A multi-joint enabled robot requires extensive mathematical calculations to be done so the end-effector's position can be determined with respect to the other connective joints involved and their respective frames in a specific coordinate system. If a control algorithm employs fewer constraints than the cases necessary to explicitly determine the leg's position, the robot is generally underconstrained. Consequently, only a subset of the end effector's degree of freedom (DoF) can be assigned for the robot's leg position for pose and trajectory estimation purposes. This paper introduces a fully functional algorithm to consider all the cases of the robot's leg position in a coordinate system so the robot's degree of freedom is not limited. Mathematical derivation of the joint angles is derived with forward and inverse kinematics, and Python-based simulation has been done to verify and simulate the robot's locomotion. Using Python-based code for serial communication with a micro-controller unit makes this approach more effective for demonstrating its application on a prototype leg.

翻译：多关节机器人需要复杂的数学计算才能确定末端执行器在特定坐标系中相对于其他连接关节及其各自坐标系的位姿。若控制算法使用的约束条件少于显式确定腿部位置所需的必要条件，则机器人通常处于欠约束状态。因此，在姿态与轨迹估计过程中，仅能为机器人腿部位姿分配末端执行器部分自由度（DoF）。本文提出一种全功能算法，能够在坐标系中考虑机器人腿部所有位姿情况，从而避免机器人自由度受限。通过正逆向运动学推导关节角度的数学模型，并采用基于Python的仿真验证机器人的运动特性。结合基于Python的串口通信代码与微控制器单元，该方法能更有效地在原型腿上展示其应用效果。