Stability Control of an Autonomous Vehicle in Overtaking Manoeuvre Using Wheel Slip Control

Abstract

Advanced driver assistance systems (ADAS) has been introduced to address driver-related accidents. One of the advantages of ADASs is that they can provide autonomous control and tracking in overtaking manoeuvres via GPS through the designed trajectory. In this study, using adaptive sliding mode control, an integrated longitudinal and lateral control of 4-DOF vehicle’s nonlinear dynamic model, in presence of uncertainties, has been proposed. Adaptive control law is utilized for switching gain based on the variations in the sliding surface. Furthermore, a sliding mode control is designed in order to control the longitudinal slip of front wheels. Simulation results show proper tracking for dry roads and acceptable tracking in low adherence roads (wet roads) in overtaking manoeuvres.

Appendices

Appendix 1

Table 1 The vehicle parameters in this study

Appendix 2

In order to prove the stability of the sliding mode control, the coefficient of switching surface can be obtained as follows [22]:

$$ {\displaystyle \begin{array}{c}{k}_x\ge {\varGamma}_x\left({\overline{F}}_x+{\eta}_x+\left({\varGamma}_x-1\right)\right)\left|{\hat{F}}_x\right|\\ {}{k}_y\ge {\varGamma}_y\left({\overline{F}}_y+{\eta}_y\right)+\left({\varGamma}_y-1\right)\left|\hat{\delta}\right|\end{array}} $$

where,

$$ {\displaystyle \begin{array}{l}{\overline{F}}_x={\overline{F}}_1\\ {}{\overline{F}}_y={\overline{F}}_2+{\overline{F}}_1{e}_2+{\overline{G}}_1{F}_x{e}_2+d{\overline{F}}_3\end{array}} $$

And the upper and lower bounds of uncertainties for controller design are:

$$ {\displaystyle \begin{array}{l}{\left.{F}_i^{+}={F}_i\right|}_{\mu ={\mu}_{\mathrm{max}}}\\ {}{\left.{F}_i^{-}={F}_i\right|}_{\mu ={\mu}_{\mathrm{min}}}\\ {}\begin{array}{c}{\left.{G}_i^{+}={G}_i\right|}_{\mu ={\mu}_{\mathrm{max}}}\\ {}{\left.{G}_i^{-}={G}_i\right|}_{\mu ={\mu}_{\mathrm{min}}}\\ {}\mathrm{I}=1,2,3,4\end{array}\end{array}}\kern0.5em {\displaystyle \begin{array}{c}{\hat{F}}_i=\frac{F_i^{+}+{F}_i^{-}}{2}\\ {}{\hat{G}}_i=\sqrt{G_i^{+}{G}_i^{-}}\\ {}\begin{array}{c}{\overline{G}}_i={\hat{G}}_i-{G}_i^{-}\\ {}{\overline{F}}_i={\hat{F}}_i-{F}_i^{-}\end{array}\end{array}}\kern0.5em {\displaystyle \begin{array}{c}\begin{array}{c}{\varGamma}_x=\sqrt{\frac{b_x^{+}}{b_x^{-}}}=\sqrt{\frac{\mu_{\mathrm{max}}}{\mu_{\mathrm{min}}}}\\ {}{\varGamma}_y=\sqrt{\frac{b_y^{+}}{b_y^{-}}}=\sqrt{\frac{\mu_{\mathrm{max}}}{\mu_{\mathrm{min}}}}\end{array}\\ {}{\varGamma}_s=\sqrt{\frac{b_s^{+}}{b_s^{-}}}=1\end{array}} $$

Appendix 3

In order to prove the stability of the proposed controller, the candidate Lyapunov function has been chosen as:

$$ {V}_x=\frac{1}{2}{S}_x^2+\frac{b{\hat{b}}_x^{-1}}{2}{\overset{\sim }{B}}_x^2 $$

where,

$$ {\displaystyle \begin{array}{l}{\overset{\sim }{B}}_x={B}_1-{k}_x\\ {}\;{\hat{b}}_x={\hat{G}}_1\end{array}} $$

In order to evaluate the Lyapunov function, the Lyapunov candidate function has been derived as:

$$ {\displaystyle \begin{array}{l}{\dot{V}}_x={S}_x\left\{{F}_1+{G}_1{F}_x-{\ddot{x}}_{ref}+{\lambda}_1\dot{x}-{\lambda}_1{\dot{x}}_{ref}\right\}+{b}_x{\hat{b}}_x^{-1}\left({B}_1-{k}_x\right){\gamma}_x\left|{S}_x(t)\right|\\ {}{\dot{V}}_x={S}_x\left\{{F}_1-{\ddot{x}}_{ref}+\lambda \dot{x}-{\lambda}_1{\dot{x}}_{ref}+{b}_x{b}_x^{-1}\right(-{\hat{F}}_1+{\ddot{x}}_{ref}-{\lambda}_1\dot{x}+{\lambda}_1{\dot{x}}_{ref}\\ {}-{B}_1{\gamma}_x tgh\left({S}_x/{\phi}_x\right)\left)\right\}+{b}_x{\hat{b}}_x^{-1}\left({B}_1-{k}_x\right){\gamma}_x\left|{S}_x(t)\right|\end{array}} $$

