On the Kinematics of Undulator Girder Motion

The theory of rigid body kinematics is used to derive equations that govern the control and measurement of the position and orientation of undulator girders. The equations form the basis of the girder matlab software on the LCLS control system. The equations are linear for small motion and easily inverted as desired. For reference, some relevant girder geometrical data is also given. Equations 6-8 relate the linear potentiometer readings to the motion of the girder. Equations 9-11 relate the cam shaft angles to the motion of the girder. Both sets are easily inverted to either obtain the girder motion from the angles or readings, or, to find the angles and readings that would give a desired motion. The motion of any point on the girder can be calculated by applying either sets of equations to the two cam-planes and extrapolating in the z coordinate using equation 19. The formulation of the equations is quite general and easily coded via matrix and vector methods. They form the basis of the girder matlab software on the LCLS control system.