A Torus Based Three Dimensional Motion Planning Model for Very Maneuverable Micro Air Vehicles

Transcription

1 A Torus Based Three Dimensional Motion Planning Model for Very Maneuverable Micro Air Vehicles Ryan D. Hurley University of Florida Rick Lind University of Florida Joseph J. Kehoe Scientific Systems Company, Inc. Micro Air Vehicles are highly maneuverable aircraft that possess the ability to fly in modes that conventional manned aircraft cannot attain. Advancements in the miniaturization of guidance, navigation, and control hardware have opened up the world of autonomous flight for unmanned air systems of all types. The melding of these two topics, MAVs and Autonomy, has been the focus of a great deal of research. Many have utilized the Dubins Car, an optimal 2-dimensional path, as the basis for their 3-dimensional path generation. This paper introduces a novel approach to the 3-dimensional path generation problem. Consideration of the flight regime of the MAV opens up maneuvers that the vehicle can perform such that it can orient itself for travel to locations outside the ability of conventional aircraft. The path generation theory is presented and utilized in a larger motion planning algorithm that produces flight through an obstacle rich environment. I. Introduction The utilization of unmanned aerial vehicles, or UAVs, by military and police has been steadily increasing, and this trend is expected to continue for the foreseeable future. Amongst the many possible applications of this technology is aerial reconnaissance within cities. A smaller subclass of UAVs named Micro Aerial Vehicles, or MAVs, are now technologically advanced enough and of appropriate size and airspeed to enable flight within urban environments. To traverse these obstacle rich environments, a 3-dimensional flight regime is required for successful obstacle avoidance. An advanced trajectory planning scheme will be critical to take advantage of the significant maneuvering capabilities of these aircraft. Traditional path planning techniques, such as the implementation of waypoints, become more difficult to utilize when considering flight in close-proximity to obstacles. Unless some guarantee of feasible maneuvering is provided between these waypoints, this method may not be suitable. In trajectory planning, the incorporation of dynamically-feasible motions is typically treated in either a direct or a decoupled fashion. 1 In direct planning methods, such as optimal control, a Graduate Student, fgdonpat@ufl.edu Associate Professor, ricklind@ufl.edu Senior Systems Engineer, joseph.kehoe@ssci.com 1 of 15

2 representation of the vehicle dynamics is considered in the formulation of the planning problem and the optimal system inputs are resolved. While optimal trajectories are produced, this method is often unmanageable for realistic problems. Alternatively, decoupled methods implement a simple vehicle motion model to plan a reference path and then smooth the path to satisfy vehicle dynamics using methods such as feedback control. This method often exhibits tractable complexity properties but with a lack of optimality. A considerable amount of work has been completed by researchers directly including dynamics in the planning process Much of this work utilizes the concept of basic maneuvers, or motion primitives, into feasible-path motion planning. The foundation for much of this work was established by Dubins for a 2-dimensional car. 13 The Dubins Car provides a closed-form solution for optimal trajectories and has been used for many types of planning such as the traveling-salesman problem. 14 While this foundation is used for several studies into aircraft motion, the process limits that motion to a 2-D plane Path planning techniques in 3-D are being developed by several different groups around the world. One approach expands the original 2-D Dubins formulation into a 3-D framework but does not deal with constraints in the climb rate or specific values of these climb rates as is the case with motion primitives. 19 Previous research efforts have translated the 2-D Dubins Car into 3-D by assuming the discontinuous transition between the straight climbing flight and turning level flight to be negligible. 2 A complete analogue to the Dubins car in 3-D is being developed to account for the shortest path between two points with associated heading constraints. 21 Another has created a more developed 3-D Dubins Car and with associated heading and flight-path angle constraints 22 but adds additional maneuvers at the initial and final conditions such that a constant climb rate Dubins Car is utilized in-between. Another approach implements heading and flight-path angle constraints but utilizes the 2-D Dubins car in 3 different planes to achieve 3-D motion. 23 While some 3-D curving path generation research has been completed in the needle steering field, 24, 25 this research considers the flight regime of a MAV in the design of motion primitives that, when connected sequentially, form a path from an initial condition to a final condition. This paper presents a novel approach to the 3-D path planning problem, one that utilizes curving primitives that lie outside the level flight plane. The primary addition to this research is the recognition that maneuvering in any plane rotated around the flight path vector produces a horn torus. The full 3- primitive sequence path that is produced is constrained by initial and final heading, initial and final flight path angle, relative distance, and maximum maneuvering rate. This motion planning model is then demonstrated as part of a comprehensive motion planning algorithm for flight through an obstacle rich environment. II. Two-Dimensional Dubins Car Model The concept of motion primitives leads to a useful framework for the simplification of complicated dynamic models. This framework involves combining sequences of compatible primitives to represent more complicated trajectories. A set of compatibility conditions are detailed in the literature. 9 This research utilizes path combinations, or motion primitives, consisting of straight paths, left turning paths, and right turning paths. Motion primitive theory is defined and described in more detail in the literature. 26 A standard model that utilizes motion primitives is known as the Dubins car This simple 2-dimensional vehicle motion model operates in a configuration space, or C-space, spanned by two Euclidean position variables, p x and p y, and an angle describing heading, ψ. Vehicle movement is restricted to driving straight or turning to either the left or right according to some turn radius. The straight motion is constrained to a constant velocity while the turns are constrained to both a constant velocity and a constant turn rate. As such, the motion of the Dubins car is described by 2 of 15

