Real time path planning via alternating minimisation through image information

IET Cyber-Systems and RoboticsVolume 3, Issue 3 p. 245-255 ORIGINAL RESEARCH PAPEROpen Access Real time path planning via alternating minimisation through image information Zheng Chen, Zheng Chen Hainan Institute of Zhejiang University, Sanya, China Ocean College, Zhejiang University, Zhoushan, ChinaSearch for more papers by this authorMinjie Zhang, Minjie Zhang Ocean College, Zhejiang University, Zhoushan, ChinaSearch for more papers by this authorJiang Zhu, Corresponding Author Jiang Zhu jiangzhu16@zju.edu.cn orcid.org/0000-0002-7646-2776 Ocean College, Zhejiang University, Zhoushan, China Correspondence Jiang Zhu, Ocean College, Zhejiang University, Zhoushan 316021, China. Email: jiangzhu16@zju.edu.cnSearch for more papers by this authorShiqiang Zhu, Shiqiang Zhu Ocean College, Zhejiang University, Zhoushan, ChinaSearch for more papers by this author Zheng Chen, Zheng Chen Hainan Institute of Zhejiang University, Sanya, China Ocean College, Zhejiang University, Zhoushan, ChinaSearch for more papers by this authorMinjie Zhang, Minjie Zhang Ocean College, Zhejiang University, Zhoushan, ChinaSearch for more papers by this authorJiang Zhu, Corresponding Author Jiang Zhu jiangzhu16@zju.edu.cn orcid.org/0000-0002-7646-2776 Ocean College, Zhejiang University, Zhoushan, China Correspondence Jiang Zhu, Ocean College, Zhejiang University, Zhoushan 316021, China. Email: jiangzhu16@zju.edu.cnSearch for more papers by this authorShiqiang Zhu, Shiqiang Zhu Ocean College, Zhejiang University, Zhoushan, ChinaSearch for more papers by this author First published: 17 August 2021 https://doi.org/10.1049/csy2.12030AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract Real time path planning from image information is of vital importance in the fields of robots as it has various applications in real time navigation, autonomous driving, robot arm manipulation and human robot cooperation and so on. To achieve this task, a two stage approach is proposed. At the first stage, a novel change point detection approach is proposed to process the image to extract the shape of the obstacles. At the second stage, several novel approximations are adopted to make the path planning problem tractable. Firstly, the irregular shapes of the obstacles in the environment are approximated as lines and circles, which simplify the distance constraint significantly. Secondly, the non-convex path planning problem is iteratively decomposed as a sequence of subproblems and alternating minimisation method is proposed to efficiently solve the subproblem. To improve the quality of the solution, good initial points obtained by A* algorithm is provided. Both numerical experiments and real experiments are conducted to demonstrate the effectiveness of the proposed algorithm. 1 INTRODUCTION With the development of robot technology, various robots with more functions have been developed and put into use. Among them, the mobile robot has a simple structure, convenient control, and has the ability to move freely in a plane environment. As a result, it has been widely used in industry, agriculture, military, aerospace and many other fields. As robots aim to complete more complex tasks, path planning becomes a challenging problem, and advanced path planning algorithms are required [1-3]. The purpose of path planning for mobile robot is to find a collision free path from the starting point to the end point in a given working environment [4], and several optimisation criteria such as least time consuming, least movement distance, and least energy consumption have been proposed. In the past few decades, many researchers have invented many methods for effective path planning, including Rapid exploration Random Tree (RRT) algorithm [5-7], artificial potential field method [8, 9], A* (A-star) algorithm, convex optimisation algorithm, etc. The RRT algorithm establishes nodes in a map by random sampling, builds paths between nodes to construct an undirected graph, and search for a relatively optimal path in the undirected graph as the final path. This method can be applied in both two-dimensional planes, and high-dimensional spaces, and the time consumption is low when the robot works in a spacious space. In a narrow or complex space, it takes a lot of time to confirm whether there are obstacles in the path between nodes, leading to especially low efficiency. Compared with other path planning algorithms, the