AI 中文总结
本文提出纤维归一化可操作性，作为跨任务纤维轨迹优化的内在目标，解决行列式可操作性对坐标和度量的依赖，并通过平面机械臂和气动分配示例验证其不变性与优化效果。
AI 中文摘要
本工作确立了基于行列式的可操作性在固定任务冗余优化中是一个精确的目标函数，尽管它依赖于任务坐标以及用于体积测量的任务空间度量的选择。在每个正则任务纤维上，行列式代理、其在任何任务图表中的表示以及每个度量完备的可操作性仅相差正常数。因此，它们诱导相同的完全排序、约束极值、梯度方向、临界点和局部最优性分类。对于跨任务纤维的比较和轨迹优化，本工作引入了纤维归一化可操作性：内部状态所达到的能力除以同一纤维上可获得的最佳能力。所得的无量纲标量在内部状态和任务流形上的坐标变换下不变，并且独立于任务空间度量。其相关的损失函数为物理上可接受的跨纤维轨迹提供了内在目标，包括具有规定或自由任务演化以及时间耦合的问题。平面机械臂和冗余气动分配示例展示了固定纤维等价性、任务相关的跨纤维差异以及不变的纤维归一化轨迹优化。
英文摘要
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.
