Adaptive soft computing paradigm for a full-car active suspension system with driver biodynamic vibration damping control

Abstract

In the current car automobile industry, vehicle ride comfort is the most important design aspect due to its direct effect on the health and efficiency of human beings. The available literature on vehicle active suspension systems to improve vehicle performance (ride comfort, handling, road holding and suspension deflection, etc.) is somewhat sparse, and there is a major lack of research on seated driver biodynamic comfort analysis and enhancement. In this paper, a nonlinear full-car active suspension system with seated driver biodynamics [19 degrees of freedom (DoF)] model is presented. The effects of biodynamic vibrations on different parts of the driver’s body are examined. A NeuroFuzzy adaptive control paradigm is applied to the full-car active suspension system to damp the vehicle low-frequency vibrations, which can cause health problems and fatigue, resulting in fatal accidents. The effectiveness of the proposed control strategy to damp vehicle low-frequency and biodynamic vibrations is validated by comparing the performance with conventional (passive) and PID-controlled suspension systems.

Appendices

Appendix A

1.1 A.1 State vector

The state vector is defined as follows:

$$\begin{aligned} \begin{array}{c} \left[ y\right] _{38\times 1} \end{array} =\left[ \begin{array}{ccc} y_{1} \ y_{2} \ \ldots \ y_{19} \ y_{20} \ y_{21} \ \ldots \ y_{38} \end{array}\right] ^T_{38\times 1} \end{aligned}$$

(26)

1.2 A.2 State matrices

The nonlinear deferential matrix, input matrix, dry friction matrix, and input excitation are as follows:

$$\begin{aligned} \begin{array}{c} \left[ F_\mathrm{car,1}(y)\right] _{8\times 1} \end{array} =\left[ \begin{array}{ccc} y_{20} \ y_{21} \ y_{22} \ y_{23} \ y_{24} \ y_{25} \ y_{26} \ y_{27} \end{array}\right] ^T_{8\times 1} \end{aligned}$$

(27)

