Unmanned aerial vehicle three-dimensional path planning method based on improved artificial bee colony algorithm

Abstract

An unmanned aerial vehicle three-dimensional path planning method based on an improved artificial bee colony algorithm comprises the following steps that S1, a GPS module and a high-definition camera are installed on an unmanned aerial vehicle to send position and environment information of the unmanned aerial vehicle in real time, and a three-dimensional flight model of the area is established based on the collected position and environment information; and S2, judging the position of the unmanned aerial vehicle in the three-dimensional flight model according to the real-time position information of the unmanned aerial vehicle, and performing three-dimensional path planning on the unmanned aerial vehicle through an improved artificial bee colony algorithm. The unmanned aerial vehicle three-dimensional path planning method based on the improved artificial bee colony algorithm solves the technical problems that in the prior art, a traditional artificial bee colony algorithm is prone to falling into local optimum, an unmanned aerial vehicle flight path optimal solution cannot be rapidly and accurately obtained, three-dimensional modeling is difficult, and the unmanned aerial vehicle flight path planning efficiency is high. And real-time environment information cannot be accurately obtained.

Description

Technical Field

[0001] The present invention relates to the technical field of UAV three-dimensional path planning, and in particular to a UAV three-dimensional path planning method based on an improved artificial bee colony algorithm. Background Technique

[0002] In recent years, unmanned aerial vehicles (UAVs) have shown great potential in fields such as power inspection, geological exploration, and military exercises. The demand for UAVs for autonomous driving is also increasing day by day. Path planning technology is one of the key technologies for realizing UAV autonomous driving. The long endurance requirement of UAVs is for a flight path with low energy consumption, no threat, and short distance. Specifically, it means that the UAV quickly and accurately plans an optimal path with a short path length, low path curvature, and low path risk in the air environment from the current position. Although the path planning of robots can be used as a reference, the path planning problem of UAVs is three-dimensional, with an additional degree of freedom in the vertical direction, and it is difficult to create a three-dimensional flight model, which makes it difficult for traditional path planning methods to well meet the three-dimensional path planning of UAVs.

[0003] The artificial bee colony algorithm is a multi-objective optimization method proposed by imitating the foraging behavior of bee colonies. It simulates the foraging elements of employed bees, follower bees, and scout bees, and also simulates the foraging behaviors such as species transformation, abandoning nectar sources, and searching for new nectar sources. Traditional artificial bee colony algorithms have advantages such as few parameters, simple control, and fast convergence speed, but also have disadvantages such as being easily trapped in local optima.

[0004] In summary, the artificial bee colony algorithm has advantages such as fast convergence speed, simple control, and high accuracy in the path planning problem of UAVs. However, it also has disadvantages such as being easily trapped in local optima and lacking an accurate three-dimensional flight model.

[0005] The patent document with the application publication number CN111626500A discloses a path planning method based on an improved artificial bee colony algorithm. First, a new initialization strategy is adopted to obtain a high-quality initial population and reduce the number of optimization iterations. Then, different search equations are used in the three stages of employed bees, follower bees, and scout bees in the traditional artificial bee colony algorithm to enhance the local search ability and avoid premature convergence in the later optimization process. Finally, the improved artificial bee colony algorithm is applied to the path planning problem. Its defects are as follows: 1) When modeling the flight of UAVs, only map information is used, which is not applicable to complex terrains or terrains lacking map information, and only a two-dimensional flight environment is modeled, reducing one degree of freedom for UAV flight, which may lead to the failure to find the optimal path; 2) When constructing the fitness function, only two aspects of the path length and path risk are considered, without considering the influence of the path curvature, which may lead to a large turning angle of the UAV, thus increasing the possibility of falling and the additional fuel consumption for turning. 3) The method does not perform mutation processing on the nectar sources with low fitness, which may result in a low convergence rate and insufficient accuracy. Summary of the Invention

[0006] The technical problem to be solved by the present invention is to provide a three-dimensional path planning method for UAVs based on an improved artificial bee colony algorithm, so as to solve the problems existing in the prior art that the traditional artificial bee colony algorithm is prone to falling into local optimum, unable to quickly and accurately obtain the optimal solution of the UAV flight path, and the three-dimensional modeling is difficult, and the real-time environmental information cannot be accurately obtained.

