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Luca Bruzzone Subscribe SciFeed Recommended Articles Related Info Link Google Scholar More by Authors Links on DOAJ Shu, M. Zheng, X. Li, F. Wang, K. Li, Q. on Google Scholar Shu, M. Zheng, X. Li, F. Wang, K. Li, Q. on PubMed Shu, M. Zheng, X. Li, F. Wang, K. Li, Q. /ajax/scifeed/subscribe Article Views Citations - Table of Contents Altmetric share Share announcement Help format_quote Cite question_answer Discuss in SciProfiles thumb_up ... Endorse textsms ... Comment Need Help? Support

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Get Information clear JSmol Viewer clear first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open AccessArticle Numerical Simulation of Time-Optimal Path Planning for Autonomous Underwater Vehicles Using a Markov Decision Process Method by

Mingrui Shu

Mingrui Shu Scilit Preprints.org Google Scholar 1,2 ORCID

Xiuyu Zheng

Xiuyu Zheng Scilit Preprints.org Google Scholar 1,2,

Fengguo Li

Fengguo Li Scilit Preprints.org Google Scholar 1,2 ORCID

Kaiyong Wang

Kaiyong Wang Scilit Preprints.org Google Scholar 1,2 and

Qiang Li

Qiang Li Scilit Preprints.org Google Scholar 1,2,* ORCID

1 Southern Marine Science and Engineering Guangdong Laboratory, Guangzhou 511458, China 2 Shenzhen Key Laboratory of Marine IntelliSense and Computation, Shenzhen International Graduate School, Tsinghua University, Shenzhen 518000, China * Author to whom correspondence should be addressed. Appl. Sci. 2022, 12(6), 3064; https://doi.org/10.3390/app12063064 Received: 10 February 2022 / Revised: 9 March 2022 / Accepted: 10 March 2022 / Published: 17 March 2022 (This article belongs to the Special Issue New Trends in the Control of Robots and Mechatronic Systems) Download Download PDF Download PDF with Cover Download XML Download Epub Browse Figures Versions Notes

