Feedback stabilization of a nonholonomic system with potential fields: application to a two-wheeled mobile robot among obstacles

Abstract

In this paper, a feedback controller for a nonholonomic system with three states and two inputs is derived using an artificial potential function that has no local minima, and the stability of equilibria of the system is analyzed. Although the system with the controller has an infinite number of equilibria due to the nonholonomic constraint, those equilibria except the critical points of the potential function are unstable because of a skew-symmetric component of the controller. When the potential function has critical points of saddle type, the saddles may be stable equilibria in addition to the stable equilibrium at the minimum of the function. The controller is applied to a two-wheeled mobile robot among obstacles and modified by using a time-varying potential function in order to avoid convergence to the saddles. As a result, with the controller, the mobile robot converges to a desired position and orientation without collision with obstacles.

Appendices

Appendix 1: Region of attraction of point $S_i^*$

In the neighborhood of a saddle point $S_i^*$, from properties (d) and (f), the function $V({\varvec{z}})$ can be approximately expressed as

$$\begin{aligned} V({\varvec{z}})= \frac{1}{2} \bar{{\varvec{z}}}^\mathrm{T} \left[ \begin{array}{c@{\quad }c@{\quad }c} p_{11} &{} p_{12} &{} 0 \\ p_{12} &{} p_{22} &{} 0 \\ 0 &{} 0 &{} p_{33} \end{array} \right] \bar{{\varvec{z}}} + V({\varvec{z}}_{si}), \end{aligned}$$

(60)

where $\bar{{\varvec{z}}} = [{\bar{z}}_1 , {\bar{z}}_2 , \theta ]^\mathrm{T} = {\varvec{z}}-{\varvec{z}}_{si}$ and, from the condition (50), $p_{11} \ge 0$. In the following analysis, we assume that $p_{11} > 0$ and that the controller (42) is applied to the system. We can choose new state variables as $[{\hat{z}}_1 , {\bar{z}}_2 , \theta ]^\mathrm{T}$ where ${\hat{z}}_1=p_{11}{\bar{z}}_1 + p_{12}{\bar{z}}_2$. Neglecting smaller terms under the assumption that the state variables are small, the following equations are obtained:

$$\begin{aligned}&{\dot{{\hat{z}}}}_1 = -\alpha p_{11} {\hat{z}}_1-{\hat{\beta }}p_{33}(p_{11}+p_{12}\theta )\theta , \end{aligned}$$

(61)

$$\begin{aligned}&{\dot{{\bar{z}}}}_2 = \theta (-\alpha {\hat{z}}_1 -{\hat{\beta }}p_{33}\theta ), \end{aligned}$$

(62)

$$\begin{aligned}&{\dot{\theta }} = {\hat{\beta }} \left\{ {\hat{z}}_1+\theta \left( \frac{p_{12}}{p_{11}}{\hat{z}}_1+\left( p_{22}-\frac{p_{12}^2}{p_{11}} \right) {\bar{z}}_2 \right) \right\} \nonumber \\&\quad -\alpha p_{33}\theta . \end{aligned}$$

(63)

Noting that $\nabla V = (\partial V/\partial {\varvec{z}})^\mathrm{T} = (\partial V/\partial \bar{{\varvec{z}}})^\mathrm{T}$, the coefficient ${\hat{\beta }}$ defined by (10) is approximately calculated from (32), (43) and (60) as

$$\begin{aligned} {\hat{\beta }} = \frac{\beta }{\epsilon _g V({\varvec{z}}_{si})\sqrt{g}} \left( \frac{p_{12}}{p_{11}}{\hat{z}}_1+\left( p_{22}-\frac{p_{12}^2}{p_{11}} \right) {\bar{z}}_2 \right) . \end{aligned}$$

(64)

In order to express the magnitude of ${\hat{z}}_1$ and $\theta$, we introduce the variable ${\hat{r}}$ as ${\hat{r}}=\sqrt{{\hat{z}}_1^2+ p_{11} p_{33} \theta ^2}$. From (61) and (63), the variable ${\hat{r}}$ satisfies approximately the following equation:

$$\begin{aligned} \dot{{\hat{r}}} = - \frac{\alpha p_{11}}{{\hat{r}}} ({\hat{z}}_1^2+p_{33}^2\theta ^2) + k_{\beta }\frac{p_{11} p_{33} \theta ^2}{{\hat{r}}\sqrt{g}} {\bar{z}}_2^2 , \end{aligned}$$

