Adaptive Event Triggered Control of Nonholonomic Mobile Robots

Abstract

In this paper, the design of adaptive regulation control of mobile robots in the presence of uncertain robot dynamics and with event-based feedback is presented. Two-layer neural networks (NN) are utilized to represent the uncertain nonlinear dynamics of the mobile robots, which is subsequently employed to generate the control torque with event-sampled measurement update. Relaxing the perfect velocity tracking assumption, control torque is designed to minimize the velocity tracking error, by explicitly taking into account the dynamics of the robot. The Lyapunov’s stability method is utilized to develop an event-sampling condition and to demonstrate the regulation performance of the mobile robot. Finally, simulation results are presented to verify theoretical claims and to demonstrate the reduction in the computations with event-sampled control execution.

Appendix

Proof of Theorem 1:

Consider the Lyapunov candidate

$$ L = \frac{1}{2}\left( {\rho^{2} + \rho \left( {\alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right)} \right) + \frac{1}{2}e_{v}^{T} \overline{M}e_{v}^{T} + 0.5tr\{ \tilde{\Theta }^{T} \Lambda \tilde{\Theta }\} . $$

(A1)

Then, the derivative of (A1) is calculated to be

$$ \begin{array}{*{20}c} {\dot{L} = \rho \dot{\rho } + \dot{\rho }\left( {\alpha ^{2} + {\text{k}}_{{{\upbeta }}} \beta ^{2} } \right) + \rho \left( {2\alpha \dot{\alpha } + 2{\text{k}}_{{{\upbeta }}} \beta \dot{\beta }} \right) + } \\ { - e_{v}^{T} \left( \begin{gathered} K_{v} e_{v} - {{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\upgamma } }}\left( e \right) + 0.5\,e_{v}^{T} (\dot{\bar{M}} - 2\bar{V}_{m} )e_{v} + \bar{\tau }_{d} \hfill \\ + \tilde{f}\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right) + K_{v} e_{{ET}} \hfill \\ \end{gathered} \right) + [f\left( z \right) - f\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right)]) + tr\{ \tilde{\Theta }^{T} \Lambda \dot{\tilde{\Theta }}\} .} \\ \end{array} $$

(A2)

Next, applying the definitions of λ(ρ_i, α_i, β_i) and γ(e_i1, e_i2, e_i3)$\upgamma \left({\mathrm{e}}_{\mathrm{i}1}, {\mathrm{e}}_{\mathrm{i}2},{\mathrm{e}}_{\mathrm{i}3}\right)$ defined in (13) and (22) as well as (kinematics), respectively, gives

$$ \begin{aligned} & \dot{L} = \left( {\rho + \alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right)\left( { - k_{\rho } \rho \cos^{2} {\upalpha } + e_{v1}^{R} \cos \alpha - \cos \alpha \varepsilon_{v} } \right) \\ & + \rho \left( \begin{gathered} 2\alpha \left( { - k_{\alpha } \alpha - k_{\rho } \left( {\frac{{\sin \alpha { }\cos \alpha }}{\alpha }} \right)k_{\beta } \beta + e_{v2}^{R} - \frac{\sin \alpha }{\rho }e_{v1}^{R} + \frac{{\sin \alpha \varepsilon_{v} }}{\rho } - \varepsilon_{\omega } } \right) \hfill \\ + 2{\text{k}}_{{\upbeta }} \beta \left( {k_{\rho } \sin \alpha \cos {\upalpha } - \frac{\sin \alpha }{\rho }e_{v1}^{R} + \frac{{\sin \alpha \varepsilon_{v} }}{\rho }} \right) \hfill \\ \end{gathered} \right) \\ & - e_{v}^{T} \left( \begin{gathered} K_{v} e_{v} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\upgamma }} + 0.5\,e_{v}^{T} (\dot{\overline{M}} - 2\overline{V}_{m} )e_{v} + \overline{\tau }_{d} \hfill \\ + \tilde{f}\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right) + K_{v} e_{ET} \hfill \\ \end{gathered} \right) \\ & + \,[f\left( z \right) - f\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right)]) + tr\{ \tilde{\Theta }^{T} \Lambda \dot{\tilde{\Theta }}\} . \\ \end{aligned} $$

(A3)

After using the skew symmetry property, $\dot{L}$ becomes

$$ \begin{aligned} & \dot{L} = \left( {\rho + \alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right)\left( { - k_{\rho } \rho \cos^{2} {\upalpha } + e_{v1}^{R} \cos \alpha - \cos \alpha \varepsilon_{v} } \right) \\ & + \,\rho \left( \begin{gathered} 2\alpha \left( { - k_{\alpha } \alpha - k_{\rho } \left( {\frac{{\sin \alpha { }\cos \alpha }}{\alpha }} \right)k_{\beta } \beta + e_{v2}^{R} - \frac{\sin \alpha }{\rho }e_{v1}^{R} + \frac{{\sin \alpha \varepsilon_{v} }}{\rho } - \varepsilon_{\omega } } \right) \hfill \\ + 2{\text{k}}_{{\upbeta }} \beta \left( {k_{\rho } \sin \alpha \cos {\upalpha } - \frac{\sin \alpha }{\rho }e_{v1}^{R} + \frac{{\sin \alpha \varepsilon_{v} }}{\rho }} \right) \hfill \\ \end{gathered} \right) \\ & - \,e_{v}^{T} \left( {K_{v} e_{v} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\upgamma }} + \overline{\tau }_{d} + \tilde{f}\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right) + K_{v} e_{ET} ) + [f\left( z \right) - f\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right)]} \right) + tr\{ \tilde{\Theta }^{T} \Lambda \dot{\tilde{\Theta }}\} . \\ \end{aligned} $$

