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Abstract

A decentralized four-dimensional (4D) trajectory generation method for unmanned aerial vehicles (UAVs), which uses the improved intrinsic tau gravity (tau-G) guidance strategy, is presented in this paper. Based on general tau theory, the current tau-G strategy can only generate the 4D trajectory with zero initial and final velocities, which is not appropriate for decentralized applications. By adding an initial velocity to the intrinsic movement of tau-G strategy, the improved tau-G strategy can synchronously guide the position and velocity to the desired values at the arrival time. In our decentralized 4D trajectory generation method, the improved tau-G strategy is used to plan the 4D trajectories for UAVs. To deal with environmental uncertainty and communication limitations, the receding horizon optimization driven by both sampling time and conflict events is utilized to renew trajectory parameters continually. The simulation results of challenging time-constrained tasks demonstrate that the proposed method can efficiently provide safer and lower-cost 4D trajectories.

Keywords

General Tau Theory Improved Tau-G Strategy Decentralized 4D Trajectory Generation Multiple UAVs

1 Introduction

In cooperative missions of UAVs, such as simultaneous attack [1] and formation aggregation [2], the members of a UAV group need to arrive at the prescribed destination simultaneously or sequentially. 4D trajectory generation, which adds the time dimension into the three-dimensional (3D) trajectory and accurately controls the arrival time at each waypoint, can dramatically lower trajectory uncertainty and reliably accomplish the time-critical tasks.

Compared with a great number of works on 4D trajectory generation in civil aviation, limited research for UAVs is available in recent literature. As the trajectories for aerobuses are often retrieved from the flight database and planned long in advance, they are inappropriate for UAVs to cope with complex missions and online replanning. The existing 4D trajectory generation methods in UAV fields include tensor field guidance [3], the Bezier polynomial function [4], the method based on the one-at-a-time strategy [5] and the Multistep A* algorithm [6]. However, to meet the arrival time, many approaches adopt speed assignments after planning flyable 3D trajectories. Without considering speed dynamics, these assignments will lower the flyability of trajectories. Furthermore, it is difficult to optimize some complicated trajectory descriptions for multiple UAVs. In the end, most of the methods in the literature are centralized ones, which require global knowledge of the multi-UAV system and lots of time for optimization.

Recently, researchers have found that the 4D trajectory generation, based on the bio-inspired general tau theory, can overcome the drawbacks of the existing approaches. General tau theory, based on the action principle of the optical tau variable in animal movements [7–9], is proposed, while several tau guidance strategies are designed to utilize the bionic knowledge of the tau variable in trajectory generation. Exactly at the desired arrival time, tau strategies can synchronously plan the time-variant position and velocity, while considering the transient process of speed regulation. Hence, tau strategies can provide more accurate and flyable guidance for UAVs. Meanwhile, the succinct form of the tau strategy-based trajectory will benefit optimization for multi-UAVs. Reference [10] was the first to use general tau theory in 4D movement guidance of an octocopter, while more test results were stated in [11]. New Mexico State University applied tau strategies to guide a quadrotor perching as a bird [12] and generate a 4D trajectory for point-to-point movement [13]. In the case of multiple UAVs, reference [14] firstly proposed a multi-UAV 4D trajectory generation method based on tau theory. This method uses the intrinsic tau-G guidance strategy to describe 4D trajectories, formulate a centralized trajectory planning problem and solve the problem by particle swarm optimization (PSO). Reference [15] proposed the tau harmonic (tau-H) guidance strategy, which was used in centralized trajectory generation for UAVs. As the tau-H strategy has a stronger trajectory shape adjustment capability, the trajectory generation achieves a better conflict resolution performance.

However, to the best of our knowledge, tau strategies have not been used in decentralized applications of multi-UAVs. This is because the online planning of trajectories in decentralized tasks always need to cope with the movements starting or ending with non-static motion states. However, existing tau guidance strategies cannot perform well in the guidance of movements with non-zero initial and final velocities, especially when the UAV is initially going further away from the destination. This drawback tremendously limits the further use of tau strategies in cooperative missions, such as formation flight with a virtual structure and persistent tracking of moving targets.

The main contribution of this paper is two-fold. Firstly, to meet the requirement of decentralized 4D trajectory generation for UAVs, an improved intrinsic tau-G strategy is proposed. By adding an initial velocity to the intrinsic movement of the original tau-G strategy, the improved tau-G strategy can synchronously guide both position and velocity to meet their expectations exactly at the desired time. Therefore, the 4D trajectories generated by the improved tau-G strategy can guide the movement starting or terminating with non-static motion states, even the position gap is initially expanding. Secondly, in order to solve the challenging 4D trajectory generation problem for multi-UAVs, a comprehensive decentralized 4D trajectory generation method is designed. In particular, this method uses the improved tau-G strategy to generate 4D trajectory for every vehicle, detects conflicts by a geometric approach, which is based on the protected airspace zone (PAZ) [16], and formulates a decentralized optimization problem for 4D trajectory generation. Decentralized receding horizon optimization (DRHO), which has not been used in 4D trajectory generation, is applied in order to obtain the optimal trajectories. To deal with environmental uncertainty, DRHO is driven not only by time-sampling, but also conflict events. The dynamics simulation results show that our decentralized trajectory generation method can efficiently provide safer and lower-cost trajectories for UAVs.

The remainder of this paper is structured as follows. In Section 2, the decentralized 4D trajectory problem for multi-UAVs is stated. In Section 3, we present the improved tau-G strategy, analyse its validity and calculate the extreme velocity during guidance. Then, the whole decentralized 4D trajectory generation method is described in Section 4, while Section 5 presents the simulation results and analysis in contrast with decentralized model predictive control (DMPC). Finally, we draw a conclusion in relation to our trajectory generation method.

2 Problem Statement

The problem considered in this paper is to generate optimal 4D trajectories for a decentralized multi-UAV system consisting N homogeneous vehicles. The 4D trajectory of the i^th UAV is described by the time-variant movement states S_i(t)={x_i(t), y_i(t), z_i(t), ẋ_i(t), ẏ_i(t), ż_i(t)}, in which (x_i, y_i, z_i) is the position in the north-east-down frame. For simplicity, the orientation of UAV can be omitted in trajectory generation and considered in trajectory tracking. The trajectory can guide the UAV moving from the arbitrary initial states S _i(t₀) to the goal states S _i(t₀+T_i) exactly at the desired arrival time of T_i. The dynamics of the UAV are:

{\begin{cases} {\dot{x}}_{i} = u_{x i} \\ {\dot{y}}_{i} = u_{y i} \\ {\dot{z}}_{i} = u_{z i} \end{cases}

(1)

The velocities along each coordinate axes u _i={u_xi, u_yi, u_zi} are used as the inputs of the trajectory tracker, which can be directly utilized for rotorcrafts. Meanwhile, u _i can be easily transformed to velocity and attitude commands for fixed-wing UAVs.

