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Precise motion control of tractor-trailer wheeled mobile structures via a newly observed key motion law

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Abstract

We present a curvature tracking approach for a tractor-trailer wheeled mobile structure (TTWMS), such that the trailer can track a desired trajectory curve accurately. A key motion law related to the curvature functions of trajectory curves is discovered for the first time, in terms of the tractor and the trailer with nonholonomic constraints. Then, based on this key motion law, the target trajectory curve of the trailer is converted to a dynamical tracking target matching the dynamics equation of the TTWMS, so as to transform the original motion task into a common tracking control problem. Finally, LQR optimal control and integral sliding mode control are introduced to design a robust tracking controller for the TTWMS. Simulation results show that the proposed method can make the TTWMS follow a given target trajectory curve accurately, even if the forward and yaw rotational speed errors are divergent.

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Acknowledgements

The authors would like to thank the Associate Editor and all anonymous reviewers for their valuable comments and suggestions which have improved the quality of this paper. This work has received the financial support of NSF of China under Grant (11802065, 11702227), the Science and Technology Program of Guizhou Province ([2018]1047), the fund Project of Key Laboratory of Advanced Manufacturing Technology, Ministry of Education, Guizhou University (KY[2018]478).

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  1. School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, China

    Yusheng Zhou & Xiangrong Wen

  2. School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, 610031, China

    Qi Xu

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  1. Yusheng Zhou
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Correspondence to Yusheng Zhou.

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Appendix

Appendix

1.1 Appendix A: Coefficient matrixes of Eq.  (13)

$$\begin{aligned}&{\mathbf {A}}(\mathbf{q })=\left[ \begin{array}{cccccc} a_{11} &{} 0 &{} a_{13} &{} a_{14} &{} 0 &{} 0 \\ 0 &{} a_{22} &{} a_{23} &{} a_{24} &{} 0 &{} 0 \\ a_{31} &{} a_{32} &{} a_{33} &{} a_{34} &{} 0 &{} 0 \\ a_{41} &{} a_{42} &{} a_{43} &{} a_{44} &{} 0&{} 0\\ 0 &{} 0 &{} 0 &{} 0&{} a_{55} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0&{} 0 &{} a_{66} \\ \end{array} \right] ,\\&\quad {\mathbf {U}}(\text {q},\dot{\text {q}})=\left[ \begin{array}{c} u_{11}\\ u_{21}\\ 0\\ u_{41}\\ 0\\ 0\\ \end{array} \right] , \end{aligned}$$

where

$$\begin{aligned} a_{11}&=a_{22}=M_f+M_B+(2M_\mathrm{w}+\frac{2I_\mathrm{w}}{r^2}+M_r)\mathrm{cos}^2\varphi ,\\ a_{13}&=a_{31}=-a_{14}=a_{41}=\frac{1}{2}M_B a\mathrm{sin}(\theta -\varphi ),\\ a_{23}&=a_{32}=-a_{24}=a_{42}=-\frac{1}{2}M_B a\mathrm{cos}(\theta -\varphi ),\\ a_{33}&=4I_{wd}+I_{fd}+I_{rd}+I_B+\frac{a^2}{4}M_B+\frac{d^2}{2}(M_\mathrm{w}+\frac{I_\mathrm{w}}{r^2}),\\ a_{34}&=-a_{43}=a_{44}=-I_{rd}-I_B-2I_{wd}\\&\quad -\, \frac{a^2}{4} M_B-\frac{d^2}{2}(M_\mathrm{w}+\frac{I_\mathrm{w}}{r^2}),\\ a_{55}&=a_{66}=M_\mathrm{w} r^2+I_\mathrm{w},\\ u_{11}&=-\frac{4 av{\dot{\varphi }}\mathrm{cos}^2\varphi \mathrm{sin}\varphi (M_r r^2+ 2 M_\mathrm{w} r^2 + 2 I_\mathrm{w})}{2 a r^2}\\&\quad -\, \frac{ M_B v^2 r^2 \mathrm{cos}(\theta - \varphi )\mathrm{sin}^{2}\varphi }{2 a r^2},\\ u_{21}&=-\frac{4av{\dot{\varphi }}\mathrm{sin}^2\varphi \mathrm{cos}\varphi ( M_r r^2+ 2 M_\mathrm{w} r^2+ 2 I_\mathrm{w})}{2 a r^2}\\&\quad -\, \frac{ M_B v^2 r^2 \mathrm{sin}^2\varphi \mathrm{sin}(\theta - \varphi )}{2 a r^2},\\ u_{41}&=-(2M_\mathrm{w}+\frac{2I_\mathrm{w}}{r^2}+M_r)v^2\mathrm{sin}\varphi \mathrm{cos}\varphi . \end{aligned}$$

