Global Path Planning for Unmanned Surface Vehicle Based on Improved Quantum Ant Colony Algorithm

As a tool to monitor marine environments and to perform dangerous tasks instead of manned vessels, unmanned surface vehicles (USVs) have extensive applications. Because most path planning algorithms have difficulty meeting the mission requirements of USVs, the purpose of this study was to plan a global path with multiple objectives, such as path length, energy consumption, path smoothness, and path safety, for USV in marine environments. A global path planning algorithm based on an improved quantum ant colony algorithm (IQACA) is proposed. The improved quantum ant colony algorithm is an algorithm that benefits from the high efficiency of quantum computing and the optimization ability of the ant colony algorithm. The proposed algorithm can plan a path considering multiple objectives simultaneously.The simulation results show that the proposed algorithm’s obtainedminimum was 2.1–6.5% lower than those of the quantum ant colony algorithm (QACA) and ant colony algorithm (ACA), and the number of iterations required to converge to the minimumwas 11.2–24.5% lower than those of the QACAandACA. In addition, the optimized path for the USV was obtained effectively and efficiently.


Introduction
An unmanned surface vehicle (USV) is a kind of autonomous marine vehicle.Determining the path of a USV is an important problem associated with its safety and efficiency [1].Depending on whether the environmental information is obtained from a digital map or sensors, path planning is divided into global and local stages [2].In this paper, a USV global path planning study is presented.Global path planning is the process of planning a path to connect the starting and destination points under a given planning space from a digital map and constraints according to the mission requirements.The indices for evaluating a path can be path length, energy consumption, path smoothness, and path safety.
Obtaining a short path from the starting point to the destination point is one of the main objectives of global path planning.Planning the shortest path is an NP-hard problem [3].Existing methods take the path length as a single objective of the path planning, and neither energy consumption nor other indices are considered.
The energy consumption during sailing determines the USV's endurance and the duration of the mission.Since the environmental loads such as wind, waves, and ocean currents influence the performance of the USV, the calculation of the USV's energy consumption is complex.Niu et al. [4] considered the effect of the ocean current on the energy consumption of USVs. Lee et al. [5] found a more economical path by considering the shallow water effect as well as tidal currents and wind for surface ship navigation.Most calculations of energy consumption have considered the effects of ocean currents on the USV without considering wind and waves.
The smoothness of a path depends on the size and number of the turns that the USV makes while sailing along the planned path.The smoother path allows the USV to make fewer turns along the path, which reduces the mechanical wear on the steering actuators, such as rudders.Smooth paths can reduce unnecessary curvature discontinuities and possible stops.In a previous report [6], the smoothness of a path was evaluated by summing the angles of each turn on the path that the vehicle follows.Ma et al. [7] evaluated the turn angle set by adopting the maximum value of the turn angle set to assess the path smoothness for the USV.
Obstacles such as islands and reefs affect the safety of the USV.Path safety means that the USV cannot collide with any obstacles while sailing.Ma et al. [7] used circles that just covered the obstacles to identify the safe area.
Since USV global path planning involves optimization algorithms, environmental models, and marine craft hydrodynamics, existing path planning algorithms have difficulty meeting the mission requirements.Intelligent optimization algorithms are widely used in global path planning, such as the genetic algorithm [8], particle swarm algorithm [9], NSGA-II [10], and ant colony algorithm [11].With the development of quantum technology, the idea of combining quantum computing with intelligent optimization algorithms has been developed.Narayanan and Moore combined quantum mechanics principles and evolutionary computing methods for the first time [12].A quantum bit and superposition of states were proposed to solve the knapsack problem by a quantum-inspired evolutionary algorithm (QEA) [13].Based on the QEA with a quantum rotation gate strategy, an adaptive evolution-based quantum-inspired evolutionary algorithm (AEQEA) introduces an adaptive evolution mechanism [14].A new improved quantum evolution algorithm (IQEA) with a mixed local search procedure was proposed [15].Li et al. [16] proposed a quantum ant colony algorithm (QACA) that combined quantum computing and the ant colony algorithm for continuous space optimization.You et al. [17] proposed a novel parallel ant colony optimization algorithm based on a quantum dynamics mechanism (PQACO).An improved quantum ant colony algorithm was proposed for the optimization of evacuation paths from dangerous areas to safe areas [18].The quantum ant colony algorithm was used to determine campus path navigation [19].
In this paper, a global path planning algorithm for USV based on the improved quantum ant colony algorithm (IQACA) is proposed.The main contributions of the proposed approach are as follows: (1) At present, most USV global path planning algorithms only search for a feasible path for one objective [3][4][5][6].In this paper, path planning was considered with multiple simultaneous objectives, which were path length, energy consumption, path smoothness, and path safety.(2) The IQACA is a new optimization algorithm that combines quantum-inspired computing with the ant colony algorithm (ACA).The quantum bit (Q-bit) is used to encode the pheromone in the ACA to obtain the quantum pheromone, and the ant movement is determined based on the concentration of the quantum pheromone on the path.Compared to the existing QACA [16][17][18][19], the phase of the quantum ant colony is transformed by an adaptive quantum rotation gate, and the quantum pheromone is updated by local and global update rules in the IQACA.
Simulation experiments in a complex environment with wind, waves, and ocean currents verified the effectiveness of the objective model, and we obtained a desired path based on the IQACA.
The paper is organized as follows.In Section 2, the USV path planning problem is established, and the USV kinetic model, environmental loads, and cost function of the path planning are described.In Section 3, the principles of the IQACA are provided, and we apply the IQACA to USV global path planning.In Section 4, the simulations for USV global path planning using the IQACA are presented.Conclusions are provided in Section 5.

