  In this paper, we present an iterative Model Predictive Control (MPC) design
for piecewise nonlinear systems. We consider finite time control tasks where
the goal of the controller is to steer the system from a starting configuration
to a goal state while minimizing a cost function. First, we present an
algorithm that leverages a feasible trajectory that completes the task to
construct a control policy which guarantees that state and input constraints
are recursively satisfied and that the closed-loop system reaches the goal
state in finite time. Utilizing this construction, we present a policy
iteration scheme that iteratively generates safe trajectories which have
non-decreasing performance. Finally, we test the proposed strategy on a
discretized Spring Loaded Inverted Pendulum (SLIP) model with massless legs. We
show that our methodology is robust to changes in initial conditions and
disturbances acting on the system. Furthermore, we demonstrate the
effectiveness of our policy iteration algorithm in a minimum time control task.
