Zhide ZHANG,Zhengjie WANG,Jin YU

School of Mechatronical Engineering,Beijing Institute of Technology,Beijing 100081,China

KEYWORDS Collision avoidance;Dynamic system;Guidance;Modulation;Unmanned Aerial Vehicles(UAVs)

Abstract In this paper,a novel real-time obstacle avoidance method based on Dynamic System(DS),is proposed.The proposed method ensures the impenetrability of multiple convex obstacles by online modulating the original velocity field of the DS.It can be applied to perform obstacle avoidance in the state space of the DS with both autonomous and non-autonomous DS-based controllers.While realizing the obstacle avoidance,the equilibrium points of the original DS can be saved.The modulation matrix form is extended based on the earlier dynamic system modulation methods of the literature.The asymmetric modulation provided by this method allows the modulated DS to satisfy the dynamic constraints of a class of DSs.In addition,the proposed method has the inherent ability of multiple-obstacle avoidance and the direction selectivity of avoidance maneuver.Moreover,to solve the simultaneous guidance and obstacle avoidance problem,a guidance law for Unmanned Aerial Vehicles (UAVs) based on the proposed method,is designed.Finally,a numerical simulation is performed to analyze the performance of the proposed method and the obstacle avoidance guidance law.

1.Introduction

In the past decades,several developments have been proposed in the area of Unmanned Aerial Vehicles(UAVs).1In particular,the application domain of surveillance and tracking with aerial vehicles currently receives great interest from industry and consumers.However,the problems of real-time processing and high-velocity obstacle avoidance are still standing.2In this paper,we focus on the high-velocity obstacle avoidance problem within the UAV guidance task in complex environment.

The current obstacle avoidance methods can be divided into two groups;path planning and reactive motion generation approaches.Generally,path planning is able to perform obstacle avoidance in arbitrarily complex environments.Path planning approaches include the Voronoi-diagram method,3the Astar algorithm,4and other graph search based,5or sampling based6,7approaches.More advanced methods of this group have developed path planning into the high-dimensional configuration space.This is also referred to as kinodynamic motion planning.8–10However,these methods still face challenges,such as the high computation power required by the mapping and graph searching as well as the gradual convergence which degrades their performance in the high-velocity scenarios,for example.Thus,the path planning approaches are desired to be more computationally efficient for the fast guidance task of the UAV.

The reactive obstacle avoidance approaches are divided into geometric and potential field methods.These approaches include the velocity obstacle method,11the collision cone method,12the vector field histogram,13,14etc.In contrast to the path planning approaches that generate a path to be followed,the reactive obstacle avoidance approaches usually look for a desired motion of local obstacle avoidance or a modified control command according to a pre-designed strategy.They have higher computational efficiency than the path planning methods.However,they lack the way to optimize the trajectory and the guarantee of global convergence.

Several researches tackled the simultaneous guidance and obstacle avoidance problem.For instance,based on the optimal guidance theory,the obstacle avoidance guidance laws of both interception and rendezvous are presented in Ref.15,in which the form of the guidance law is similar to the Augmented Proportional Navigation (APN).However,considering the error caused by the linearization model and the remaining calculation time,this method has poor ability to deal with maneuvering obstacle avoidance scenarios.Based on the similar derivation,the problem of simultaneous guidance with fixed interception angle and multiple obstacle avoidance,is solved in Ref.16,while the same problem is also studied in Ref.17 using the RRT*algorithm.The method consists of sampling in the state space of the guidance system,and designing a heuristic function based on the optimal guidance law.

Collision avoidance approaches have also been involved in the simultaneous guidance and obstacle avoidance studies.For instance,in Ref.18,the authors aim to provide researchers with a state-of-the-art overview of various approaches for multi-UAV collision avoidance.In Ref.19,some improvements on the three-dimensional velocity obstacle algorithm are proposed.In Ref.20,a fast finite-time convergent guidance law with a nonlinear disturbance observer is proposed for the problem of UAV collision avoidance.An active disturbance rejection control guidance law is proposed in Ref.21 This law can achieve collision avoidance in the presence of sensor noise,an unknown acceleration of an obstacle,and wind disturbance.In Ref.22 a sub-optimal feedback control for spacecraft collision avoidance in proximity operations,is presented.In Ref.23,the authors propose a new method based on visibility graphs.This method is computationally efficient for online use,and it is also suitable for actual applications.

