Three-Degrees-of-Freedom Simulations

During preliminary design, system characteristics are very often not known in detail. The aerodynamics can only be given in trimmed form, and the autopilot structure can be greatly simplified. Fortunately, the trajectory of the c. m. of the vehicle is usually of greater interest than its attitude motions, and, therefore, the simple three-DoF simulations are very useful in the preliminary design of aerospace vehicles.

Newton’s second law governs the three translational DoF. The aerodynamic, propulsive, and gravitational forces must be given. In contrast to six-DoF simula­tions, Euler’s law is not used to calculate body rates and attitudes; therefore, there is no need to hunt for the aerodynamic and propulsive moments.

Suppose we build a three-DoF simulation for a hypersonic vehicle. We use the J2000 inertial frame of Chapter 3 for Newton’s law. The inertial position and velocity components are directly integrated, but the aerodynamic forces of lift and drag are given in velocity coordinates. Therefore, we also need a TM of velocity wrt inertial coordinates to convert the forces to inertial coordinates.

The equations of motion are derived from Newton’s law, Eq. (5.9):

mD’vg = fap + mg (5.28)

where m is the vehicle mass and vlB is the velocity of the missile c. m. В wrt the inertial reference frame I. Surface forces are aerodynamic and propulsive forces fa p, and the gravitational volume force is mg. Although v’B is the inertial velocity, we also need the geographic velocity vB to compute lift and drag. Let us derive a relationship between the two velocities.

The position of the inertial reference frame I is oriented in the solar ecliptic, and one point I is collocated with the center of Earth. The Earth frame E is fixed with the geoid and rotates with the angular velocity u)EI. By definition the inertial velocity is v! B = D’sfii. where sBi is the location of the vehicle’s c. m. wrt point I. To introduce the geographic velocity, we change the reference frame to E

D1Sbi = DE$bi + EIe, Sbi (5.29)

and introduce a reference point E on Earth (any point), Sg/ = Sbe + %/, into the first right-hand term

DEsBi — Desbe + Desei = Desbe = vf

where Desei is zero because sei is constant in the Earth frame. Substituting into Eq. (5.29), we obtain a relationship between the inertial and geographic velocities

vb ~ vb +MEIsbi (5.30)

For computer implementation Eq. (5.28) is converted to matrices by introducing coordinate systems. The left side is integrated in inertial coordinates ];, while the aerodynamic and propulsive forces are expressed in velocity coordinates and the gravitational acceleration in geographic coordinates ]G. The details of obtaining the TMs are given in Chapter 3. We just emphasize here that we have to distinguish the two velocity coordinate systems. The one associated with the inertial velocity vB is called Jy, and the geographic velocity coordinate system is |v. With these provisions we have the form of the translational equations of motion:

= lTl’G([T]GUlfa, p]u + m[g]G) (5.31)

These are the first three differential equations to be solved for the inertial velocity components [Vg]1. The second set of differential equations calculates the inertial position

= K]’ (5-32)

Both equations are at the heart of a three-DoF simulation. You can find them implemented in the CADAC GHAME3 simulation of a hypersonic vehicle.

If you stay closer to Earth, like flying in the Falcon jet fighter, you can simplify your simulation by substituting Earth as an inertial frame. In Eqs. (5.31) and (5.32) you replace frame I and point I by frame E and point E. The distinction between inertial and geographic velocity disappears, and the geographic coordinate system is replaced by the local-level system ]L:

L

= [T]LV[fa, p]v +rn[g]L (5.33)

These equations are quite useful for simple near-Earth trajectory work.

You will find more details in Chapter 8 with other useful information about the aerodynamic and propulsive forces. To experience an actual computer implemen­tation run the CAD AC GHAME3 simulation.