results of RRT have greater randomness, and the path quality generated by it is usually low, and there are a lot of sharp angles, which are far from the optimal path. The classic idea of the artificial potential field method is to regard the movement of the robot in the environment as the movement in the virtual artificial force field. The target point generates gravitational force on the robot to form an attraction field, while the obstacles in the environment generate repulsive force on the robot to form a repulsive force field. The artificial potential field is the resultant force field formed by the superposition of the attraction field and the repulsive force field. In the resultant force field, the robot moves along the direction of the resultant force, and finally reaches the end. This method has a simple structure and good real-time performance, but it has certain shortcomings: in the vicinity of several obstacles, the complex potential field may cause a jitter path, and the path may even fall into a local minimum, which makes mobile robots unable to move forward [1]. For the A* algorithm based on a rasterised image, its core idea is to use heuristic valuation function to assess the path. The evaluated function combines the idea of depth-first search (DFS) algorithms and breadth-first search (BFS) algorithms. It can evaluate the cost of the distance travelled and the estimated cost of reaching the target, thereby obtaining the optimal path and reducing the number of search grid points. In the rasterised environment, the A* algorithm is the most effective direct path search algorithm and can find the optimal solution in a rasterised environment, so it is widely used in various path planning fields. However, the A* algorithm has some limitations in a rasterised environment. On one hand, the rasterised path has only 8 directions, which leads to sharp angles in the path. On the other hand, rasterising the scene enlarged the impassable area, making the result too conservative, and the robot has to travel long distances to avoid obstacles. Therefore, numerous works for the improvement of A* algorithm have been studied. Reference [10] proposes an improved A* algorithm which takes the grid boundary instead of the grid centre as nodes, to make the planned paths smoother. Reference [11] uses two different directions of hexagonal grids to re-rasterise the environment, and at the same time uses the computer's dual core to design parallel algorithm in the hexagonal environment, which improves the efficiency of A* operation. The convex optimisation algorithm is much more flexible as it introduces the cost function and constraints which characterise the smoothness of the path, the dynamics and the modelling of the obstacles. In a small range trajectory optimisation, the convex optimisation method has been widely used [12, 13]. However, when obstacles exist, solving the shortest path is a non-convex problem, and it is difficult to obtain the global minimum. Reference [14] proposed the sequential quadratic programming method (SQP) to solve the optimisation problem by transforming the non-convex problem into a convex problem. Literature [15, 16] used the convex feasible set (CFS) algorithm to transform the optimisation problem into a series of convex subproblems. However, the convex optimisation method relies heavily on the initial path. Without good initial points, the convergence speed of the convex optimisation is slow and its solution maybe bad. This problem is more serious in the path planning of larger maps. The convex optimization algortihm work well when good initial points are provided, and A* algorithm is a simple method to obtain initial points with high quality [17, 18]. In practice, the information of the obstacles are often obtained through image, and the shape of the obstacles make the constraints complicated. Consequently, this study proposes a systematic approach to optimise the path of the robots, which consists of two stages. In the first stage, image processing is adopted to extract the shape and location of the obstacles. In the second stage, efficient alternating minimisation algorithm (AMM) is proposed. To solve the path planning problem, the complicated constraint due to the irregular shape of the obstacles is approximated by approximating the boundaries of the obstacles as line or arcs. Then the convex optimisation approach is adopted by approximating the non-convex constraints as convex constraints. Finally, AMM is proposed to solve the path planning problem efficiently. 