Cooperative Control of a Team of Unmanned Vehicles Using Smoothed Particle Hydrodynamics

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1 Cooperative Control of a Team of Unmanned Vehicles Using Smoothed Particle Hydrodynamics Doug Lipinski and Kamran Mohseni A procedure is outlined for trajectory planning based on Lagrangian coherent structures (LCS) and using a fluid cooperative control algorithm. Smoothed particle hydrodynamics (SPH) was used to obtain a reduced model of the fluid cooperation. LCS provide insight in flow transport and allow for plausible trajectory planning while fluid based control provides a simple and uniform control mechanism with built in collision avoidance. The resulting near optimal trajectories are compared to truely optimal trajectories that are found by numerically solving an optimal control problem given a cost function which combines fuel and time costs. The LCS based trajectories with fluid control provide good results, but fail to reproduce the optimal trajectories in terms of pure fuel savings or pure speed. Finally, a hybrid approach is proposed. This approach solves the optimal control problem for a single vehicle and then uses SPH control to allow a swarm of vehicles to follow this single trajectory while maintaining spacing and avoiding collisions. Nomenclature T Integration time for LCS Φ Flow map Cauchy-Green deformation tensor λ max Maximum eigenvalue of σ Finite time Lyapunov exponent ψ Streamfunction for double gyre velocity field A Amplitude of double gyre flow ɛ Amplitude of double gyre perturbation ω Angular frequency of oscillating perturbation to double gyre flow u, v Velocity of the double gyre flow v max Maximum velocity of the double gyre flow X, Y Coordinates in physical space Ẋ, Ẏ Vehicle velocities L Domain scale in meters x, y Rescaled coordinates used to compute double gyre velocity C Cost function to be optimized C time Coefficient telling relative time cost C fuel Coefficient telling relative fuel cost t start time end time t f I. Introduction In the unmanned aerial vehicle (UAV) and unmanned underwater vehicle (UUV) communities, it is often desired to deploy a swarm of vehicles from a departure point with the objective of traveling through their fluid medium to a goal location. Possible applications of such a cooperative control problem include atmospheric research 8 and chemical plume detection 5. There are many possibilities for trajectory planning and obstacle/collision avoidance in such situations. Here, we will explore the use of a fluid based control method using smoothed particle hydrodynamics (SPH) for control and collision avoidance for a swarm of UUVs. We will also employ Lagrangian coherent structures (LCS) for trajectory planning. Smoothed particle hydrodynamics (SPH) was developed as a means of simulating complex problems in astrophysics. The method uses discrete, Lagrangian particles to simulate fluids. Each particle is assigned physical properties such as mass which are applied using a smoothing kernel that may be Gaussian or, more commonly, a compactly supported spline function. The Lagrangian nature of SPH makes this method very well suited to problems Ph.D. Candidate, Applied Mathematics, University of Colorado - Boulder, AIAA Student Member Associate Prof., Aerospace Engineering Sciences, University of Colorado - Boulder, AIAA Associate Fellow of 7