$$\begin{aligned} \begin{array}{c} \left[ F_\mathrm{car,2}(y)\right] _{8\times 1} \end{array} = \left[ \begin{array}{ccc} (1/m_1)[k_{s1}(y_6+a\sin y_7-c\sin y_8-y_1)-k_{t1}(y_1)\\ +c_{s1}(y_{25}+ay_{26}\cos y_7-cy_{27}\cos y_8-y_{20})]\\ (1/m_2)[k_{s2}(y_6+a\sin y_7+d\sin y_8-y_2)-k_{t2}(y_2)\\ +c_{s2}(y_{25}+ay_{26}\cos y_7+dy_{27}\cos y_8-y_{21})]\\ (1/m_3)[k_{s3}(y_6-b\sin y_7-c\sin y_8-y_3)-k_{t3}(y_3)\\ +c_{s3}(y_{25}-by_{26}\cos y_7-cy_{27}\cos y_8-y_{22})]\\ (1/m_4)[k_{s4}(y_6-b\sin y_7+d\sin y_8-y_4)-k_{t4}(y_4)\\ +c_{s4}(y_{25}-by_{26}\cos y_7+dy_{27}\cos y_8-y_{23})]\\ (1/m_5)[k_{s5}(y_6+e\sin y_7+f\sin y_8-y_5)+c_{s5}(y_{25}\\ +ey_{26}\cos y_7+fy_{27}\cos y_8-y_{24})+k_9(y_9-\\ y_5)+c_9(y_{28}-y_{24})]\\ (1/m_6)[k_{s1}(y_1-a\sin y_7+c\sin y_8-y_6)+c_{s1}(y_{20}\\ -ay_{26}\cos y_7+y_{27}\cos y_8-y_{25})+k_{s2}(y_2-a\sin y_7\\ -d\sin y_8-y_6)+c_{s2}(y_{21}-ay_{26}\cos y_7-dy_{27}\cos y_8\\ -y_{25})+k_{s3}(y_3+b\sin y_7+c\sin y_8-y_6)+c_{s3}(y_{22}\\ +by_{26}\cos y_7+cy_{27}\cos y_8-y_{25})+k_{s4}(y_4+b\sin y_7\\ -d\sin y_8-y_6)+c_{s4}(y_{23}+by_{26}\cos y_7-dy_{27}\cos y_8\\ -y_{25})+k_{s5}(y_5-e\sin y_7-f\sin y_8-y_6)+c_{s5}(y_{24}\\ -ey_{26}\cos y_7-fy_{27}\cos y_8-y_{25})]\\ (1/I_{\theta })[[k_{s1}(y_1-a\sin y_7+c\sin y_8-y_6)+c_{s1}(y_{20}\\ -ay_{26}\cos y_7+cy_{27}\cos y_8-y_{25})](a\cos y_7)+[k_{s2}(y_2\\ -a\sin y_7-d\sin y_8-y_6)+c_{s2}(y_{21}-ay_{26}\cos y_7\\ -dy_{27}\cos y_8-y_{25})](a\cos y_7)+[k_{s3}(y_3+b\sin y_7\\ +c\sin y_8-y_6)+c_{s3}(y_{22}+by_{26}\cos y_7+cy_{27}\cos y_8\\ -y_{25})](-b\cos y_7)+[k_{s4}(y_4+b\sin y_7-d\sin y_8-y_6)\\ +c_{s4}(y_{23}+by_{26}\cos y_7-dy_{27}\cos y_8-y_{25})](-b\cos y_7)\\ +[k_{s5}(y_5-e\sin y_7-f\sin y_8-y_6)+c_{s5}(y_{24}-ey_{26}\\ \cos y_7-fy_{27}\cos y_8-y_{25})](e\cos y_7)]\\ (1/I_{\alpha })[[k_{s1}(y_1-a\sin y_7+c\sin y_8-y_6)+c_{s1}(y_{20}\\ -ay_{26}\cos y_7+cy_{27}\cos y_8-y_{25})](-c\cos y_8)+[k_{s2}(y_2\\ -a\sin y_7-d\sin y_8-y_6)+c_{s2}(y_{21}-ay_{26}\cos y_7-dy_{27}\\ \cos y_8-y_{25})](d\cos y_8)+[k_{s3}(y_3+b\sin y_7+c\sin y_8\\ -y_6)+c_{s3}(y_{22}+by_{26}\cos y_7+cy_{27}\cos y_8-y_{25})]\\ (-c\cos y_8)+[k_{s4}(y_4+b\sin y_7-d\sin y_8-y_6)+c_{s4}(y_{23}\\ +by_{26}\cos y_7-dy_{27}\cos y_8-y_{25})](d\cos y_8)+[k_{s5}(y_5\\ -e\sin y_7-f\sin y_8-y_6)+c_{s5}(y_{24}-ey_{26}\cos y_7\\ -fy_{27}\cos y_8-y_{25})](f\cos y_8)] \end{array}\right] _{8\times 1} \end{aligned}$$

(28)

$$\begin{aligned} \begin{array}{c} \left[ F_\mathrm{driver,3}(y)\right] _{11\times 1} \end{array} =\left[ \begin{array}{ccc} y_{28} \ y_{29} \ y_{30} \ y_{31} \ y_{32} \ y_{33} \ y_{34} \ y_{35} \ y_{36} \ y_{37} \ y_{38} \end{array}\right] ^T_{11\times 1} \end{aligned}$$

