Analysis of the kinematics of mechanisms can provide a technique for rational design of manipulators and workspaces. Mechanical arrangements for robot manipulators vary widely among operational robots, the most common configurations being best described in terms of their coordinate features: cartesian, spherical, and articulated. In a cartesian robot, a wrist is mounted on a rigid framework to permit linear movement along three orthogonal axes, rather like a gantry crane or (for two axes) a graph plotter; the resulting workspace is box-shaped. The cylindrical robot has a horizontal arm mounted on a vertical column which is fixed to a rotating base. The arm moves in and out; a carriage moves the arm up and down along the column, and these two components rotate as a single element on the base; the workspace is a portion of a cylinder. The spherical robot is similar to the turret of a tank: the arm moves in and out, pivots vertically, and rotates horizontally about the base; the workspace is a portion of a sphere. An articulated robot is more anthropomorphic: an upper arm and forearm move in a vertical plane above a rotating trunk. The limbs are connected by revolute joints; the workspace approximates a portion of a sphere. For all robots, additional degrees of freedom are provided at the extremity of the arm, at the wrist. Wrists generally allow rotation in two or three orthogonal planes. To make proper use of a robot arm, transformations between encoded axis values (joint angles, etc) and more convenient coordinate systems must be computed at high speed. Transforming a set of axis values to a position and orientation in space is called the forward kinematics transformation. The reverse transformation is used to convert a desired position and orientation in space into commanded axis values.