Journal of Intelligent & Robotic Systems

, Volume 93, Issue 1–2, pp 193–211 | Cite as

Bi-level Flight Path Planning of UAV Formations with Collision Avoidance

  • Egidio D’AmatoEmail author
  • Massimiliano Mattei
  • Immacolata Notaro


This paper deals with the problem of generating 3D flight paths for a swarm of cooperating Unmanned Aerial Vechicles (UAVs) flying in a formation having a prespecified shape, in the presence of polygonal obstacles, no-fly zones and other non cooperative aircraft. UAVs are modeled as Dubins flying vehicles with bounds on the turning radius and flight path climb/descent angle. A Reduced Visibility Graph (RVG) based method, connecting selected nodes by means of circular arcs and segments, is adopted to minimize the length of each path. Then, to keep as much as possible the formation shape while flying between obstacles, the RVG is refined with the addition of so called Rendez-Vous Waypoints (RVWs). These are placed between groups of obstacles where it is impossible to maintain the desired formation. Waypoints locations and UAVs paths are optimized using a bi-level game theoretic approach based on the leader-follower Stackelberg model, where the lower level and upper level problems are the search of the shortest paths and the optimal locations of waypoints respectively. Such an approach allows to fly between obstacles, dispersing the formation and forcing UAVs to recompose it at given waypoints (RVWs) beyond groups of obstacles. Collision avoidance among UAVs and possible non-cooperative aircrafts, called intruders, is then achieved solving a set of linear quadratic optimization problems based on an original geometric based formulation. The effectiveness of the proposed approach is shown by means of numerical simulations where RVWs positions are optimized via a genetic algorithm.


Path planning Obstacle avoidance Collision avoidance Guidance Swarm of UAVs Formation flight Bi-level optimization Stackelberg game 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



A short version of this paper was presented in ICUAS 2017 [16].


