  The importance of humanoid robots in today's world is undeniable, one of the
most important features of humanoid robots is the ability to maneuver in
environments such as stairs that other robots can not easily cross. A suitable
algorithm to generate the path for the bipedal robot to climb is very
important. In this paper, an optimization-based method to generate an optimal
stairway for under-actuated bipedal robots without an ankle actuator is
presented. The generated paths are based on zero and non-zero dynamics of the
problem, and according to the satisfaction of the zero dynamics constraint in
the problem, tracking the path is possible, in other words, the problem can be
dynamically feasible. The optimization method used in the problem is a
gradient-based method that has a suitable number of function evaluations for
computational processing. This method can also be utilized to go down the
stairs.
