In this work, we start with a generic mathematical framework for the equations of motion (EOM) in flight mechanics with six degrees of freedom (6-DOF) for a general (not necessarily symmetric) fixed-wing aircraft. This mathematical framework incorporates (1) body axes (fixed in the airplane at its center of gravity), (2) inertial axes (fixed in the earth/ground at the take-off point), wind axes (aligned with the flight path/course), (3) spherical flight path angles (azimuth angle measured clockwise from the geographic north, and elevation angle measured above the horizon plane), and (4) spherical flight angles (angle of attack and sideslip angle). We then manipulate these equations of motion to derive a customized version suitable for inverse simulation flight mechanics, where a target flight trajectory is specified while a set of corresponding necessary flight controls to achieve that maneuver are predicted. We then present a numerical procedure for integrating the developed inverse simulation (InvSim) system in time; utilizing (1) symbolic mathematics, (2) explicit fourth-order Runge-Kutta (RK4) numerical integration technique, and (3) expressions based on the finite difference method (FDM); such that the four necessary control variables (engine thrust force, ailerons' deflection angle, elevators' deflection angle, and rudder's deflection angle) are computed as discrete values over the entire maneuver time, and these calculated control values enable the airplane to achieve the desired flight trajectory, which is specified by three inertial Cartesian coordinates of the airplane, in addition to the Euler's roll angle. We finally demonstrate the proposed numerical procedure of flight mechanics inverse simulation (InvSim).

翻译：在本研究中，我们首先建立了一个通用的飞行力学六自由度（6-DOF）运动方程（EOM）数学框架，适用于一般（不一定对称）固定翼飞机。该数学框架包含：（1）机体轴系（固定于飞机重心）、（2）惯性轴系（固定于地面/起飞点）、风轴系（与飞行航迹/航线对齐）、（3）球面飞行航迹角（方位角从地理北向顺时针测量，仰角测量于地平面上方）以及（4）球面飞行角度（攻角与侧滑角）。随后，我们通过推导运动方程，得到适用于逆仿真飞行力学的定制版本，其中指定目标飞行轨迹，同时预测实现该机动所需的一组对应飞行控制量。接着，我们提出了一种数值方法，用于对开发的逆仿真（InvSim）系统进行时间积分；该方法综合运用（1）符号数学、（2）显式四阶龙格-库塔（RK4）数值积分技术以及（3）基于有限差分法（FDM）的表达式；从而计算出四个必要控制变量（发动机推力、副翼偏转角、升降舵偏转角及方向舵偏转角）在整个机动时间内的离散值，这些计算得到的控制值使飞机能够实现期望的飞行轨迹——该轨迹由飞机的三个惯性笛卡尔坐标及欧拉滚转角共同定义。最后，我们演示了所提出的飞行力学逆仿真（InvSim）数值流程。