Artificial lateral line based relative state estimation between an upstream oscillating fin and a downstream robotic fish

Figure 1 shows the configurations of the robotic fish used in the experiments. It is a boxfish-inspired robotic fish whose size (length × width × height) is about 40 cm × 14.1 cm × 13.2 cm. It consists of a sealed shell, a pair of pectoral fins, and a caudal fin. The electrical system, which includes a credit card-sized micro-computer called Raspberry Pi serving as the main processor, rechargeable batteries, steering engines, circuit boards, and multiple onboard sensors, is wrapped in the sealed shell. As shown in figure 2, the onboard sensors include an inertial measurement unit (IMU) serving for attitudes monitoring, a camera serving for environment capturing, an infrared sensor serving for obstacle avoidance, and a nine pressure sensors (MS5803-01BA, TE Connectivity Ltd) based ALLS serving for external HPVs measurement. More about the ALLS can be found in [21]. Multiple 32-bit micro controllers (STM32F405 and STM32F103) on the circuit boards serve the functions of sensor data acquisition and steering engine control. Through controlling the steering engines connected with the paired pectoral fins and the caudal fin with given oscillating amplitude, frequency, and offset parameters, the fins are able to generate propulsive forces which actuate the robotic fish for realizing multiple swimming patterns including rectilinear motion, turning motion, gliding motion, spiral motion, yawing motion, rolling motion, pitching motion, etc. Such multiple swimming patterns enable two adjacent robotic fish to form multiple relative positions and attitudes, as shown in figure 3.

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As shown in figure 3, in the experiments, the pectoral fins and caudal fin of the downstream robotic fish have been removed. The two objects are located in a flume with a specific velocity of 0.175 m s⁻¹, with given relative states. The positions and attitudes of the two objects are controlled by controllable steering engines linked with them. Thus the relative states can be set, and there are 7 relative states including d_vertical, f, A, ϕ, α, β, and γ in total. For each relative state, we investigate p experimental parameters, and p = 7, 16, 6, 13, 19, 9, and 11 for d_vertical, A, f, ϕ, α, β, and γ, respectively. Details of the p experimental parameters mentioned above are defined in table 1. As we have mentioned in detail, this article mainly focuses on investigating the sensitivity of each pressure sensor, the insufficiency and redundancy of the pressure sensors, regression modeling, and model based relative state estimation using the measure experimental data, to avoid the repetition, more details about such a simplification, the control of relative states between the two adjacent objects, the experimental conditions, the experimental platform, and the method of measuring data can be found in [21]. The experimental scenes can be found in supplementary video (https://stacks.iop.org/JPBB/16/016012/mmedia). In the following part, we will use the ALLS-measured HPVs to estimate the above-mentioned relative states using the RF algorithm, BPNN, SVR method, and REG method.

Because of the hardware deficiency of the pressure sensors and the background noise in the environment, there exist significant fluctuations of the HPVs measured by the pressure sensors. In this case, we have smoothed the raw HPVs data using a Gaussian smoothing window function. Figure 4 shows the raw HPVs and the smoothed HPVs data. For the data in investigating d_vertical, α, β, and γ, the oscillating amplitude, frequency, and offset are set as 15°, 1 Hz, and 0. For the data in investigating ϕ, the above-mentioned parameters are 15°, 1 Hz, and 30°. For the data in investigating f, the above-mentioned parameters are 15°, 1.0 Hz, and 0. For the data in investigating A, the above-mentioned parameters are 30°, 1 Hz and 0. The sampling rate of the sensors is 50 Hz. In each recording, 250 samples of the HPVs are selected for forming the original sample set O used for regression model analysis. Taking the regression analysis for d_vertical for example, there are seven experimental parameters of d_vertical, varying from −45 mm to 45 mm, with an interval of 15 mm. For each experimental parameter, the HPVs recording is repeated five times. The final samples in the original sample set O for d_vertical have a size of 7 × 5 × 250 = 8750.