[0007] To solve the above technical problems, the technical solution adopted by the present invention is: a three-dimensional path planning method for UAVs based on an improved artificial bee colony algorithm, including the following steps: S1. Install a GPS module and a high-definition camera on the UAV to send the position and environmental information of the UAV in real time, and establish a three-dimensional flight model of the area based on the collected position and environmental information. S2. Then, judge the position of the UAV in the three-dimensional flight model according to the real-time position information of the UAV, and perform three-dimensional path planning on it through an improved artificial bee colony algorithm.

[0008] Preferably, in the step S1, the establishment of the three-dimensional flight model is specifically as follows: Taking the longitude as the x-axis, the latitude as the y-axis, and the terrain height as the z-axis, setting the UAV starting point as the origin, decomposing the space into cubes according to a certain length, width and height, and then comparing the UAV flight height with the actual terrain height corresponding to the cube. If the UAV flight height is lower than the actual terrain height corresponding to the cube, it is regarded as an obstacle grid.

[0009] Preferably, in the step S2, the steps of performing three-dimensional path planning on it through an improved artificial bee colony algorithm are as follows: S2-1. Construct a fitness function, which is used to evaluate the quality of path planning, including path length, energy loss and path risk, and the fitness is the weighted sum of path length, path curvature and path risk. S2-2. Generate a chaotic sequence through the Logistic mapping formula, and introduce the chaotic sequence to improve the nectar source initialization process. S2-3. Use employed bees to correspond to all nectar sources one by one, search near the nectar sources, and calculate their fitness values. S2-4. Use the Gaussian mutation mechanism to improve the accuracy of local search in the artificial bee colony algorithm. Select several honey sources with the worst fitness values among all honey sources for Gaussian mutation to obtain new honey sources; S2-5. Optimize the path selection using the greedy selection strategy. Compare the fitness values of the honey sources and their nearby ones one by one, and retain the one with a higher fitness value. The path corresponding to this honey source is the employed bee flight path, that is, the old flight path; S2-6. Use the onlooker bees to search near the honey sources selected in the roulette wheel strategy. The result of each search is the onlooker bee flight path, that is, the new flight path, and calculate its fitness value; S2-7. Compare the fitness values of the old flight path and the new flight path; if the fitness value of the new flight path is greater than that of the old flight path, retain the new flight path, otherwise, retain the old flight path; S2-8. During the search process, if a better honey source is not found after multiple iterations, it is determined that it has entered the local optimum, that is, abandon this honey source. The employed bee corresponding to this honey source will be transformed into a scout bee, and the scout bee will search for a new honey source and update it; S2-9. After this iteration is completed, record the current optimal solution and return to S2-3 to continue the iteration; S2-10. After the number of iterations reaches the upper limit, output the optimal flight path.

[0010] Preferably, in the step S2-1, the fitness function includes path length, path curvature, and path risk; The path length is the total length calculated from the coordinate points in the three-dimensional flight model, and its expression is as follows: ; ; Where is the path length fitness function value, represents the coordinate distance between two adjacent points, , , represent the three-dimensional coordinates of point i; The path curvature is the total curvature calculated from three adjacent points in the three-dimensional flight model, and its expression is as follows: ; ; Where is the path curvature fitness function value, , , represent the three-dimensional coordinates of point i, represents the coordinate distance from the (i - 1)-th point to the i-th point; The path risk is the sum of the number of obstacles passed by the path in the three-dimensional flight model and the distance from the obstacle center, and its expression is as follows: ; ; ; where is the path risk fitness function value, represents the risk coefficient of the UAV at the t-th obstacle, represents the radius of the obstacle, represents the safety radius of the UAV, represents the distance from the UAV to the center of the t-th obstacle, , , represent the three-dimensional coordinates of the center of the t-th obstacle; The path fitness function is as follows: ; ; where f is the weighted sum of the path fitness function, is the weighting function.

[0011] Preferably, the step S2-2 is specifically as follows: The nectar source of the traditional artificial bee colony algorithm is generated by the following expression: ; ; ; where is the j-th dimension variable of the i-th nectar source, N represents the number of nectar sources, d represents the problem dimension, and are respectively the upper and lower bounds of, is a random number within; When the Logistic mapping formula is used to generate the chaotic sequence, the above expression is improved to obtain: ; ; ; ; where is the value of the algorithm at the k-th iteration. When the initial value is not 0, 0.25, 0.5, 0.75, 1, the sequence is in a chaotic state. is the growth rate. When is 4, the system is in a fully chaotic state, which can expand the search range of the algorithm.

