  In this paper, a class of nonlinear driftless control-affine systems
satisfying the bracket generating condition is considered. A gradient-free
optimization algorithm is developed for the minimization of a cost function
along the trajectories of the controlled system. The algorithm comprises an
approximation scheme with fast oscillating controls for the nonholonomic
dynamics and a model-free extremum seeking component with respect to the output
measurements. Exponential convergence of the trajectories to an arbitrary
neighborhood of the optimal point is established under suitable assumptions on
time scale parameters of the extended system. The proposed algorithm is tested
numerically with the Brockett integrator for different choices of generating
functions.