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RF, proposed by Breiman L, is a decision-tree-based ensemble learning algorithm which can be used for classification and regression [32]. In this article, RF is used for regression. A RF can be defined as $R=\left\{{T}_{1}\left(X\right),\dots ,{T}_{N}\left(X\right)\right\}$, consisting of N decision trees which are defined as T_i (i = 1, ..., N). $X=\left\{X\left(1\right),\dots ,X\left(p\right)\right\}$ is a p-dimensional state vector. In a regression task, by importing X into the RF R, N estimated values ${\hat{Y}}_{i}\left(i=1,\dots ,N\right)$ of Y are obtained, where ${\hat{Y}}_{i}$ indicates the estimated value of Y obtained by the tree T_i . $\hat{Y}=\frac{{\hat{Y}}_{1}+\cdots +{\hat{Y}}_{N}}{N}$ is the estimated result of Y using multiple classifiers in the RF. The classifiers can learn and predict the results using the inputs. By importing a sample data set D = {(X₁, Y₁), ..., (X_n , Y_n )} for training the RF model, a model which links X_j (j = 1, ..., n) to Y_j can be obtained [33]. In this article, each sample in the original sample set consists of the HPVs measured by the nine pressure sensors of the ALLS and value of the relative state. Each sample can be defined as $\left\{S,{P}_{0},{P}_{{L}_{1}},{P}_{{L}_{2}},{P}_{{L}_{3}},{P}_{{L}_{4}},{P}_{{R}_{1}},{P}_{{R}_{2}},{P}_{{R}_{3}},{P}_{{R}_{4}}\right\}$, where S indicates value of the relative state, and other variables indicate the ALLS-measured HPVs. In this article, for each sample, the state vector X includes nine HPVs measured by the ALLS, and $X=\left\{{P}_{0},{P}_{{L}_{1}},{P}_{{L}_{2}},{P}_{{L}_{3}},{P}_{{L}_{4}},{P}_{{R}_{1}},{P}_{{R}_{2}},{P}_{{R}_{3}},{P}_{{R}_{4}}\right\}$. Y is the relative state S.

Figure 5 shows the flow of the RF for the regression task. It includes four steps described as follows [32, 33].

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Step 1: obtain a bootstrap sample set of size n by randomly sampling with replacement from the original sample set O of size n, as a training set D of the established RF. The data samples which have not been sampled into the bootstrap sample set are called out-of-bag (OOB) samples and form a validation set V.

Step 2: randomly select m variables from M variables of the original sample set as branch variables at each node of each regression tree. Then determine the optimum splitting attributes according to the goodness of split criterion. In general, m = M/3 ± 1. If M < 3, m is set as 1. In this article, the variables of the original sample indicate nine ALLS-measured HPVs, so M = 1 ∼ 9.

Step 3: repeat Step 1 and 2 for N times, for obtaining N training sets and N validation sets. Then respectively establish an initial RF consisting of N regression trees T_i (i = 1, ..., N) for the above training sets and validation sets.

Figure 6 shows the mean of squared residuals with N for d_vertical, f, A, ϕ, α, β, and γ. It can be seen that the mean of squared residuals gradually decreases with the increasing N, and it converges to the minimum when N exceeds 400. In this article, N is set as 500.

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Step 4: calculate mean absolute error (MAE) and coefficient of determination (R²) of the estimated results using the original sample set O.

The artificial neural network has been widely used for modeling the relationship between the input signal and output signal. It has exhibited excellent performance in tasks of data classification, data speculation, and pattern identification [34]. BPNN is one of the neural networks which has the widest application. The information in a BPNN propagates forward, and the error propagates backward. The basic theory of BPNN is using gradient descent methods to make the mean square error between the actual output and the expected output least. For a BPNN, it consists of one input layer, one or more hidden layers, and one output layer. In each layer, there are several nodes [35]. In this article, the activation function of BPNN is the sigmoid function. And we have used L2 regularization (weight decay). A three-layer BPNN has been respectively structured for each experiment, as shown in figure 7. The BPNN has one input layer of which the number of nodes p = 9 and the nine HPVs in each sample are used as the node data. Besides, there is one output layer of which the number of nodes q = 1 and the relative state in each sample is used as the node data. Besides, there is one hidden layer of which the number of nodes m is determined as follows.

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Figure 8 shows the training time and coefficient of determination R² with respect to the number of the nodes m in the hidden layer of BPNN for the estimation of β. reported R² score is from a training set. It can be seen that the training time increases with m, and R² varies little when the number of nodes exceeds 6. In this case, we have determined m as 6 for the estimation of β. Similarly, m has been determined as 11, 10, 9, 6, 13, 6, and 10 for the estimation of d_vertical, A, f, ϕ, α, β, and γ, respectively, according to figure 9.

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As shown in figure 10, the training time increases with iterations of BPNN, and R² varies little when iterations exceeds 400. In this case, iterations of BPNN for α is determined as 400. reported R² score is from a training set. Figure 11 shows the coefficient of determination R² with iterations of BPNN for d_vertical, A, f, ϕ, α, β, and γ. Similarly, iterations of BPNN for d_vertical, A, f, ϕ, α, β, and γ are determined as 150, 250, 150, 200, 400, 150, and 300, respectively.

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SVR is used to describe regression using support vector methods. It is developed from support vector machine (SVM) by introducing an alternative loss function [36]. In this article, SVR with eps-regression and radial basis function as the kernel is used to investigate the regression relationship between the relative state and the ALLS-measured HPVs.