[0012] Preferably, in the step S2-3, the expression for the employed bee to search near the nectar source is; ; ; where is the newly searched nectar source near the nectar source, represents the i-th nectar source, is a random number.

[0013] Preferably, in the step S2-4, the method of using Gaussian mutation is: After each iteration, the fitness function values of the nectar sources are sorted, and the with the worst fitness value among all nectar sources are selected for Gaussian mutation to obtain new nectar sources, where n is the number of nectar sources, is the mutation ratio, and the expression for the new nectar source is as follows: ; where is the position of the mutated nectar source, represents the mean value, represents the standard deviation.

[0014] Preferably, in the step S2-5, the expression for the fitness value of the nectar source is as follows: ; where, is the function value of the problem to be solved.

[0015] Preferably, in the step S2-6, the roulette wheel strategy expression for the follower bee is: ; where is the selection probability for the nectar source i, and N represents the number of nectar sources.

[0016] Preferably, in the step S2-8, the method for the scout bee to update the new nectar source is as follows: During the search process, if a better nectar source has not been found after multiple iterations, it is determined that it has entered a local optimum, that is, the nectar source is abandoned, and the employed bee corresponding to this nectar source will be transformed into a scout bee. The scout bee will search for a new nectar source and update it, and the expression is as follows: ; where is the new nectar source found by the scout bee, t represents the number of iterations after falling into the local optimum, Indicates the upper limit of the number of iterations, and are respectively the upper and lower bounds of is a random number within

[0017] The present invention provides a three-dimensional path planning method for unmanned aerial vehicles based on an improved artificial bee colony algorithm, which has the following beneficial effects: 1. The present invention installs a GPS module and a high-definition camera on the unmanned aerial vehicle to detect environmental information in real time, and builds a three-dimensional flight model based on this. Different from building a model using map information, this method is more accurate and has instantaneity. It will not affect the accuracy of the three-dimensional flight model due to the change or error of map information, and can accurately obtain real-time environmental information.

[0018] 2. The present invention effectively avoids falling into local optimum by improving the artificial bee colony algorithm, and can quickly and accurately obtain the optimal solution of the unmanned aerial vehicle flight path. By adding a chaos sequence strategy during the initialization of the nectar source, the situation that the spatial position distribution of the nectar source is uneven due to random initialization in the traditional artificial bee colony algorithm is solved, effectively improving the speed and quality of the solution; by adding a Gaussian mutation strategy during the search for the nectar source, the situation that the quality of the new nectar source deteriorates due to the poor quality of the nectar source in the traditional artificial bee colony algorithm is solved, effectively improving the local and overall search capabilities, and can find the optimal path more accurately and effectively. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] The present invention will be further described below with reference to the drawings and embodiments: Figure 1 is the flowchart of the method of the present invention; Figure 2 is the comparison of the iteration curves of the improved artificial bee colony algorithm of the present invention and other algorithms. DETAILED DESCRIPTION OF THE INVENTION

[0020] As Figure 1 shown, a three-dimensional path planning for unmanned aerial vehicles based on an improved artificial bee colony algorithm includes the following steps: Step S1, install a GPS module and a high-definition camera on the unmanned aerial vehicle to send the position and environmental information of the unmanned aerial vehicle in real time. After a test flight in a given area, establish a three-dimensional flight model of the area based on the collected position and environmental information.

[0021] Specifically, step S1 includes: Taking longitude as the x-axis, latitude as the y-axis, and terrain height as the z-axis, set the starting point of the drone as the origin. Decompose the space into cubes with a certain length, width, and height. Then compare the flight height of the drone with the actual terrain height corresponding to the cube. If the flight height of the drone is lower than the actual terrain height corresponding to the cube, it is regarded as an obstacle grid; The path planning of the drone on the cube is given by the following expression: Let the current position information of the drone be , where A and B are the abscissa and ordinate of the drone respectively. In the two-dimensional space, the distance between the drone and its surrounding nodes is: ; Then find the several nodes closest to the distance . The expressions of the nodes are: ; ; Among them, , are the abscissa and ordinate of the ij-th node respectively, are the abscissa and ordinate of the next node on the path respectively. After determining the abscissa and ordinate, determine the position of the vertical coordinate according to the GPS positioning, that is, determine the three-dimensional coordinates of the next node of the drone path.

