A theorem of target points for the ground reaction force in a planar serial-linked walking limb

An important energy expense in the legged locomotion of both animals and robots is the mechanical antagonism between muscles/actuators, which is the positive mechanical work of some muscles opposed by the simultaneous negative work of the others. One known way to minimize the mechanical antagonism is the proper employment of the redundant degrees of freedom of the limb. Here, I present and analyze a generalized model of a planar serial-linked limb composed of any number of segments and conclude that the minimization of the inter-actuator antagonism requires fixation of all the joint angles except the two defined by simple geometric considerations. So, regardless of the number of joints, the limb should optimally act as a two-joint system. Which of the redundant joints to fix or move depends on the instantaneous position of the joints relative to the vertical line through the center of the foot contact with the ground. Subsequently, for the first time, I pose and solve the following problem: how to eliminate the inter-actuator antagonism by the adjustment of the horizontal component of the ground reaction force. The solution is that, during the contact phase, the ground reaction force vector should be redirected so as to maintain alignment with the limb joints in the order in which they attain the smallest angular deflection from the vertical line through the foot. As the joints pass the vertical line through the point of limb contact with the ground one by one, abrupt changes of the horizontal ground force component from positive to negative should occur. Mammals cannot follow this algorithm exactly due to muscular actuator limitations and they tend to align the ground reaction force with some compromise target point above the hip or scapula. The suggested principles of the optimal choice of redundant degrees of freedom and direction of the ground reaction force can be implemented in robotics to achieve a lower cost of transport than is known for animals.