Distributed Flow Estimation for Autonomous Underwater Robots Using Proper Orthogonal Decomposition-Based Model Reduction Free

Autonomous underwater robots attract increasing scientific attention for their agile, long-range operation and great human-labor efficiency [1–5]. Their scope of application crosses various fields including environmental monitoring, search and rescue, surveillance and security, scientific research, and public education. In recent years, underwater robotics has achieved many advances in terms of locomotion efficiency [2,5], actuation mechanism [3], battery power endurance [4], etc., however, there is still one unsolved fundamental research problem—the flow estimation, which plays an essential role for control and navigation of autonomous underwater robots.

Flow estimation is challenging for underwater robots because of the complex and dynamic fluid environment. Scientists and engineers have been making great efforts in improving flow estimation capability of underwater robots over the past years. There are two main methods to sense the flow field: (1) using flow sensors to measure flow fields directly and (2) assimilating other sensor measurements (e.g., pressure) through flow estimation algorithms to estimate the flow field.

Various types of flow sensors have been designed to estimate the flow field directly. Ships use acoustic instruments such as acoustic Doppler current profilers to measure water velocity. However, acoustic Doppler current profilers are expensive and have a dead zone in proximity to the underwater vehicle [6,7]. Based on thermal principles, Fan et al. [8] and Yang et al. [9] designed a micromachined hot-wire flow sensor. This sensor was tested in a manually varied laminar flow to show the ability of detecting flow rate. In Ref. [10], a flow sensor based on torque transfer was presented where a static turbine converted the volume flow into a torque measurement. Optical flow sensors were used to obtain hydrodynamic information in Refs. [11] and [12]. In Ref. [11], the authors used optical flow sensors to measure the flow motion inside artificial canals and then quantified the pressure gradient. In Ref. [12], an optical feedback interferometry flow sensor was designed and used to measure the local flow velocity. A crystal polymer micro-electro-mechanical system sensor was designed for flow estimation in Refs. [13,14]. This sensor was able to detect the velocity of towed underwater objects [13] and flow speed and direction [14]. Recently, more and more flow sensors have been developed, which have high spatial resolution and short response time. However, these designs are mostly customized and ad hoc solutions with a long design cycle and relatively high cost. These disadvantages impede broad real-world applications in autonomous underwater robots.

Parallel to designing flow sensors to directly measure local flow velocity, more and more research efforts have been put into using multiple low-cost, commercially available sensors, and a flow estimation algorithm to estimate the whole flow field. Inspired by the high performance of fish lateral line, scientists and engineers have been making great efforts in developing similar flow-sensing systems [15–22]. The lateral line helps aquatic vertebrates sense surrounding flow fields and assists with flow relative behaviors, such as rheotaxis, predation, and schooling [23]. Some papers use linear regression to approximate the flow field [22], while others use nonlinear estimation algorithms including extended Kalman filter [16], Bayesian filter [15,17–19], unscented Kalman filter [20], and particle filter [21] to assimilate distributed (pressure and velocity) sensor measurements for flow sensing. For example, in our prior work [18,19], co-author Zhang studied flow estimation for a Joukowski hydrofoil locomoting in a uniform flow using a Bayesian filter. Although estimation performances were satisfactory, our prior designs like most existing flow sensing algorithms have very serious limitation that they only apply to specific flow fields. We will discuss the causes in detail later.

The flow estimation algorithms typically require a mathematical flow model, which is obtainable through one of the many methods such as analytical flow modeling [15–19], computational fluid dynamics (CFD) simulation [20], and towing tank experiments [21,22]. The analytical flow modeling method is easy-to-implement and cost-effective. However, this approach is only suitable for special-shaped robots. For example, the potential flow model in Refs. [15] and [19] can only be used with the same foil shape described by a conformal mapping function. CFD simulation on the other hand can provide accurate flow models for underwater robots with any designed shape. However, even with a simplified CFD model such as the panel method used in Ref. [20], the flow model still cannot be used in real-time flow estimation due to the computational cost. Towing tank experiments are commonly used in a well-controlled lab environment. With the help of flow visualization methods such as particle image velocimetry [21], the flow model can be experimentally established. However, it is not suitable for autonomous underwater robots that typically operate in open water.

