Toward Real-Time Team Cooperative Control Steve Rogers, 25 March 2006

Abstract Many studies on the subject of team cooperative control have been performed. Recently, there has been a dramatic upswing of interest in the subject. This study has to do with demonstration of a heuristic solution to the problem of accomplishing stated sequential goals with respect to geographically dispersed targets or objective locations. As an example, such sequential goals may include object classification, attack, and destruction verification. These tasks require sequential time-critical order of accomplishment and cooperation among the team members. This paper shows a potential real-time heuristic sub-optimal O(n) approach to the solution of this problem and compares the answer to published optimal results.

Introduction When targets are discovered multiple tasks must be prosecuted sequentially subject to timing constraints. These tasks are performed in a fixed order to accomplish a mission or tour. When multiple targets are discovered the problem may become extremely complex -. This type of problem is coupled multiple-assignment task allocation. The optimization model is run so that all of the known targets or objectives are prosecuted by the allocation in the specific sequence. For a simple problem the network state transition diagram is shown as:

Figure 1

State Transition Diagram for n=2 Targets, w=3 Team Members

Schumacher etal, devised a MILP (mixed integer linear program) approach to solve the problem. Their approach relies on a cost function and numerous constraints to enforce a usable solution. The following table shows the increase of constraints with increasing size of the problem statement. 1

n (targets)

w (vehicles)

2 2 3

3 4 4

Binary decision variables 33 44 88

Continuous decision variables 8 9 11

Linear equality constraints 13 16 22

Linear inequality constraints 76 100 194

Note that the number of constraints and variables is linear in the number of vehicles, but quadratic in the number of targets. The approach developed for this paper is based on the Johnson Algorithm ,, which was designed for solving job shop or machine flow shop problems. Johnson’s rule may be stated as below for a two-machine flow shop makespan problem with dynamic availability but static arrivals. Step 1: Schedule the group of jobs U that are shorter on the first machine than the second; U = { j | p j1 ≤ p j 2 } as the first priority group. Step 2: Schedule the group of jobs V that are shorter on the second machine; U = { j | p j 2 ≤ p j1 } as the second priority group. Step 3: Schedule within U by SPT on the first machine. Step 4: Schedule within V by LPT on the second machine. An example will be useful to explain the application to autonomous vehicle team cooperative control. The initial condition of a problem is defined by the relative distances, or flight times, between the nodes. Let the time matrix T for w = 3 vehicles and n = 2 targets be: 0 5.831 7.071 1 7.280 1 9.219 5

5.831 0 7.2111 5 3.6056

The start node i is indexed down the rows, and the end node j is indexed over the columns. Thus, the time for a vehicle to fly from node 2 to node 1, T 2,1 = 5.831. The time for a vehicle to fly from node 3 to node 2, T 3,2 = 7.2111. The diagonal elements where i = j corresponds with a vehicle starting and ending a task at the same node, which is not allowed. The target nodes are the columns and the vehicle nodes are rows 3 – 5. The tasks are to classify/attack and battle damage assessment. Attacks are considered suicide missions and following attacks the vehicle is eliminated from the minimization problem. We assume that each team member may prosecute each task equally well. We eliminate the vehicles by setting the appropriate row to infinity.

2

Now we shall apply the Johnson Rule to the above minimization problem, which is shown in Figure 2. Since the vehicles are at nodes 3 â€“ 5 we first find the minimum of the matrix 7.071 1 7.280 1 9.219 5

7.2111 5 3.6056

Figure 2 State Transition Diagram for n=2 Targets, w=3 Team Example

The minimum is T5,2 = 3.6056, which means that vehicle 3 at node 5 classifies/attacks target 2 at 3.6056 seconds. We now update the T matrix where vehicle 3 is eliminated. 7.071 1 7.280 1 inf

7.2111 5 inf

We then minimize the T matrix to get vehicle 2 at node 4 verifies target 2 at 5 seconds. We now update the T matrix where target 2 is eliminated and vehicle 2 is now located at target 2. 7.0711 13.111 1 inf

inf inf inf

We again minimize the T matrix to get vehicle 1 at node 3 classifies/attacks target 1 at 7.0711 seconds. We now update the T matrix where vehicle 1 is eliminated and vehicle 2 is now located at target 2. inf

3

inf

13.111 1 inf

inf inf

We again minimize the T matrix to get vehicle 2 at node 4 verifies target 1 at 13.1111 seconds. The tasking is now complete. For such a small problem the solution matches the published answers, exactly. Note that minimization of the T matrix requires that the iterations be the number of tasks times the number of targets, which is O(n). This is fast enough to make possible real-time processing of team cooperative control. Of course, the T matrix is not static between each iteration in the above procedure. This is the same as in the MILP approach,.

Simulation Results The Johnsonâ€™s rule approach was implemented in matlab and compared to the published results. The matlab script file for testing is in Figure 3.

Figure 3 Team Cooperative Control Test Script

The Johnsonâ€™s rule is coded in Figure 4 below.

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Figure 4 Johnsonâ€™s Rule matlab script

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A time comparison between the above and the published times is shown below. Since this is a heuristic approach the solutions for targets in excess of 3 do not match the published results. The solutions are suboptimal and produce flyable trajectories. # vehicles # targets # tasks Published w N Per target computational time (sec) 3 2 2 0.05 3 2 3 1.06 5 2 2 0.5 5 2 3 8.4 4 3 2 6.07 4 3 3 1620

Proposed computational time (sec) 0.03 0.033 0.035 0.037 0.038 0.04

The matlab scripts were executed on a latitude laptop. The times are comparable for small problem sizes, however, the O(n) computational time method shows its advantages for larger problem sizes. Again, since the proposed approach outlined in the paper is heuristic and suboptimal the tours do not match with the optimal published results when the target number is three or larger. In the above table the number of tasks per target is varied from two to three. The total tasks for this analysis is classify, attack, and battle damage assessment or verify. To get two tasks classify and attack are combined to one task. To get three tasks classify and attack are not combined.

Conclusions An approach that shows promise for real-time solution of team cooperative control problems based on the Johnson’s Rule or Johnson’s algorithm was given. It was applied to a published problem and compared to published computational times. Although it is heuristic and suboptimal, a flyable trajectory is virtually guaranteed.

References 1. Schumacher, C. etal, ‘Constrained Optimization for UAV Task Assignment,’ AIAA 2004-5352, AIAA GNC Conference, 16-19 August 2004, Providence, RI. 2. Schumacher, C. etal, ‘Task Allocation for Wide Area Search Munitions via Iterative Network Flow,’ Proceedings of the 2002 AIAA GNC Conference. 3. Schumacher, C. etal, ‘Task Allocation for Wide Area Search Munitions with Variable Path Length,’ Proceedings of the 2003 AIAA American Control Conference. 4. Schumacher, C. etal, ‘UAV Task Assignment with Timing Constraints,’ Proceedings of the 2003 AIAA GNC Conference. 5. Alighanbardi, M., etal, ‘Coordination and Control of Multiple UAVs with Timing Constraints and Loitering,’ Proceedings of the 2003 American Control Conference. 6. Johnson, S. M., ‘Optimal Two- and Three-Stage Production Scheduling with Setup Times Included,’ Naval Research Logistics Quarterly 1, 61-68, 1954. 6

7. Morton, T. etal, Heuristic Scheduling Systems, Wiley, 1993, ISBN 0-471-57819-3. 8. Research & Education Association, Problem Solvers - Operations Research, 1996, ISBN 0-87891-548-8.

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