[0022] Step S2. Then, based on the real-time position information of the drone, judge the position of the drone in the three-dimensional flight model, and perform three-dimensional path planning through an improved artificial bee colony algorithm.

[0023] The method for three-dimensional path planning through an improved artificial bee colony algorithm includes: Step S2-1. Construct a fitness function. The fitness function is used to evaluate the quality of path planning, including path length, energy loss, and path risk. The fitness is the weighted sum of path length, path curvature, and path risk; The path length is the total length calculated from the coordinate points in the three-dimensional flight model, and its expression is as follows: ; ; Among them is the value of the path length fitness function, represents the coordinate distance between two adjacent points, , , represent the three-dimensional coordinates of the i-th point; The path curvature is the sum of the curvature calculations of three adjacent points in the three-dimensional flight model, and its expression is as follows: ; ; where is the value of the path curvature fitness function, , , represent the three-dimensional coordinates of the i-th point, represents the coordinate distance from the (i - 1)-th point to the i-th point; The path risk is the sum of the number of obstacle areas passed by the path in the three-dimensional flight model and the calculation of the distance from the obstacle center, and its expression is as follows: ; ; ; where is the value of the path risk fitness function, represents the risk coefficient of the UAV at the t-th obstacle, represents the radius of the obstacle, represents the safety radius of the UAV, represents the distance from the UAV to the center of the t-th obstacle, , , represent the three-dimensional coordinates of the center of the t-th obstacle; The path fitness function is as follows: ; ; where f is the weighted sum of the path fitness function, is the weighting function.

[0024] Step S2-2, generate a chaotic sequence through the Logistic mapping formula and introduce the chaotic sequence to improve the nectar source initialization process; During initialization, the nectar source of the traditional artificial bee colony algorithm is generated by the following expression: ; ; ; where is the j-th dimension variable of the i-th nectar source, N represents the number of nectar sources, d represents the problem dimension, and are respectively the upper and lower bounds of, is Random numbers within When the Logistic mapping formula is used to generate the chaotic sequence, the above expression can be improved to obtain: ; ; ; ; where is the value of the algorithm at the k-th iteration. When the initial value is not 0, 0.25, 0.5, 0.75, or 1, the sequence is in a chaotic state. is the growth rate. When is 4, the system is in a fully chaotic state, which can expand the search range of the algorithm.

[0025] Step S2-3: Use the employed bees to correspond to all the nectar sources one by one, search near the nectar sources, and calculate their fitness values; The expression for the employed bees to search near the nectar sources is; ; ; where is the newly searched nectar source near the nectar source, represents the i-th nectar source, is a random number.

[0026] Step S2-4: Use the Gaussian mutation mechanism to improve the accuracy of the local search of the artificial bee colony algorithm. Take the few nectar sources with the worst fitness values among all the nectar sources for Gaussian mutation to obtain new nectar sources; After each iteration, sort the fitness function values of the nectar sources. Take the nectar sources with the worst fitness values among all the nectar sources for Gaussian mutation to obtain new nectar sources, where n is the number of nectar sources, is the mutation ratio. The expression for the new nectar source is as follows: ; where is the position of the mutated nectar source, represents the mean value, represents the standard deviation.

[0027] Step S2-5: Use the greedy selection strategy to optimize the path selection. Compare the fitness values of the nectar sources and their nearby ones one by one, and select the one with a higher fitness value to retain. The path corresponding to this nectar source is the flight path of the employed bees, that is, the old flight path; The expression for the fitness value of the nectar source is as follows: ; Among them, is the function value of the problem to be solved.

[0028] Step S2-6: Search near the nectar source selected by the follower bee in the roulette wheel strategy. The result of each search is the flight path of the follower bee, that is, the new flight path, and calculate its fitness value; The roulette wheel strategy expression of the follower bee is: ; Among them is the selection probability of the nectar source i, and N represents the number of nectar sources.

