For both non-redundant and redundant systems, the inverse kinematics (IK) calculation is a fundamental step in the control algorithm of fully actuated serial manipulators. The tool-center-point (TCP) position is given and the joint coordinates are determined by the IK. Depending on the task, robotic manipulators can be kinematically redundant. That is when the desired task possesses lower dimensions than the degrees-of-freedom of a redundant manipulator. The IK calculation can be implemented numerically in several alternative ways not only in case of the redundant but also in the non-redundant case. We study the stability properties and the feasibility of a tracking error feedback and a direct tracking error elimination approach of the numerical implementation of IK calculation both on velocity and acceleration levels. The feedback approach expresses the joint position increment stepwise based on the local velocity or acceleration of the desired TCP trajectory and linear feedback terms. In the direct error elimination concept, the increment of the joint position is directly given by the approximate error between the desired and the realized TCP position, by assuming constant TCP velocity or acceleration. We investigate the possibility of the implementation of the direct method on acceleration level. The investigated IK methods are unified in a framework that utilizes the idea of the auxiliary input. Our closed form results and numerical case study examples show the stability properties, benefits and disadvantages of the assessed IK implementations.

中文翻译：

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冗余机械手的速度和加速度水平逆运动学计算替代方案

对于非冗余和冗余系统，反向运动学 (IK) 计算是全驱动串行机械手控制算法中的基本步骤。工具中心点 (TCP) 位置已给定，关节坐标由 IK 确定。根据任务的不同，机器人操纵器在运动学上可能是多余的。那就是当所需任务的维度低于冗余机械手的自由度时。不仅在冗余的情况下，而且在非冗余的情况下，IK 计算都可以以多种替代方式在数值上实现。我们研究了稳定性特性和跟踪误差反馈的可行性，以及在速度和加速度水平上 IK 计算的数值实现的直接跟踪误差消除方法。反馈方法基于所需 TCP 轨迹的局部速度或加速度和线性反馈项逐步表示关节位置增量。在直接误差消除概念中，假设 TCP 速度或加速度恒定，关节位置的增量直接由期望和实现 TCP 位置之间的近似误差给出。我们调查了在加速度水平上实施直接方法的可能性。所研究的 IK 方法统一在一个利用辅助输入思想的框架中。我们的封闭形式结果和数值案例研究示例显示了评估的 IK 实现的稳定性属性、优点和缺点。