Kinematics of X-Y Pedestals with Joint and Pointing Misalignment

New inverse kinematics solutions are presented for X-Y antenna pedestals with unideal geometry caused by joint and pointing misalignment, which might arise from issues in manufacturing, assembly, or mechanical wear. For certain cases of misalignments, such as flexure of the joints, they may be variable and therefore the inverse kinematic solutions are required to compensate for it in real-time. For such imperfections, as long as they can be measured, the new parameters can be plugged into the synthesized inverse kinematic functions for each produced joint angle without needing to reproduce the functions. Solutions are synthesized analytically and their validity is tested using simulations of discrete rotations as well as simulations of tracking with real satellite positions. As expected of an analytical solution, for positions inside the workspace of the misaligned antenna, no error that could be differentiated from floating-point inaccuracies (i.e. position error bigger than 10
⁻¹⁴
degrees) is produced. In addition, an inequality relation is defined for checking if the target position produces a real-valued joint angle for the pedestal. The effects of misalignments on the workspace of the pedestal are also analyzed. For target positions outside the workspace of the pedestal, the inverse kinematics functions produce a complex joint angle where the real part of the angles correlate to the nearest position inside the workspace.