Optimal Control Problems with Symmetry Breaking Cost Functions

Abstract: Symmetry reduction is a well studied subject in geometric mechanics, where symmetries are usually described as an invariance under an action of a Lie group. Symmetry breaking is also common in several physical contexts, from classical mechanics to particle physics and in certain cases, it is still possible to carry out symmetry reduction. The simplest example is the heavy top dynamics (the motion of a rigid body with a fixed point in a gravitational field), where due to the presence of gravity, we get a Lagrangian that is SO(2)-invariant but not SO(3)-invariant, contrary to what happens for the free rigid body. Based on the ideas of symmetry reduction studied in geometric mechanics, symmetry reduction of optimal control problems (OCPs) for left-invariant control systems on Lie groups has been studied extensively over the past couple of decades and by exploiting these symmetries, the system can be reduced to a lower-dimensional one or decoupled into subsystems. In this talk, I will discuss symmetry reduction of OCPs for left-invariant control affine systems on Lie groups with partially broken symmetries, more specifically, cost functions that break some but not all of the symmetries. I will illustrate the theory with the motion planning problem of a controlled unicycle (a popular model used in robotics) in the presence of an obstacle, where the symmetry breaking appears naturally in the form of a barrier function.