Problem Statement
2.1.USV Kinetic Model.The kinetic model of a USV accounts for the forces, such as the control force and environmental loads, which cause USV motion.For the USV, the control force is mainly the thrust of each propeller.The environmental loads on the USV are generated by wind, waves, and ocean currents.The kinetic model of the USV, which was proposed previously [20], is as follows: where  is the system inertia matrix, (]) is the Corioliscentripetal matrix, (]) ∈  3×3 is damping matrix.  ,  V , and   are wind, wave, and ocean current forces acting on the USV, respectively, and  ℎ is the thrust generated by the USV propulsion system.The generalized velocity ] = [, V, ]  is obtained by (1), where the first two components (, V) are the linear velocities of the surge and sway, and  is the angular velocity of the yaw.

Models of Environmental Loads.
When planning a global path for USVs, it is necessary to consider the environmental effects on the vehicles.Thus, we need to analyze the impacts of wind, waves, and ocean currents on the USV.The planned area is a confined sea with some static obstacles, and the mission execution time is short.Therefore, it can be assumed that the environmental loads are basically stable in limited time and space.

Wind Forces.
The wind acts directly on the superstructure of the hull.As reported previously [21], the wind forces are written as follows: where   is the density of air,   and   are the frontal and lateral projected areas,   (  ),   (  ), and   (  ) are the empirical force coefficients,   is the angle between the wind and the heading of the vessel,  is the length of the vessel,   is the relative wind speed, and   ,   , and   are the wind forces during the surge, sway, and yaw, respectively [22].

Wave Forces.
When a vehicle is sailing on the sea, the interference of wave forces is complicated.The wave forces acting on the hull are first-and second-order wave forces.The second-order wave forces, which impact the heading and path of the USV, are proportional to the square of the wave height [22].The wave forces are simplified as follows: where   ( = 1, 2, 3) are Gaussian white noise processes, and  V ,  V , and  V are the wave forces during the surge, sway, and yaw, respectively.The amplitudes of  V ,  V , and  V are adjusted by choosing the constants   ( = 1, 2, 3), while the spectra are parameterized in terms of the pairs   and   ( = 1, 2, 3).The wave drift forces   ( = 1, 2, 3) are usually modeled as slowly varying bias terms: where   ( = 4, 5, 6) are Gaussian white noise processes [22].

Ocean Current Forces.
The ocean currents cause vessels sailing on the sea to change their positions and postures.The ocean current forces are given as follows: where  is the density of the seawater,   and   are the frontal and lateral projected areas below the waterline, respectively,   ,   , and   are the empirical force coefficients,   is the relative current speed,  is the angle between the ocean current and the heading of the vessel,  is the length of the vessel, and   ,   , and   are the ocean current forces during surge, sway, and yaw, respectively [22].

Path Representation by Grids.
The real task area is partitioned to reduce the modeling complexity.Visibility graphs [23], Voronoi diagrams [24], and grid maps [25] are the most commonly used path planning algorithms.The grid mapbased path planning algorithm is powerful in that it generates a path with the shortest computation time [25].To facilitate the calculation, the planned path is represented on grids.The area under consideration is discretized into grids.The information, such as the relative speed and direction of the wind, the amplitude and direction of the waves, the relative speed and direction of ocean current, and the position of the obstacles, is discretized in each grid.Stationary obstacles are encoded in a binary format on the grids.We assigned weights of 1 to all obstacle grids and weights of 0 to all free neighbor grids of them.

