Formation Tracking and Transformation Control of Nonholonomic AUVs Based on Improved SOM Method LI Xin, ZHU Daqi, CHEN Yangyang, LIU Qingqin Lab of Underwater Vehicles and Intelligent Systems, Shanghai Maritime University, Shanghai 201306, P. R. China E-mail:
[email protected]
Abstract: Formation tracking and transformation are the key problems in formation control of multi-AUV (autonomous underwater vehicle) system. In this paper, an improved self-organizing map (SOM) neural network method is proposed for solving the formation issues of a group of autonomous underwater vehicles (AUVs). All the AUVs in the formation are treated equal to be the leaders or the followers. The desired locations are set as input vectors of SOM neural network. Self-organizing competitive calculations are carried out with workload balance taken into account. Output vectors of the SOM network are the corresponding AUVs’ coordinates, so that a group of AUVs are controlled to reach the designated locations. This method hold the followers’ positions in the formation when the formation moves as a whole along pre-planned trajectories. Moreover, the formation could change its shape as needed in the procedure. Formation transformations are efficient and reasonable using this strategy. Finally, due to the characteristics of SOM neural network, adaption and fault tolerance can be achieved. Simulation results demonstrate the effectiveness of the proposed approach. Key Words: Multi-AUV, SOM, Formation Control, Formation Tracking, Formation Transformation
1 INTRODUCTION With the increasing activities of the human in the ocean, it is necessary to carry out underwater missions which are beyond the capacity of a single AUV (autonomous underwater vehicle). In recent years, multi-AUV cooperation control has become a hot issue in the relative research filed [1, 2]. Formation control is a fundamental problem in multiAUV cooperation control [3]. Various formation structures have been proposed for formation control, including leaderfollower, artificial potential field, virtual structure, etc. Leader-follower approach [4] divides the formation into several groups and each group has two AUVs, one is the leader and the other is the follower. The follower AUV maintains a certain angle and distance with the leader AUV to achieve formation control. Based on the relative position between the leader AUV and follower AUV, the formation would be different. When the environment changes or encounters the obstacles, the formation could transforms to another formation style to come through the obstacles’ area. A follower in one group could be the leader in another group. The reference trajectory of the follower is generated as the leader cruises. A virtual follower is designed to track the reference trajectory. The position tracking control is formulated for the follower to track the virtual vehicle’s trajectory. The leader-follower approach is widely used to supply a framework for robots’ formation control; however, the leader-follower scheme encounters a critical issue in actual applications: when a leader AUV breaks down, its followers also fail to continue moving. If the broken AUV happens to be the formation’s leader, the whole team will fail, which may cause disasters [6–9]. K. Yoshida et al. [10] proposed a This project is supported by the National Natural Science Foundation of China (51279098, 51575336), and Creative Activity Plan for Science and Technology Commission of Shanghai (14JC1402800).
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decentralized structure for formation control, but this method is not for totally distributed formation. Literatures [11] also propose distributed formation methods. In [11], large swarms are achieved based on bathymetric maps, but the inner positions of vehicles are not explicit and the formation shape can not change freely neither. In this paper, an improved SOM method is proposed for the distributed formation control of multi-AUV system. The contributions of the proposed method can be summarized as follows. 1) The SOM-based approach is firstly applied to formation control. Although the proposed approach is based on leader-follower framework, the AUVs in the formation are randomly distributed and each AUV is treated equal to be a leader or a follower. Whether an AUV becomes a leader or a follower in the formation only depends on the algorithm itself. 2) Adaption and fault tolerance are important characteristics of the proposed method. This method deals with the formation transformation in a more efficient way and handles the situation that some AUVs break down in the formation tracking process. This method overcomes the shortcoming of the traditional leader-follower scheme, i.e., in the leader-follower formation, when a leader AUV breaks down, its followers also fail to continue moving. The rest of the paper is organized as follows. The problem of AUV underwater formation control is explained in Section II. Section III presents the SOM-based algorithm for formation tracking and transformation control. Simulation results with discussions are provided in Section V. The concluding remarks are given in Section VI.
