Derivation of the constraint equations in the kinematic modeling of legged robots with rigid semi-spherical feet

Using semi-sphere rigid feet in legged robots reduces the sinking of a foot into soft terrains. However, since the foot rolls on the ground during the stance phase, the assumption of fixed contact points with the ground is no longer valid. Therefore, the rolling effect must be accounted for in the kinematics. We present a method here to derive the constraint equations in the kinematic model of the robot in presence of pure rolling. The position of the contact point in the initial configuration changes as the foot rolls. First, we obtain its new position in terms of the generalized coordinates of body and the support leg. Then, we formulate the constraint equations in a similar way to the case of fixed contact point. The method is not limited to a particular design of the leg or the number of leg joints. Both prismatic and revolute joints can exist with arbitrary arrangement. A kinematic simulation of a quadruped robot in stance phase is provided to demonstrate the application of the proposed method.