Inverted pendula

The first chapter of Part II illustrates all these aspects by considering a coaxially coupled inverted pendula system that is unstabilisable when torque actuation is applied but is controllable in case of velocity actuation although there is a right-half plane zero. 

Coaxially Coupled Inverted Pendula Bond Graph-Based Modelling, Design and Control... 

Abstract A bond graph method is used to examine qualitative aspects of a class of unstable under-actuated mechanical systems. It is shown that torque actuation leads to an unstabilisable system, whereas velocity actuation gives a controllable system which has, however, a right-half plane zero. The fundamental limitations theory of feedback control when a system has a right-half plane zero and a right-half plane pole is used to evaluate the desirable physical properties of coaxially coupled inverted pendula. An experimental system which approximates such a system is used to illustrate and validate the approach. 

Figure 5.1a shows two inverted pendula mounted on a common pivot - rather like the hands of a clock. An actuator is mounted on the same axis so that each pendulum has an equal and opposite torque ro this is termed coaxial coupling. It is assumed that the angles from the vertical of the two pendula (0i and O2) can be measured. The equations of motion are... 

Fig. 5.1 Coaxially coupled inverted pendula (a) general and (b) repaiameterised...

A class of under-actuated systems, coaxially coupled inverted pendula, have been introduced and their qualitative properties analysed using established bond graph approaches. In particular, it is shown that torque actuation leads to an uncontrollable system, whereas angular velocity actuation gives a controllable system. 

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