3 the differential system shown in Equation 1. ṗ x ṗ ẏ ψ = cos ψ sin ψ ω (1) The discrete set of values assumed by the turn rate ω is shown by Equation 2. ω { 1,, +1} /sec (2) The Dubins car is an especially interesting model in that a closed-form solution for optimal trajectories has been derived. 13 This solution notes the path from any initial point (x o, y o ) and heading (ψ o ) to any final point (x f, y f ) and heading (ψ f ) can be expressed using only the proper sequence of only 3 primitives. There are 6 combinations of these 3 primitive sequences. The 6 combinations are divided into Turn-Straight-Turn and Turn-Turn-Turn categories, as detailed in Table 1. Such a result allows strategies using optimal control to be directly compared with a global minimum for evaluating techniques of path planning. Turn-Straight-Turn Left-Straight-Left Right-Straight-Right Right-Straight-Left Left-Straight-Right Turn-Turn-Turn Right-Left-Right Left-Right-Left Table 1. Dubins car primitive sequences. The transformations describing the left turn, right turn, and straight ahead motions are shown as Equations 3, 4, and 5, respectively. The closed-form expressions for the trim durations τ are presented in the literature. 26, 27 L(p x, p y, ψ, τ) = (p x + sin(ψ + τ) sin ψ, p y cos(ψ + τ) + cos ψ, ψ + τ) (3) R(p x, p y, ψ, τ) = (p x sin(ψ τ) + sin ψ, p y + cos(ψ τ) cos ψ, ψ τ) (4) S(p x, p y, ψ, τ) = (p x + τ cos ψ, y + τ sin ψ, ψ) (5) Examples of solutions from an initial configuration, C o, defined by an initial position and initial heading, to a final configuration, C f, defined by a final position and final heading, are presented in Figure 1 for a Dubins car. The four Turn-Straight-Turn Dubins car combinations are utilized and plotted. The Turn-Turn-Turn combination is only utilized for instances when the final configuration is located and oriented in a manner such that the vehicle cannot orient itself for a Turn-Straight- Turn sequence due to the limitation in the minimum turn radius The optimal solution, which corresponds to the lowest travel time, is highlighted. 3 of 15

4 3 Y (ft) 4 X (ft) Figure 1. Possible ( ) and optimal ( ) 2-D Dubins path solutions. In addition to the 3-primitive sequences listed above, there is also a family of 2-primitive sequences utilizing only the Turn-Straight methodology (Left-Straight and Right-Straight). This combination will provide a path from any initial configuration C o, defined by the point (x o, y o ) and heading (ψ o ), to any final point (x f, y f ), but note that the final heading (ψ f ) cannot be guaranteed using the 2-primitive sequence in view of the fact that a third primitive is necessary to provide the desired heading. Examples of solutions from an initial position and heading to a final position are shown in Figure 2 for a 2-primitive Turn-Straight solution sequence. There are 3 examples of both the Left-Straight and Right-Straight paths, each with a different turn rate ω. 6 4 Y (ft) X (ft) Figure 2. Turn-straight solution sequences. 4 of 15

