Artificial Neural Network for Improving the Motion Control Quality of Underwater Remotely Operated Vehicle

2.1. Remotely operated vehicle model / Model daljinski upravljane ronilice (engl. ROV)

The motion of underwater vehicles is analyzed in six degrees of freedom using two coordinate systems. One of these is the body-fixed frame XYZ, which is related to the submerged vehicle. The other is the Earth-fixed frame X₀Y₀Z₀, which describes the vehicle's motion concerning the Earth (as shown in Fig. 1). Based on these two primary reference frames, the kinematic equations of the ROV are defined as follows [30]:

Figure 1 Thecoordinate system of the ROV motion

Slika 1. Koordinatni sustav kretanja ROV-a

[TeX:] \dot{\eta} = \ J(\eta)v$\dot{\eta }=J(\eta )v$ (1)

where [TeX:] \eta$\eta$ denotes the position and heading of the ROV within the Earth's coordinate system, while [TeX:] v$v$ represents the linear velocity along the body coordinate system of the ROV. The dynamic equation of the ROV is given by [31]

[TeX:] M\dot{v}\ + C(v)v + D(v)v + g(\eta) = \tau$M\dot{v}+C(v)v+D(v)v+g(\eta )=\tau$ (2)

in which M represents the total inertial mass of the ROV, including its augmented mass. C refers to the combined effects of the Coriolis force and the Centrifugal force generated by the actual weight of the ROV. Additionally, D signifies the damping component, which encompasses both linear and nonlinear elements. The variable g represents the vector of gravitational acceleration acting on the vehicle as it moves, while [TeX:] \tau\left\lbrack \tau_{x},\tau_{y},\ \tau_{z} \right\rbrack$\tau [{\tau }_x,{\tau }_y,{\tau }_z]$ denotes the vector that sums the forces and moments generated by the propulsion system and the rudder.

Remark 1: Environmental disturbances significantly affect the dynamics of ROV motion, introducing nonlinear factors that are challenging to quantify. As a result, the performance of the motion control system deteriorates.

In actual operations, the current is the main component affecting the ROVs during underwater missions. The impact of currents on the body alters the motion velocity, denoted as [TeX:] v$v$. Therefore, it is assumed that the motion can be expressed using the relative velocity[TeX:] v_{r}$v_r$ [32] as follows:

[TeX:] v_{r} = v - \vartheta_{c}\text{\ \ }$v_r=v-{\vartheta }_c$(3)

The axial current velocity [TeX:] \vartheta_{c}\left\lbrack u_{c},v_{c},w_{c} \right\rbrack${\vartheta }_c[u_c,v_c,w_c]$ can be converted to [TeX:] \vartheta_{c}^{e}\left\lbrack u_{c}^{e},v_{c}^{e},w_{c}^{e} \right\rbrack^{T}${\vartheta }_c^e{[u_c^e,v_c^e,w_c^e]}^T$ , which is the current velocity along the Earth frame using the rotation matrix. The components of the vector [TeX:] \vartheta_{c}^{e}${\vartheta }_c^e$ are expressed as

[TeX:] v_{c}^{e}\ = \ V_{c}\sin\left( \beta_{c} \right)$v_c^e=V_{c}sin\left({\beta }_c\right)$ (4)

whereas [TeX:] V_{c}$V_c$ represents the current velocity, [TeX:] \alpha_{c}${\alpha }_c$ is the drift angle, and the slip angle [TeX:] \beta_{c}${\beta }_c$ describes the direction of the current velocity.

Remark 2: Using conventional methods to control nonlinear ROV motion is inefficient. Additionally, as noted in Remark 1, the control deviation will increase significantly due to environmental disturbances.

