Adaptive Control of Human-Interacted Mobile Robots with Velocity Constraint

Abstract

In this paper, we present an adaptive control for mobile robots moving in human environments with velocity constraints. The mobile robot is commanded to track the desired trajectory while at the same time guarantee the satisfaction of the velocity constraints. Neural networks are constructed to deal with unstructured and unmodeled dynamic nonlinearities. Lyapunov function is employed during the course of control design to implement the validness of the proposed approach. The effectiveness of the proposed framework is verified through simulation studies.

A Appendix

1.1 A.1 Proof of Theorem 1

(i) Multiplying (35) by $\text {e}^{\rho _2 t}$ yields

$$\begin{aligned} \frac{d}{dt}(V_3(t)\text{ e }^{\rho _2 t})\le \varsigma _2\text{ e }^{\rho _2 t} \end{aligned}$$

(40)

Integrating (40) over [0, t] yields

$$\begin{aligned} 0\le V_3(t)\le (V_3(0)-\frac{\varsigma _2}{\rho _2})\text{ e }^{-\rho _2 t}+\frac{\varsigma _2}{\rho _2} \end{aligned}$$

(41)

Further, it is easily found that

$$\begin{aligned} 0\le V_3(t)\le (V_3(0)-\frac{\varsigma _2}{\rho _2})\text{ e }^{-\rho _2 t}+\frac{\varsigma _2}{\rho _2}\le V_3(0)+\frac{\varsigma _2}{\rho _2} \end{aligned}$$

(42)

Then, we can conclude that $e_z$, $\{\tilde{W}_M\}$ and $\{\tilde{W}_P\}$ are all bounded.

(ii) From (42), we have

$$\begin{aligned} V_3(0)+\frac{\varsigma _2}{\rho _2}\ge \left\{ \begin{array}{c} \frac{1}{2}\log \frac{k_{1,i}^2(t)}{k^2_{1,i}(t)-e_{z,i}^2},~0<e_{z,i}<k_{1,i}\nonumber \\ \frac{1}{2}\log \frac{k_{2,i}^2(t)}{k^2_{2,i}(t)-e_{z,i}^2},~-k_{2,i}<e_{z,i}\le 0 \end{array} \right. \end{aligned}$$

Taking exponentials on both sides of the above inequality, it can be easily obtained that

$$\begin{aligned} e_{z,i}^2\le \left\{ \begin{array}{c} k^2_{1,i}(1-\text{ e }^{-2(V_3(0)+\frac{\varsigma _2}{\rho _2})}),~0<e_{z,i}<k_{1,i}\nonumber \\ k^2_{2,i}(1-\text{ e }^{-2(V_3(0)+\frac{\varsigma _2}{\rho _2})}),~-k_{2,i}<e_{z,i}\le 0 \end{array} \right. \end{aligned}$$

Taking square root of both sides of the above inequality will lead to

$$\begin{aligned} \underline{D}_i(t)\le e_{z,i}\le \overline{D}_i(t)~\forall t>0,~i=1,2 \end{aligned}$$

(43)

(iii) Since $z_i=e_{z,i}+z_{c,i}$ and $-k_{1,i}(t)\le e_{z,i}\le k_{2,i}(t)$, for $i=1,2$, we infer that

$$\begin{aligned} -k_{2,i}(t)+z_{c,i}\le z_i\le k_{1,i}(t)+z_{c,i} \end{aligned}$$

(44)

for all $t>0$. From the definition of $k_1$ and $k_2$ in (9), we conclude that $\mid z_{i}\mid \le k_{ai},~i=1,2,~\forall t>0$.