Once a target is located, location must be maintained, and the mission shifts from a strategic mode to a tactical mode. Tactical problems are smaller in dimension and can change more rapidly. As such, more precise path determination and following is required at this level. As the target may be stationary or moving, both loitering and dynamic paths are used. A concurrent requirement to target following is UAV position coordination. This is important not only for collision avoidance, but also to take advantage of simultaneously sensing the target from multiple UAVs with fixed relative position. Five different, minimal heuristic methods are considered and compared for performance and stability.

The throttle and elevator are hold the altitude and airspeed. Limited ability to vary speed is assumed available. In addition, side-slip will be zeroed via turn coordination. This kinematically relates the turn rate and bank angle. Bank angle hard limits and rate limits are imposed. First order dynamics relate speed and bank angle commands to aircraft response.

The first method is an extension of path following as developed here by Rysdyk [1]. However, regaining the path consists of stabilizing two degrees of freedom to zero: course angle error, and cross track distance. "Good Helmsman" behavior is implemented to relate the commanded course to the cross track error and thus resolve the problem of controlling an under-actuated system.

For two UAV coordination, both the lead and follower aircraft adjust speed and stand-off distance, oppositely, to control relative rotational rate about the target. This provides the necessary additional degree of freedom to achieve a desired angular separation between the two UAVs.

Figure 1. Two UAV Target Following, Maintaining 75 Degree Separation, Using Method 1

In Figure 1, the target makes a turn to the north. The target is moving at 20% of UAV speed. The combined effect of wind is shown in the lower right. Here, the wind blows due west at a speed equal to 20% that of the UAV nominal speed.

The second method builds upon the work done by Lawrence [2]. The guidance of the UAV to the observation 'orbit' is determined by building a vector field that has a stable limit cycle centered on the target position. Our work extends this approach by including a dynamic model for the UAV and allowing the target to move while simultaneously requiring the UAVs to maintain a constant angular separation with respect to the each other.

For two UAV coordination, the follower aircraft vector field is modified as follows. The stand-off radius of the limit cycle orbit is increased for positive clock angle separation error. This produces a heading angle command that tends to increase the UAV distance to target. In turn, the UAV's rotational rate about the target slows. This slow down, relative to the other UAV, results in increasing the clock angle separation and so decreases the positive error. A negative clock angle separation error will cause the opposite effect to decrease a negative error. In addition, the speed of each UAV is adjusted, slightly, and in opposite directions (1 up and 1 down) to reduce the oscillation magnitude.

Figure 3. Two UAV Target Following, Maintaining Clock Angle Separation, Using Method 2

In Figure 3, the same target and wind profiles are in effect. Note the amplitude of the oscillation is about five (5) times that of Method 1. The effect of wind is again considerable, causing the oscillation amplitude to double.

The third method builds upon the work done by Klein & Morgansen [3]. In [3], the guidance of unit speed planar kinematic unicycles is commanded such that the centroid of the 'flock' of unicycles corresponds with the moving target position. Our extension on this work applies this method to UAV dynamics, again attempting to impose a constant angular separation between two UAVs.

In [3], it is shown that this method is not valid for 2 UAVs. To compensate, a 'swarming' function is used to align the UAVs' headings to drive (or swarm) the UAVs towards the target. This causes the centroid to also drive to the target. In Figure 4 (left), two UAVs cannot keep up with a target, even though the target is only moving at 10% of the UAV speed. On the right, the 'swarming' function is used and the UAVs can keep up. Note that in this result, the oscillation model, as described in [3], is used for the aircraft.

Figure 4. Centroid Method Compensation Using a 'Swarming' Function

In Figure 5, the same target profile, as above, is in effect. The resulting clock angle separation oscillation is +/- 25 degrees, or 5 times greater than the Lyapunov method. Note that the oscillation is drifting as the centroid (in the red track) has drifted southward of the target. Also note that, as opposed to the first two methods, there is no direct control of clock angle separation (i.e. cannot specify). Rather, the separation is always 180 degrees from the centroid (for two vehicles). The clock separation with respect to the target depends on where the centroid is with respect to the target.

Figure 5. Two UAV Target Following, Maintaining Clock Angle Separation, Using Method 3

Model Predictive Control uses 'truth' models to predict future states based on current states given control inputs. To use this idea in real time, the model must accurately predict future states over a time period that exceeds that which is necessary to calculate the control inputs. See Figure 6.

