An enlarged polygon method without binary variables for obstacle avoidance trajectory optimization

Rouhe ZHANG, Zihan XIE, Changzhu WEI, Naigang CUI

School of Astronautics, Harbin Institute of Technology, Harbin 150090, China

KEYWORDS Enlarged polygon method;Hypersonic glide vehicles;Mixed-integer programming;No-fly zone;Obstacle avoidance;Trajectory optimization;Unmanned aerial vehicles

Abstract In this article,an Enlarged Polygon/Polyhedron(ELP)method without binary variables is proposed to represent the Convex Polygonal/Polyhedral Obstacle Avoidance(CPOA)constraints in trajectory optimization.First,the equivalent condition of a point outside the convex set is given and proved rigorously.Then, the ELP condition describing the CPOA constraints equivalently is given without introducing binary variables, and its geometric meaning is explained.Finally, the ELP method is used to transform the CPOA trajectory optimization problem into an optimal control problem without binary variables.The effectiveness and validity of ELP method are demonstrated through simulations with both simple linear dynamic model (unmanned aerial vehicle)and complex nonlinear dynamic model (hypersonic glide vehicle).Comparison indicates the computational time of ELP method is only 1%-20%of that of the traditional Mixed-Integer Programming (MIP) method.

1.Introduction

Obstacle avoidance / No-Fly Zone (NFZ) constraints are a kind of common constraints, which exist widely in the path planning / trajectory design of various aircraft.1–6For example, Unmanned Aerial Vehicle (UAV) avoids buildings,Hypersonic Glide Vehicle (HGV) avoids no-fly zones, cruise missile avoids complex terrains, etc.In this article, these areas where the aircraft is not allowed to pass are collectively referred to as obstacles, and the corresponding constraints are referred to as obstacle avoidance constraints.

In the past few decades, many effective methods have emerged to solve the problem of obstacle avoidance path planning / trajectory design in the field of UAVs and robots, such as A* method, V-graph method and Rapidly-exploring Random Trees (RRT) method.7–9These methods usually do not consider dynamic model,or convert the model and constraints into simple geometric constraints, and then obtain geometric paths connecting the start and terminal points,meeting obstacle avoidance constraints and other constraints through different ideas.These methods are simple in principle and computationally efficient, but are usually difficult to describe the motion process comprehensively and accurately.

Another way to solve the trajectory design problem is to establish a trajectory optimization model including dynamic equation, performance index and various constraints from the perspective of optimal control, and then solve the optimal trajectory through analytical or numerical methods.10,11With the rapid progress of numerical optimization, numerical method has become the mainstream of trajectory optimization.12The widely used algorithms include pseudospectral method,13,14convex program,15,16heuristic method,17,18etc.However, when the trajectory optimization problem is highly nonlinear/non-convex and with complex constraints, it is still challenging to deal with the model and constraints or transform the origin problem to a problem which can be solved efficiently.

For the convenience of modeling and solving trajectory optimization problems considering obstacle avoidance constraints,many researches represented obstacles as circles,cylinders, spheres or cones.Zhao and Zhou19expressed NFZ as a circle, and used multi-phase Gauss pseudospectral method to solve the entry trajectory optimization problem with multiple constraints.Liu et al.3used linearization technique to deal with circular NFZ, and transformed entry trajectory optimization problem into a Second-Order Cone Programming (SOCP)problem through various convexification methods.Yao et al.20used Interfered Fluid Dynamical System (IFDS) and receding horizon technology to solve UAV trajectory optimization problems with cylindrical, conical and spherical obstacles.Zhang et al.21used sequential convex optimization to solve the UAV trajectory design problem with spherical obstacles,and proved the convergence of the method.Jiang and Liu22transformed the non-convex and highly nonlinear UAV three-dimensional trajectory planning problem into a SOCP problem while considering cylindrical and spherical obstacles.Szmuk et al.23used lossless convexification and sequential convexification technique to solve the on-board trajectory optimization of quad-rotor considering circular obstacles.Dueri et al.24considered inter-sample obstacle avoidance constraints while solving the cylindrical obstacle avoidance trajectory using convex optimization.Qu et al.25used a hybrid grey wolf optimizer algorithm to study the UAV trajectory design problem with cylindrical obstacles.Bai et al.26used a cone to replace irregular obstacles on planet, and solved the minimum-fuel trajectory of the powered-descent phase.

