Formalism for solving the first problem of the dynamics of walking and wheel-walking machines

The aim is to solve the first problem of the dynamics of machines moving their bodies with the help of legs, i.e., to calculate the control actions of the leg actuators that provide the given dynamics of the absolute motion of the body and satisfy the constraints, when fulfilled, there is no sliding of the supporting legs relative to the supporting surface. The research methods are related to system analysis, mechanics of body systems and robotics. We consider walking and wheel-walking machines with legs suspended from the body, in which the last kinematic pair is translational, there is no controlled foot, and the foot and the supporting surface interact at a single point. The resultsof the study contain analytical formulas for calculating dynamic reactions at the points of contact between the legs and the supporting surface, as well as driving forces and moments of forces in the drives of the supporting and carrying legs, providing a given motion of the body relative to the supporting surface, as well as its corresponding relative motion of the bodies of the supporting legs. A formalism for solving such problems is described, based on inverse vector recurrence formulas for calculating forces and moments of forces of dynamic reactions in kinematic pairs of tree-like systems of bodies with closure of the last body of the branch on the supporting surface. This formalism is extended to wheeled, wheel-walking and walking machines, which is demonstrated by appropriate examples. Analytical conditions for realizing a step without slipping of the support legs relative to the support surface are written down. Two simple wheel-walking machines, consisting of a linear electric drive and a driven (passive) wheelset, are proposed for the experimental determination of sliding and rolling friction coefficient values. In analytical forms of deduced calculation formulas (equations and inequalities) the structural, geometric, inertial and kinematic parameters of the investigated machines are explicitly expressed. The proposed formalism is demonstrated on the solutions of four problems from simple to complex through the reuse of formulas.