2 PROBLEM SETUP Consider a 2 dimensional convex region of interest R with the four vertices named V1, V2, V3, V4, listed in counterclockwise order. The coordinates of the four vertices are [ x V 1 , y V 1 ] , [ x V 2 , y V 2 ] , [ x V 3 , y V 3 ] and [ x V 4 , y V 4 ] , respectively, and are known to the robot. There exists several solid colour obstacles named { O i } i = 1 M o in the region. The coordinates of the centroid of the obstacles { [ x O i , y O i ] } i = 1 M o are known to the robot, while the shapes of the obstacles are unknown. Let S i denote the region of the ith obstacle, which are unknown to the robots. Define rt ≜ [xt, yt] as the position of the robot at the tth instant. To ensure that the robot avoids all obstacles in the map, a necessary condition is that the minimum distance between the robot and the obstacles is greater than a tolerance dmin, that is, the squared distance f ( x t , y t ; x , y ) ≜ ( x t − x ) 2 + ( y t − y ) 2 between [xt, yt] and [ x , y ] ∈ S i should satisfy m i n [ x , y ] ∈ S i f ( x t , y t ; x , y ) ≥ d min 2 . (1) Given the reference path { r ∼ t } t = 1 T which can be obtained by the A* algorithm, the path of the robot { r t } t = 1 T should not deviate far away from the reference path { r ∼ t } t = 1 T . In order to ensure the following accuracy of the robot, the path should be smooth, and the length of each step should be as small as possible. Considering the above two conditions, we construct a path planning objective function L(r) as L ( r ) = 1 2 ‖ r − r ∼ ‖ 2 2 + λ 2 ∑ t = 2 T ‖ r t − r t − 1 ‖ 2 2 , (2)where r = [r1, r2, …, rT] and r ∼ = [ r ∼ 1 , r ∼ 2 , … , r ∼ T ] , λ is a regularisation parameter to control the tradeoff between the similarity of the path given by A* and the length of the optimised path. In addition, the mobile robot needs to move from a given starting point rstart ≜ [xstart, ystart] to the given end point rgoal ≜ [xgoal, ygoal], and avoid all obstacles in the map. Therefore, the path planning optimisation problem can be formulated as follows: m i n i m i z e r L ( r ) , (3a) s u b j e c t t o f ( r t , s ) ≥ d min 2 , ∀ s ∈ S i , ∀ t (3b) r 1 = r start , (3c) r T = r goal . (3d) To achieve the path planning, the task is decomposed as the four key steps: Coordinate transformation, obstacles identification, obstacle constraints formulation and algorithm design, see Figure 1. In the following, these details are presented. FIGURE 1Open in figure viewerPowerPoint Steps of path planning and optimisation 3 OBSTACLE BOUNDARY IDENTIFICATION For path planning, the information of obstacles is required. However, the obstacles captured by the camera will deform into twisted shapes. Therefore, this section starts with image conversion which transforms the area in the image into a real scene. Then, apply Laplace operator to identify the boundary of obstacles. 3.1 Image conversion In order to realise the image conversion, the first step is to establish the relationship of the coordinates of the point between the image and the real scene. Then, the coordinates of the point in the image can be mapped to the corresponding point in the real scene. Figure 2 shows the relationship between the point in the real scene plane and its corresponding point in the viewing plane. Let P I = [ u P , v P ] be the coordinates of a point P in the viewing plane I, which corresponds to the point P R = [ x P , y P ] . According to perspective transformation method [19], the coordinate relationship between a pair of corresponding points is x P = a 1 u P + a 2 v P + a 3 c 1 u P + c 2 v P + 1 , (4) y P = b 1 u P + b 2 v P + b 3 c 1 u P + c 2 v P + 1 , (5)where { a i } i = 1 3 , { b i } i = 1 3 and { c i } i = 1 2 are perspective parameters. Note that the four points { [ x k , y k ] } k = 1 4 and { [ u k , v k ] } k = 1 4 are one to one mapping. According to (4), the perspective parameters can be obtained through solving the 8 linear equations. Then, the point P R = [ x P , y P ] in the real scene can be obtained through x P ′ y P ′ z = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 1 u P v P 1 . (6)where z = c 1 u P + c 2 v P + 1 and x P = x P ′ z , y P = y P ′ z . Consequently, the whole real scene can be constructed through the image information. To achieve the high resolution real scene, the bilinear interpolation algorithm can be applied. FIGURE 2Open in figure viewerPowerPoint Perspective transformation