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Panel Method Based Motion Planning for Swarming MAVs with Probabilistic Target Tracking

Ilkay Yavrucuk1 and Oguz Uzol.2

Department of Aerospace Engineering, Middle East Technical University (METU),

Ankara, Turkey

This paper presents a novel methodology for motion path planning of swarming Micro Aerial Vehicles (MAVs) operating in an urban environment. The method makes use of the potential field panel method commonly used in fluid dynamics, and guarantees obstacle avoidance as well as collision avoidance, while allowing for tracking and scanning probable target locations in a complex urban environment. Vehicle velocities on each trajectory are related to the calculated streamline velocities and are adjusted to match vehicle performance. A probabilistic approach allows maximum area coverage and preferred area concentrations in presence of large number of MAVs. The method allows for easy handling of urban environments filled with complex shaped structures. Demonstrations are given using a synthetic urban environment as well as the geometrical layout of the downtown area of an actual city.

Nomenclature dqmn = velocity induced at m by a small vortex element at n. Kmn = coefficient matrix of vortex strengths rmn = distance between the vortex element and the point at which the velocity is induced sn = length of the surface element U∞ = x-component of the freestream velocity V∞ = y-component of the freestream velocity W∞ = freestream velocity vector x = x coordinate y = y coordinate α∞ = angle of attack β = angle between the surface element and the horizontal γ = vortex strength on a surface element

I. Introduction The problem of navigation and coordination of multiple unmanned vehicles is the topic of many recent research efforts. The problem becomes specialized when large numbers of vehicles are to navigate in swarms in complex environments. Moreover, when Micro Aerial Vehicles (MAVs) are considered, limited onboard sensor requirements make tasks like path planning, obstacle avoidance, collision avoidance, etc. a challenging task. Commonly this problem is divided into two parts: One is to solve the problem of navigation and obstacle avoidance. The other is to allow the MAVs to move in a swarm-like behavior, i.e. avoid collision and allow some sort of

1 Dr., Department of Aerospace Engineering, Middle East Technical University, 06531, Ankara, Turkey, AIAA member 2 Assistant Professor, Department of Aerospace Engineering, Middle East Technical University, 06531, Ankara, Turkey

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coordination. Hence the solution of this problem can quickly grow into a big computational intensive optimization problem hard to solve in real-time, sometimes even hard to deal with off-line.

While, a lot of research is done to solve this problem through classic large optimization problems or local awareness concepts of independent MAVs, some have attempted to propose this as a coordination problem of multi-agents. In this paper we borrow the concept of streamlines used to describe a flow behaviour in fluid dynamics. Previously a number of researchers have used similar potential field concepts to characterize the motion of multi-agents. The potential field approach for obstacle avoidance was first proposed by Khatib1 in 1986. In this work, goals were modeled as attractors and obstacles as repellers of an artificial potential field in which each vehicle moves. A more rigorous application of this idea to swarm robots and UAVs was proposed a few years ago2,3,4,5 and also more recently6. Though these methods did describe an application of potential field theory to swarm behavior using concepts of fluid dynamics, the relationship to the physical capacity of MAVs was left out. In particular, it was not clear if the unmanned systems would actually be able to follow the proposed trajectories.

In this paper we apply the idea of a potential field, using a more specific tool borrowed from the fluid

dynamics domain, namely the Potential Field Panel Method. The velocity field of an irrotational flow of an ideal fluid around a geometry can be obtained as the gradient of a “velocity potential” and therefore these flows are known as “potential flows” 7,8. The stream functions and streamlines that are obtained as a part of the potential flow solution are used for solving the problem of navigation with obstacle avoidance by previous researchers3,4,5. These streamlines can be obtained analytically by using and combining the complex potentials of flows like uniform flow, sink and vortex. However analytical solutions can only be used to generate obstacle avoiding trajectories around relatively simple geometries. In order to obtain streamlines (trajectories) around complex geometries, numerical methods known as “panel methods” are more appropriate. These panel methods are previously used for robot motion planning9 and real-time obstacle avoidance10. Since these methods allow us to obtain streamline-like trajectories around very complex shapes, we apply it here for calculation of MAV trajectories in complex urban environment. Moreover we also use the information of velocities that come with the solution of streamlines and attach a velocity profile to each trajectory, such that when a MAV rides on one of the trajectories, it both avoids obstacles, as well as is guaranteed not to hit another MAV, as no streamline intersects with another and each MAV on the same streamline will have a pre-defined velocity. The velocity profile calculated for each streamline is adjusted based on the performance of the MAVs. Streamline trajectories that cannot be followed are eliminated as not being fit for that particular type of MAV. Moreover, the generation of streamlines can be increased in some areas of the trajectory maps, if for instance it is desired to have more MAVs passing through a certain area. Here we use a probabilistic approach to generate more streamlines over a certain area. This way both maximum coverage around the total coverage area is enabled with statistically more MAVs passing through a preferred area. Two representative demonstrations will be presented: one using a synthetic urban environment, and another one using the geometrical layout of the downtown area of an actual city.

