Abstract
This paper studies theoretical properties of the positive semidefinite cone lift (PSD lift) and utilizes the PSD lift to solve the constrained attitude control problem for a rigid body. We first introduce the PSD lift and the mapping between the PSD lift to the convex hull of the rotation matrices conv SO(3), commonly used in attitude estimation and control, as well as in areas such as computer vision. We next study conditions under which this mapping defines an affine isomorphism. We focus particularly on the set of extreme points of the PSD lift and show that this set is in fact a group; connections with SO(3) and unit quaternions H are also explored. Moreover, the differentiation operation on this group, which is related to the angular velocity of the rigid body, will be discussed. Lastly, we apply the PSD lift in the constrained attitude control problem and approach it via a semidefinite program (SDP). Furthermore, we provide insights into the physical realization and the optimization results through an example.
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