Dynamics and optimal control of multibody systems using fractional generalized divide-and-conquer algorithm
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ORIGINAL PAPER
Dynamics and optimal control of multibody systems using fractional generalized divide-and-conquer algorithm Arman Dabiri · Mohammad Poursina · J. A. Tenreiro Machado
Received: 29 June 2019 / Accepted: 10 September 2020 © Springer Nature B.V. 2020
Abstract In this paper, a new framework is presented for the dynamic modeling and control of fully actuated multibody systems with open and/or closed chains as well as disturbance in the position, velocity, acceleration, and control input of each joint. This approach benefits from the computed torque control method and embedded fractional algorithms to control the nonlinear behavior of a multibody system. The fractional Brunovsky canonical form of the tracking error is proposed for a generalized divide-and-conquer algorithm (GDCA) customized for having a shortened memory buffer and faster computational time. The suite of a GDCA is highly efficient. It lends itself easily to the parallel computing framework, that is used for the inverse and forward dynamic formulations. This technique can effectively address the issues corresponding to the inverse dynamics of fully actuated closed-chain systems. Eventually, a new stability criterion is proposed to obtain the optimal torque control using the new fractional Brunovsky canonical form. It is shown that A. Dabiri (B) Department of Mechanical and Mechatronics Engineering, Southern Illinois University Edwardsville, Edwardsville, IL, USA e-mail: [email protected] M. Poursina Department of Engineering and Science, University of Agder, Grimstad 4879, Norway J. A. T. Machado Department of Electrical Engineering, Institute of Engineering, Polytechnic Institute of Porto, Rua Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal
fractional controllers can robustly stabilize the system dynamics with a smaller control effort and a better control performance compared to the traditional integerorder control laws. Keywords Multibody dynamics · Closed-chain system · Open-chain system · LQR · Optimal control · Divide and conquer algorithm · Computed-torque control · Fractional control · PID · Parallel computing
1 Introduction Fractional controllers are introduced by incorporating fractional operators in classical integer-order control laws, resulting in a few more tuning parameters [1– 3]. The class of fractional PIα Dβ controllers is an example with the extra control parameters α and β, that can be tuned along the three conventional PID gains [3,4]. These additional parameters cause fractional controllers generally outperform their integerorder counterparts with a finite number of control parameters [5–9]. Furthermore, an equivalent linear integer-order controller obtains similar control performance if and only if it comprises numerous control parameters, which can generally be realized by comparing their transfer functions. The fractional controller’s transfer function includes fractional powers of the Laplace variable s in the frequency domain (or sdomain), which can be expanded with a series expansion. This appr
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