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Constrained control of flexible-joint lever arm based on uncertainty estimation with data fusion for correcting measurement errors

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Abstract

This study presents a novel uncertainty estimation algorithm that aims to enhance the accuracy of a reduced-order dynamic model for a flexible-joint lever arm (FJLA). The model uncertainties are compensated by adding a complementary term to the reduced-order model, upgrading it to the real model. The gyroscope bias in measuring the angular velocity is corrected by fusing the information of encoder/gyroscope. The effective data fusion is provided by adaptive adjustment of the estimator coefficient. The constructed model, with low order but rich content, is reliable for use in the position controller of FJLA, developed by a continuous predictive method. The control system adapts itself to real conditions and is cost-effective due to the use of fewer sensors. The saturation of motor torque, as the control input, is modeled within the structure of a constrained optimization problem solved by Karush–Kuhn–Tucker Theorem. The boundedness of the mean and covariance of tracking error and its derivative are demonstrated by stochastic analysis. The results of simulations and experimental implementations demonstrate the high efficiency of the proposed system in controlling the position of the FJLA under different trajectories. Moreover, the comparative results with the other methods show a great performance for the suggested control system in rejecting disturbances and other uncertainties under saturated control input.

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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Samiei, Mirzaei and Rafatnia. The first draft of the manuscript was written by Samiei; Mirzaei and Rafatnia commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Mehdi Mirzaei.

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Appendix A: Extended-state observer algorithm

Appendix A: Extended-state observer algorithm

The common algorithm of ESO defines the term including the uncertainty as an additional state as \(x_3=f(x)=-\dfrac{G}{I}\sin x_1+\Delta \). Therefore, the augmented state space equations are written as

$$\begin{aligned} \begin{aligned} {\dot{x}}_1&=x_2,\\ {\dot{x}}_2&=x_3+\frac{1}{I}\tau ,\\ {\dot{x}}_3&=\Upsilon (t),\\ y&=x_1. \end{aligned} \end{aligned}$$
(A1)

where \(\Upsilon (t)={\dot{f}}(x)\) is assumed unknown and bounded. In order to estimate the extended states, the ESO is expressed as

$$\begin{aligned} \begin{aligned} \dot{{\hat{x}}}_1&={\hat{x}}_2-\phi _1({\hat{x}}_1-y), \\ \dot{{\hat{x}}}_2&={\hat{x}}_3+\frac{1}{I}\tau -\phi _2({\hat{x}}_1-y),\\ \dot{{\hat{x}}}_3&=-\phi _3({\hat{x}}_1-y). \end{aligned} \end{aligned}$$
(A2)

in which \(\phi _{i\ (i=1,2,3)}\) are dedicated to the observer gains. These gains are properly selected for the ESO to give a good estimate for the additional state \(x_3\) together with of other states of the system. Subtracting (A1) from (A2) leads to the error dynamics for the ESO:

$$\begin{aligned} \begin{aligned} {\dot{e}}_1&=e_2-\phi _1e_1,\\ {\dot{e}}_2&=e_3-\phi _2e_1,\\ {\dot{e}}_3&=-\Upsilon (t)-\phi _3e_1. \end{aligned} \end{aligned}$$
(A3)

in which \(e_i={\hat{x}}_i-x_i \ (i=1,2,3)\) are dedicated to the estimation errors. It is found that the estimation error dynamics defined by Eq. (A3) is linear. By selecting \(\phi _1=3/\epsilon \), \(\phi _2=3/\epsilon ^2\) and \(\phi _3=1/\epsilon ^3\), in which \(\epsilon \) is a free parameter, the error dynamics (A3) will be bounded provided that \(\Upsilon (t)\) is bounded [42].

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Samiei, S.K., Mirzaei, M. & Rafatnia, S. Constrained control of flexible-joint lever arm based on uncertainty estimation with data fusion for correcting measurement errors. Nonlinear Dyn 112, 11147–11166 (2024). https://doi.org/10.1007/s11071-024-09637-1

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