The Artificial Potential Field (APF)24is one of the most popular obstacle avoidance methods.It uses a potential function to assign a virtual attractive force to the goal,and to shape the obstacles as repulsive forces,so as to reach the target avoiding obstacles.A potential field approach allowing to track a moving target is proposed in Ref.25 In this method,the interception is guaranteed while avoiding moving obstacles.However,the local minimum26and strong chattering problems in the narrow and long channels,27are inherent in the artificial potential field method.In addition,due to the limitation of the UAV maneuverability,dynamic constraints should be considered in the obstacle avoidance control.The artificial potential field method cannot directly deal with those limitations.

The Dynamic System Modulation (DSM) method is proposed in Ref.28 This method retains the low computation of the artificial potential field method and overcomes its local minimum problem.The dynamic system velocity field is modulated using the modulation matrix,so that the penetration component of the velocity near the obstacle surface is gradually eliminated.It retains the convergence characteristics of the system brought by the original control input,while ensuring obstacle avoidance.A DSM-based obstacle avoidance method that uses the relative distance between the robot and the obstacle,instead of the geometric information of the obstacles,is presented in Ref.29 The Lyapunov function vector field,based on the relative position of the UAV and the target,is modulated to yield the desired tracking velocity field.30As for the trajectory optimization,the Model Predictive Control(MPC) is also added into the DSM control system.31In Ref.32,the original velocity field of the UAV is pre-designed to ensure the convergence of the target position and then modulate it using a similar method to the DSM.Since the predesigned velocity field is necessary,this method consists in planning the motion rather than reactively controlling the UAV.

Despite its several advantages,the DSM still has some drawbacks.Firstly,although the dynamic system of the DSM has a generic form,the modulated trajectory,such as the integral of the modulated dynamics,is not always realizable.This problem is faced because of the symmetrical modulation on each state dynamics.Consequently,to guarantee the existence of the control input which produces the modulated dynamics,each of the DS state dynamics that will be modulated,should be able to be freely tuned.This also means that the DS is fully-actuated and has a full degree of freedom.This issue will be further explained,with a specific case,in Section 4.

To overcome this limitation,this paper presents the Extended Dynamic System Modulation (EDSM).This proposed method generates asymmetrical modulation to extend the early method application by making the modulated DS satisfy the dynamic constraints of a class of DSs.Simultaneously,it has the ability of inherent multiple obstacle modulation,and it provides a variable obstacle avoidance behavior.This method can be also considered as a linear transformation of the velocity field,based on the state feedback.Based on the EDSM,we also propose an obstacle avoidance guidance law,which can be applied to the UAV in order to realize the simultaneous guidance and obstacle avoidance in a complex environment.

The remainder of this paper is organized as follows.Section 2 presents a brief review of the DSM obstacle avoidance method.The EDSM is then detailed in Section 3.Section 4 describes the obstacle avoidance guidance law.The numerical simulation results are shown in Section 5.Finally,conclusions are drawn and prospects are provided in Section 6.

2.Dynamic system modulation

The Dynamic System Modulation (DSM) method was first proposed by Khansari-Zadeh and Aude Billard.28This method can be used to solve an obstacle avoidance problem in an environment containing multiple convex obstacles.Note that a review on the DSM is given below,before proposing enhancements to the present study.Further details about the DSM can be found in Ref.28

A dynamic system with an original controller can be expressed as:

By expanding the obstacle,any state of the system can be located on the surface of the expanded obstacle.The normal vector of the local deflection hyper-plane is given by:

The illustration of the tangential hyper-plane and the deflection hyper-plane of a 2-D object is shown in Fig.1.A group of vectors on the deflection hyper-plane at the same state,can then be chosen:

Fig.1 Illustration of the tangential hyper-plane and the deflection hyper-plane of a 2-D object.