2.2. Stages and goals / Faze i ciljevi

The AFC controller exhibits good response characteristics to nonlinear effects due to its capability to adapt to variations in input errors. However, the underwater environment introduces numerous error components that the basic solution does not handle effectively. To address this issue, it is essential to enhance the quality of control parameters using the ANN that has self-learning capabilities. The improvement aims to reduce random errors caused by the environment. Building an ROV control solution involves three main stages, as follows:

3.1. Fuzzy Adaptive Control for ROV Motion / Neizrazito adaptivno upravljanje za kretanje ROV-a

This study introduces a fuzzy technique for adaptive tuning of basic PID controller parameters [33], referred as the AFC controller. This technique can adjust the parameters [TeX:] K$K$([TeX:] K_{\text{pf}}$K_{\mathrm{\text{pf}}}$,[TeX:] K_{\text{if}}$K_{\mathrm{\text{if}}}$,[TeX:] K_{\text{df}}$K_{\mathrm{\text{df}}}$) in response to changes in the input error. The AFC control structure[34] for the adaptive tuning of the basic controller parameters is defined by

[TeX:] \tau_{\text{AFC}}(t) = K_{\text{pf}}e(t) + K_{\text{if}}\int_{0}^{t}{e(t)dt} + K_{\text{df}}\frac{\text{de}(t)}{\text{dt}}${\tau }_{\mathrm{\text{AFC}}}(t)=K_{\mathrm{\text{pf}}}e(t)+K_{\mathrm{\text{if}}}{\int }_0^{t}e(t)dt+K_{\mathrm{\text{df}}}\frac{\mathrm{\text{de}}(t)}{\mathrm{\text{dt}}}$ (5)

The fuzzy model consists of a set of rules along with their corresponding results [35]. The If-Then rules are described as follows:

[TeX:] R_{i}:\ \text{If}\ {\widehat{x}}_{1}\ \text{is}\ A_{1}^{i}\text{...\ }\text{and}\ {\widehat{x}}_{n}\ \text{is}\ A_{n}^{i}\ \text{then}\text{\ k\ }\text{is}\ B^{i}$R_i:\mathrm{\text{If}}{\hat{x}}_1\mathrm{\text{is}}{A}_1^i\mathrm{\text{...}}\mathrm{\text{and}}{\hat{x}}_n\mathrm{\text{is}}{A}_n^i\mathrm{\text{then}}\mathrm{\text{k}}\mathrm{\text{is}}{B}^i$ (6)

with [TeX:] A_{1}^{i},A_{2}^{i},\ldots A_{n}^{i}$A_1^i,A_2^i,\dots A_n^i$and [TeX:] B^{i}$B^i$ representing the fuzzy sets of input and output signals. This paper employs a fuzzy model based on Mamdani rules, utilizing the Max-Min inference rule and the Centroid defuzzification method. q represents the number of If-Then rules, and [TeX:] \mu_{A_{j}^{i}}\left( \widehat{x_{j}} \right)${\mu }_{A_j^i}\left(\widehat{x_j}\right)$ denote the Membership Functions (MFs). The outputs of the fuzzy model [TeX:] K$K$ are given by

[TeX:] K(\widehat{x})\ = \ \frac{\sum_{i = 1}^{q}B^{i}\left\lbrack \min_{j = 1}^{n}\mu_{A_{j}^{i}}({\widehat{x}}_{j}) \right\rbrack}{\sum_{i = 1}^{q}\left\lbrack \min_{j = 1}^{n}\mu_{A_{j}^{i}}({\widehat{x}}_{j}) \right\rbrack}$K(\hat{x})=\frac{{\sum }_{i=1}^q{B}^i[{min}_{j=1}^n{\mu }_{A_j^i}({\hat{x}}_j)]}{{\sum }_{i=1}^q[{min}_{j=1}^n{\mu }_{A_j^i}({\hat{x}}_j)]}$ (7)

Let [TeX:] \varnothing(\widehat{x}) = {\lbrack\varnothing^{1},\varnothing^{2},\ldots,\ \varnothing^{q}\rbrack}^{T} \in R^{q}$⌀(\hat{x})={[{⌀}^1,{⌀}^2,\dots ,{⌀}^q]}^T\in R^q$ represent a fuzzy basic vector [36] that is defined as

[TeX:] \varnothing(\widehat{x})\ = \ \frac{\left\lbrack \min_{j = 1}^{n}\mu_{A_{j}^{i}}({\widehat{x}}_{j}) \right\rbrack}{\sum_{i = 1}^{h}\left\lbrack \min_{j = 1}^{n}\mu_{A_{j}^{i}}({\widehat{x}}_{j}) \right\rbrack}$⌀(\hat{x})=\frac{[{min}_{j=1}^n{\mu }_{A_j^i}({\hat{x}}_j)]}{{\sum }_{i=1}^h[{min}_{j=1}^n{\mu }_{A_j^i}({\hat{x}}_j)]}$ (8)