These researches used different methods to solve the trajectory optimization problem with circular/cylindrical/spherical/conical obstacle avoidance constraints and achieved good results.However,circle/cylinder/sphere/cone can only describe some special obstacles.Since shapes of real-world obstacles are various, they have strong limitations to be applied to general scenes.

In actual flying environment, the shape of obstacle may be irregular,as shown in Fig.1.Hence,approximating real-world obstacles with polygon/polyhedron seems to be more accurate than approximating with circle/cylinder/sphere/cone.Since a general polygon/polyhedron can be represented as multiple convex polygons/polyhedrons, only Convex Polygonal/Polyhedral Obstacle(CPO)is considered in this article.When polygon/polyhedron is mentioned hereinafter, we refer to convex polygon/polyhedron in particular.

Fig.1 Actual flight environment.27

Currently, the mainstream method to describe CPOA constraints is the Big M method.28Big M method introduces binary variables to represent CPOA constraints, and transforms the corresponding trajectory design problem into a MIP problem.Richards et al.29introduced binary variables to represent obstacle and collision avoidance constraints, and transformed the fuel optimal trajectory optimization problem of spacecraft into a Mixed-Integer Linear Programming (MILP) problem.Earl and D’Andrea30considered obstacle avoidance constraints of robots, and transformed the cooperative control problem of multiple robots into a MILP problem.Ademoye and Davari31solved the optimal maneuvering trajectories of multiple unmanned systems using MILP method considering collision and obstacle avoidance constraints.Maia and Galva?o32combined Model Predictive Control (MPC) and MIP, and gave an additional constraint to satisfy intersample avoidance constraints.Richards and Turnbull33considered inter-sample avoidance constraints, and reduced the number of additional constraints of MIP model through the geometric relationship.Afonso et al.34presented a method to reduce the number of binary variables in MIP while considering the inter-sample avoidance.Radmanesh and Kumar35considered moving obstacles and collision avoidance constraints,and adopted a fast dynamic MILP method to solve the trajectory optimization problem of multi-UAV formation flight.Blackmore et al.36used MIP method to describe CPOA constraints when solving obstacle avoidance trajectory planning problem with chance constraints.Chai et al.37used MIP method to solve the path-length optimal trajectory optimization problem of UAV with chance constraints.

It is effective to use MIP method to solve the trajectory optimization problem with obstacle avoidance constraints,but the introduction of binary variables may lead to a significant increase in the difficulty of solving the problem.As the complexity and scale of the problem increase, the computational burden and computational time of MIP method may be unbearable.28,38.

To solve the above problems, this article presents an ELP method which can equivalently represent CPOA constraints without introducing binary variables to avoid transforming the obstacle avoidance trajectory optimization problem into a MIP problem.The equivalent condition of a point outside the convex set is given first.Then, ELP condition describing CPOA constraints is given, and its geometric meaning is explained.The ELP method is simple in principle and clear in geometric meaning.Using ELP method,the trajectory optimization problem with CPOA constraints can be transformed into an optimal control problem without binary variables,which can be solved by commonly used optimal control algorithms.

The main contributions of this article are given as follows.(A) For a general convex set, the equivalent condition of a point outside the convex set is given and proved rigorously.(B) Without introducing binary variables, an ELP condition is given to describe CPOA constraints equivalently.ELP condition is concise in form and clear in geometric meaning.(C)Obstacle avoidance trajectory optimization model without binary variables is established by ELP method, and the threedimensional and two-dimensional ELP conditions are verified through different trajectory optimization problems.

The structure of this article is shown as follows.Section 1 presents general trajectory optimization model and CPOA constraints.Section 2 gives the equivalent condition that a point is outside the convex set.Section 3 introduces ELP condition and its geometric meaning,and establishes the trajectory optimization model.Simulation uses trajectory optimization problems of UAV and HGV respectively to verify ELP condition.

2.Problem formulation

In this section, the general trajectory optimization model is established, and the description of CPOA constraints is given.

2.1.General trajectory optimization model

The trajectory optimization problem of aircraft is essentially a kind of optimal control problem with multiple constraints.In the modeling process, dynamic equations, path constraints,initial and final state constraints and other conditions need to be considered.

The dynamic equation of aircraft can be expressed as

where x is usually the state variable representing physical quantities such as the position and velocity of the aircraft, u is the control variable, t is time.

There are following constraints on state variable and control variable:

2.2.CPOA constraints

The CPO can be expressed as

where Rpis a CPO, apiand bpiare the coefficients of the i th edge/face, and npis the number of the edges/faces.