II. Potential Flow Vortex Panel Method Flow of an ideal irrotational fluid past a two-dimensional body of arbitrary shape, which is approximately

represented by straight line segments, can be obtained by distributing a vorticity sheet, initially of unknown strength, covering the entire surface of the body (Figure 1). The velocity induced at m by a small vortex element at n can be calculated using Biot-Savart law as7,8,

( )n n

mnmn

s dsdq

2 rγ

=π

(1)

The total induced velocity at m is obtained by adding up all the induced velocities due to all available surface vortex panels. This calculation is repeated for the finite number of line segments on the body surface and then the Dirichlet boundary condition that the velocity component normal to the body surface should be zero is applied. Application of this condition results in a system of linear equations as,

[ ][ ] [ ]mn n mK RHSγ = (2)

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where,

( ) ( )

( ) ( )m n m m n mn

mn 2 2m n m n

y y Cos x x SinsK2 x x y y

⎧ ⎫− β − − β∆ ⎪ ⎪= ⎨ ⎬π − + −⎪ ⎪⎩ ⎭ (3)

m m mRHS U Cos V Sin∞ ∞= − β − β (4) and γn are the vortex strengths to be solved. After solving the system of equations given in (2), the vortex element strengths, γn, are determined, and the

streamlines and the velocities in the flow domain can be calculated. The number of panels that is used for representing a geometry depends on the complexity of the geometry itself. For highly complex shapes, one needs many panels and this increases the computational time, especially during the matrix inversion process. In this study we use either a direct or an iterative solver depending on the number of panels needed to represent a complex urban environment. More details about the vortex-panel method can be found in Ref. 8.

III. Results

A. Collision and Obstacle Avoidance In order to demonstrate the technique, a two-dimensional geometrical layout of a synthetic urban

environment is generated, as presented in Figure 2a. The layout is composed of cross-sections of various geometrical structures that could be found in a modern city. These structures could essentially be of any arbitrary and complex shape since they are basically represented by a collection of straight panels as described above. The total number of panels can be selected as needed.

Figure 2b shows the calculated streamlines using the vortex-panel method described above. These streamlines are to be used as the trajectories of a swarm of MAVs that are released at varying y coordinates at a specified x location. Figure 2b shows 20 sample trajectories with release locations uniformly distributed along y. As is evident, due to the inherent nature of the fluid flow characteristics, the trajectories do not cross each other and automatically avoid any type of obstacles. Once the geometrical layout of an urban area is known, these trajectories can be pre-calculated, and the MAVs just have to choose and follow a streamline without having to locate and avoid obstacles with complex sensor devices and other hardware. Furthermore, for small aircraft like MAVs, which generally has their biggest dimension less than 15 cm, carrying on-board collision avoidance sensors is not usually possible or practical. Therefore this type of trajectory calculation could be useful for swarming MAV systems.

Figure 1. Surface representation of an arbitrary geometry by straight line segments and the velocity induced at m by a surface vortex element at n.

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B. Probabilistic Target Tracking The calculated MAV trajectories could easily be diverted to any specific region of a city for target tracking

and location purposes. The probability of a target existing within a search area could be specified either as a uniform or a Gaussian distribution with the most probable target location (mean) and a standard deviation. If the distribution is specified as uniform, then the MAV trajectories are calculated one by one starting from each and every grid point within the search area, as long as the grid points are not within a building structure. On the other hand if the Gaussian distribution is specified, then the trajectories are calculated randomly, based on the mean and standard deviation of the Gaussian distribution. As a result, more MAV trajectories are generated passing over the most probable location of the target.

Figure 3 presents two samples for MAV trajectory diversion for two different size uniform probability search areas at two different parts of the synthetic city. Figure 3a has a small target search area in the upper part of the city (marked by the red line) and Figure 3b has a larger search area at the lower part. Note how the trajectories are passing through the search areas.