of 2D image. The interested region R in the real scene plane is mapped to the viewing plane, and transformed into region I 3.2 Boundary extraction Boundary extraction is a common technique in image processing. For solid colour obstacles, their internal colour often changes smoothly, and there will be a colour jump at the boundary, and this allows us to use second-order difference operator for identification. Laplace operator is a common second order differential operator in image processing, it is sensitive to colour mutation and less sensitive to linear colour change caused by the change of illumination. Furthermore, Laplace operator is an isotropic operator, that is, it can identify colour mutation in all directions. Consequently, Laplace operator is used to detect colour jumps and determine the boundary of obstacles. In a matrix with the size of 3 × 3, the Laplace operator can be expressed as 0 1 0 1 − 4 1 0 1 0 . (7) The process of boundary extraction is shown schematically in Figure 3, and the specific steps are as follows: Convolve each pixel in the graph R with the Laplace operator, and receive a new graph R C . Apply binary algorithm to graph R C , the boundary points are set to 1, other points are set to 0, and receive a new graph R B . Starting from the centroid of each obstacle, find the boundary of obstacles, receive image R O , which is the map of boundaries of all the obstacles. FIGURE 3Open in figure viewerPowerPoint Steps of boundary extraction In step 3, the connected block algorithm can traverse and distinguish the boundaries of different obstacles in image R O . The steps of breadth first connected the block algorithm are as follows: For each obstacle, set the centroid of the i-th obstacles [ x O i , y O i ] as the beginning point. Add this point to open list, here open list stores the points to be visited. Visit the first point in open list. Search all adjacent pixels around this pixel, add all unvisited adjacent pixel with value 0 to open list. Clear current point from open list, repeat step 2–4 until open list is empty. Then mark all the visited points as i-th obstacle, and receive the boundary of the obstacle. 4 ALGORITHM DESIGN The previous section extracts the boundaries { S i } i = 1 M o of the obstacles via image processing methods, and the obstacle constraints Equation (3b) are obtained. In this section, an efficient algorithm is proposed to try to solve the path planning problem Equation (3). Note that the objective function of Equation (3) is quadratic, which is simple. The constraint (3b) is complicated, as for a given i, it is equivalent to the following optimisation problem m i n i m i z e s ∈ S i f ( r t , s ) ≥ d min 2 . (8) Such a constraint indeed consists of infinite constraints and is difficult to optimise. To simplify the above constraint the boundaries { S i } i = 1 M o of the obstacles are approximated as arcs or line segments. As shown later, the constraint Equation (3b) is approximated as a single constraint, which simplifies the constraint significantly. An efficient iterative algorithm is proposed to solve Equation (3). For the current iteration point, the ensuring non-convex constraints are approximated as convex constraints, and convex programming is implemented to obtain the global solution of the resulting problem. Note that the optimisation problem has a block structure, and block coordinate is adopted. As a result, we embed the block coordinate to solve the optimisation problem iteratively. In the following, we detail both the boundaries of the obstacles approximation and the algorithm procedures. 4.1 Obstacle constraint simplification This subsection consists of three parts: First, divide the boundary into several structures, meanwhile decide whether the structures are approximated as arcs or line segments. Second, propose a method to estimate the parameters of the arc or line segment. Thirdly, simplify the constraint via eliminating the minimising operator, that is, finding the optimal point of Equation (8). 