Without loss of generality, the obstacle avoidance constraint can be expressed as

where p is the position vector of the aircraft, Riis the set of points inside the i th obstacle, and m is the number of obstacles.

For a CPO,the position of the aircraft only needs to be outside any edge/face of the obstacle to satisfy the CPOA constraint.That is, there exists i=1,2,...,npsuch that

By introducing positive number M and binary variable bi,the expression of CPOA is established.We note that due to the introduction of binary variables in the CPOA constraint Eq.(9), the corresponding trajectory optimization problem is transformed into a MIP problem.This increases the complexity of the problem and the computational time.

Aiming at the above problems,we firstly give the equivalent condition that the point is outside the convex set,and then give an ELP condition describing CPOA constraints equivalently without introducing binary variables.

3.Equivalent condition of point outside convex set

4.ELP method

This section gives ELP method describing CPOA constraints equivalently.Starting from Theorem 1,we give ELP condition and explain its geometric meaning.Then, the trajectory optimization model is established using ELP method.The ELP method does not introduce binary variables and avoids transforming the original problem into a MIP problem.

4.1.ELP condition

Theorem 2.For a given convex polygon/polyhedron S and any point P,select a point O in S that is not on the edge/face of S and does not coincide with P.The sufficient and necessary condition that P is outside S is.

We call Eq.(18) the ELP condition, and call the method that uses ELP condition to express the CPOA constraints the ELP method.

4.2.Geometric meaning

In order to understand Theorem 2 intuitively, we explain the meaning of ELP condition from the geometric relationship.

Taking a convex polyhedron as an example, the geometric meaning of ELP condition is shown in Fig.2.Select a point O inside S that is not on the face of S and does not coincide with P.Let O be the zoom center.We call the convex polyhedron enlarged from S as ELP.The point P outside S is equivalent to the existence of a unique ELP and the corresponding magnification ρ, so that P is exactly on the face of ELP enlarged from S.We call this ELP the Exact ELP (EELP).

It can be seen intuitively that for any point in space, when the magnification ρ is large enough, the point must be inside the ELP.However,the condition that points inside ELP cannot guarantee that the point is outside the obstacle.If P is exactly on the face of an ELP, then P must be outside the obstacle.

For the convenience of understanding,we organize Eq.(18)into the following form:

Similarly, for the convex polygon obstacles, the geometric meaning of ELP condition is shown in Fig.3.

Furthermore, from the perspective of ELP condition, we can think the circular obstacles as regular polygons with infinite edges.Then the circular obstacles avoidance constraints can be expressed as

The above ideas and conclusions are also applicable to spheric obstacles.

4.3.Trajectory optimization model without binary variables

According to ELP condition given above, this subsection formulates the trajectory optimization model without binary variables.

Fig.3 Geometric meaning of two-dimensional ELP condition.

Therefore, the trajectory optimization problem with obstacle avoidance constraints can be established as the following optimal control problem without binary variables:

where ～h（x,～u,t) is the path constraint describing the obstacle avoidance constraints using ELP condition, and other constraints in ～h（x,～u,t) are the same as h（x,u,t).

The only difference between P-O and P-E in this article is the description of the CPOA constraints.Because the equivalence of Eq.(9) and Eq.(18) has been proved by Theorem 2,it is obvious that P-O and P-E are equivalent.

Fig.2 Geometric meaning of three-dimensional ELP condition.

Table 1 Parameters of GPOPS mesh.

Table 2 Constraints of UAV trajectory optimization.

So far, we have established the trajectory optimization problem model considering CPOA constraints without binary variables.Then, the problem Eq.(28) can be solved by the commonly used optimal control algorithm.

5.Simulations

In this section, ELP condition in three-dimensional and twodimensional form are demonstrated, respectively.Section 5.1 takes UAV trajectory optimization as an example of the simple linear system and analyzes the three-dimensional ELP condition.Section 5.1.1 verifies the effectiveness of ELP method,Section 5.1.2 tests the validity and accuracy of ELP conditions,and Section 5.1.3 compares ELP method with MIP method.Then, Section 5.2 displays a simple application of the twodimensional ELP condition for complex nonlinear system,taking the entry trajectory optimization with polygonal NFZs as an example.

The CPU used in the simulation is Intel Core i5-11300H(3.10 GHz).ELP trajectory is solved by the software GPOPS II using the solver SNOPT, and MIP trajectory is solved by Gurobi 9.1.2.Other parameters of the structure mesh are shown in Table 1.