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2 PROBLEM FORMULATION
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Figure 1: Typical geometric formations: (a)Line (b)Triangle (c)Diamond (d)Wedge (e)Polygon
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Formation control is the foundation of coordination and cooperation of a group of AUVs in performing a task. Formation tracking and transformation has become a main challenge in multi-AUV system’s formation working. Kalantar and Zimmer proposed a method considering the formation control into two decoupled problems: the boundary and the interior [11], but this method is only suitable for large scale swarms. It is also too complicated to use when the formation includes a small amount of AUVs, for example, the number of AUVs is less than 10. The positions of AUVs in the interior layer are not fixed, either. In this paper, assuming a group of AUVs working in 2-D Cartesian space, these AUVs are distributed in a bounded area and they are expected to form a corresponding formation style. The leader and the followers are randomly chosen and each AUV has a position in the formation. After the formation is formed, the positions of AUVs are relatively fixed. The whole formation moves without distortion or deformation as the task needs. The workspace is assumed to be an ideal bounded area without any obstacles or ocean currents influence. As mentioned above, the formation could be any shape as needed, such as triangle, diamond or polygon. Some regular formation type is shown in Fig. 1. We assume any formation shape comprise of points which are corresponding to the AUVs. What the SOM approach deals with is to assure that every point has an AUV to reach. What’s more, the AUVs occupy all the target points within a minimal total expense and minimum energy consuming. Here, for each AUV, the expense is evaluated by its total traveling distance. A typical workspace with the AUVs and the desired formation is shown in Fig. 2, where the 3 red diamonds represent randomly distributed AUVs and the black edged diamonds are the points that form the triangular formation. In addition, we assume that the AUVs are homogeneous and have basic capabilities for navigation so that the kinematic constraints are free of considering here in this paper. The AUVs are expected to reach the positions (where the black edged diamonds represent) to form a triangular formation. The formation then moves as a whole along desired trajectories.
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Figure 2: Triangular formation and 3 randomly distributed AUVs in 2-D Cartesian workspace
3 MAIN ALGORITHM BASED ON IMPROVED SOM The SOM neural network method was first introduced by T. Kohonen in the 1980s and extended later [12–14]. It is based on the idea that there is a special order of processing units in the mammalian brain. Each unit is dedicated to a specific task and each group of neurons is sensitive to a particular type of input signals. The units are determined by parameters that can be changed in certain processes to produce meaningful organizations. This algorithm soon becomes a valuable tool and used to solve many kinds of problems. For example, it solves task assignment problem of multi-robot system in [15– 17], whereas the workload functions are not perfect. The SOM approach could deal with the formation control problem. The idea comes from the similar characteristics and phenomena between a multi-AUV system and a SOM network. First, a multi-AUV system could be considered to be a self-organizing system that changes its basic structure. Second, the SOM algorithm has the competitive, cooperative and self-organizing characteristics that are attractive for a multi-AUV system. Thus, the SOM network could be adapted to an unordered team of AUVs to automatically achieve the designated locations with cooperation and competition. The designated locations are the corresponding points in the expected formation. The AUV formation could be then achieved in this way. In this paper, an improved workload function is proposed, with the limited traveling length of a single vehicle taken into consideration. In order to further explain how the SOM method applied on formation control of multiple AUV system, mathematical models and formulas with discussions are given in this section. 3.1 Winner Selection Rule At the initial state, J AUVs are randomly distributed in the workspace. L points are set at the key positions of the expected formation shape. For instance, in Fig. 2, there are 3 points to form a triangle, so that L = 3. The SOM model has two layers of neurons. The first layer is the input layer including L neurons, which represent the Cartesian coordinates of the points in the 2-D workspace. T is an 2 × L input matrix to the first layer. T denotes the coordinates of the target points as input neurons. Each input neuron is a vector value (xl , yl ) according to the coordinates of a target. The