5 III. Three-Dimensional Torus Model The translation of the 2-dimensional (2-D) Dubins Car into 3-dimensions (3-D) has been approached by various methods The main objective is to create a feasible 3-D path consisting of primitives, either straight or curving, from an initial condition to a final condition. For the 2-D Dubins Car, the initial configuration C o is defined by the initial point (x o, y o ) and the initial heading (ψ o ). Likewise, the final configuration C f is defined by the final point (x f, y f ) and final heading (ψ f ). For a 3-D path, the initial and final configurations also include the elements found in the 2-D definitions, but also include both the z component of the position and a parameter defining flight path angle, γ. This relationship is demonstrated in Table 2. 2-D: C o = (x o, y o, ψ o ) C f = (x f, y f, ψ f ) 3-D: C o = (x o, y o, z o, ψ o, γ o ) C f = (x f, y f, z f, ψ f, γ f ) Table 2. Initial and Final Configurations for 2-D and 3-D paths. The development of the approach utilized in this paper was conducted by analyzing the geometry of the primitives utilized in the Dubins Car path. The turn primitive utilized in the Dubins Car is defined such that the vehicle remains in this turning maneuver until the vehicle is prescribed to transition into either a Straight primitive or to a different Turn primitive. If there is no second primitive for the vehicle to proceed onto, the vehicle will continue turning, eventually returning to its initial location, thus creating a circular path. It is plain to see this is true for turning primitives to both the left or right, as demonstrated in Figure 3. Note that the left and right turns in the Dubins Car can be considered the same maneuver rotated 18 degrees about the initial direction vector. 15 Y (ft) X (ft) Figure 3. Demonstration of the Left and Right Turning Primitives. As the Dubins Car is a 2-D path, comparisons are easily made to a ground vehicle which can only travel forward, turn left, or turn right. Since the 3-D path being developed is intended for a 5 of 15

6 vehicle which is not limited to only turning left or right, further considerations must be made for utilization of the Dubins Car theory in 3-D. An extremely aerobatic MAV can essentially turn, or maneuver, in any direction via the combination of roll rotation followed by a pitch rotation. This then leads to the recognition that this maneuver can be conducted at any angle rotated about the initial flight path vector, or maneuver angle, m. A visualization of this statement produces a horn torus. This is demonstrated in Figure 4. (a) 15 5 Z X Y (b) Figure 4. Rotation of a Turn primitive about the flight path vector: A) At maneuver angles of o, 9 o, 18 o, and 27 o. B) At all maneuver angles o 36 o with highlighted primitives at intervals of 45 o. 6 of 15

7 As stated before, the turning primitives utilized by the Dubins Car are defined by the maximum turn rate, ω, and consequently, the minimum turning radius of the ground vehicle, R t. The torus is similarly defined by the maximum maneuver rate, ω 3, which defines the minimum maneuvering radius, R m. For simplicity, a constant maximum maneuver rate is assumed regardless of the flight path vector or maneuver angle that the MAV is at when conducting the maneuver. Considerations about the MAVs maximum maneuvering rate at various configurations can be made if necessary. The addition of a straight primitive at the end of a turn primitive provides the opportunity to create a 2-D path from any initial configuration C o to any 2-D goal point (x f, y f ). This is accomplished by the vehicle traveling a prescribed distance along the turning primitive to orient itself for transfer to a straight primitive for travel to the desired 2-D position. This is demonstrated in Figure Y (ft) X (ft) Figure 5. The Turn-Straight primitive combination provides paths from any 2-D configuration to any 2-D location desired. Several example paths are demonstrated. Analogous to the 2-D Turn-Straight path, the attachment of a straight primitive at the end of a maneuver primitive produces the circumstances for 3-D path generation from any 3-D initial configuration C o to any 3-D goal point (x f, y f, z f ). Execution of this path generation is dependent upon proper travel along the surface of the torus, using a combination of maneuver angle and distance traveled along the circumference. Successful completion of this maneuver provides the MAV the opportunity to transfer onto a straight primitive for travel to any 3-D position desired. This is illustrated in Figure 6. 7 of 15

8 15 5 Z Y 3 4 X (a) 15 5 Z 5 15 X 4 3 Y 3 (b) Figure 6. The Maneuver-Straight primitive combination provides paths from any 3-D configuration to any 3-D location desired: A) Several example paths are demonstrated. B) Another view of the same set of examples. 8 of 15