5.1.Simple linear system:UAV trajectory optimization in urban environment

This subsection verifies the effectiveness and validity of ELP condition by simple linear system.We consider UAV obstacle avoidance trajectory optimization with linear dynamic model as an example.

Table 3 Constraints of UAV trajectory optimization.

Fig.4 Flight trajectory of Scenarios 1–3.

In this subsection, the obstacles for UAV are polyhedron and the three-dimensional ELP condition is used.In addition,ELP method is compared with the traditional MIP method.

The linear dynamic model of UAV21is

where p=［px,py,pz]Tis position,v=［vx,vy,vz]Tis velocity and a=［ax,ay,az]Tis acceleration.

J=∫｜｜v｜｜dt is taken as the performance index.In this way,the UAV flies through the shortest distance trajectory to conserve air traffic resources as much as possible.

To limit the change rate of the acceleration, ˙a is chosen to be the control variable.The constraints are shown in Table 2.

Table 3 shows four scenarios with different number of obstacles.Other conditions and parameters used in these scenarios are the same.To satisfy the inter-sample obstacle avoidance constraint, we set ρs=1.1, that is, ELP condition needs to meet ρ>ρs=1.1.The number of discrete nodes in each scenario is set to N=81.

5.1.1.Effectiveness verification

This subsection verifies whether ELP method can guarantee UAV satisfying CPOA constraints.According to the given scenarios,ELP method is used to solve the optimal trajectories in different scenarios, so as to verify the effectiveness of ELP method.

Fig.4 shows the flight trajectories of Scenarios 1–3.Fig.5 shows the flight trajectory and ground track of Scenario 4.In different scenarios, the trajectories generated by ELP method avoid obstacles successfully and reach the terminal position accurately.During the entire flight process, the trajectories do not collide with obstacles, which satisfies the inter-sample obstacle avoidance constraints.

Fig.7 Acceleration change rate.

Fig.5 Trajectory and ground track of Scenario 4.

Fig.6 Curves of velocity and acceleration in Scenario 3.

In addition to meeting the obstacle avoidance constraints,we also pay attention to whether state variable and control variable satisfy other constraints.Taking Scenario 3 as an example,Fig.6 shows the velocity and acceleration curves during the flight.It can be seen that the UAV starts to move from a stationary state, and the terminal velocity is accurately reduced to zero.During the flight,the velocity and acceleration in each direction are always kept within the given range, and the change processes are continuous and stable.The change process of acceleration change rate,which is the actual control variable,is displayed in Fig.7.The acceleration change rate in each direction changes within the boundary of the control during the whole flight, and reaches the boundary for some periods time.

Therefore, the generated obstacle avoidance trajectory satisfies the capability of UAV, which is beneficial to realization of control system.

Simulations in the above scenarios show that the trajectory obtained by ELP method can ensure UAV successfully avoiding obstacles and meeting other constraints during the flight.

In order to further verify the validity and effectiveness of ELP method, we also need to check whether ELP conditions are accurately satisfied.

5.1.2.ELP condition test

To illustrate the unique characteristic of ELP method, we check and analyze ELP condition in more detail.The variables tested include magnification ρ and relaxation variable ci.According to ELP condition,magnification ρ should be always greater than ρs.Relaxation variable cishould always be less than or equal to 0,and product Пcishould always be equal to 0.

Fig.8 shows the magnification curve of each obstacle in Scenarios 1–4.It can be seen that during the flight, magnification of the ELP relative to the original obstacle changes dynamically with the position of UAV.When UAV approaches obstacle gradually, the corresponding magnification decreases.When UAV is away from obstacle, the magnification increases gradually.The above process shows that magnification of the ELP is determined by the position of UAV relative to obstacle,which is consistent with the geometric meaning of ELP condition.

At the same time,the magnification ρ of each ELP is always greater than ρsduring the flight,which ensures that the trajectory satisfies the inter-sample obstacle avoidance constraints.

In addition, we take Scenario 2 as an example to analyze the change process of relaxation variables, as shown in Fig.9.Among them, Figs.9 (a)-(c) are relaxation variables ci,and Fig.9(d)is product.It can be seen from Figs.9(a)-(c)that each relaxation variable ciis less than or equal to 0,and at least one ci=0 holds.

Fig.8 Curves of magnification.

At the same time,it can be seen from Fig.9(d)that the product of relaxation variables always approaches to 0 during the flight, and the maximum absolute value is about 1.5×10-4.Therefore, within a sufficiently high computational accuracy,the product of relaxation variables can always be considered to be 0.We noticed that some of the time the value of product is less than 0.This phenomenon is caused by the calculation error in numerical calculation process.

