Mirror-Descent Inverse Kinematics for Box-constrained Joint Space (2101.07625v2)

Abstract: To control humanoid robots, the reference pose of end effector(s) is planned in task space, then mapped into the reference joints by IK. By viewing that problem as approximate quadratic programming (QP), recent QP solvers can be applied to solve it precisely, but iterative numerical IK solvers based on Jacobian are still in high demand due to their low computational cost. However, the conventional Jacobian-based IK usually clamps the obtained joints during iteration according to the constraints in practice, causing numerical instability due to non-smoothed objective function. To alleviate the clamping problem, this study explicitly considers the joint constraints, especially the box constraints in this paper, inside the new IK solver. Specifically, instead of clamping, a mirror descent (MD) method with box-constrained real joint space and no-constrained mirror space is integrated with the Jacobian-based IK, so-called MD-IK. In addition, to escape local optima nearly on the boundaries of constraints, a heuristic technique, called $\epsilon$-clamping, is implemented as margin in software level. Finally, to increase convergence speed, the acceleration method for MD is integrated assuming continuity of solutions at each time. As a result, the accelerated MD-IK achieved more stable and enough fast tracking performance compared to the conventional IK solvers. The low computational cost of the proposed method mitigated the time delay until the solution is obtained in real-time humanoid gait control, achieving a more stable gait.

• The paper introduces MD-IK, integrating Mirror Descent with Jacobian-based methods to overcome numerical instability from box constraints.
• It applies an epsilon-clamping strategy and adapted Nesterov acceleration to ensure fast convergence and smoother joint trajectories.
• Numerical results demonstrate reduced tracking errors and lower computational overhead, enabling robust real-time control in humanoid robotics.

Overview of "Mirror-Descent Inverse Kinematics with Box-constrained Joint Space"

The paper "Mirror-Descent Inverse Kinematics with Box-constrained Joint Space" by Taisuke Kobayashi and Takanori Jin presents an innovative approach to solving the inverse kinematics (IK) problem for humanoid robots, emphasizing efficiency and robustness under joint space constraints. The research offers a novel solution through the Mirror Descent (MD) method, primarily addressing the typical shortcomings encountered in conventional Jacobian-based IK methods.

Technical Contributions

The core innovation in this paper is the integration of the Mirror Descent method with Jacobian-based inverse kinematics, referred to as MD-IK. The authors explicitly tackle the challenges imposed by box constraints within joint spaces, often neglected or crudely handled in traditional methods. Key contributions of the paper are as follows:

1. Nonlinear Mapping Design: The authors propose a nonlinear mapping between unconstrained mirror space and box-constrained joint space, ensuring a smooth transition without the numerical instability typical of clamping operations.
2. Heuristic Escape from Local Optima: An $\epsilon$-clamping strategy is introduced to avoid local optima near constraint boundaries. This method acts as a margin at the software level, ensuring stable control performance without violating constraints.
3. Acceleration Method Integration: An adapted Nesterov acceleration technique is combined with the proposed IK method. This adaptation is designed to swiftly converge while maintaining solution smoothness, leveraging prior results for continuous solutions.

Numerical Results

Empirical evaluations, conducted through simulation with different robotic models, showcase the efficiency of the MD-IK method. The results highlighted are as follows:

• Convergence and Tracking Performance: The proposed MD-IK showcases improved convergence, leading to smaller tracking errors and smoother trajectories compared to conventional Jacobian-based methods like Levenberg-Marquardt and advanced solvers like OSQP.
• Computational Efficiency: MD-IK manifested lower computational overhead, which is critical for maintaining real-time control in robotic applications such as humanoid gait patterning. The method consistently operated within milliseconds, contrasting with longer delays observed in other solvers that could destabilize robot motion.
• Real-time Application: In a dynamic walking task simulation, MD-IK achieved superior performance by enabling consistent, stable locomotion without the adverse effects of computation-induced delays that were evident with OSQP.

Implications and Future Directions

The implications of this research are significant for robotics where real-time control under constraints is necessary. By addressing the clamping issues and enhancing convergence speed, MD-IK sets a new benchmark for joint space control efficiency. The method holds potential for broad application in humanoid robotics where dynamic and adaptable motion planning is critical.

Future research could expand on the generality of MD-IK by developing mappings that accommodate more complex constraints, such as coupling effects and self-collision avoidance. This extension would increase the applicability of MD-IK in diverse robotic systems and complex environments.

In conclusion, this research presents a compelling case for using the Mirror Descent method in constrained IK problems, offering a promising pathway to agile and reliable humanoid robot control while managing computational costs effectively.