REG refers to investigating a linear regressive relationship between a dependent variable and two or more independent variables. In this article, we establish a model for linking the ALLS-measured HPVs to the relative states, as follows:

$$\begin{equation}Y={a}_{0}+\sum {a}_{k}X\left(k\right)+\varepsilon \end{equation} \tag{ 1 }$$

where Y is the relative state, a₀ is the intercept, a_k is the regression coefficient corresponding to the variable X(k), k = 1, 2, ..., 9, X(k) indicates the HPVs measured by the nine pressure sensors, and ɛ is the residual error. F-test is conducted for verifying the rationality of the model. To be specific, the model has the rationality to explain the linear relationship between the dependent variable and the independent variables only when all of the regression coefficients do not equal to zero at the same time.

In order to study the insufficiency and redundancy of the pressure sensors used for regression analysis, we have first proposed two criteria for measuring the sensitivity of the HPVs measured by the pressure sensors to the variations of relative states. The determination of the two criteria is described as follows.

Step 1: calculating the variations of the HPVs (ΔHPV) with the variations of the experimental parameters (ΔE) for each pressure sensor. The above experimental parameters include d_vertical, A, f, ϕ, α, β, and γ.

$$\begin{equation}{\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}=\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i+1}\left(k\right)}-\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}\end{equation} \tag{ 2 }$$

where i = 1, 2, ..., p − 1. p is the number of the experimental parameters, and p = 7, 16, 6, 13, 19, 9, and 11 for d_vertical, A, f, ϕ, α, β, and γ, respectively. $\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}$ indicates the mean HPVs of 500 samples in the ith experimental parameter for X(k).

Step 2: calculating the difference ${\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}$ between the maximum and the minimum of $\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}$.

$$\begin{equation}mm\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}=\mathrm{max}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}-\mathrm{min}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}\end{equation} \tag{ 3 }$$

Step 3: nondimensionalize the ${\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}$.

$$\begin{equation}{{\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}}^{\prime }=\frac{{\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}}{mm\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}}\end{equation} \tag{ 4 }$$

Step 4: calculate the mean of ${{\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}}^{\prime }$ and the mean of ${\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}$.

$$\begin{equation}{C}_{{k}_{1}}=\frac{{\sum }_{i=1}^{p-1}{{\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}}^{\prime }}{p-1}\end{equation} \tag{ 5 }$$

$$\begin{equation}{C}_{{k}_{2}}=\frac{{\sum }_{i=1}^{p-1}{\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}}{p-1}\end{equation} \tag{ 6 }$$

${C}_{{k}_{1}}$ and ${C}_{{k}_{2}}$ are the two criteria that are respectively used for measuring the sensitivity of X(k) to the variations of relative state. In the experiments, fish head of the downstream robot directly faces to the vortex wake generated by the upstream oscillating caudal fin, so the vortex wake has a more significant effect on the fish head. And thus, the HPVs measured by the pressure sensors around the fish head are significantly bigger than HPVs measured by pressure sensors at the posterior. In order to ensure the comparability of different pressure sensors, we have nondimensionalized the ${\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}$ and then calculate ${C}_{{k}_{1}}$. On the other hand, we have also calculated calculate ${C}_{{k}_{2}}$ using the ${\Delta}\bar{{\mathrm{H}\mathrm{P}\mathrm{V}}_{i}\left(k\right)}$, which has not been nondimensionalized.

Table 2 and 3 show ${C}_{{k}_{1}}$ and ${C}_{{k}_{2}}$ of each pressure sensor in each experiments. A bigger value of ${C}_{{k}_{1}}$ or ${C}_{{k}_{2}}$ means a bigger sensitivity. We have sorted the HPVs measured by the pressure sensors from biggest to smallest according to ${C}_{{k}_{1}}$ and ${C}_{{k}_{2}}$, respectively. Table 4 and 5 show the order of the HPVs measured by the pressure sensors from biggest to smallest based on the value of ${C}_{{k}_{1}}$ and ${C}_{{k}_{2}}$. It can be seen that there are significant differences between the two orders of the HPVs.