This paper proposes a general flow estimation approach using distributed pressure measurements of autonomous underwater robots via integrating proper orthogonal decomposition (POD)-based flow models and recursive Bayesian estimation. Proper orthogonal decomposition is a data-driven model reduction method that is often used to analyze fluid fields. It was first introduced by Lumley [24] in 1979 and has been widely used since then. This approach can be applied to arbitrary robot designs and reduce the computational cost. At the same time, this paper adopts a Bayesian filter to assimilate the distributed pressure measurements for flow estimation. The likelihood calculation of the Bayesian estimator is integrated with a reduced-order POD flow model.

The contribution of this paper lies in (1) a novel real-time flow estimation algorithm that assimilates distributed pressure measurements by integrating recursive Bayesian estimation and POD-based reduced-order flow modeling; and (2) quantitative analysis of flow model reduction accuracy and sensor placement through proposing performance metrics utilizing POD modeling percent error and empirical observability analysis, respectively. There are three main advantages of the proposed method. First, POD-based flow modeling reduces the computational complexity thus making real-time flow estimation possible. Second, the POD-based flow estimation using a Bayesian filter does not rely on any analytical flow models, thus applicable to any underwater robot with arbitrary shape designs. Third, our flow estimation method is based on low-cost commercially available pressure sensors, thus especially useful in small and networked autonomous underwater robots.

The remainder of the paper is organized as follows. Section 2 presents the POD flow model reduction algorithm and the Bayesian filter approach that estimates the flow field by assimilating distributed pressure measurements. Section 3 discusses the POD flow model reduction accuracy, relationship between the POD model and conventional flow parameters of interest, and sensor placement strategy. In Sec. 4, two simulation examples illustrate the proposed flow estimation method with a cylinder shaped robot and a Joukowski-foil-shaped robot. In Sec. 5, the closed-loop angle-of-attack regulation is investigated in simulation which utilizes the proposed flow estimation algorithm. Finally, conclusion remarks are presented in Sec. 6.

In this section, we present a general flow estimation approach that assimilates distributed pressure measurements of autonomous underwater robots via integrating POD reduced-order flow models and recursive Bayesian estimation.

Proper orthogonal decomposition or POD is a model reduction method, which decomposes a nonlinear and high-dimensional or infinite-dimensional system into a lower dimensional system using finite number of basis POD modes based on a large set of data. The optimal set of POD modes to represent the system is determined based on L₂ norm optimization using the POD algorithm [24].

In fluid dynamics, potential flow is used to describe the velocity field as the gradient of a scalar function—the velocity potential [25]. As a result, a potential flow is characterized by an irrotational velocity field. In the case of an incompressible flow, the velocity potential satisfies Laplace's equation, and potential theory is applicable.

In this paper, the POD algorithm deals with flow field snapshots that represent the velocity or pressure field, obtained from either CFD simulation, theoretical calculation, or towing tank experiments. Optimal POD modes and corresponding coefficients are calculated and used to model the flow field. The procedure of POD calculation is as follows.

First, obtain M flow field snapshots $U1,U2,…,UM$⁠, each of which is generated under certain values of flow parameters of interest, such as the flow speed and the angle of attack. Every snapshot includes i rows and j columns of points that represent the local flow velocity. Each snapshot is described by an a × b dimensional matrix.

Second, reshape each flow field snapshot matrix into a column vector. Use vector $ui$ to describe the flow field snapshot $Ui$⁠. Concatenating all the column vectors into a new matrix, we get the flow field snapshot matrix $U=[u1,u2,…,uM]$ whose dimension is a × b by M.

where the POD coefficients are calculated by the inner product of the flow field vector $ui$ and the POD mode vector $vj$⁠.

where O, less than q, is the selected number of POD modes in the reduced-order flow model, and $Ûi$ represents the reconstructed ith flow field snapshot in a vector form.

With the flow field modeled by the reduced-order POD model, we use a Bayesian filter to assimilate distributed pressure measurements and estimate the coefficients of POD modes. This section presents our distributed flow estimation method that coalesces a Bayesian filter and POD flow model reduction.

where C is a constant, F is the complex potential of the flow field, w is the local flow velocity, and ρ is the flow density.

where $p̃i$ is the actual pressure sensor measurement at sensor position z_i, and ε_i is the Gaussian with a mean of zero and a variance of $σi2$⁠, i.e., $εi∼(0,σi2)$⁠.