According to Appendix 2,

$$ {\displaystyle \begin{array}{l}{\dot{V}}_x\le \left|{\overline{F}}_x\right|\left|{S}_x(t)\right|+\left|1-{b}_x{\hat{b}}_x^{-1}\right|\left|{\hat{F}}_x\right|\left|{S}_x(t)\right|-{b}_x{b}_x^{-1}{B}_1{\gamma}_x\left|{S}_x(t)\right|\\ {}+{b}_x{\hat{b}}_x^{-1}\left({B}_1-{k}_x\right){\gamma}_x\left|{S}_x(t)\right|\\ {}{\dot{V}}_x\le {\overline{F}}_x\left|{S}_x(t)\right|+\left|1-{\varGamma}_y^{-1}\right|\left|{\hat{F}}_x\right|\left|{S}_x(t)\right|-{\varGamma}_x^{-1}\left({\varGamma}_x\left({\hat{F}}_x+{\eta}_x\right)+\left({\varGamma}_x-1\right)\left|{\hat{F}}_x\right|\right){\gamma}_x\left|S(t)\right|\end{array}} $$

Since η_y is a positive constant, the derivative of the Lyapunov function becomes negative only if γ_x is greater than 1, therefore:

$$ {\gamma}_x\ge 1\to {\dot{V}}_x\le 0 $$

Hence, by choosing γ_x greater than or equal to 1, we can guarantee the system’s stability based on the Lyapunov stability.

Appendix 4

In order to prove the stability of the proposed controller, the candidate Lyapunov function has been chosen as:

$$ {V}_y=\frac{1}{2}{S}_y^2+\frac{b_y{\hat{b}}_y^{-1}}{2}{\overset{\sim }{B}}_y^2 $$

where,

$$ {\displaystyle \begin{array}{c}{\overset{\sim }{B}}_y={B}_2-{k}_y\\ {}{b}_y={\hat{G}}_2+d{\hat{G}}_3\end{array}} $$

In order to evaluate the Lyapunov function, the Lyapunov candidate function has been derived as:

$$ {\dot{V}}_y={S}_y{\dot{S}}_y+{b}_y{\hat{b}}_y^{1-}{\overset{\sim }{B}}_y{\dot{\overset{\sim }{B}}}_y $$

$$ {\displaystyle \begin{array}{c}{\dot{V}}_y={S}_y\left\{{F}_2+{G}_2\delta +\left({F}_1+{G}_1{F}_x\right){e}_2+\dot{x}{\dot{e}}_2+d{F}_3+d{G}_3\delta \right\}\\ {}+{b}_y{\hat{b}}_y^{-1}\left({B}_2-{k}_y\right){\gamma}_y\left|{s}_y\right|\end{array}} $$

According to Appendix 2,

$$ {\displaystyle \begin{array}{l}{\dot{V}}_y\le \left|{\overline{F}}_y\right|\left|{S}_y(t)\right|+\left|1-{b}_y{b}_y^{-1}\right|\left|\hat{\delta}\right|\left|{S}_y(t)\right|-{b}_y{b}_y^{-1}{B}_2{\gamma}_y\left|{S}_y(t)\right|+{b}_y{\hat{b}}_y^{-1}\left({B}_2-{k}_y\right){\gamma}_y\left|{S}_y(t)\right|\\ {}{\dot{V}}_y\le {\overline{F}}_y\left|{S}_y(t)\right|+\left|1-{\varGamma}_y^{-1}\right|\left|\hat{\delta}\right|\left|{S}_y(t)\right|-{\varGamma}_y^{-1}\left({\varGamma}_y\left({\overline{F}}_y+{\eta}_y\right)+\left({\varGamma}_y-1\right)\left|\hat{\delta}\right|\right){\gamma}_y\left|S(t)\right|\\ {}{\dot{V}}_y\le \left(1-{\gamma}_y\right)\left(F+\left|1-{\varGamma}_y^{-1}\right|\left|\hat{\delta}\right|\right)\left|{S}_y(t)\right|-{\eta}_y{\varGamma}_y\left|{S}_y(t)\right|\end{array}} $$

Since η_y is a positive constant, the derivative of the Lyapunov function becomes negative only if γ_y is greater than 1, therefore:

$$ {\gamma}_y\ge 1\to {\dot{V}}_y\le 0 $$

Hence, by choosing γ_y greater than or equal to 1, we can guarantee the system’s stability based on the Lyapunov stability.