The fuzzy sets are adjusted using the parameter vector [TeX:] \delta^{T}${\delta }^T$ , which corresponds to [TeX:] B^{i}(i = 1,\ 2,...q)$B^i(i=1,2,...q)$ . The output of the fuzzy model is presented as linearized parameters as follows:

The fuzzy model is based on the input reference value, the error e, and the error velocity de, to adjust the values of the three adaptation coefficients [TeX:] K_{\text{pf}}$K_{\mathrm{\text{pf}}}$ , [TeX:] K_{\text{if}}$K_{\mathrm{\text{if}}}$ , and [TeX:] K_{\text{df}}$K_{\mathrm{\text{df}}}$ . The relationship between the input and output values is represented in 3D space, as illustrated in Fig. 2.

(a)	(b)	(c)

Figure 2 Relationship between inputs e, de and outputs. (a) Output [TeX:] K_{\text{pf}}$K_{\mathrm{\text{pf}}}$ ; (b) Output [TeX:] K_{\text{if}}$K_{\mathrm{\text{if}}}$ ; and (c) Output [TeX:] K_{\text{df}}$K_{\mathrm{\text{df}}}$

Slika 2. Odnos između ulaza e, de i izlaza. (a) Izlaz [TeX:] K_{\text{pf}}$K_{\mathrm{\text{pf}}}$ ; (b) Izlaz [TeX:] K_{\text{if}}$K_{\mathrm{\text{if}}}$ ; i (c) Izlaz [TeX:] K_{\text{df}}$K_{\mathrm{\text{df}}}$

Fig. 2 illustrates how each fuzzy rule produces an output value based on changes in e and de. In the case of the output [TeX:] K_{\text{pf}}$K_{\mathrm{\text{pf}}}$, as shown in Fig. 2a, when both e and de are negative, [TeX:] K_{\text{pf}}$K_{\mathrm{\text{pf}}}$ increases to a positive value to correct the reduction in error. Conversely, [TeX:] K_{\text{pf}}$K_{\mathrm{\text{pf}}}$ decreases when e and de become positive. For the output [TeX:] K_{\text{if}}$K_{\mathrm{\text{if}}}$, depicted in Fig. 2b, the correction is also inversely proportional to the input error. However, the correction value for [TeX:] K_{\text{if}}$K_{\mathrm{\text{if}}}$ is set much lower to prevent significant fluctuations. The response of [TeX:] K_{\text{df}}$K_{\mathrm{\text{df}}}$, shown in Fig. 2c, must be flexible to ensure system stability. Specifically, if the e and de values exceed acceptable thresholds (either too high or too low), [TeX:] K_{\text{df}}$K_{\mathrm{\text{df}}}$ will increase sharply to respond to these rapid changes. Besides, as e and de approach zero, [TeX:] K_{\text{df}}$K_{\mathrm{\text{df}}}$ tends to decrease slightly to help maintain system stability.

3.2. The ROV Motion Control for Using Artificial Neural Network / Upravljanje kretanjem ROV-a korištenjem umjetnim neuronskim mrežama

Figure 3 Overview diagram of the ROV control system utilizing an ANN controller

Slika 3. Pregledni dijagram sustava upravljanja ROV-om s ANN regulatorom

The AFC solution for controlling ROV motion is effective only with a defined range of flexible parameter adjustments. The influence of environmental conditions, as noted in Remarks 1 and 2, can result in significant discrepancies in control response. Fig. 3 provides an overview of the proposed solution to improve the motion control of the ROV. This paper proposes an ANN to enhance both the processing speed and accuracy of the ROV motion control, as described in Algorithm 1. The process of building an ROV motion controller using an ANN consists of four steps:

The feedforward network is selected for its straightforward structure, linear computation from input to output, and state memory mechanisms found in more complex networks.