(29)

$$\begin{aligned} \begin{array}{c} \left[ F_\mathrm{driver,4}(y)\right] _{11\times 1} \end{array} =\left[ \begin{array}{ccc} (1/m_9)[k_{9}(y_5-y_9)+k_{10}(y_{10}-y_9)+k_{16}(y_{16}-y_9)+c_{9}\\ (y_{24}-y_{28})+c_{10}(y_{29}-y_{28})+c_{16}(y_{35}-y_{28})]\\ (1/m_{10})[k_{10}(y_9-y_{10})+k_{11}(y_{11}-y_{10})+c_{10}(y_{28}-y_{29})\\ +c_{11}(y_{30}-y_{29})]\\ (1/m_{11})[k_{11}(y_{10}-y_{11})+k_{12}(y_{12}-y_{11})+c_{11}(y_{29}-y_{30})\\ +c_{12}(y_{31}-y_{30})]\\ (1/m_{12})[k_{12}(y_{11}-y_{12})+k_{13}(y_{13}-y_{12})+c_{12}(y_{30}-y_{31})\\ +c_{13}(y_{32}-y_{31})]\\ (1/m_{13})[k_{13,12}(y_{12}-y_{13})+k_{13,17}(y_{17}-y_{13})+k_{14}(y_{14}-\\ y_{13})+c_{13,12}(y_{31}-y_{32})+c_{13,17}(y_{36}-y_{32})+c_{14}(y_{33}\\ -y_{32})]\\ (1/m_{14})[k_{14}(y_{13}-y_{14})+k_{15}(y_{15}-y_{14})+c_{14}(y_{32}-y_{33})\\ +c_{15}(y_{34}-y_{33})]\\ (1/m_{15})[k_{15}(y_{14}-y_{15})+c_{15}(y_{33}-y_{34})]\\ (1/m_{16})[k_{16}(y_{9}-y_{16})+k_{17}(y_{17}-y_{16})+c_{16}(y_{28}-y_{35})\\ +c_{17}(y_{36}-y_{35})]\\ (1/m_{17})[k_{17}(y_{16}-y_{17})+k_{13,17}(y_{13}-y_{17})+k_{18}(y_{18}-\\ y_{17})+c_{17}(y_{31}-y_{36})+c_{13,17}(y_{32}-y_{36})+c_{18}(y_{37}-y_{36})]\\ (1/m_{18})[k_{18}(y_{17}-y_{18})+k_{19}(y_{19}-y_{18})+c_{18}(y_{36}-y_{37})\\ +c_{19}(y_{38}-y_{37})]\\ (1/m_{19})[k_{19}(y_{18}-y_{19})+c_{19}(y_{37}-y_{38})] \end{array}\right] _{11\times 1} \end{aligned}$$

(30)

$$\begin{aligned} \begin{array}{c} \left[ B_\mathrm{car}\right] _{16\times 4} \end{array} =\left[ \begin{array}{ccccc} &{} &{}\,\quad[0]_{8\times 4}&{} &{}\\ 1/m_1 &{}\,\quad 0 &{}\,\quad 0 &{}\,\quad 0\\ 0 &{}\,\quad 1/m_2 &{}\,\quad 0 &{}\,\quad 0\\ 0 &{}\,\quad 0 &{}\,\quad 1/m_3 &{}\,\quad 0\\ 0 &{}\,\quad 0 &{}\,\quad 0 &{}\,\quad 1/m_4\\ 0 &{}\,\quad 0 &{}\,\quad 0 &{}\,\quad 0\\ 1/m_6 &{}\,\quad 1/m_6 &{}\,\quad 1/m_6 &{}\,\quad 1/m_6\\ (a/I_{\theta })\cos y_7 &{}\,\quad (a/I_{\theta })\cos y_7 &{}\,\quad (-b/I_{\theta })\cos y_7 &{}\,\quad (-b/I_{\theta })\cos y_7\\ (-c/I_{\alpha })\cos y_8 &{}\,\quad (d/I_{\alpha })\cos y_8 &{}\,\quad (-c/I_{\alpha })\cos y_8 &{}\,\quad (d/I_{\alpha })\cos y_8 \end{array}\right] _{16\times 4} \end{aligned}$$

(31)

$$\begin{aligned} \begin{array}{c} \left[ B_\mathrm{driver}\right] _{22\times 4} \end{array} =\left[ \begin{array}{c} 0 \end{array}\right] _{22\times 4} \end{aligned}$$

