Abstract
In solving specific problems, physical laws and mathematical theorems directly express the connections between variables with equations/inequations. At times, it could be extremely hard or not viable to solve these equations/inequations directly. The PE (principle of equivalence) is a commonly applied pragmatic method across multiple fields. PE transforms the initial equations/inequations into simplified equivalent equations/inequations that are more manageable to solve, allowing researchers to achieve their objectives. The problem-solving process in many fields benefits from the use of PE. Recently, the ZE (Zhang equivalency) framework has surfaced as a promising approach for addressing time-dependent optimization problems. This ZEF (ZE framework) consolidates constraints at different tiers, demonstrating its capacity for the solving of time-dependent optimization problems. To broaden the application of ZEF in time-dependent optimization problems, specifically in the domain of motion planning for redundant manipulators, the authors systematically investigate the ZEF-I2I (ZEF of the inequation-to-inequation) type. The study concentrates on transforming constraints (i.e., joint constraints and obstacles avoidance depicted in different tiers) into consolidated constraints backed by rigorous mathematical derivations. The effectiveness and applicability of the ZEF-I2I are verified through two optimization motion planning schemes, which consolidate constraints in the velocity-tier and acceleration-tier. Schemes are required to accomplish the goal of repetitive motion planning within constraints. The firstly presented optimization motion planning schemes are then reformulated as two time-dependent quadratic programming problems. Simulative experiments conducted on the basis of a six-joint redundant manipulator confirm the outstanding effectiveness of the firstly presented ZEF-I2I in achieving the goal of motion planning within constraints.