dynamics, and bimanual coordination.
  • Exploration coverage: The QP expands feasible exploration while baseline and LPF remain clustered because strict-bound guarantees force conservative covariances.The diagnostic compares sampling strategies under velocity, acceleration, jerk, and initial-velocity constraints.
  • Empirical validation: 100× and 45× are the QP effective exploration-radius advantages over baseline and LPF, respectively.The QP samples fill the feasible envelope and reach ±1 rad/s velocity, ±2 rad/s2 acceleration, and ±4 rad/s3 jerk bounds without exceeding them.

2) Compact QP Formulation:

The compact QP stacks position, velocity, acceleration, and jerk inequalities with an initial-velocity boundary condition. Its slack-variable form preserves trajectory continuity at the MPC boundary while constraining derivative behavior through inequalities.

  • Constraint system: Stacked inequalities jointly enforce position, velocity, acceleration, and jerk bounds on the projected trajectory.A nonnegative slack variable converts the inequalities into the equality form used by the solver.
  • Boundary condition: The equality constraint encodes the initial-velocity boundary condition at the current replanning state.The projected velocity trajectory remains continuous with the command applied at the MPC boundary.

3) ADMM Solver:

PRISM solves the projection QP with an ADMM-inspired Bregman-splitting scheme. Alternating slack, primal, and multiplier updates separate simple nonnegative projection from constrained trajectory minimization.

  • Solver structure: The solver alternates nonnegative slack projection, constrained primal trajectory minimization, and a residual-driven Bregman multiplier update.The augmented objective uses a positive penalty parameter δ.
  • Slack update: The slack update is element-wise clipping and requires no matrix factorization.This makes the first iteration step computationally lightweight.
  • Primal and multiplier updates: The primal update obtains the projected velocity trajectory by minimizing the augmented objective with respect to the trajectory.The Bregman update then maps constraint residuals back through G⊤ for the next primal minimization.

4) Computational Properties:

The custom solver is designed for GPU execution through reusable factorization, joint-wise separability, and batched sample processing. Boundary updates change right-hand sides rather than the factorized KKT matrix, and convergence is observed within 10–20 iterations.

  • Reusable factorization: A factorization of the constant KKT matrix can be computed once and reused across samples, joints, and MPC iterations.The sampled trajectory and projected variable do not affect the matrix factorization.
  • Joint-wise separability: Joint-wise separability decomposes the bimanual projection into independent horizon-wise QPs.For two UR5e arms, the implementation exploits 12 joint-wise subproblems per sampled trajectory.
  • Batch parallelism: Batched GPU operations process all sampled trajectories using matrix-vector products, element-wise operations, and solves with the pre-factorized KKT matrix.The matrices remain fixed while raw trajectories and right-hand sides vary across samples.
  • Boundary updates: Boundary-value changes require only recomputing right-hand sides, leaving the KKT matrix unchanged.Updating h and b is negligible compared with precomputing the factorization.
  • Convergence: 10–20 iterations are sufficient in practice to obtain low constraint residuals for the considered bimanual tasks.The convex QP and affine constraints support convergence to the global projection optimum under standard assumptions.

E. Bayesian Optimization

Bayesian optimization automatically tunes task-cost weights by evaluating candidate configurations over randomized full episodes. A Gaussian-process surrogate and expected-improvement acquisition guide data-efficient search, with the lowest observed episode cost selected after the budget is exhausted.

  • Manual cost-weight tuning is labor-intensive, non-systematic, and often fails to generalize across varying initial conditions.
  • Bayesian optimization samples candidate weight vectors with expected improvement and assigns them to the planner’s cost function.A Gaussian-process surrogate models the mapping from cost weights to task performance.
  • Each candidate configuration is evaluated by running a full episode of up to T MPC steps in a randomized simulation environment.Robot configuration, object pose, and target pose are randomized before evaluation.
  • The episode objective combines dense stepwise costs with a sparse terminal cost encoding task success and final pose error.Unsuccessful executions receive a substantially larger penalty because ℓpose ≪ ℓfail.
  • After B iterations, the weight vector with the lowest observed cost is returned, enabling systematic and data-efficient tuning.The procedure is reported to improve performance, robustness, and generalization over manual tuning.

IV. COST FUNCTION DESIGN

PRISM uses shared and task-specific cost components to guide sampling-based optimization across challenging bimanual tasks. These costs encode collision avoidance, joint regularization, coordination, object transport, and phase-dependent manipulation objectives.

  • The evaluation covers challenging PerAct2 variants: Moving Tray, Lifting and Transporting Ball, Hand-Over Cubes, and Lifting and Transporting Box.
  • Cost formulation: Task costs are horizon-aggregated and combined as weighted components for rollout evaluation.The total cost includes shared components and phase-specific weights.
  • Shared cost components: Shared costs address collision avoidance, deviation from the home configuration, and relative end-effector velocity during coupled manipulation.The relative-velocity cost prevents stretching or compression during rigid-body transport.
  • Moving Tray: Moving Tray uses pick and move phases with alignment, grasp pose, arm separation, tray pose, and end-effector coupling costs.The tray is indirectly controlled through the manipulators because it is not actuated.
  • Moving Tray: Phase weights switch between pick and move objectives, while remaining weights may be manually tuned or optimized through Bayesian optimization.