4.1.1 Obstacle approximation The key to divide the boundary into these two structures is to locate connection points of the structures. To find these connection points, we use the method of decomposition and reorganisation. Firstly, divide the boundary into a large number of short segments. For the i-th obstacle, take nP points evenly along the obstacle boundary, marked as p i ( j ) j = 1 n P , here nP is a constant determined by the length of boundary. Start from pi(1), connect each point and its next point, form nP short segments, and mark them as q i ( j ) j = 1 n P . Then, through angle relationship, we can determine the location of connection points, and the shape of structures. If several adjacent segments form a straight line, then the angles between adjacent short segments are 0, and the derivatives of angle sequence are 0; If several adjacent segments form an arc, then the angles between adjacent short segments are constant, and the derivatives of angle sequence are 0. Therefore, the next step is to reformulate the short segments into structures. Starting from the first short segment qi(1), calculate the angle between each short segment and its next short segments, marked as d i ( j ) j = 1 n P . The derivative of d i ( j ) j = 1 n P is marked as h i ( j ) j = 1 n P . Check the value of each hi(j). If all the absolute values of hi(j) are greater than γ which is a constant, it is regarded as the connection point of structures. Next, identify the structures. For each structure, calculate the sum of all di(j). If this value is greater than a certain constant, then this structure is regarded as an arc of circles, otherwise it is regarded as a line segment. 4.1.2 Parameter estimation The next step is to estimate the parameters of the approximated lines or arcs. Suppose the endpoints of a certain structure are point qi(A) and qi(B). For a line segment, it is easy to calculate the analytic formula through the coordinates of point qi(A) and qi(B). For an arc, firstly take the midpoint of the structure, take integer D closest to ( A + B ) 2 , suppose the coordinates of centre is [xR, yR], and coordinates of point qi(A), qi(B) and qi(D) are [xA, yA], [xB, yB] and [xD, yD], respectively. Solve the following equation ( x A − x R ) 2 + ( y A − y R ) 2 = R 2 , (9) ( x B − x R ) 2 + ( y B − y R ) 2 = R 2 , (10) ( x D − x R ) 2 + ( y D − y R ) 2 = R 2 (11)to obtain the centre [xR, yR] and the radius of the arc R. 4.1.3 Constraint simplification Let the origin of the coordinate system be point O. For the line segments, suppose the coordinates of the endpoints of a segment are A = [xA, yA], B = [xB, yB], then the coordinates of any point P = [xP, yP] in the segment can be expressed as P = λA + (1 − λ)B, where 0 ≤ λ ≤ 1. Let point R be the projection point of the interested point Q = [xQ, yQ] on straight line AB, and point Q can be expressed as: Q = θ O A + ( 1 − θ ) O B + R Q , (12)where, θ depends on the position of point R, and it can be any value. The minimum distance between point Q and a point on the segment can be expressed as min 0 ≤ λ ≤ 1 f ( λ ) = | P Q | = | ( θ − λ ) ( O A − O B ) + R Q | = ( θ − λ ) 2 | B A | 2 + | R Q | 2 (13) When |θ − λ| is at minimum value, function f(λ) also reaches the minimum value. According to the different value of θ, the plane is divided into 3 parts according to λ, as shown in Figure 4. When 0 < θ < 1, point Q is located in the blue area of case (a), and fmin(λ) = f(θ) = |RQ|, which is the distance between point Q and straight line AB. When θ > 1, point Q is located in the blue area of case (b), and fmin(λ) = f(1), which is the distance between point Q and A. When θ < 0, point Q is located in the blue area of case (c), and fmin(λ) = f(0), which is the distance between point Q and B. In this way, the infinite constraint is transformed into a single constraint, the constraint can be written as ( x − x P ) 2 + ( y − y P ) 2 ≥ d min 2 , (14) FIGURE 4Open in figure viewerPowerPoint Three cases of the position of point P. The blue area shows the feasible position of point Q when (a) 0 < θ < 1, (b) θ > 1, (c) θ < 0 Let the coordinate of the centre and the radius of the circle arc be R = [xR, yR] and rc, respectively. The coordinate of the endpoints of circle arc are A = [xR + rc cos a, yR + rc sin a], B = [xR + rc cos b, yR + rc sin b], where a and b are the angles of the vector RA and RB. Then the coordinates of any point P in the arc AB can be expressed as P = [xR + rc cos p, yR + rc sin p], p = λa + (1 − λ)b, where 0 < λ < 1. The interested point Q in the plane can be expressed as Q = [xR + r2 cos q, yR + r2 cos q], where θ depends on the position of point R, and it can take any value. The distance between point Q and a point on the circle arc can be expressed as f ( λ ) = | P Q | = r 1 2 + r 2 2 − 2 r 1 r 2 cos ( ( λ − θ ) ( a − b ) ) (15) As shown in Figure 5. When |θ− λ| is at minimum value, function f(λ) also reaches the minimum value. When 