[0029] Step S2-7: Compare the fitness values of the old flight path and the new flight path; if the fitness value of the new flight path is greater than that of the old flight path, keep the new flight path, otherwise, keep the old flight path; Step S2-8: During the search process, if a better nectar source has not been found after multiple iterations, it is determined that it has entered the local optimum, that is, abandon the nectar source, and the employed bee corresponding to the nectar source will be transformed into a scout bee, and the scout bee will search for a new nectar source and update; The method for the scout bee to update the new nectar source is as follows: During the search process, if a better nectar source has not been found after multiple iterations, it is determined that it has entered the local optimum, that is, abandon the nectar source, and the employed bee corresponding to the nectar source will be transformed into a scout bee, and the scout bee will search for a new nectar source and update. The expression is as follows: ; Among them is the new nectar source found by the scout bee, t represents the number of iterations after falling into the local optimum, represents the upper limit of the number of iterations, and are respectively the upper and lower bounds of, is a random number within.

[0030] After this iteration is completed, record the current optimal solution and return to S23 to continue the iteration; Step S2-10: After the number of iterations reaches the upper limit, output the optimal flight path.

[0031] Figure 2 For the fitness value comparison, the improved artificial bee colony algorithm of the present invention is compared with the traditional artificial bee colony algorithm, particle swarm algorithm, and gray wolf algorithm, and the number of iterations is set to 30. It can be seen that the improved artificial bee colony algorithm has a faster iteration speed and the best fitness value finally obtained.

[0032] The above embodiments are only the preferred technical solutions of the present invention and should not be regarded as limitations on the present invention. The protection scope of the present invention shall be the technical solutions recorded in the claims, including the equivalent replacement solutions of the technical features in the technical solutions recorded in the claims. That is, equivalent replacement improvements within this scope are also within the protection scope of the present invention.

Claims

1. A three-dimensional path planning method for unmanned aerial vehicles based on an improved artificial bee colony algorithm, characterized in that: The following steps are involved: S1. Install a GPS module and a high-definition camera on the drone to send the drone's location and environmental information in real time, and build a three-dimensional flight model of the area based on the collected location and environmental information; S2. The position of the UAV in the three-dimensional flight model is determined based on the real-time position information of the UAV, and a three-dimensional path planning is performed using an improved artificial bee colony algorithm.

2. The method for three-dimensional path planning of a UAV based on an improved artificial bee colony algorithm according to claim 1, characterized in that: In step S1, the three-dimensional flight model is established as follows: Longitude is used as the x-axis, latitude as the y-axis, and terrain height as the z-axis. The starting point of the drone is set as the origin, and the space is decomposed into cubes according to a certain length, width and height. Then, the flight altitude of the drone is compared with the actual terrain height corresponding to the cube. If the flight altitude of the drone is lower than the actual terrain height corresponding to the cube, it will be regarded as an obstacle grid.

3. The method for three-dimensional path planning of an unmanned aerial vehicle based on an improved artificial bee colony algorithm according to claim 1, characterized in that: The steps of performing three-dimensional path planning by using the improved artificial bee colony algorithm in step S2 are as follows: S2-1. Construct a fitness function, which is used to evaluate the quality of path planning, including path length, energy loss and path hazard. The fitness is the weighted sum of path length, path curvature and path hazard. S2-2, generate chaotic sequence through Logistic mapping, introduce chaotic sequence to improve honey source initialization process; S2-3, use hired bees to correspond to all nectar sources one by one, search near the nectar sources, and calculate their fitness values; S2-4, use the Gaussian mutation mechanism to improve the accuracy of local search of the artificial bee colony algorithm, select the few with the worst fitness values ​​from all nectar sources for Gaussian mutation, and obtain new nectar sources; S2-5. Optimize path selection using the greedy selection strategy, compare the fitness values ​​of the nectar source and its vicinity one by one, select the one with the higher fitness value and keep it. The path corresponding to the nectar source is the flight path of the employed bees, that is, the old flight path; S2-6, using a follower bee to search near the nectar source selected in the roulette strategy, and the result of each search is the flight path of the follower bee, that is, the new flight path, and its fitness value is calculated; S2-7, comparing the fitness values ​​of the old flight path and the new flight path; if the fitness value of the new flight path is greater than the fitness value of the old flight path, the new flight path is retained, otherwise, the old flight path is retained; S2-8. During the search process, if a better nectar source is not found after multiple iterations, it is determined that it has reached a local optimum, that is, the nectar source is abandoned, and the employed bees corresponding to the nectar source are transformed into scout bees, which search for new nectar sources and update; S2-9, after the iteration is completed, record the current optimal solution and return to S2-3 to continue iterating; S2-10: After the number of iterations reaches the upper limit, the optimal flight path is output.

