Guidance Based Lane-Changing Control in High-Speed Vehicle for the Overtaking Maneuver

Abstract

Vehicle lane-changing and overtaking are fundamental technology in autonomous driving. This paper proposes an approach that combines guidance navigation and image based visual detection for road following and overtaking maneuver. Two critical issues are considered: Firstly, it is suggested to design a robust constrained road following based line of sight guidance in the presence of unknown time-varying sideslip angle and under unknown dynamic model. Secondly, to trigger the autonomous overtaking manoeuvre. An overtaking procedure is defined as a high-priority task while the road following as a low-priority task. The objective is then to properly switching between the two guidance modes according to the camera vision reading. The adopted control strategy consists of a constrained along and cross-track control law and a nonlinear RISE control law. The closed-loop dynamics of both kinematic errors and the dynamical errors are analyzed in detail using nonlinear interconnected systems theory and the overall system is shown to be input to state stable (ISS). The proposed approach for road following and overtaking are validated in simulations.

Appendices

Appendix: A

To prove Lemma 1, we will introduce the following tan-type BLF [22], in order to take into account the constraint requirement on the safety distance D(t) that separates the host vehicle from the preceding vehicle:

$$ V_{G}=\frac{D^{2}}{\pi}\tan\left( \frac{\pi s^{2}}{2D^{2}}\right)+\frac{1}{2}e^{2} $$

(40)

Taking the time derivative of V_G with respect to time, by letting $s_{\cos \limits }=\frac {s}{\cos \limits ^{2}\left (\frac {\pi s^{2}}{2D^{2}}\right )}$ we obtain

$$ \begin{array}{@{}rcl@{}} \dot{V}_{G}&=&s_{\cos}\dot{s}+\frac{2D\dot{D}}{\pi}\tan\left( \frac{\pi s^{2}}{2D^{2}}\right) -\left( \frac{\dot{D}}{D}\right) ss_{\cos}+e\dot{e}\\ &=& s_{\cos}\left( -k_{1}s+e\dot{\psi}_{p}-\zeta\right)+e\left( -\frac{U e}{\sqrt{(e+{\Delta} \hat{\beta})^{2}+{\Delta}^{2}}}-\dot{\psi}_{p} s\right)\\ &&+eg\tilde{\psi}+\frac{2D\dot{D}}{\pi}\tan\left( \frac{\pi s^{2}}{2D^{2}}\right) -\left( \frac{\dot{D}}{D}\right) ss_{\cos}\\ &=&-\frac{k_{2}D^{2}}{\pi}\tan\left( \frac{\pi s^{2}}{2D^{2}}\right)-k_{1}s_{\cos}s-\frac{U e^{2}}{\sqrt{(e+{\Delta} \hat{\beta})^{2}+{\Delta}^{2}}} \end{array} $$

(41)

$$ \begin{array}{@{}rcl@{}} &&+eg\tilde{\psi}+\frac{2D\dot{D}}{\pi}\tan\left( \frac{\pi s^{2}}{2D^{2}}\right) -\left( \frac{\dot{D}}{D}\right) ss_{\cos} \end{array} $$

(42)

by choosing the control gains (22) and (23), we obtain

$$ \dot{V}_{g}=-(k_{2}-2({k_{1}^{2}}-k_{c}))\frac{D^{2}}{\pi}\tan\left( \frac{\pi s^{2}}{2D^{2}}\right)-\frac{U e^{2}}{\sqrt{(e+{\Delta} \hat{\beta})^{2}+{\Delta}^{2}}}+eg\tilde{\psi} $$

(43)

Using the following inequalities:

$$ \frac{|\cos(\tilde{\psi}-1)|}{|\tilde{\psi}|}\leq \Big|\sin(\frac{\tilde{\psi}}{2})\Big|\leq1, \quad \Big\vert \frac{\sin(\tilde{\psi})}{\tilde{\psi}}\Big\vert\leq 1 $$

