Typically, a robotic task is specified in terms of the T matrix. Since the drive variables are the joint variables, for the purposes of actuating and controlling a robot, it is necessary to solve Equation and determine the joint motion vector q corresponding to a specified T. This is the inverse-kinematics problem associated with a robot.
Since six coordinates are needed to specify a rigid body (or a body frame) in the three-dimensional space, T is specified using six independent quantities (typically three position coordinates and three angles of rotation).
It follows that Equation, in general, represents a set of six algebraic equations. These equations contain highly nonlinear trigonometric functions (of coordinate transformations) and are coupled. Hence a simple and unique solution for the joint coordinate vector q might not exist even in the absence of redundant kinematics (Note: If n=6, the robot does not have redundant kinematics in the 3-D space).
Some simplification is possible by proper design of robot geometry. For example, by using a spherical wrist so that three of the six degrees off reedom are provided by three revolute joints whose axes coincide at the wrist of the end effector, it is possible to decouple the six equations in Equation into two sets of three simpler equations. In general, however, one Must resort to numerical approaches to obtain the inverse-kinematics solution.
In the presence of redundant kinematics (n>6), an infinite set of solutions would be possible for the inverse kinematics problem.
In this case, it is necessary to employ a useful set of constraints for joint motions, or minimize a suitable cost function, in order to obtain a unique solution.