  1. 1.
    Anderson, M., Robbins, A.: Formation flight as a cooperative game. In: Guidance, navigation, and control conference and exhibit, p. 4124 (1998)Google Scholar
  2. 2.
    Ariola, M., Mattei, M., D’Amato, E., Notaro, I., Tartaglione, G.: Model predictive control for a swarm of fixed wing Uavs. In: 30Th Congress of the international council of the aeronautical sciences, ICAS 2016 (2016)Google Scholar
  3. 3.
    Babel, L.: Curvature-constrained traveling salesman tours for aerial surveillance in scenarios with obstacles. Eur. J. Oper. Res. 262(1), 335–346 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bellingham, J., Tillerson, M., Richards, A., How, J.P.: Multi-task allocation and path planning for cooperating Uavs. In: Cooperative control: models, applications and algorithms, pp. 23–41. Springer (2003)Google Scholar
  5. 5.
    Blake, W., Multhopp, D.: Design, Performance and Modeling Considerations for Close Formation Flight. In: 23Rd atmospheric flight mechanics conference, p. 4343 (1998)Google Scholar
  6. 6.
    Blasi, L., Barbato, S., D’Amato, E.: A mixed probabilistic-geometric strategy for uav optimum flight path identification based on bit-coded basic manoeuvres. Aerosp. Sci. Technol. 71(Supplement C), 1–11 (2017)CrossRefGoogle Scholar
  7. 7.
    Bortoff, S.A.: Path planning for Uavs. In: American control conference, vol. 1, pp. 364–368 (2000)Google Scholar
  8. 8.
    Burns, R., McLaughlin, C.A., Leitner, J., Martin, M.: Techsat 21: formation design, control, and simulation. In: Aerospace conference proceedings, 2000 IEEE, vol. 7, pp. 19–25. IEEE (2000)Google Scholar
  9. 9.
    Camacho-Vallejo, J.F., Cordero-Franco, A.́E., González-ramírez, R.G.: Solving the bilevel facility location problem under preferences by a stackelberg-evolutionary algorithm. Mathematical Problems in Engineering (2014)Google Scholar
  10. 10.
    Chandler, P., Rasmussen, S., Pachter, M.: Uav cooperative path planning. In: AIAA guidance, navigation, and control conference and exhibit, p. 4370 (2000)Google Scholar
  11. 11.
    Chen, X., Zhang, J.: The three-dimension path planning of Uav based on improved artificial potential field in dynamic environment. In: 2013 5Th international conference on intelligent human-machine systems and cybernetics (IHMSC), vol. 2, pp. 144–147. IEEE (2013)Google Scholar
  12. 12.
    Chichka, D.F., Speyer, J.L.: Solar-powered, formation-enhanced aerial vehicle systems for sustained endurance. In: American control conference, 1998. Proceedings of the 1998, vol. 2, pp. 684–688. IEEE (1998)Google Scholar
  13. 13.
    la Cour-Harbo, A., Bisgaard, M.: State-control trajectory generation for helicopter slung load system using optimal control. In: AIAA guidance, navigation, and control conference, p. 6296 (2009)Google Scholar
  14. 14.
    D’Amato, E.: Multiobjective evolutionary-based optimization methods for trajectory planning of a quadrotor UAV 3DTech (2012)Google Scholar
  15. 15.
    D’Amato, E., Daniele, E., Mallozzi, L., Petrone, G.: Equilibrium strategies via ga to stackelberg games under multiple follower’s best reply. Int. J. Intell. Syst. 27(2), 74–85 (2012)CrossRefGoogle Scholar
  16. 16.
    D’Amato, E., Notaro, I., Silvestre, F., Mattei, M.: Bi-level flight path optimization for Uav formations. In: 2017 international conference on unmanned aircraft systems (ICUAS), pp. 690–697 (2017)Google Scholar
  17. 17.
    Della Vecchia, P., Daniele, E., D’Amato, E.: An airfoil shape optimization technique coupling parsec parameterization and evolutionary algorithm. Aerosp. Sci. Technol. 32(1), 103–110 (2014)CrossRefGoogle Scholar
  18. 18.
    Dever, C., Mettler, B., Feron, E., Popovic, J., McConley, M.: Nonlinear trajectory generation for autonomous vehicles via parameterized maneuver classes. J. Guid. Control. Dyn. 29(2), 289–302 (2006)CrossRefGoogle Scholar
  19. 19.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische mathematik 1(1), 269–271 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Duan, H., Li, P.: Bio-inspired computation in unmanned aerial vehicles (2014)Google Scholar
  21. 21.
    Dubins, L.E.: On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. Am. J. Math. 79(3), 497–516 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Eun, Y., Bang, H.: Cooperative control of multiple unmanned aerial vehicles using the potential field theory. J. Aircr. 43(6), 1805–1814 (2006)CrossRefGoogle Scholar
  23. 23.
