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J Intell Robot Syst (2014) 73:783–795DOI 10.1007/s10846-013-9948-x

Collision-Free 4D Trajectory Planning in UnmannedAerial Vehicles for Assembly and Structure Construction

D. Alejo · J. A. Cobano · G. Heredia · A. Ollero

Received: 1 September 2013 / Accepted: 13 September 2013 / Published online: 1 October 2013© Springer Science+Business Media Dordrecht 2013

Abstract This paper presents a new system forassembly and structure construction with multipleUnmanned Aerial Vehicles (UAVs) which au-tomatically identifies conflicts among them. Thesystem proposes the most effective solution con-sidering the available computation time. Afterdetecting conflicts between UAVs, the systemresolves them cooperatively using a collision-free4D trajectory planning algorithm based on a sim-ple one-at-a-time strategy to quickly compute afeasible but non-optimal initial solution and astochastic optimization technique named ParticleSwarm Optimization (PSO) to improve the ini-tial solution. An anytime approach using PSOis applied. It yields trajectories whose qualityimproves when available computation time in-creases. Thus, the method could be applied inreal-time depending on the available computa-tion time. The method has been validated with

D. Alejo · J. A. Cobano (B) · G. Heredia · A. OlleroRobotics, Vision and Control Group EngineeringSchool, University of Seville, 41092 Seville, Spaine-mail: [email protected]

D. Alejoe-mail: [email protected]

G. Herediae-mail: [email protected]

A. Olleroe-mail: [email protected]

simulations in scenarios with multiple UAVs in acommon workspace and experiment in an indoortestbed.

Keywords Aerial robotics · Trajectory planning ·Real-time applications

1 Introduction

Multiple UAVs present important advantages tocarry out different cooperative missions such assurveillance [1], mapping, data collection [2], firedetection and monitoring, etc. In these missions isimportant to maintain as much as possible the tra-jectories planned, and meet the Estimated Timeof Arrival (ETA) of each UAV to avoid collisionsbetween the UAVs and obstacles in the environ-ment. Therefore a system to plan collision-free4D trajectories is required to ensure the safety ofthe mission.

Cooperative missions are being performed inthe ARCAS FP7 European Project [3]. Thisproject is developing a cooperative free-flyingrobot system for assembly and structure con-struction (see Fig. 1). The ARCAS system usehelicopters and quadrotors with multi-link manip-ulators for assembly tasks [4]. The aerial vehiclescarry structure parts that will be assembled at thetarget destination. An important part in ARCASis cooperative assembly planning and safe trajec-

784 J Intell Robot Syst (2014) 73:783–795

Fig. 1 Assembly and structure construction in the ARCASproject

tory generation to perform the coordinated mis-sions, assuring that neither the aerial vehicles northe manipulators or the objects carried collidewith each other. The work presented is related tothis task.

General concepts and methods on path plan-ning could be applied in order to solve theproblem. A review of these methods with a com-prehensive mathematical discussion is presentedin [5]. In practical applications the most popularmethods include potential fields [6], graph searchlike A* and D* [7] and Rapidly-exploring Ran-dom Trees (RRT) [8]. Other kind of methodshave been proposed such as evolutionary tech-niques [9–11], particle swarm optimization [12]and multi-objective evolutionary algorithms [13].

The problem of trajectory planning is NP-hard[14, 15]. Sampling-based techniques, as opposedto combinatorial planning, are usually preferred inthese NP-hard problems. These planning schemesare appropriated when the solution space is hardto model or unknown a priori because of itsdynamic nature.

Furthermore, planning optimal collision-freetrajectories for multiple UAV leads to optimiza-tion problems with multiple local minimum inmost cases and, thus, local optimization meth-ods as gradient-based techniques are not wellsuited to solve it. The application of evolutionarytechniques or particle swarm optimization is anefficient and effective alternative for this problem,since they are global optimization methods.

