Fiber-Normalized Manipulability and Determinant Proxies: Intrinsic Redundancy Optimization Across and Within Task Fibers

2026-09-25

AI Summary

This study addresses the coordinate- and metric-dependence of manipulability indices in cross-task fiber optimization for redundant robots. Drawing upon differential geometry and manifold theory, this work proves the equivalence of manipulability indices within a single fiber and establishes determinant-based manipulability as the exact optimization objective for a fixed task. The core innovation lies in introducing a novel fiber-normalized scalar that eliminates metric dependence, thereby formulating a coordinate-invariant intrinsic optimization criterion across fibers. Through case studies involving planar manipulators and pneumatic allocation, the proposed method is validated in terms of its equivalence properties, as well as the coordinate invariance and effectiveness of the resulting trajectory optimization.

Abstract

This work establishes that determinant-based manipulability is an exact objective for fixed-task redundancy optimization, despite its dependence on task coordinates and the choice of task-space metric used for volume measurement. On every regular task fiber, the determinant proxy, its representation in any task chart, and every metric-completed manipulability differ only by positive constants. They consequently induce the same complete ordering, constrained extrema, gradient directions, critical points, and local optimality classifications. For comparisons and trajectory optimization across task fibers, this work introduces fiber-normalized manipulability: the capability attained at an internal state divided by the best capability available on the same fiber. The resulting dimensionless scalar is invariant under coordinate changes on the internal-state and task manifolds and independent of the task-space metric. Its associated loss provides an intrinsic objective for physically admissible cross-fiber trajectories, including problems with prescribed or free task evolution and temporal coupling. Planar-manipulator and redundant aerodynamic-allocation examples demonstrate fixed-fiber equivalence, task-dependent cross-fiber differences, and invariant fiber-normalized trajectory optimization.

Problem

• redundancy optimization
• manipulability
• task fibers
• trajectory optimization
• determinant proxies

Innovation

• Fiber-Normalized Manipulability
• Determinant Proxies
• Redundancy Optimization
• Task Fibers
• Trajectory Optimization