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Research Papers

Control of Uncertain Nonlinear Multibody Mechanical Systems

[+] Author and Article Information
Firdaus E. Udwadia

Professor of Aerospace and
Mechanical Engineering, Civil Engineering, Mathematics, and Information and
Operations Management,
410 K Olin Hall,
University of Southern California,
Los Angeles, CA 90089-1453 
e-mail: fudwadia@usc.edu

Thanapat Wanichanon

Lecturer
Department of Mechanical Engineering,
Mahidol University,
25/25 Puttamonthon,
Nakorn Pathom 73170, Thailand
e-mail: thanapat.wan@mahidol.ac.th

Manuscript received September 27, 2012; final manuscript received August 30, 2013; accepted manuscript posted September 12, 2013; published online December 16, 2013. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 81(4), 041020 (Dec 16, 2013) (11 pages) Paper No: JAM-12-1461; doi: 10.1115/1.4025399 History: Received September 27, 2012; Revised August 30, 2013

Descriptions of real-life complex multibody mechanical systems are usually uncertain. Two sources of uncertainty are considered in this paper: uncertainties in the knowledge of the physical system and uncertainties in the “given” forces applied to the system. Both types of uncertainty are assumed to be time varying and unknown, yet bounded. In the face of such uncertainties, what is available in hand is therefore just the so-called “nominal system,” which is our best assessment and description of the actual real-life situation. A closed-form equation of motion for a general dynamical system that contains a control force is developed. When applied to a real-life uncertain multibody system, it causes the system to track a desired reference trajectory that is prespecified for the nominal system to follow. Thus, the real-life system's motion is required to coincide within prespecified error bounds and mimic the motion desired of the nominal system. Uncertainty is handled by a controller based on a generalization of the concept of a sliding surface, which permits the use of a large class of control laws that can be adapted to specific real-life practical limitations on the control force. A set of closed-form equations of motion is obtained for nonlinear, nonautonomous, uncertain, multibody systems that can track a desired reference trajectory that the nominal system is required to follow within prespecified error bounds and thereby satisfy the constraints placed on the nominal system. An example of a simple mechanical system demonstrates the efficacy and ease of implementation of the control methodology.

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Figures

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Fig. 1

Triple pendulum with the datum at the origin O

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Fig. 2

Trajectory of mass m3 in the XY-plane (meter) of the triple pendulum shown for a duration of 10 s. The trajectory starts at the circle and ends at the square.

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Fig. 3

Energies in N-m: (a) E1; (b) E = E2 + E3

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Fig. 4

(a) Control force applied to mass m1 of the nominal system to satisfy E = E2 + E3; (b) magnitude of the control force. The control forces on masses m2 and m3 are zero.

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Fig. 5

Trajectory of mass m3 of the actual system over a period of 10 s. The masses are m1 = 1.1 kg (δm1 = 0.1), m2 = 1.8 kg (δm2 = −0.2), and m3 = 3.3 kg (δm3 = 0.3). The system satisfies the energy constraint in Eq. (2.15).

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Fig. 6

The block diagram of the controlled actual system. Note that the compensating controller uses the mass matrix of the nominal system.

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Fig. 7

The three masses mi ± δmi, i = 1, 2, 3, of the actual system lie somewhere in the box shown. The figure shows 1014 uniformly distributed random points generated from a Monte Carlo simulation.

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Fig. 8

Probability density function of ‖δq··‖ at each time t using Eq. (3.18) for the 1014 simulation points in which the masses have ±10% uncertainties

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Fig. 10

Tracking errors between the controlled nominal system and the controlled actual system (ei(t):=θi(t)-θci(t),i=1,2,3) in radians of the masses m1,m2,and m3

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Fig. 9

Trajectory response (meter) of mass m3 over a period of 10 s of the controlled actual system when the uncertainties in the masses are prescribed as δm1 = 0.1 kg, δm2 = −0.2 kg, and δm3 = 0.3 kg and the uncertainty bound in Eq. (4.3) is chosen to be Γ(t) = 40

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Fig. 11

Control forces (newtons) on the controlled actual system. The solid line shows the total control force, QT, and the dashed line shows the additional force, Qu, needed to compensate for uncertainties in the actual system.

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