(65)

where $k_{\beta }=\beta (p_{12}^2/p_{11}-p_{22})^2/(\epsilon _g V({\varvec{z}}_{si}))$. Since $|{\dot{{\bar{z}}}}_2| \ll |\dot{{\hat{r}}}|$ from (62) and (65), we obtain

$$\begin{aligned}&{\dot{{\bar{z}}}}_2 = 0, \end{aligned}$$

(66)

$$\begin{aligned}&\dot{{\hat{r}}} \le - \frac{\alpha l_{m}}{p_{33}} {\hat{r}} + k_{\beta }\left( \frac{p_{11} p_{33}}{l_{m}} \right) ^{\frac{1}{4}} {\hat{r}}^{\frac{1}{2}}{\bar{z}}_2^2 , \end{aligned}$$

(67)

where $l_{m} = \min \{p_{11}p_{33} , p_{33}^2\}$ and the following equation was used:

$$\begin{aligned} \frac{l_m}{p_{11}p_{33}}{\hat{r}}^2 \le {\hat{z}}_1^2 + p_{33}^2 \theta ^2 \approx g^2 . \end{aligned}$$

(68)

Equation (67) shows that the system moves into the region ${\hat{r}} \le {\mathcal {O}}({\bar{z}}_2^4)$.

On the other hand, from (60),

$$\begin{aligned} V({\varvec{z}}) = V({\varvec{z}}_{si}) + \frac{1}{2}\left( \frac{1}{p_{11}}{\hat{r}}^2 -\frac{1}{p_{11}}(p_{12}^2-p_{11}p_{22}){\bar{z}}_2^2\right) , \end{aligned}$$

(69)

where, as shown in “Appendix 2,” $p_{12}^2-p_{11}p_{22} > 0$ because $S_i^*$ is a saddle point. Therefore, (67) means that the system moves into the region $V({\varvec{z}})<V({\varvec{z}}_{si})$ for nonzero but sufficiently small ${\bar{z}}_2$. Once the system enters the region, it cannot approach $S_i^*$ because $\dot{V} \le 0$. In other words, the region of attraction of $S_i^*$ can be approximated as a two-dimensional space that satisfies ${\bar{z}}_2=0$ (Fig. 7).

Fig. 7

Approximate behavior of the system near saddle point $S_i^*$

The behavior of the system in the region ${\hat{r}} \le {\mathcal {O}}({\bar{z}}_2^4)$ can be approximately analyzed in a similar way as in [36, 39]. The state variables oscillate with a high frequency around the set of equilibria ${\mathcal {H}}$ and move slowly along ${\mathcal {H}}$ increasing the amplitude of oscillation (Fig. 7). For simplicity, we assume that $p_{11}=1$, $p_{12}=0$, $p_{22}=-1$ and $p_{33}=1$. The variable $g$ in (12) is approximated as $g= \sqrt{{\hat{z}}_1^2+ \theta ^2 } = {\hat{r}}$, and the approximate solution of ${\varvec{z}}$ in the region ${\hat{r}} \le {\mathcal {O}}({\bar{z}}_2^4)$ is summarized as follows. Neglecting smaller terms in (61), (62), (63) and (65), we obtain

$$\begin{aligned} {\dot{{\hat{z}}}}_1=-{\hat{\beta }}\theta , \ \ {\dot{\theta }}={\hat{\beta }}{\hat{z}}_1, \ \ {\dot{{\bar{z}}}}_2=0, \ \ {\dot{g}}=\dot{{\hat{r}}}=0 . \end{aligned}$$

(70)

From (70), the solutions for ${\hat{z}}_1$ and $\theta$ are given as ${\hat{z}}_1=g\cos ({\hat{\beta }}t+\phi )$ and $\theta =g\sin ({\hat{\beta }}t+\phi )$ where $\phi$ is a constant. Substituting the solutions into (62) and (65), averaging them with the period $2\pi /{\hat{\beta }}$ and neglecting smaller terms, we obtain

$$\begin{aligned}&{\dot{g}}=-\alpha g+\frac{1}{2}k_{\beta }g^{\frac{1}{2}}{\bar{z}}_2^2 , \end{aligned}$$

(71)

$$\begin{aligned}&{\dot{{\bar{z}}}}_2=\frac{1}{2}k_{\beta }g^{\frac{3}{2}}{\bar{z}}_2 . \end{aligned}$$

(72)

Because $|{\dot{{\bar{z}}}}_2| \ll |{\dot{g}}|$ from the above equations, $g$ converges to $k_{\beta }^2{\bar{z}}_2^4/(4\alpha ^2)$ exponentially, while ${\bar{z}}_2$ is constant. Substituting $g=k_{\beta }^2{\bar{z}}_2^4/(4\alpha ^2)$ into (72), we obtain

$$\begin{aligned} {\dot{{\bar{z}}}}_2=\frac{1}{16\alpha ^3}k_{\beta }^4{\bar{z}}_2^7 . \end{aligned}$$

(73)

From the above equation, $|{\bar{z}}_2|$ becomes large as time advances and the system goes away from the point $S_i^*$. It should be noted that we can make the escape from the saddle faster by changing the choice of $h(g)$ [36].