Inspired from the lateral line of fish, we use pressure difference between sensor pair as the flow measurement function [23], i.e., $△p̃ij(t)=p̃i(t)−p̃j(t)$⁠. In this paper, we assume that flow fields change slowly compared to the fast sampling from pressure sensors. Thus, we ignore the unsteady effect in flow estimation and the flow measurement function becomes $△p̃ij=p̃i(t)−p̃j(t)=12ρ|w(zj)|2−12ρ|w(zi)|2$

Here, we use all the pairs of pressure difference between all the sensors rather than mimicking the lateral line that uses only pressure difference between adjacent sensors. Using all possible pairs will help reduce the estimation error considering the averaging effect over more noisy measurements, thus improving estimation accuracy.

where κ is the coefficient that ensures the probability of the entire sample space equals one, $Pr(Dc|∧)$ is the likelihood function of new measurement $Dc$ given the coefficients of the POD modes $∧, Pr(∧|D(t))$ is the posterior probability density function, and $Pr(∧|D(t−△t))$ is the prior probability density function.

where $Ds=[△p12,△p13,…,△p(N−2)N−2,△p(N−1)N]T$ represents the theoretical values of the flow measurements calculated using the POD flow model (9).

where $Pr(∧(t)|∧(t−△t))$ represents a general motion model, the solution of which typically requires solving ordinary/partial differential equations with advection/diffusion.

The process of the distributed flow estimation method for underwater robots is summarized as follows. First, we obtain the optimal reduced-order POD modes that model the flow field around the underwater robot from the snapshots of the flow field. When the robot receives new sensor measurements, the likelihood function $Pr(Dc|∧)$ is computed using the reduced-order flow model. Given a prior probability density function $Pr(∧(t)|D(t−△t))$⁠, the posterior probability density function is updated based on the Bayes' rule (10). Through the posterior probability density function $Pr(∧(t)|D(t))$⁠, we determine the current optimal POD coefficients $∧$⁠, selected as the point in the estimation state space with the highest joint posterior PDF. The Chapman–Kolmogorov equation predicts the prior PDF $Pr(∧(t)|D(t−△t))$ at time t based on the posterior PDF $Pr(∧(t−△t)|D(t−△t))$ at time $t−△t$⁠.

This paper uses a POD reduced-order flow model to compute the likelihood function $Pr(Dc|∧)$ and estimates the flow parameters $∧$ through distributed pressure measurements $Dc$⁠. The adoption of the reduced-order flow model is expected to significantly improve the computational efficiency and the application scope.

In this section, we analyze the proposed flow estimation approach, in terms of the accuracy of the POD flow model reduction, the mapping between POD coefficients and conventional flow parameters, and the sensor placement strategy.

Performance indices are proposed for studying model reduction and sensor placement. The proposed performance indices depend on many design factors such as number of pressure measurements, the size of the snapshot, and flow conditions. The focus of this section is to provide a general approach for researching model reduction design and sensor placement strategy with quantified measures, therefore, we have chosen to minimize the discussions on the influences of different factors. Interested readers are encouraged to refer to sensitivity analysis [29] and use tools therefrom to explore the influences of different designs.

where $Û$ is the reconstructed flow field snapshot matrix using POD model reduction and U is the flow field snapshot matrix. $E¯$ represents the averaged modeling percent error given a certain number of POD modes.

Table 1 shows the POD modeling percent error $E¯$ given different numbers of POD modes for the same uniform flow past a Joukowski-foil-shaped robot used in the case study as in Fig. 1. The modeling percent error $E¯$ is about 2.57% with one POD mode and reduces to 0.18% with two POD modes. The proposed performance index $E¯$ quantifies the reduced-order modeling error and clearly shows that more POD modes lead to higher modeling accuracy. Considering the balance between computational cost and model reduction accuracy, a minimum number of POD modes will be selected to meet the flow estimation design requirements.

Some conventional flow parameters, for example, the flow speed and the angle of attack, are very important variables in determining/estimating the hydrodynamics and control/navigation of autonomous underwater robots. However, most of the flow parameters like the angle of attack are very difficult if not impossible to measure with on-board sensors as they require the flow field information. Therefore, once we are able to estimate the flow field described by the POD reduced-order model (Sec. 2), we proceed to extract or calculate the relevant flow parameters using the estimated flow field.