Table 2.  ${C}_{{k}_{1}}$ of each pressure sensor in each experiment.

	dvertical	A	f	ϕ	α	β	γ
P0	0.3165	0.0667	0.2000	0.1633	0.1691	0.2446	0.4535
${P}_{{L}_{1}}$	0.3117	0.1109	0.2000	0.1340	0.1268	0.2586	0.1495
${P}_{{L}_{2}}$	0.1672	0.2608	0.4971	0.2757	0.1420	0.1713	0.1205
${P}_{{L}_{3}}$	0.2676	0.3224	0.3555	0.2394	0.1336	0.2439	0.1000
${P}_{{L}_{4}}$	0.4260	0.3172	0.3354	0.2143	0.1113	0.3339	0.2280
${P}_{{R}_{1}}$	0.3280	0.0968	0.2025	0.1427	0.1205	0.2199	0.1708
${P}_{{R}_{2}}$	0.1723	0.2442	0.5251	0.2713	0.1276	0.1606	0.1167
${P}_{{R}_{3}}$	0.3130	0.3174	0.4818	0.2009	0.1346	0.3175	0.1000
${P}_{{R}_{4}}$	0.2750	0.3012	0.5028	0.2162	0.1030	0.2462	0.1009

Table 3.  ${C}_{{k}_{2}}$ of each pressure sensor in each experiment.

	dvertical	A	f	ϕ	α	β	γ
P0	11.3143	3.4602	2.5864	3.8769	5.2647	6.9166	0.9537
${P}_{{L}_{1}}$	2.3996	1.6968	1.7521	2.4489	4.4564	3.0227	1.6151
${P}_{{L}_{2}}$	1.4778	1.3393	1.1334	1.1984	2.0292	1.6177	1.0072
${P}_{{L}_{3}}$	0.6341	1.2951	0.8205	1.3811	1.1658	1.6542	0.5840
${P}_{{L}_{4}}$	1.0560	1.4444	0.4281	1.1442	1.2445	1.6735	0.2736
${P}_{{R}_{1}}$	2.2620	1.4399	1.5111	2.1011	4.1315	2.9065	1.4136
${P}_{{R}_{2}}$	1.7180	1.2739	0.8769	1.1659	2.0737	1.9250	0.9149
${P}_{{R}_{3}}$	0.7541	1.3676	0.6749	1.1011	1.2411	1.0540	0.4223
${P}_{{R}_{4}}$	0.6398	1.2407	0.9156	0.9262	1.2121	1.5360	0.1482

Table 4. Sorting the order of the HPVs measured by the pressure sensors according to ${C}_{{k}_{1}}$ (from biggest to smallest).

Experiments	Order
dvertical	${P}_{{L}_{4}}$, ${P}_{{R}_{1}}$, P0, ${P}_{{R}_{3}}$, ${P}_{{L}_{1}}$, ${P}_{{R}_{4}}$, ${P}_{{L}_{3}}$, ${P}_{{R}_{2}}$, ${P}_{{L}_{2}}$
A	${P}_{{L}_{3}}$, ${P}_{{R}_{3}}$, ${P}_{{L}_{4}}$, ${P}_{{R}_{4}}$, ${P}_{{L}_{2}}$, ${P}_{{R}_{2}}$, ${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, P0
f	${P}_{{R}_{2}}$, ${P}_{{R}_{4}}$, ${P}_{{L}_{2}}$, ${P}_{{R}_{3}}$, ${P}_{{L}_{3}}$, ${P}_{{L}_{4}}$, ${P}_{{R}_{1}}$, P0, ${P}_{{L}_{1}}$
ϕ	${P}_{{L}_{2}}$, ${P}_{{R}_{2}}$, ${P}_{{L}_{3}}$, ${P}_{{R}_{4}}$, ${P}_{{L}_{4}}$, ${P}_{{R}_{3}}$, P0, ${P}_{{R}_{1}}$, ${P}_{{L}_{1}}$
α	P0, ${P}_{{L}_{2}}$, ${P}_{{R}_{3}}$, ${P}_{{L}_{3}}$, ${P}_{{R}_{2}}$, ${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, ${P}_{{L}_{4}}$, ${P}_{{R}_{4}}$
β	${P}_{{L}_{4}}$, ${P}_{{R}_{3}}$, ${P}_{{L}_{1}}$, ${P}_{{R}_{4}}$, P0, ${P}_{{L}_{3}}$, ${P}_{{R}_{1}}$, ${P}_{{L}_{2}}$, ${P}_{{R}_{2}}$
Γ	P0, ${P}_{{L}_{4}}$, ${P}_{{R}_{1}}$, ${P}_{{L}_{1}}$, ${P}_{{L}_{2}}$, ${P}_{{R}_{2}}$, ${P}_{{R}_{4}}$, ${P}_{{L}_{3}}$, ${P}_{{R}_{3}}$

Table 5. Sorting the order of the HPVs measured by the pressure sensors according to ${C}_{{k}_{2}}$ (from biggest to smallest).

