An Efficiently Solvable Quadratic Program for Stabilizing Dynamic Locomotion (1311.1839v2)

Abstract: We describe a whole-body dynamic walking controller implemented as a convex quadratic program. The controller solves an optimal control problem using an approximate value function derived from a simple walking model while respecting the dynamic, input, and contact constraints of the full robot dynamics. By exploiting sparsity and temporal structure in the optimization with a custom active-set algorithm, we surpass the performance of the best available off-the-shelf solvers and achieve 1kHz control rates for a 34-DOF humanoid. We describe applications to balancing and walking tasks using the simulated Atlas robot in the DARPA Virtual Robotics Challenge.

• The paper presents a novel quadratic program formulation that exploits sparse constraint structures to enable robust dynamic locomotion control at 1kHz.
• It introduces a custom active-set solver that reuses active inequality constraints across control steps to drastically reduce computational overhead.
• Simulation results on a 34-DOF humanoid validate the method's superior performance over traditional solvers in dynamic and challenging environments.

An Efficiently Solvable Quadratic Program for Stabilizing Dynamic Locomotion

The paper "An Efficiently Solvable Quadratic Program for Stabilizing Dynamic Locomotion," authored by Scott Kuindersma, Frank Permenter, and Russ Tedrake, presents a novel approach to controlling dynamic locomotion in high-dimensional humanoid systems. This work focuses on the design of a whole-body dynamic walking controller, formulated as a convex quadratic program (QP), which efficiently stabilizes walking in these systems.

The primary contribution of the paper is the exploitation of problem structure in the QP to achieve significant computational efficiency. Specifically, the authors introduce a custom active-set method that leverages the sparsity and temporal consistency of active inequality constraints over successive control steps. By doing so, they enable real-time control rates at 1kHz for a 34-degree-of-freedom (DOF) humanoid, outperforming existing state-of-the-art solvers like CVXGEN and Gurobi by a factor of five.

Key Contributions and Results

1. QP Formulation and Solution: The controller effectively handles the complex dynamics of legged locomotion through the efficient formulation of dynamic, input, and contact constraints. The objective is to minimize a quadratic motion cost while respecting these constraints, leading to stable and optimal control inputs for the full robot dynamics. The QP framework accounts for linearized Zero-Moment Point (ZMP) dynamics, enabling the integration of a predefined ZMP trajectory to achieve dynamic stability.
2. Exploiting Model Structure: The authors demonstrate that by employing a time-varying linear quadratic regulator (TVLQR) approach, the cost-to-go for simple unconstrained models can be computed efficiently. This solution informs a constrained optimization problem where inputs are computed for the full system.
3. Active-set Solver Design: The development of a custom active-set solver is a critical component of the research. This solver exploits the fact that active sets of inequality constraints remain largely invariant across solution steps, requiring minimal computational effort to solve the resulting QP in most cases. The detailed design allows for efficient computation by focusing on only necessary constraints, often needing just a single linear equation solve.
4. Simulation Validation: The effectiveness of the method is validated on the simulated Atlas robot used in the DARPA Virtual Robotics Challenge. The controller demonstrated robust performance in various challenging environments, such as walking over uneven terrain and through simulated mud, while maintaining constrained stability with sporadic updates.

Implications and Future Directions

The paper’s methodology has significant implications for real-time robotic control, especially in high-dimensional systems where traditional optimization techniques may be computationally prohibitive. By showing substantial improvements in solve time and performance during dynamic tasks, these results could influence future developments in model predictive control (MPC) techniques and their application across different domains of robotic control.

There are potential extensions of this research that could be explored:

• Application to Physical Robots: Implementing this control strategy on a physical humanoid robot could validate the approach's robustness and real-world applicability, potentially adding considerations for sensory noise and calibration uncertainties.
• Adaptation to Varying Models: While demonstrated on bipedal humanoids, the QP formulation and solver approach may be adaptable to other robotic systems, such as quadrupeds or multi-limbed robots, with appropriate modifications to the dynamic model.
• Hybrid Control Systems: Exploring integration with other control paradigms, such as reinforcement learning or adaptive control strategies, could enhance the capability and versatility of the proposed controller, particularly in dynamic or unstructured environments.

In conclusion, this paper contributes a substantial methodological advance in the field of dynamic locomotion in robotics, providing a foundation for further exploration and refinement in both theoretical and practical aspects of control systems engineering.