1.2 Appendix B: Related expressions of Eq. (15)

$$\begin{aligned} \varDelta _1&=\alpha _4\mathrm{cos}^{2} \varphi +\alpha _5,\\ \varDelta _2&=\alpha _1 v \dot{\theta } \mathrm{sin}\varphi \mathrm{cos}\varphi \\&\quad +\, \alpha _2 \dot{\varphi } v \mathrm{sin}\varphi \mathrm{cos}\varphi +\alpha _3 v^2 \mathrm{sin}^2\varphi \mathrm{cos}\varphi ,\\ \varDelta _3&= \beta _{10}\mathrm{cos}^2\varphi +\beta _{11},\\ \varDelta _4&=\beta _1v \dot{\theta }\mathrm{cos}^3\varphi +\beta _2v\dot{\theta }\mathrm{cos}\varphi \\&\quad +\, \beta _3\dot{\varphi } v\mathrm{cos}^3\varphi +\beta _4\dot{\varphi }v\mathrm{cos}\varphi \\&\quad +\, \beta _5v^2\mathrm{sin}\varphi \mathrm{cos}^3\varphi +\beta _6v^2\mathrm{sin}\varphi \mathrm{cos}\varphi +\beta _9\mathrm{sin}\varphi u_2, \\ \varDelta _5&= \beta _7\mathrm{cos}^2\varphi +\beta _8.\\ \end{aligned}$$