Experiments	Order
dvertical	P0, ${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, ${P}_{{R}_{2}}$, ${P}_{{L}_{2}}$, ${P}_{{L}_{4}}$, ${P}_{{R}_{3}}$, ${P}_{{R}_{4}}$, ${P}_{{L}_{3}}$
A	P0, ${P}_{{L}_{1}}$, ${P}_{{L}_{4}}$, ${P}_{{R}_{1}}$, ${P}_{{R}_{3}}$, ${P}_{{L}_{2}}$, ${P}_{{L}_{3}}$, ${P}_{{R}_{2}}$, ${P}_{{R}_{4}}$
F	P0, ${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, ${P}_{{L}_{2}}$, ${P}_{{R}_{4}}$, ${P}_{{R}_{2}}$, ${P}_{{L}_{3}}$, ${P}_{{R}_{3}}$, ${P}_{{L}_{4}}$
Φ	P0, ${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, ${P}_{{L}_{3}}$, ${P}_{{L}_{2}}$, ${P}_{{R}_{2}}$, ${P}_{{L}_{4}}$, ${P}_{{R}_{3}}$, ${P}_{{R}_{4}}$
Α	P0, ${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, ${P}_{{R}_{2}}$, ${P}_{{L}_{2}}$, ${P}_{{L}_{4}}$, ${P}_{{R}_{3}}$, ${P}_{{R}_{4}}$, ${P}_{{L}_{3}}$
Β	P0, ${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, ${P}_{{R}_{2}}$, ${P}_{{L}_{4}}$, ${P}_{{L}_{3}}$, ${P}_{{L}_{2}}$, ${P}_{{R}_{4}}$, ${P}_{{R}_{3}}$
Γ	${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, ${P}_{{L}_{2}}$, P0, ${P}_{{R}_{2}}$, ${P}_{{L}_{3}}$, ${P}_{{R}_{3}}$, ${P}_{{L}_{4}}$, ${P}_{{R}_{4}}$

Then we have successively selected the first (M) sensors in the information-criterion-ordered lists, for establishing the regression model and thus ensuring that the pressure sensors used are not redundant or insufficient. For example, when M = 3, we used the first M pressure sensors for regression analysis. The comparisons of regression results corresponding to ${C}_{{k}_{1}}$ and ${C}_{{k}_{2}}$ have been conducted in the following part.

Importance measurement is conducted for evaluating the importance of the HPV measured by a pressure sensor (variable X(k)(k = 1, ..., p) in the state vector X) to the relative state in RF-based analysis. The detailed processes of conducting importance measurement are as follows.

Step 1: obtain an initial RF model, including N trees. Then respectively calculate the mean-square error (MSE) of the estimated results using N OOB samples. And the obtained MSEs can be defined as MSE_i (i = 1, ..., N), taking the form as

$$\begin{equation}{\mathrm{M}\mathrm{S}\mathrm{E}}_{i}=\frac{{\sum }_{j=1}^{n}{\left({\hat{Y}}_{i}\left(j\right)-{Y}_{i}\left(j\right)\right)}^{2}}{n}\end{equation} \tag{ 7 }$$

where ${\hat{Y}}_{i}\left(j\right)$ and Y_i (j) indicate the jth estimated value and actual relative state value in the ith OOB sample, respectively.

Step 2: randomly permute X(k)(k = 1, ..., p) of the state vector X in the OOB samples, for obtaining new OOB samples. Then calculate the MSE_i (k)(i = 1, ..., N; k = 1, ..., p) of the estimated results using the new OOB samples. Based on the above processes, an MSE matrix can be obtained, taking the form as

$$\begin{equation}\left(\begin{matrix}\hfill {\mathrm{M}\mathrm{S}\mathrm{E}}_{1}\left(1\right)\hfill & \hfill {\mathrm{M}\mathrm{S}\mathrm{E}}_{2}\left(1\right)\hfill & \hfill \dots \hfill & \hfill {\mathrm{M}\mathrm{S}\mathrm{E}}_{N}\left(1\right)\hfill \\ \hfill {\mathrm{M}\mathrm{S}\mathrm{E}}_{1}\left(2\right)\hfill & \hfill {\mathrm{M}\mathrm{S}\mathrm{E}}_{2}\left(2\right)\hfill & \hfill \dots \hfill & \hfill {\mathrm{M}\mathrm{S}\mathrm{E}}_{N}\left(2\right)\hfill \\ \hfill {\vdots}\hfill & \hfill {\vdots}\hfill & \hfill {\vdots}\hfill \\ \hfill {\mathrm{M}\mathrm{S}\mathrm{E}}_{1}\left(p\right)\hfill & \hfill {\mathrm{M}\mathrm{S}\mathrm{E}}_{2}\left(p\right)\hfill & \hfill \dots \hfill & \hfill {\mathrm{M}\mathrm{S}\mathrm{E}}_{N}\left(p\right)\hfill \end{matrix}\right)\end{equation} \tag{ 8 }$$