C. Lifting and Transporting Ball

The ball transport task requires both manipulators to cooperatively lift and move an unattached ball through obstacles. Its phase-specific costs regulate contact, planar alignment, end-effector geometry, and target transport.

  • Both manipulators apply contact forces through their end-effectors to lift and transport the ball while avoiding obstacles.The task is divided into pick and move phases.
  • Phase-specific costs: During the move phase, ball–manipulator collisions are excluded so the end-effectors can press against the ball’s surface.
  • Phase-specific costs: The planar-alignment cost requires both end-effectors to apply contact forces at the same location in the yz-plane.
  • Geometric constraints: Orientation and distance costs regulate target orientations and maintain a fixed end-effector offset around the ball.
  • Task objectives: Additional costs guide the end-effectors toward the ball during picking and the ball toward its desired target position during transport.

E. Lifting and Transporting Box

The box task replaces the spherical payload with an asymmetric rigid box, making stable contact and orientation control more difficult. Statistical evaluation shows strong performance for most tasks, while box lifting has lower success and higher computation time.

  • Lifting and Transporting Box: The asymmetric box is susceptible to rotational slipping and misalignment, requiring continuous, precisely coordinated contact forces from both manipulators.
  • Lifting and Transporting Box: Box-specific costs align each end-effector with designated contact locations on opposing faces and regulate orientation during the move phase.No additional obstacles are introduced because of the task’s increased complexity.
  • Runtime phase transitions: Runtime phase transitions occur when all relevant positional and rotational costs fall below predefined acceptable thresholds.Transitions therefore depend on the state rather than fixed time steps.
  • Statistical validation: 65% at batch size 500 and nearly 100% at batch sizes ≥1250 are reported for ball lifting, while tray moving remains around 85–95%.The evaluation randomizes goal poses and computes statistics over 20 runs per setting.
  • Statistical validation: Box lifting saturates around 55–60% success and reaches up to 130 ms computation time, whereas ball, tray, and hand-over remain below ∼80 ms.The higher box cost is attributed to its increased task complexity and additional optimization iterations.
  • Task completion time: Successful box-lifting episodes take approximately 8–15 s, while hand-over execution times remain approximately 85–95 s.Reported task completion times include only successful episodes.

D. Effect of Bayesian Optimization

Bayesian optimization improved success rates across all four tasks while preserving comparable computation time and execution speed. The results support BO as a systematic approach for tuning cost weights across randomized task configurations.

  • Optimization setup: 3–9 hours are required for a full 1,000-episode BO run, depending on task complexity and episode length.Weights were optimized over randomized robot configurations, object poses, and target locations.
  • Task-level results: Success increased from 75% to 100% for ball lifting, 60% to 95% for handover, 45% to 90% for tray moving, and 45% to 80% for box lifting.The comparison evaluates BO-optimized weights against the corresponding unoptimized configurations.
  • Computational efficiency: Average computation time remained comparable between optimized and unoptimized approaches across all tasks.Differences across tasks were attributed to the number of CEM iterations and planning-horizon length.
  • Execution speed: Successful episodes using optimized weights had reduced or similar completion times, with a significant decrease in the handover task.For the other tasks, completion-time differences were marginal.
  • Overall effect: Overall, BO provided a systematic, data-efficient way to tune cost weights while maintaining computational efficiency and execution speed.The reported benefit is improved robustness and generalization of the sampling-based planner.

1) Comparison Methods:

PRISM compares projection-based sampling with filtering and baseline CEM methods, combining feasible exploration with online physics rollouts. Across four bimanual tasks, it generally improves success and execution while retaining comparable computation, though results depend on task structure and manual cost-function design.

  • Results and Analysis:: 45%: Bayesian-optimized cost weights increase success in tray moving without additional online computational overhead.Figure 10 compares manually tuned and Bayesian-optimized weights across success rate, computation time, and task completion time.
  • Comparison Methods:: QP Projection minimally corrects samples to satisfy strict position, velocity, acceleration, and jerk bounds while enforcing boundary conditions.Unlike LPF and SGF, it decouples exploration from kinematic feasibility.
  • Practical Considerations:: Baseline CEM, LPF, and SGF required additional modifications because their samples lacked strict kinematic feasibility under high-exploration settings.The resulting issue was jerky, dynamically inconsistent behavior.
  • Results and Analysis:: The QP method achieves the highest or joint-highest success rate in three of four tasks, including 95% for ball lifting and tray moving and 90% for handover.LPF slightly outperforms QP in box lifting, 90% versus 80%.
  • Practical Considerations:: QP projection adds negligible overhead relative to MuJoCo rollouts, which dominate the computational budget.Observed timing differences mainly reflect optimization iterations and planning-horizon length.
  • Results and Analysis:: QP trajectories achieve faster or comparable task completion across tasks by exploiting permissible velocities and accelerations without filtering attenuation.Filtering-based methods can trade derivative suppression for slower motion.
  • Results and Analysis:: LPF achieves higher box-lifting success but substantially longer completion times, whereas QP bounds derivatives without attenuating feasible signals.The trade-off reflects the slower execution induced by filtering-based attenuation.
  • Comparison Methods:: PRISM’s QP solver exploits joint-wise separability and precomputed factorizations, making projection cheap relative to physics rollouts.The framework was evaluated across tray moving, ball lifting and transport, cube handover, and box lifting, with reported operation at 10–16 Hz.
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