where

$$\begin{aligned} \alpha _1&= 2a^2(-M_Br^2+4M_r r^2+8M_\mathrm{w} r^2+8 I_\mathrm{w}),\\ \alpha _2&= (3M_B a^2\!+\!2 M_\mathrm{w} d^2\!+\!4 I_B\!+\!4 I_{rd}\!+\!8 I_{wd})r^2\!+\!2 I_\mathrm{w} d^2,\\ \alpha _3&= -2a(M_Br^2+6 M_r r^2+12 M_\mathrm{w} r^2+12 I_\mathrm{w}),\\ \alpha _4&= M_Ba^2 r^2\!+\!4 M_r a^2 r^2\!+\!8 M_\mathrm{w} a^2 r^2\!\\&\quad +\, \!2 M_\mathrm{w} d^2 r^2\!+\!4 I_B r^2 +4 I_{rd} r^2+8 I_\mathrm{w} a^2+2 I_\mathrm{w} d^2+8 I_{wd}r^2, \\ \alpha _5&= 3M_Ba^2r^2+4M_fa^2r^2+8M_\mathrm{w} a^2 r^2-2 M_\mathrm{w} d^2 r^2\\&\quad -\, 4 I_B r^2-4 I_{rd} r^2+8 I_\mathrm{w} a^2-2 I_\mathrm{w} d^2-8 I_{wd} r^2,\\ \beta _1&= a^2 ((4 M_B^2 a^2+16 M_r M_\mathrm{w} d^2+32 M_\mathrm{w}^2 d^2\\&\quad +\, 32 I_B M_r +32 I_{rd} M_r+64 I_B M_\mathrm{w}+64 I_{rd} M_\mathrm{w}+64 I_{wd} M_r \\&\quad +\, 128 I_{wd} M_\mathrm{w}) r^4+(16 I_\mathrm{w} M_r d^2+64 I_\mathrm{w} M_\mathrm{w} d^2+64 I_B I_\mathrm{w}\\&\quad +\, 64 I_{rd} I_\mathrm{w}+128 I_\mathrm{w} I_{wd})r^2+32 I_\mathrm{w}^2 d^2),\\ \beta _2&= (4M_B^2+8M_B M_f+8 M_B M_r+32 M_B M_\mathrm{w}) r^4 a^4 \\&\quad -\, (16 M_r M_\mathrm{w} d^2+32 M_\mathrm{w}^2 d^2+32 I_B M_r+64 I_B M_\mathrm{w} \\&\quad +\, 32 I_{rd} M_r+64 I_{rd} M_\mathrm{w}+64 I_{wd} M_r+128 I_{wd} M_\mathrm{w}) r^4 a^2\\&\quad -\, (I_\mathrm{w} M_r d^2 -32 I_\mathrm{w} M_B a^216+64 I_\mathrm{w} M_\mathrm{w} d^2\\&\quad +\, 64 I_B I_\mathrm{w} +64 I_{rd} I_\mathrm{w}+128 I_\mathrm{w} I_{wd}) a^2 r^2-32 I_\mathrm{w}^2 a^2 d^2,\\ \beta _3&= -(4 M_B^2+4 M_B M_r+8 M_B M_\mathrm{w}) r^4 a^4-8 a^4 I_\mathrm{w} r^2 M_B \\&\quad -\, (8 M_r M_\mathrm{w} d^2+16 M_\mathrm{w}^2 d^2+16 I_B M_r+32 I_B M_\mathrm{w}\\&\quad +\, 16 I_{rd} M_r+32 I_{rd} M_\mathrm{w}+32 I_{wd} M_r+64 I_{wd} M_\mathrm{w}) r^4 a^2\\&\quad -\, (8 I_\mathrm{w} M_r d^2+32 I_\mathrm{w} M_\mathrm{w} d^2+32 I_B I_\mathrm{w}+32 I_{rd} I_\mathrm{w}\\&\quad +\, 64 I_\mathrm{w} I_{wd}) r^2 a^2-16 I_\mathrm{w}^2 a^2 d^2,\\ \beta _4&= -\!(4 M_B M_f\!+\!8 M_B M_\mathrm{w}) r^4 a^4\!\\&\quad -\, \!8 a^4 I_\mathrm{w} r^2 M_B\!-\!(8 M_B M_\mathrm{w} d^2 +\!8 M_f M_\mathrm{w} d^2\!\\&\quad +\, \!16 M_\mathrm{w}^2 d^2\!+\!16 I_B M_B\!+\!16 I_B M_f\!+\!32 I_B M_\mathrm{w} +\!16 I_{rd} M_B\!\\&\quad +\, \!16 I_{rd} M_f\!+\!32 I_{rd} M_\mathrm{w}\!+\!32 I_{wd} M_B\!+\!32 I_{wd} M_f \\&\quad +\, \!64 I_{wd} M_\mathrm{w}) r^4 a^2\!-\!(8 I_\mathrm{w} M_B d^2\!+\!8 I_\mathrm{w} M_f d^2\!