Similarly, Figs.10 (a) and (b) show the product of relaxation variables of Scenarios 3 4, respectively.The products in these two scenarios approach to 0 within a high accuracy in whole process, which means that ELP condition is satisfied accurately.

The above simulation and analysis show that ELP conditions are strictly satisfied during the entire flight process.The UAV is always on the ELP, so as to ensure that flight trajectory can successfully avoid obstacles.Therefore, the obstacle avoidance trajectory optimization problem can be transformed into an optimal control problem without binary variables by ELP method,and solved by a commonly used optimal control algorithm.

5.1.3.Comparison with MIP

This subsection compares ELP method and MIP method under the conditions given earlier.For both methods,ρs=1.1.

Fig.9 Curves of relaxation variable in Scenario 2.

Fig.10 Product of relaxation variables in Scenarios 3 and 4.

Under Scenarios 1 and 2,the results of the two methods are shown in Table 4.The flight trajectory comparisons are shown in Fig.11.

It can be seen from Fig.11 that flight trajectories generated by the two methods can avoid obstacles.The trajectories basically coincide, with almost the same performance indicators.

However, there is a huge difference in the computational time between the two methods.In Scenarios 1 and 2,the computational time of MIP method reaches 5.0/126.0 times that of ELP method, respectively.This shows that under the same conditions, the computational efficiency of ELP method has obvious advantages.

Under Scenario 3 and Scenario 4, the results of the two methods are shown in Table 5.It should be noted that ELP method can obtain the optimal trajectory faster when N=81.However, due to the increase of the complexity,MIP method fails to obtain a solution within 1000 s.Therefore, for MIP method, only the results with N=41 are given.It is obvious from Table 5 that, although ELP method uses more nodes, the computational time is still much lower than that of MIP method.

Taking Scenario 3 as an example,trajectory comparison of the two methods is shown in Fig.12.It can be seen that although the trajectory obtained by MIP method with N=41 can satisfy the obstacle avoidance constraint at each discrete node,the connections between the discrete points pass through the obstacles and cannot satisfy the inter-sample obstacle avoidance constraints.

Therefore, the trajectory of MIP method when N=41 is actually infeasible, and more nodes are needed to satisfy the inter-sample obstacle avoidance constraints.However, the computational time of MIP method has reached 364.7 s, and it is inefficient to increase the nodes.

In contrast, the trajectory of ELP method with N=81 avoids obstacles throughout the flight, and the computationaltime is only 4.8 s,which is about 1.3%of that of MIP method with N=41.To sum up, the computational time of MIP method is much longer than that of ELP method while the number of nodes is only half of that of the latter.

Table 4 Results of MIP and ELP method in Scenarios 1 and 2.

Table 5 Results of MIP and ELP method in Scenarios 3 and 4.

Therefore, compared with MIP method, ELP method has higher computational efficiency and is more applicable to relatively complex scenarios.

5.2.Complex nonlinear system: HGV trajectory optimization with NFZs

This subsection presents an application of ELP condition for complex nonlinear system.HGV trajectory optimization considering polygonal NFZ constraints is taken as an example.The NFZ can be regarded as a flat polygon so that twodimensional ELP condition is applied.

Through brief simulation, analysis and comparison, ELP condition is verified in HGV trajectory optimization with complex nonlinear dynamic model.

The dynamics model of HGV without considering the earth’s rotation3is

where v,γ,ψ are the velocity, flight-path angle and heading angle, respectively.r,θ,φ are the distance from the vehicle to the center of the earth, longitude and latitude, respectively.α,σ are the Angle of Attack (AOA) and bank angle, respectively.D,L are the aerodynamic drag and lift, respectively.

Fig.11 Comparison of ELP and MIP trajectories in Scenarios 1 and 2.

Fig.12 Comparison of ELP and MIP trajectories in Scenario 3.

Fig.13 AOA profile.

The bank angle rate is used as the actual control variable and is limited in ［-5,5]o（)/s.In this way, σ becomes a state variable.A fixed AOA profile is used, as shown in Fig.13.

Consider a scenario with two rectangular NFZs.The performance index is set to J=∫｜˙σ｜dt, which is conducive to make the curve of bank angle more stable.In this way, the total energy consumed by the control during the entry process can be considered to be minimized.The initial longitude and latitude are 0o, attitude is 70 km, velocity is 7000 m/s, and flight-path angle is 0o.The terminal longitude and latitude are 85oand -15orespectively, attitude is 25 km, and velocity is 2000 m/s.The vehicle model used is CAV-H.Other conditions are shown in Table 6.