(A4)

By following the similar steps done in 11 without the blending functions yields (A4)

$$ \begin{gathered} \dot{L} = - \rho^{2} \cos^{2} {\upalpha } - \rho \alpha^{2} - {{\rho \cos^{2} {\upalpha }\left( {\alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right)} \mathord{\left/ {\vphantom {{\rho \cos^{2} {\upalpha }\left( {\alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2} - \cos \alpha \left( {\rho + \alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right)\varepsilon_{v} \hfill \\ + 2\alpha \sin \alpha \varepsilon_{v} - 2\alpha \rho \varepsilon_{\omega } + \sin \alpha \varepsilon_{v} - e_{v}^{T} (K_{v} e_{v} - \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\upgamma }} + \overline{\tau }_{d} + \tilde{f}\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right) + K_{v} e_{ET} ) \hfill \\ \end{gathered} $$

(A5)

$$ + [f\left( z \right) - f\left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right)]) + tr\left\{ {\tilde{\Theta }^{T} \Lambda \dot{\tilde{\Theta }}} \right\}. $$

Using defining of the bound $\left\| {\psi \left( z \right)} \right\| \le \psi_{M}$, we obtain

$$ \begin{aligned} & \dot{L} \le - \rho^{2} \cos^{2} {\upalpha } - \rho \alpha^{2} - {{\rho \cos^{2} {\upalpha }\left( {\alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right)} \mathord{\left/ {\vphantom {{\rho \cos^{2} {\upalpha }\left( {\alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2} - \cos \alpha \left( {\rho + \alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right)\varepsilon_{v} \\ & + 2\alpha \sin \alpha \varepsilon_{v} - 2\alpha \rho \varepsilon_{\omega } + \sin \alpha \varepsilon_{v} - \left( {K_{v} - 0.5} \right)\left\| {e_{v} } \right\|^{2} + e_{v}^{T} \tilde{\Theta }^{T} \psi \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right) + e_{v}^{T} K_{v} e_{ET} \\ \end{aligned} $$

(A6)

$$ + 2\psi_{M}^{2} \left\| \Theta \right\|^{2} + e_{v}^{T} \left( {\overline{\tau }_{d} } \right) + tr\left\{ {\tilde{\Theta }^{T} \Lambda \dot{\tilde{\Theta }}} \right\}. $$

Utilizing the parameter adaptation rule defined in (21) and the definition of the NN weights estimation error, we have

$$ \begin{aligned} & \dot{L}_{i} \le - \rho ^{2} \cos ^{2} {{\upalpha }} - \rho \alpha ^{2} - {{\rho \cos ^{2} {{\alpha }}\left( {\alpha ^{2} + {\text{k}}_{{{\upbeta }}} \beta ^{2} } \right)} \mathord{\left/ {\vphantom {{\rho \cos ^{2} {{\alpha }}\left( {\alpha ^{2} + {\text{k}}_{{{\upbeta }}} \beta ^{2} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2} - \cos \alpha \left( {\rho + \alpha ^{2} + {\text{k}}_{{{\beta }}} \beta ^{2} } \right)\varepsilon _{v} \\ & + 2\alpha \sin \alpha \varepsilon _{v} - 2\alpha \rho \varepsilon _{\omega } + \sin \alpha \varepsilon _{v} . - \left( {K_{v} - 0.5} \right)\left\| {e_{v} } \right\|^{2} + e_{v}^{T} \tilde{\Theta }^{T} \psi \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right) + e_{v}^{T} K_{v} e_{{ET}} \\ & + 2\psi _{M}^{2} \left\| \Theta \right\|^{2} + e_{v}^{T} \left( {\bar{\tau }_{d} } \right) - tr\left\{ {\tilde{\Theta }^{T} \Lambda \left( {\Lambda _{1} \psi \left( z \right)e_{v}^{T} - \Lambda _{1} \kappa \hat{\Theta }} \right)} \right\}. \\ \end{aligned} $$

