The problem of “gymnast” motion control in free flight and during the transition in the state of contact with the support

The study from the position of theoretical mechanics discusses some aspects of the jump down of the “athlete-gymnast” (or the “robot-gymnast”) from the “crossbar”. We discuss all phases of the jump in the planar case. “Gymnast” body is modeled as three-links physical pendulum, however, after departing from the "crossbar" and lowering the “hands” model the dynamic system becomes two-links pendulum. For a two-links model in the regime of “kinematic” control, we deal with the process of lowering hands, free flight, the output of the “legs” on the frame (absolutely inelastic impact) and holding operation in support phase with the aim of vertical stabilize of the pendulum system. Mode “kinematic control” means the ability to instantly change the angle between the links of the body (in some limits). For each of the phases of movement, we found a convenient form of describing dynamical equations. These equations are based on the use as a variable momentum of the system relative to various points of the body or space. The order of this system of equations is lower than for the full order system. The stage of calm “gymnast” that occurs after the foot on the surface of the support, will also be examined on the basis of special system of equations of this type. It was shown, that we can use numerical analysis to build the region of controllability for transition of the two-links model in a state of stabilization, corresponding to the equality of the horizontal coordinate of the support foot and the center of gravity of the two-links pendulum system. The algorithm of stabilizing control is designed. The results of presented of the analysis of the problem allow us to construct a convenient approximate model of the phenomenon as a whole, and to use it to control a robotic counterpart. As an example, it is considered one of the cases of motion that corresponds to the anthropomorphic model.