Diversity of Servo-Constraint Problems for Underactuated Mechanical Systems: A Case Study Illustration

Abstract:

Underactuated mechanical systems are systems with fewer control inputs than degrees of freedom. Determination of an input control strategy that forces an underactuated system to complete specified in time outputs (servo-constraints), whose number is equal to the number of inputs, can be a challenging task. Diversity of the servo-constraint problems is discussed here using a simple spring-mass system mounted on a carriage (two degrees of freedom, one control input, and one specified in time output). A formulation of underactuated system dynamics which includes the output coordinates is motivated, with the governing equations arising either as ODEs (ordinary differential equations) or DAEs (differential-algebraic equations). Solutions to the servo-constraint problem are then discussed with reference to so-called non-flat systems (with internal dynamics) and differentially flat systems (no internal dynamics). Some computational issues related to the ODE and DAE formulations are finally discussed, and relevant simulation results for the sample case study are reported.

• Periodical: Solid State Phenomena (Volume 198)
• Pages: 473-482
• DOI: https://doi.org/10.4028/www.scientific.net/SSP.198.473
• Online since: March 2013
• Authors: Wojciech Blajer, Robert Seifried, Krzysztof Kołodziejczyk
• Keywords: Inverse Trajectory Control, Servo-Constraints, Underactuated Systems

References

[1] M.W. Spong, Underactuated mechanical systems, in: B. Siciliano, K. P. Valavanis (Eds. ), Control Problems in Robotics and Automation, Springer, London, 1998, pp.135-150.

[2] I. Fantoni, R. Lozano, Nonlinear Control for Underactuated Mechanical Systems, Springer, London, (2002).

[3] W. Blajer, K. Kołodziejczyk, A geometric approach to solving problems of control constraints: theory and a DAE framework, Multibody System Dynamics, 11(2004) 343-364.

DOI: 10.1023/b:mubo.0000040800.40045.51

[4] W. Blajer, K. Kołodziejczyk, Control of underactuated mechanical systems with servo-constraints, Nonlinear Dynamics, 50 (2007) 781-791.

DOI: 10.1007/s11071-007-9231-4

[5] R. Seifried, Two approaches for feedforward control and optimal design of underactuated multibody systems, Multibody System Dynamics, 27 (2012) 75-93.

DOI: 10.1007/s11044-011-9261-z

[6] R. Seifried, Integrated mechanical and control design of underactuated multibody systems, Nonlinear Dynamics, 67 (2012) 1539-1557.

DOI: 10.1007/s11071-011-0087-2

[7] W. Blajer, K. Kołodziejczyk, A DAE formulation for the dynamic analysis and control design of cranes executing prescribed motions of payloads, in J.C. García Orden, J.M. Goicolea, J. Cuadrado (Eds. ), Multibody Dynamics. Computational Methods and Applications, Series: Computational Methods in Applied Sciences, Vol. 4, Springer, Dordrecht, 2007, pp.91-112.

DOI: 10.1007/978-1-4020-5684-0_5

[8] W. Blajer, J. Graffstein, M. Krawczyk, Modeling of aircraft prescribed trajectory flight as an inverse simulation problem, in J. Awrejcewicz (Ed. ), Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems, Springer, Netherlands, 2009, pp.153-162.

DOI: 10.1007/978-1-4020-8778-3_14

[9] A. Isidori, Nonlinear Control Systems, Springer, London, (1995).

[10] M. Fliess, J. Lévine, P. Martin, P. Rouchon P., Flatness and defect of nonlinear systems: introductory theory and examples, International Journal of Control, 61 (1995) 1327-1361.

DOI: 10.1080/00207179508921959

[11] H. Sara-Ramirez, S.K. Agrawal, Differentially Flat Systems, Marcel Dekker, New York, (2004).

[12] P. Rouchon, Flatness based control of oscillators, ZAMM, 85 (2005), 411-421.

DOI: 10.1002/zamm.200410194