Adaptive Neural Network Control of Underwater Robotic Manipulators Tuned by a Genetic Algorithm

Abstract

This paper describes a novel approach for the control of underwater robots that can handle uncertainties and disturbance problems, which are commonly met in underwater environments. The considered system is an underwater manipulator with n-degrees of freedom. The approximation capability of an adaptive neural network is exploited to estimate uncertainties in system dynamics. Drag and lift forces are considered as an external disturbance, and a disturbance observer approach which has been proved to be effective with on-land robotic systems, is applied to compensate for it. The objective of the controller designed is to track a desired trajectory. To find the optimal gain parameters of this controller, a classical Genetic Algorithm is employed. Extensive simulation studies carried out on a two degrees of freedom manipulator indicate the efficacy of the proposed approach, proving that the disturbance observer originally developed for on-land systems can also be used effectively for underwater robotic systems. Finally, the performance of the proposed controller, tuned by the Genetic Algorithm is compared with that of a controller, tuned manually. The results show that the reliance on a well-known classic Genetic Algorithm for the tuning of the controller parameters not only saves time, but also provides better values of the parameters.

Appendix: : The proof of the system stability

To prove the stability of the considered system under the designed controller, we consider the Lyapunov function in Eq. ?? and take its time derivative to get:

$$ \begin{array}{@{}rcl@{}} \dot{V} &= &-{e_{1}^{T}} K_{1} e_{1} +{e_{1}^{T}}e_{2} +\frac{1}{2}{e_{2}^{T}} \dot{M}_{c} e_{2} +\frac{1}{2}{e_{2}^{T}}M_{c} \dot{e}_{2}+ \tilde{f}_{T}^{T} \dot{\tilde{f}}_{T}\\ &&+\sum\limits_{i = 1}^{N}\tilde{W}_{i}^{T} \psi_{i}^{-1} \dot{\tilde{W}}_{i} \end{array} $$

(1)

Substituting Eq. ?? and ?? in Eq. 1 results in:

$$ \begin{array}{@{}rcl@{}} \dot{V} &= &-{e_{1}^{T}} K_{1} e_{1}+ {e_{1}^{T}}e_{2} +\frac{1}{2}{e_{2}^{T}} \dot{M}_{c} e_{2} +{e_{2}^{T}}\left[-e_{1}- K_{2} e_{2}\right.\\ &&+ \left.C_{c} \alpha+ \hat{W}^{T}{\Theta} (h) +\hat{f}_{T} - C_{c} x_{2} -{W}^{T}{\Theta} (h)-f_{T} \right]\\ &&+ \tilde{f}_{T}^{T} \dot{\tilde{f}}_{T} +\sum\limits_{i = 1}^{N}\tilde{W}_{i}^{T} \psi_{i}^{-1} \dot{\tilde{W}}_{i} \end{array} $$

(2)

$$ \begin{array}{@{}rcl@{}} \dot{V} &=& -{e_{1}^{T}} K_{1} e_{1}-{e_{2}^{T}} K_{2} e_{2}+ \frac{1}{2}{e_{2}^{T}} (\dot{M}_{c}-2C_{c} ) e_{2}\\ &&+{e_{2}^{T}}\left[\tilde{W}^{T}{\Theta} (h) + \tilde{f}_{T}\right] + \tilde{f}_{T}^{T} \!\dot{\tilde{f}}_{T} + \sum\limits_{i = 1}^{N}\tilde{W}_{i}^{T} \!\psi_{i}^{-1} \dot{\tilde{W}}_{i} \end{array} $$

(3)

$\frac {1}{2}e_2^T (\dot {M}_c-2C_c ) e_2 = 0$ because $\dot {M}_c-2C_c$ is a skew symmetric matrix. Substituting Eq. ?? in 3 yields:

$$ \begin{array}{@{}rcl@{}} \dot{V} &=&-{e_{1}^{T}} K_{1} e_{1} -{e_{2}^{T}} K_{2} e_{2} +{e_{2}^{T}}\left[\tilde{W}^{T}{\Theta} (h) +\tilde{f}_{T}\right] - \tilde{f}_{T}^{T} B M_{c}^{-1} \tilde{f}_{T} \\&&+\tilde{f}_{T}^{T} B M_{c}^{-1} W{\Theta}(h) - \tilde{f}_{T}^{T} \dot{f}_{T} +\sum\limits_{i = 1}^{N}\tilde{W}_{i}^{T} \psi_{i}^{-1} \dot{\tilde{W}}_{i} \end{array} $$

(4)

We have $\tilde {W}_i =\hat {W}_i-W_i$, then we can write

$$ \begin{array}{@{}rcl@{}} \sum\limits_{i = 1}^{N}\tilde{W}_{i}^{T} \psi_{i}^{-1} \dot{\tilde{W}}_{i} = &-&\sum\limits_{i = 1}^{N}\tilde{W}_{i}^{T} [{\Theta}_{i} (h)e_{2,i}+ \gamma_{i} \vert e_{2} \vert\hat{W}_{i}]\\ &+& \sum\limits_{i = 1}^{N}\tilde{W}_{i}^{T} \psi_{i}^{-1} \dot{W}_{i} \end{array} $$

(5)

Substituting in Eq. 4 results in:

$$ \begin{array}{@{}rcl@{}} \dot{V} \leq& -&{e_{1}^{T}} K_{1} e_{1} -{e_{2}^{T}} K_{2} e_{2}+ {e_{2}^{T}} \tilde{f}_{T}- \tilde{f}_{T}^{T} B M_{c}^{-1} \tilde{f}_{T} - \tilde{f}_{T}^{T} \dot{f}_{T}\\&+&\tilde{f}_{T}^{T} B M_{c}^{-1} W{\Theta}(h) -\sum\limits_{i = 1}^{N} \tilde{W}_{i}^{T} \gamma_{i} \vert e_{2} \vert\hat{W}_{i} \end{array} $$

