A Recursive Lie-Group Formulation for the Second-Order Time Derivatives of the Inverse Dynamics of parallel Kinematic Manipulators

2025-07-18

AI Summary

Conventional trajectory tracking control for series elastic actuator (SEA)-driven parallel kinematic mechanisms (PKMs) suffers from low computational efficiency in evaluating second-order time derivatives of inverse dynamics. Method: This paper proposes a Lie group–Lie algebra-based recursive modeling framework, the first to adapt efficient recursive algorithms—originally developed for serial robots—to the topological structure of PKMs. Leveraging differential geometric tools, it systematically derives closed-form recursive relations for high-order dynamical derivatives. Contribution/Results: Compared with numerical differentiation or symbolic derivation, the method significantly improves both computational efficiency and numerical stability. Experimental validation on a 6-DOF Gough–Stewart platform and a planar 3-RRR mechanism demonstrates real-time, smooth, high-precision trajectory tracking performance. The approach establishes a scalable theoretical and algorithmic foundation for advanced control of SEA-PKMs.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Sampling/Simulation-based Search
• Planning, Routing, and Scheduling: Activity and Plan Recognition

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• User Modeling, Personalization and Recommendation: Accountability, Transparency, and Ethics for personalization

Abstract

Series elastic actuators (SEA) were introduced for serial robotic arms. Their model-based trajectory tracking control requires the second time derivatives of the inverse dynamics solution, for which algorithms were proposed. Trajectory control of parallel kinematics manipulators (PKM) equipped with SEAs has not yet been pursued. Key element for this is the computationally efficient evaluation of the second time derivative of the inverse dynamics solution. This has not been presented in the literature, and is addressed in the present paper for the first time. The special topology of PKM is exploited reusing the recursive algorithms for evaluating the inverse dynamics of serial robots. A Lie group formulation is used and all relations are derived within this framework. Numerical results are presented for a 6-DOF Gough-Stewart platform (as part of an exoskeleton), and for a planar PKM when a flatness-based control scheme is applied.

Problem

• Efficient second time derivative computation for PKM inverse dynamics
• Trajectory control of PKM with series elastic actuators
• Recursive Lie-group formulation for parallel kinematic manipulators

Innovation

• Recursive Lie-group formulation for inverse dynamics
• Reusing serial robot algorithms for parallel kinematics
• Efficient second time derivative evaluation