Step 3: calculate the difference between elements in ${\left[{\mathrm{M}\mathrm{S}\mathrm{E}}_{1},\dots ,{\mathrm{M}\mathrm{S}\mathrm{E}}_{N}\right]}^{\mathrm{T}}$ and ${\left[{\mathrm{M}\mathrm{S}\mathrm{E}}_{i}\left(1\right),\dots ,{\mathrm{M}\mathrm{S}\mathrm{E}}_{i}\left(p\right)\right]}^{\mathrm{T}}$, and the difference can be defined as

$$\begin{equation}{\Delta}{\mathrm{M}\mathrm{S}\mathrm{E}}_{i}\left(k\right)={\mathrm{M}\mathrm{S}\mathrm{E}}_{i}-{\mathrm{M}\mathrm{S}\mathrm{E}}_{i}\left(k\right)\end{equation} \tag{ 9 }$$

where i = 1, ..., N, and k = 1, ..., p.

Step 4: calculate the importance of variable X(k)(k = 1, ..., p). The importance value I_k is defined as

$$\begin{equation}{I}_{k}=\frac{{\sum }_{i=1}^{N}{\Delta}{\mathrm{M}\mathrm{S}\mathrm{E}}_{i}\left(k\right)}{N\cdot {\mathrm{S}\mathrm{E}}_{k}}\end{equation} \tag{ 10 }$$

where SE_k is the standard error of ΔMSE_i (k)(i = 1, ..., N), taking the form as

$$\begin{equation}{\mathrm{S}\mathrm{E}}_{k}=\sqrt{\frac{{\sum }_{i=1}^{N}{\left({\Delta}{\mathrm{M}\mathrm{S}\mathrm{E}}_{i}\left(k\right)-\bar{{\mathrm{M}\mathrm{S}\mathrm{E}}_{k}}\right)}^{2}}{N}}\end{equation} \tag{ 11 }$$

where $\bar{{\mathrm{M}\mathrm{S}\mathrm{E}}_{k}}$ is the mean of ΔMSE_i (k)(i = 1, ..., N). A bigger I_k indicates that X(k) is more important in the regression model.

Mean absolute error (MAE) and coefficient of determination (R²) are used for evaluating the accuracy of the above-trained regression model. Smaller MAE and bigger R² indicate a more accurate model.

$$\begin{equation}\mathrm{M}\mathrm{A}\mathrm{E}=\frac{{\sum }_{j=1}^{n}\left\vert \hat{{y}_{j}}-{y}_{j}\right\vert }{n}\end{equation} \tag{ 12 }$$

$$\begin{equation}{R}^{2}=1-\frac{{\sum }_{j=1}^{n}{\left({y}_{j}-\hat{{y}_{j}}\right)}^{2}}{{\sum }_{j=1}^{n}{\left({y}_{j}-\bar{y}\right)}^{2}}\end{equation} \tag{ 13 }$$

where $\hat{{y}_{i}}$ and y_i indicates the estimated value and actual relative state value in the ith sample. $\bar{y}$ indicates the mean of y_j (j = 1, ..., n).

In the investigation of insufficiency and redundancy of the pressure sensors, we aim to find the least number of pressure sensors, with which the accuracy of estimation is big enough. Besides, when the number used for estimation increases, the accuracy of estimation has not increased much more. Such a characteristic means that the pressure sensors are neither insufficient nor redundant. Figure 12 shows the mean absolute error MAE with the used number of pressure sensors (M). It can be seen that MAE decreases with the increasing M as a whole. Figure 13 shows the coefficient of determination R² with the used number of sensors. ${R}_{{d}_{\text{vertical}}}^{2}$, ${R}_{A}^{2}$, ${R}_{f}^{2}$, ${R}_{A}^{2}$, ${R}_{\phi }^{2}$, ${R}_{\alpha }^{2}$, ${R}_{\beta }^{2}$, and ${R}_{\gamma }^{2}$ indicates R² in the experiments of investigating d_vertical, A, f, ϕ, α, β, and γ. It can be seen that R² increases with the number of M as a whole in each experiment when using the four methods. Comparing the MAE and R² obtained based on the order of HPVs sorted by ${C}_{{k}_{1}}$ and ${C}_{{k}_{2}}$, it can be seen that regression models obtained according to order of HPVs sorted by ${C}_{{k}_{2}}$ have better effects. From this perspective, ${C}_{{k}_{2}}$ is more reasonable for measuring the sensitivity of the HPVs measured by pressure sensors to the variations of relative states. In the following part, we mainly focus on the results corresponding to ${C}_{{k}_{2}}$.