Considering the nonlinear and complex relationship between the flow parameters like the angle of attack and estimated flow field parameterized by POD coefficients, we propose to use the neural network technique to establish the mapping. Neural network is a computing system that is usually organized in layers and can approximate nonlinear systems based on sampled training data [30–32]. Neural networks, composed of parallel-working nodes, imitate the biological neural system. The most common structure of a neural network includes an input layer, single/multiple hidden layer(s), and an output layer as shown in Fig. 2. The input and output layers of the neural network in our application are decided by the number of POD modes and flow parameters. The original flow field snapshots and the corresponding calculated POD mode coefficients are used to train the neural network.

Sensor placement affects the flow field estimation because different sensor positions contain a different volume of flow field information. We propose to optimize sensor placement using the concept of system observability [17,33]. While the observability is difficult to capture or theoretically compute, we will leverage the empirical observability Gramian to quantify the observability of the estimation states or the POD coefficients and design a sensor placement strategy thereby.

Here, $△psum=|△p̃12|+|△p̃13|+⋯+|△p̃(N−1)N|$ is the sum of absolute value of all the pairwise pressure differences with pressure sensor positioned at certain locations given the POD coefficients $∧=[c1,c2,c3,…,cO]T$⁠. ε_i is a small perturbation of the ith POD coefficient, $△psum+i$ is the sum of absolute value of all the pairwise pressure differences when the POD coefficients are equal to $∧=[c1,…,ci+εi,…,cO]T, and 〈x,y〉$ denotes the inner product of complex numbers x and y [15].

where $δpci$ is the change of $△psum$ when the ith POD mode coefficient has a perturbation ±ε_i about c_i and $△ci=(ci+εi)−(ci−εi)=2εi$ is the overall perturbation of the ith POD mode coefficient.

From the empirical observability Gramian, we compute characteristic indices (I₁, I₂, I₃) as follows:

An an example, Fig. 3 shows the simulation results of the three indices evaluated at different sensor positions that are represented by polar angles in the same case study with a Joukowski-foil-shaped robot as in Fig. 1. We consider four sensors for flow estimation, located symmetrically on two sides of the robots. We also consider the dimension of the sensors and define a minimal separation distance to avoid sensor overlapping. Here, the simulation results are based on flow speed at 0.3 m/s and angle of attack at 0 deg which are considered to be one of the nominal working points in the robot state space. The optimal sensor placement for observing POD coefficient c₁ is one sensor near the polar angle 160.21 deg and the second sensor near 159.39 deg, while the optimal sensor placement for observing c₂ is one sensor near 168.15 deg and the second sensor near 153.42 deg. The selection of sensor positions near the polar angle 169.04 deg and 168.15 deg minimizes index I₃.

where l₁, l₂, and l₃ are the design coefficients that balance the weights of performance indices. In sensor placement design, we will use simulation/experiment to search for optimal sensor positions that minimize the composite index to achieve maximum observability. The proposed strategy facilitates the design process through providing a convenient quantitative metric.

The proposed distributed flow estimation method will be illustrated by two simulation examples, one with a uniform flow past a circular-shaped underwater robot and the other with a uniform flow past a Joukowski-foil-shaped underwater robot.

A circular-shaped underwater robot is used to show the effectiveness of our flow estimation method. The robot rotates about the center point of the circle in a uniform flow. The flow velocity relative to the robot is denoted as Q. We define, as shown in Fig. 4, an inertial reference frame I whose horizontal axis, x^I, is aligned with the direction of the flow velocity Q and its vertical axis, y^I, is perpendicular to x^I. The origin of the inertial reference frame is arbitrarily chosen to be a fixed point in the flow field. We define a body-fixed reference frame B that is attached with the underwater robot. The origin of the body-fixed frame is defined to be the center point of the circle. The horizontal axis, x^B, is along the direction that points from the head of the robot to the tail of the robot. The vertical axis, y^B, is perpendicular to x^B. The rotation angle from the x^B axis to the x^I axis is defined as the angle of attack with the counter-clockwise direction as positive.

where Q is the relative flow speed, ζ = x + iy is a complex number that represents the point (x, y) in the body-fixed reference frame, ζ_c is the center of the circle, R is the radius of the circle, and α is the angle of attack. The value of Γ can be found by imposing the Kutta condition that requires the trailing edge to be a stagnation point [36].