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second layer is the output layer. Neurons in the output layer represent the coordinates of the AUVs and their corresponding waypoints. There are K neurons as the waypoints in one group to establish an path for a single AUV, and J AUVs are used, so that there are K × J neurons in all. The connection strength of every output neurons, i.e. the weight Wjk of that neuron, is given by a 2-D vector, which is initialized as the coordinates of the initial AUV position. The network is initialized with the weight vector Rjk = (wjkx , wjky ), where j = 1, 2, ..., J and k = 1, 2, ..., K. Rjk changes with Wjk during every computation iteration. After the iterations, we get a 2 × J × K matrix R which denotes the trajectories of the AUVs in formation. The points in the formation would attract output neurons to form a formation for the AUVs. The iterations end until all of the AUVs reach the points. The first step in the SOM approach is to select the winner node. For an input neuron (the target point in formation), the output neurons compete to become the winner node. Let [Nj ] represent the jth neuron from the kth group and it is the selected winner to the lth input node, where k ∈ [1, K], j ∈ [1, J], and l ∈ [1, L]. Dkjl is a weight value at an instant related to Euclid distance between the correlated two neurons. S is the safety length a single AUV can travel without considering running out of energy and Smax is the maximal distance that a single AUV can travel. The weight variable Dkjl is given by ⎧ 0 ≤ Pj < S ⎨ |Tl − Rjk | , |Tl − Rjk | (1 + V ) , S ≤ Pj < Smax . (1) Dkjl = ⎩ ∞, Pj ≥ Smax |Tl − Rjk | gives the Euclid distance between Tl and Rjk . Here, Tl = (xl , yl ) denotes the location of lth input neural node; Rjk = (wjkx , wjky ) is the kth neuron from the jth group of the output neurons, which represents the location of a certain AUV at a certain instant. The work load balance control variable V is introduced in (1), which guarantees that each AUV is capable to move in the task without running out energy. The benefit of work load balance control is that the energy consumption of each AUV could be equalized, which could makes the maximum use of the AUV team as a whole formation. Workload balance control is mainly determined by the AUVs’ actual moving length and the safety length preset. Let Pj be the actual moving length of the jth AUV in the certain process of formation and v be the average moving length of the team of AUV in a certain task. Then the work load balance control variable V is calculated by Pj − v V = . (2) S+v Under this workload balance strategy, the proposed method could achieve a good balance between the shortest total traveling distance and equity of the group of AUV under the premise that the energy is sufficient. That is an improvement to conventional SOM neural network method. After the winner neuron is chosen, the next step is to design the neighborhood function to decide the neighbors’ weights. The neighborhood should be a circle with radius γ. The center of the neighborhood circle is the winner node. The neighborhood function determines the influence of the input target on the winner neuron and its neighbor nodes. The
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winner was put on the highest attractive force which is diminishing as the distances of the neighbor neurons increase to the winner. This function should have no effect on the neurons outside the neighborhood. The neighborhood function f (·) is described as 2 2 e−dm /G (t) , dm < γ f (dm , G) = (3) 0, others where dm is the distance between the mth output neuron Nm and the winner Nj from the corresponding group of the output neurons. G(t) = (1 − μ)t Go is a nonlinear function, where t is the number of iterations; μ and Go are constants that affect the computation time and computational accuracy through adjusting step length. 3.2 Neural Weights Updating Rule After the winner neuron and its neighbors are selected, the next step is to move the winner neuron and its neighbors toward the input neuron (the target), whereas the other neurons stay still. The update rule is defined as ⎧ Dkjl < Dmin ⎨ Tl , Rjk + α · f (dj , G) Rj(k+1) = . (4) ⎩ · (Tl − Rjk ) , others Dmin is the termination condition of operation, which could reduce computation time. The weight value’s revision is not only decided by the position of the winner neuron and the neighboring neurons, but also decided by the neighborhood function and the network learning speed α. Considering l = j in the SOM algorithm and once the lth target is achieved by an AUV, this target should not be visited again by any other AUVs. In this case, α is set to zero so that the selected AUV and its neighbors will not move at all. Using this scheme, there is no need to delete the achieved target from the matrix, and the iteration is bounded to end after an acceptable times. In other cases, when some targets should be achieved by a relatively faster or slower speed, using the parameter α is an adaptive method to implement the solution. 3.3 Formation Tracking and Transformation Scheme After all the AUVs reach their positions in the formation, the formation moves along an arbitrary trajectory and the AUVs in the formation also move to keep their positions. This procedure is named as formation tracking. In order to implement the formation tracking based on the SOM approach, we assume a virtual formation which is totally the same as the actual one. The virtual formation has the same key points as the real formation. As we implement the formation tracking, the geometrical shape of the formation could be changed. The we adjust the virtual formation as needed and use the SOM based method to track the new formation shape. Let Tl denotes the lth AUV’s location in the virtual formation, we have the moving rule as bellow:
Tl (n) = Tl (n − 1) + ΔT(n), n = 1, 2, ..., Nstep . (5) In (5), Nstep is the total moving steps of the formation. ΔT(n) is an arbitrary 2-D vector satisfying 0 < |ΔT(n)| ≤ λ. The formation could track an arbitrary path with the single step length no more than λ, which is a reasonable constant
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triangular formation. After the formation is completed, the whole formation moves to a specified target or changes its shape to avoid obstacles. For example, as shown in Fig. 3b, the virtual formation moves n steps where the step length is limited. The procedure in Fig. 4(a) is implemented again after every moving step. The moving formation is kept in this way until the whole formation reaches the destination.