Objectives of USV Global Path Planning.
Since USV global path planning is a multiobjective optimization problem, we should analyze the interrelated objectives and discuss the importance of each objective based on the requirements of the mission.A cost function can be constructed as a weighted sum of the objective functions.Finally, the cost function is used to evaluate the quality of the planned path.

Path Length.
Since the task area is modeled by grids, the planned path is represented on a rectangular grid.The path passes the centers of the grids.Thus, the distance  ,+1 between two adjacent waypoints   = (  ,   ) and  +1 = ( +1 ,  +1 ) is equal to the Euclidean distance between the centers of the grids as follows: The positions of   and  +1 are shown in Figure 1.If   and  +1 are adjacent in the horizontal or vertical direction,  ,+1 = 1.If   and  +1 are adjacent in the diagonal direction,  ,+1 = √ 2.
Therefore, the total length of the path  is the sum of the distances between the adjacent waypoints: where  is the number of path segments.

Energy Consumption.
In this paper, the energy consumption of the USV while sailing is derived from the propulsion system.Thus,  is the sum of the energy consumption of each segment along the entire path: Supposing that the USV is sailing at a constant velocity between   and  +1 , the energy consumption  ,+1 between   and  +1 equals the work done by the propulsion system to overcome the environmental loads, such that where  is the time for the USV to sail in  ,+1 .
where |  → V V | is the magnitude of the velocity  → V V generated by the USV propulsion system,  V is the resultant force of the environmental loads, and |  → V  | is the magnitude of the velocity  → V  of the USV moving in the horizontal plane.
Since the headings of the USV in the grid are several fixed values, as shown in Figure 1, the angular velocity  caused by the yaw motion can be ignored when solving where  and V are obtained by (1).It is known from (10) The difference   between  +1 and   is ,  +1 , and   , are shown in Figure 2. Therefore, the cost function of the path smoothness  ℎ is where  is the number of differences   .

Path Safety.
Using the safety cost of the nodes on the grids cannot accurately represent the threat impact of each path segment.First, three sampling points are selected on a path segment and the average Euclidean distance between the three sampling points and the center of the obstacle is calculated.The schematic diagram of the calculation of the path safety is shown in Figure 3.   is the length of the th path segment between the waypoint   and  +1 .For the th path segment, three sampling points are taken at   /6,   /2, and 5  /6, respectively.The average Euclidean distance between the three sampling points and the center of the obstacle is where   , () is the Euclidean distance from the sampling point on the th path segment to the center of the obstacle   .
The path safety cost between waypoint   and  +1 denoted as  ,+1  is where  is the distance between the USV and the obstacle's center,   max is the radius of the obstacle's affected area, and   min is the radius of the no-sail zone. ,+1 , is obtained using (17).
Thus, the entire path safety cost function   is where  is the number of the waypoints of the planned path.

Cost Function.
In summary, the cost function of the USV global path planning was established as where , ,  ℎ , and   are obtained by ( 8), ( 9), ( 16), and where  max is the maximum voyage distance of the USV and |V max | is the maximum speed of the USV.

Optimization Algorithm
In this section, we will introduce the optimization algorithm for the USV global path planning-the IQACA.The IQACA is a new optimization algorithm that combines quantuminspired computing with ant colony optimization algorithm.We will introduce quantum code and a quantum rotation gate from quantum-inspired computing.Some rules based on the ant colony optimization algorithm are presented.
where   = ( 1 , ⋅ ⋅ ⋅ ,   ) and   = ( 1 , ⋅ ⋅ ⋅ ,   ) are the two sets of solutions for individual   .Therefore, after quantum coding, every individual has two sets of solutions and the search space is doubled.In the IQACA, the quantum pheromone is obtained by encoding the pheromone left by the ants on the path in the ACA by the Q-bits.The transfer direction of the ants is selected by the quantum pheromone concentration on the path.Thus, the quantum pheromone concentration value    of the th ant on the th point in the th iteration is expressed as 3.2.Adaptive Quantum Rotation Gate.In the quantum optimization algorithm, a quantum rotation gate is used to update the Q-bits.The update rule of a Q-bit is as follows: where [   ,    ]  represents the probability amplitude of the Q-bits in the th iteration.(  ) is the quantum rotation gate in the th iteration where   is the rotation angle in the th iteration.In a previous paper [13], the rotation angle was obtained by looking it up in a table.In another paper [26], the local and global updates of the pheromone concentration increments in the ACA were added to the rotation angle step function.In the IQACA, an adaptive adjustment strategy for the rotation angle is obtained by comparing the current solution and the global optimal solution currently being searched.Thus, the rotation angle   in the th iteration is where − sgn(  ) is the direction of the rotation angle and Δ  is the size of the rotation angle.  is where  0 and  0 are the probability amplitudes of the quantum pheromone corresponding to the global optimal solution currently searched and  1 and  1 are the probability amplitudes of the quantum pheromone corresponding to the current solution.Δ  is where   is the cost value of ant  in the current solution,   is the cost value of the global optimal solution currently searched, and  max is the maximum number of iterations.