In simulation, we generate M = 231 velocity field snapshots using Eq. (21) for POD reduced-order flow modeling. The flow speed Q used in generating the snapshots ranges from 0.2 m/s to 0.4 m/s with a 0.02 m/s increment, and the angle of attack α ranges from –20 deg to 20 deg with a 2 deg increment. Using these flow field snapshots, we calculate the optimal POD modes of the flow field. Considering jointly the computational cost and modeling accuracy, two POD modes are finally selected in the reduced-order flow model for the flow field around the robot.

The circular robot rotates about its center point. The angle of attack follows a sinusoidal function $α=A sin(2πft)$ with the rotation amplitude A and frequency f. Designed to maximize the observability of estimated states (Sec. 3.3), four pressure sensors are distributed on the circular robot, shown as white dots in Fig. 4 with polar angles of 170 deg and 160 deg on both sides. Each sensor measures the real-time local pressure. A Bayesian filter (Sec. 2.2) assimilates all the incoming sensor measurements and recursively estimates the two coefficients of the POD modes of the reduced-order flow model. The motion model $Pr(∧(t)|∧(t−△t))$ adopts a diffusion process for the convenience of computation. Equation (9) then gives the flow field estimation.

Figure 5 shows the simulation results of the flow estimation for the circular-shaped underwater robot rotating in a uniform flow. In simulation, the rotation amplitude A is equal to 5 deg and the rotation frequency f is equal to 0.2 Hz. The uniform flow velocity is 0.3 m/s. The diameter of the circular robot is 5.8 cm. The sensor position vector is $z=[−3.225+0.99i;−3.357+0.4991i;−3.357−0.4991i;−3.225−0.99i]$ and the flow measurement vector at time t is $Dc=[△p̃12,△p̃13,△p̃14,△p̃23,△p̃24,△p̃34]T$⁠. The left column shows the actual velocity flow field and the right column shows the estimated velocity flow field described by the POD reduced-order flow model.

Table 2 shows the POD modeling percent error $E¯$ at different time instants. The modeling percent error $E¯$ is about 3.53% at first and reduces to around 1% after 1.25 s.

From Fig. 5 and Table 2, we see that the estimated flow field approximates the actual flow field with an increasing accuracy along with time. The Bayesian filter assimilates the temporal and spatial measurements to update the probability density function of the POD coefficients. More measurement data of the flow lead to better estimation accuracy and the estimation error converges to 1% eventually. With the existence of flow model reduction errors and sensor measurement noises, we consider the flow estimation results satisfactory.

In this section, we apply the proposed distributed flow estimation approach on a Joukowski-foil-shaped underwater robot. The shape is defined as the output image of the Joukowski transformation, which is a conformal mapping commonly used in airfoil design [37]. Through the potential flow theory, we calculate the complex potential of the flow field as a function of the flow speed and the angle of attack [2,16,18,24,34,38,39]. The gradient of the complex potential gives the flow velocity.

In simulation, we use a robot with a length of 19.2 cm and a width of 3.8 cm. The underwater robot is equipped with four pressure sensors with two on each side, shown as white dots in Fig. 6. The sensor placement follows the strategy as discussed in Sec. 3.3. The sensor position vector is $z=[−4.8572+0.4889i;−5.0357+0.2495i;−5.0357−0.2495i;−4.8572−0.4889i]T$⁠. The robot rotates in a uniform flow with its angle of attack following a sinusoidal function α = Asin(2πft). We generate M = 231 velocity field snapshots using potential flow theory and calculate the optimal POD modes for the flow field. The flow speed used to generate the flow field snapshot ranges from 0.2 m/s to 0.4 m/s with a 0.02 m/s discretization step. The angle of attack ranges from –20 deg to –20 deg with a 2 deg discretization step. Two POD modes are selected for the reduced-order flow model. The motion model $Pr(∧(t)|∧(t−△t))$ adopts a diffusion process.