Figure 4 shows two examples for target search when the possible target location is specified as a Gaussian distribution. In this case, the x location of the target is specified at x=10 and the y location has a mean of y=5 (most probable target location) with a standard deviation of 0.75. A set of 100 trajectories are generated randomly using this Gaussian distribution. Notice that the trajectories cluster together around the mean value and spread around according to the standard deviation value. Therefore most of the MAVs in the swarm pass over the most probable target location, which in turn increase the probability of locating the target. The release location of the MAVs can also be changed according to the need and still the MAVS can be diverted to focus on the most probable target location, as seen in Figures 4a and b.

C. MAV Performance Matching The panel method potential flow calculation not only gives the trajectories, but the results could also be

extended to obtain the vehicle velocities along the trajectories as well. This is achieved first by relating the calculated fluid velocities to vehicle velocities and then restricting the maximum centripetal acceleration along the trajectory to the maximum permissible vehicle acceleration. As a result of this technique, the vehicle can go as fast as it can along a selected trajectory in a safe manner, flying below the permissible limits. Furthermore, since the vehicle will be flying at a scaled value of the fluid velocity, there will be no collisions among the vehicles that choose the same trajectory (at different times) since all of them will be traveling like fluid particles with the same velocity vector at a corresponding position, hence making it impossible for them to catch one another.

This is what we refer to as the “performance matching” technique and it is demonstrated in Figure 5. A sample single MAV trajectory is shown in Figure 5a. Figure 5b shows the calculated fluid velocity (VFluid) along this trajectory (green line). Notice the change in velocity as the fluid moves between the obstacles. In narrow regions, for example between x=0 and x=2, the fluid accelerates and then subsequently decelerates in the wider parts. Instead of

(a) (b)

Figure 2. (a) A sample synthetic urban environment and (b) streamlines calculated by the potential flow vortex element panel method, to be used as MAV trajectories. Collision and Obstacle Avoidance are automatically guaranteed. Motion from left to right.

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Figure 3. MAV trajectories (streamlines) can easily be diverted and MAV release locations can be changed to cover possible target search areas: a wider target search area in the lower city (a) and a smaller target search area in the upper city (b).

(a) (b)

Figure 4. Target locations can be specified as probability distributions and the MAV trajectories are diverted accordingly. Above, Gaussian distribution at x=10 and around y=5 with a standard deviation of 0.75 in y. 100 trajectories are generated randomly according to this distribution. (a) and (b) show different release locations.

(a) (b)

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using this fluid velocity variation directly as the vehicle velocity, we choose to use 1/ VFluid to ensure that the vehicle actually slows down in narrow passages and speeds up in wider passages, keeping in mind that this would be beneficial in terms of vehicle safety. This, i.e. Vvehicle=1/ VFluid , is what we refer as the “initial” vehicle velocity in Figure 5b (red line). Next step is to check whether the vehicle could actually travel along this trajectory with this Vvehicle. Here, we use the maximum permissible centripetal acceleration that the vehicle can handle during a maneuver along the trajectory as the criteria. For this purpose the centripetal acceleration calculated using the trajectory curvature and vehicle initial velocity is plotted in Figure 5c (red line). We see that the maximum centripetal acceleration occurs at about x=7 with a value about a=2.8. If this value is higher than the maximum permissible vehicle acceleration, then the acceleration variation is scaled down such that the maximum value will be the maximum permissible vehicle value. As an example, this value is chosen to be avehicle=2.2, and the acceleration variation is scaled down accordingly and called the “performance matched” value (Figure 5c, blue line). The corresponding “performance matched” vehicle velocity variation along the trajectory is presented in Figure 5b (blue line). Therefore the vehicle will be able to fly with the performance matched velocity along the trajectory. The highest centripetal acceleration it will experience during maneuvering will be within its design limits. It will fly safe and slow in narrow passages and as fast as it can in the wider areas, and should two different vehicles choose the same trajectory at different times, they will not be able to catch each other so a possible collision will be avoided (of course we assume that all the vehicle within the swarm are identical in performance). Of course, a matched acceleration profile might aerodynamically be not achievable, for instance if the trajectory would require a lower speed than allowable. In that case, the trajectory would simply be not included in the usable set of trajectories. Moreover, it might be possible to rise the velocity profile to match the maximum performance of the vehicle, instead of reducing it for safety.

Figure 5. Vehicle velocities along the trajectories are determined from the calculated fluid velocities along the streamlines, and are rescaled to guarantee that the maximum vehicle centripetal acceleration along the trajectory is less than the permissible vehicle centripetal acceleration (performance matching). (a) A single MAV trajectory, (b) Fluid and Initial and Performance Matched Vehicle Velocities, (c) Initial and Performance Matched Vehicle Accelerations, along the trajectory on the left

(a) (b)

(c)

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Figure 6. (a) Downtown Phoenix, AZ obtained from Google Earth. Marked region (red rectangle) covers a four-block area; (b) calculated swarm-MAV trajectories (streamlines) within the marked area in (a), and (c) a close-up of the trajectories between buildings in the lower downtown area. Colors indicate the vehicle velocities.