(32)

$$\begin{aligned} \begin{array}{c} \left[ Q_\mathrm{car}\right] _{16\times 4} \end{array} =\left[ \begin{array}{ccccc} &{} &{}\, \quad [0]_{8\times 4}&{} &{}\\ 1/m_1 &{}\, \quad 0 &{}\, \quad 0 &{}\, \quad 0\\ 0 &{}\, \quad 1/m_2 &{}\, \quad 0 &{}\, \quad 0\\ 0 &{}\, \quad 0 &{}\, \quad 1/m_3 &{}\, \quad 0\\ 0 &{}\, \quad 0 &{}\, \quad 0 &{}\, \quad 1/m_4\\ 0 &{}\, \quad 0 &{}\, \quad 0 &{}\, \quad 0\\ -1/m_6 &{}\, \quad -1/m_6 &{}\, \quad -1/m_6 &{}\, \quad -1/m_6\\ (-a/I_{\theta })\cos y_7 &{}\, \quad (-a/I_{\theta })\cos y_7 &{}\, \quad (b/I_{\theta })\cos y_7 &{}\, \quad (b/I_{\theta })\cos y_7\\ (c/I_{\alpha })\cos y_8 &{}\, \quad (-d/I_{\alpha })\cos y_8 &{}\, \quad (c/I_{\alpha })\cos y_8 &{}\, \quad (-d/I_{\alpha })\cos y_8 \end{array}\right] _{16\times 4} \end{aligned}$$

(33)

$$\begin{aligned} \begin{array}{c} \left[ Q_\mathrm{driver}\right] _{22\times 4} \end{array} =\left[ \begin{array}{c} 0 \end{array}\right] _{22\times 4} \end{aligned}$$

(34)

$$\begin{aligned} \begin{array}{c} \left[ G_\mathrm{car}\right] _{16\times 4} \end{array} =\left[ \begin{array}{ccccc} &{} &{}\,\quad[0]_{8\times 4}&{} &{}\\ k_{t1} &{}\,\quad 0 &{}\,\quad 0 &{}\,\quad 0\\ 0 &{}\,\quad k_{t2} &{}\,\quad 0 &{}\,\quad 0\\ 0 &{}\,\quad 0 &{}\,\quad k_{t3} &{}\,\quad 0\\ 0 &{}\,\quad 0 &{}\,\quad 0 &{}\,\quad k_{t4}\\ &{} &{}\,\quad[0]_{4\times 4}&{} &{} \end{array}\right] _{16\times 4} \end{aligned}$$

(35)

$$\begin{aligned} \begin{array}{c} \left[ G_\mathrm{driver}\right] _{22\times 4} \end{array} =\left[ \begin{array}{c} 0 \end{array}\right] _{22\times 4} \end{aligned}$$

(36)

1.3 A.3 Vehicle parameters

$$\begin{array}{ll} m_6=1100\, \mathrm{kg}, I_\theta =1848 \,\mathrm{kg\,m^2}, I_\alpha =550\, \mathrm{kg \, m^2}, \\ m_1=m_2=25\, \mathrm{kg}, m_3=m_4=45 \,\mathrm{kg}, m_5=20 \,\mathrm{kg}\\ k_{s1}=k_{s2}=15000\,\mathrm{N/m}, k_{s3}=k_{s4}=17000\,\mathrm{N/m},\\ k_{s5}=15000\,\mathrm{N/m}\\ c_{s1}=c_{s2}=c_{s3}=c_{s4}=2500\,\mathrm{N\,s/m}, c_{s5}=150\,\mathrm{N\,s/m}\\ k_{t1}=k_{t2}=k_{t3}=k_{t4}=250000\,\mathrm{N/m}\\ a=1.2\,\mathrm{m},b=1.4\,\mathrm{m},c=0.5\,\mathrm{m},d=1.0\,\mathrm{m},e=0.3\,\mathrm{m}, \\ f=0.25\,\mathrm{m} \end{array}$$