9 As with the Dubins Car, the inclusion of an additional curving primitive after the straight primitive provides the ability to create a path such that the final orientation of the vehicle can be prescribed as well. Utilizing a second torus centered and oriented at the final configuration C f = (x f, y f, z f, ψ f, γ f ), the primitive sequence of maneuver-straight-maneuver produces the 3-D analogue to the 2-D Dubins Car. The first torus provides the visualization of the maneuver primitive from the initial configuration to the beginning of the straight primitive. The second torus provides the visualization of the maneuver primitive from the end of the straight primitive to the final configuration. Note that for simplicity, the solution presented here assumes that the initial and final configurations are sufficiently far enough away from one another such that the maneuvermaneuver-maneuver primitive combination defined by Dubins will not need to be utilized. The problematic issue is determining which maneuver angles, m o and m f, to utilize for both the initial maneuver and the final maneuver. As stated in Table 1, the Dubins Car utilizes a total of six possible primitive sequences to calculate the shortest distance path. This is due to the limiting factor of the vehicle being able to maneuver in only two directions: left or right. With this 3-D approach, the maneuver angle can be any value between o and 36 o for both the initial configuration and the final configuration. For this problem, the initial maneuver and the final maneuver each lie on their own planes, defined as M o and M f. These two planes can be calculated using any three points along each of the curving maneuvers respectively. The intention is to determine the proper combination of maneuvers which will define planes that are actually coplanar, as in the case of the 2-D Dubins Car. In most cases though a coplanar combination cannot be determined due to the orientations of the initial and final flight path vectors, i.e. the initial heading and flight path angles (ψ o, γ o ) and the final heading and flight path angles (ψ f, γ f ). In this situation a solution can be formulated from a combination of planes M o and M f that intersect such that the line of intersection between the two planes is the line on which the straight primitive lies along. Using this line of intersection, and the two respective maneuvers that define the two planes, the straight primitive can be calculated using the maneuver primitive end points as straight primitive end points such that they are tangent to the surface of the tori. In either case, iterating over a variety of maneuver angle combinations and subsequently refining the angle combinations will eventually provide the solution. An example of this is demonstrated in Figure 7. A MAV is initially configured in Euclidean space at an initial position of (4, 4, 3), with an initial heading of o (due north), and an initial flight path angle of o (level). The MAV has a known maximum maneuvering rate of ω 3 = 3 o /sec, which corresponds to a minimum maneuvering radius of R m = 76.4 feet. The MAV leaves its initial configuration C o at a maneuver angle m o = 129 o and pitches up, creating the first maneuver primitive along the outside of the visualized initial configuration torus. The MAV then leaves the surface of the torus and continues onto the straight primitive, diving to the southwest (decreasing x, decreasing y, decreasing z). Eventually, the MAV transitions onto the second maneuver primitive, connecting with the surface of the visualized final condition torus. Banking to a m f = 18 o and pitching up, the MAV finally reaches its prescribed goal configuration defined in Euclidean space at the location (,, 5), with the heading 27 o (due west), and a flight path angle of 45 o (nose down). 9 of 15

10 Z Y 6 4 (a) X Z X (b) Y Figure 7. A) A demonstration of the Maneuver-Straight-Maneuver primitive combination path. This combination produces paths from any initial 3-D configuration to any goal 3-D configuration. B) Another view of the same path. 1 of 15

11 IV. Example A. Motion Planning Algorithm This Three-Dimensional torus based motion primitive model is demonstrated by means of its utilization in a motion planning algorithm that utilizes random dense trees with three-dimensional motion primitives in an obstacle rich environment. 2, 26, 28 The approach uses the tree to expand nodes into the environment while utilizing trajectory primitives as branches that connect the nodes This approach accounts for obstacles using a pruning algorithm and 2-primitive path segments until the vehicle reaches a node from which a 3-primitive sequence can connect to the final configuration. The initial configuration is given as C o C 5 and the final configuration is given as C f C 5. The algorithm logic is as follows: 1. Select a Node: A point, C i+1 C 3, is randomly selected from the subspace of the feasibility space which is spanned by the position variables. This node is considered an extension beyond the closest node, C i C 5, of the current tree as determined by a distance metric. 2. Extend a Branch: A branch is generated to connect the current configuration, C i, with the next node, C i+1, in the tree. This branch is generated using a 2-primitive sequence composed of a three-dimensional maneuver primitive followed by a three-dimensional straight primitive. 3. Obstacle Avoidance: A pruning method is used to ensure obstacle avoidance. This method does not directly consider the location of the obstacles to optimize tree growth. The algorithm prunes any branches that intersect with an obstacle in the environment back to a safe node closer to the start of the tree. The growth of the tree occurs such that a node, C i+1, is valid if neither that node nor a path to that node intersect any obstacles. 4. Check for Solutions: The solution path from the new tree node to the final configuration is determined using the 3-primitive Maneuver-Straight-Maneuver sequence that connects two configurations in C 5 and the solution upper bound is updated, if necessary. This 3-primitive solution represents the trajectory between those configurations so it is assumed that additional nodes, and their associated sub-optimal 2-primitive solutions, will only increase the total cost of the motion planning. The expansion process evaluates if such a 3-primitive sequence exists between every tree node and the final configuration that does not intersect any obstacles. A crucial feature of the tree s expansion is the utilization of only the 3-D 2-primitive sequences (Maneuver-Straight) to connect the nodes C i to C i+1. While it is understood that these sections are not able to guarantee a desired heading or flight path angle at C i+1, it should be stressed that these constraints are only defined at the initial configuration C o and the final configuration C f. The nodes of the tree are are not associated with any heading or flight path constraints and, as such, a 2-primitive sequence is therefore computationally faster at generating node connections. B. Vehicle and Environment In this example, a simulated MAV needs to traverse an urban environment. The vehicle has the dynamic properties given in Table 3, which are based on measurements from a class of MAVs developed at the University of Florida. The MAV is regulated to a maximum flight path angle of 45 o to avoid any stall characteristics while traversing from node to node. Also, a distinct set of values for maneuver rate are chosen. While the minimum maneuver rate will produce the minimum distance from an initial configuration to another location, the path can benefit from the options in maneuver rate due to the inclusion of obstacles in the environment. 11 of 15