(44)

to show that |g|≤ U and the fact that $ge\tilde {\psi }\leq \frac {U}{2\varepsilon _{1}}\vert e\vert ^{2}+\frac {\varepsilon _{1}U}{2}|\tilde {\psi }|^{2}$, where ε₁ is any arbitrary value such that $1-\frac {1}{2\varepsilon _{1}}>0$, the following holds

$$ \begin{array}{@{}rcl@{}} \dot{V}_{g}&\leq&-(k_{2}-2({k_{1}^{2}}-k_{c}))\frac{D^{2}}{\pi}\tan\left( \frac{\pi s^{2}}{2D^{2}}\right)-\frac{U (1-\frac{1}{2\varepsilon_{1}})e^{2}}{\sqrt{(e+{\Delta} \hat{\beta})^{2}+{\Delta}^{2}}}+\frac{\varepsilon_{1}U}{2}|\tilde{\psi}|^{2}\\ &\leq&-CV_{G}+\frac{\varepsilon_{1}U}{2}|\tilde{\psi}|^{2} \end{array} $$

(45)

where $C=\min \limits \{k_{2}-2({k_{1}^{2}}-k_{c}), 2\max \limits (U)\}$. From (45), it can be inferred that for all $\mu =(s,e)^{\top } \in {\mathscr{M}}:=\{\mu \in \mathbb {R}^{2}~|~\frac {1}{2}CV_{G}\geq \gamma (|\psi |)\}$, with γ(.) is $\mathcal {K}_{\infty }$ function, we have

$$ \begin{array}{@{}rcl@{}} \dot{V}_{G}&\leq&-\frac{1}{2}CV_{G}+\Bigg(\gamma(|\tilde{\psi}|)-\frac{1}{2}CV_{G}\Bigg)\\ &\leq&-\frac{1}{2}CV_{G} \end{array} $$

(46)

By the comparison lemma in [23], it follows immediately that for all $\mu _{0} \in {\mathscr{M}}$, there exists a $\mathcal {K}{\mathscr{L}}$ function ς such that V_G(μ) ≤ ς(V_G(μ₀), t). Also for all μ such that $\frac {1}{2}CV_{G}\leq \gamma (|\psi |)$, we have $V_{g}(\mu )\leq \frac {1}{C}\left (2\gamma (\sup _{\lambda \in [t_{0},t]}|\psi (\lambda )|)\right )$, combining the two inequalities, we arrive at $V(\mu (t))\leq \max \limits \{\varsigma (V_{G}(\mu _{0}),t),\frac {1}{C}\left (2\gamma (\sup _{\lambda \in [t_{0},t]}|\psi (\lambda )|)\right )\}$, which shows that V_G is an ISS-Lyapunov function and thus the closed-loop kinematic tracking error (25) is ISS w.r.t to the entry $\tilde {\psi }$.

Appendix: B

To prove Lemma 2, first the open-loop tracking error dynamics is developed by taking the time derivative of the filtering error r and using the dynamics (27) and (28) to obtain

$$ \begin{array}{@{}rcl@{}} \dot{r}&=&\ddot{e}+c_{e} \dot{e}=C[\dot{A}_{0}(t)\chi+B_{0} \dot{e}=C\left[\dot{A}_{0}(t)\chi+B_{0} \dot{e}\right]\\ &=&\tilde{N}+N_{d}+CB_{0}\dot{\delta}_{ref}-e \end{array} $$

(47)

where the auxiliary terms $\tilde {N}(\chi ,\dot {\chi },e,\chi _{r},\dot {\chi }_{r})\in \mathbb {R}^{4}$ and $N_{d}(\chi _{r},\dot {\chi }_{r},\dot {u})\in \mathbb {R}^{4}$ are defined as

$$ \tilde{N}=CA_{0}(t)(\dot{\chi}-\dot{\chi}_{r})+C\dot{A}_{0}(\chi-\chi_{r})+c_{e} \dot{e}+e $$

(48)

and

$$ N_{d}=C\dot{f}(t)-CB_{r}\dot{u}-CA_{r}\dot{\chi}_{r}+CA_{0}(t)\dot{\chi}_{r}+C\dot{A}_{0}(t)\chi_{r} $$

(49)

The segregated terms (48) and (49) are motivated by the fact that these terms can be upper bounded by state-dependent terms and upper bounded by constants respectively [11] as follows:

$$ \Vert \tilde{N} \Vert \leq \rho_{0} \Vert \boldsymbol{z}\Vert, \quad \Vert N_{d}\Vert\leq \rho_{1} $$

(50)

where $\boldsymbol {z}=[\boldsymbol {e}^{\top },\boldsymbol {r}^{\top }]^{\top } \in \mathbb {R}^{8}$ and ρ₀, ρ₁ are positive upper bound constants. Substituting the reference steering angle δ_ref designed as (29) in the open-loop dynamics (47), yields the following closed-loop dynamics:

$$ \dot{\boldsymbol{r}}=\tilde{N}+N_{d}-k_{3}\left( \frac{k_{s}}{k_{3}}+1\right)\boldsymbol{r}-k_{r}\text{sign}(\boldsymbol{r})-\boldsymbol{e} $$