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Take the experiment of investigating d_vertical for example, a more careful inspection reveals that R² and MAE vary little when M exceeds 4. This characteristic demonstrates that 4 pressure sensors which specifically refer to P₀, ${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, and ${P}_{{R}_{2}}$ here are enough for regression analysis. In other words, five or more pressure sensors are redundant. In this way, the reasonable numbers (defined as M_r) of pressure sensors used for regression analyses of d_vertical, A, f, ϕ, α, β, and γ are determined as 4, 1, 3, 4, 7, 4, and 5, respectively.

In order to have a fair comparison of all of the regression methods, similar cross-validations have been conducted when we conducting regression analyses. In each experiment, R²s obtained by the RF algorithm is bigger than R²s obtained by BPNN, REG, and SVR. A further inspection has revealed that R² obtained by RF has the best performance though M is small. Based on the above analyses, we can conclude that the RF algorithm has better regression effects than the other three methods. Besides, R² obtained by the RF algorithm varies smoothly whereas the R² obtained by the other three methods have more significant variations with M. Such a characteristic demonstrates that RF has better noise-resistibility. On the above analyses, a conclusion can be obtained that RF algorithm has the best regression effect.

Figure 14 shows the estimated and actual relative states obtained by the RF algorithm with M_r = 4, 1, 3, 4, 7, 4, and 5 for the experiment of d_vertical, A, f, ϕ, α, β, and γ, respectively. On the whole, it can be seen that the established RF based regression models have good performances for describing the relative states. Specifically, the combination of the R², MAE, and M_r corresponding to the best regression effect is defined as (R², MAE, M), which refers to (0.972, 3.250 mm, 4), (0.975, 1.119°, 1), (0.949, 0.030 Hz, 3), (0.958, 2.467°, 4), (0.952, 5.778°, 7), (0.952, 1.836°, 4), and (0.985, 1.915°, 5) for the experiment of d_vertical, A, f, ϕ, α, β, and γ, respectively.

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Figure 15 shows the order of the variables X(k) sorted by the importance I_k . A bigger I_k indicates that the variable X(k) is more important to the relative state in the RF-based regression analysis. The combinations of the M_r most important variables for d_vertical, A, f, ϕ, α, β, and γ are (P₀, ${P}_{{R}_{2}}$, ${P}_{{L}_{2}}$, ${P}_{{L}_{1}}$), (P₀), (${P}_{{R}_{1}}$, ${P}_{{L}_{1}}$, P₀), (${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, P₀, ${P}_{{L}_{3}}$ ), (${P}_{{R}_{1}}$, ${P}_{{R}_{2}}$, ${P}_{{L}_{1}}$, ${P}_{{R}_{4}}$, P₀, ${P}_{{L}_{2}}$, ${P}_{{L}_{4}}$), (P₀, ${P}_{{R}_{2}}$, ${P}_{{L}_{3}}$, ${P}_{{L}_{2}}$), and (${P}_{{R}_{2}}$, ${P}_{{L}_{2}}$, ${P}_{{L}_{1}}$, ${P}_{{R}_{1}}$, ${P}_{{L}_{3}}$), respectively. That is to say, we have determined the M_r pressure sensors used for estimation. A further investigation has revealed that M_r pressure sensors in the above combinations are almost the first M_r pressure sensors in table 5 for each experiment. Such a characteristic demonstrates also proves that it is reasonable for sorting the pressure sensors according to ${C}_{{k}_{2}}$ because the first M_r pressure sensors in table 5 happen to be the most important in the regression analyses. Besides, there exist differences between concrete orders of pressure sensors in the above combinations and table 5. Such a characteristic demonstrates that the sensitivity of the pressure sensors-measured HPVs to the relative states is not the same thing as the importance of the HPVs.