    Frazzoli, E., Dahleh, M.A., Feron, E.: Real-time motion planning for agile autonomous vehicles. In: American control conference, vol. 1, pp. 43–49. IEEE (2001)Google Scholar
  24. 24.
    Gill, P.E., Wong, E.: Methods for convex and general quadratic programming. Math. Program. Comput. 7, 71–112 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Girard, A.R., De Sousa, J.B., Hedrick, J.K.: An overview of emerging results in networked multi-vehicle systems. In: Proceedings of the 40th IEEE conference on decision and control, 2001, vol. 2, pp. 1485–1490. IEEE (2001)Google Scholar
  26. 26.
    Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput. 13(5), 1194–1217 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Harada, M., Nagata, H., Simond, J., Bollino, K.: Optimal trajectory generation and tracking control of a single coaxial rotor Uav. In: AIAA Guidance, navigation, and control (GNC) conference, p. 4531 (2013)Google Scholar
  28. 28.
    Jeyaraman, S., Tsourdos, A., Zbikowski, R., White, B.: Kripke modelling of multiple robots with decentralized cooperation specified with temporal logic. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 219(1), 15–31 (2005)CrossRefGoogle Scholar
  29. 29.
    Kitamura, Y., Tanaka, T., Kishino, F., Yachida, M.: 3-D path planning in a dynamic environment using an octree and an artificial potential field. In: Proceedings. 1995 IEEE/RSJ international conference on intelligent robots and systems 95.’human robot interaction and cooperative robots’, vol. 2, pp. 474–481. IEEE (1995)Google Scholar
  30. 30.
    Kuriki, Y., Namerikawa, T.: Consensus-based cooperative formation control with collision avoidance for a multi-Uav system. In: American control conference (ACC), 2014, pp. 2077–2082. IEEE (2014)Google Scholar
  31. 31.
    Latombe, J.-C.: Robot motion planning. Kluwer Academic Publishers, Norwell, MA, USA (1991). ISBN: 079239206XCrossRefzbMATHGoogle Scholar
  32. 32.
    Lian, F.L., Murray, R.: Real-time trajectory generation for the cooperative path planning of multi-vehicle systems. In: Proceedings of the 41st IEEE conference on decision and control, 2002, vol. 4, pp. 3766–3769. IEEE (2002)Google Scholar
  33. 33.
    Lin, Y., Saripalli, S.: Path planning using 3D dubins curve for unmanned aerial vehicles. In: 2014 international conference on unmanned aircraft systems (ICUAS), pp. 296–304. IEEE (2014)Google Scholar
  34. 34.
    Liu, P., Yu, H., Cang, S.: Geometric analysis-based trajectory planning and control for underactuated capsule systems with viscoelastic property. Transactions of the Institute of Measurement and Control p. 0142331217708833 (2017)Google Scholar
  35. 35.
    Loridan, P., Morgan, J.: A theoretical approximation scheme for stackelberg problems. J. Optim. Theory Appl. 61(1), 95–110 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Maini, P., Sujit, P.B.: Path planning for a Uav with kinematic constraints in the presence of polygonal obstacles. In: 2016 international conference on unmanned aircraft systems (ICUAS), pp. 62–67 (2016)Google Scholar
  37. 37.
    Mansouri, S.S., Nikolakopoulos, G., Gustafsson, T.: Distributed model predictive control for unmanned aerial vehicles. In: Workshop on research, education and development of unmanned aerial systems (RED-UAS), pp. 152–161. IEEE (2015)Google Scholar
  38. 38.
    Mattei, M., Blasi, L.: Smooth flight trajectory planning in the presence of no-fly zones and obstacles. J. Guid. Control. Dyn. 33, 454 (2010)CrossRefGoogle Scholar
  39. 39.
    Mattei, M., Scordamaglia, V.: Task priority approach to the coordinated control of a team of flying vehicles in the presence of obstacles. IET Control Theory Appl. 6(13), 2103–2110 (2012)MathSciNetCrossRefGoogle Scholar
  40. 40.
    McKinsey, J.C.C.: Introduction to the theory of games courier corporation (2012)Google Scholar
  41. 41.
    Notaro, I.: Guidance navigation & control of a fleet of fixed wing UAVs. Aracne (2016)Google Scholar
  42. 42.
    Owen, M., Beard, R.W., McLain, T.W.: Implementing dubins airplane paths on fixed-wing Uavs. In: Handbook of unmanned aerial vehicles, pp. 1677–1701. Springer (2015)Google Scholar
  43. 43.
    Pachter, M., D’Azzo, J.J., Proud, A.W.: Tight formation flight control. J. Guid. Control Dynam. 24(2), 246–254 (2001)CrossRefGoogle Scholar
  44. 44.
    Pehlivanoglu, Y.V.: A new vibrational genetic algorithm enhanced with a voronoi diagram for path planning of autonomous uav. Aerosp. Sci. Technol. 16(1), 47–55 (2012)CrossRefGoogle Scholar
