Conclusions

The most obvious conclusion from this research is that a vehicle may be guided to—and landed upon—a wire using stochastic, delayed, bearing-only measurements. Future systems that may benefit from energy harvesting are sensor limited, and most

systems of such size have only a monocular camera. Additional sensors may be

desirable for landing on power lines, but are not required. Furthermore, this study provides support that the Unscented Kalman Filter is a suitable estimation tool for such applications, and that real-time optimal control may be applied to direct a path that will acquire the level of target position certainty necessary to commit to a landing maneuver.

In the realm of trajectory optimization, several conclusions can be drawn from these efforts. The first is that there is a fundamentally different way to approach the localization and dual control problems that is more suitable and effective than traditional methods. The ubiquitous technique of optimizing a cost functional com-

prised of a scalar approximation of a multi-dimensional certainty metric has several disadvantages that are overcome with the methods developed in this work.

In the new approach, a user retains the directional information that was formerly approximated by a scalar, dispensing with difficulties of randomly odd-shaped uncer- tainty ellipsoids and other effects of information compression. This allows shaping of the uncertainty to match the physical requirements of the system, such as the actual shape of an arresting hook on a sUAS. Furthermore, previous methods minimized current uncertainty as much as possible, vice to a specific level. This research has now provided a way to prescribe the final uncertainty, which is the true requirement for mission accomplishment. Without this ability, a vehicle will maneuver as much as it can until an arbitrary time, or perhaps will balance the amount of maneuvering based on some arbitrary weight on the current certainty. Either way, it will not know whether it will achieve the necessary amount of target information, or whether it has wasted effort collecting too much information until the vehicle reaches the point where the information is required, when it is too late.

Early efforts provided a shooting solution which would allow a user to prescribe a final covariance. Trial solutions would be checked for the expected final covari- ance, iterating the weighed cost functional until the path produced yielded the right size and shape final certainty. This method was eclipsed by an elegant, single-shot solution that simultaneously handles the optimal control desires without weighting adjustments while meeting the physical information needs of the system. The single- shot solution was made possible by augmentation of the system state vector with states that contain an estimation in the polynomial space of the knowledge gained by the constellation of discrete measurements normally expressed by the Fisher In- formation Matrix. Dynamics were developed for these information states, and with care to avoid singularities, boundary conditions were enforced to ensure that by the

time the system arrives at the desired final state, it will have collected the appro- priate measurements, from the necessary angles, to finish the flight with the desired certainty in the target location estimate.

A further conclusion drawn is that the requirement of fixing a final time in the dual control problem can now be changed to a free final time. This was previously done either explicitly, or implicitly, through methods such as fixing a final distance with a given closure, fixing the total number of measurements with a given update rate, or by fixing an allowable travel proportion of the estimated distance to the target (with a constant speed). A fixed final time is a significant limitation for application to real systems beyond simulation. In reality, the time that will elapse during maneuvers not yet solved for is unknown, as is the number of measurements that will be required to meet the final mission requirements. The proportion of distance relative to the initial unknown distance is obviously also unknown. Choosing any of these, or more directly just choosing the fixed final time ends up being a primary driver of the characteristics of the solution trajectory. A solution that is truly optimal must be able to vary the problem geometry to get the required number of measurements from the necessary angles to accomplish the mission without limiting the set of possible solutions to those paths which end at a particular final time.

Several conclusions can also be drawn in the area of RTOC. The successful appli- cation of a recursive algorithm with pseudospectral methods as the engine working sequentially with a UFK receiving measurements from a bearing-only source is of great benefit. It validated the theory of disturbance rejection and the ability to use the speed of the pseudospectral methods to produce solutions that can guide in real- time. Applying the theory to real hardware produced several tools that were not required in previous PSM RTOC simulations, such as an intermediate function to ad-

dress the asynchronous timing loops between a control system and an unpredictable optimal solver.

This work clearly demonstrated that allowing the calculation time of the optimal solver to vary has great value, increasing the flexibility and responsiveness of the system by increasing the rate of available optimal solutions. The structure necessary to address the potential discontinuities that result from achieving this benefit was also designed and implemented, using a blending solution to ensure smooth and accurate control with the most current data from both the path planner and the estimation filter.

From a systematic perspective of basic RTOC implementation, this research showed that the trend in the RTOC community of equating closed-loop feedback control with a fast, recursive optimal solution is insufficient. This conclusion has developed over time as a byproduct of most of the RTOC applications being limited to simulation. Non-zero mean and time-correlated biases will cause steady-state errors that will be unaccounted for by a purely feed-forward solution. Though such a system will reach the final condition, the optimality of the path it takes is more of a mathematical construct than an operational reality. A more comprehensive and effective method must account for the anticipated future effects of disturbances and model inaccura- cies. This can be done through classical integration of the error between the expected and actual paths, and through feedback of disturbance estimates into the dynamical model for each optimal solver epoch. The ideal RTOC structure is to accomplish both, updating the model with estimates, and applying a total control solution that is a combination of the open-loop optimal solution and an integrated error feedback component.

Finally, this research provides a planning tool that may be used to develop heuris- tics for a suboptimal approach to landing on a power line that may be sufficient for

systems with significant computational limitations, as may be the case for many sUAS platforms. Re-creating the single-shot solution with the particular system dynamics and limitations would provide the characteristics common to optimal solution paths.