IMPACT: An Implicit Active-Set Augmented Lagrangian for Fast Contact-Implicit Trajectory Optimization

Robotic systems operating in contact-rich environments require precise planning and control strategies to handle nonsmooth dynamics and discrete contact mode changes. Early approaches relied on predefined contact sequences, simplifying the optimization but limiting adaptability. Recent advances introduced contact-implicit methods, modeling contact as complementarity constraints, which better reflect physical reality. However, these formulations are inherently nonconvex and nonsmooth, leading to numerical difficulties. Traditional smoothing or penalty methods improve robustness but often degrade accuracy and control quality. The challenge remains to develop scalable, reliable solvers that can handle the true nonsmooth structure while providing theoretical guarantees. IMPACT addresses this by integrating augmented Lagrangian techniques with active-set identification, enabling efficient and stable solutions for complex contact scenarios.

The core issue is efficiently solving MPCCs that model contact dynamics without resorting to smoothing or relaxation, which can compromise fidelity. Existing solvers often face numerical instability, slow convergence, or sensitivity to initialization, especially in high-dimensional, multi-contact tasks. The difficulty lies in handling the complementarity constraints that induce nonconvexity and nonsmoothness, making standard nonlinear programming techniques ineffective. Achieving real-time, reliable trajectory optimization in such settings is critical for advancing autonomous manipulation and locomotion but remains an open problem due to the mathematical and computational complexities involved.

IMPACT introduces a novel combination of augmented Lagrangian methods with an implicit active-set strategy tailored for MPCCs. Its key innovations include:

1) Explicitly maintaining the original complementarity structure, avoiding smoothing or penalization that can cause numerical issues.

2) Dynamically identifying contact modes during optimization, eliminating the need for predefined contact sequences.

3) Developing a block coordinate descent (BCD) solver with closed-form updates for the auxiliary variables enforcing complementarity, ensuring convergence to stationary points.

4) Theoretical guarantees under standard regularity conditions, validated through rigorous analysis.

5) Efficient C++ implementation optimized for trajectory workloads, enabling significant speedups over existing methods.

• �� Formulate the contact-implicit trajectory optimization as a MPCC with complementarity constraints.
• �� Use augmented Lagrangian (AuLa) framework to handle equality and inequality constraints, keeping complementarity as explicit constraints.
• �� Introduce auxiliary slack variables (Y, Z) to reformulate the MPCC into a smooth problem with explicit complementarity conditions.
• �� Design a block coordinate descent (BCD) algorithm:
• X-update: damped Gauss-Newton step with globalization for trajectory variables.
• (Y, Z)-update: closed-form solutions based on current multipliers, selecting between two candidate updates.
• �� Implement a safeguarded outer loop that adaptively updates penalties and multipliers, ensuring convergence.
• �� Theoretically prove stationarity guarantees under regularity assumptions, with convergence to feasible, stationary points.

Experiments involve benchmarking IMPACT on the CRISP long-horizon planning suite and a dexterous manipulation suite (CI-MPC). Metrics include computational speed, trajectory quality, and control accuracy. Baselines include smoothing-based solvers and ADMM approaches. Hyperparameters such as penalty factors and convergence tolerances are tuned for each task. Ablation studies analyze the impact of active-set identification and solver components. Hardware tests with Allegro Hand and Panda robots validate real-world applicability, focusing on tasks like pushing and grasping with multiple contact points. Results show consistent speedups and improved robustness across scenarios.

IMPACT outperforms baselines with speedups from 2.9× to 70×, averaging 13.8× across benchmarks. It maintains high trajectory fidelity, with reduced contact switching errors and smoother control signals. In simulation, control errors decreased by 20%, and contact mode transitions were more accurate. Hardware experiments confirmed stable task execution, such as pushing a T-shaped object, demonstrating real-time capability. The results confirm the method’s scalability, robustness, and potential for complex robotic tasks.

Applicable in autonomous manipulation, legged locomotion, industrial assembly, and rescue robots where contact interactions are frequent and complex. The algorithm requires accurate contact models and system dynamics, making it suitable for high-precision tasks. Its efficiency enables real-time control in dynamic environments, facilitating deployment in industry and service sectors. Future integration with learning modules could further improve initialization and scalability, broadening its impact.

Current limitations include sensitivity to initial guesses and difficulty handling highly nonlinear or deformable contact models. Computational costs remain high for systems with many degrees of freedom, requiring further optimization. Extreme contact frequency or rapid mode changes can challenge numerical stability. Extending the framework to soft or deformable contacts and multi-robot coordination presents additional challenges. Future work aims to address these issues through hardware acceleration, adaptive schemes, and hybrid smoothing strategies.