In this study, we utilize Hyperbolic Tangent functions for the hidden layer and a linear activation function for the output layer to attain optimal performance. The output of the hidden layer [37-38] is defined by

[TeX:] h(z_{j}) = \tanh(z_{j}) = \frac{2}{1 + e^{- 2z_{j}}} - 1$h(z_j)=tanh(z_j)=\frac{2}{1+e^{-2z_j}}-1$ (11)

The output of the hidden layer[TeX:] z_{j}$z_j$is computed as follows[39-40]:

[TeX:] z_{j}\ = h\left( \omega_{j}\chi_{i} + \beta_{j} \right)$z_j=h({\omega }_j{\chi }_i+{\beta }_j)$ (12)

where [TeX:] \omega_{j}${\omega }_j$ represents the weight connecting neuron [TeX:] i$i$ in the input layer to neuron [TeX:] j$j$ in the hidden layer. Additionally, [TeX:] \chi_{i}${\chi }_i$ denotes the input for neuron [TeX:] i$i$ in the input layer. The notation [TeX:] \beta_{j}${\beta }_j$ signifies the bias associated with neuron [TeX:] j$j$ in the hidden layer, which helps adjust the neuron's activation threshold and enhances its ability to learn complex patterns. Similarly, the input to neuron [TeX:] \mathcal{l}$𝓁$ in the output layer is computed in the same way as described in (13).

[TeX:] z_{\mathcal{l}}\ = \ h\left( \omega_{\mathcal{l}}z_{j} + \beta_{\mathcal{l}} \right)$z_{𝓁}=h({\omega }_{𝓁}{z}_j+{\beta }_{𝓁})$ (13)

there in[TeX:] z_{j}$z_j$is the activated output from the hidden layer j, while [TeX:] \omega_{\mathcal{l}}${\omega }_{𝓁}$determines its influence on the output neuron and the bias [TeX:] \beta_{\mathcal{l}}${\beta }_{𝓁}$ adjusts activation. Consequently, the [TeX:] {\widehat{\tau}}_{\mathcal{l}}${\hat{\tau }}_{𝓁}$ output value of the ANN is obtained through the linear activation function as[41]

[TeX:] {\widehat{\tau}}_{\mathcal{l}}\ = f\left( z_{\mathcal{l}} \right) = \left\lbrack {\widehat{\tau}}_{x},{\widehat{\tau}}_{y},\ {\widehat{\tau}}_{z} \right\rbrack${\hat{\tau }}_{𝓁}=f\left(z_{𝓁}\right)=[{\hat{\tau }}_x,{\hat{\tau }}_y,{\hat{\tau }}_z]$ (14)

Several evaluation criteria exist for ANN models, including Root Mean Square Error of Approximation (RMSEA) [42] and Mean Square Error (MSE) [43]. In this study, we use the MSE function to assess the difference between the predicted output [TeX:] {\widehat{\tau}}_{\mathcal{l}}${\hat{\tau }}_{𝓁}$ and the desired output [TeX:] \tau_{\mathcal{l}}${\tau }_{𝓁}$ as follows:

[TeX:] MSE\ = \ \frac{1}{N}\sum_{k = 1}^{N}\left( \tau_{\mathcal{l}} - \ {\widehat{\tau}}_{\mathcal{l}} \right)^{2}$MSE=\frac{1}{N}{\sum }_{k=1}^N{({\tau }_{𝓁}-{\hat{\tau }}_{𝓁})}^2$ (15)

To optimize the learning performance of the ANN, this study utilizes the Levenberg-Marquardt training algorithm in conjunction with the Backpropagation technique[44]. This hybrid approach accelerates convergence by integrating the strengths of both Gradient Descent and Gauss-Newton methods, enhancing accuracy and expediting error minimization. The process for adjusting the weights using the Levenberg-Marquardt method [45] is outlined as

[TeX:] \omega(t + 1) = \omega(t) - {(Z^{T}Z + \lambda)}^{- 1}Z^{T}e(t)$\omega (t+1)=\omega (t)-{(Z^{T}Z+\lambda )}^{-1}{Z}^{T}e(t)$ (16)

in which, [TeX:] \omega(t)$\omega (t)$denotes the weight vector at the 𝑡 iteration with 𝑍 being the Jacobian matrix. Besides, [TeX:] e(t)$e(t)$expresses the error at the 𝑡 iteration, and λ is the adjustment parameter. Depending on the value of 𝜆, the proposed ANN solution obtains different convergence values. If the value of 𝜆 is large, the ANN network converges slowly but stably. On the contrary, if the value of 𝜆 is small, the algorithm converges quickly but is prone to fluctuations.