Due to the complex model of the reentry trajectory optimization, it is difficult to solve it by MIP method.Therefore,this subsection compares ELP method with the Circle Approximation Method(CAM).39The idea of CAM is to use a circular NFZ to approximately replace the polygonal NFZ.There is no fixed criterion for the selection of circular NFZ(cNFZ),but it is generally necessary to ensure that the original NFZ(oNFZ) is covered by cNFZ.As a comparison method, the CAM method in this article replaces the oNFZ with the circumcircle of the rectangular oNFZ.The center and radius of the cNFZ are shown in Table 6.

The ground tracks obtained by ELP and CAM method are shown in Fig.14.It can be seen that taking a trajectory passing through two NFZs as the initial guess,trajectories meeting the oNFZ constraints can be obtained by both ELP and CAM method.

Table 6 Condition of HGV trajectory optimization.

Fig.15 Bank angle and its change rate history.

It can be seen from the ground tracks that the change trends of ELP and CAM trajectory are relatively close.The difference is that CAM trajectory avoids the cNFZs, while ELP trajectory cuts across the cNFZs while avoiding the oNFZs.

It can be seen from Table 6 that the performance index of ELP trajectory is significantly better than that of CAM trajectory,with a reduction of 19%.This is because cNFZ generates an additional NFZ while covering oNFZ, which increases the area of NFZ and reduces the feasible region of the optimal control problem.Therefore, the trajectory optimization problem described by ELP method obtains a larger feasible region.As a result, ELP trajectory is closer to oNFZs, thus obtaining better performance index.

The control variable and bank angle history of ELP and CAM are displayed in Fig.15.These curves of the two methods are closer, which corresponds to the ground tracks.

It can be seen from Fig.15(a) that the amplitude of bank angle change rate of CAM is larger in the first 50 s, while the bank angle change process of ELP is relatively slow.This is because CAM trajectory needs to avoid a larger NFZ, so the aircraft will increase the amplitude of bank angle earlier to adjust flight direction.

Another significant difference occurred after 1300 s.At this time, the bank angle of ELP trajectory increases slowly, while the bank angle of CAM trajectory remains at a small level,and increases rapidly in the final stage with a larger change rate.This is because the lateral positions of ELP and CAM trajectories relative to the target are different.At this time, CAM trajectory is almost aimed at the target, while ELP trajectory needs a continuous turning process to reach the target.

Fig.16 Altitude history.

The altitude history is shown in Fig.16.It can be seen that ELP trajectory is more stable than CAM trajectory.This is because the feasible region of ELP trajectory is larger, and it is easier to keep the trajectory more stable.

The path constraints of ELP trajectories are shown in Fig.17.It is obvious that all of the path constraints are satisfied during the whole flight.This means that the trajectory can satisfy common path constraints when considering the PNFZ constraints described by ELP condition.

According to the unique characteristic of ELP method,Fig.18 shows the change process of relaxation variables of ELP condition.For each oNFZ,all of the relaxation variables are less than or equal to 0 and at least one relaxation variable equals to 0 throughout the flight.This shows that ELP condition is strictly satisfied.

Fig.17 Path constraints of ELP trajectory.

Fig.18 Curves of relaxation variable.

Therefore, the two-dimensional ELP condition is feasible and effective for HGV trajectory optimization,which has complex nonlinear dynamic model.

6.Conclusions

In this article,the ELP method is used to transform the trajectory optimization problem considering CPOA constraints into an optimal control problem without binary variables.Traditional method formulates this kind of problem into a MIP problem, but the introduction of binary variables may cause huge computational burden in complex scenarios.To remove binary variables, an equivalent condition of a point outside a general convex set is given firstly, and then ELP condition is proposed.In this way,CPOA trajectory optimization problem can be solved by common optimal control algorithms.Simulations with both simple linear dynamic model(unmanned aerial vehicle) and complex nonlinear dynamic model (hypersonic glide vehicle) demonstrate the validity and feasibility of ELP method.Computational time of ELP method is about 1%-20% of that of MIP method, which indicates significantly higher computational efficiency of ELP method.In future research,general polygon/polyhedron obstacle avoidance constraints and more complex dynamic models may be taken into account using ELP method.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No.52232014) and the National Natural Science Foundation of China Joint Fund (No.U2241215).