In the diagonal matrix D,each variable can be calculated according to the following expression:

Let the original system dynamic function f multiplied by M.The modulated dynamics is then given by:

Theorem 1.28Consider that the convex manifoldencloses a staticn-dimensional obstacle,and its reference center is located at point xoinside the obstacle.The movement｛x｝tstarting from the initial point Γ（｛x｝0）≥1 outside the obstacle,can realize obstacle avoidance under the dynamic shown in Eq.(11),i.e.Γ（｛x｝t）≥1,t≥0.

Proof.cf.28

3.Extended DSM method

In this paper,we focus on a class of systems that can be written as:

where fUand fLare continuous.

Compared to Eq.(1),this expression can be divided into two parts (upper partand lower partThe state xUwhich do not contain control inputs in its dynamics,is then distinguished from the other states.We assume that the lower part is fully-actuated so that an input u producing the desired response exists for any desired.In this type of system,the dynamics of the upper part should be consistent to ensure the modulated dynamics is realizable.Therefore,the problem resides in the modulation of the system dynamics without changing the upper part,while retaining the obstacle avoidance realization.

ConsideringNconvex obstacles in the state space,the obstacle set can be defined similarly to Eq.(3) as follows:

By choosingmvectors among them,the following matrix can be defined:

Considering the geometric meaning,the avoidance direction can be chosen in the null space of matrix A.Consequently,m＜nis a required condition,which means A should have at least 1-dimensional null space.Since the priority of obstacle avoidance can be determined according to the relative Euclidean distance,a nearest strategy is used to determine matrix A,that is,a group of normal vectors n1,n2,...,nmcorresponding to the obstacles that are the nearest to the current state.

The conditionm＜nis not enough to rule the limitation ofm,because we are not only concerned about the obstacle that causes a constraint on the velocity direction,but also the consistent dynamics causing the constraint.Considering the upper part of the system,rewritten as:

Correspondingly,a vector is defined:

Note that n1,n2,...,nmshould be linearly independent ofthat is,l+m≤n.In other words,when constructing the obstacle space,it is necessary to ensure that its surface normal is not orthogonal to the controllable dynamic system.If this condition cannot be satisfied by just reducing the number of nearest obstaclesm,an expanded obstacle space can be constructed to change the obstacle boundary so as to change the normal vector of the local tangent hyper-plane and make it possible to meet this condition.In summary,sincel＜nis always met,and the minimum value ofmis 1,then we have 1 ≤m≤n-l.

Moreover,there exist a set ofn-m-l∈[0,n-1] linearly independent vectors e1,e2,...,en-m-l,combined with n and eto obtainnlinearly independent vectors in Rn.Finally,the formulation of the extended modulation matrix G is given by:

where,

where ξ and η are scale factors,the specific forms of which are variable.However,the following conditions should be met:for ξ,ξi∈[0,1 ]and ξi→0 when Γi→1,ξi→1 when Γi→∞;for η,ηi≠0 when Γi→1,ηi→1 when Γi→∞.For ease of reference,this condition is referred to as a scale factor condition.Note that Eq.(10) presents a feasible form.However,other feasible forms also exist.We can easily verify,by calculation,that the scale factor will affect the characteristics of the obstacle avoidance path.

Proof.cf.Appendix A.

As mentioned earlier,since we only require e1,e2,...,en-m-lto be linearly independent of n and,it is still usually free to determine their exact values based on the local obstacle information and the desired behavior of the system.Note that vectors e are referred to as‘‘free vectors”.It will be shown,in the simulation results,that different free vectors result in significantly various obstacle avoidance paths.

It should be emphasized that the modulated dynamics Gf is only a reference.For realization,we need to approximately design the real control input.Ideally,it can be obtained by solving the following equation:

where [·]Lrefers to the lower part corresponding to xL.This equation always has a solution since we assumed that the lower part of system (Eq.(12)) is fully-actuated.However,it is usually not easy to solve it.We can also design the controller using another method,and use the modulated dynamics as a reference to infer the obstacle avoidance performance of the resulting dynamics.In fact,for the systems of Eq.(12) that do not satisfy the partial full-actuated assumption,the modulated dynamics can also help us to design the obstacle avoidance controller.This will be detailed in the next section.

