In this paper, a new application of Lyapunov exponents is introduced for stability analysis of contact task control in robotics. The dynamic model is derived including a nonlinear model of the contact which allows bouncing. The system is controlled by a discontinuous controller composed of laws for free and constrained motions that are switched based on detection of the contact force. The stability analysis of such a nonsmooth system using Lyapunov's direct method is extremely difficult. A model based algorithm for the calculation of Lyapunov exponents in nonsmooth systems is employed to prove the overall stability of the system and the required equations and transition conditions are obtained. Numerical results demonstrate the stability and provide valuable insight into the effect of approach velocity and controller gains on it.