(51)

To show the stability of the closed-loop dynamics (51), let V_D(e, r) be a continuously differentiable, positive-definite function defined as

$$ V_{D}=\frac{1}{2}\boldsymbol{e}^{\top} \boldsymbol{e}+\frac{1}{2} \boldsymbol{r}^{\top} \boldsymbol{r} $$

(52)

Taking the time derivative of (52) along the solutions of (51) and using the upper bounds (50), to obtain

$$ \begin{array}{@{}rcl@{}} \dot{V_{L}}&=&-c_{e} \boldsymbol{e}^{\top} \boldsymbol{e}-k_{3}\left( \frac{k_{s}}{k_{3}}+1\right)\boldsymbol{r}^{\top} \boldsymbol{r}+\boldsymbol{r}^{\top} \tilde{N}+\boldsymbol{r}^{\top} N_{d}-k_{r}\boldsymbol{r}^{\top} \text{sign}(\boldsymbol{r})\\ &\leq&-c_{e} \Vert \boldsymbol{e} \Vert^{2}-k_{s}\Vert \boldsymbol{r} \Vert^{2}-\left( k_{3} \Vert \boldsymbol{r}\Vert^{2}-\rho_{0}\Vert\boldsymbol{r}\Vert \Vert\boldsymbol{z}\Vert\right)-(k_{r}-\rho_{1})\Vert \boldsymbol{r}\Vert \end{array} $$

(53)

If k_r satisfies the sufficient gain condition in (31), then the last term in (53) becomes strictly negative, which implies that $\dot {V}_{D}$ can be upper bounded as

$$ \dot{V_{L}} \leq -c_{e} \Vert \boldsymbol{e} \Vert^{2}-k_{s}\Vert \boldsymbol{r} \Vert^{2}-\left( k_{3} \Vert \boldsymbol{r}\Vert^{2}-\rho_{0}\Vert\boldsymbol{r}\Vert \Vert\boldsymbol{z}\Vert\right) $$

(54)

By completing the squares, the upper bound in (54) can be expressed in a more convenient form. To this end, the term $\frac {{\rho _{0}^{2}}}{4{k_{3}^{2}}}\Vert \boldsymbol {z}\Vert ^{2}$ is added then subtracted to the right-hand side of (54), yields

$$ \begin{array}{@{}rcl@{}} \dot{V_{L}} &\leq& -c_{e} \Vert \boldsymbol{e} \Vert^{2}-k_{s}\Vert \boldsymbol{r} \Vert^{2}-k_{3}\left( \Vert \boldsymbol{r}\Vert-\frac{\rho_{0}}{2k_{3}} \Vert\boldsymbol{z}\Vert\right)^{2}+\frac{{\rho_{0}^{2}}}{4{k_{3}^{2}}}\Vert \boldsymbol{z}\Vert^{2} \\ &\leq&-\boldsymbol{z}^{\top} \text{diag}\{c_{e} I_{4\times 4}, k_{r}I_{4\times 4}\}\boldsymbol{z}+\frac{{\rho_{0}^{2}}}{4{k_{3}^{2}}}\Vert \boldsymbol{z}\Vert^{2}\\ &\leq&-\underbrace{\left( \min\{c_{e},k_{r}\}-\frac{{\rho_{0}^{2}}}{4{k_{3}^{2}}}\right)}_{\lambda}\Vert \boldsymbol{z} \Vert^{2} \end{array} $$

(55)

where the fact that z^⊤diag{z} is positive provided that the condition on the gain k₃ is satisfied and verifies (31), from the definition of V_D in (52) using (55), it follows that $\dot {V}_{D}\leq -\lambda V_{D}$, hence it can be concluded that $\Vert \boldsymbol {e}\Vert \leq \Vert \boldsymbol {z}(0)\Vert \exp \left (-\frac {\lambda }{2}t\right ), \forall t\in [0,\infty )$. This implies that U_y converges to U_yd as time goes to infinity. Also $\dot {\psi }$ converges to $\dot {\bar {\psi }}$ which consequently implies that ψ converges to ψ_d as time goes to infinity.