Six-Degrees-of-Freedom Simulation

We have assembled all of the gear we need to embark on the ultimate trip of full six degrees of freedom. In Chapter 5 you learned to model the three translational degrees of freedom, and Chapter 6 taught you the secrets of three rotational degrees of freedom. Newton and Euler should have become your favorite companions.

You also have become familiar with simple simulations. Chapter 8 instructed you how to build a three-DoF simulation and under what circumstances it can satisfy your needs. Then you moved up one notch and added two more degrees of freedom in Chapter 9 to arrive at that peculiar form of pseudo-five-DoF simulations. Both have important roles to play during the development cycle of an aerospace vehicle. As an engineer, you have to adjust your tools to the task at hand and the resources available. Start simple! It is better to have a reliable three-DoF simulation ready in time than a nonvalidated and tardy six-DoF simulation.

Eventually, the time will come when you have to strike out and go for maximum fidelity. But count your cost, and let your sponsor know the extensive resources he has to set aside. You will need lots of data: full aerodynamic and thrust tables— trimmed data will no longer suffice; a complete flight control design; mass pa­rameters that include moments of inertia; and other subsystems, like sensors and guidance logic. You also will need lots of time. I have yet to finish a simulation, completely verified and validated, within the projected period. From experience. I know that once a simulation runs reasonably well it takes another third of the development time to produce a reliable product.

The challenge is great, but very rewarding. I will help you with the details. We develop together two types of equations of motion. For in-atmosphere, near-Earth flight we assume the flat Earth to be the inertial frame. Tactical missiles and aircraft fall into this category, and I will add spinning missiles and Magnus rotors to the collection. Quaternions will serve us to compute the attitude angles.

Other types of equations are required for hypersonic and orbital vehicles. They are based on an inertial frame aligned with the solar ecliptic and convecting with the center of the Earth—the inertial frame J2000 you encountered in Chapter 3. We shall study the elliptical geometry of Earth, the gravitational field, and introduce the geodetic coordinate system. Then I shall formulate the equations of motion relative to the inertial and geographic frames. The attitude angles will be computed from the direction cosine matrix equations.

Next, we have to flesh out the right-hand side of the equations of motion. I present aerodynamic models for airplanes, hypersonic vehicles, and missiles. The autopilot for six-DoF simulations requires an entirely new treatment with such options as rate damping loop, roll autopilot, heading angle, and flight-path-angle controller, acceleration autopilot, and altitude hold autopilot. Actuators are now also part of the control loop.

All of the seeker and guidance options of five-DoF simulations can directly be integrated into your six-DoF. I shall just expand your horizons. The modeling of INS errors becomes now important; extensions of proportional navigation and the advanced guidance law will improve the intercept performance of your missile, and a gimbaled IR sensor provides you with a modem missile seeker.

This chapter will not satisfy all of your needs. Its purpose is mainly to furnish enough examples so that you can develop the specialized components for your own simulation. It should also help you decipher six-DoF simulations when you are asked to evaluate or to modify them. For that reason, I refer you also to the simula­tion examples on the CADAC CD, which enrich this chapter; theFALCON6 aircraft simulation, the GHAME6 hypersonic vehicle, and the SRAAM6 short – range air-to-air missile.