2 with large deformations and multiple phases of matter with greatly differing properties, such as astrophysics and solid mechanics problems. This paper, along with previous work 6,4, proposes to use SPH as a fluid based control method for cooperative control of a swarm of vehicles. Other authors have previously used SPH control for a 2D swarm of robots 9,2. By treating each vehicle as a fluid particle, collision and obstacle avoidance is built into the control system. Additionally, goals may be created by using particles with negative density, that act as attractors for the regular vehicle particles 4. The swarm will move toward a goal region while automatically avoiding collisions and obstacles. Additionally, the SPH computations in these simulations are very efficient since a relatively small number of particles is required (one for each vehicle or goal) and a compactly supported kernel is used so vehicles need only consider other vehicles nearby. In fact, SPH based collision avoidance has been used in several UAV test flights where all SPH computations were performed in realtime using onboard electronics. To aid in trajectory planning, Lagrangian coherent structures (LCS) will be used to provide insight into the behavior of the background flow field. In any situation where vehicles must travel through a moving fluid, it is advantageous to use the existing flow to aid in traveling. Doing so may result in significant time and energy savings. LCS have seen increasingly wide application in the fluid dynamics community over the past several years. Application range from swimming jellyfish to three-dimensional turbulence 3. Since LCS act as barriers to transport, they also reveal the mechanisms for transport and are commonly applied to system where transport is of significant interest. By taking advantage of the underlying transport in a background flow, UUVs or UAVs may move more efficiently through a fluid. In fact, Inanc et al. have previously explored the correlation between LCS and optimal trajectories for a single vehicle in oceanic flows 7. In this paper the use of LCS as the basis for ad hoc trajectory planning is explored and combined with SPH control. The resulting trajectories are compared to optimal trajectories based on minimizing a cost function via a nonlinear programming (NLP) problem. It is concluded that LCS do indeed provide a good understanding of relatively simple flows and one may construct a vehicle trajectory based on LCS trajectory planning with SPH based collision avoidance and control. II. Smoothed Particle Hydrodynamics Although there are many possible implementations of fluid based control algorithms, we have chosen to use smoothed particle hydrodynamics. This method also provides a simple, efficient means of controlling a swarm of vehicles and includes built in collision avoidance capabilities. Each vehicle is treated as a particle in the SPH computation and assigned appropriate material properties based on the desired vehicle speed and spacing. Goal regions are created by placing a particle in the chosen area and assigning a negative density to that particle. Given the current state of the system, all current particle positions, velocities and properties, the force on a particle may be computed by summing force terms over all other particles within the domain of support of the particle kernel (see Monaghan and the references therein for an overview of SPH techniques).the vehicle corresponding to that particle must then accelerate in the appropriate direction, based on the net force and particle mass. In real world applications, this computation may be carried out by each individual vehicle, updating speed and heading appropriately at fixed time intervals. In this investigation, SPH calculations are incorporated into a fourth order Runge-Kutta time integrator to update vehicle positions. If the vehicles are traveling through a background flow, one must also account for the motion of the fluid medium. In practice, the fluid and the vehicle interact in complex ways depending on vehicle orientation and relative velocity difference. For simplicity, it is assumed here that all vehicles would normally travel as passive drifters with the background velocity in the absence of propulsion. The SPH particle velocities are computed in the absence of the background flow and particle positions are then updated based on the superposition of SPH velocity and background flow velocity. This is an appropriate approximation as long as v h << where h is the width of the smoothing kernel used in SPH computations. In all flows used here, v h O( 4 ) so this is a valid approximation. III. LCS based trajectory planning LCS based on the finite time Lyapunov exponent (FTLE) field were proposed by Haller and Yuan 4 and further developed by Shadden et al. 3. LCS have several important properties, most notably, there is negligible fluid flux through LCS 3. For completeness, several definitions relevant to LCS from Shadden et al. 3 are repeated here. LCS are based on the FTLE field, which is computed by examining the flow map that maps initial particle positions to final particle positions. The gradient of the flow map reveals stretching in the flow and can reveal important information about 2 of 7

3 transport. The FTLE is defined as t+t Φt t+t (x) = x(t ) + = ( dφ dx t ) v(x(t))dt () (2) ) ( dφ dx σ T t (x) = T ln λ max ( ) (3) where T is the integration time, Φ is the flow map, is the finite time Cauchy-Green deformation tensor, and σ is the FTLE. The LCS are then defined as ridges in the FTLE field and may either be explicitly extracted or (more commonly) visualized by viewing a contour plot of the FTLE field. Additionally, there are two types of LCS, forward and backward, since one may consider both positive and negative T. It is well known that LCS act as barriers to transport and so reveal the underlying transport in a fluid flow. By carefully examining the LCS, it is possible to develop an understanding of a how a fluid flow will transport particles and gain insight into the most efficient means of transport. This may be as simple as traveling clockwise around a clockwise rotating vortex, but it may be much more complicated as well, involving careful timing and precise movement to take advantage of the existing transport. IV. Problem set up In this investigation, a simple time dependent double gyre flow will be considered and the LCS will be used to develop a planned trajectory that uses the underlying flow. This double gyre example has been used extensively in other LCS applications and is chosen because it is well understood and bears resemblance to ocean gyres. The flow is given by the stream function ψ(x, y, t) = A sin(πf(x, t)) sin(πy) (4) where The velocity field is then given by f(x, t) = a(t)x 2 + b(t)x, a(t) = ɛ sin(ωt), (5) b(t) = 2ɛ sin(ωt). u = ψ y, (6) v = ψ x. This results in two counter-rotating gyres in the domain [, 2] [, ] with an oscillating perturbation. The amplitude of the perturbation is controlled by ɛ. Typical the parameters for this system are A =., ɛ =. and ω = 2π/. However, have chosen to rescale the domain and parameters while maintaining the same transport structure. This is done by choosing physical parameters L = m, (7) v max = m/s corresponding to a 2km by km domain with maximum flow speed of m/s. The velocity field is then computed by using x = X/L, y = Y/L, A = v max /π, ɛ =., (8) ω = 2v max L. where X and Y are the physical variables used in the simulation. The vehicles used are also given a maximum speed of m/s. These velocities and spatial scales are typical of UUVs in the ocean. All simulations involve a vehicle or vehicles moving from a departure location of X = km, Y = km to a final position of X = 5km, Y = 5km. In fluid based control multi-vehicle simulations, the vehicle spacing was set to 2km, much larger than would typically be required, but necessary so that individual vehicles are visible at this scale. Finally, the departure time was chosen to be t = 2hrs 28min, a time when one of the LCS lobes passes near the departure point, although trajectories using the truly optimal departure time were also computed as described below. 3 of 7