Abstract: Many path planning algorithms developed for land or air based autonomous vehicles no longer apply under the water. A time-optimal path planning method for autonomous underwater vehicles (AUVs), based on a Markov decision process (MDP) algorithm, is proposed for the marine environment. Its performance is examined for different oceanic conditions, including complex coastal bathymetry and time-varying ocean currents, revealing advantages compared to the A* algorithm, a traditional path planning method. The ocean current is predicted using a regional ocean model and then provided to the MDP algorithm as a priori. A computation-efficient and feature-resolved spatial resolution are determined through a series of sensitivity experiments. The simulations demonstrate the importance to incorporate ocean currents in the path planning of AUVs in the real ocean. The MDP algorithm remains robust even if the ocean current is complex. Keywords: path planning; autonomous underwater vehicles; Markov decision process; ocean current 1. IntroductionAutonomous underwater vehicles (AUVs) are deployed in the ocean to carry out specific missions, such as ocean observations, target identification, etc., due to their excellent advantages in maneuverability. In order to complete the mission, the AUV is often required to move to a target location under some constraints, such as the shortest time, shortest distance, and least energy consumption. Avoiding risks or obstacles in the ocean is also required. Aiming at different constraints, relevant path planning algorithms are developed. In this paper, the time-optimal is prioritized.The path planning algorithms for mobile robots can be divided into two categories: discrete-grid-based and sampling-based planning algorithms [1]. The former one is established on a gridded map. For example, the A* algorithm is transformed from the Dijkstra algorithm by adding a heuristic cost to enhance the computational efficiency. The A* algorithm can be modified, aiming at speeding the convergence of the A* algorithm, such as with iterative deepening A*, lifelong planning A*, and bidirectional A* algorithms [2,3]. Liu et al. [4] introduced an improved A* algorithm for generating the procedure of the normal path and berthing path when considering obstacles with currents and marine traffic. The sampling-based planning algorithm does not directly calculate global optimization on a gridded map. It uses the random scattering of particles on the map to extract map-assisted planning. The probabilistic roadmap [5] and rapidly-exploring random tree algorithms [6] are two typical examples. Recently, some clustering algorithms are also applied to path planning, for instance, the ant colony algorithm, genetic algorithm, etc. [7]. The level-set method is also an important branch of robotic path planning [8].With the rapid development of artificial intelligence, reinforcement learning is applied to robotic path planning. Julien et al. [9] presented an MDP-based planning method for a robot with wheels. In order to resolve the impact of their surrounding environment, Lou et al. [10] modeled the robotic motion as a Markov process and proposed a probabilistic-model-checking method to seek the optimal path. Singh [11] applied the object-oriented Markov decision process for indoor robots, which greatly simplifies the problem of the so-called “curse of dimensionality”. It reduces the state spaces by making the MDP properties as objects. Pereira et al. [12] used a minimum expected risk planner and a risk-aware Markov decision process to improve the reliability and safety of AUV operation in coastal regions.Different from the land or air based robots, the AUVs have their special characters due to the underwater environment. (1)Low positioning accuracy and high positioning costLand robots can directly acquire real-time positions through the global positioning system (GPS) with relatively high positioning accuracy (about 0.5 m [13]). Because of the absorption of electromagnetic waves by seawater, satellite signals cannot be directly received underwater by AUVs. Alternatively, the inertial navigation system (INS) is commonly used for underwater positioning. However, it is expensive while providing low positioning accuracy, which produces about 100 m errors after traveling by 1 km [14]. Its positioning error accumulates over time, which must be frequently calibrated for long-term underwater deployment. (2)Low cruising speed and high fault toleranceCompared with land or air based robots, the maximum speed of underwater robots is much slower, which ranges from about 3 to 10 knots [15]. Except in the harbors, marine traffic is limited. The marine morphology, such as the coastline, islands, and seamounts, is relatively fixed. Therefore, more tolerance is allowed for AUVs to tip or roll over in the ocean.(3)High impacts by the marine environmentIn the ocean, the current velocity, and the speed of AUVs are usually in the same order of magnitude. If the ocean current is neglected, it will lead to an obvious deviation between the planned and practical paths. In contrast, if a real-time robust control is applied to eliminate these deviations, it will cost extra power and computational resources.Several works have been done to solve the AUV path planning problem. Garau et al. [16] used the A* searching procedure to determine the optimal path with consideration to the ocean currents and their spatial variabilities. Zeng et al. [17] introduced a quantum-behaved particle swarm optimization algorithm for solving the optimal path planning problem of an AUV operating in environments with ocean static currents. Witt et al. [18] described a novel optimum path planning strategy for long-duration operations in environments with time-varying ocean currents. Kularatne et al. [19] presented a graph-search-based method to compute energy-optimal paths for AUVs in two-dimensional (2-D) time-varying flows. Subramani et al. [20] integrated data-driven ocean modeling with the stochastic dynamically orthogonal level-set optimization methodology to compute and study energy-optimal paths. Lolla et al. [21] predicted