![]()
Part I: Devices containing a single-link pendulum
Many unstable mechanical systems have links that can be described as inverted pendulums. For example, an individual transportation device like “Segway” [165], platforms on paired coaxial wheels, bipedal walking machines [19, 20, 55, 65]. In a complex problem of controlling unstable pendulum systems, the elementary problems of controlling a single-link pendulum and stabilizing its unstable top equilibrium can be called the core ones. They are considered classical in theoretical mechanics and control theory. In most cases, such problems are solved by controlled movement of the pivot. In studies [93, 149] an inverted pendulum is stabilized by vertical movements of its pivot. When stabilized in such a way, it is often called Kapitza’s pendulum. Study [137] considers stabilization of a multi-link pendulum, and a single-link pendulum as a most simple case, by horizontal movements of its pivot. Such a device can bemounted on a cartmoving on a horizontal plane (see [29]). It iswell known that one can hold a vertical rod standing on an open palm of his hand and prevent it from falling by moving his hand. Study [74] discusses a robot that can maintain its balance on a cylinder. See also article [118] on the control of unstable mechanical systems in particular on pendulum, and article [153] on the stabilization of linear systems with bounded control parameters.
This part introduces several problems related with controlling a single-link, plane physicalpendulum[22–24, 54, 55, 57, 58, 64, 111, 112, 122, and 123].First, a regularphysical pendulum is considered, with control torque applied to it in its pivot point. The absolute value of this control torque is assumed limited. Then, a pendulum with its pivot located in the center of a wheel is discussed (see [57, 58, 64]). Here the wheel rolls over a straight horizontal line. The control torque is again applied at the pivot point of the pendulum. Therefore, it turns the wheel at the same time as it turns the pendulum. The task is to stabilize the pendulum in its topmost point. Yet another device considered in this part is a pendulumwith a flywheel attached to its end [7, 22–24, 64], or, in other words, an “inertia wheel pendulum” [1, 25, 49, 147]. A control algorithm is suggested for relocating the pendulum from its bottom position, which is a stable equilibrium, into the top position. The suggested algorithm can stabilize the top equilibrium that is naturally unstable. Further, this part discusses the problem of controlling the motion of a wheel by means of a pendulum attached to its center. The conditions for the wheel to be able to move uphill are formulated. The last chapter of this part, once again a simple single-link physical pendulum is considered, with control torque applied at its stationary pivot point [54, 55]. The problem investigated there concerns transferring the pendulum to its bottom – stable – equilibrium, and to its top – unstable – equilibrium. Among all possible control algorithms, the optimal ones are chosen, that provide minimal energy consumption for the translation process. These optimal algorithms are proved to be pulse-based.
One more interesting device that is investigated in stabilization studies is worth mentioning here. It is not fully related to pendulum systems, though. In article [31] and patent article [32] a long body is discussed; in particular, it can be a beam. This beam is being compressed by forces acting along its axial direction. The beam is connected to a support by a joint. Tomaintainin stability of the static position of the beam, a stabilizing torque is suggested that is applied in the joint. This stabilizing torque depends on the beamstrainmeasured in some of its points. The ...
