  Bipedal robots have received much attention because of the variety of motion
maneuvers that they can produce, and the many applications they have in various
areas including rehabilitation. One of these motion maneuvers is walking. In
this study, we presented a framework for the trajectory optimization of a
5-link (planar) Biped Robot using hybrid optimization. The walking is modeled
with two phases of single-stance (support) phase and the collision phase. The
dynamic equations of the robot in each phase are extracted by the Lagrange
method. It is assumed that the robot heel strike to the ground is full plastic.
The gait is optimized with a method called hybrid optimization. The objective
function of this problem is considered to be the integral of torque-squared
along the trajectory, and also various constraints such as zero dynamics are
satisfied without any approximation. Furthermore, in a new framework, there is
presented a constraint called impact invariance, which ensures the periodicity
of the time-varying trajectories. On the other hand, other constraints provide
better and more human-like movement.