4 SIMULATION RESULTS AND DISCUSSIONS
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Figure 3: Working process of the proposed approach, (a) the SOM computational process in 2-D workspace (b) virtual formation moves n steps along an arbitrary trajectory. as the maximal single step length of the formation. After the virtual formation moves or transforms, Tl is updated. The virtual points in the formation is set as the input vector of the SOM network by Tl ⇐ Tl (n), where n = 1, 2, ..., Nstep After the value of Tl is updated, the SOM algorithm is executed again. The AUVs move to the new positions and the actual formation also changes as the virtual formation changes. 3.4 The Whole Procedure The detailed working process of the proposed approach is shown in Fig. 3. The red diamonds represent the 3 AUVs and the black edged diamonds represent the point positions as input neurons. AUV1 is the winner as it is the nearest one to the first input, which is the top point P1 of the triangular formation; according to the rule in (3), the neighbors are selected, where AUV2 is the only neighbor of the winner, as shown in Fig. 3a. Both AUV1 and AUV2 move toward P1 by changing the weight vectors, whereas the other neurons do not move. It is easy to find that the winner moves a greater distance than its neighbor. The closer the neuron is to the winner, the greater distance the neuron will move. That’s exactly the rule in (4) and (5). The loop repeats until all of the points are achieved by an AUV. All the AUVs move to its corresponding position in the formation spontaneously. The procedure is done through iterations introduced above to calculate all the paths to form a
This section is to demonstrate the performance and effectiveness of the algorithm described in this paper. Simulations are carried out with a model of the 2-D workspace with several AUVs randomly distributed. Based on the approach proposed before, programs in MATLAB codes are implemented. Parameters are modified during our simulation to get the best effects. 4.1 Formation Tracking of 6-AUV Team Firstly, the proposed method is applied to a case where the workspace has 6 AUVs to form a triangular shape and move along an arbitrary path. In this case, the learning rate α = 0.2; radius of neighborhood γ = 3; the workspace is a bounded square where (x, y) ∈ [0, 50m]. Parameter Smax depends on the characteristics of AUV. In this simulation, Smax could be set to the length of diagonal of the square workspace. The average moving length v is set to zero at initial state, because no AUV moves at the beginning. The value of v increases as the number of iterations increases. All the AUVs’ moving lengths are recorded in the corresponding registers. In the path tracking process, the formation shape is kept of 6-AUV team after the formation is set up. As mentioned above, when the formation is moving, the steps are changeable so that the trajectory is arbitrary. To approve the proposed approach, the formation is expected to move along a random trajectory as illustrated in Fig. 4a. Let vst be the leader’s step, or the instantaneous velocity. The norm and direction of vector vst are not fixed. Without loss of generality, we assume 0 ≤ |vst| = |ΔT(n)| ≤ λ, and 0 ≤ ∠vst < π. To track this path for the AUV team in formation, we choose Dmin = 0.1. Other parameters are not modified. Note that as the learning rate parameter α increases, the SOM network performs faster. We give α a relative bigger value when the formation’s moving step is larger, so that the convergence could be guaranteed and the iteration times would be in an acceptable number range, which means the neural network performs fast enough. Since the neural network performs fast enough and the computational consumption is limited, the proposed algorithm keeps its practical significance. Fig. 4b shows the formation tracking result of the 6-AUV team. Some intermediate states are not shown in the figure. The formation is kept as the desired trajectory is tracked at a fast enough speed. 4.2 Formation Transformation The proposed approach also works well for the formation transformation, in which situation the formation shape is changed. For instance, when the AUV team has to go through a relative narrow place, they can’t keep the triangular formation. A linear shape is better in this situation.