0 < θ < 1, point Q is located in the blue area of case (a), and f min ( λ ) = f ( θ ) = r 1 2 + r 2 2 − 2 r 1 r 2 = | r 2 − r 1 |. When θ > 1 or θ < 0 (case b), the minimum value is fmin(λ) = f(0). When θ > 1 (case c), the minimum value is fmin(λ) = f(1). In this way, the infinite constraint is transformed into a single constraint ( x Q − x P ) 2 + ( y Q − y P ) 2 ≥ d min . (16) FIGURE 5Open in figure viewerPowerPoint Three cases of position of point P. The blue area shows the feasible position of point Q when (a) 0 < θ < 1, (b) θ > 1, (c) θ < 0 4.1.4 Convex approximation of constraints The obstacle constraints Equation (3b) are simplified as Equations (14) or (16) according to the shape of the boundary of the obstacles. However, this is still a non-convex constraint and the global solution is not guaranteed to be found. In the following, the non-convex constraints are further approximated as convex constraints. As a result, the iterative algorithm is guaranteed to find the global solution of the approximated problem. Below we present the details. Note that the feasible point sets of both Equations (14) and (16) are outside the circle with radius dmin, see the shaded region in Figure 6. Since we will apply an iterative algorithm to solve the problem, the constraint set is iteratively approximated as a convex set. Let Q = [xQ, yQ] be the iteration point, as shown in Figure 6. We apply Taylor-series expansion at point R to obtain x Q ( 2 x R − 2 x P ) + y Q ( 2 y R − 2 y P ) ≥ d min 2 − ( x R 2 + y R 2 ) + ( x P 2 + y P 2 ) (17)which is a halfspace, as shown in Figure 6b. Consequently, the constraint formed by the t-th point on the path rt and the i-th structure of the obstacle is denoted as11 Note that (17) is a linear inequality constraint and is abstracted as (18). A i , t r t ≥ b i , t (18) FIGURE 6Open in figure viewerPowerPoint Feasible solution area before (a) and after (b) applying first-order Taylor approximate. The hyperplane determining feasible solution area is the tangent of the circle at point R 4.2 Alternating minimisation The original problem is iteratively approximated as a quadratic programming (QP) problem with linear inequalities. Also, the objective function and constraint sets are linearly separable with respect to the variables { r t } t = 1 T . Therefore, in each iteration, AMM is adopted. For the k th iteration, let r ( k ) = [ r 1 ( k ) , r 2 ( k ) , … , r T ( k ) ] be the optimised result. We apply Taylor-series expansion base on the location of point r(k) to obtain linear constraints. Then for i = 1, …, T, we compute r t ( k + 1 ) = argmin r t L ( r 1 ( k + 1 ) , … , r ( t − 1 ) ( k + 1 ) , r t , r t + 1 ( k ) , … , r T ( k ) ) , s u b j e c t t o A i , t ( k − 1 ) r t ≥ b i , t ( k − 1 ) . Once we obtain r(k+1), we go on approximating the constraints of the original problem and use AMM. When the difference between the current iteration and the previous iteration of the objective function is less than a tolerance, we stop. 5 SIMULATION AND EXPERIMENT 5.1 Scene construction and simulation results In order to test the robustness of our algorithm, we constructed six binary maps. The simulation was run in Matlab on a Dell Notebook computer of 2.2 GHz using Intel Core i7. The size of each image is 800 × 600 pixels. The environments of the first three maps are simple, while the obstacles of the last three maps are relatively complicated. For each map, the path starts from the upper left corner and reaches the lower right corner through the obstacle area in the map. The safety distance between path and obstacles is set as 20 pixels, and the information about obstacles is unknown. The results of path planning in the first three maps and the last three maps are shown in Figures 7 and 8, respectively. The blue and red solid lines denote the referenced path (obtained by A* algorithm) and the optimised algorithm (obtained by the alternating minimisation approach). It can be seen that all the optimised paths are smoother than the paths before optimisation, and the proposed method to approximate the boundary of the obstacles is accurate. FIGURE 7Open in figure viewerPowerPoint Three simple maps for simulation. The pictures in the left column show the binary maps and path planning results, and the pictures in the right column show the obstacle approximation result and path planning result FIGURE 8Open in figure viewerPowerPoint Three complex maps for simulation. Same as Figure 7 the pictures in the left