A large class of problems in robotics, e.g., tracking with obstacle avoidance, compliant motion control, and complex assembly, can be formulated as a least-squares tracking problem on the Euclidean group subject to constraints on the state and/or control. In this paper we develop a general, mathematically rigorous optimal control framework for this class of problems, and derive a simple closed-form analytic solution. Our formalism can be viewed as generalization to the Euclidean group of the linear quadratic regulator (LQR) subject to state equality constraints. Examples from force-guided complex assembly and tracking with obstacle avoidance are given.

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