  45. 45.
    Proud, A., Pachter, M., D’Azzo, J.: Close formation flight control. In: Guidance, navigation, and control conference and exhibit, p. 4207 (1999)Google Scholar
  46. 46.
    Ren, W., Beard, R.W.: Distributed consensus in multi-vehicle cooperative control. Springer, London, UK (2008)CrossRefzbMATHGoogle Scholar
  47. 47.
    Richards, A., How, J.: Decentralized model predictive control of cooperating uavs. In: 43rd IEEE conference on Decision and control, IEEE, pp. 4286–4291 (2004)Google Scholar
  48. 48.
    Sastry, S., Meyer, G., Tomlin, C., Lygeros, J., Godbole, D., Pappas, G.: Hybrid control in air traffic management systems. In: Proceedings of the 34th IEEE conference on Decision and Control, 1995, vol. 2, pp. 1478–1483. IEEE (1995)Google Scholar
  49. 49.
    Scherer, S., Singh, S., Chamberlain, L., Elgersma, M.: Flying fast and low among obstacles: methodology and experiments. Int. J. Robot. Res. 27, 549–574 (2008)CrossRefGoogle Scholar
  50. 50.
    Schøler, F., Cour-Harbo, A., Bisgaard, M.: Configuration space and visibility graph generation from geometric workspaces for Uavs. In: AIAA guidance, navigation, and control conference. AIAA (2011)Google Scholar
  51. 51.
    Schøler, F., la Cour-Harbo, A., Bisgaard, M.: Generating approximative minimum length paths in 3D for Uavs. In: Intelligent vehicles symposium (IV), 2012 IEEE, pp. 229–233. IEEE (2012)Google Scholar
  52. 52.
    Schumacher, C., Singh, S.: Nonlinear control of multiple Uavs in close-coupled formation flight. In: AIAA Guidance, navigation, and control conference and exhibit, p. 4373 (2000)Google Scholar
  53. 53.
    Shorakaei, H., Vahdani, M., Imani, B., Gholami, A.: Optimal cooperative path planning of unmanned aerial vehicles by a parallel genetic algorithm. Robotica 34(4), 823–836 (2016)CrossRefGoogle Scholar
  54. 54.
    Sinha, A., Malo, P., Deb, K.: Evolutionary algorithm for bilevel optimization using approximations of the lower level optimal solution mapping. Eur. J. Oper. Res. 257(2), 395–411 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Smith, J.M.: Evolution and the Theory of Games. In: Did Darwin Get It Right?, pp. 202–215. Springer (1988)Google Scholar
  56. 56.
    Tartaglione, G., D’Amato, E., Ariola, M., Rossi, P.S., Johansen, T.A.: Model predictive control for a multi-body slung-load system. Robot. Auton. Syst. 92, 1–11 (2017)CrossRefGoogle Scholar
  57. 57.
    Tsourdos, A., White, B., Shanmugavel, M.: Cooperative path planning of unmanned aerial vehicles, vol. 32. Wiley (2010)Google Scholar
  58. 58.
    Vicente, L., Savard, G., Júdice, J.: Descent approaches for quadratic bilevel programming. J. Optim. Theory Appl. 81(2), 379–399 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Wu, J., Yi, J., Gao, L., Li, X.: Cooperative path planning of multiple Uavs based on ph curves and harmony search algorithm. In: 2017 IEEE 21St international conference on computer supported cooperative work in design (CSCWD), pp. 540–544. IEEE (2017)Google Scholar
  60. 60.
    Xu, N., Kang, W., Cai, G., Chen, B.M.: Minimum-time trajectory planning for helicopter Uavs using computational dynamic optimization. In: 2012 IEEE international conference on systems, man, and cybernetics (SMC), pp. 2732–2737. IEEE (2012)Google Scholar
  61. 61.
    Yan, F., Liu, Y.S., Xiao, J.Z.: Path planning in complex 3d environments using a probabilistic roadmap method. Int. J. Autom. Comput. 10(6), 525–533 (2013)CrossRefGoogle Scholar
  62. 62.
    Yang, Y., Polycarpou, M.M., Minai, A.A.: Multi-uav cooperative search using an opportunistic learning method. J. Dyn. Syst. Meas. Control. 129(5), 716–728 (2007)CrossRefGoogle Scholar
  63. 63.
    Yao, P., Wang, H., Su, Z.: Cooperative path planning with applications to target tracking and obstacle avoidance for multi-uavs. Aerosp. Sci. Technol. 54, 10–22 (2016)CrossRefGoogle Scholar
  64. 64.
    Yin, Y.: Genetic-algorithms-based approach for bilevel programming models. J. Transp. Eng. 126(2), 115–120 (2000)CrossRefGoogle Scholar
  65. 65.
    Yu, H., Meier, K., Argyle, M., Beard, R.W.: Cooperative path planning for target tracking in urban environments using unmanned air and ground vehicles. IEEE/ASME Trans. Mechatron. 20(2), 541–552 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of EngineeringUniversity of Campania L. VanvitelliAversaItaly

Personalised recommendations