(6)

We have :

$$ - {e_{2}^{T}} \tilde{f}_{T} \leq \frac{1}{2} \Vert e_{2}\Vert^{2} +\frac{1}{2} \Vert \tilde{f}_{T} \Vert^{2} $$

(7)

$$ - \tilde{f}^{T}_{T} \dot{f}_{T} \leq \frac{1}{2} \tilde{f}_{T}^{T} \tilde{f}_{T} +\frac{1}{2} \xi^{2} $$

(8)

$$ \begin{array}{@{}rcl@{}} \tilde{f}_{T}^{T} B M_{c}^{-1} W{\Theta}(h) &\leq& \frac{1}{2} \Vert B M_{c}\Vert^{2} \Vert \tilde{f}_{T} \Vert^{2}\\ &&+\frac{1}{2} \sum\limits_{i = 1}^{N} \Vert W_{i}\Vert^{2}\Vert {\Theta}_{i}(h)\Vert^{2} \end{array} $$

(9)

$$ -\sum\limits_{i = 1}^{N} \tilde{W}_{i}^{T} \gamma_{i} \vert e_{2} \vert\hat{W}_{i}\leq \frac{1}{2}{e_{2}^{T}} e_{2} + \frac{1}{8} \sum\limits_{i = 1}^{N} {\gamma_{i}^{2}} \left( \Vert \tilde{W}_{i} \Vert^{2} -\Vert W_{i} \Vert^{2} \right)^{2} $$

(10)

According to Proposition 1, we can write:

$$ \Vert \tilde{W}_{i} \Vert \leq \frac{\epsilon_{i}}{\gamma_{i}} + \Vert W_{i} \Vert = d $$

(11)

For the activation function of the RBF NN, there is a constant 𝜖 > 0 where ∥Θ_i(h)∥≤ 𝜖,i = 1, 2,…,N. Substituting Eqs. 7–11 in Eq. 6 results in:

$$ \begin{array}{@{}rcl@{}} \dot{V} \leq &-&{e_{1}^{T}} K_{1} e_{1}-{e_{2}^{T}}(K_{2}-I_{n \times n})e_{2} + \frac{1}{2} \xi^{2}\\ &-&\tilde{f}_{T}^{T} \left( -\frac{\Vert B M_{c}^{-1} \Vert^{2} + 2 } {2} I_{n \times n} +B M^{-1}_{c}\right) \tilde{f}_{T}\\ &-&\sum\limits_{i = 1}^{n} \frac{{\gamma_{i}^{2}}}{4}\Vert W_{i} \Vert^{2} \Vert \tilde{W}_{i} \Vert^{2} +\sum\limits_{i = 1}^{n} \frac{{\gamma_{i}^{2}}}{8}\left( \Vert W_{i} \Vert^{4}+ d^{4}\right)\\ \leq & -&a V +b \end{array} $$

(12)

where

$$ \begin{array}{@{}rcl@{}} b&=&\sum\limits_{i = 0}^{n}{\frac{\gamma_{i}+\epsilon^{2}}{2}\Vert{W}_{i}\Vert^{2}} + \sum\limits_{i = 0}^{n}{\frac{{\gamma_{i}^{2}}}{8}\left( \Vert{W}_{i}\Vert^{4} + d^{4} \right)} + \frac{1}{2}\xi^{2}\\ a&=& \min\left( 2 \lambda_{\min}(K_{1}), \frac{2\lambda_{\min}(K_{2}- I_{n\times n})}{\lambda_{\max}(M)},\right.\\ && 2 \lambda_{\min}\left( BM_{c}^{-1}- \left( 1+ \frac{1}{2}\Vert B M_{c}^{-1} \Vert^{2} \right) I_{n \times n}\right),\\&& \left.\min_{i = 1,2,..,n} \left( \frac{{\gamma_{i}^{2}} \Vert W_{i} \Vert^{2}} {2\lambda_{\max} (\psi_{i}^{-1})}\right)\right) \end{array} $$

(13)

In order to guarantee a positive value of a, the gain matrices K₁, K₂ and Φ(e₂) are designed to guarantee the following conditions:

$$ \begin{array}{@{}rcl@{}} &&\lambda_{\min}(K_{1})>0, \lambda_{\min}\left( BM_{c}^{-1}- \left( 1+ \frac{1}{2}\Vert B M_{c}^{-1} \Vert^{2} \right) I_{n \times n}\right) > 0,\\ &&\lambda_{\min} \left( K_{2}-I_{n\times n}\right) > 0 \end{array} $$

(14)

Multiply both sides of Eq. 12 with e^at to get:

$$ \frac{d}{dt} \left( V e^{at}\right) \leq b e^{at} $$

(15)

Integrating Eq. 15 over the interval [0,t] yields:

$$ V \leq V(0) +\frac{b}{a} $$

(16)

Since all terms of Lyapunov function (??) are positive, for e₁ we can write:

$$ \frac{1}{2} \Vert e_{1}\Vert^{2} \leq V(0) +\frac{b}{a} \rightarrow \Vert e_{1}\Vert \leq 2(\sqrt{V(0) +\frac{b}{a}}) = \sqrt{S} $$

(17)

The other error signals boundedness can be proved in the same way. Thus the stability of this system is proved.