Defining the communication topology as the edge-weighted directed graph changes along with the movements of UAVs. In the graph, the vertices depict the positions of UAVs, while the directed edge e_ij in edge set E refers to the information flow from UAVi to UAVj. The adjacent matrix A = [a_ij]_NxN ∈ R^NxN is the weight matrix of E , in which a_ij is defined as a linear changed weight:

a_{i j} = {\begin{cases} \frac{d_{i j} - R_{c}}{R_{c} - R_{s a f e}} d_{i j} \leq R_{c} \\ 0 d_{i j} > R_{c} \end{cases} i, j = 1 \dots N

(2)

In this problem, a_ij=a_ji⩽0. The notation R_safe refers to the radius of PAZ, which is the minimum safe distance between UAVs. R_c is the valid communication distance. When the distance between UAVi and UAVj meets the condition d_ij⩽R_c (i, j = 1… N), valid communication can be established. The two UAVs are referred to as neighbours of each other.

Next, the Laplacian matrix [17] L = [l_ij]_NxN ∈ R^NxN of the graph can be defined as follows:

l_{i j =} {\begin{array}{l} - a_{i j} & i f e_{i j} \in E a n d j \neq i \\ - \sum_{j = 1, j \neq i}^{N} a_{i j} & i f e_{i j} \in E a n d j = i i, j = 1... N \\ 0 & i f e_{i j} \notin E \end{array}

(3)

As the decisions of the nearer UAVs should be considered preferentially in order to avoid potential conflicts, the non-negative edge weight l_ij indicates the influence of UAVi upon the trajectory generation of UAVj.

The architecture of the decentralized 4D trajectory generation is shown in Figure 1. In the decentralized system without a central planning unit, the vehicles exchange their movement states S _i and trajectory strategies u _i through wireless communication. The initial states of targets $\cup_{i = 1}^{N} S_{i} (t_{0 i} + T_{i})$ and the arrival time set T are provided by the task planner. Meanwhile, in the execution of tasks, we assume that every UAV can obtain the states for its own target and the target dynamics should be limited to the flyability of the UAV.

The optimization problem of the whole system is divided into N local problems, which will be solved by every vehicle according to its own states and the information of its neighbours. The local optimal objective is to find the feasible parameters, with minimum cost, for the 4D trajectory constrained by simultaneous arrival, being collision-free, maximum velocity etc. The whole multi-UAV system is organized and coordinated through the task objective and constraints. This decentralized optimization problem is detailed in Section 4.2.

Figure 1.

Decentralized 4D trajectory generation architecture of multi-UAV system

2 Improved Intrinsic Tau-G Strategy

3.1 Overview of existing tau guidance strategies

General tau theory [18] is put forward based on the research relating to the bio-inspired tau theory [19]. In general tau theory, the tau variable represents the time-to-contact (TTC) of the goal-directed movement, which can guide the UAV to close the gaps between current motion states and the goals. The tau variable is defined as follows:

τ_{χ} (t) = {\begin{cases} \frac{χ (t)}{\dot{χ} (t)} | \dot{χ} (t) | \geq {\dot{χ}}_{\min} \\ sgn (\frac{χ (t)}{\dot{χ} (t)}) τ_{\max} | \dot{χ} (t) | < {\dot{χ}}_{\min} \end{cases}

(4)

in which the motion gap χ can be the gap of any state in S (t); for instance, τ_y(t) is the tau variable for movement along the y-axis. When $\dot{χ} (t)$ approaches zero, the value of τ_χ(t) should be limited. Generally, τ_χ(t) is set as negative when the gap is closing. So far, five tau guidance strategies have been proposed for the purpose of utilizing general tau theory in movement guidance.

1 Tau-dot (tau-D) Strategy

According to tau-D strategy, if a UAV needs to stop at the destination on time, it should retain a derivative of the tau variable $\dot{τ}$ as a constant k. Substituting the tau variable in Eq. (4) for $\dot{τ}$ =k, the time-variant expressions of χ and its derivatives can be solved as follows:

{\begin{cases} χ = χ_{0} {(1 + k \frac{{\dot{χ}}_{0}}{χ_{0}} t)}^{1 / k} \\ \dot{χ} = {\dot{χ}}_{0} {(1 + k \frac{{\dot{χ}}_{0}}{χ_{0}} t)}^{1 / k - 1} \\ \ddot{χ} = \frac{{\dot{χ}}_{0}^{2}}{χ_{0}} (1 - k) {(1 + k \frac{{\dot{χ}}_{0}}{χ_{0}} t)}^{1 / k - 2} \end{cases}

(5)

in which χ₀ and ${\dot{χ}}_{0}$ refer to the initial position gap and velocity, respectively. Analysing the relations between (χ, $\dot{χ}$ , $\ddot{χ}$ ) and k, the UAV will stop at the destination steadily at time T=τ₀/k if k ∈ (0, 0.5), in which τ₀=χ₀/ ${\dot{χ}}_{0}$ is the initial value of tau. The tau-D strategy can be used to guide the UAV to break with τ₀<0, while it is helpless in the tasks relating to the initially expanding gap τ₀>0 or accelerating to a certain speed. Meanwhile, the arrival time T of the task cannot always ensure that 0<k<0.5 for steady guidance.

2 Tau Coupling Strategy

If the tau variables of different motion gaps maintain a non-zero constant ratio k during the closing process, these gaps will be closed synchronously. Ratio k is called the coupling coefficient. Taking the two motion gaps χ and μ as an example, by solving τ_χ=kτ_μ, the relation between the states of the two movements is obtained as follows:

{\begin{cases} χ = C μ^{1 / k} \\ \dot{χ} = \frac{C}{k} \dot{μ} μ^{1 / k - 1} \\ \ddot{χ} = \frac{C}{k} μ^{1 / k - 2} (\frac{1 - k}{k} {\dot{μ}}^{2} + μ \ddot{μ}) \end{cases} (C = \frac{χ_{0}}{μ_{0}^{1 / k}})

(6)

If k ∈ (0, 0.5), (χ, $\dot{χ}$ , $\ddot{χ}$ )→0 when μ→0. The tau coupling strategy can play an important role in the synchronous guidance of 3D spatial gaps.

