  In this work, we propose a trajectory generation method for robotic systems
with contact force constraint based on optimal control and reachability
analysis. Normally, the dynamics and constraints of the contact-constrained
robot are nonlinear and coupled to each other. Instead of linearizing the model
and constraints, we directly solve the optimal control problem to obtain the
feasible state trajectory and the control input of the system. A tractable
optimal control problem is formulated which is addressed by dual approaches,
which are sampling-based dynamic programming and rigorous reachability
analysis. The sampling-based method and Partially Observable Markov Decision
Process (POMDP) are used to break down the end-to-end trajectory generation
problem via sample-wise optimization in terms of given conditions. The result
generates sequential pairs of subregions to be passed to reach the final goal.
The reachability analysis ensures that we will find at least one trajectory
starting from a given initial state and going through a sequence of subregions.
The distinctive contributions of our method are to enable handling the
intricate contact constraint coupled with system's dynamics due to the
reduction of computational complexity of the algorithm. We validate our method
using extensive numerical simulations with a legged robot.