The Conflict Detection and Resolution (CDR)problem has also been studied extensively anddifferent types of techniques have been proposed[16]. One method to resolve conflicts is basedon the speed assignment [17]. In this method thespeed profile for all the aerial vehicles involvedin a collision is computed in a centralized wayto solve the conflict. A method based on mixed-integer linear programming (MILP) is presentedin [18]. It resolves the conflict by changing speedto a large number of aerial vehicles subject tovelocity change constraints, but some conflictscannot be solved. Other method resolves pairwiseconflicts [19] but do not consider more UAVs.More methods based on MILP to avoid collisionsare presented in [20] and [21]. The method formultiple-UAV conflict avoidance proposed in [22]assumes that UAVs fly at constant altitude withvarying velocities and that conflicts are resolvedin the horizontal plane using heading change, ve-locity change, or a combination thereof. Methodsbased on Ant Colony Optimization (ACO) algo-rithms have also been proposed [23]. In [24], theapplication of a game theory approach to airborneconflict resolution is presented. These techniquespresent a disadvantage: they are not well suitedfor applications that require a high level of scala-bility for the application to many UAVs.

This paper presents a system for collision-free4D trajectory planning. The system automaticallyidentifies conflicts between multiple UAVs andproposes the most effective solution consideringthe available computation time. The conflicts aredetected by using a detection algorithm based onaxis-aligned minimum bounding box. The 4D tra-jectory planning algorithm is based on a stochas-tic global optimization technique named ParticleSwarm Optimization (PSO) and resolves cooper-atively the conflicts. Moreover, a simple one-at-a-time strategy is considered to quickly compute afeasible but non-optimal solution, which is takenas the initial point. PSO presents two advantageswith respect to other evolutionary techniques[25, 26]: its low computational overheads andfaster solution convergence. The approach tosolve the conflicts is to add an intermediate way-point to the initial trajectory and/or change thespeed to meet the ETA. Preliminary results werepresented in [27].

J Intell Robot Syst (2014) 73:783–795 785

On the other hand, computing time of most op-timization methods is not deterministic, and mostpublished works either do not consider computingtime or simply show that in average it is muchless than the available time to get a solution, thuswasting available computing time. The proposedsystem adopts an anytime approach to resolve theconflicts. The aim is to provide a flyable solutionat any time. This solution will be more or less closeto the optimum depending on the available time.Thus, valid trajectories whose quality improveswhen available computation time increases areyielded.

The paper is organized into six sections. Theformulation of the problem is presented inSection 2. The proposed system is described inSection 3. Sections 4 and 5 present the simulationsand experiments performed, respectively. Finallythe conclusions are detailed in Section 5.

2 Multi-UAV Trajectory Planning

The problem of collision-free 4D trajectory plan-ning of multiple UAVs to perform the coordi-nated missions proposed by the ARCAS projectis considered in this paper. First the initial spa-tial trajectories are computed to implement theassembly planning. The deviations of UAVs fromthese trajectories could be significant due to per-turbations (i.e. wind). The system should com-pute collision-free 4D trajectories when a conflictis detected during the execution of the mission.A trajectory is defined by a sequence of way-points and the ETA to the last waypoint. UAVs

share a common workspace and the separationbetween them should be greater than a givensafety distance. It is assumed that heading andspeed changes are allowed to solve the conflictsand to meet the ETA. This latter is importantto meet it in coordinated missions. The proposedsystem detects potential conflicts and computescooperatively a collision-free 4D trajectory foreach UAV. The information that the system needsin order to solve the problem is the following:

1) Initial spatial trajectory of each UAV.2) Parameters of the model of each UAV3) Location of each UAV4) ETA of each UAV5) The available computation time to solve the

conflicts

The objective is to find collision-free 4D tra-jectories that minimize the probability of havinga collision while minimizing the changes of way-points and speed for each UAV. Moreover, theETA will be met to perform the mission of theARCAS project.

3 Description of the System

This section describes the blocks of the pro-posed system to resolve the detected conflicts(see Fig. 2).

3.1 Axis-Aligned Minimum Bounding Boxfor Detection

The detection algorithm is based on axis-alignedminimum bounding box presented in [26] with

Fig. 2 Description of theproposed system toidentify and resolveconflicts


some modifications to adapt to the characteristicof the ARCAS project. This technique presents asadvantages the low execution time and the needof few parameters to describe the system.