3 Intrinsic Tau-G Guidance Strategy

To avoid external disturbance, μ in the tau coupling strategy can be designed as a virtual movement called the intrinsic movement. In tau-G strategy, the intrinsic movement gap is a free fall motion represented by G(t)=g(T²-t²)/2. If the gap χ maintains appropriate ratios with τ_G, χ will be closed at the desired arrival time T. By solving τ_χ=kτ_G, χ, $\dot{χ}$ and $\ddot{χ}$ can be yielded as follows:

{\begin{cases} χ (t) = \frac{χ_{0}}{T^{2 / k}} {(T^{2} - t^{2})}^{1 / k} \\ \dot{χ} (t) = \frac{- 2 χ_{0} t}{k T^{2 / k}} {(T^{2} - t^{2})}^{1 / k - 1} \\ \ddot{χ} (t) = \frac{2 χ_{0}}{k T^{2 / k}} (\frac{2 - k}{k} t^{2} - T^{2}) {(T^{2} - t^{2})}^{1 / k - 2} \end{cases}

(7)

As with tau coupling strategy, if k ∈ (0, 0.5), (χ, $\dot{χ}$ , $\ddot{χ}$ )→0 when t→T. As $\dot{χ}$ (0)=0, tau-G strategy can be used to generate the 4D trajectory with static initial and final states; namely, the UAV takes off from the starting point, flies to the destination and arrives at the destination at time T.

4 Tau Jerk (tau-J) Strategy

The tau-J guidance strategy uses the constant jerk motion gap J(t)=j(T³- t³)/6 as the intrinsic movement [13]. The letter j refers to the constant jerk. The relations between movement states and the k value and the applications of this strategy are the same as tau-G strategy.

5 Tau-H Strategy

Utilizing the harmonic motion H(t)=H₀ [sin(2πt/T)/(2π)-t/T+1] as the intrinsic movement gap, if the coupling coefficient k ∈ (0, 1], tau-H strategy [15] can accomplish (χ, $\dot{χ}$ , $\ddot{χ}$ )→0 when t→T. With the expansion of the value range of k, the tau-H strategy has a stronger shape adjustment capability for movement guidance with initial and final velocity, which is helpful for deconfliction in a multi-UAV system.

Above all, these existing tau guidance strategies cannot be ideally applied to the tasks of decentralized trajectory generation, including velocity regulation, goal-directed movement with $\dot{χ} (0) \neq 0$ and $\dot{χ} (T) \neq 0$ , and movement with the initially opening motion gap τ₀>0.

3.2 Improved tau-G strategy

3.2.1 Improvement of tau-G strategy

In this section, the improved tau-G strategy is proposed to overcome the deficiencies of the existing strategies. According to $\dot{χ}$ of the tau coupling strategy shown in Eq. (5), $\dot{μ} (0)$ should be non-zero if $\dot{χ} (0)$ ≠0 is needed. Therefore, an initial velocity V_G is added to the intrinsic movement G(t) of tau-G strategy, formulating a new intrinsic guidance movement represented by G_v(t):

{\begin{cases} G_{v} (t) = - \frac{1}{2} g t^{2} + V_{G} t + G_{0} \\ {\dot{G}}_{v} (t) = - g t + V_{G} \\ {\ddot{G}}_{v} (t) = - g \end{cases}

(8)

in which G₀ refers to the initial intrinsic gap.

When taking the movement along the x-axis from time 0 to T as an example, the position gap is Δx= x_T -x, and the velocity gap is Δẋ =ẋ_T -ẋ, in which x_T and ẋ_T denote the goal states at time T. According to the tau coupling strategy in Eq. (6), the velocity ẋ is defined as follows:

\dot{x} = {\dot{x}}_{T} - \frac{χ_{x 0}}{k_{x} G_{0}^{1 / k_{x}}} {\dot{G}}_{v} G_{v}^{1 / k_{x} - 1}

(9)

It is important to note that χ_x0=ƒΔẋ(t)dt |_t=0 is not equal to the initial position gap Δx₀, as ẋ_T may not be zero. Integrating ẋ and taking the terminal position x_T into account, the movement states can be expressed as follows:

{\begin{cases} x (t) = x_{T} + {\dot{x}}_{T} (t - T) - \frac{χ_{x 0}}{G_{0}^{1 / k_{x}}} G_{v}^{1 / k_{x}} \\ \dot{x} (t) = {\dot{x}}_{T} - \frac{χ_{x 0}}{k_{x} G_{0}^{1 / k_{x}}} {\dot{G}}_{v} G_{v}^{1 / k_{x} - 1} \\ \ddot{x} (t) = - \frac{χ_{x 0}}{k_{x} G_{0}^{1 / k_{x}}} G_{v}^{1 / k_{x} - 2} (\frac{1 - k_{x}}{k_{x}} {\dot{G}}_{v}^{2} + G_{v} {\ddot{G}}_{v}) \end{cases}

(10)

Substituting G_v for Eq. (10), one can yield χ_x0=x_T -x₀-ẋ_TT and V_G=k_xΔẋ₀G₀/χ_x0, in which Δẋ₀ is the initial velocity gap. Then, by taking the expressions of χ_x0 and V_G into G_v(t), as shown in Eq. (8), G₀ and V_G are calculated as follows:

{\begin{cases} G_{0} = \frac{χ_{x 0} g T^{2}}{2 (χ_{x 0} + k_{x} Δ {\dot{x}}_{0} T)} \\ V_{G} = \frac{k_{x} Δ {\dot{x}}_{0} g T^{2}}{2 (χ_{x 0} + k_{x} Δ {\dot{x}}_{0} T)} \end{cases}

(11)

As such, G₀ and V_G can be seen as the bond between the actual and intrinsic movements. Analysing the movement states in Eq. (10), the final movement states depending on k_x are shown in Table 1. If k_x ∈ (0, 0.5), (χ, $\dot{χ}$ , $\ddot{χ}$ )→(x_T, ẋ_T,0) when t→T, the position and velocity can be steadily guided to the expected values.

