By Amal Choukchou-Braham, Brahim Cherki, Mohamed Djemaï, Krishna Busawon
This monograph offers readers with instruments for the research, and keep watch over of structures with fewer regulate inputs than levels of freedom to be managed, i.e., underactuated structures. The textual content offers with the implications of an absence of a basic concept that may permit methodical remedy of such platforms and the advert hoc method of keep watch over layout that regularly effects, implementing a degree of association every time the latter is lacking.
The authors take as their start line the development of a graphical characterization or keep an eye on move diagram reflecting the transmission of generalized forces in the course of the levels of freedom. Underactuated platforms are categorized in line with the 3 major constructions wherein this can be discovered to happen—chain, tree, and remoted vertex—and regulate layout systems proposed. The approach is utilized to a number of recognized examples of underactuated structures: acrobot; pendubot; Tora process; ball and beam; inertia wheel; and robot arm with elastic joint.
The textual content is illustrated with MATLAB®/Simulink® simulations that show the effectiveness of the tools detailed.
Readers drawn to airplane, automobile keep watch over or quite a few kinds of jogging robotic should be capable of study from Underactuated Mechanical Systems easy methods to estimate the measure of complexity required within the regulate layout of a number of sessions of underactuated structures and continue directly to additional generate extra systematic regulate legislation in response to its equipment of analysis.
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Additional resources for Analysis and Control of Underactuated Mechanical Systems
The exploitation of such property has permitted to reduce the complexity of the control synthesis for UMSs. In effect, under some change of coordinates, the initial system is transformed into two cascaded subsystems where the first subsystem is nonlinear and the second one is linear, often in the form of a chain of integrators. The analytical tools allowing such reduction are the generalized momenta and their integrals computed from the Lagrangian. Nevertheless, several benchmarks and real systems does not possess integrable generalized momentums.
9 Examples of Underactuated Mechanical Systems 29 Fig. 3 The Tora System The Tora system2 depicted in Fig. 4, is composed of an oscillatory platform which is controlled via a central mass . The inertia matrix M(q) and the potential energy V (q) are given by M(q) = m1 + m2 m2 r cos q2 m2 r cos q2 m2 r 2 + I2 1 and V (q) = k1 q12 + m2 gr cos q2 2 where q1 is the platform displacement, q2 is the pendulum angular, m1 is the mass of the cart, m2 the mass of the eccentric mass, r the radius of the rotation, k the spring constant, I2 the inertia of the arm, g the gravity acceleration, and τ is the torque input.
The general model for UMSs with two degrees of freedom is of the form  m11 (q2 )q¨1 + m12 (q2 )q¨2 + m11 (q2 )q˙1 q˙2 + m12 (q2 )q˙22 − g1 (q1 , q2 ) = τ1 1 1 m21 (q2 )q¨1 + m22 (q2 )q¨2 − m11 (q2 )q˙12 + m22 (q2 )q˙22 − g2 (q1 , q2 ) = τ2 2 2 where gi (q1 , q2 ) = −∂V (q)/∂qi , i = 1, 2, and denotes d/dq2 . Class I are those for which q2 is actuated (τ1 = 0). Class II are those for which q2 is not actuated (τ2 = 0). It is shown that every underactuated system of Class I can be transformed into a strict feedback form.