the time-optimal paths of autonomous vehicles navigating in any continuous, strong, and dynamic ocean currents through solving an accurate partial differential equation. Rhoads et al. [22] presented a numerical method for minimum time heading control for the underwater vehicle moving at a fixed speed in known time-varying and two-dimensional flow fields. The MDP method is suitable for the AUV underwater path planning. The MDP seeks a globally optimal solution through a value iterative method. The optimal paths of all state points in the whole domain to the target are computed only once. This is more efficient than the traditional A* algorithm [23,24], which has to repeat similar computations for every step. Because AUVs move underwater with a low cruising speed and a high fault tolerance, the actions of AUVs are limited, and thus, suitable for establishing the MDP model. Otherwise, computational difficulties will increase exponentially with increasing robotic actions. In our application, the ocean current is predicted from an oceanic forecast model, therefore the ocean currents can be regarded as fully observable. This information is provided to the AUV as a priori or in real-time through acoustic communications so that the parameters used in the MDP model can be updated. The fully observable MDP model has a faster convergence rate than the partially observable MDP models.The paper is organized as follows. The principle of the MDP path planning and its numerical algorithm for applications in the AUV navigation are introduced in Section 2. In Section 3, the efficiency of the MDP algorithm is examined and its performance is compared with the traditional A* algorithm. Then, the ocean currents predicted by a regional ocean model are incorporated into the MDP model to evaluate the performance of the MDP algorithm in a ‘real’ oceanic environment. The conclusions are presented in Section 4. 2. Path Planning Algorithm Based on the Markov Decision Process (MDP) 2.1. Markov Decision ProcessThe target region is first divided into multiple orthogonal grids. For path planning, an action that the AUV takes only depends upon the present state, not on the previous ones that it has experienced. Here, the state specifically refers to the appearance of the AUV inside a grid, rather than the movement process of the AUV itself. Since each grid usually ranges from several hundreds of meters to several kilometers, which is much larger than the actual size of the AUV (usually 1 to 10 m), the AUV has enough time and space to adjust its state inside the grid. Therefore, the AUV’s movement can be treated as a Markov process. A tuple including five parameters (S, A, { P s , s ′ a } , γ, R) is used to describe this process. Here, S is a state set, providing information of the position and velocity of the AUV. A denotes an action set, which the AUV takes to move from its present grid to neighbors. { P s , s ′ a } denotes a transition probability matrix, defined as: P s , s ′ a = P [ S t + 1 = s ′ | A t = a , S t = s ] , in which P s , s ′ a is the probability when the AUV changes from its current state s to its successor state s′ by taking an action a. Thus: ∑ s ′ P s , s ′ a = 1 and P s , s ′ a , γ ∈ [ 0 , 1 ] is a discount factor, which is used to adjust the proportion between the present and future values. In our simulation, we choose the discount factor γ = 0.95. R is a reward function which is a function of S and A, i.e., R : S × A → R . R S a denotes the reward for the AUV to take an action a in the present state s. Its value is predicted by A and S at step t, i.e., R S a = E [ R t + 1 | A t = a , S t = s ] , in which Rt+1 is the reward that the AUV gets at the next step t + 1. Thus, the Markov decision process is described as: s 0 → a 0 s 1 → a 1 s 2 → a 2 s 3 → a 3 s 4 → a 4 … The goal is to find an optimal strategy from all possible actions to maximize the expectation of the system’s total reward, i.e., max E [ R ( s 0 ) + γ R ( s 1 ) + γ 2 R ( s 2 ) + … ] . The Bellman equation [25] is used to iteratively solve the Markov decision process. When the AUV takes an action in the present state s following a strategy π, the expectation of total reward functions become: V ( s ) = E [ R ( s 0 ) + γ R ( s 1 ) + γ 2 R ( s 2 ) + … | s 0 = s , π ] = E [ R ( s 0 ) + γ ( R ( s 1 ) + γ R ( s 2 ) + … ) | s 0 = s , π ] = E [ R ( s 0 ) + γ V π ( s 1 ) | s 0 = s , π ] , in which R(s) indicates the reward in the current state s. 2.2. Value Iteration MethodThe general solution of the MDP algorithm can be obtained through the value iteration or strategy iteration [26]. The value iteration method can give the optimal strategy from an arbitrary point to the target, which is suitable for the real-time control of AUVs. Its abbreviated process is described below.In Algorithm 1, we use the greedy strategy to compute Vn(s′), i.e., assuring the maximum reward in every iteration. And then the Bellman Equation (5) is used to solve V(s). π(s) is solved using the value iteration method. It turns out that the solution is convergent to the optimal strategy π*(s) [27]. Algorithm 1. The Value Iteration in the MDP AlgorithmINPUT: MDP five-tuple (S, A, { P s , s ′ a } , γ, R), max iteration number N, deviation εITERATION: ∀ s ∈ S , V 0 ( s ) = 0 for n in range (1, N):for each state s do: V n + 1 ( s ) = max a ∈ A ∑ s ′ P ( s ′ | s , a ) ( R ( s , a ) + γ V n ( s ′ ) ) if ∀ s | V n + 1 ( s ) − V n ( s ) | CrossRef]Egbert, G.D.; Erofeeva, S.Y. Efficient inverse modeling of barotropic ocean tides. J. Atmos. Ocean. Technol. 2002, 19, 183–204. [Google Scholar] [CrossRef][Green Version]Fu, B.; Chen, L.; Zhou, Y.; Zheng, D.; Wei, Z.; Dai, J.; Pan, H. An improved A* algorithm for the industrial robot path planning with high success rate and short length. Robot. Auton. Syst. 2018, 106, 26–37. [Google Scholar] [CrossRef]Bai, A.; Wu, F.; Chen, X. Online planning for large MDPs with MAXQ decomposition. In Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems-Volume 3, International Foundation for Autonomous Agents and Multiagent Systems, Valencia, Spain, 4–8 June 2012; pp. 1215–1216. [Google Scholar] Applsci 12 03064 g001 550