Table 1.

Relations between final movement states and k_x, guided by the improved tau-G strategy

k _x	x	ẋ	ẍ
k _x <0	x→x _T	ẋ →ẋ _T	ẍ →0
0< k _x<0.5	x → x _T	ẋ → ẋ _T	ẍ →0
0.5⩽ k _x <1	x → x _T	ẋ → ẋ _T	ẍ →∞
k _x =1	x → x _T	ẋ = Ġ_v	ẍ = G̈_v
k _x >1	x → x _T	ẋ →∞	ẍ →∞

Figure 2 displays the relations between (x, ẋ, ẍ) and k_x, guided by our improved tau-G strategy. This task, with x₀=0, x_T=5, ẋ₀ =-0.4, ẋ_T =0.1 and T=10s (τ _0x>0), cannot be finished by any of the existing tau strategies. For curves k_x=0.1 and k_x=0.4 when t→T, (x, ẋ, ẍ)→(5,-0.4,0).

Figure 3 shows the performance comparison between the improved tau-G strategy and tau-D strategy in two breaking tasks. The improved tau-G strategy performs well in two tasks, while the tau-D strategy fails to guide the closure of the gap with τ_0x>0. Additionally, if τ_0x/T>k_x (k_x =0.2), the gap with τ_0x<0, guided by tau-D strategy, is not exactly closed at time T. Therefore, for the definite task, k_x of the tau-D strategy does not make any allowance for optimization. These results indicate that our improved tau-G strategy can guide the motion with initial and final velocities to the expected states, even when the position gap is initially expanding (τ_0x>0). Meanwhile, in the tasks with ẋ₀ =0 and ẋ_T =0, the initial velocity V_G of G_v (t) equals zero. The improved tau-G strategy is the same as for the original tau-G strategy. The comparison of all tau guidance strategies and their applications is summarized in Table 2. Above all, the drawbacks of the existing tau strategies can be overcome by our improved tau-G strategy.

Figure 2.

Relations between the movement states and k_x values with τ_0x>0

Figure 3.

Comparison between the improved tau-G strategy and tau-D strategy in breaking tasks

3.2.2 Validity of the improved tau-G strategy

In the 4D movement guidance of a UAV, the following content should be noted in order to insure the validity of the improved tau-G strategy:

According to the expressions of G₀ and V_G in Eq. (11), k_x = -χ_x0 / (Δẋ₀T) is the singularity in which G₀ and V_G are invalid. If χ_x0/Δẋ₀<0 and -χ_x0 / (Δ ẋ₀T) <0.5, the singularity will appear in the value range of k_x. At the singularity, V_G=-G₀/T, the intrinsic movement G_v as shown in Eq. (8), turns into a uniform rectilinear motion as follows:

{\begin{cases} G_{v} (t) = - G_{0} t / T + G_{0} \\ {\dot{G}}_{v} (t) = - G_{0} / T \\ {\ddot{G}}_{v} (t) = 0 \end{cases}

(12)

This motion can be used as the intrinsic movement at the singularity.

In terms of Table 1, if k_x∈ (0, 0.5), x, ẋ and ẍ approach their desired value when G(t)→0. To enable the gap to close exactly at the arrival time T, the function G(t)=0 should have the unique solution t=T in the time range (0, T]; otherwise, the gap will be closed in advance. Solutions of G(t)=0 are calculated as follows:

{\begin{cases} t_{1} = T \\ t_{2} = \frac{- χ_{x 0} T}{χ_{x 0} + k_{x} Δ {\dot{x}}_{0} T} \end{cases}

(13)

To ensure that t₁=T is the unique solution, t₂ should meet t₂ ∈(-∞, 0]∪[T, ∞), which is equivalent to k_xΔẋ₀T/χ_x0⩾-2. Therefore, the value range of k_x can be yielded as follows:

{\begin{cases} 0 < k_{x} < 0.5 i f \frac{χ_{x 0}}{Δ {\dot{x}}_{0}} \geq 0 o r \frac{χ_{x 0}}{Δ {\dot{x}}_{0}} \leq - 0.25 T \\ 0 < k_{x} \leq - \frac{2 χ_{x 0}}{T Δ {\dot{x}}_{0}} i f - 0.25 T < \frac{χ_{x 0}}{Δ {\dot{x}}_{0}} < 0 \end{cases}

(14)

Table 2.

Summary of tau guidance strategies

Tau guidance strategies		Guidance formulas	Applications
Tau-D strategy		$\dot{τ} (t)$ =k	Breaking guidance with ẋ(0)≠0, ẋ(T)=0 and τ ₀<0
Tau coupling strategy		τ_χ(t)=kτ_μ (t)	Synchronous closure of different gaps
Intrinsic tau guidance strategy (based on tau coupling strategy)	Intrinsic movement gap
	Tau-G strategy	G(t)=g(T²-t²)/2	Point to point movements with ẋ(0)=0, and ẋ(T)=0
	tau-J strategy	J(t)=j(T³-t³)/6
	tau-H strategy	H(t)= H₀[1/2π sin(2πt/T)-t/T + 1]
	Improved tau-G strategy	G_v(t)=-1/2_gt²+V_Gt + G₀	Movements with arbitrary ẋ(0) and ẋ(T), whatever the sign of τ₀ is

3.2.3 Extreme velocity during guidance

As the maximum velocity constraint is necessary in 4D trajectory generation, the velocity extremum conditions of our improved tau-G strategy need to be analysed.

When χ_x0=0, ẋ equals ẋ₀ all the time. Therefore, the velocity extremum ẋ_ex does not exist.

At the singularity χ_x0=-k_xΔẋ₀T, if Δẋ₀≠0, ẍ(t)≠0 in the period of t ∈ (0, T). Hence, there is no extreme velocity.

In other conditions, by solving ẍ (t)=0, the time when ẋ_ex comes out can be yielded as follows:

\begin{array}{l} {\begin{cases} t_{1} = \frac{(β_{0} + 1) k_{x} Δ {\dot{x}}_{0} T^{2} + 2 χ_{x 0} T}{2 β_{0} (χ_{x 0} + k_{x} Δ {\dot{x}}_{0} T)} \\ t_{2} = \frac{(β_{0} - 1) k_{x} Δ {\dot{x}}_{0} T^{2} - 2 χ_{x 0} T}{2 β_{0} (χ_{x 0} + k Δ {\dot{x}}_{0} T)} \end{cases} \\ β_{0} = \sqrt{2 / k_{x} - 1} \end{array}

(15)

When considering t₁ and t₂∈ (0, T), the conditions to justify the extremum of ẋ at t₁ and t₂ are shown in Table 3.

