Abstract
This paper shows how the recently developed formulation of conformal geometric algebra can be used for analytic inverse kinematics of two six-link industrial manipulators with revolute joints. The paper demonstrates that the solution of the inverse kinematics in this framework relies on the intersection of geometric objects like lines, circles, planes and spheres, which provides the developer with valuable geometric intuition about the problem. It is believed that this will be very useful for new robot geometries and other mechanisms like cranes and topside drilling equipment. The paper extends previous results on inverse kinematics using conformal geometric algebra by providing consistent solutions for the joint angles for the dierent congurations depending on shoulder left or right, elbow up or down, and wrist ipped or not. Moreover, it is shown how to relate the solution to the Denavit-Hartenberg parameters of the robot. The solutions have been successfully implemented and tested extensively over the whole workspace of the manipulators.
Highlights
Analytical inverse kinematics is a well-developed problem in robotics
The formulation extends the 3-dimensional Euclidean space with 2 extra dimensions resulting in a homogeneous space including the point at infinity. In this formalism, the inverse kinematics has been previously solved for a robot with 5 revolute joints in terms of spheres, planes and lines, and the intersection of these geometric objects [Hildenbrand (2013); Hildenbrand et al (2005); Hildenbrand et al (2006); Zamora and Bayro-Corrochano (2004)]. These inverse kinematic solutions have primarily been developed for graphical rendering, as the focus has been on the link configurations, whereas the joint angles are only given in terms of the cosines of the angles, which means that there is no systematic way of determining the right quadrant of the joint angles
In this work we extend the existing solutions for analytic inverse kinematics based on conformal geometric algebra to obtain a systematic way of calculating the signs and quadrants of the joint angles
Summary
Analytical inverse kinematics is a well-developed problem in robotics. Solutions are available as text-book material for revolute robots with a spherical wrist, or with three consecutive parallel axes [Siciliano et al (2009); Spong et al (2006)]. The formulation extends the 3-dimensional Euclidean space with 2 extra dimensions resulting in a homogeneous space including the point at infinity In this formalism, the inverse kinematics has been previously solved for a robot with 5 revolute joints in terms of spheres, planes and lines, and the intersection of these geometric objects [Hildenbrand (2013); Hildenbrand et al (2005); Hildenbrand et al (2006); Zamora and Bayro-Corrochano (2004)]. In this work we extend the existing solutions for analytic inverse kinematics based on conformal geometric algebra to obtain a systematic way of calculating the signs and quadrants of the joint angles This includes the calculation of consistent solutions corresponding to shoulder left and right, elbow up or down and wrist flipped or not. The implementation of the analytic inverse kinematics is presented for the Agilus R900 Sixx and the UR5 robot
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