column show the binary maps and path planning results, and the pictures in the right column show the obstacle approximation result and path planning result The quality of the optimised path is also evaluated and is compared with the interior point method (IPM). Results are shown in Tables 1 and 2 that after optimisation, the total length of the path has been shortened, with an average reduction of 7.45%. The maximum corner decreases significantly and the path becomes smoother. It is shown that the path from two optimisation methods have almost the same quality, but AMM is effective in terms of path quality and running time. TABLE 1. The length of the optimised path for the six maps in Figure 7 Path length/Pixel Reduced length Runtime/Sec A* IPM AMM IPM AMM IPM AMM map1 1129.1 1076.9 1077.5 4.62% 4.57% 0.663 0.060 map2 1094.0 1040.0 1040.6 4.94% 4.88% 1.127 0.348 map3 1152.6 1073.3 1073.8 6.88% 6.84% 1.738 0.534 map4 2343.7 2179.0 2179.6 7.03% 7.00% 4.391 1.285 map5 1465.7 1330.6 1331.3 9.22% 9.17% 3.229 1.016 map6 1364.3 1250.5 1251.0 8.34% 8.30% 0.791 0.250 Abbreviations: AMM, alternating minimisation algorithm; IPM, interior point method. TABLE 2. The angles of the optimised path for each of the six maps in Figure 8 Average angle/degree Maximum angle A* IPM AMM A* IPM AMM map1 19.15 4.657 4.657 45 12.35 12.35 map2 10.71 4.053 4.052 45 13.25 13.25 map3 10.00 4.056 4.056 45 16.45 16.45 map4 7.059 5.876 5.875 45 27.32 27.49 map5 14.75 6.582 6.582 45 21.34 21.34 map6 12.41 6.339 6.338 45 17.95 17.95 Abbreviations: AMM, alternating minimisation algorithm; IPM, interior point method. 5.2 Experiment On this basis, we designed a set of experiments, all the steps proposed in this study are carried out completely. Figure 9 shows the field of the environment, the size of the field is 2.0 m × 2.0 m, and the red lines show the boundary of the region. There are three obstacles in the field. The mobile robot moves from the bottom left area to the top left area, and keeps a distance of more than 20 cm from obstacles. FIGURE 9Open in figure viewerPowerPoint Experiment environment. The mobile robot is supposed to move from left bottom to left top The image is shown in Figure 10a. We transform the region to a map with 2000 × 2000 pixels. Then we apply boundary extraction algorithm and obstacle approximation algorithm to the map, the result is shown in Figure 10b. Then we apply path planning algorithm to obtain the reference path and the optimised path, which are shown as Figure 11. The blue line and the red line denote the reference path, and the red line refers to the optimised path. It can be seen that the optimised path is smoother than A* path, and has 14.3% less length than A* path. Meanwhile, the optimised path can effectively keep a safe distance from obstacles. FIGURE 10Open in figure viewerPowerPoint The experimental field (a), and the obstacle approximation results (b) FIGURE 11Open in figure viewerPowerPoint Trajectory of mobile robot in A* path (a) and optimised path (b) 6 CONCLUSION This study solves the path planning problem from image information via efficient alternating minimisation methods. To improve the path quality of the alternating minimisation algorithm, the solution obtained by A* algorithm is used as the initial point. To make the path planning problem tractable, novel approximations which approximate the boundaries of the obstacles as lines and arcs are proposed. In addition, the structure of the problem is further utilised to design efficient alternating minimisation algorithms. Both numerical simulations and real experiments are conducted to further validate the effectiveness of the proposed approach. ACKNOWLEDGEMENTS This work was supported in part by the National Science Foundation of China under Grant 92048302. CONFLICT OF INTEREST We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled. Open Research DATA AVAILABILITY STATEMENT All data, models, or code generated or used during the study are available from the corresponding author by request. Supporting Information Filename Description csy212030-sup-0001-suppl-data.docx16.3 KB Supplementary Material 1 Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article. 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Sens Actuators, A. 272, 301– 309 (2018) CrossrefCASWeb of Science®Google Scholar 1 Note that (17) is a linear inequality constraint and is abstracted as (18). Volume3, Issue3September 2021Pages 245-255 FiguresReferencesRelatedInformation