A Conformal Geometric Algebra Method for Inverse Kinematics Analysis of 6R Robotic Arm

Abstract

Addressing the issue of angle multiplicity in solving joint angles using cosine expressions within the context of conformal geometric algebra for inverse kinematics analysis, an extended method for joint angle determination has been developed. The core of this method lies in the precise definition of the rotation plane for each joint and its corresponding rotation vector. By accurately calculating the angle between these two vectors and introducing a sign function as a coefficient, the correct sign for each joint angle is ensured, thereby determining the unique and correct value of the joint angle. Based on this foundation, we applied this improved joint angle solution method to the inverse kinematics analysis framework of conformal geometric algebra. To verify the effectiveness and accuracy of this method, we conducted detailed case studies using a 6R robotic arm with three orthogonally oriented axes and another with three parallel axes. The results showed that for the positional inverse solution of the three-axis orthogonal robotic arm, all eight possible solutions were successfully obtained; for the positional inverse solution of the three-axis parallel robotic arm, all four possible solutions were found. Furthermore, compared to the traditional Denavit–Hartenberg (D–H) parameter method, the conformal geometric algebra approach exhibits significant advantages in complex mechanisms. Especially when performing inverse kinematics analysis for three-axis parallel robotic arms, the conformal geometric algebra method achieves operational efficiency that is more than double that of the D–H parameter method, fully demonstrating its remarkable superiority in high-performance computing.