[Paper Review] Inverse Kinematics with Forward Dynamics Solvers for Sampled Motion Tracking

Stefan Scherzinger, Arne Roennau+1 more|arXiv (Cornell University)|Aug 17, 2019

Robotic Mechanisms and Dynamics11 references17 citations

TL;DR

This paper proposes a dynamics-based enhancement to the Jacobian transpose method for inverse kinematics (IK) in sampled motion tracking, using a virtual, conditioned twin robot's mass matrix to improve convergence and smoothness. By integrating forward dynamics into the IK solver, the method achieves superior path accuracy and interpolation quality over the standard Jacobian approach, validated on a UR10 robot with real-time performance.

ABSTRACT

Tracking Cartesian motion with end~effectors is a fundamental task in robot control. For motion that is not known in advance, the solvers must find fast solutions to the inverse kinematics (IK) problem for discretely sampled target poses. On joint control level, however, the robot's actuators operate in a continuous domain, requiring smooth transitions between individual states. In this work, we present a boost to the well-known Jacobian transpose method to address this goal, using the mass matrix of a virtually conditioned twin of the manipulator. Results on the UR10 show superior convergence and quality of our dynamics-based solver against the plain Jacobian method. Our algorithm is straightforward to implement as a controller, using common robotics libraries.

Motivation & Objective

• To address the challenge of generating smooth, feasible joint trajectories between sparsely sampled Cartesian target poses in real-time robot control.
• To improve convergence and path quality in inverse kinematics solvers for motion tracking applications where targets are not known in advance.
• To provide a simple, intuitive, and efficient extension to the Jacobian transpose method that leverages manipulator dynamics without requiring full dynamic model knowledge.
• To offer a plug-and-play ROS-compatible controller implementation for immediate use in robotics research and applications.

Proposed method

• The method enhances the standard Jacobian transpose IK by replacing the constant gain matrix K with the inverse of the inertia matrix (H⁻¹) of a virtually conditioned twin of the robot manipulator.
• It formulates the IK as a forward dynamics problem: ¨q = H⁻¹JTf, where f is the Cartesian error vector (xd − g(q)), and H is the virtual mass matrix.
• The virtual twin's dynamics are simplified by omitting Coriolis and gravitational terms, focusing only on the inertia matrix to ensure computational efficiency and intuitive behavior.
• The algorithm uses explicit time integration (forward Euler) with a fixed time step ∆t to compute joint accelerations and update joint positions iteratively.
• The controller gain kp scales the overall response, allowing users to tune between smooth, delayed tracking and fast, precise convergence.
• The approach is implemented as a ROS-control compatible controller, enabling direct integration into existing robotic systems.

Experimental results

Research questions

• RQ1 Can a dynamics-inspired modification to the Jacobian transpose method improve convergence speed and path quality in sampled motion tracking?
• RQ2 How does using a virtual, conditioned mass matrix instead of a constant gain matrix affect the smoothness and feasibility of intermediate joint trajectories?
• RQ3 To what extent does the proposed method outperform the standard Jacobian transpose method in terms of error reduction and interpolation accuracy?
• RQ4 Can the method be efficiently implemented in real-time robotic control systems without significant computational overhead?

Key findings

• The proposed method achieved significantly faster convergence than the standard Jacobian transpose method, reducing the number of iterations required to reach target poses.
• The method maintained rotational errors nearly constant throughout the path, while the Jacobian transpose method showed increasing rotational error over time.
• In interpolation experiments, the proposed method generated goal-directed, smooth Cartesian paths, whereas the Jacobian transpose method produced distorted and non-optimal trajectories.
• For motion tracking of a moving target at 0.2 m/s, the method with kp = 5 provided smooth, accurate tracking with minimal overshoot, while higher gains (kp = 50) enabled faster, more precise convergence.
• The method demonstrated robustness and real-time feasibility, with no performance degradation even at 10 kHz solver rates on standard hardware.
• The use of a virtual, non-physical mass matrix—deliberately mismatched from the real robot—still yielded stable and high-quality solutions, confirming the method's robustness to parameter mismatch.