12 Property forward velocity, V maximum flight path angle, γ max maneuver rates, ω 3 Value 4 ft/s 45 o {, ±15, ±2, ±3} deg/s Table 3. Vehicle properties for examples. The vehicle s initial configuration C o is defined in Euclidean space by the starting position of (,, ), the initial heading of 3 o, and the initial flight path angle of o. It is required to travel to a final configuration C f defined in Euclidean space by the position (5, 5, ), a heading of 9 o, and a flight path angle of o. These values are detailed in Table 4. 2-D: C o = (x o, y o, z o, ψ o, γ o ) = (,,, 3 o, o ) 3-D: C f = (x f, y f, z f, ψ f, γ f ) = (5, 5,, 9 o, o ) Table 4. Initial and Final Configurations utilized for the example. Positions are defined by feet. The path between these configurations must avoid obstacles consisting of 2 large towers, 1 small tower, a covered walkway, and an elevated bridge. The details of the obstacles are presented in Table 5. Obstacle Coordinates of Center dx dy dz North Tower (5,25,) Covered Walkway (175,275,25) Northeast Tower (275,275,) Elevated Bridge (275,175,175) East Tower (25,5,) Table 5. Tower, walkway, and bridge dimensions and locations. All units in feet. C. Results A tree is grown to compute a sub-optimal trajectory as a combination of a series of 2-primitive sequences followed by a 3-primitive sequence. A solution is identified with a travel time of 28.4 s. The vehicle rotates up slightly with the first maneuver, then continues to climb to the northeast with the first straight primitive to a node on the tree. The MAV then rolls to approximately 6 o and pulls up creating a climbing right maneuver towards the east, continuing with a climbing straight primitive under the bridge. At this point a 3-primitive sequence can be conducted without obstacle interference. The MAV rolls to approximately 9 o and pulls to a northern heading where transition the straight primitive occurs. The MAV continues towards the north until transition to the final maneuver primitive is utilized to bring the MAV to its goal configuration. The complete path is presented in Figure 8. Three different viewing angles of the path and environment are plotted for improved visibility. 12 of 15

13 (a) (b) (c) Figure 8. The utilization of the 3-D torus based model in the motion planning algorithm is demonstrated. Three different views of the path are presented. The tori produce a visualization of the initial and final configurations. V. Conclusion Several approaches to the 3-D path generation problem have been made by researchers. The 3-D torus based motion primitive model presented here, designed for a specific class of aircraft, utilizes the highly maneuverable nature of a MAV for path planning. The legitimacy of the path is enhanced due to the consideration of the dynamics of the vehicle prior to the creation of the path. This torus based model, demonstrated here with the random dense tree motion planning algorithm, 13 of 15

