A Framework for Combining Optimization-Based and Analytic Inverse Kinematics

TL;DR

The paper addresses solving inverse kinematics (IK) by unifying analytic IK and optimization-based approaches. It treats a known analytic IK solution as a smooth change of variables, enabling optimization over the end-effector pose and self-motion parameters while computing joint angles via the analytic map, which linearizes the IK constraint. Across multiple robots and tasks, the approach yields higher success rates than the standard formulation and remains practical for real-time use, even with challenging constraints like collision avoidance and humanoid stability. The work demonstrates significant robustness gains and provides detailed comparisons with three solver paradigms and with baselines like Global-IK, highlighting practical benefits for complex robotic manipulation and planning under redundancy.

Abstract

Analytic and optimization methods for solving inverse kinematics (IK) problems have been deeply studied throughout the history of robotics. The two strategies have complementary strengths and weaknesses, but developing a unified approach to take advantage of both methods has proved challenging. A key challenge faced by optimization approaches is the complicated nonlinear relationship between the joint angles and the end-effector pose. When this must be handled concurrently with additional nonconvex constraints like collision avoidance, optimization IK algorithms may suffer high failure rates. We present a new formulation for optimization IK that uses an analytic IK solution as a change of variables, and is fundamentally easier for optimizers to solve. We test our methodology on three popular solvers, representing three different paradigms for constrained nonlinear optimization. Extensive experimental comparisons demonstrate that our new formulation achieves higher success rates than the old formulation and baseline methods across various challenging IK problems, including collision avoidance, grasp selection, and humanoid stability.

Figures (5)

• Figure 1: Setup for our robot simulation experiments. (b) shows three poses which are feasible for one instance of the grasp selection experiment, with the shelves hidden for better visibility. (e) shows the nominal stable configuration used in the Hubo experiments. (f) shows the 3d scaling setup, where arms with different numbers of joints are used for the same IK problem.
• Figure 2: Optimal joint centering cost found by new and old formulation divided by the optimal costs found through sampling. When a solution is found (with either formulation) in the same self-motion manifold as the global optimum, it is generally that global optimum, with the new formulation obtaining costs close to global optimum more often. When the solution is in a different self-motion manifold, both formulations have similar performance. The cost difference seen in \ref{['tab:optimal_costs']} is due to the new formulation's restriction to producing solutions from a prespecified self-motion manifold (unlike the sampling approach). This difference disappears if the sampling approach is restricted to the same self-motion manifold as the new formulation.
• Figure 3: Diverse solutions from the Hubo experiments, from the different formulations and stability constraint representations. All solutions were obtained with IPOPT.
• Figure 4: Results for old and new formulation of the 2-dimensional link-chain IK.
• Figure 5: Results for old and new formulation of the 3-dimensional link-chain IK.