Table 3.

Extreme velocity conditions at t₁ and t₂

Extremum conditions	ẋ(t₁)	ẋ(t₂)
Δẋ₀ =0, ẋ_T =0 & χ_x0 >0	Max	-
Δẋ₀ =0, ẋ_T =0 & χ_x0 <0	Min	-
χ_x0Δẋ₀>0	Max	-
Δẋ₀[χ_x0 + (β₀ + 1)k_xΔẋ₀T/2]<0	Min	-
χ_x0Δẋ₀ > 0 & Δẋ₀[χ_x0-(β₀-1)k_xΔẋ₀T/2]<0	-	Min

The expression of ẋ_ex at t₁ is:

\begin{array}{l} {\dot{x}}_{e x} (t_{1}) = {\dot{x}}_{T} + \frac{γ}{β_{0} k_{x} T} {[\frac{(β_{0}^{2} - 1) γ^{2} /2}{2 β_{0}^{2} χ_{x 0} (γ - χ_{x 0})}]}^{1 / k_{x} - 1} \\ γ = k_{x} Δ {\dot{x}}_{0} T + 2 χ_{x 0} \end{array}

(16)

The value of ẋ_ex (t₁) increases when k_x is closer to 0 and the extreme velocity at t₂ is ẋ_ex(t₂)=2ẋ_T -ẋ_ex(t₁). When there is no extreme velocity in the period of (0, T), the maximum velocity will appear at t=0 or t=T.

4 Decentralized 4D Trajectory Generation Based on Improved Tau-G Strategy

4.1 4D trajectory guided by improved tau-G strategy

In 4D trajectory generation for UAVs, the 3D movements guided by the improved tau-G strategy are:

{\begin{cases} \dot{x} = {\dot{x}}_{T} - \frac{χ_{x 0}}{k_{x} G_{0}^{1 / k_{x}}} {\dot{G}}_{v} G_{v}^{1 / k_{x} - 1} \\ \dot{y} = {\dot{y}}_{T} - \frac{χ_{y 0}}{k_{y} G_{0}^{1 / k_{y}}} {\dot{G}}_{v} G_{v}^{1 / k_{y} - 1} \\ \dot{z} = {\dot{z}}_{T} - \frac{χ_{z 0}}{k_{z} G_{0}^{1 / k_{z}}} {\dot{G}}_{v} G_{v}^{1 / k_{z} - 1} \end{cases}

(17)

in which the coupling coefficient set k = {k_x, k_y, k_z} should meet the validity conditions in Eq. (14).

To achieve the spatial function of the trajectory, Eq. (17) can be rewritten as follows:

\begin{array}{l} {\begin{cases} f_{y} = \frac{χ_{y 0}}{χ_{x 0}^{k_{x} / k_{y}}} f_{x}^{k_{x} / k_{y}} \\ f_{z} = \frac{χ_{z 0}}{χ_{x 0}^{k_{x} / k_{z}}} f_{x}^{k_{x} / k_{z}} \end{cases} \\ f_{μ} = {\dot{μ}}_{T} (t - T) + μ_{T} - μ, μ = x, y, z \end{array}

(18)

Therefore, the 3D shape of the trajectory is determined by the coupling coefficient set k ={k_x, k_y, k_z}. The trajectory shapes, with k_x=0.17, k_z=0.17 and different k_y, are shown in Figure 4 in order to demonstrate the shape adjustment by the selection of k .

Figure 4.

Trajectories generated by the improved tau-G strategy with k_x=0.17, k_z=0.17 and different k_y, in which ‘S’ is the starting point and ‘D’ is the destination

4.2 Decentralized 4D trajectory optimization

4.2.1 Decentralized optimization problem

In decentralized 4D trajectory generation, the optimization problem is divided into N local problems according to the number of UAVs. For every vehicle, the local problem is independently solved based on its own states and information from its neighbours. When comprehensively considering the trajectory length L_i, the conflicts between UAVs and obstacles and the maximum velocity v_max, the objective function of the i^th UAV is designed as a weighted sum:

J_{i} = ω_{1} L_{i} + \sum_{e_{i j} \in E} l_{i j} \int_{t = 0}^{T} \frac{ω_{2}}{{‖ p_{i} (t) - p_{j} (t) ‖}^{2} + α} d t + ω_{3} \sum_{j = 1}^{N} C u_{i j} + ω_{4} \sum_{k = 1}^{N_{o b s}} C o_{i k} + ω_{5} {‖ v_{\max, i} ‖}^{2}

(19)

in which l_ij is the element of the Laplacian matrix, p _i is the central position of UAVi, α is a smaller positive constant to prevent the fraction value going to infinity, Cu_ij refers to conflicts between UAV i and j, N_obs is the number of obstacles, and Co_ik represents where UAVi has collisions with the k^th obstacle.

Since UAVs cannot be regarded as point masses in trajectory generation, an efficiently geometric approach is adopted for conflict detection. In this paper, we assume that every UAV equips the distance sensor with the view range of R_view. If the obstacles or other vehicles are located in the view range, their nearest points p _c=(x_c, y_c, z_c) from the UAV can be sensed. The conflict detection approach assigns the spherical PAZ [20] for every vehicle as Figure 5. If Eq. (20) is fulfilled, a conflict is detected. The coordinate (x_o, y_o, z_o) refers to the centre of the UAV.

{(x_{c} - x_{o})}^{2} + {(y_{c} - y_{o})}^{2} + {(z_{c} - z_{o})}^{2} < R_{s a f e}^{2}

(20)

Figure 5.

Spherical PAZ and conflict detection

The objective of the local trajectory generation problem is to find the optimal k _i for the i^th tau trajectory, which can minimize J_i:

\begin{array}{l} \min_{k_{i}} J_{i} \\ s . t . {\begin{cases} s (p_{i}, k_{i}) = 0 \\ g_{l} (k_{i}) \leq 0 \\ g_{u} (k_{i}) \leq 0 \end{cases} \end{array}

(21)

The constraints include the trajectory descriptions s( p _i, k _i), as shown in Eq. (17), the lower bound g _l( k _i) and the upper bound g_u( k _i), as expressed in Eq. (14).