(a)

(b) (c)

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D. Swarm-MAV Trajectory Planning in an Actual City The method described above is applied to the layout of an actual city. For this purpose downtown Phoenix,

AZ is chosen as an example, since it has nicely structured avenues and a well organized city plan. Figure 6a shows a satellite image of the downtown area obtained from Google Earth11. Main building structures are marked as white areas on this image and the actual scale of the image is given in the lower left-hand corner. The calculation procedure starts with marking the boundaries of the building structures directly on the image obtained from the Google Earth software. A utility code converts these marked points to surfaces and prepares an input file for the solver. These surfaces are divided into pre-determined number of panels in the potential flow solver and the panel vortex strengths are calculated. Once the vortex strengths are known, the velocities and streamlines are calculated on a background grid covering the entire calculation region.

The four-block area marked by the red rectangle in Figure 6a is arbitrarily chosen for the calculation of MAV trajectories. Figure 6b and 6c show the MAV trajectories (streamlines) calculated within this area (Figure 6c is a close-up of the lower right-hand area within the selected region), and colors indicate the vehicle velocities. As is evident, the trajectories automatically avoid the building structures and pass through the streets and roads between them, while the vehicles adjust their velocities by speeding up in wider areas and slowing down in relatively narrower zones.

IV. Conclusions In this paper we presented a potential flow panel method to generate trajectories for multiple MAVs

swarming through a city in low altitude. The method is used to generate trajectories as well as velocities for the MAVs, such that buildings and other obstacles are avoided and as each point of the trajectory has a unique velocity they will not collide. Moreover the trajectories are updated based on the performance characteristic of the vehicle.

The benefit of the method is seen in that the computations are relatively fast and allow the MAVs to move statistically more frequently to a specifically defined area. The concept is applied to a synthetic urban environment and to the downtown area of an actual city. Using simple mapping tools it is possible to generate obstacles of major city downtowns and allow the MAV swarms to move through streets and open areas. This idea of using streamlines and performance based velocity profiles allows an easy, fast and in many ways an optimized solution to the multi-agent swarming problems.

The downside one might claim is that we still need the full information of a city that is going to be “swarmed” and the information has to be uploaded to each MAV. However, for MAVs with little computation power and limited onboard sensing, information sharing will be an essential part. Also, the method in its current form is two-dimensional and does not present a way to overcome disturbances that might be present in real environments (e.g. wind turbulence). These will be investigated in our future studies.

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of Robotics Research, 5(1):90–98, 1986. 2. Waydo, S., and R.M. Murray, "Vehicle Motion Planning Using Stream Functions", in IEEE International

Conference on Robotics and Automation 2003. 3. Sullivan, J., S. Waydo and M. Campbell, "Using Stream Functions for Complex Behavior and Path

Generation", AIAA Guidance, Navigation and Control Conference 2003. 4. G.Ye ¤ H. O. Wang ¤ K. Tanaka,” Coordinated Motion Control of Swarms With Dynamic Connectivity in

Potentail Flows.” www.rc.mce.uec.ac.jp/wangh.pdf 5. Karin Sigurd and Jonathan How, “UAV Trajectory Design Using Total Field Collision Avoidance” AIAA

Guidance, Control and Navigation Confenerce, 2003 Austin, TX 6. Eun, Y., Bang, H., 2006, "Cooperative Control of Multiple Unmanned Air Vehicles Using the Potential

Field Theory," Journal of Aircraft, Vol. 43, No. 6, pp. 1805-1814. 7. Katz, J., Plotkin, A., 2001, Low Speed Aerodynamics, Cambridge University Press, Cambridge, UK,

Anderson, J.D., 2001, Fundamentals of Aerodynamics, McGraw-Hill 8. Lewis, R.I., 1991, Vortex Element Methods for Fluid Dynamic Analysis of Engineering Systems,

Cambridge University Press, Cambridge, UK 9. Zhang, Y., Valavanis, K.P., 1997, "A 3-D Potential Panel Method for Robot Motion Planning," Robotica,

Vol. 15, pp. 421-434. 10. Kim, J.O., Khosla, P., 1991, "Real-Time Obstacle Avoidance Using Harmonic Potential Functions,"

Proceedings of the IEEE International Conference on Robotics and Automation, Sacramento, California. 11. http://earth.google.com/