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Research Papers

The Use of Servo-Constraints in the Inverse Dynamics Analysis of Underactuated Multibody Systems

[+] Author and Article Information
Wojciech Blajer

Faculty of Mechanical Engineering,
Institute of Applied Mechanics
and Power Engineering,
University of Technology
and Humanities in Radom,
ul. Krasickiego 54 26-600 Radom, Poland
e-mail: w,blajer@uthrad.pl

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received August 22, 2013; final manuscript received October 27, 2013; published online July 11, 2014. Assoc. Editor: Javier Cuadrado.

J. Comput. Nonlinear Dynam 9(4), 041008 (Jul 11, 2014) (11 pages) Paper No: CND-13-1204; doi: 10.1115/1.4025855 History: Received August 22, 2013; Revised October 27, 2013

Underactuated mechanical systems have fewer control inputs than degrees of freedom. The specified in time outputs, equal in number to the number of inputs, lead to servo-constraints on the system. The servo-constraint problem is then a specific inverse simulation problem in which an input control strategy (feedforward control) that forces an underactuated system to complete the partly specified motion is determined. Since mechanical systems may be “underactuated” in several ways, and the control forces may be arbitrarily oriented with respect to the servo-constraint manifold, this is, in general, a challenging task. The use of servo-constraints in the inverse dynamics analysis of underactuated systems is discussed here with an emphasis on diverse possible ways of the constraint realization. A formulation of the servo-constraint problem in configuration coordinates is compared with a setting in which the actuated coordinates are replaced with the outputs. The governing equations can then be set either as ordinary differential equations (ODEs) or differential-algebraic equations (DAEs). The existence and nonexistence of an explicit solution to the servo-constraint problem is further discussed, related to so-called flat systems (with no internal dynamics) and nonflat systems (with internal dynamics). In case of nonflat systems, of paramount importance is stability of the internal dynamics. Simple case studies are reported to illustrate the discussion and formulations.

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Figures

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Fig. 1

Geometry of passive constraint realization (a) and servo-constraint realization (b)

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Fig. 2

Structure of control for underactuated mechanical systems with orthogonal realization of servo-constraints

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Fig. 3

Structure of control for underactuated mechanical systems with any realization of servo-constraints, based on the DAE formulations

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Fig. 4

Rotational arm with one active and one passive joint

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Fig. 5

Values of Y defined in Eq. (37) in terms of θ1 and θ2, obtained for l1 = l2, m1 = m2, JC1 = JC2, and then si = li/2, JCi = mili2/12 (i = 1,2), and sP = sE = l2

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Fig. 6

Illustration of the action of τ on link 2

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Fig. 7

Illustration of possible design variants for altering the center of mass location of link 2

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Fig. 8

Simulation results for the rotational arm (Fig. 6), with sP = l2/2 (P→C2), and the links modeled as identical homogeneous bars

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Fig. 9

The spring-mass system mounted on a movable carriage

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Fig. 10

Simulation results for α = 40 deg (orthogonal realization), with damping d = 2 Ns/m (dashed lines) and without damping (solid lines)

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Fig. 11

Simulation results for α = 0 deg (tangential realization), with damping d = 2Ns/m (dashed lines) and without damping (solid lines)

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