\\&\quad +\, \!32 I_\mathrm{w} M_\mathrm{w} d^2 +32 I_B I_\mathrm{w}+32 I_{rd} I_\mathrm{w}\\&\quad + 64 I_\mathrm{w} I_{wd}) r^2 a^2-16 I_\mathrm{w}^2 a^2 d^2,\\ \beta _5&= (2M_B^2+14M_B M_r+28 M_B M_\mathrm{w}+8 M_r^2+32 M_r M_\mathrm{w} \\&\quad +\, 32 M_\mathrm{w}^2) r^4 a^3+(28 I_\mathrm{w} M_B+32 I_\mathrm{w} M_r+64 I_\mathrm{w} M_\mathrm{w}) r^2 a^3 \\&\quad +\, \!32 I_\mathrm{w}^2 a^3\!-\!(4 M_B M_\mathrm{w} d^2\!+\!20 M_r M_\mathrm{w} d^2\!\\&\quad +\, \!40 M_\mathrm{w}^2 d^2\!+\!8 I_B M_B \\&\quad +\, 40 I_B M_r+80 I_B M_\mathrm{w}+8 I_{rd} M_B+40 I_{rd} M_r+80 I_{rd} M_\mathrm{w} \\&\quad +\, 16 I_{wd} M_B+80 I_{wd} M_r+160 I_{wd} M_\mathrm{w}) r^4 a-(4 I_\mathrm{w} M_B d^2 \\&\quad +\, 20 I_\mathrm{w} M_r d^2+80 I_\mathrm{w} M_\mathrm{w} d^2+80 I_B I_\mathrm{w}+80 I_{rd} I_\mathrm{w}\\&\quad +\, 160 I_\mathrm{w} I_{wd}) r^2 a-40 I_\mathrm{w}^2 a d^2,\\ \beta _6&=-(2M_B^2+6 M_B M_r+12 M_B M_\mathrm{w}\\&\quad -\,8 M_f M_r-16 M_f M_\mathrm{w} -16 M_r M_\mathrm{w}\! -\, \!32 M_\mathrm{w}^2) r^4 a^3\\&\quad -(12 I_\mathrm{w} M_B-16 I_\mathrm{w} M_f-16 I_\mathrm{w} M_r \\&\quad -\, 64 I_\mathrm{w} M_\mathrm{w}) r^2 a^3+32 I_\mathrm{w}^2 a^3+40 I_\mathrm{w}^2 a d^2+(4 M_B M_\mathrm{w} d^2\\&\quad -\, 20 M_r M_\mathrm{w} d^2\!+\!40 M_\mathrm{w}^2 d^2\!\\&\quad +\, \!8 I_B M_B\!-\!40 I_B M_r\!+\!80 I_B M_\mathrm{w} +8 I_{rd} M_B\!\\&\quad +\, \!40 I_{rd} M_r\!+\!80 I_{rd} M_\mathrm{w}\!+\!16 I_{wd} M_B\!+\!80I_{wd} M_r \\&\quad +\, 160 I_{wd} M_\mathrm{w}) r^4 a+(4 I_\mathrm{w} M_B d^2-20 I_\mathrm{w} M_r d^2\\&\quad +\, 80 I_\mathrm{w} M_\mathrm{w} d^2-80 I_B I_\mathrm{w}+80 I_{rd} I_\mathrm{w}+160 I_\mathrm{w} I_{wd}) r^2 a,\\ \beta _7&= (M_B a^3 d+4 M_r a^3 d+8 M_\mathrm{w} a^3 d+2 M_\mathrm{w} a d^3+4 I_B a d\\&\quad +4 I_{rd} a d+8 I_{wd} a d) r^3 +\, (8 I_\mathrm{w} a^3 d+2 I_\mathrm{w} a d^3)r,\\ \beta _8&= (3M_B a^3 d+4 M_f a^3 d+8 M_\mathrm{w} a^3 d-2 M_\mathrm{w} a d^3-4 I_B a d\\&\quad -\, 4 I_{rd} a d-8 I_{wd} a d) r^3+(8 I_\mathrm{w} a^3 d-2 I_\mathrm{w} a d^3) r,\\ \beta _9&= (4M_B a^4\!-\!8 M_\mathrm{w} a^2 d^2\!-\!16 I_B a^2\!-\!16 I_{rd} a^2\!-\!32 I_{wd} a^2) r^3\\&\quad -\, 8 I_\mathrm{w} a^2 d^2 r,\\ \beta _{10}&= (M_\mathrm{w} a d^2 r^2+2 I_{fd} a r^2+I_\mathrm{w} a d^2+4 I_{wd} a r^2) (M_B a^2 r^2\\&\quad +\, 4 M_r a^2 r^2+8 M_\mathrm{w} a^2 r^2+2 M_\mathrm{w} d^2 r^2+4 I_B r^2+4 I_{rd} r^2\\&\quad +\, 8 I_\mathrm{w} a^2+2 I_\mathrm{w} d^2+8 I_{wd} r^2),\\ \beta _{11}&= (M_\mathrm{w} a d^2 r^2+2 I_{fd} a r^2+I_\mathrm{w} a d^2+4 I_{wd} a r^2) (3 M_B a^2 r^2\\&\quad +\, 4 M_f a^2 r^2+8 M_\mathrm{w} a^2 r^2-2 M_\mathrm{w} d^2 r^2-4 I_B r^2-4 I_{rd} r^2\\&\quad +\, 8 I_\mathrm{w} a^2-2 I_\mathrm{w} d^2-8 I_{wd} r^2). \end{aligned}$$