Figure 6 shows the simulation results of distributed flow estimation of the Joukowski-foil-shaped underwater robot rotating in a uniform flow with a constant flow velocity of 0.3 m/s. Table 3 shows the POD modeling percent error $E¯$ at different time instants. From 0.5 s to 2.5 s, $E¯$ is nearly 1% and grows larger at 3 s to about 3.44% that is when the fish rotates through zero angle of attack. After 4 s, $E¯$ returns to about 1% and maintains at that level. We conjecture the growing in the error comes from a rapid rotating movement of the robot around zero angle of attack, which challenges the assumption of the quasi-steady flow conditions. With the existence of flow model reduction errors and sensor measurement noises, we consider the estimation error satisfactory.

This section presents the application of the proposed flow estimation method in the closed-loop control of underwater robots. The design idea is illustrated by the simulation results of the angle-of-attack regulation of a Joukowski-foil-shaped robot.

Using their lateral line systems, fish tend to turn against flow currents for station holding to save energy [23,40]. In this paper, we will use the angle-of-attack regulation as an example to demonstrate the application of the proposed flow estimation approach in closed-loop control of underwater robots. The control objective is to regulate a Joukowski-foil-shaped underwater robot to stay at zero degree of angle of attack in a uniform flow using distributed flow estimation feedback [19,41].

We adopt a proportional–integral–derivative (PID) controller for the angle of attack regulation. The block diagram of the closed-loop system is shown in Fig. 7.

where T_c is the control torque used to control the rotation motion of underwater robot and e = α_des − α_e is the difference between the desired and estimated angle of attack. K₁, K₂, and K₃ are the PID controller coefficients. The feedback angle of attack is obtained from estimated flow using the neural network model discussed in Sec. 3.2.

In simulation, we used four POD modes to model the flow field following the POD mode selection analysis in Sec. 3.1 for a balance between modeling error and computation effort in angle-of-attack regulation. Figure 8 shows the simulated four optimal POD modes of the reduced-order flow model. The first mode represents the major component of the flow field around underwater robot. The remaining ones add more detailed features to the flow field.

These four optimal POD modes are used to construct the POD reduced-order flow model, based on which we calculate the likelihood function of the Bayesian filter and estimate the flow parameters. The angle of attack estimation is then calculated using the neural network introduced in Sec. 3.2. We adopt a three-layer neural network with ten nodes in the hidden layer to establish the relationship between POD mode coefficients and the angle of attack. The original 231 flow field snapshots and their POD mode coefficients are used to train the neural network. Figure 9 shows that the estimated angle of attack follows the trajectory of the actual angle of attack reasonably well in a testing case study where the actual angle of attack follows a sinusoidal function.

Using a PID controller (25), we simulated the closed-loop angle-of-attack regulation with the initial condition of the angle of attack at 5 deg. The parameters of the dynamic system used in simulation are C_p = 1 kg and K_p = 1 kg.

In simulation, we assume that the sensor noise level is 10% of pressure measurements. Figure 10 shows the simulation results on the trajectory of the angle of attack. The color map shows the marginal probability density of the estimated angle of attack. The estimated value is selected to be the point in the estimation space with the highest probability density. First, the results show that the estimated angle of attack matches with the actual value sufficiently well. Second, we see that the angle of attack converges to near 0 deg within 7 s with the closed-loop control. We consider the angle-of-attack regulation satisfactory, especially considering the sensor measurement noise and the modeling error in the POD flow model reduction.

This paper proposed a new method of distributed flow estimation for autonomous underwater robots, which integrates a Bayesian filter and a POD-based reduced-order flow model. We quantitatively analyzed and discussed the POD flow model reduction accuracy, relationship between the POD coefficients and conventional flow parameters, and the distributed sensor placement strategy. A cylinder-shaped underwater robot and a Joukowski-foil-shaped underwater robot were used as examples in simulation to demonstrate the proposed flow estimation approach. A closed-loop angle-of-attack regulation system was also studied using the estimated flow as feedback. The simulation results showed that the angle-of-attack regulation was satisfactory which further validated the effectiveness of the proposed flow estimation in real-time control of autonomous underwater robots.

In future work, we will experimentally evaluate the proposed model-reduction-based flow estimation method with a custom-designed flow sensing underwater robot.

This work was supported in part by the Woodrow W. Everett, Jr. SCEEE Development Fund in cooperation with the Southeastern Association of Electrical Engineering Department Heads (SCEEE-18-04).

• George Mason University (Grant No. 2017 MDR Seed Grant; Funder ID: 10.13039/100006369).