Equations of Motion Extension to Suborbital Flight

Suborbital flight is flight within the atmosphere but at extremely high altitudes. In this regime, flight speeds are very high, and the curving of constant-altitude flighttrajectories around the earth’s surface adds appreciable centrifugal force to wing lift. Bryan’s equations

of rigid-body motion are for flight over a flat earth. Flat-earth equations of motion generally are inadequate for airplanes that operate in a suborbital mode.

A derivation of nonlinear airplane equations of motion for the spherical-earth case can be found in Etkin (1972). The main distinction between the spherical – or oblate-earth cases and the classical Bryan flat-earth equations lies in additional kinematic (nondifferential) equations. As in the ordinary flat-earth equations, 12 state equations must be integrated. In the Etkin approach, linear accelerations are integrated in airplane body axes, producing the usual inertial velocity and angle of attack and sideslip variables. However, this is only one of several possible choices for the linear accelerations. The angular acceleration equations of motion are integrated in airplane body axes, as for the flat-earth case. This is the only practical choice, since airplane moments and products of inertia are constant only in body axes.

Full nonlinear equations of airplane motion about a spherical or oblate rotating earth were produced somewhat later at Rockwell International in connection with the Space Shuttle Orbiter and still later for studies of the National Aerospace Plane (NASP). The earliest set is found in Rockwell Report SD78-SH-0070, whose authors we have been unable to identify. Six distinct reference axes systems are used. The Rockwell set integrates linear accelerations and velocities in an earth-centered inertial axis system, making transformations to the other axes, such as the body and airport reference sets.

Still another approach was followed at the NASA Dryden Flight Research Center (Powers and Schilling, 1980, 1985) for the Space Shuttle Orbiter, in order to build on an earlier flat-earth 6-DOF computer model. A heading coordinate frame is centered at the orbiter’s center of gravity, with the Z-axis pointed to the earth’s center and the X-axis aligned with the direction of motion. X and Z define the orbit plane through the geocenter. Linear accelerations and velocities are integrated in heading coordinate and earth axes frames, respectively Vehicle vertical and horizontal velocities in the orbit plane and body axis heading relative to the orbit plane replace the ordinary body axis velocity coordinates in the airplane’s state vector. Altitude above a reference sphere of equatorial radius, latitude, and longitude replace the ordinary altitude, downrange and cross-range position coordinates in the airplane’s state vector. High precision data, such as FORTRAN double precision with 15 significant figures, are needed.

Attitude deviations from the Rockwell/Dryden heading coordinate frame produce Euler angles in the classical sense: yaw, then pitch, then roll. Use of this particular heading coordinate system also for space or re-entry vehicles would produce a consistent set of aerospace flight mechanics axes, which would seem to be an advantage.

The oblate earth version of the equations of airplane motion is sometimes used even when there is no question of hypersonic or suborbital flight operations. This is in flight simulators when one wishes to have only one set of airplane equations for both flying qualities and long-range navigation studies. A single, unified airplane mathematical model for both purposes avoids duplication of costly manned flight simulators and the problem of keeping current two different data bases during airplane development. For simulated flights lasting on the order of hours, correct latitude and longitude coordinates can be calculated as inputs to flight data computers.

The almost incredible capacity of modern digital computers makes it feasible to expend computing capacity by including high-frequency airplane dynamics terms in the flight simulation of an hours-long navigational mission, as compared with spending engineering time to develop a special simulation without the high-frequency terms. This was the route chosen for the Northrop B-2, according to our best information.