4 V. Optimal trajectories As a basis for comparison, the truly optimal trajectories for the double gyre flow were computed. To define optimal trajectories, the cost function C = C time (t f t ) + C fuel tf t ( (Ẋ u)2 + (Ẏ v)2) dt (9) was used. This represents a balance between the fuel needed for the trajectory and desirability of arriving quickly. The departure and arrival times, t and t f, may also be parameters of optimization. This cost function assumes that fuel use is proportional airspeed (or waterspeed) squared, although the constant of proportionality is unknown and will depend on vehicle dynamics and environment. Depending on the needs of a given mission, the importance of conserving fuel versus conserving time may be adjusted relative to one another by adjusting C time and C fuel relative to one another. This optimization problem is solved by using the OPTRAGEN 2. toolbox for MATLAB, interfaced with the free student version of SNOPT 2. OPTRAGEN translates optimal control problems to nonlinear programming (NLP) problems by parameterizing trajectories as splines. The NLP problem is then solved numerically via SNOPT. It is important to note that minimization process used to solve the NLP problems may converge to a local (not global) minimum and can be sensitive to the initial guesses for the trajectory and other parameters. However, the results achieved here agree over a wide range of initial guesses as well as intuition Figure. Fluid based control and truly optimal trajectories. Videos of all trajectories can be seen at edu/ mohseni/tmp/aiaagnc2.html The blue and red curves are forward and backward LCS respectively. Each trajectory begins at the green and ends at the red. Images are: Fluid control with one vehicle (top left), Fluid control with six vehicles (top right), Optimal fuel consumption (middle left), Optimal fuel consumption with optimal departure time (middle right), Optimal travel time (bottom left), Optimal travel time with optimal departure time (bottom right) 4 of 7

5 Total energy usage =9965 Total energy usage = x 5 Total energy usage = x 5 Total energy usage = x 5 Total energy usage = x 5 Total energy usage = x x 5 Figure 2. Time dependent for fluid based control and truly optimal trajectories. Power consumption is known only to a constant of proportionality so quantities shown are unitless. The power for multi-vehicle trajectories is the average per vehicle. Images are: Fluid control with one vehicle (top left), Fluid control with six vehicles (top right), Optimal fuel consumption (middle left), Optimal fuel consumption with optimal departure time (middle right), Optimal travel time (bottom left), Optimal travel time with optimal departure time (bottom right) 5 of 7

6 VI. Results Videos of all trajectories reported here can be viewed online at mohseni/tmp/aiaagnc2.html As seen in Fig., all trajectories begin in the bottom left of the domain (at the green ), travel up and to the right, then back down toward the destination point. The time optimized trajectories (logically) follow a more direct path to the destination, while the fuel optimized trajectories loop around the destination before arriving. The fluid control trajectories were planned using the LCS seen in the flow. The blue LCS curve (lobe) seen at the left side of the domain periodically moves fluid from left to right. The trajectory was created by placing a negative density particle in this lobe to attract vehicles into the lobe. Once the vehicles reach the negative density particle, it is removed from the simulation and the vehicles drift passively with the flow. This is clearly seen in Fig. 2 as the power consumption drops to zero during the period of passive drifting. Fluid control is still active, but serves only to prevent collisions at this point. This accounts for the small spikes seen in the middle time range of the six vehicle trajectory with fluid control. These quick spikes in fuel consumption result from the SPH control enforcing the desired vehicle spacing. Finally, when the vehicles reach the right half of the domain, a negative density particle is added at the destination and the vehicles are drawn to the goal. The power consumption for the vehicles (Fig. 2) clearly reflects the type of trajectory used. In the case of multiple vehicle trajectories, the average power consumption is reported. Time optimized trajectories use maximum power for the entire trajectory to reach the goal as quickly as possible. On the other hand, fuel optimized trajectories take a long time to reach the goal, but use very little fuel. The fluid based control trajectories lie somewhere between these two extremes. Table shows the results of the LCS based trajectory planning with fluid control, as well as several optimal trajectories with different optimization parameters. Optimal trajectories have been computed to minimize travel time as well or fuel cost while departing at the same time as the fluid controlled trajectories. Additionally, optimal trajectories have been computed using a flexible departure time. Table. Results of SPH control and optimal trajectories, fuel use is per vehicle Description C time C fuel Departure Travel Normalized Time Time Fuel Usage vehicle SPH - - 2hrs 28min 97hrs min. 4 vehicle SPH - - 2hrs 28min 98hrs 26min vehicle SPH - - 2hrs 28min 98hrs 8min vehicle SPH - - 2hrs 28min 98hrs 58min.98 Optimized time 2hrs 28min 38hrs 4min Optimized time with optimal t 22hrs 2min 37hrs 59min Optimized fuel 2hrs 28min 77hrs min.4 Optimized fuel with optimal t 2hrs 35min 78hrs 3min.23 Note that the fuel usage (averaged over all vehicles) is not significantly effected by adding more vehicles to the swarm while using fluid control. In fact, much of the variability in the fuel usage for the fluid controlled trajectories is due to uncertainty in the end time since the entire swarm is never exactly on the goal. On the other hand, trajectories optimized for fuel efficiency use only 2-4% as much fuel as the fluid based control, but require about.8 times as much travel time (7.5 days versus 4). Meanwhile, time-optimized trajectories require only 38 hours of travel time, but use over 6.8 times as much fuel. VII. Conclusions We have seen that LCS based trajectory planning with fluid based control is a viable option for trajectory planning. The LCS based trajectories result in moderate fuel usage while maintaining reasonable travel times. However, without numerical optimization techniques, it is nearly impossible to achieve truly optimal travel times or minimal fuel usage. In many situations it is impractical to use numerical optimization, especially with large swarms of vehicles. The computational costs quickly become prohibitive for large systems. On the other hand, while LCS can provide some guidance in trajectory planning, the end results are far from optimal in terms of fuel usage and travel time. An ideal system would combine the best of each technique applied here to achieve nearly optimal trajectories for large swarms of vehicles without large computational costs. 6 of 7

