A generic optimal tracking control is developed where the optimal control iscalculated by optimizing a universal quadratic penalty function. Several attitudeerror representations are presented for describing the tracking orientationerror kinematics. Compact forms of attitude error equation are derived for eachcase. The attitude error is initially defined as the quaternion (rotation) error betweenthe current and the reference orientation. Transformation equations arepresented that enable the development of nonlinear kinematic models that arevalid for arbitrarily large relative rotations and rotation rates. The nonlinear errorfor the equation of motion is retained, yielding a tensor-based series solutionfor the Co-State as a function of error dynamics. By utilizing several attitudeerror kinematics to describe the spacecraft rotation error, we introduce auniversal quadratic penalty function of tracking errors that is consistent in eachof the coordinate choices-i.e. a quadratic penalty on the MRPs error is clearlynot “the same” physically as a quadratic penalty on the classical Rodrigues parameters.We utilize this universal attitude error measure expressed through approximatetransformations as a positive function of each of the coordinatechoices. This allows for a universal solution to many spacecraft optimal controlproblems and removes the dependency on the attitude coordinate choice.
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