The modulation effect on the convergence of the original dynamic system,is also discussed in this section.Assume that system (Eqs.(1),(12)) is globally stable,the equilibrium point is located at the target point x*,that is,f（x*）=0 for autonomous dynamic systems,while limt→∞f（x*,t）=0 for nonautonomous dynamic systems.When the dynamic modulation matrix G is used to modulate f,x*remains an equilibrium point because the speed will vanish there,that is,Gf（x*）=0 for an autonomous dynamic system,and limt→∞Gf（x*,t）=0 for a non-autonomous dynamic system.However,in some cases,the target may not be the only equilibrium point of the system.Due to the existence of the modulation term,other system equilibrium points may occur.These points can be calculated by observing the null space of G.For allmatrix G is full rank,thus x*will still be the only equilibrium point.On the boundary of obstaclesG is not full rank,and spurious equilibrium points will then be produced.These spurious equilibrium points are generated when the original velocity at the surface is collinear with the normal vector of the tangent hyperplane of the obstacle space:

According to the original dynamic function f,these equilibrium points may be saddle points or local minima.

Finally,to illustrate the relationship between EDSM and DSM,as well as the relationship between EDSM and the state velocity stabilization control,two specific cases of modulation matrix design are given:

In this case,matrix H is orthogonal.If the number of obstacle spaces is further restricted tom=1,and the scale factor function is designed according to Eq.(10),EDSM and DSM are equivalent.In this case,the difference between EDSM and DSM resides in the free vector design.While maintaining the obstacle avoidance ability,the EDSM can realize a designable path.DSM can be considered as a specific form of EDSM.

In this case,by ignoring the calculation process,the expression of the modulation matrix G is given by:

It can be noticed that the modulation effect retains the original dynamics of the previous (n-1)-order system,and when ξ=0:

Furthermore,if the original system is ann-order series integral system,i.e.1=x2,2=x3,...,n=u,then Eq.(26) can also be written as:

In this case,the obstacle avoidance modulation is equivalent to the stabilization control ofThis conclusion is helpful to understand the relationship between obstacle avoidance control and state velocity stabilization control near the obstacle set.

4.EDSM application in obstacle avoidance guidance

In this section,the UAV guidance in the 2-D obstacle environment,is studied.As shown in Fig.2,the UAV is denoted as a particle,wherevis its velocity,ais the normal acceleration and σ is the heading angle,i.e.the angle between the velocity vector and thex-axis of the inertial coordinate with the counterclockwise direction taken as the positive direction.We assume that the tangential velocity change of the UAV during the guidance is negligible,i.e.vis constant.

Fig.2 Missile–target–obstacle engagement geometry.

Similarly,the target is also denoted as a particle,wherevtis its velocity and σtis the heading angle,both of which are time variables.We represent the space occupied by the obstacles as set Ω ?R2.For each obstaclei,the velocity isvoiand the heading angle is σoi,both of which are time variables.

The relative kinematic variables of the UAV,the target and the obstacle,are as follows;the distancerand the line-of-sight angleqbetween the UAV and the target,the distancediand the line-of-sight angle θibetween the UAV and the obstacle.Based on the geometric relationship,the parallel relative kinematics of the obstacle-target-UAV can be established as follows:

System (Eq.(28)) is a time-varying nonlinear system,while its input is the UAV accelerationa.When we focus on the obstacle avoidance,the relative motion state of the target is ignored.By differentiating all thein system (Eq.(28)) with respect to time until the system inputaappears in the expression,and regarding all θias the time-varying parameters,an equivalent state space dynamic description is obtained:

Assumption 1.There exists a fniite number of points（xi,yi）located on the motion planeR2,and a circular area is generated with each point as the center.The following formula holds:

Assumption 1 indicates that there is a set O composed of a finite number of circular obstacles,which can contain the space occupied by actual obstacles,as shown in Fig.3.Therefore,we can transform the obstacle avoidance problem of arbitrary shape obstacles into the obstacle avoidance problem of the circular obstacle,so that the obstacle movement can be expressed by the movement of the circle obstacle center point set.

Fig.3 Illustration of Assumption 1.