In our model, the 'Leader' will follow the Lyapunov vector field, as in Method 2, but with constant velocity. The target will have a constant velocity profile, based on the last known velocity information, and the 'Follower' will have a constant heading / constant velocity profile. It is not possible to maintain constant clock angle separation if the 'Follower' has a constant heading since the Lyapunov vector field headings change continuously. The optimization is thus to find the 'Follower' constant heading command that minimizes the divergence in clock angle separation.

Figure 7. Two UAV Target Following, Maintaining Clock Angle Separation, Using Method 4

Figure 7 demonstrates the utility and drawback of real time optimization. Note that, in general, the method calculates a command that indeed keeps the clock angle separation divergence low. However, as this method produces less optimal commands, the time to calculate the command also increases (see Figure 8). The worst divergence is associated with the longest calculation time. In other words, real time optimization works as long as the initial conditions for the optimization are near the optimal conditions.

Evolution-Based Cooperative Planning System (ECoPS) was developed at AFSL as a strategic, cooperative planner. See Cooperative Planning for Teams of Heterogeneous Autonomous Vehicles for details. ECoPS uses evolutionary-based computation to determine, in real time, a feasible vehicle path based on a selection and mutation scheme of a plan consisting of multiple potential paths. Our goal here is to evolve ECoPS to be able to transition to the tactical problem of target tracking and back to the strategic mission phase, when the tactical mission is over. By using the same structural framework and evolutionary ideas already in place, we are increasing the modularity of ECoPS to be able to generate paths for multiple mission types, simply by logically switching to an appropriate set of evolutionary plan types and fitness methods.

Unlike a standard MPC problem, ECoPS utilizes dynamic replanning. Predictive models are still used; however, the optimality of the solution depends largely on how long is given to calculate. But, a solution is always available, no matter how short the calculation time given. This method is preferrable in a environment that either is too complex or changes too rapidly for the MPC method.

The key factor that makes dynamic replanning work is that all solutions given by the optimization routine are feasible. In ECoPS, no path is created as a solution that is outside of the performance limitations of the UAV (turn rate, max speed, accelearation capability, etc.). This is combined with evolutionary techniques:

given a set of N paths

The solution, at any time, is just the highest ranked path. To make the optimality dynamic, the ranking is done with parameters that depend on future predictions based on current conditions.

Figure 9 shows the inadequacy of ECoPS to determine paths necessary for the tactical orbit problem. Here, the target is stationary. ECoPS in a strategic mode has a basic mission of locate and evade. When the UAV nears the target, it determines that the probability of locating the target is high and thus plans to head toward the goal. However, as the UAV passes near and heads away, the task is not complete, so the algorithm replans. The result is an endless loop.

To transition to the tactical tracking mission, it is important to keep the time and distance scale decrease in mind. The time scale decrease is especially important as we are solving an optimization problem and, as described above, the extent to which the solution is optimized depends largely on the time alloted. To this end, reducing the time necessary to reach an optimized solution amounts to reducing the calculation time of the evolutionary loop. Computational simplifications such as the flat earth approximation is appropriate given the UAV maximum speed and the short time scale. It is also reasonable to assume that good solutions stay 'near' the target. The distance scale has also decreased and so we can say that the path lengths must also decrease accordingly.

Another major difference in the tactical problem is that there is no defined 'goal' or end state to the problem. Where the UAV will be when the tracking mission is determined over depends on factors that cannot be accurately guessed that far into the future, such as target position. As such, tactical paths have a 'receding time horizon'. In other words, every potential path in the evolution has the same length, as measured in time projected into the future. As time passes, the path end time lengthens to keep the total path time constant.

Figure 10. Target Tracking Using Tactical ECoPS

Figure 10 demonstrates that ECoPS in a tactical formulation can perform target tracking. We are now pursuing the cooperative target tracking problem by sharing 'best paths' between UAVs to keep the clock angle separation constant.

1. Rysdyk, R., “UAV Path Following for Constant Line of Sight”, 2nd AIAA “Unmanned Unlimited” Systems, Technologies and Operations Conference, Sept 15-18 2003, San Diego, CA
2. Lawrence, D., "Lyapunov Fields for UAV Flock Coordination", 2nd AIAA “Unmanned Unlimited” Systems, Technologies and Operations Conference, Sept 15-18 2003, San Diego, CA
3. D. Klein and K. Morgansen., "Controlled Collective Motion for Trajectory Tracking", Submitted to the American Control Conference, 2005