Transfer Rule and Transition
Probability.The ant colony optimization algorithm is a bionic intelligent algorithm inspired by the foraging behavior of ant colonies [27].During the foraging, ants produce a substance called a pheromone.
The concentration of the pheromone, which is related to the path length, will determine the movement of other ants.If the path is shorter, the concentration of the pheromone left on the path is larger.
To achieve multiobjective path planning, multiple pieces of heuristic information are used to determine the ant's transfer rules and transition probabilities.The transfer rule of ant  from point  to point  is where  is a random number in the range [0, 1]. 0 is a constant within [0, 1]. is the set of points that ant  may reach by point .s is the target waypoint selected by the following equation: where    () is the pheromone on the path from point  to point  in the th iteration and  ( > 0) is the pheromone index.   () is the multiple inspiration information on the path from point  to point  in the th iteration,  ( > 0) is the index of multiple inspiration information,    () is the quantum information strength on the path from point  to point  in the th iteration, which is expressed as    () = 1/|   ()| 2 , and  ( > 0) is the index of the quantum information strength.
The multiple pieces of heuristic information include the path length heuristic information    (), energy consumption heuristic information    (), path smoothness heuristic information    (), and path safety heuristic information    (). [ where   ,   ,   ℎ , and    are obtained by ( 8), ( 9), ( 16), and (19), respectively., , , and  are the indices of the path length heuristic information, energy consumption heuristic information, path smoothness heuristic information, and path safety heuristic information, respectively.

Update Rules of Pheromone.
After every ant completes a one-transfer, the pheromone on the path it passes is locally updated to avoid falling into a local optimum.When the current point of the ant is   and the next point is   , the pheromone local updating rule is where (  ) is the pheromone of the current point, (  ) is the pheromone of the next point,  1 (0 <  1 < 1) is the pheromone local updating coefficient, and Δ  is the pheromone that every ant leaves on the path from   to   in this iteration, expressed as follows: where  is a constant and   is the cost value of the th ant's path.After all the ants complete an iteration, the pheromone is globally updated to increase the pheromone concentration on the optimized path.The rules are as follows: , ) belongs to the optimal path in this cycle where  is a constant,   is the cost value of the optimal path in this iteration,  2 (0 <  2 < 1) is the pheromone global updating coefficient, and s is the global optimal solution currently being searched.

Start
Place n ants at the starting point.
Determine the transfer rule of the ants by Eq. ( 30), and select the target waypoint by Eq. (31).
Update the phases of Q-bits by Eq. ( 25).
Whether all n ants completed the search Update the pheromone globally by Eq. (37).
Output the waypoints and cost value of the optimal path.

No
Output the candidate solution selected by the ants and calculate the path cost.The main steps are as follows:

t ⩾ N max
Step 1 (initialize the parameters).The number of the ants in the colony is .The maximum number of iterations is  max .The initial quantum pheromone concentration value of the th ant on the th waypoint is expressed as  , where  = 0; Step 2. The ants are placed at the starting point.The transfer rule of the ants is determined by (30), and the target waypoint is selected by (31).
Step 3. The phases of the Q-bits are updated by (25).Step 4. The pheromone is locally updated by (34).
Step 5.After all the ants have passed by all the points in an iteration, the pheromone is globally updated by (37).
Step 6.The candidate solution selected by the ants is output and the path cost is calculated.
Step 7. If the iteration  >  max , the algorithm moves to Step 8; otherwise, it returns to Step 2.
Step 8.The waypoints of the optimized solution and the cost value of the path are output.The global optimized path is obtained by the waypoints of the optimized solution.
Step 9.The algorithm ends.

Simulation Studies
In this section, the effectiveness and efficiency of the IQACA are validated.The section consists of two parts.The first subsection compares the performance of the ACA, QACA, and IQACA with the Traveling Salesman Problem (TSP).The second subsection deals with the USV global path planning based on the IQACA.To validate the proposed algorithm, simulations were conducted.