We define the obstacle surface as=｛x∈X|Γi（x）=ki（di-Ri）+=0｝,where x=and X is the state space.Since distanced-Ris always positive,according to the surface function,when the distance decreases,the approach velocitydecreases atktimes the distance and disappears atd=R.In addition,not crossing the boundary also means that the approach velocity is not greater thanki（di-Ri）at any time.

As shown in Fig.4,the trajectory ofx1andx2in system(Eq.(29)) can be drawn in a 2D plane denoting the R2state space.Constrained by=x2,the horizontal velocity is forced to be positive on the positivex2half-plane,and negative on the negativex2half-plane,which results in the detour shown in Fig.4.If the original DSM is used to modify the dynamics,when the original velocity is pointing downwards relative to the normal vector,the modulated velocity will hold this relative direction and finally break the dynamic constraint (cf.red line in Fig.4).This result is improper,because we are not able to realize such a state trajectory in system=f（x,u）,with any control input.

Fig.4 Illustration of the DSM trajectory realization problem.

As an improvement of the original DSM,EDSM aims to solve this problem.By dividing system (Eq.(29)) into the upper and lower parts as shown in Eq.(12),and according to EDSM,we obtain:

Letm=n,substituting the obstacle surface expression into Eq.(14),we obtain:

The scalar factors are designed as:

where the safety parameter α and the sensitivity parameter τ are designable.

The results of these tuning parameters will be given in Section 5.Based on the above formulas,the modulation matrix can be constructed as:

where ?denotes the Kronecker (tensor) product.

By substituting it in system (Eq.(29)),the modulated dynamics of the system is obtained:

It can be verified that Eq.(33)meets the scale factor conditions.Therefore,according to Theorem 2,the modulated system,defined in Eq.(35),can guarantee the obstacle avoidance.

Finally,we go back to the problem of realization of DSM trajectory,previously mentioned in this section.With the EDSM,the modulated system remains the upper part of the original system (Eq.(12)).Therefore,the dynamic constraints defined by the upper part are satisfied.Whenx2is negative and the obstacle surface is near the state,≈-kx2since ξ is small.The velocity of state is then close to [x2,-kx2],which is orthogonal to the normal vector of the obstacle surface,and pointing upwards.This result is consistent with the direction of the ideal trajectory denoted by the green line in Fig.4.

Since system (Eq.(29)) does not satisfy the assumption of partial full-actuated,we cannot directly solve the controller by solving the equation mentioned in Section 3.However,we can still use the modulated dynamics of system (Eq.(35))as a reference to derive our controller.The control input is given by:

Proof.cf.Appendix B.

5.Simulation results

In this section,the EDSM method and the proposed obstacle avoidance guidance law are tested in three ways:

(1) On a set of theoretical dynamic systems,the theoretical performance of the EDSM is verified.Simultaneously,the impact of the free vector on the behavior of the obstacle avoidance is studied.

(2) On a set of cases of the moving target guidance in a multiple obstacle environment,the trajectories of the UAV and the detailed information of the system states are presented to verify the performance of the obstacle avoidance guidance law.

(3) On a set of modulation parameter pairs,the trajectories are compared to demonstrate the efficiency of the modulation parameters in trajectory optimization.

5.1.Theoretical simulation results

In this section,the Original State Trajectories (OSTs)are generated from the original dynamics,while the Modulated State Trajectories (MSTs) are realized from the EDSM.By reviewing the construction of the modulation matrix,it is clear that for each case,the modulation matrix can be resolved with the specific conditions given in Table 1 and the local obstacle surface normal vector.Thus,once the initial state is determined,the OST and the MST can be obtained by respectively integrating the original dynamics and the modulated dynamics.

Table 1 Conditions of dynamic system modulation for theoretical simulation.

The obtained results are shown in Fig.5 and Fig.6.It can be observed that all the MSTs achieve the obstacle avoidance,while the OSTs mostly intersect the obstacles.Furthermore,all the convergence characteristics of the original equilibrium points (except Case 3 where those points do not exist) remain as expected.

Fig.5 Simulation results of the dynamic system modulation in both 2D and 3D systems.

