This means that the follower only needs to keep an appropriate relative position and direction from the leader. This paper takes follower as origin to establish reference frame on follower[ 31 — 34 ] to show the relationship between leader and follower, as Fig 1 shows. In follower reference frame, x w , y w , z w refers the distance between leader and follower. V l , V w refers velocity of leader and follower.

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The motion of UAV is controlled by autopilot, its mathematical model[ 35 — 37 ] as Eq 1 :. According to geometrical relationship in Fig 1 , the coordinate in inertial frame of leader can display as follow:. The primary purpose of this algorithm is to let the robot move within an unknown environment in a way similar to how a beetle crawls around and uses its antennae to avoid obstacles.

Indeed, the basic idea consists of using a set of virtual antennae called tentacles that probe an ego-centred occupancy grid.

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According to Ref. Based on the information from a sensor, the environment around an intelligent vehicle can be described through a binary image as Fig 2A.

Each tentacle is a potential path and 16 sets of tentacles are used. Fig 2B shows the 81 tentacles in one speed set. The support area of a tentacle in the occupancy grid is used to determine whether a tentacle is drivable; its geometric definition is illustrated in Fig 2C. The Rule A of obstacle detection is as follows:. Divide a tentacle to n sections as Fig 2D , and each point c i in support area corresponds to a position k i ;.

Use an array v [ n ] to count the number of the black points in each section k i ;. Use a sliding window to determine the position of the first obstacle. If the sum of binary values within this window exceeds a threshold n o , an obstacle is detected and the position of the sliding window yields the distance l o to the first obstacle. Set the crash distance l c , which depends on the speed v , a deceleration a and a security distance l s :. In the end, one best tentacle is selected as the expected trajectory with the three functions: v clearance , v flatness , v trajectory.

The v clearance depends on the distance to the first obstacle l o , the v flatness has the goal to prefer tentacles leading over smooth terrain, and the v trajectory pushes the vehicle towards following a given trajectory. The three function can be combined with different weight:. The Rule B of tentacle selection is as follows:. The flow chart of the basic tentacle algorithm is as Fig 3. Problem 1: It is noticed that all the parameters are related to wheel force and steering angle. This means that once a group of wheel forces and steering angles is certain, the trajectory of the vehicle may be fixed.

Therefore, a group of wheel forces and steering angles refer to each tentacle. So the vehicle can be controlled to follow the selected tentacle with the two parameters. This method pre-computes all the corresponding groups of wheel forces and steering angles through creating a steady state as shown in Fig 4 according to each tentacle by curvature matching, and then stores them so that once a tentacle is selected, the corresponding wheel force and steering angle can be acquired at the same time.

Although this method is correct in theory, some problems may cause the tentacle algorithm invalid because when the radius of a tentacle is too large, the matching between this tentacle and the corresponding steady state costs plenty of time and still worst a steady state cannot be achieved. Problem 2: The basic tentacle algorithm in Ref. The velocity of the UAV is much greater than the autonomous robot. So the radius of each tentacle should be greater to match the velocity. Meanwhile, this change results that most tentacles gather in a small area, so that many neighbouring tentacles will provide the same curvature.

Thus, the number of tentacles in one speed set should be reduced so as to enhance the ability of real time computation for UAVs. We use the XOY plane here to show how the modify tentacle algorithm solves the two problems above. In the basic tentacle algorithm, once the radius of each tentacle is fixed, the corresponding control instruction only can be obtained by matching a steady state with the same curvature, but it is hard to match when the tentacle has large radius.

For solving this problem, different from determining the radius firstly for each tentacle, we consider using an inverse derivation method, firstly determine the manoeuver capacity for an UAV. So the radius of each tentacle r k in one speed set j can be computed by:. It should be noted that, both of the control instruction and the radius of tentacle are transferred to the functions with tentacle number k. It means once a tentacle is selected, instead of curvature matching in basic algorithm, the corresponding control instruction can be computed directly through reading the variable k.

Thus, the modification can avoid the curvature matching in basic tentacle algorithm, and the Problem 1 can be solved in theory. Fig 5B shows the 41 tentacles in one speed set.

It can be seen that each tentacle divide the occupancy grid averagely. It proves the Problem 2 is solved in theory.

## Flight Formation Control

The obstacle detection is similar to the basic tentacle algorithm, which, therefore, can be used for the UAV. The obstacle can be detected through Rule A.

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It should be noted that, for an UAV in its formation, all the other UAVs are considered to be the dynamic obstacles in the occupancy grid, so this detection method may eliminate impact on the manoeuver of multiple UAVs. The crash distance l c also is given by Eq 12 , and the three function v clearance , v flatness , v trajectory in Eq 13 are given by the following equations respectively. This means that the smaller v clearance shows the corresponding tentacle with a large distance from the next obstacle, so that the tentacles with small v clearance are preferable.

Similar to v clearance , v trajectory also follow a normal distribution:. The flow chart of the modified tentacle algorithm is as Fig 6. Problem 1: Different from computing the steering command according to the radius of each tentacle with the basic tentacle algorithm, by using the inverse derivation, the modified tentacle algorithm divides the control instruction into some uniform sections, and then computes the radius of each tentacle with Newton's second law. It ensures each tentacle corresponds with only one overload instruction, so the curvature matching in basic algorithm become unnecessary.

Thus, the data calculation problem Problem 1 can be solved. Problem 2: Because the speed sets and tentacles in one speed set are reduced and reconstructed, the influence of computation load in real time computation become insignificant. It ensures the modified tentacle algorithm meet the requirement of the real time computation for UAVs Problem 2.

## Formation Control of UAVs with a Fourth-Order Flight Dynamics

In order to verify the performance of the modified algorithm, a complex scenario with 3 obstacles and 5 UAVs is created in this paper. The leader-follower formation model contains one leader and four followers. Table 2 shows the relative distance in three dimensions between each follower and leader:. Table 3 gives the information on obstacles during the simulation, where d s represents the requirements for safe distances in each tentacle.

The simulation will be terminated when the leader UAV reach m in x direction. The simulation step is 0. It can be observed that all the static obstacles are successfully avoided. These prove our modified algorithm can avoid static and dynamic obstacles effectively.

Both of them show that each follower can recover its original formation position after the avoidance. The successful multiple large-radius manoeuver of 5 UAVs prove our inverse derivation solution to the data calculation problem Problem 1 is credible. The results show that all UAVs can maintain their relative distances farther than the safe distances from other followers and obstacles.

These prove that the collision avoidance method proposed in this paper has a high performance. Meanwhile, the high-performance collision avoidance method proves that our reduction and reconstruction solution to the application problem Problem 2 can be solved, and the modified tentacle algorithm can be successfully applied into the collision avoidance of multiple high-speed UAVs.

With the modified tentacle algorithm and other classical collision avoidance methods, Table 6 compares the computational load of path planning and environment modelling in 5 continuous time sets. The modified tentacle algorithm are validated in two other scenarios, and the corresponding simulation results are shown in S1 File.

This paper proposes the modified tentacle algorithm for the formation flight and collision avoidance of multiple UAVs.