Appendix 2: Derivation of (58)

From property (f), $(\partial {\varvec{B}}^\mathrm{T}\nabla V/\partial {\varvec{z}})|_{{\varvec{z}}={\varvec{z}}_{si}}$ can be calculated as

$$\begin{aligned} \left. \frac{\partial {\varvec{B}}^\mathrm{T}\nabla V}{\partial {\varvec{z}}} \right| _{{\varvec{z}}={\varvec{z}}_{si}} = \left[ \begin{array}{c@{\quad }c@{\quad }c} \frac{\partial ^2 V}{\partial z_1^2} &{} \frac{\partial ^2 V}{\partial z_1 \partial z_2} &{} 0 \\ 0 &{} 0 &{} \frac{\partial ^2 V}{\partial \theta ^2} \end{array} \right] . \end{aligned}$$

(74)

Therefore, we can choose ${\varvec{t}}$ as

$$\begin{aligned} {\varvec{t}} = [\partial ^2 V/\partial z_1 \partial z_2 , -\partial ^2 V/\partial z_1^2 , 0]^\mathrm{T} . \end{aligned}$$

(75)

Then, $\delta V = V({\varvec{z}}_{si}+ a{\varvec{t}})-V({\varvec{z}}_{si})$ is calculated as

$$\begin{aligned} \delta V = a (\nabla V({\varvec{z}}_{si}))^\mathrm{T}{\varvec{t}} + \frac{a^2}{2}{\varvec{t}}^\mathrm{T} {\varvec{P}}({\varvec{z}}_{si}){\varvec{t}} + O(a^3) . \end{aligned}$$

(76)

From property (f) and the definitions of $V_n$ and $V$ in (36) and (40), the matrix ${\varvec{P}}({\varvec{z}}_{si})$ is calculated as

$$\begin{aligned} {\varvec{P}}= & {} \left[ \begin{array}{c@{\quad }c@{\quad }c} \frac{\partial ^2 V}{\partial z_1^2} &{} \frac{\partial ^2 V}{\partial z_1 \partial z_2} &{} 0 \\ \frac{\partial ^2 V}{\partial z_1 \partial z_2} &{} \frac{\partial ^2 V}{\partial z_2^2} &{} 0 \\ 0 &{} 0 &{} \frac{\partial ^2 V}{\partial \theta ^2} \end{array}\right] \nonumber \\&\quad = \left[ \begin{array}{c@{\quad }c@{\quad }c} \frac{\partial ^2 V_n}{\partial z_1^2} &{} \frac{\partial ^2 V_n}{\partial z_1 \partial z_2} &{} 0 \\ \frac{\partial ^2 V_n}{\partial z_1 \partial z_2} &{} \frac{\partial ^2 V_n}{\partial z_2^2} &{} 0 \\ 0 &{} 0 &{} \frac{\partial ^2 V}{\partial \theta ^2} \end{array}\right] . \end{aligned}$$

(77)

Since $\partial ^2 V/\partial \theta ^2 > 0$ from property (f) and the function $V_n$ has no local maxima, ${\varvec{P}}$ has one negative eigenvalue for the saddle ${\varvec{z}}={\varvec{z}}_{si}$. Therefore, we obtain

$$\begin{aligned} \left( \frac{\partial ^2 V}{\partial z_1 \partial z_2}\right) ^2 - \frac{\partial ^2 V}{\partial z_1^2}\frac{\partial ^2 V}{\partial z_2^2} \!=\! \left( \frac{\partial ^2 V_n}{\partial z_1 \partial z_2}\right) ^2 - \frac{\partial ^2 V_n}{\partial z_1^2}\frac{\partial ^2 V_n}{\partial z_2^2} \!>\! 0. \end{aligned}$$

(78)

For the saddle $S_i^*$ that is stable in the analysis in Sect. 3.2, the condition (50) also holds. Noting that $\nabla V({\varvec{z}}_{si}) =0$, $\delta V$ is calculated from (50), (75), (77) and (78) as

$$\begin{aligned} \delta V\approx & {} \frac{a^2}{2}{\varvec{t}}^\mathrm{T} {\varvec{P}}({\varvec{z}}_{si}){\varvec{t}} \nonumber \\= & {} -\frac{a^2}{2}\frac{\partial ^2 V}{\partial z_1^2} \Bigl \{ \Bigl (\frac{\partial ^2 V}{\partial z_1 \partial z_2}\Bigr )^2 - \frac{\partial ^2 V}{\partial z_1^2}\frac{\partial ^2 V}{\partial z_2^2} \Bigr \} \le 0.\nonumber \\ \end{aligned}$$

(79)