Control of mobile robot by switching traveling direction and control gain

Abstract

The control method based on the time-state control form has been proposed to stabilize the chained system, which is a canonical-form nonlinear system. In this study, the control method is used for controlling a mobile robot in auto-parking situations. The proposed controller includes a parameter that is allowed to switch at arbitrary times without loss of the stability of the system. The robot employing the proposed controller reaches the target position by switching its traveling direction to avoid collisions with obstacles. However, the shape of the robot gives a problem. We resolve this by using the switchable parameter included in the proposed controller, and show the availability of switching the parameter. Furthermore, the appropriate switching of the traveling direction and the parameter enables the robot to reach the target faster. Thus, we search the appropriate values of the parameter and the switching points of the traveling direction using the genetic algorithm. In the auto-parking experiments that incorporate the search results, the robot can reach the target position faster.

Appendix

In the method based on the time-state control form, a chained system is transformed into two linear systems: the state-control part, which is an nth order linear system, and the time-control part, which is a first-order linear system. We generally provided the conditions for asymptotically converging the state of the nth order linear system under arbitrary switching of the time axis [18]. The two-wheeled car version of the proof is given below.

Under the assumption that |θ| < π/2, we use the nonlinear coordinate transformation

$$\varvec{z} = (y,\tan \theta )^{T}$$

(A1)

and the input transformation

$$\mu = \frac{{v_{2} }}{{v_{1} \cos^{3} \theta }}$$

(A2)

for Eq. (1). The state x increases with respect to time t when v ₁ > 0; therefore, the state x can be regarded as an alternative time scale. Since the derivative of y with respect to x is

$$\begin{aligned} \frac{dy}{dx} & = \frac{dy}{dt}\frac{dt}{dx} \\ & = \tan \theta , \\ \end{aligned}$$

(A3)

the state equation of z with the time scale x is obtained as

$$\frac{d}{dx}{\varvec{z}} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ 0 & 0 \\ \end{array} } \right]{\varvec{z}} + \left[ {\begin{array}{*{20}c} 0 \\ 1 \\ \end{array} } \right]\,\mu .$$

(A4)

The state x is controlled by the input v ₁.

$$\frac{dx}{dt} = v_{1} \cos \theta$$

(A5)

Eq. (A4) is the controllable canonical form. A state feedback controller for stabilizing the system given by Eq. (A4) can be easily designed using the linear control theory. When z → 0 as x → ∞ by using the stabilizing feedback controller, θ also converges to 0 because of the assumption.

When v ₁ < 0, x decreases with respect to the actual time t. Therefore, we have to introduce another time scale x′ = − x which increases with respect to t for analyzing the stability of the system. The state equation of z with x′ is obtained as

$$\frac{d}{dx'}{\varvec{z}} = \left[ {\begin{array}{*{20}c} 0 & { - 1} \\ 0 & 0 \\ \end{array} } \right]{\varvec{z}} + \left[ {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } \right]\,\mu .$$

(A6)

Now, a new time scale τ is defined as

$$\tau = \int_{0}^{\,t} {\left| {\,\frac{dx}{dt}} \right|\,dt} .$$

(A7)

The time scale τ is the distance traveled along the x-axis. The two systems (A4) and (A6) are described as a switched system by using the time scale τ [17].

$$\frac{d}{d\tau }{\varvec{z}} = \left\{ {\begin{array}{*{20}c} {\left[ {\begin{array}{*{20}c} 0 & {\,\,1\,\,} \\ 0 & {\,\,0\,} \\ \end{array} } \right]{\varvec{z}} + \left[ {\begin{array}{*{20}c} 0 \\ {\,\,1\,\,} \\ \end{array} } \right]\,\mu \,\,\,\,\,\,\,\,\,(v_{1} > 0)} \\ {\left[ {\begin{array}{*{20}c} 0 & { - 1} \\ 0 & 0 \\ \end{array} } \right]{\varvec{z}} +\left[ {\begin{array}{*{20}c} 0 \\ { - 1} \\ \end{array} } \right]\,\mu \,\,\,\,\,\,\,\,\,(v_{1} < 0)} \\ \end{array} } \right.$$

(A8)

The input function μ in the controller (2) can be rewritten as

$$\mu = \left\{ \begin{aligned} - \,\,\left[ {\,\begin{array}{*{20}c} {k_{1} } & {\,\,\,\,\alpha k_{2} } \\ \end{array} } \right]\,\,{\varvec{z}}\,\,\,\,\,\,\,\,\,(v_{1} > 0) \hfill \\ - \,\,\left[ {\begin{array}{*{20}c} {\,k_{1} } & { - \alpha k_{2} } \\ \end{array} } \right]\,\,{\varvec{z}}\,\,\,\,\,\,\,\,\,(v_{1} < 0) \hfill \\ \end{aligned} \right..$$

(A9)

By substituting (A9) into (A8), the closed-loop system becomes

$$\frac{d}{d\tau }{\varvec{z}} = \left\{ \begin{aligned} &\tilde{A}\,\,{\varvec{z}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(v_{1} > 0) \hfill \\ &E_{2} \tilde{A}\,E_{2}\, {\varvec{z}}\,\,\,\,\,\,\,\,\,(v_{1} < 0) \hfill \\ \end{aligned} \right.,$$

(A10)

where

$$\tilde{A} = \left[ {\begin{array}{*{20}c} 0 & 1 \\ { - k_{1} } & { - \alpha k_{2} } \\ \end{array} } \right]\,\,,\,\,\,\,E_{2} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & { - 1} \\ \end{array} } \right].$$

(A11)

Now, consider a Lyapunov candidate function as

$$V = {\varvec{z}}^{T} P\,{\varvec{z}},\quad P = \left[ {\begin{array}{*{20}c} {k_{1} k_{2} }& 0 \\ 0 &{k_{2} } \\ \end{array} } \right].$$

(A12)

If k ₁ and k ₂ > 0, V > 0 (z ≠ 0). The derivative of V with respect to τ for both v ₁ > 0 and v ₁ < 0 is calculated as follows:

$$\begin{aligned} \frac{d}{d\tau }V & = \left\{ {\begin{array}{*{20}c} {{\varvec{z}}^{T} \left( {\tilde{A}^{T} P + P\tilde{A}} \right)\,{\varvec{z}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(v_{1} > 0)} \\ {{\varvec{z}}^{T} \left( {E_{2} \tilde{A}^{T} E_{2} P + PE_{2} \tilde{A}E_{2} } \right)\,{\varvec{z}}\,\,\,\,\,\,\,\,\,\,\,\,(v_{1} < 0)} \\ \end{array} } \right. \\ & = {\varvec{z}}^{T} \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & { - 2\alpha k_{2}^{2} } \\ \end{array} } \right]\,{\varvec{z}} \\ & = - \alpha \,{\varvec{z}}^{T} {\varvec{c}}^{T} {\varvec{c}}\,{\varvec{z}}\,\,, \\ \end{aligned}$$

(A13)

where

$${\varvec{c}} = [\begin{array}{*{20}c} 0 & {\sqrt 2 } \\ \end{array} k_{2} ].$$

(A14)

From the condition α > 0, we have dV/dt ≤ 0. Furthermore, both $({\varvec{c}},\tilde{A})$ and $({\varvec{c}},\,\,E_{2} \tilde{A}E_{2} )$ are observable pairs; therefore, it is guaranteed that z → 0 as τ → ∞ under the arbitrary switching of α and the sign of v ₁ by Lyapunov’s stability theory.