By comparing Case 1 (Fig.5(a)) and Case 2 (Fig.5(b)),it can be observed that the biased free vectors significantly change the obstacle avoidance path.With the local obstacle surface normal as a deviation added to the orthogonal direction (i.e.e=n+t,where n ⊥t),the obstacle avoidance path in Case 2 shows a clear tendency to avoid on a specific side of the obstacles,while this phenomenon is completely absent in Case 1.

By comparing Case 4 (Fig.6(a)) and Case 5 (Fig.6(b)),it can be seen that,when the system has a partial dynamics to be hold during the modulation,the modulated path obtained by the DSM method will break the constraints of the system(=y),while the obstacle avoidance path obtained by the EDSM method strictly hold the dynamics of the statex.

Fig.6 Simulation results of the dynamic system modulation with differential constraint.

5.2.Simulation results of the UAV obstacle avoidance guidance

The performance of the proposed obstacle avoidance guidance law is demonstrated in a multiple dynamic obstacle environment with a maneuvering target.The simulation settings are as follows: the environment contains five moving obstacles which motion parameters are listed in Table 2,while all the safety radii are 10 m;the UAV motion parameters,the original guidance law (PN guidance law) and the modulation conditions are given in Table 3;the target motion parameters in both cases are shown in Table 4.

The simulation results of Case 1 and Case 2 are shown in Fig.7 and Fig.8,respectively.In Fig.7(b) and Fig.8(b),the relative distance between the UAV and the target converges at the end time.Thus,the UAV intercepts the target successfully.In Fig.7(c)and Fig.8(c),the maximum overload produced by the UAV controlling during the entire interception is less than 2.5 g,which represents a realistic trajectory characteristic.It can be seen from Fig.7(d) and Fig.8(d) that the minimum relative distance between the UAV and all the obstacles is greater than the safety radius,i.e.the UAV completely avoids the obstacles in both cases.

Fig.7 Simulation results of the UAV obstacle avoidance guidance with a target of constant velocity.

Fig.8 Simulation results of the UAV obstacle avoidance guidance with a maneuvering target.

5.3.Simulation results of the modulation with different parameters

There are two variable parameters (α and τ) in Eq.(33),to be determined for modulation design.A good selection of this pair of parameters will significantly improve the characteristics of the obstacle avoidance trajectory and the robustness of the obstacle avoidance,under the impact of uncertainty.Roughly speaking,τ is related to sensitivity,while α is related to safety.The smaller τ is,the farther the obstacle avoidance action occurs,while the smaller α is,the smaller the minimum distance is.If the value is too large or too small,the trajectory performance may be significantly reduced.It can be deduced from the experimental results that τ is usually better in the range of 0.1–3,while α design needs to weigh security and the ability to pass through dense obstacles.To evaluate the trajectory varying with the parameters,a numerical simulation was performed in an environment of static single obstacle and static target with different parameters,in order to prevent the coupling caused by the obstacle and target change.

The simulation is performed in two cases;with fixed α and variable τ,with fixed τ and variable α.The corresponding values are shown in Table 5.The simulation conditions of each object in the environment are shown in Table 6.It can be seen from Fig.9(b)that the trajectory is significantly better for values of τ equal to 1.8 or 2.2 than for other values,because of the less energy consumption and the shorter interception time.The same conclusion can also be deduced from Fig.9(a)where the values of τ equal to 1.8 and 2.2 lead to the smoothest trajectories.In fact,the adjustment of parameter τ can be used to minimize the obstacle avoidance energy consumption.

Fig.9 UAV obstacle avoidance guidance with fixed α and varying τ.

Table 2 Initial conditions and motion parameters of the moving obstacles.

Table 3 Initial conditions and motion parameters of the UAV.

Table 4 Initial conditions and motion parameters of the target for two cases.

Table 5 Modulation parameters for two cases.

Table 6 Initial conditions and motion parameters of all objects.

On the other hand,the expansion parameter α determines the minimum distance between the UAV and the obstacle,as shown in Fig.10(a).More precisely,the larger α,the greater the ‘‘repulsion” of the obstacle.In addition,Fig.10(b) shows that the change in α rarely affects the overall characteristics of the overload.It only alters the peak time and the interception time.

Fig.10 UAV obstacle avoidance guidance with fixed τ and varying α.

