串联机构运动学反解的D-H四元数方法

D-H Quaternion Method for Inverse Kinematics of Serial Mechanisms

普通四元数方法在串联机构运动学反解时存在方程数量不足和求解困难的问题，为了解决这些问题并建立新的串联机构运动学反解方法，提出串联机构运动学反解的D—H四元数方法。首先给出了包含D—H参数的四元数变换通用方程式，提出将四元数变换方程式分离为位置和姿态两个方程式，这两个方程式可构造出含有7个方程的方程组，使方程数量满足4R以上串联机构运动学反解的要求。为了降低方程组的求解难度，提出了取姿态方程中三角函数的一半组成新的姿态方程，将方程次数降低为原来的一半。采用所提出的D—H四元数方法对PUMA机器人进行运动学反解分析，得到了该机器人的8组反解。根据所求得的8组解，建立了PUMA机器人的8个位姿的三维模型，并测量了PUMA机器人三维模型的末端位姿数值，与所给末端位姿数值完全相同，验证了所提出的D—H四元数方法的正确性和有效性。

The normal quaternion method of inverse kinematics of serial mechanisms has the limitation of lacking equations and is difficult to solve. In order to solve these problems and put forward a new method of inverse kinematics of serial mechanisms, a D - H quaternion method for inverse kinematics of serial mechanisms is proposed. The general equation of quaternion transformation including D - H parameters was given first. Two equations of position and posture were obtained by separating the general equation of quaternion transformation. By these two equations, an equation system with seven equations was constructed, which met the number requirement of the equations for inverse kinematics of serial mechanisms with more than four degrees of freedom. In order to lower the difficulty in solving equations, the degree of posture equation was reduced to half by taking half of the trigonometric function in the original posture equation to construct a new posture equation. By using the proposed D - H quaternion method, the inverse kinematics of PUMA robot was analyzed, and eight groups of inverse solutions were obtained. Three dimensional models of PUMA robot were established based on the eight groups of inverse solutions. Measured results of end positions and postures in the three dimensional models are consistent with the given values. The example of PUMA robot shows the correctness and validity of the proposed D- H quaternion method.