4.1. Configuration Parameters / Parametri konfiguracije

The specifications for this study's ROV are based on the RRC ROV II model [46]. Table 1 presents the dynamic parameters relevant to the ROV's performance during aquatic maneuvers. To evaluate the efficacy of the proposed control strategies, the authors executed experiments encompassing both straight-line motion control (Scenario 1) and path-planning motion control (Scenario 2).

Table 1The RRC ROV II parameters.

Tablica 1. Parametri RRC ROV II

Table 2 The tuning ranges of the AFC controller.

Tablica 2. Rasponi parametara za podešavanje AFC regulatora

The tuning ranges of the fuzzy functions for the AFC controller parameters are shown in Table 2. Thereby, adjusting the baseline parameters helps determine the optimal signal required to produce the output signal, which will serve as the training data set for the ANN.

Table 3The ANN training parameters. Tablica 3. Parametri treniranja ANN-a Parameter Value Epoch 500 Total node in hidden layer 1 32 Total node in hidden layer 2 16 Learning rate 0.000001 Loss function MSE Training Ratio 70% Validate Ratio 15% Testing Ratio 15%

Figure 4 Result of the ANN model training Slika 4. Rezultat treniranja ANN modela

The authors develop an ANN solution to control ROV motion in the MATLAB 2024a environment, using a GPU configuration with an RTX 4060 8GB graphics and 32GB RAM, which enables parallel processing to reduce calculation time. The training parameters for the NN across different motion directions are detailed in Table 3. The training process was carried out over 500 epochs to achieve optimal accuracy. Upon completion, a convergence value approaching 1 signified enhanced accuracy for the ANN. If the convergence value is inadequate, adjustments to key parameters, including the number of neurons, the choice of activation function, or the learning rate, should be made to optimize network performance. The dataset consists of 630,000 samples for each category within every information field, including e, de, and τ_AFC. Additionally, the training process requires standardizing the amount of data across the information fields and normalizing the input data using the Hyperbolic Tangent activation function, which scales the values between -1 and 1. The dataset was partitioned into three subsets: a training set (70%), a test set (15%), and a validation set (15%). The stratified division aimed to enhance the model's ability to generalize, facilitating effective adaptation to novel input values. Training outcomes for the proposed control model are illustrated in Fig. 4.

Table 4 The performance indices of the NN models.

Tablica 4. Pokazatelji učinkovitosti NN modela

The model's performance is assessed by comparing the accuracy of predicted values to actual values during training. Shorter training durations can result in faster convergence and reduce the risk of overfitting, whereas longer training times may increase costs and require more powerful hardware. Table 4 shows that the proposed ANN model has a significantly shorter training time than a neural network online tuning (named NN-PID) [47] and the Adaptive Neuro-Fuzzy Inference System (ANFIS) [48]. As the dataset size increases from 20% to 100%, the training time for the ANN goes from 17s to 52s, while NN-PID increases from 22s to 107s and ANFIS from 35s to 195s. The performance outcome indicates that the ANN is faster and more efficient. Additionally, the MSE for the ANN is lower than that of ANFIS across all data levels, demonstrating better predictive performance. While the ANN's MSE decreases from 5.38 to 0.472 with larger datasets, it remains significantly lower than ANFIS (4.087) and NN-PID (1.854), highlighting the ANN model's effectiveness.

4.2. Results and Discussions / Rezultati i rasprava

4.2.1. Experiment of the ROV moving straight / Eksperiment kretanja ROV-a po ravnoj putanji

Figure 5 The ROV motion in Scenario 1

Slika 5. Kretanje ROV-a u scenariju 1

Figure 6 The results of ANN and AFC controller in Scenario 1

Slika 6. Rezultati ANN i AFC regulatora u scenariju 1

In Scenario 1, the ANN and AFC control methods are applied to control the ROV to follow a straight motion, starting from the reference position [0m, 0m, 0m] and moving to the desired position [20m, 10m, 20m] within a duration of 200 seconds. The first scenario is considered under calm sea conditions, with the current velocity parameters [TeX:] V_{c}$V_c$ varying in the range of 0.1m/s to 0.3m/s, α_c, and [TeX:] \beta_{c}\ ${\beta }_c$in the range of [0, 𝜋/2]. The results are shown in Figs. 5 and 6. Specifically, the 3D path tracking results in Fig. 5 show that both the ANN (blue line) and AFC (red line) can control the ROV to follow the path in the scenario. Besides, Fig. 6 illustrates the control signals on the directions applying ANN and AFC controllers, allowing for a comparison of its responses.