The ARCAS project uses autonomous quadro-tors or helicopters with a multi-link arm attachedto the bottom of the aerial robot to carry out themanipulation and assembly tasks. These UAVsshould grasp different structural parts and trans-port them to the corresponding assembly location.To avoid collisions between the UAVs (includingthe arms and the grasped objects), a security en-velope is defined around each one. Each UAVsecurity envelope is approximated by two boxesjoined together: a horizontal box that covers theaerial robot and its rotors and a vertical box, whichsurrounds the arm and the transported object (seeFig. 3). Each box is defined by the intersectionof three intervals, one by axis. The measurementof the horizontal box is related to the minimumhorizontal separation between UAV consideringthe arm and the grasped object, and the verticalbox is related to the vertical separation. There-fore, the minimum separation, S, between twoUAVs is defined by the dimension of both joinedboxes. A collision is detected when there is anoverlapping between the intervals that define eachbox (see Fig. 3). Thus, the 3D problem is reducedto three problems of overlapping, one in eachcoordinate axis. Let us consider the intervals inone coordinate A = [Ai, Ae] and B = [Bi, Be].

Fig. 3 Detection algorithm based on axis-aligned mini-mum bounding box: A and B overlap (collision)

The condition of overlapping for this coordinateis given by:

(Ae > Bi) � (Ai < Be) (1)

3.2 4D Trajectory Planning Algorithm

The collision-free 4D trajectory planning algo-rithm is based on obtaining a feasible initial so-lution very fast, and then optimizing the solutionusing the PSO method, which is an heuristic globaloptimization algorithm. PSO is iterative, and thesolution improves with time. Thus, it is guaranteedthat a feasible solution is available at any time, andthat this solution will improve its quality if there ismore execution time available.

The initial solution can be obtained with a sim-ple one-at-a-time strategy: when there is a possi-ble conflict between several UAVs, one of themmoves to its destination point while the others stayhovering at the initial position, then the next UAVgoes to its target position, and so on. By movingonly one UAV at a time, the conflict is avoided.On the other hand, the solution is far from theoptimum since the total time will be much higherthan the ETA. Other velocity planning methodsto quickly compute initial solutions have beenalso implemented using the two velocities and thegreedy approach described in [28].

There are some situations in which the conflictcannot be solved only by changing the speedsof the involved UAVs. This is the case when afrontal collision is detected, for example. In thesecases, the path must be changed and the round-about technique can be a good candidate [29].The main idea is to make the involved aircraftscircle the conflict in the same direction. The radiusof the circle should be long enough to ensurecollision-free trajectories and to provide flyabletrajectories.

The PSO algorithm was first proposed in [30]. Itis developed from swarm intelligence and is basedon the research of bird and fish flock movementbehavior. It works by maintaining a swarm ofparticles that move around in the search-spaceinfluenced by both the improvements discoveredby the other particles (social behavior) and theimprovements made by the particle so far (greedybehavior). Its main advantages are its simplicity,


easy implementation and the existence of fewparameters to tune when comparing with otherevolutionary algorithms.

In this paper, we consider that the initial po-sition is known and the final waypoint of eachUAV should be the same as the one in the initialtrajectory. Moreover, the ETA to the final way-point should be met. Therefore, the goal of thisalgorithm is to obtain collision-free 4D trajecto-ries by adding one intermediate waypoint in thetrajectory of each UAV and changing the speedto meet the ETA while minimizing the followingcost function:

J =N∑

i=1

(Li + k�v2i ) + ωc (2)

where N is the total number of UAVs in thesystem, Li is the total length of ith trajectory, �vi

is the change of speed of each UAV to meet itsETA, k is a factor to convert to distance, and ωc isthe collision penalty that will be added if the newtrajectories still lead to collisions in the systemor if at least one unfeasible plan is generated.This function can be easily modified in order totake into account energy analysis and other oper-ational costs. Note that the approach consideredis centralized.