1.3 Appendix C: Another proof of Eq. (24)

As can be seen in Fig. 1, the positional relationship of point Q and P can be expressed as

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} x=x_0+a \mathrm{cos}\theta _0, \\ y=y_0+a \mathrm{sin}\theta _0. \end{array}\right. } \end{aligned} \end{aligned}$$
(45)

Taking the first and second derivatives of Eq. (45) with respect to t, respectively, one has

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} {\dot{x}}={\dot{x}}_0-a{\dot{\theta }}_0\mathrm{sin}\theta _0, \\ {\dot{y}}={\dot{y}}_0+a{\dot{\theta }}_0 \mathrm{cos}\theta _0, \end{array}\right. } \end{aligned} \end{aligned}$$
(46)

and

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} \ddot{x}=\ddot{x}_0-a\ddot{\theta }_0 \mathrm{sin}\theta _0-a{\dot{\theta }}_0^2 \mathrm{cos}\theta _0, \\ \ddot{y}=\ddot{y}_0+a\ddot{\theta }_0 \mathrm{cos}\theta _0-a {\dot{\theta }}_0^2 \mathrm{sin}\theta _0. \end{array}\right. } \end{aligned} \end{aligned}$$
(47)

By substituting Eqs. (46) and (47) into Eq. (18), the forward speed and yaw rotation speed of point P can be given by

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} {\dot{v}}=\sqrt{{\dot{x}}_0^2+{\dot{y}}_0^2+a^2 {\dot{\theta }}_0^2},\\ {\dot{\theta }}= \frac{a \ddot{\theta }_0({\dot{x}}_0 \mathrm{cos}\theta _0+{\dot{y}}_0 \mathrm{sin}\theta _0)-a {\dot{\theta }}_0 (\ddot{x}_0 \mathrm{cos}\theta _0+\ddot{y}_0 \mathrm{sin} \theta _0)}{{\dot{x}}_0^2+{\dot{y}}_0^2+a^2 {\dot{\theta }}_0^2}\\ \qquad +\, \frac{a^2 {\dot{\theta }}_0^3+{\dot{x}}_0 \ddot{y}_0-\ddot{x}_0 {\dot{y}}_0}{{\dot{x}}_0^2+{\dot{y}}_0^2+a^2 {\dot{\theta }}_0^2} \end{array}\right. } \end{aligned} \end{aligned}$$
(48)

Similar to Eq. (4) of the tractor, and together with (18), the trailer also satisfies the following equations, described by

$$\begin{aligned} {\left\{ \begin{array}{ll} -{\dot{x}}_0\mathrm{sin}\theta _0+ {\dot{y}}_0\mathrm{cos}\theta _0 = 0, \\ v_0=\sqrt{{\dot{x}}_{0}^2+{\dot{y}}_{0}^2}={\dot{x}}_0\mathrm{cos}\theta _0+{\dot{y}}_0\mathrm{sin}\theta _0, \\ {\dot{\theta }}_0=\frac{{\dot{x}}_0\ddot{y}_0-\ddot{x}_0{\dot{y}}_0}{{\dot{x}}_{0}^2+{\dot{y}}_{0}^2}. \end{array}\right. } \end{aligned}$$
(49)

By substituting Eq. (49) into Eq. (48) results

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} v = \sqrt{v_0^2+a^2{\dot{\theta }}_0^2}, \\ {\dot{\theta }}=\frac{a \ddot{\theta }_0 v_0-a {\dot{\theta }}_0 {\dot{v}}_0+a^2 {\dot{\theta }}_0^3+{\dot{\theta }}_0 v_0^2}{v_0^2+a^2{\dot{\theta }}_0^2}, \end{array}\right. } \end{aligned} \end{aligned}$$

which is exactly the same as Eq. (24).

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Zhou, Y., Wen, X. & Xu, Q. Precise motion control of tractor-trailer wheeled mobile structures via a newly observed key motion law. Nonlinear Dyn 103, 833–848 (2021). https://doi.org/10.1007/s11071-020-06162-9

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