Since the two parameters have a relatively clear correspondence with the trajectory characteristics,they can be tuned when the motion trajectory characteristics need to be roughly adjusted.However,when considering a dynamic environment and maneuvering targets,the parameter tuning process may require a more complex adaptive strategy.

5.4.Computational complexity analysis

The computation in obstacle avoidance guidance control based on the EDSM consists of two parts;the calculation of modulation related parameters and the calculation of control input.The former needs the information of the selected obstacle bundle chosen from the detected obstacles in a single control loop,and the matrix inversion shown in Eq.(18).Usually,as the form of matrix H (Eq.(19)) is fixed,a pre-computation can be done to obtain the matrix inversion with undetermined variables,and so as to the undetermined modulation matrix.Then-dimensional modulated dynamics is then solved in an expected running time ofO（n）.Therefore,the calculation of the related modulation parameters has an expected running time ofO（n+m）,wheremdenotes the number of obstacles contained in the obstacle bundle.On the other hand,in the proposed obstacle avoidance guidance law,which presents an application of the developed EDSM,the expected running time isO（m）,since the control input is the sum ofmparts,where each part corresponds to the information of a single obstacle.The simulation results (Fig.11),based on ROS and Gazebo (v9.16.0),show that the proposed method can be calculated at 100 Hz.

Fig.11 Simulation scenario based on Linux ROS and Gazebo (v9.16.0).

6.Conclusions

In this paper,a new Extended Dynamic System Modulation(EDSM) obstacle avoidance control method is proposed.Based on this EDSM method,we present a new obstacle avoidance guidance law,which can be applied to the UAV to realize simultaneous guidance and obstacle avoidance in a complex environment.The main conclusions are as follows:

(1) The EDSM is a real-time obstacle avoidance method.Compared with the earlier dynamic modulation method,the proposed EDSM can be applied to the dynamic system of wider forms,and leads to a realizable modulated obstacle avoidance trajectory.

(2) EDSM has an inherent multiple-obstacle avoidance modulation ability and a designable obstacle avoidance behavior.This feature is brought by the permission of directly using the multiple-obstacle surface normal and the free vector group,when calculating the modulation matrix.

(3) Theory and numerical simulation results prove the obstacle avoidance performance of the proposed EDSM method and the efficiency of the obstacle avoidance guidance law.The obstacle avoidance guidance law achieves the simultaneous guidance and obstacle avoidance,in a complex dynamic obstacle environment.By designing the parameters of the modulation function,the UAV path can be further optimized.

In practice,since solving the modulation matrix only requires the relative motion information of the agent and the obstacle,the EDSM method can be applied to the obstacle avoidance control task with the egocentric sensing mechanism.This method can also be used as an auxiliary module of the original motion controller,in order to enhance the robustness of the system in a complex environment with unpredictable disturbance.

In future work,we will further discuss the influence of the designable variables included in the EDSM method,and study the modulation technology considering the path optimization.In addition,we also need to take the system uncertainty and sensor error into consideration,in order to improve the performance of the proposed 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.

Acknowledgement

This study was supported by the National Key R&D Program of China (No.2018YFF01013403).

Considering the disorder of the obstacle sets,anykobstacle sets can be represented by the firstkobstacle sets.If the normal velocity on the boundary of all thesekobstacles disappears,the impenetrability of all the obstacles can be ensured:

Since ?i=1,2,...,k,Γi（xb）=1,with function ξ satisfying the scaling factor condition,i.e.Γi=1 ?ξi=0,we can finally conclude that=0.

Appendix B.Proof of Theorem 3: By substituting Eq.(36) in Eq.(29),for anyi∈[1,n],we obtain:

In summary,using Eq.(36),the modulated dynamics form of Eq.(21)can be established on the relative distance dynamics of each obstacle,while the original obstacle set is a subset of the new obstacle set.It can be concluded from Theorem 2 that,any movement ｛x｝tfrom an arbitrary initial point satisfying?i∈1,2,...,n,ki[di（0）-Ri]+≥0 realizes obstacle avoidance,i.e.?i∈1,2,...,n,ki（di-Ri）+i≥0.