The implemented algorithm is based on [31].Let S be the number of particles in the swarm,each particle is defined by a state vector xi in thesearch-space and a velocity vector vi. This statevector contains the information about the locationof the intermediate waypoint and the velocity inthe first sector of the trajectory of each UAV.Note that the speed in the second sector is calcu-lated so the ETA to the final waypoint is the sameas in the original trajectory.

In first place, the swarm is initialized by ran-domly assigning initial locations and velocitieswith an uniform distribution. Then a special par-ticle containing the initial solution is added toensure the existence of one conflict-free solutionat any time.

Let pi be the best known state vector of particlei and let g be the best known state vector of theentire swarm. These are recalculated whenever anew iteration is obtained.

Then the exploration loop is executed. In eachiteration, both the state vector and the veloc-ity of each particle are updated by applying theexpressions indicated in steps 10 and 11 (seeAlgorithm 1).

The most important parameters in this formulaare the social weight, φg, and the local weight, φp.ω is the inertia weight. rg and rp are vectors whereeach component is generated at randomly withan uniform U(0, 1) distribution. Local and globalbest state vectors are also updated if necessary(steps 13–15).

The exploration loop can be finished by usingmany different termination criteria. Among thesecriteria a timeout condition and a convergencecondition (most of the individuals lay in to a tight

Algorithm 1 Basic PSO algorithm1. for Each particle do2. Initialize each particle’s state vector xi

with the desired probability function3. Initialize particle best state vector pi ←

xi

4. If f (pi) < f (g) update the swarm beststate vector g ← xi

5. Initialize each particle’s velocity vector vi.An uniform distribution is usually used.

6. end for7. repeat8. for Each particle do9. Pick random numbers rg rp with

U(0, 1)

10. Update the particle’s velocity:vi ← ωvi + φprp(pi − xi) + φgrg(g −xi)

11. Update the particle’s state vector:xi ← xi + vi

12. if f (xi) < f (pi) then13. Update the particle’s best

known state vector14. if f (xi) < f (g) then15. Update the swarm’s best

known state vector g ← xi

16. end if17. end if18. end for19. until A termination criterion is met


region of the search space) are the common ap-proaches. In this paper, the algorithm concludeswhen the available computation time is reached.

The parameters φg and φp have been tuned byperforming several tests with the same conditionsand only changing one parameter at a time. Theseparameters are usually selected in the interval[0, 1]. In our case, the best values found wereφg = 0.9 and φp = 0.1.

4 Simulations

A comprehensive set of simulations have beencarried out to check the properties the proposedsystem. Also, a random generation process of thetest considered has been performed to evaluatethe system.

The definition of a metric plays an importantrole to evaluate the results. In this paper, a test sethas been developed in a given scenario to validatethe system and to provide a way to measure theproperties of the system regarding time of exe-cution, optimization and level of scalability withnumber of vehicles. The considered scenario hasa base of 10 × 10 dimensional units. The designmethodology is based on [26].

The test set consists of 40,000 different prob-lems grouped by the number of vehicles involved,from 2 to 5, in subsets of 10,000 tests. This clas-

sification, using the number of vehicles, is useful tostudy the scalability characteristics of the method.The number of vehicles is from 2 to 5 becauseis the number used in the experiments of AR-CAS project. Moreover, the tests have been per-formed in the same computer and under the sameconditions.

The algorithms have been run in a PC with a2GHz Dual Core processor and 2 GB of RAM.The operating system used was Kubuntu Linuxwith kernel 2.6.32. The code was written in C++language and compiled with the gcc-4.4. 1.

The minimum horizontal and vertical separa-tion between UAVs is shown in Fig. 3. Theseseparation distances have to be calculated accord-ing to the dimension of each vehicle and theirlocalization and control errors. The dimensionsof the scenario are 10 m × 10 m. Two hundredtests have been performed for each subset. Thenumber of intermediate waypoints, M, is set to 1.Therefore, each solution trajectory is composed oftwo segments. Each UAV should meet its ETAto perform the coordinated mission. The allowedspeed for each UAV is between 0.3 m/s and 2 m/s,and k = 5. The speed in the first segment is chosenrandomly and in the second one is computed tomeet the ETA. If a particle does not meet theETA, it is penalized.