7 As seen in Table, the fuel usage and travel time are relatively insensitive to the number of vehicles in a swarm. This raises the possibility of using numerical optimization to generate a single optimal trajectory and then guiding a swarm of vehicles along this trajectory using SPH control. Once the optimal trajectory is generated, the swarm may be guided along this path by moving a negative density particle along the trajectory. By constraining the optimal trajectory s speed to be slightly less than that of an individual vehicle, the entire swarm will be able to keep up. This approach should provide an excellent compromise between efficient motion and cheap computational cost. Acknowledgements D.L. would like to thank Apratim Shaw for his help in using the SPH code described in Apratim and Mohseni 4. References R. Bhattacharya. OPTRAGEN: A MATLAB toolbox for optimal trajectory generation. In 45th IEEE Conference on Decision and Control, pages , San Diego, CA, USA, December P.E. Gill, W. Murray, and M.A. Saunders. Snopt: An sqp algorithm for large-scale constrained optimization. SIAM Review, 47():99 3, M. A. Green, C. W. Rowley, and G. Haller. Detection of Lagrangian coherent structures in 3d turbulence. J. Fluid Mech., 572:, G. Haller and G. Yuan. Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D, 47:352, 2. 5 A.B. Hasan, B. Pisano, S. Panichsakul, P. Gray, J. Huang, R. Han, D.A. Lawrence, and K. Mohseni. SensorFlock: A mobile system of networked micro-air vehicles. Technical Report Technical Report CU-CS-8-6, Department of Computer Science, University of Colorado, Boulder, CO, December S. Huhn and K. Mohseni. Cooperative control of a team of AUVs using smoothed particle hydrodynamics with restricted communication. In Proceedings of the ASME 28th International Conference on Ocean, Offshore and Arctic Engineering, number OMAE , Honalulu, HA, May 3-June T. Inanc, S.C. Shadden, and J.E. Marsden. Optimal trajectory generation in ocean flows. In 24th American Control Conf., pages , Portland, Oregon, USA, D.A. Lawrence, K. Mohseni, and R. Han. Information energy for sensor-reactive UAV flock control. AIAA paper , Chicago, Illinois, 2-23 September 24. 3rd AIAA Unmanned Unlimited Technical Conference, Workshop and Exhibit. 9 R.C. Mesquita L.C.A. Pimenta, M.L. Mendes and G.A.S. Pereira. Fluids in electrostatic fields: An analogy for multirobot control. IEEE Transactions on Magnetics, 43(4), 27. D. Lipinski and K. Mohseni. Flow structures and fluid transport for the hydromedusae Sarsia tubulosa and Aequorea victoria. J. Exp. Biology, 22:2436, 29. J. Monaghan. Smoothed particle hydrodynamics. Annual Review of Astronomy and Astrophysics, 3: , L.C.A. Pimenta, N. Michael, R.C. Mesquita, G.A.S. Pereira, and V. Kumar. Control of swarms based on hydrodynamic models. In IEEE International Conference on Robotics and Automation, May S. C. Shadden, F. Lekien, and J. E. Marsden. Definition and properties of Lagrangian coherent structures. Physica D, 22(3-4):27, A. Shaw and K. Mohseni. A fluid based coordination of a wireless sensor network of unmanned aerial vehicles: 3d simulation and wireless communication characterization. IEEE Sensors Journal, Issue on Cognitive Sensor Networks, 2. accepted for publication. 7 of 7