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The above work has shown that RF method has the best regression effect. In this part, we validate the effectiveness of RF method in relative state estimation. Considering that the number of the investigated experimental parameters, d_vertical, A, f, ϕ, α, β, and γ, is 7, 16, 6, 13, 19, 9, and 11, respectively. Here, we select the experiments of investigating A and α for validation works because more experimental parameters have been considered. Specifically, 80% data in the data set O form a training set for training the RF-based models which are different from the models in the section 4.2, and the remaining 20% data form a test set, for validating the effect of the RF-based models in estimating the relative yaw angle and the oscillating amplitude. Figure 16 shows the results of RF-based oscillating amplitude and relative yaw angle estimation. R² and MAE of the regression model for oscillating amplitude estimation is 0.975 and 1.091°, respectively. And the MAEs of the oscillating amplitude obtained by test set are 2.171°, 3.163°, and 3.211° for A = 4°, 14°, 28°, respectively. R² and MAE of the regression model for the relative yaw angle is 0.956 and 5.513°, respectively. And the MAEs of the relative yaw angle obtained by test set are 9.284°, 7.506°, 7.694°, and 45.435° for α = −60°, −30°, 30°, 60°, respectively. It can be seen that the regression model has a bad performance when the relative yaw angle is big enough. However, it performs well when the relative yaw angle is small. The estimation errors may mainly result from the low noise-signal ratio of the HPVs measured by the pressure sensors. Though the pretreatment of the HPVs has been conducted, the hardware deficiency of the pressure sensors has reduced the estimation accuracy. In this case, improving the ALLS is necessary and urgent for improving the data quality. Besides, further data pretreatment by fusing ALLS data and IMU data may also provide a potential way for improving the estimation accuracy. The above-mentioned will be conducted in the following researches on multiple adjacent freely-swimming robotic fish.

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As shown in figure 17, the relationship between the HPV and the relative lateral distance varies with the different relative longitudinal distances. There is a common qualitative regularity existing in the HPV curves under a short-range relative longitudinal distance. However, when the relative longitudinal distance exceeds a specific value, the above-mentioned qualitative regularity disappears. This may be because the vortex wake generated by the upstream oscillating fin scatters and disappears with the increasing propagation distance. To acquire significant regularities between the HPVs and the relative states, we have mainly focused on close-range sensing (local sensing), and the relative longitudinal distance of the upstream oscillating fin and the downstream robotic fish has been fixed at 10 cm in [21].

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Such a relative longitudinal distance is relatively small when compared with the size of the robot. So the downstream robotic fish is not just a passive observer, but it actively changes the vortex wake generated by the upstream oscillating caudal fin. Figure 18 shows the instantaneous vortex structure by Q-criterion around the robotic fish body and HPVs on the surface of the robotic fish body. They are obtained by the computational fluid dynamics simulation software called Fluent. More details about the Fluent based simulation can be found in [21]. As shown in figure 18, the vortices disperse after they propagate over the fish head. As a result, the HPVs measured by the pressure sensors at the posterior of the downstream robotic fish have small variations.

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The differences include three parts, such as the following:

First, in the experiments, the upstream oscillating caudal fin and the downstream robotic fish played roles as 'vortex signal generating source' and 'vortex signal detector', respectively. So, the difference between investigating the relative yaw angle of the downstream robotic fish and investigating the oscillating offset of the upstream oscillating caudal fin was the difference between changing the 'vortex signal generating source' and changing 'vortex signal detector'. Specifically, in the experiment of investigating the oscillating offset, the nonzero oscillating offset resulted in the deviation of the shedding vortices' propagating direction from the direction of the flow. Meanwhile, the scope of the backward reaction also deviated from the robotic fish in the downstream region. However, the downstream robotic fish that served the function of signal detector never changed. In the experiment of investigating the relative yaw angle, the pressure sensors mounted on the downstream robotic fish who served the function of hydrodynamic signal detector changed its orientation to the incoming flow as its attitude changed. Meanwhile, the body of the robotic fish may also cause flow separation [37] around itself and resulted in the self-generating vortices [38]. However, the upstream oscillating caudal fin who served as the hydrodynamic signal generating source never changed.

Second, in the experiment of investigating the oscillating offset of the upstream oscillating caudal fin, because the shedding vortices' propagating direction deviated from the direction of the flow, the vortex wake generated by the oscillating caudal fin may be dispersed by the flow. Thus it may be not as well-formed and stable as the vortex wake generated in the experiment of investigating the relative yaw angle.

Third, for the experiment of investigating the oscillating offset of the upstream oscillating caudal fin, the hydrodynamic pressure-bearing area of the downstream robotic fish was concentrated on the head of its body. Moreover, with the increasing oscillating offset, the above-mentioned area gradually decreased. This may result from the deviation of the vortex wake, as described in the second point. By comparison, the hydrodynamic pressure-bearing area can be almost distributed on the whole body of the robot with the changing orientation of the robot to the incoming flow in the experiment of investigating the relative yaw angle. The above-mentioned differences have resulted in the different qualitative and quantitative regularities of HPVs with respect to the experimental parameters. Specifically, in the experiment of investigating the oscillating offset, the HPVs measured by P₀, ${P}_{{L}_{1}}$ and ${P}_{{R}_{1}}$ change apparently with the oscillating offsets, whereas the changes of the HPVs measured by other pressure sensors are not apparent with the changing oscillating offsets. By comparison, in the experiment of investigating the relative yaw angle, the HPVs measured by all the pressure sensors have apparent changes with the changing relative yaw angles.