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before which transforms from triangle into line. After the formation changes from triangle into line, the AUVs’ workloads are recorded, as shown in Table 3. With the traditional SOM approach, AUV4’s workload is obviously much heavier than the other ones while AUV3’s is a little bit lighter. This situation may cause breakdown of AUVs or other problems. The improved adaptive SOM method handles the job, where all the AUVs’ workloads are better balanced. We use standard deviation σ to evaluate the two methods. σ is much smaller in the simulation using the proposed method, which means the workload balance is guaranteed. Finally, fault tolerance can be achieved with the SOM based approach. In the traditional leader-follower formation, when a leader AUV breaks down, its followers also fail to continue moving. This method overcomes this shortcoming. For example, in a large team where the AUVs are redundant, when a leader AUV stops working, by simply removing the corresponding point in the formation shape, the rest AUVs continue moving until a substitutive AUV takes the broken ones position. As a result, it can be concluded that this method overcomes the shortcoming of the traditional leader-follower scheme in the situation when the leader AUV breaks down.
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Figure 4: Formation tracking of 6-AUV team along a straight line: (a) expected formation moving path, (b) AUVs’ tracking trajectory in formation.
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Simulations are implemented on the base of previous section. The triangular formation moves a certain distance and change to a linear formation. Base on the proposed method, all the AUVs went to the new positions in the formation during a competitive computation procedure. As the SOM strategy take effect, the total expense or the energy consumption for the transformation is minimal. Fig. 5 shows the simulation results. In Fig. 5b , the AUVs finally achieve a linear formation which is depicted by the blue line, just as planned in Fig. 5a before.
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Firstly, we compare the proposed method with traditional leader-follower approach on formation transformation in Table 1. The total computational costs are measured by number of loops and total travelling length. Simulation result shows the SOM based method has advantages due to its distributed and adaptive characteristics. It needs fewer loops (NS ) than the traditional approach (NT ) in the transformation procedure. The total travelling length (LS ) of the AUV team using the proposed method is also shorter than that using the traditional approach (LT ) in this experiment. Secondly, we compare the proposed method with traditional SOM approach without workload balance. In the comparison, the AUVs’ workloads are measured by the traveling distance in the 2-D workspace. Table 2 records the traveling length representing the workload for each AUV in the formation
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Figure 5: Formation transformation from triangular shape to linear shape: (a) desired formation transformation, (b) formation transforms as expected.
5 CONCLUSION In this paper, a SOM-based neural network approach is proposed for multi-AUV system’s formation tracking and
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Table 1: Comparison of Improved Som Method with the Traditional Approach for Formation Transformation
Improved SOM Method Traditional Leader-follower Approach Comparison
Number of Loops
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Table 2: Workload of Each AUV Before the Formation Transforms into Line AUV1 Workload 41m
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Table 3: Workload of Each AUV After the Formation Transforms into Line
AUV1 AUV2 AUV3 AUV4 AUV5 AUV6 σ
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Improved Adaptive SOM Method
50.7m 50m 49.2m 54.9m 50.1m 50.3m 2.036
51.2m 50.7m 51.5m 51m 50.1m 50.3m 0.537
transformation in 2-D Cartesian workspace. The calculation process involves special definition of the initial neural weights of the network, the rule to select the winner, the computation of the neighborhood function, workload balance law, the method to update the weights, and the formation tracking scheme. Based on the self-organizing map neural network method, formation keeping as well as formation tracking and transformation could be implemented. This adaptive approach could guide the AUVs to move to the expected destinations in the formation along optimized paths and form a desired shape. In addition, it can deal with complicated cases such as the formation moves along arbitrary path or the formation changes its geometrical shape. The total traveling length and balance of workloads of AUVs are rationally adjusted on the premise of sufficient energy. Simulation results validate the proposed method. Simulations in this paper assume that the 2-D workspace is ideal, but in real world there are many uncertainties such as moving obstacles and ocean currents, which would be studied in our future work.
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