$$ \begin{aligned} & \dot{L}_{i} \le - \rho ^{2} \cos ^{2} {{\upalpha }} - \rho \alpha ^{2} - {{\rho \cos ^{2} {{\alpha }}\left( {\alpha ^{2} + {\text{k}}_{{{\upbeta }}} \beta ^{2} } \right)} \mathord{\left/ {\vphantom {{\rho \cos ^{2} {{\alpha }}\left( {\alpha ^{2} + {\text{k}}_{{{\upbeta }}} \beta ^{2} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2} \\ & + \,\left[ {\begin{array}{*{20}c} { - \cos \alpha \left( {\rho + \alpha ^{2} + {\text{k}}_{{{\beta }}} \beta ^{2} } \right) + 2\alpha \sin \alpha + \sin \alpha } & { - 2\alpha \rho } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varepsilon _{v} } \\ {\varepsilon _{\omega } } \\ \end{array} } \right] \\ & - \,\left( {K_{v} - 0.5} \right)\left\| {e_{v} } \right\|^{2} + e_{v}^{T} \tilde{\Theta }^{T} \psi \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{z} } \right) + e_{v}^{T} K_{v} e_{{ET}} + 2\psi _{M}^{2} \left\| \Theta \right\|^{2} \\ & + \,e_{v}^{T} \left( {\bar{\tau }_{d} } \right) - tr\left\{ {\tilde{\Theta }^{T} \Lambda \left( {\Lambda _{1} \psi \left( z \right)e_{v}^{T} - \Lambda _{1} \kappa \hat{\Theta }} \right)} \right\}. \\ \end{aligned} $$

Combining the similar terms and using the Young’s inequality once again yields

$$ \dot{L} \le - {\rm K}{\rm E} + K_{ET} {\rm E}_{ET} + B $$

(A7)

where $B = 2\psi_{M}^{2} \left\| \Theta \right\|^{2}$ with $\psi_{M} > \psi \left( t \right)$ is the upper bound for the activation function, $\Xi_{i} = [\rho^{2} \,\,\alpha^{2} \,\,\,\,\left\| {e_{v} } \right\|^{2} \,\,\,\left\| {\tilde{\Theta }} \right\|^{2} ],\,$ $\,{\rm K}_{i} = \left[ {\begin{array}{*{20}c} {\overline{\rm k}_{1} } & {\overline{\rm k}_{2} } & {\overline{\rm k}_{3} } & \kappa \\ \end{array} } \right]$ with $\overline{\rm k}_{1} = \cos^{2} {\upalpha }$ $\,\overline{\rm k}_{2} = \rho$$\overline{\rm k}_{3} = K_{v}$. $K_{ET} = \left[ {\begin{array}{*{20}c} \begin{gathered} - \cos \alpha \left( {\rho + \alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right) \hfill \\ + 2\alpha \sin \alpha + \sin \alpha \hfill \\ \end{gathered} & { - 2\alpha \rho } \\ \end{array} } \right],E_{ET} = \left[ {\begin{array}{*{20}c} {\varepsilon_{v} } \\ {\varepsilon_{\omega } } \\ \end{array} } \right]$. Using the assumption that the measurement errors are bounded, $\left\| {E_{ET} } \right\| \le \,\overline{B}_{ETM}$ we can claim that the regulation and velocity tracking errors and NN weight estimation errors are bounded.

Proof of Theorem 2:

Consider the Lyapunov candidate in (A1) follow the similar steps done in the proof of the first theorem and obtain

$$ \dot{L} \le - {\rm K}{\rm E} + K_{ET} {\rm E}_{ET} + B, $$

(A8)

where $B = 2\psi_{M}^{2} \left\| \Theta \right\|^{2}$ with $\psi_{M} > \psi \left( t \right)$ is the upper bound for the activation function, $\Xi_{i} = [\rho^{2} \,\,\alpha^{2} \,\,\,\,\left\| {e_{v} } \right\|^{2} \,\,\,\left\| {\tilde{\Theta }} \right\|^{2} ],\,$ $\,{\rm K}_{i} = \left[ {\begin{array}{*{20}c} {\overline{\rm k}_{1} } & {\overline{\rm k}_{2} } & {\overline{\rm k}_{3} } & \kappa \\ \end{array} } \right]$ with $\overline{\rm k}_{1} = \cos^{2} {\upalpha }$, $\,\overline{\rm k}_{2} = \rho$, $\overline{\rm k}_{3} = K_{v}$:

$$ K_{ET} = \left[ {\begin{array}{*{20}c} \begin{gathered} - \cos \alpha \left( {\rho + \alpha^{2} + {\text{k}}_{{\upbeta }} \beta^{2} } \right) \hfill \\ + 2\alpha \sin \alpha + \sin \alpha \hfill \\ \end{gathered} & { - 2\alpha \rho } \\ \end{array} } \right],E_{ET} = \left[ {\begin{array}{*{20}c} {\varepsilon_{v} } \\ {\varepsilon_{\omega } } \\ \end{array} } \right]. $$

Using the event-sampling condition (11) in (A7), we get $\dot{L} \le - {\rm K}{\rm E} + K_{ET} {\upmu }{\rm E} + B$ Choosing ${\upmu } = {1 \mathord{\left/ {\vphantom {1 {{\rm K}_{ET} }}} \right. \kern-\nulldelimiterspace} {{\rm K}_{ET} }},$ the Lyapunov derivative is further simplified such that $\dot{L} \le - \left( {{\rm K} - 1} \right){\rm E} + B.$ It can be seen that the regulation and velocity tracking errors and NN weight estimation errors are bounded during the inter-event period since the unknown NN weights are not updated, they remain constant during the inter-event period.