Coordinate transformations, orthographic projections, and robot kinematics

Humans do not consciously think of moving each of their joints while they move their hands from one place to another. Likewise, robot arms can be commanded to move about in cartesian space without the need to address the individual joints. To do this, the direct and inverse kinematic equations of any robot arm must be derived. The direct kinematic equations uniquely transform the joint positions into the position (and orientation) of the hand, whereas the inverse kinematic equations transform the position (and orientation) of the hand into joint positions. The derivation of the inverse kinematic equations for any particular robot is a difficult problem which may have more than one solution. In this paper, these equations are derived for a six degree of freedom robot arm. A combination of matrix operations to perform coordinate rotations, and trigonometry within the appropriate orthographic projects to perform coordinate translations is employed. This complementary approach yields a solution which is more easily obtained, and also more easily visualized. The resulting solution was programmed into a real-time computer as a part of a higher level software system to control the motion of the arm