N15A-T006 TITLE: Pseudospectral Optimal Control for Flight Trajectory Optimization

TECHNOLOGY AREAS: Weapons

OBJECTIVE: Develop a software application that applies pseudospectral optimal control theory to guided aerial vehicle applications such as missiles, bombs, targets, and unmanned aerial systems for the purpose of optimizing system performance within given constraints.

DESCRIPTION: Any flying vehicle has a limited amount of energy with which to perform its mission, whether it is an air defense missile maximizing range and terminal maneuverability, a guided bomb maximizing speed at impact, or an unmanned aerial vehicle maximizing endurance on station. A common challenge for these systems is developing guidance algorithms that manage the energy to achieve maximum kinematic performance within various constraints on the vehicle's size, weight, payload, cost, or other factors. Changing warfighter requirements also often bring needs for longer range, enhanced maneuverability, or trajectory shaping for survivability. Further compounding these difficulties is budget pressures that demand making do with existing systems and adapting them to meet evolving needs instead of developing entirely new weapons, targets, or unmanned aerial systems (UAS).

Recent developments in optimal control theory may provide a significant technology to address these challenges. Optimal control has long been an attractive means of developing algorithms to maximize flight performance. However, until recently the theory was difficult to apply in practice because of the computation time needed to search the space of possible solutions. New mathematical breakthroughs and today�s computation power offer the potential of practical solutions that can be used real time aboard guided vehicles. Approaches such as the pseudospectral optimal control method have been demonstrated on spaceflight applications and show considerable improvements in optimizing energy management to achieve vehicle maneuvers. The goal of this effort is to extend these techniques to the more challenging non-linear dynamics of atmospheric flight.

The purpose of this topic is to develop a software application that applies optimal control methods to the autopilot of an unmanned guided aerial vehicle (such as a weapon, target, or UAS) in real time. The optimization method will be integrated with a generic but representative autopilot and set of guidance algorithms. The integrated algorithms will be tested and implemented in a hardware test environment to provide a real-time guidance solution for the vehicle. Such a solution obviates the need for methods such as pre-stored lofting tables or a multi-phase guidance implementation (i.e., boost/midcourse/terminal). The software provided shall demonstrate the capability to optimize trajectory characteristics such as: launch envelope, range, time on target, terminal velocity, terminal g-capability, impact angle, and trajectory shaping.

The implementation must allow for re-computable solutions to address physical phenomena that vary over the course of a flight, such as changes in wind speed and direction or thrust misalignment induced by a rocket motor. The algorithmic solution must be fast enough to re-compute the guidance solution at five times the time constant of the guidance and control system.

PHASE I: Develop a conceptual design for a pseudospectral optimal control solution method applicable to a generic guided aerial vehicle that can feasibly run real time onboard the vehicle.

PHASE II: Develop prototype optimization software in an all-digital six degree of freedom simulation environment that demonstrates real-time computational ability to perform the following tasks: (1) increase the launch acceptable envelope, (2) provide simultaneous arrival time on target (through platform pre-coordination), (3) increase standoff range by computing optimal loft trajectory, (4) optimize terminal conditions to achieve a desired range of impacts angles or speeds or g-capability, and (5) re-compute the optimal trajectory solution for flight path deviations experienced after launch caused by phenomena such as winds or thrust misalignments induced by a rocket motor.

PHASE III: Demonstrate integrated performance of the optimal control solver on guidance and control hardware via software-in-the-loop (SWIL) or hardware-in-the-loop (HWIL) simulations. Successful completion of this task should demonstrate maturity to transition to a program of record.

PRIVATE SECTOR COMMERCIAL POTENTIAL/DUAL-USE APPLICATIONS: This technology can also be employed to advantage in any application requiring autonomous operation, such as robotics, unmanned aerial ground vehicles, and spacecraft.

REFERENCES:
1. Fahroo, F. & Ross, I. M. (2008, August). Advances in pseudospectral methods for optimal control. Proceedings of the AIAA Guidance, Navigation, and Control Conference. Paper number AIAA 2008-7309. Paper presented at AIAA Guidance, Navigation, and Control Conference 2008, Honolulu, Hawaii. Retrieved from http://www.elissarglobal.com/wp-content/uploads/2012/04/Advances-in-Pseudospectral-Methods-for-Optimal-Control.pdf

2. Ross, I. M. (2006). Space trajectory optimization and L1-optimal control problems. In Gurfil, P. (Ed.), Modern astrodynamics (pp. 155 188). Oxford, UK: Elsevier Ltd.

3. Fahroo, F. & Ross, I. M. (2002). Direct trajectory optimization by a Chebyshev pseudospectral method. Journal of Guidance, Control, and Dynamics, Vol. 25, No. 1, 160�166.

4. Benson, D.A., Huntington, G.T., Thorvaldsen, T.P., & Rao, A.V. (2006, November�December). Direct trajectory optimization and costate estimation via an orthogonal collocation method. Journal of Guidance, Control, and Dynamic, Vol. 29, No. 6, 1435�1440.

5. Rao, A. V., Benson, D. A., Darby, C. L., Patterson, M. A., Francolin, C., Sanders, I., & Huntington, G. T. (2010, April�June). Algorithm 902: GPOPS, a matlab software for solving multiple-phase optimal control problems using the gauss pseudospectral method. ACM Transactions on Mathematical Software, Vol. 37, No. 2, Article 22, 22:1-22:39.

KEYWORDS: Optimization; Weapon; Guidance and Control; Optimal Control; Trajectory Shaping; Missile

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