Figure 1. The numbers in each grid indicate the different states, and the vectors represent the actions for the 2D case. Figure 1. The numbers in each grid indicate the different states, and the vectors represent the actions for the 2D case. Applsci 12 03064 g001

Figure 2. Same as Figure 1, but for the 3D case. Figure 2. Same as Figure 1, but for the 3D case. Applsci 12 03064 g002

Figure 3. The MDP algorithm is tested in Daya Bay. The coastline is acquired from GSHHG, and the bathymetry is from a nautical chart. Figure 3. The MDP algorithm is tested in Daya Bay. The coastline is acquired from GSHHG, and the bathymetry is from a nautical chart. Applsci 12 03064 g003

Figure 4. The tidal current in the experimental area at 17:00:00 6 October 2013 (UTC) was predicted using the FVCOM. It is in an ebb tidal course, in which seawater flows out of Daya Bay. Figure 4. The tidal current in the experimental area at 17:00:00 6 October 2013 (UTC) was predicted using the FVCOM. It is in an ebb tidal course, in which seawater flows out of Daya Bay. Applsci 12 03064 g004

Figure 5. The comparison of the optimal paths for different spatial resolutions. The thick black lines are the optimal paths predicted by the MDP algorithm. The triangles indicate the starting point, and the squares are the target. The grid size of 2000 m is used for the top panels, 1000 m for the middle, and 200 m for the bottom. The left, middle and right panels represent Case 1, 2, and 3, respectively. Figure 5. The comparison of the optimal paths for different spatial resolutions. The thick black lines are the optimal paths predicted by the MDP algorithm. The triangles indicate the starting point, and the squares are the target. The grid size of 2000 m is used for the top panels, 1000 m for the middle, and 200 m for the bottom. The left, middle and right panels represent Case 1, 2, and 3, respectively. Applsci 12 03064 g005

Figure 6. The comparison using the MDP (top) and the A* (bottom) algorithms for Cases 1 (left), 2 (middle), and 3 (right). The triangle is the starting point, and the square is the target. Figure 6. The comparison using the MDP (top) and the A* (bottom) algorithms for Cases 1 (left), 2 (middle), and 3 (right). The triangle is the starting point, and the square is the target. Applsci 12 03064 g006

Figure 7. The comparison of the optimal paths with (top) and without (bottom) consideration of steady ocean currents is shown in Figure 4. The AUV moves against the ocean current. The symbol ‘×’ indicates where the AUV stops moving forward because it cannot overcome the counter-current. The panels from left to right are for Cases 1, 2, and 3. Figure 7. The comparison of the optimal paths with (top) and without (bottom) consideration of steady ocean currents is shown in Figure 4. The AUV moves against the ocean current. The symbol ‘×’ indicates where the AUV stops moving forward because it cannot overcome the counter-current. The panels from left to right are for Cases 1, 2, and 3. Applsci 12 03064 g007

Figure 8. Same as Figure 7, but the AUV moves along the ocean current. Figure 8. Same as Figure 7, but the AUV moves along the ocean current. Applsci 12 03064 g008

Figure 9. The predicted paths with consideration of time-varying currents for Cases 1 (a), 2 (b), and 3 (c). Figure 9. The predicted paths with consideration of time-varying currents for Cases 1 (a), 2 (b), and 3 (c). Applsci 12 03064 g009

Figure 10. The optimal path predicted by the 3D-MDP algorithm with (black solid) and without (magenta dash) consideration to the 3D ocean currents. Figure 10. The optimal path predicted by the 3D-MDP algorithm with (black solid) and without (magenta dash) consideration to the 3D ocean currents. Applsci 12 03064 g010