Figure 4 shows the relation between the timeof execution and the number of iterations per-

Fig. 4 Time of executionvs. number of iterationsdepending on the numberof vehicles: two UAVs(black), three UAVs(red), four UAVs (blue)and five UAVs (green).Solid line shows the meantime and dotted line showsthe standard deviation


Fig. 5 Median ofminimum cost throughoutsuccessive iterationsdepending on the numberof vehicles: two UAVs(black), three UAVs(red), four UAVs (blue)and five UAVs (green).Solid line shows the meantime and dotted line showsthe standard deviation

formed. Mean time and its standard deviation areshown each ten iterations. One hundred iterationsare considered in each test. The dependence withthe number of UAVs shows the scalability charac-teristics of the method.

Figure 5 shows the evolution of the mean costand its standard deviation with the number ofiterations considering different number of UAVs.The proposed system finds a better solution astime passes.

The speed should be changed in order to meetthe ETA of each UAV. Figure 6 show the in-formation on the speed of each UAV involvedwhen five UAVs are considered. Each box of thefigure depicts statistics of the two hundred testsperformed for a given number of UAVs. The

central mark is the median, the edges of each boxare the 25th and 75th percentiles, and the whiskersextend to the extreme data points.

Note that the mean change of speed of eachUAV is low and similar.

The median of the minimum costs computedin all the tests has been chosen as statistical in-dicator. This indicator indicates how much timeit would cost to achieve a solution with certainlevel of optimality. This relates the cost in a giveniteration to the obtained minimum cost in thecorresponding problem.

Figure 7 shows a normalization of the costagainst the number of iterations. A line that marksthe required number of iterations to compute fora 90 % level of optimality is drawn. If the test

Fig. 6 Speeds with fiveUAVs by considering200 tests


Fig. 7 Normalized costthroughout successiveiterations. The line marksthe 90 % optimality

set is executed in the same computer where theuser has installed the proposed method, Fig. 4 willprovide an estimation of the time needed for thatnumber of iterations, and therefore, that level ofoptimality.

For the cost normalization, a linear transforma-tion, f (x) = ax + b , is applied to the actual costvalues to set them in the range [0,1]. Therefore aand b are chosen in such a way that the maximumcost equals to 1 and the minimum cost equals to 0.Therefore,

a = 1Costmax − Costmin

(3)

b = Costmin

Costmin − Costmax(4)

Table 1 Evolution of the solution for two UAVs: time ofexecution and cost

Time (s) Improvement ofthe cost (m)

0.009 38.010.251 23.440.416 23.010.826 22.241.439 21.081.622 20.981.987 20.782.798 20.36

Table 2 Evolution of the solution for three UAVs: time ofexecution and cost


0.010 60.470.135 47.560.353 40.010.804 38.341.246 36.232.157 33.273.021 32.154.403 30.01

Table 3 Evolution of the solution for four UAVs: time ofexecution and cost


0.014 79.890.198 66.730.390 63.710.771 61.121.190 58.262.410 49.943.780 47.355.132 44.85


Table 4 Evolution of the solution for five UAVs: time ofexecution and cost


0.016 115.340.215 103.150.413 99.970.680 96.021.284 87.542.495 70.273.327 67.705.364 61.40

Fig. 8 Hummingbird quadrotor from Ascending technolo-gies used in the experiments

Depending on the number of UAVs, a solutionof great quality, 90 %, is computed between 20or 47 iterations. This characteristic is importantto apply this collision-free 4D trajectory planningalgorithm in real-time applications. Moreover, thealgorithm based on PSO presents better resultsthan the algorithm based on genetic algorithmspresented in [26].

The method can be applied in real-time appli-cations by considering an anytime approach. Ityields trajectories whose quality improves whenavailable computation time increases. Tables 1,2, 3, and 4 show how the quality of the solutionimproves as the time increases by using PSO. Theadvantage of this approach can be noted becausea good quality of solution is achieved on two sec-onds. Obviously, the quality of solution improvesas the time increases. Moreover, a first solution iscomputed in few milliseconds.

