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Research Papers

# Control Constraint Realization for Multibody Systems

[+] Author and Article Information
Alessandro Fumagalli1

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano 20156, Italyalessandro.fumagalli@polimi.it

Pierangelo Masarati2

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano 20156, Italypierangelo.masarati@polimi.it

Marco Morandini

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano 20156, Italymarco.morandini@polimi.it

Paolo Mantegazza

Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Milano 20156, Italypaolo.mantegazza@polimi.it

The index of a DAE in the form $f(x,ẋ,u,t)=0$, where $u$ is the vector of algebraic variables, is here defined as one plus the minimum number of times that all or part of the original problem must be differentiated with respect to time in order to determine $ẋ$ as a continuous function of $x$, $t$(12-15).

1

Also at Altran Italia.

2

Corresponding author.

J. Comput. Nonlinear Dynam 6(1), 011002 (Sep 27, 2010) (8 pages) doi:10.1115/1.4002087 History: Received May 27, 2009; Revised November 10, 2009; Published September 27, 2010; Online September 27, 2010

## Abstract

This paper discusses the problem of control constraint realization applied to generic underactuated multibody systems. The conditions for the realization are presented. Focus is placed on the tangent realization of the control constraint. An alternative condition is formulated, based on the practical observation that differential-algebraic equations need to be integrated using implicit algorithms, thus naturally leading to the solution of the problem in form of matrix pencil. The analogy with the representation of linear systems in Laplace’s domain is also discussed. The formulation is applied to the solution of simple, yet illustrative problems, related to rigid and deformable bodies. Some implications of considering deformable continua are addressed.

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## Figures

Figure 1

Projection of the input vector u on the constraint manifold

Figure 2

Figure 3

Overhead crane control inputs: trolley force F (top) and winch moment M (bottom)

Figure 4

Two masses connected by a spring

Figure 5

Two masses connected by a spring: motion x1(t) (top) and input force F (bottom)

Figure 6

Control of deformable one-dimensional continuum: problem (top); spatial discretization (bottom)

Figure 7

Deformable continuum discretized using N=2, 3, and 4 elements: actuator displacement x1 (top), computed force F (bottom)

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