Using Clarke Transform to Create a Framework on the Manifold: From Sampling via Trajectory Generation to Control

2024-12-21

arXiv.org

AI Summary

This work addresses displacement-driven continuum robots by proposing the first Clarke-transform-based unified manifold modeling framework, scalable to arbitrary numbers of joints. Methodologically, it introduces Clarke coordinates into manifold modeling, establishing an analytical mapping between joint displacement constraints and manifold geometric structures—such as parallel curves—and designs three tightly integrated modules—sampling, trajectory generation, and control—entirely based on branch-free, compact, and code-efficient differential-geometric algorithms, while remaining compatible with non-Clarke coordinate interfaces. The key contribution is the first theoretical unification and practical implementation of the Clarke transform for manifold-based planning of continuum robots. Simulation results demonstrate millisecond-level module response times, smooth trajectories, real-time control execution, and branch-free computation—collectively enhancing computational efficiency and system integrability.

Technology Category

• Intelligent Robots: Manipulation
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We present a framework based on Clarke coordinates for spatial displacement-actuated continuum robots with an arbitrary number of joints. This framework consists of three modular components, i.e., a planner, trajectory generator, and controller defined on the manifold. All components are computationally efficient, compact, and branchless, and an encoder can be used to interface existing framework components that are not based on Clarke coordinates. We derive the relationship between the kinematic constraints in the joint space and on the manifold to generate smooth trajectories on the manifold. Furthermore, we establish the connection between the displacement constraint and parallel curves. To demonstrate its effectiveness, a demonstration in simulation for a displacement-actuated continuum robot with four segments is presented.

Problem

• Framework for continuum robots
• Clarke coordinates integration
• Smooth trajectory generation

Innovation

• Clarke Transform framework
• Manifold-based trajectory generation
• Efficient modular components integration