Table 1. The action set partitioning in the 3D space. Table 1. The action set partitioning in the 3D space. Action NumberAction115→17215→14315→11415→18515→15615→12715→19815→16915→131015→51115→25 Table

Table 2. The coordinates of starting and target points in the three cases. Table 2. The coordinates of starting and target points in the three cases. CaseStartTarget1114.6787° E, 22.7375° N114.6787° E, 22.4315° N2114.5813° E, 22.7375° N114.9123° E, 22.4675° N3114.7176° E, 22.7195° N114.9317° E, 22.6295° N Table

Table 3. Algorithm performances for different spatial resolutions. Table 3. Algorithm performances for different spatial resolutions. Grid Size (m) Grid NumberCase 1Case 2Case 3CPU Time (s)Path Length (m)CPU Time (s)Path Length (m)CPU Time (s)Path Length (m)20003864.51563.4000 × 1043.57814.8770 × 1043.50004.7314 × 10410001591184.59603.4000 × 104212.73864.7598 × 104205.21934.6142 × 104200379285.3942 × 1043.4000 × 1044.6778 × 1044.6778 × 1046.7776 × 1044.4874 × 104 Table

Table 4. The performance comparison between MDP and A* algorithms. Table 4. The performance comparison between MDP and A* algorithms. AlgorithmNodesCase 1Case 2Case 3CPU Time (s)Path Length (m)CPU Time (s)Path Length (m)CPU Time (s)Path Length (m)MDP3864.51563.4000 × 1043.57814.8770 × 1043.50004.7314 × 104A*2710.27853.4000 × 1040.23135.3452 × 1040.23485.2968 × 104 Table

Table 5. The comparison of the travel time (h) along the optimal paths between with (MDP − Current) and without (MDP without Current) consideration of ocean currents. In this case, the AUV moves against the ocean current. ∞ means that the AUV cannot finish the predicted path due to strong counter-currents. Table 5. The comparison of the travel time (h) along the optimal paths between with (MDP − Current) and without (MDP without Current) consideration of ocean currents. In this case, the AUV moves against the ocean current. ∞ means that the AUV cannot finish the predicted path due to strong counter-currents. Case 1Case 2Case 3MDP − Current40.4790112.8243108.2778MDP No Current40.4790∞∞ Table

Table 6. Same as Table 5, but the AUV moves along the ocean currents. Table 6. Same as Table 5, but the AUV moves along the ocean currents. Case 1Case 2Case 3MDP + Current13.330814.461517.9979 MDP No Current13.330814.727819.4622 Table

Table 7. The travel times in the three cases with time-varying ocean currents. Table 7. The travel times in the three cases with time-varying ocean currents. Case123Travel time (h)14.952117.969819.5373 Table

Table 8. The coordinates and depths of starting point and target point in the 3D-MDP algorithm simulation. Table 8. The coordinates and depths of starting point and target point in the 3D-MDP algorithm simulation. Longitude (°E)Latitude (°N)Depth (m)Start114.951222.431515Target114.620322.73751 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). Share and Cite MDPI and ACS Style

Shu, M.; Zheng, X.; Li, F.; Wang, K.; Li, Q. Numerical Simulation of Time-Optimal Path Planning for Autonomous Underwater Vehicles Using a Markov Decision Process Method. Appl. Sci. 2022, 12, 3064. https://doi.org/10.3390/app12063064

AMA Style

Shu M, Zheng X, Li F, Wang K, Li Q. Numerical Simulation of Time-Optimal Path Planning for Autonomous Underwater Vehicles Using a Markov Decision Process Method. Applied Sciences. 2022; 12(6):3064. https://doi.org/10.3390/app12063064

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Shu, Mingrui, Xiuyu Zheng, Fengguo Li, Kaiyong Wang, and Qiang Li. 2022. "Numerical Simulation of Time-Optimal Path Planning for Autonomous Underwater Vehicles Using a Markov Decision Process Method" Applied Sciences 12, no. 6: 3064. https://doi.org/10.3390/app12063064

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