4. The method for three-dimensional path planning of a UAV based on an improved artificial bee colony algorithm according to claim 3, characterized in that: In step S2-1, the fitness function includes path length, path curvature and path hazard; The path length is the sum of the lengths calculated from the coordinate points in the three-dimensional flight model, and its expression is as follows: ； ； in is the path length fitness function value, Represents the coordinate distance between two adjacent points. , , Represents the three-dimensional coordinates of point i; The path curvature is the sum of the curvatures of three adjacent points in the three-dimensional flight model, and its expression is as follows: ； ； in is the path curvature fitness function value, , , represents the three-dimensional coordinates of point i, Represents the coordinate distance from the i-1th point to the i-th point; The path hazard is the sum of the number of obstacle areas passed by the path in the three-dimensional flight model and the distance from the obstacle center. Its expression is as follows: ； ； ； in is the path hazard fitness function value, represents the risk factor of the drone at the tth obstacle, represents the radius of the obstacle, represents the safety radius of the drone, represents the distance from the drone to the center of the tth obstacle, , , Represents the three-dimensional coordinates of the center of the t-th obstacle; The path fitness function is as follows: ； ； Among them, f is the weighted sum of the path fitness function, is the weighting function.

5. The method for three-dimensional path planning of a UAV based on an improved artificial bee colony algorithm according to claim 3, characterized in that: The step S2-2 is specifically as follows: The honey source of the traditional artificial bee colony algorithm is generated by the following expression: ； ； ； in is the j-th dimension variable of the i-th nectar source, N represents the number of nectar sources, d represents the dimension of the problem, and They are The upper and lower bounds of for Random numbers within When the Logistic mapping formula is used to generate chaotic sequences, the above expression is improved to obtain: ； ； ； ； in is the value of the algorithm at the kth iteration, when the initial value When it is not 0, 0.25, 0.5, 0.75, or 1, the sequence is in a chaotic state. is the growth rate, when When it is 4, the system is in a completely chaotic state, which can expand the search range of the algorithm.

6. The method for three-dimensional path planning of a UAV based on an improved artificial bee colony algorithm according to claim 3, characterized in that: In step S2-3, the expression for employed bees to search for the vicinity of the nectar source is: ； ； in is the newly found nectar source near the nectar source. represents the i-th nectar source, Is a random number.

7. The method for three-dimensional path planning of a UAV based on an improved artificial bee colony algorithm according to claim 3, characterized in that: In step S2-4, the method of using Gaussian variation is: After each iteration, the fitness function values ​​of the nectar sources are sorted, and the one with the worst fitness value among all the nectar sources is selected. Gaussian mutation is performed to obtain new nectar sources, where n is the number of nectar sources. is the mutation ratio, and the new nectar source expression is as follows: ； in is the location of the nectar source after mutation, represents the mean, Represents standard deviation.

8. The method for three-dimensional path planning of a UAV based on an improved artificial bee colony algorithm according to claim 3, characterized in that: In step S2-5, the fitness value of the nectar source is expressed as follows: ； in, is the function value of the problem being solved.

9. The method for three-dimensional path planning of a UAV based on an improved artificial bee colony algorithm according to claim 1, characterized in that: In step S2-6, the roulette strategy expression of the follower bee is: ； in is the probability of selecting nectar source i, and N represents the number of nectar sources.

10. The method for three-dimensional path planning of a UAV based on an improved artificial bee colony algorithm according to claim 3, characterized in that: In step S2-8, the method for the scout bees to update new nectar sources is as follows: During the search process, if a better nectar source is not found after multiple iterations, it is determined that it has entered a local optimum, that is, the nectar source is abandoned, and the employed bees corresponding to the nectar source are transformed into scout bees. The scout bees will search for new nectar sources and update them. The expression is as follows: ； in is the new nectar source found by the scout bees, t is the number of iterations after falling into the local optimum, represents the upper limit of the number of iterations, and They are The upper and lower bounds of for Random numbers within.

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