Performance Evaluation of IQACA.
To validate the effectiveness of the IQACA presented in this paper, we compared the algorithm performance between ACA, QACA, and IQACA with the TSP.In this paper, RAND100 was selected from the TSPLIB standard library for the simulations.The maximum number of iterations  max = 200, the number of the ants  = 100,  = 3,  = 1,  = 2, and  1 =  2 = 0.8.The obtained values are shown in Table 1, and the iterations are shown in Figure 5.
The known optimal value of RAND100 is 7891.From Table 1, it was concluded that the path length of RAND100 obtained by the IQACA was 2.12% lower than that obtained by the QACA and 6.58% lower than that obtained by the ACA.The number of iterations required for the IQACA to converge to the minimum was 11.27% lower than the QACA and 24.55% lower than the ACA.The results show that the IQACA was superior to the QACA and the ACA in both the path length and iteration number.Since the algorithm uses the pheromone local and global updates, and the phases of the Q-bits are updated by the adaptive quantum rotation gate, the IQACA can avoid the local optimal solution.Because the pheromone is encoded by Q-bits, the search space is doubled, and the convergence speed is faster.Thus, the IQACA is an effective and efficient algorithm.The coordinates of the starting point are (0.5, 24.5), and the coordinates of the destination point are (29.5, 0.5).The length of the side of a grid is 1 .We assumed that the wind, waves, and ocean current act on the vessel from the same direction, since in most cases the ocean current is the most significant environmental disturbance on the vessel.In this simulation, the direction of the disturbances is assumed as 240 ∘ in the Northeast coordinate system.The relative wind speed was 7.5 /.The wave height was 2.5 .The relative ocean current was 2.0 /.The thrust of the USV propulsion system was 500.The coefficients of the USV in this simulation are shown in Table 2.The maximum number of iterations was 500, the number of the ants was 100,  = 3,  = 2, and  1 =  2 = 0.8.The safety boundary of the obstacle was represented by a dotted red circle whose radius was 1.3 times the distance between the vertex and the center of the obstacle.The optimal path is represented by a solid blue line.The objectives of the global path planning were determined based on their respective weights.We considered four scenarios in this simulation.In the first scenario, we highly weighted  3. The results indicate that the proposed algorithm can plan feasible paths for the USV considering different objectives simultaneously.Moreover, by adjusting the weights of different objectives, the proposed algorithm can generate paths for different purposes.

Conclusion
This paper proposed a global path planning algorithm for the USV based on an improved quantum ant colony algorithm (IQACA).The IQACA is an optimization algorithm that combines quantum computing with the ACA.In IQACA, using Q-bits to encode the pheromone of the ants, the search space is doubled when the number of the ants is the same.The simulation results show that the proposed algorithm's obtained minimum was 2.1-6.5% lower than those of the quantum ant colony algorithm (QACA) and ant colony algorithm (ACA), and the number of iterations required to converge to the minimum was 11.2-24.5% lower than those of the QACA and ACA.Based on the model of the kinetics of the USV and the marine environment, we defined the objectives of the path planning: the path length, energy consumption, path smoothness, and path safety.The simulation results showed that the proposed algorithm can consider several optimization objectives and generate paths satisfying these requirements.
In the future, the following studies should be conducted in depth.First, the correlation between the multiple objectives should be calculated to determine the weight of each objective in the cost function to meet the actual mission requirements.Moreover, the kinetic and kinematic constraints of the USV should be added to the cost function.Finally, more practical environmental loads should be applied to calculate their effects on the path energy consumption of the USV.

Figure 2 :
Figure 2: The angles of the path segments.

Figure 3 :
Figure 3: Schematic diagram of the calculation of the path safety.

Figure 4 :
Figure 4: The flowchart of the global path planning algorithm based on the IQACA.

3. 5 .
Global Path Planning Algorithm Based on IQACA.The flowchart of global path planning algorithm based on the IQACA is shown in Figure 4.

Figure 6 :Figure 7 :
Figure 6: Planned path focusing on the path length.

Figure 8 :
Figure 8: Planned path focusing on the path smoothness.

Figure 9 :
Figure 9: Planned path focusing on the path safety.
It is assumed that the current waypoint of the USV is   = (  ,   ), the previous waypoint is  −1 = ( −1 ,  −1 ), and the next waypoint is  +1 = ( +1 ,  +1 ).Thus, the angle  +1 of the vector   →    +1 and the angle   of the vector   →  −1   are and (11) that the energy consumption is proportional to |  → V V | and 1/|  → V  |, when the thrust  ℎ generated by the propulsion system is a fixed value.To reduce the energy consumption, it is necessary to adjust the USV's heading to take advantage of the environmental loads to increase |  → V  |. 2.4.3.Path Smoothness.

Table 1 :
The obtained values from the simulations.

Table 3 :
Path data for different objectives.