4.2.2 Decentralized optimization based on receding horizon optimization

If the UAVs are initially located beyond the communication range, the 4D trajectories generated without information from neighbours cannot ensure flight safety. Meanwhile, the states of moving targets, the environment and the communication topology are time-variant. Hence, the DRHO method is applied in order to renew the trajectory parameters step by step to correct the errors repeatedly [21].

The decentralized 4D trajectory generation method, based on the improved tau-G strategy and DRHO (I-tau-G-DRHO), is shown in Algorithm 1. At the beginning, every UAV initializes the coupling coefficient set k _i and arrival time T_i according to its own states, after which it sends k _i, starting states S _0i and goal states S _Ti to its neighbours via wireless communication. In turn, the DRHO process begins.

Algorithm 1: Decentralized 4D trajectory generation
1	Initialize T_i and k _i (i=1…N)
2	Communication: send k _i, S _0i and S _Ti out in broadcast, and receive information from neighbor UAVs
3	for in every sampling time or any event happens do
4	if the motion states of target have changed or conflicts exist
5	repeat
6	for every UAV in re-planning do
7	Calculate trajectories of neighbors according to the received information
8	Solve the local problem of Eq. (21) to get the optimal k_i
9	Communication: send k_i, S _i0 and S _iT out in broadcast, and receive information from neighbors
10	end for
11	until maximum optimization number or terminal criterion Eq. (22) is met
12	else
13	break
14	end if
15	end for

To improve the response capacity to emergencies, DRHO is driven not only by the sampling time, but also the conflict events. In the conflict event-driven mechanism, a conflict detection module works at a higher frequency than time sampling. If Eq. (20) is fulfilled, this module will raise an alarm. At every sampling instant or event alarm, I-tau-G-DRHO will firstly check the target states and conflict events. If the states of target have changed or any potential conflict has happened, the reoptimization procedures will be executed. In particular, if receding optimization happens on one UAV, its neighbours will also go into the replanning process in order to achieve optimal decisions for the group. This replanning mechanism not only lowers the computation load, but also keeps enough capability to deal with environmental uncertainty.

In replanning, I-tau-G-DRHO calculates the 4D trajectories of neighbours, solves the local optimization problem in Eq. (21) to yield optimal k _i and then exchanges the decisions and states with neighbours. It is important to note that the ultimate trajectory parameters of neighbours cannot be used in trajectory generation, as the UAVs execute the planning process synchronously. As a result, after the m^th local optimization, each UAV compares the latest decision to the last one, after which the procedures will be repeated using the newly received information until the terminal criterion, as per Eq. (22), or the maximum optimization number is met. The notation ɛ refers to the error tolerance. After replanning, the optimal k _i is carried out by UAVs, after which I-tau-G-DRHO will execute the above optimization process repeatedly at the next sampling time or event alarm, in order to drive the optimization horizon going forward.

{‖ k_{i} (m) - k_{i} (m - 1) ‖}_{2} < ε m \geq 2

(22)

As there is great difficulty in obtaining an analytical solution for the local optimization problem in Eq. (21), the numerical computation algorithms should be adopted. The commonly used algorithms include sequential quadratic programming (SQP) [22], PSO [23] and mixed integer linear programming (MILP), which is solved by CPLEX [21]. Since the 4D trajectory generated by the improved tau-G strategy is continuous, SQP and PSO are carried out in order to solve decentralized 4D trajectory generation cases for the purpose of comparison.

In these cases, 10 UAVs cooperatively set off from their starting points, pass a common middle point sequentially and arrive at the static target at time T. The error tolerance is ɛ=10⁻³, the PSO algorithm has 10 particles and each algorithm can iterate 100 times at most in one local optimization. Table 4 shows the average time cost for solving a local problem ( ${\bar{t}}_{l p}$ ) and the average number of local optimizations ( ${\bar{N}}_{l o}$ ) to meet the terminal criterion in Eq. (22). The results show that SQP costs less time and involves a smaller optimization number than PSO, while the time cost of SQP can meet the requirement of real practice. Therefore, SQP is applied in the local optimization of I-tau-G-DRHO.

Table 4.

Optimization performance comparison between SQP and PSO

No. of UAVs		1	2	3	4	5	6	7	8	9	10
${\bar{t}}_{l p}$ (s)	SQP	0.011	0.013	0.012	0.012	0.031	0.036	0.045	0.046	0.045	0.103
${\bar{t}}_{l p}$ (s)	PSO	0.080	0.120	0.147	0.161	0.201	0.266	0.368	0.466	0.625	0.582
${\bar{N}}_{l o}$	SQP	2	2	2	2	3.495	3.879	3.783	4.205	4.326	4.333
${\bar{N}}_{l o}$	PSO	2	2.327	3.514	4.122	5.898	6.077	6.021	7.254	7.000	7.086

5 Simulation Results and Analysis

In this section, a comprehensive simulation is carried out in order to demonstrate the performance of the I-tau-G-DRHO method. Since there is no benchmark for 4D trajectory generation in a multi-UAV system [24], we have designed a convincing simulation scenario where five homogeneous UAVs cooperatively generate 4D trajectories for a series of tasks in an urban area. Table 5 details these coordinated tasks. The simulation tasks mainly include formation aggregation and tracking moving targets. In formation aggregation, each UAV needs to generate two trajectories: firstly, to cross the middle point at the expected speed, and then to arrive at the aggression position on time. Next, the UAVs follow the virtual moving targets to finish the tasks, including regulating velocity, changing formation, circling the building etc.