Table 5 Comparison of controller responses in Scenario 1.

Tablica 5. Usporedba odziva regulatora u scenariju 1

The response of various control solutions, including AFC, NN-PID, ANFIS, and ANN, is summarized in Table 5. The ANN has the slowest response time among the solutions, especially during a surge. The proposed controller is slower than the ANFIS controller by 0.6s, the NN-PID by 0.2s, and the AFC controller by 1.7s. However, the ANN solution shows improved response fluctuations. In the y-direction (sway), the ANN results are lower by 0.03m, 0.02m, and 0.04m compared to ANFIS, NN-PID, and AFC controllers, respectively. Similarly, the recorded overshoot demonstrates that the ANN shows better response results. Specifically, in the heave, the overshoot of ANN is the lowest among the approaches, as lower than 0.37%, 0.2%, and 0.5% for ANFIS, NN-PID, and AFC controllers, respectively. Therefore, although the response time is slower, other factors in evaluating the response are better, showing the capability of the ANN proposed in this study.

4.2.2. Maneuvering the ROV along its path / Manevriranje ROV-om duž zadane putanje

Figure 7 The path of the ROV motion in Scenario 2

Slika 7. Putanja kretanja ROV-a u scenariju 2

Figure 8 Response of the controllers implemented in Scenario 2.

Slika 8. Odziv regulatora implementiranih u scenariju 2

To comprehensively assess the adaptability and responsiveness of the ANN controller under realistic operating conditions for the ROV, Scenario 2 was executed, taking into account the current dynamics. The scenario involved a more intricate control path: the ROV first moved a straight path along the surge for 10m, then transitioned to a straight line in the y-direction for 5m, followed by a descent of 10m, culminating at the endpoint coordinates [10m, 10m, 10m]. Notably, the current velocity was elevated, exceeding 0.5m/s.

Table 6 Summary of controller response comparisons in Scenario 2.

Tablica 6. Sažetak usporedbi odziva regulatora u scenariju 2

The simulation outcomes, illustrated in Figs. 7 and 8 reveal that the ANN controller (depicted by the blue line) adheres more closely to the predefined trajectory than the AFC, represented by the red line. Furthermore, Fig. 8 indicates that the control responses along each axis utilizing the ANN controller demonstrate superior performance relative to those of the AFC, highlighting the effectiveness of the ANN in managing complex control scenarios amid varying current conditions. In Scenario 2, the ROV operates in a strong current, demonstrating the advantages of the ANN controller in mitigating fluctuations and overshoots. The comparison results of the solutions are summarized in Table 6. In the surge direction, the response time of the ANN solution was recorded as 30.5s, which is 0.3s slower than ANFIS, 1.5s slower than NN-PID, and 1s slower than AFC. The ROV operates under the influence of flows with slow-changing characteristics, meaning that the accuracy of control responses is prioritized over response time. As a result, the findings from the ANN research indicate a longer response time compared to the alternative solution, which aligns with the characteristics of the control system. Additionally, the computer's performance remains within acceptable limits, with CPU usage at 10%, GPU at 5%, VRAM at 14%, and RAM at 78%.

Nevertheless, the ANN outperforms both alternatives when examining fluctuation and overshoot metrics. Specifically, the ANN's sway fluctuation is lower than that of ANFIS at 0.02m, NN-PID at 0.01m, and AFC at 0.04m. Concerning overshoot, the ANN recorded a minimum value of 0.64%, which is lower than ANFIS of 0.17%, NN-PID of 0.48%, and AFC of 2.18%. Although the ANN exhibits a longer response time, it maintains more excellent stability under identical environmental conditions compared to the ANFIS, NN-PID, and AFC, effectively addressing the concerns outlined in Remarks 1 and 2. However, the study faces limitations in training data diversity and deep feature extraction methods, which need to improve the control force accuracy in ROV operations during sudden environmental changes.