5 Experiments

Several experiments with the Hummingbirdquadrotors (see Fig. 8) in an indoor testbed havebeen also performed to validate the method. Thequadrotors used in these experiments are fromAscending Technologies and have 200gr payloadand up to 20 min of autonomy.

The useful volume of the indoor testbed wheretests can be conducted is a cube with a base of10 × 10 m and 3 m height. The localization systemis based on 20 VICON cameras that only needs

Fig. 9 Hummingbird quadrotor from Ascending technologies used in the experiments


Fig. 10 Separation between the QR trajectories from theinitial spatial trajectories

Fig. 11 Trajectories computed by the proposed system foreach quadrotor in the experiment

the installation of passive markers on each of theaerial vehicles. This system is able to provide, inreal-time, the position and attitude of each aerialvehicle with centimeter accuracy.

In this section, one experiment with threequadrotors is presented. Taking into account thecharacteristics of the scenarios of the ARCASproject, the minimum horizontal separation be-tween quadrotors in XY plane was Sxy = 1.0 m,and the vertical separation in Z axis, Sz = 0.45 m.

Figure 9 shows the initial spatial trajectoriesof each quadrotor. The trajectories are simu-lated to check if collisions are detected. Thequadrotors fly with constant velocity, v = 0.5 m/s.Figure 10 shows the separation between quadro-tors. Two collisions are detected between QR0-QR1 and QR0-QR2. Therefore, the proposedsystem should compute collision-free trajectories.

Figure 11 shows the solution trajectories com-puted by the system. Different changes of velocityshould be performed to avoid the collisions andmeet the ETA. These changes of velocity canbe observed in Fig. 11. The speed depends onthe proximity of the waypoints (circles): closestwaypoints indicate less speed and farther pointsindicate more speed.

Finally, the solution trajectories are flown.Figure 12 represent the real data from VICONsystem and Fig. 13 shows the horizontal and ver-tical separation between quadrotors. Note thatthe horizontal minimum separation QR0-QR1 isviolated between the time instants 26.45 s and 28s approximately but the vertical separation meetsthe vertical minimum separation in this interval.

Fig. 12 Trajectories flown by each quadrotor in the experiment


Fig. 13 Horizontalandvertical separationbetween the QRtrajectories from the realdata. Green dashed lineshows the horizontal andvertical minimumseparation

Therefore, the trajectories are safe and collisionsare avoided.

6 Conclusions

In this paper, we have presented a new system forassembly and structure construction with multipleUAVs (Unmanned Aerial Vehicles) which auto-matically identifies and resolves conflicts amongthem. The planning algorithm is based on ananytime stochastic optimization approach, that is,the system proposes the most effective solutionconsidering the available computation time. Theproposed system detects conflicts in trajectoriesof multiple UAVs using an algorithm based on

axis-aligned minimum bounding box, and resolvesthem cooperatively using a collision-free 4D tra-jectory planning algorithm based on a simpleone-at-a-time strategy to quickly compute a fea-sible but non-optimal initial solution and Parti-cle Swarm Optimization to improve incrementallythe initial solution. The system provides solution4D trajectories at any time, so it can be usedwith low available computation times, although asub-optimal solution will be obtained. Thus, theproposed system is well suited to situations withvariable available computation times, dependingon the number of UAVs and the distance topotential conflicts.

An important requirement in coordinated mis-sions of the ARCAS project is to meet the ETA.


The proposed system computes solution trajec-tories meeting the ETA of each quadrotor toperform the mission.

The system has been validated with manysimulations performed in different scenarios andseveral studies to analyze the characteristics ofthe system. A comparative to genetic algorithmsis mentioned. Moreover, real experiments havebeen carried out in an indoor testbed to show theperformance of the system.

The main advantages of the proposed systemwith respect to the one presented in [12] are: itconsiders multiple vehicles in the space and couldbe applied in real-time.

Acknowledgements This work has been supported by theARCAS Project, funded by the European Commissionunder the FP7 ICT Programme (ICT-2011-287617) andthe CLEAR Project (DPI201 1-28937-C02-01), funded bythe Ministerio de Ciencia e Innovacion of the SpanishGovernment. Also it has been partially funded by the Juntade Andalucia project P09-TEP- 5120.

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