Table 5.
Main tasks of the simulation

Time (s) Main cooperative tasks

0∼91 Take off from individual starting points, avoid the buildings and cross a common middle point (19.9646, 29.179, 15) (m) sequentially at a certain speed of (0.2824, 0.2824, 1) (m/s). Then aggregate a triangle formation as Figure 6(a) at the height of 20m with the forward speed of 1m/s in horizontal plane

91∼161 Accelerate to 1.5m/s, and change to the linear formation as Figure 6(b). Then circle around the central building

161∼215 Accelerate to 3m/s, and descend to 15m in height. Then turn right and decelerate to 2m/s. Turn right and the height changes along with the cosine function h=15+2.5cos[π(t-190)/10](m)

Time (s)	Main cooperative tasks
0∼91	Take off from individual starting points, avoid the buildings and cross a common middle point (19.9646, 29.179, 15) (m) sequentially at a certain speed of (0.2824, 0.2824, 1) (m/s). Then aggregate a triangle formation as Figure 6(a) at the height of 20m with the forward speed of 1m/s in horizontal plane
91∼161	Accelerate to 1.5m/s, and change to the linear formation as Figure 6(b). Then circle around the central building
161∼215	Accelerate to 3m/s, and descend to 15m in height. Then turn right and decelerate to 2m/s. Turn right and the height changes along with the cosine function h=15+2.5cos[π(t-190)/10](m)

For the simulation to have a definite practical meaning, a published kinematics and dynamics model of a quadrotor UAV [25] is selected in order to execute 4D trajectories. As the safe separation between UAVs is 1m, the radius of the PAZ is set as R_safe=1.5m, considering the tracking errors. The sampling time for DRHO is 0.5s. To visualize the simulation results, a 3D urban scenario with five UAVs are designed by the MATLAB Virtual Reality Toolbox, as shown in Figure 7, while these buildings are treated as cuboids and cones, as shown in Figure 8, for obstacle avoidance. Figure 8 also shows the spatial tracking results of the 4D trajectories provided by I-tau-G-DRHO. A video of this simulation can be found at https://www.youtube.com/watch?v=HDbNXdvlG1g.

Figure 6.

Formation shapes of five UAVs: (a) triangle formation and (b) linear formation

Figure 7.

Visualized 3D urban scenario and UAVs

Figure 8.

Shapes of buildings for collision detection and tracking results of trajectories given by I-tau-G-DRHO. The solid points are the starting positions of each UAV and the dotted rectangles are zoomed-in to show the formation more clearly.

The existing decentralized path-planning methods for a multi-UAV system include the artificial potential field [26], the consensus-based algorithm [27], decentralized overlapping control [28], DMPC [29] and so on. With the advantages of low computation load, good flexibility and strong capability when dealing with environmental uncertainty, DMPC has been widely studied in relation to the applications of multi-UAVs, such as conflict-free trajectory planning [29], formation flight control [30], encirclement of a target [31] etc. The comparison with DMPC can fully verify the validity and performance of our I-tau-G-DRHO method. In the simulation executed by DMPC, the prediction horizon is N_p=5, the control horizon is N_c=3 and the receding period is 0.5s. All of the simulations are designed by MATLAB/Simulink R2013a and finished with the use of a laptop with a 2.6GHz Core i5-3230M CPU and 4GB of RAM.

5.1 Simulation results of formation aggregation

Formation aggregation can examine the capabilities of conflict resolution, obstacle avoidance and the 4D guidance capacity of positions and velocities. Table 6 shows the performance during the aggregation process, in which ${\bar{N}}_{r o}$ refers to the average number of receding optimization, while ${\bar{ε}}_{n p o s}$ and ${\bar{ε}}_{nvel}$ represent the average normalized position and velocity errors, respectively. The I-tau-G-DRHO method repeatedly optimizes three times on average, as the UAVs do not establish communication links at the start time. Nevertheless, the ${\bar{N}}_{r o}$ of DMPC is 182 because DMPC is unable to generate a whole trajectory in relation to the target and continually depends on receding optimization. For ${\bar{ε}}_{n p o s}$ and ${\bar{ε}}_{n v e l}$ , the two errors of I-tau-G-DRHO are well balanced and mainly come from the tracking of UAVs. For DMPC, however, it is hard to adjust the weights in order to balance different objectives; more importantly, DMPC is helpless with regard to the condition that the orientation of goal velocity is oppositional to the direction of the position gap, such as in the case of the aggregation task of the UAV4.

Table 6.
Mean receding optimization number, mean position and velocity errors during formation aggregation

Method ${\bar{N}}_{r o}$ ${\bar{ε}}_{n p o s} (m)$ ${\bar{ε}}_{n v e l} (m/s)$

I-tau-G-DRHO 3 0.025 0.040

DMPC 182 0.022 0.875

Method	${\bar{N}}_{r o}$	${\bar{ε}}_{n p o s} (m)$	${\bar{ε}}_{n v e l} (m/s)$
I-tau-G-DRHO	3	0.025	0.040
DMPC	182	0.022	0.875

The conflicts between the most dangerous UAV and others in the aggregation process are shown in Figure 9. As the most dangerous vehicle of the two methods is different (UAV5 of I-tau-G-DRHO and UAV1 of DMPC), hence the distance curves of the two subfigures change in a different way. According to this figure, I-tau-G-DRHO safely guides UAVs to their arrival at the targets. As for trajectories generated by DMPC, UAV1 collides with its neighbour near the middle point. Considering the distance between UAVs, the triangle formation of I-tau-G-DRHO is aggregated exactly at the arrival time T=91s. However, DMPC guides the UAVs that are approaching the aggregation positions in advance and hovering there. Hence, the goal velocities are not achieved by DMPC.

Figure 9.

Distance between the most dangerous UAV and others in formation aggregation: (a) UAV5 of I-tau-G-DRHO and (b) UAV1 of DMPC

Figure 10.

Tracking results of UAV1 in formation aggregation

Figure 10 shows the 3D tracking results of the trajectories in formation aggregation. The tracking of the trajectory guided by I-tau-G-DRHO is much smoother than that of DMPC.

Collisions with buildings, which are shown in Figure 11, are identical to the results in Figure 9. The I-tau-G-DRHO method ensured the safe flight, while the UAV4 guided by DMPC collided with buildings.

Figure 11.

Horizontal distance between UAVs and buildings in formation aggregation: (a) I-tau-G-DRHO and (b) DMPC

5.2 Simulation results for tracking moving targets

The moving target tracking paths of the UAV group are shown in Figure 8. As all UAVs keep a safe distance from neighbours and buildings in these tasks, the safety results are not shown in this paper. Figure 12 shows the mean cost ratio between the 4D trajectories generated by I-tau-G-DRHO and DMPC in tracking moving targets. The cost function is shown in Eq. (19). On average, the cost of trajectories generated by I-tau-G-DRHO is 16.8% lower than that of DMPC. For the whole simulation tasks, including formation aggregation, the cost of trajectories generated by I-tau-G-DRHO is 18.1% lower.

Figure 12.

