Aircraft Equations of Motion Derivation

Understanding the fundamental equations governing an aircraft's motion is critical for flight simulation, control system design, and performance analysis. These equations form the mathematical bedrock that allows engineers to predict how an aircraft will respond to control inputs and external disturbances. This article derives the complete six-degree-of-freedom (6-DOF) nonlinear equations of motion for a rigid aircraft, establishing a framework you can later simplify for specific flight conditions like steady cruise or small disturbances.

1. Foundational Assumptions and Reference Frames

The derivation begins by establishing clear assumptions and coordinate systems. We treat the aircraft as a rigid body, meaning the distance between any two points on the airframe is constant, neglecting structural flexibility. We also assume a flat, non-rotating Earth, which is valid for most short-duration flight dynamics analysis. The two primary reference frames are the body-fixed axes (Fb) and the Earth-based inertial frame (Fi).

The body-fixed frame is attached to the aircraft, typically with its origin at the center of gravity (CG). The x-axis (xb) points forward through the nose, the y-axis (yb) points out the right wing, and the z-axis (zb) points downward, completing a right-handed system. The Newton-Euler approach is used, applying Newton's second law for translational motion relative to an inertial frame and Euler's equations for rotational motion about the body's CG. However, expressing vectors in the rotating body frame introduces additional terms, which we will account for in the next section.

2. Translational Dynamics: The Force Equations

Newton's second law states that the sum of external forces equals the time rate of change of linear momentum. For a rigid body with constant mass m, this is $\sum \vec{F}=m\frac{d\vec{V}}{dt}{|}_i$, where the derivative is taken in the inertial frame (Fi). The external forces include aerodynamic (${\vec{F}}_A$), propulsive/thrust (${\vec{F}}_T$), and gravitational (${\vec{F}}_G$) components.

The challenge is that the velocity vector $\vec{V}=[u,v,w{]}^T$ is most naturally expressed in the body-fixed frame. The derivative of a vector in a rotating frame (like Fb) relative to an inertial frame is given by the transport theorem: $$\frac{d\vec{V}}{dt}{|}_i=\frac{d\vec{V}}{dt}{|}_b+\vec{\omega }\times \vec{V}$$ Here, $\vec{\omega }=[p,q,r{]}^T$ is the angular velocity of the body frame relative to the inertial frame, expressed in body axes. Applying this theorem yields the force equations in body axes: $$\begin{array}{rl}\sum F_x& =m(\dot{u}+qw-rv)\\ \sum F_y& =m(\dot{v}+ru-pw)\\ \sum F_z& =m(\dot{w}+pv-qu)\end{array}$$ The terms like $qw-rv$ are Coriolis acceleration terms arising from the rotation of the body frame. The force summations are: $\sum F_x=X_A+X_T+X_G$, and similarly for y and z, where gravitational components must be transformed from the inertial frame using Euler angles.

3. Rotational Dynamics: The Moment Equations

Euler's equation governs rotational motion: the sum of external moments about the CG equals the time rate of change of angular momentum. For a rigid body, angular momentum is $\vec{H}=I\vec{\omega }$, where $I$ is the constant inertia tensor in body axes. The law states $\sum \vec{M}=\frac{d\vec{H}}{dt}{|}_i$.

Again, applying the transport theorem to the derivative of $\vec{H}$ gives: $$\frac{d\vec{H}}{dt}{|}_i=\frac{d(I\vec{\omega })}{dt}{|}_b+\vec{\omega }\times (I\vec{\omega })=I\dot{\vec{\omega }}+\vec{\omega }\times (I\vec{\omega })$$ Assuming the body axes are aligned with the principal axes of inertia, the inertia tensor simplifies to a diagonal matrix $I=diag(I_x,I_y,I_z)$. This leads to the scalar moment equations: $$\begin{array}{rl}\sum L& =I_x\dot{p}-(I_y-I_z)qr\\ \sum M& =I_y\dot{q}-(I_z-I_x)rp\\ \sum N& =I_z\dot{r}-(I_x-I_y)pq\end{array}$$ Here, $\sum L,\sum M,\sum N$ are the sums of rolling, pitching, and yawing moments from aerodynamic and propulsive sources. The coupled terms like $(I_y-I_z)qr$ represent gyroscopic moments.

4. Kinematic Relations: Orientation and Position

The dynamic equations describe how forces and moments change velocities ($u,v,w$) and angular rates ($p,q,r$). To know the aircraft's actual orientation and position, we need kinematic equations. Orientation is described by the Euler angles ($\varphi$, $\theta$, $\psi$)—roll, pitch, and yaw—which define the rotation from the Earth-fixed inertial frame to the body frame.

