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Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot

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Nonlinear Approaches in Engineering Applications

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Abstract

The problem of nonlinear identification and energy conservative control of balance in a monopod is addressed in this chapter. A monopod is emulated using a concave balancing mechanism, referred to as the body, mounted on an inverted pendulum, referred to as the leg, via a hip joint. The body curvature, represented by ϕ, can be altered and is elected as the design parameter of interest as it is observed that at an optimal body span angle, ϕopt, certain interesting phenomena transpire: The linearized system is transformed from nonminimum to minimum phase (MP), the conditions for feedback linearization of the nonlinear model satisfied, and minimal mechanical power required for stability of the simulated model is observed. A nonlinear gray-box system identification routine is developed and implemented within MATLAB, to estimate certain immeasurable parameters that arise within the original system dynamics. To estimate the optimal angle ϕopt, another immeasurable parameter, the identification routine is further employed for various values of ϕ. A locus of the transfer function zeros is then used to interpolate the value of ϕ at which the system achieves MP behavior. At this configuration, the magnitude of the transfer function zeros become much larger in comparison with that of the largest pole and, therefore, has a negligible contribution to the phase characteristics of the system. After the experimental identification of ϕopt, a Linear Quadratic Regulator with integral action is design and implemented to achieve the control objectives, with the design parameters set as close as physically realizable to the optimal setting. A nonlinear controller based on Feedback Linearization is also designed and implemented to take advantage of one benefit of this proposed design. The performance of the nonlinear controller is also compared against that of the LQ Regulator. Analysis and documentation of the power savings from the proposed design, in relation to existing prototypes, is a subject of further works. Furthermore, the performance of the developed controllers can be further improved via careful tuning to meet more stringent control objectives.

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Notes

  1. 1.

    In the phase plane, an equilibrium point is a singular point.

  2. 2.

    In accordance with the control objectives.

  3. 3.

    Throughout this chapter, the variables q i and x i, i = 1, 2 will be used interchangeably.

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Author information

Authors and Affiliations

  1. Software Engineering, EMS Global Tracking - a Honeywell Company, 400 Maple Grove, Ottawa, ON, K2V 1B8, Canada

    Klaus Nji

  2. Faculty of Engineering, University of Regina, Regina, SK, S4S 0A2, Canada

    Mehran Mehrandezh (P.Eng., Associate Professor)

Authors
  1. Klaus Nji
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  2. Mehran Mehrandezh
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Correspondence to Mehran Mehrandezh .

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Editors and Affiliations

  1. , Industrial Systems Engineering, University of Regina, Wascana Parkway, Regina, S4S 0A2, Saskatchewan, Canada

    Liming Dai

  2. School of Aerospace, Mechanical, and Man, RMIT University, Bandoora Campus, Building 251.03.42, Melbourne, 3083, Victoria, Australia

    Reza N. Jazar

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Nji, K., Mehrandezh, M. (2012). Energy Conservative Design and Nonlinear Control of Balance in a Hopping Robot. In: Dai, L., Jazar, R. (eds) Nonlinear Approaches in Engineering Applications. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1469-8_14

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  • DOI: https://doi.org/10.1007/978-1-4614-1469-8_14

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