Mean cost ratio between the trajectories generated by I-tau-G-DRHO and DMPC in tracking moving targets

Table 7 shows the mean normalized position error ${\bar{ε}}_{n p o s}$ and velocity error ${\bar{ε}}_{n v e l}$ during formation maintenance. The tracking errors of I-tau-G-DRHO are a bit smaller. Furthermore, the errors of triangle formation and formation changing are larger than those of linear formation. Above all, I-tau-G-DRHO achieves more minor target tracking errors with lower cost trajectories than DMPC.

Table 7.

Mean position and velocity errors during formation maintenance

Formation	Method	${\bar{ε}}_{n p o s} (m)$	${\bar{ε}}_{n v e l} (m/s)$
Triangle	I-tau-G-DRHO	0.039	0.107
Triangle	DMPC	0.045	0.185
Formation changing	I-tau-G-DRHO	0.041	0.136
Formation changing	DMPC	0.042	0.145
Line	I-tau-G-DRHO	0.030	0.059
Line	DMPC	0.030	0.057

5.3 Performance of the whole simulation

The statistical data of the performance during the whole simulation process is shown in Table 8. The time consumption ${\bar{t}}_{l p}$ and the mean local optimization number ${\bar{N}}_{l o}$ of I-tau-G-DRHO are less than those for DMPC, on average. It can be found that the time cost of larger UAV numbers is less than that for one or two UAVs. This phenomenon is related to our tasks. For the tasks executed by one or two UAVs in formation aggregation, communication is just about established and there are many potential conflicts, such that the problems are harder to solve. Meanwhile, the cooperative tasks with five UAVs are much safer and occupy more than 90% of all local problems. For the mean position and velocity errors, the results of I-tau-G-DRHO are a little smaller than those of DMPC with the same control laws for position, velocity and attitudes.

Table 8.
Performance comparison of the whole simulation (the notation ‘-’ means these data cannot be produced)

Performance I-tau-G-PSO DMPC

${\bar{t}}_{l p}$ (s) 1 UAV 0.345 1.389

2 UAVs - 1.336

3 UAVs 0.265 1.201

4 UAVs 0.031 1.208

5 UAVs 0.020 0.702

${\bar{N}}_{l o}$ 1 UAV 3 3.411

2 UAVs - 4.590

3 UAVs 3 4.174

4 UAVs 3 4.164

5 UAVs 3 3.226

${\bar{ε}}_{n p o s}$ (m) 0.035 0.037

${\bar{ε}}_{n v e l}$ (m/s) 0.068 0.075

Performance	I-tau-G-PSO	DMPC
${\bar{t}}_{l p}$ (s)	1 UAV	0.345	1.389
2 UAVs	-	1.336
3 UAVs	0.265	1.201
4 UAVs	0.031	1.208
5 UAVs	0.020	0.702
${\bar{N}}_{l o}$	1 UAV	3	3.411
2 UAVs	-	4.590
3 UAVs	3	4.174
4 UAVs	3	4.164
5 UAVs	3	3.226
${\bar{ε}}_{n p o s}$ (m)	0.035	0.037
${\bar{ε}}_{n v e l}$ (m/s)	0.068	0.075

The simulation results and the analysis above have presented comprehensive demonstrations of the performance of I-tau-G-DRHO. This proposed method obviously performs better than DMPC in the 4D position and velocity guidance, as well as safety of the task execution. Meanwhile, I-tau-G-DRHO can generate 4D trajectories with a visibly lower cost in less time.

6 Conclusion

In this paper, we propose an improved tau-G strategy based on the bio-inspired general tau theory, as well as design a decentralized 4D trajectory generation method for multi-UAVs (I-tau-G-DRHO) utilizing the proposed strategy. The main work of this paper is stated as follows:

Based on general tau theory, an improved tau-G strategy is proposed. The 4D trajectory generated by this strategy can guide both position and velocity towards the desired values at the arrival time. Furthermore, our improved tau-G strategy plays well with the initially expanding position gap (τ_0x>0), which cannot be achieved by any existing tau strategies. This improved strategy will enlarge the applications of the bio-inspired tau theory in UAV fields.

Using the improved tau-G strategy, a decentralized 4D trajectory generation method is designed for multi-UAVs. The parameters of the 4D trajectories are repeatedly corrected by a receding horizon optimization approach. To cope with environmental uncertainty and communication limitation, an event-driven mechanism is added to the receding optimization.

To demonstrate the effectiveness of the improved tau-G strategy and the I-tau-G-DRHO method, a comprehensive simulation and a visualized scenario are designed based on the MATLAB Virtual Reality Toolbox. The computation load, conflict resolution and the ability of tracking a moving target have been validated by the simulation. The comparison with DMPC shows that I-tau-G-DRHO can efficiently provide safer and lower-cost 4D trajectories.

Our future work will concentrate on performing more tests for the performance of I-tau-G-DRHO. Moreover, new tau guidance strategies should be designed in order to generate 4D trajectories with better features, which can extend the applications of general tau theory.

Footnotes

Acknowledgements

This work is supported by the National Natural Science Foundation of China under grant no. 61004066 and the Zhejiang Provincial Natural Science Foundation of China under grant no. LY15F030005. Our thanks also go to Mr. Corke from the Queensland University of Technology for his dynamics model of the quadrotor.

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Decentralized 4D Trajectory Generation for UAVs Based on Improved Intrinsic Tau Guidance Strategy

Abstract

Keywords

1 Introduction

2 Problem Statement

3.1 Overview of existing tau guidance strategies

5 Tau-H Strategy

3.2 Improved tau-G strategy

3.2.1 Improvement of tau-G strategy

4.1 4D trajectory guided by improved tau-G strategy

4.2.1 Decentralized optimization problem

Table 6. Mean receding optimization number, mean position and velocity errors during formation aggregation Method N ̄ r o ε ̄ n p o s ( m ) ε ̄ n v e l ( m/s ) I-tau-G-DRHO 3 0.025 0.040 DMPC 182 0.022 0.875

Footnotes

Acknowledgements

References

Table 6.
Mean receding optimization number, mean position and velocity errors during formation aggregation

Method ${\bar{N}}_{r o}$ ${\bar{ε}}_{n p o s} (m)$ ${\bar{ε}}_{n v e l} (m/s)$

I-tau-G-DRHO 3 0.025 0.040

DMPC 182 0.022 0.875