The relationship between the Euler angle rates and the body angular rates $\vec{\omega }$ is given by: $$\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]=\left[\begin{array}{ccc}1& \sin \varphi \tan \theta & \cos \varphi \tan \theta \\ 0& \cos \varphi & -\sin \varphi \\ 0& \sin \varphi \sec \theta & \cos \varphi \sec \theta \end{array}\right]\left[\begin{array}{c}p\\ q\\ r\end{array}\right]$$ This equation has a singularity at $\theta =\pm 90^{\circ }$ (pitch straight up/down), which is a limitation of the Euler angle representation. For position, we transform the body-axis velocity $\vec{V}$ to Earth-frame coordinates (${\dot{x}}_E,{\dot{y}}_E,\dot{h}$) using the Euler angle rotation matrix: $$\left[\begin{array}{c}{\dot{x}}_E\\ {\dot{y}}_E\\ -\dot{h}\end{array}\right]=R(\varphi ,\theta ,\psi {)}^T\left[\begin{array}{c}u\\ v\\ w\end{array}\right]$$ where $R$ is the standard 3-2-1 (yaw-pitch-roll) rotation matrix, and $h$ is altitude.

5. Simplification for Specific Flight Conditions

The complete set of 12 equations (3 force, 3 moment, 3 angular kinematic, 3 positional kinematic) is coupled and nonlinear. To analyze specific scenarios, we apply simplifying assumptions. For symmetric, steady, level flight (the "trim" condition), all angular rates and side velocity are zero ($v=p=q=r=0$), and forces/moments are balanced. This reduces the equations to algebraic conditions for trim.

For stability and control analysis, we use the small-disturbance theory. We assume the aircraft's motion consists of a known trim condition plus a small perturbation. Each state variable is written as $u=u_0+\Delta u$, and similarly for others. The nonlinear equations are then expanded in a Taylor series about the trim point, and second-order terms ($\Delta u\cdot \Delta q$, etc.) are neglected. This results in two decoupled sets of linear differential equations: longitudinal motion ($\Delta u,\Delta w,\Delta q,\Delta \theta$) and lateral-directional motion ($\Delta v,\Delta p,\Delta r,\Delta \varphi$), which are far more tractable for analysis and control design.

Common Pitfalls

1. Misapplying the Derivative: Forgetting to use the transport theorem when taking time derivatives of vectors expressed in the rotating body frame is the most common error. Always remember: $\frac{d()}{dt}{|}_i=\frac{d()}{dt}{|}_b+\vec{\omega }\times ()$.
2. Incorrect Force/Moment Summation: Gravity acts through the CG and thus only contributes to the force equations, not the moment equations. However, the gravitational force vector must be correctly resolved into body axes using the Euler angles ($X_G=-mg\sin \theta$, $Y_G=mg\cos \theta \sin \varphi$, $Z_G=mg\cos \theta \cos \varphi$).
3. Ignoring Assumptions in Simplification: Applying the decoupled longitudinal/lateral linear models to highly maneuverable or asymmetric flight (e.g., a steep turn with high roll rates) will yield invalid results. Always verify that the flight condition justifies the small-disturbance and symmetry assumptions.
4. Handling the Inertia Tensor: Using the full, non-diagonal inertia tensor complicates the moment equations significantly. While more accurate, it is often reasonable to assume principal axes alignment for preliminary analysis. If you cannot assume this, the general form $\sum \vec{M}=I\dot{\vec{\omega }}+\vec{\omega }\times (I\vec{\omega })$ must be used with the full $I$ matrix.

Summary

• The 6-DOF nonlinear equations of motion for a rigid aircraft are derived using the Newton-Euler approach in body-fixed axes, accounting for frame rotation via the transport theorem.
• The three force equations ($m(\dot{u}+qw-rv)=\sum F_x$, etc.) and three moment equations ($I_x\dot{p}-(I_y-I_z)qr=\sum L$, etc.) constitute the core dynamic model.
• Kinematic relations for Euler angles ($\varphi ,\theta ,\psi$) and position link the body-axis velocities and angular rates to the aircraft's orientation and location in space.
• Key assumptions (rigid body, flat Earth, constant mass) enable this formulation, and further simplifications (symmetric trim, small-disturbance theory) are applied to analyze specific flight conditions like stability and control.
• This complete set of equations serves as the starting point for all detailed flight dynamics analysis, simulation, and autopilot design.