14 can be utilized in several different autonomous guidance topics including target sensing, multiple waypoint navigation, and more. Further work with this model will include verifying the potential optimality of the model, as in the case with the Dubins Car model, as well as utilizing the model in various additional guidance, navigation, and control topics. References 1 Choset, H., Lynch, K., Hutchinson, S., Kantor, G., Burgard, W., Kavraki, L. and Thrun, S., Principles of Robot Motion: Theory, Algorithms, and Implementations, MIT Press, Cambridge, MA, 5, pp Faiz, N., Agrawal, S. and Murray, R., Trajectory Planning of Differentially Flat Systems with Dynamics and Inequalities, Journal of Guidance, Control, and Dynamics, Vol. 24, No. 2, 1, pp Fliess, M., Lévine, J., Martin, P. and Rouchon, P. Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples, International Journal of Control, Vol. 61, No. 6, 1995, pp Van Nieuwsadt, M.J. and Murray, R.M., Real-Time Trajectory Generation for Differentially Flat Systems, International Journal of Robust and Nonlinear Control, Vol. 8, No. 11, 1998, pp Kuwata, Y. and How, J., Three Dimensional Receding Horizon Control for UAVs, AIAA Richards, A. and How, J., Aircraft Trajectory Planning with Collision Avoidance Using Mixed Integer Linear Programming, Proceedings of the 2 IEEE American Control Conference, Anchorage, AK, May 2, pp Schouwenaars, T., How, J. and Feron, E., Receding Horizon Path Planning with Implicit Safety Guarantees, Proceedings of the 4 IEEE American Control Conference, Boston, MA, June 4, pp Frazzoli, E., Dahleh, M. and Feron, E., Real-Time Motion Planning for Agile Autonomous Vehicles, Journal of Guidance, Control, and Dynamics, Vol. 25, No. 1, January-February 2, pp Frazzoli, E., Dahleh, M. and Feron, E. Maneuver-Based Motion Planning for Nonlinear Systems with Symmetries, IEEE Transactions on Robotics, Vol. 21, No. 6, December 5, pp Hsu, D., Latombe, J.C. and Motwani, R., Path Planning in Expansive Configuration Spaces, Proceedings of the 1997 IEEE International Conference on Robotics and Automation, Albuquerque, NM, 1997, pp Kavraki, L.E., Svestka, P., Latombe, J.C. and Overmars, M.H., Probabilistic Roadmaps for Path Planning in High-Dimensional Configuration Spaces, IEEE Transactions on Robotics and Automation, Vol. 12, No. 4, August 1996, pp LaValle, S.M. and Kuffner, J.J., Randomized Kinodynamic Planning, International Journal of Robotics Research, Vol. 2, No. 5, May 1, pp Dubins, L., On Curves of Minimal Length with a Constraint on Average Curvature and with Prescribed Initial and Terminal Positions and Tangents, American Journal of Mathematics, Vol. 79, No. 1, 1957, pp Le Ny, J., and Feron, E., An Approximation Algorithm for the Curvature-Constrained Traveling Salesman Problem, Proceedings of the 43 rd Annual Allerton Conference on Communications, Control, and Computing, Monticello, IL, September 5, pp Howlett, J., Goodrich, M. and McLain, T., Learning Real-Time A Path Planner for Sensing Closely-Spaced Targets from an Aircraft, AIAA McGee, T.G. and Hedrick, J.K., Optimal Path Planning with a Kinematic Airplane Model, Journal of Guidance, Control, and Dynamics, Vol. 3, No. 2, March-April 7, pp Tang, Z. and Ozguner, U., Motion Planning for Multitarget Surveillance with Mobile Sensor Agents, IEEE Transactions on Robotics, Vol. 21, No. 5, October 5, pp Yang, G. and Kapila, V., Optimal Path Planning for Unmanned Vehicles with Kinematic and Tactical Constraints, Proceedings of the 41 st IEEE Conference on Decision and Control, Las Vegas, NV, December 2, pp Shanmugavel, M., Tsourdos, A., Zbikowski, R. and White, B. A., 3D Dubins Sets Based Coordinated Path Planning for Swarm of UAVs, AIAA Hurley, R., Lind, R. and Kehoe, J., A Mixed Local-Global Solution to Motion Planning within 3-D Environments, AIAA Guidance, Navigation and Control Conference, Chicago, IL, August 9, AIAA Chitsaz, H. and LaValle, S. M., On Time-optimal Paths for the Dubins Airplane, 7 IEEE Conference on Decision and Control New Orleans, LA, 7, pp Ambrosino, G., Ariola, M., Ciniglio, U., Corraro, F., Pironti, A. and Virgilio, M., Algorithms for 3D UAV Path Generation and Tracking, Proceedings of the 45th IEEE Conference on Design and Control, San Diego, CA, December 6, pp of 15

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