Five-Degrees-of-Freedom Simulations

If the point-mass model of an aerospace vehicle, as implemented in three-DoF simulations, does not adequately represent the dynamics, one can expand the model by two more DoF. For a skid-to-tum missile pitch and yaw attitude dynamics are added, whereas for a bank-to-tum aircraft, pitch and bank angles are used. Euler’s law could be used to formulate the additional differential equations. However, the increase in complexity approaches that of a full six-DoF simulation. To maintain the simple features of a three-DoF simulation and, at the same time, account for attitude dynamics, one adds the transfer functions of the closed-loop autopilot to the point-mass dynamics. This approach, with linearized attitude dynamics, is called a pseudo-five-DoF simulation.

The implementation uses the translational equations of motion, formulated from Newton’s law and expressed in flight-path coordinates. The state variables and their derivatives are the speed of vehicle c. m. wrt Earth: V — vEfi | and dV/dV, the heading angle and rate x and dx/dt; and the flight-path angle and rate у and dy/dt. One key variable, the angular velocity of the vehicle wrt the Earth frame a>BE, is not available directly because Euler’s equations are not solved. Therefore, it must be pieced together from two other vectors where V is the frame associated with the geographic velocity vector vf of the vehicle. The two angular velocities can be calculated because their angular rates and angles are available from the autopilot. The incidence rates are obtained from angle of attack a, sideslip angle /3, and bank angle ф

u>riv = f(a, a, p, $) skid-to-tum u>BV = f(a, d, ф, ф) bank-to-tum

and the flight-path angle rates

WVE = fix, X, r, y)

Thus, the solution of the attitude differential equations is replaced by kinematic calculations.

We formulate the translational equations for near-Earth trajectories, invoking the flat-Earth assumption and the local-level coordinate system. Application of Newton’s law yields

mDEv = /ДіР + mg

with aerodynamic, propulsive, and gravity forces as externally applied forces. The rotational time derivative is taken wrt the inertial Earth frame E. Using Euler’s
transformation, we change it to the velocity frame V

DvvEB + nVEvEB = ^-+g m

and use the velocity coordinate system to create the matrix equation

(5.35)

The rotational time derivative is simply [DvvB]v — [V 0 0]. The aerodynamic and thrust forces are given in body coordinates, thus [fa, p]v — [TRV[fa. pR, whereas the gravity acceleration is best expressed in local-level coordinates [g]v — [T]VL[g]L. With these terms and the angular velocity

—X sin у

[a>VEf = у

X cos у

we can solve Eq. (5.35) for the three state variables V, x, and у

The vehicle’s position is calculated from the differential equations

These are the translational equations of motion for pseudo-five-DoF simulations. The details, and particularly the derivation of Eq. (5.36), can be found in Chapter 9. Note that a singularity occurs at у = ±90 deg.

Pseudo-five-DoF simulations have an important place in modeling and simula­tion of aerospace vehicles. They can easily be assembled from trimmed aerody­namic data and simple autopilot designs. Surprisingly, they give a realistic picture of the translational and rotational dynamics unless large angles and cross-coupling effects dominate the simulation. Trajectory studies, performance investigations, and guidance and navigation (outer-loop) evaluations can be executed success­fully with pseudo-five-DoF simulations.

Chapter 9 is devoted to much more detail. There you find examples for aerody­namics, propulsion, autopilots, guidance, and navigation models, both for missile and aircraft. The CADAC CD offers application simulations of air-to-air and cruise missiles AIM5, SRAAM5, and CRUISE5.