1.4 A.4 Driver model parameters

$$\begin{array}{ll} m_9=27.23 \, \mathrm{kg},\,m_{10}=5.906 \, \mathrm{kg}, \,m_{11}=0.454 \, \mathrm{kg}, \\ m_{12}=1.362 \, \mathrm{kg}, \,m_{13}=32.697 \, \mathrm{kg}, \,m_{14}=5.470 \, \mathrm{kg}, \\ m_{15}=5.297 \, \mathrm{kg}, \,m_{16}=2.002 \, \mathrm{kg}, \, m_{17}=4.806 \, \mathrm{kg}, \\ m_{18}=1.084 \, \mathrm{kg}, \, m_{19}=5.445 \, \mathrm{kg}\\ k_{9}=25016 \,\mathrm{N/m}, k_{10}=k_{11}=k_{12}=k_{1317}=877 \,\mathrm{N/m}, \\ k_{1312}=52621 \,\mathrm{N/m}, \, k_{14}=k_{15}=67542 \,\mathrm{N/m},\\ k_{16}=k_{17}=k_{18}=k_{19}=52621 \,\mathrm{N/m}\\ c_{9}=370.8 \,\mathrm{N\,s/m},c_{10}=c_{11}=c_{12}=292.3 \,\mathrm{N\,s/m}, \\ c_{1317}=3581.6 \,\mathrm{N\,s/m} c_{1312}=292.3 \,\mathrm{N\,s/m},\\ c_{14}=c_{15}=c_{16}=c_{17}=c_{18}=c_{19}=3581.6 \,\mathrm{N\,s/m}\\ \end{array}$$

1.5 A.5 Dry friction Parameters

$$\begin{aligned} R=22N, \epsilon =0.0012m/s, n=3 \end{aligned}$$

1.6 A.6 Car chassis corners vertical displacement

$$\begin{aligned} x_1=y_6+a\sin y_8-c\sin y_9 \\ x_2=y_6+a\sin y_8+d\sin y_9 \\ x_3=y_6-b\sin y_8-c\sin y_9 \\ x_4=y_6-b\sin y_8+d\sin y_9 \\ \end{aligned}$$

Appendix B

1.1 B.1 Vertical displacement and ITAE comparison

See Figs. 13, 14, 15, and 16.

Fig. 13

Vertical displacements (without seat control)

Fig. 14

ITAE vertical displacements (without seat control)

Fig. 15

Vertical displacements (with seat control)

Fig. 16

ITAE vertical displacements (with seat control)

1.2 B.2 Vertical weighted RMS acceleration and ITAE comparison

See Figs. 17, 18, 19, and 20.

Fig. 17

Vertical weighted RMS acceleration (without seat control)

Fig. 18

ITAE vertical weighted RMS acceleration (without seat control)

Fig. 19

Vertical weighted RMS acceleration (with seat control)

Fig. 20

ITAE vertical weighted RMS acceleration (with seat control)

1.3 B.3 Suspension travel

See Figs. 21 and 22.

Fig. 21

Suspension travel (without seat control) a front left, b rear right, c rear left

Fig. 22

Suspension travel (with seat control) a front left, b rear right, c rear left

1.4 B.4 Tire deflection

See Figs. 23 and 24.

Fig. 23

Wheels deflection (without seat control) a front left, b rear right, c rear left

Fig. 24

Wheels deflection (with seat control) a front left, b rear right, c rear left

1.5 B.5 RMS vertical acceleration versus frequency

See Figs. 25 and 26.

Fig. 25

RMS acceleration versus frequency (without seat control)

Fig. 26

RMS acceleration versus frequency (with seat control)

1.6 B.6 Tables

See Tables 2, 3, and 4.

Table 2 Performance comparison of the ITAE vertical displacement of body organs

Table 3 Performance Comparison of the weighted rms acceleration of body organs

Table 4 Performance Comparison of the ITAE weighted rms acceleration of body organs