This work considers the problem of minimizing the path-curvature of a unicycle moving between two configurations in SE(2) with free final time. The problem is posed as an optimal control problem and the necessary conditions are derived using the Pontryagin Maximum Principle and Lie-Poisson reduction. Solutions are categorized into three cases corresponding to the value of the Casimir. A numerical solver for obtaining the optimal control is described. Experimental results are reported on a ground robot.