Analytical Solution for Joint Coupling in NAO Humanoid Hips

Usually the legs of humanoids capable of omnidirectional walking are not underactuated. In other words each one of the six degrees of freedom of the torso can be commanded independently from the leg joint angles. However the NAO humanoid robot has a coupled joint at the hips, which makes 11 degrees of freedom instead of 12 for the locomotor apparatus. As a consequence the trunk of the robot has only 5 independent degrees of freedom when the positions of both feet are fixed, and each leg cannot be commanded independently to execute walking steps. Up to now only bypass solutions have been proposed, where the coupled joint angles are not calculated exactly. This paper describes an analytical solution to determine the exact angle to be applied to the coupled joint. The method uses the positions of both foot ankles in the trunk reference frame and the angle between footprints as inputs, and calculates the yaw angle of the trunk. The solution was demonstrated in a dynamics simulator using the NAO model.