Equations of motion solvers

Background Material [MI]

Introduction

Motion models for points (particles) and rigid bodies in space and time are based on mathematical equations.

Three degrees of freedom models employ translational equations of motion. Six degrees of freedom models incorporate both translational and rotational equations of motion.

The inputs to the equations of motion are the forces and moments acting on the body; yielding body accelerations as outputs.

Nomenclature and Convention

Typically, the forces and moments on a body are resolved into components in the body coordinate system. Fig-1 shows the components of force, moment velocity, and angular rate of a body resolved in the body coordinate system. The six projections of the linear and angular velocity vectors on the moving body frame axes are the six degrees of freedom. The nomenclature and conventions for positive directions are as shown in Fig-1 and in the following Table:

Fig-1: Forces, velocities, moments, and angular rates in body reference frame

Axis	Force along axis	Moment about axis	Linear velocity	Angular displacement	Angular velocity	Moment of Inertia
\(x_b\)	\({F_x}_b\)	\(L\)	\(u\)	\(\varphi\)	\(p\)	\(I_{xx}\)
\(y_b\)	\({F_y}_b\)	\(M\)	\(v\)	\(\theta\)	\(q\)	\(I_{yy}\)
\(z_b\)	\({F_z}_b\)	\(N\)	\(w\)	\(\psi\)	\(r\)	\(I_{zz}\)

The position of the mass center of the body is given by its Cartesian coordinates expressed in an inertial frame of reference, such as the fixed-earth frame \((x, y, z)\).

The body’s angular orientation is defined by three rotations \((\psi, \theta, \varphi)\) relative to the inertial frame of reference. These are called Euler rotations, and the order of the successive rotations is important. Starting with the body coordinate frame aligned with the earth coordinate frame, the adopted order here is 3-2-1, i.e.:

1. Rotate the body frame about the \(z_b\) axis through the heading angle \(\psi\),

2. Rotate about the \(y_b\) axis through the pitch angle \(\theta\), and

3. Rotate about the \(x_b\) axis through the roll angle \(\varphi\)

The total inertial velocity \(V\) has components \(u, v\), and \(w\) on the body frame axes, and \((v_x, v_y, v_z)\) on the earth-frame axes.

Newton’s Second Law

Newton’s second law of motion establishes the foundational equation governing the relationship among force, mass, and acceleration.

in the context of Newton’s second law, the force \((F)\) acting on an object is the derivative of its momentum \((m \cdot v)\) with respect to time \((t)\):

\[F = {d(m \cdot v) \over dt}\]

Where:

• \(F\) is the total force acting on the object

• \(m\) is the mass of the object

• \(v\) is the velocity

• \(t\) is time

This equation yields the final form of the equations of linear motion. In the final form, acceleration is represented by the rate of change of the velocity:

\[F = m \cdot \dot{v}\]

Where:

• \(F\) is the total force acting on the object

• \(m\) is the mass of the object

• \(\dot{v}\) is the acceleration of the object

A direct extension of Newton’s second law to rotational motion reveals that the moment of force (torque) on a body about a given axis equals the time rate of change of the angular momentum of the paricle about that axis.

\[M = {dh \over dt}\]

Where:

• \(M\) is the total moment (torque) acting on the object

• \(h\) is the angular momentum vector of the object

Hence, the final form of the equations of angular motion is given by:

\[M = [I] \cdot \dot{\omega}\]

Where:

• \(M\) is the total moment (torque) acting on the object

• \([I]\) is the inertia matrix of the body relative to the axis of rotation

• \(\dot{\omega}\) is the absolute angular acceleration vector of the body

Translational Equations of Motion

The basis of the translational equation of motion was introduced above. The usual procedure used to solve this equation is to sum the external forces \(F\) acting on the body, express them in an inertial frame, and substitute \(F\) into the equation. Once the acceleration, namely the forces divided by the mass, is expressed in inertial coordinates, it is integrated twice to yield the translational displacement.

\[ \begin{align}\begin{aligned}dx = v_x\\dy = v_y\\dz = v_z\\dv_x = {F[0] \over m}\\dv_y = {F[1] \over m}\\dv_z = {F[2] \over m}\end{aligned}\end{align} \]

Where:

• \(dx, dy, dz\) are the changes in position in the \(x, y, z\) inertial directions, respectively

• \(dv_x, dv_y, dv_z\) are the changes in velocity in the \(x, y, z\) inertial directions, respectively

• \(v_x, v_y, v_z\) are the velocities in the \(x, y, z\) inertial directions, respectively

• \(f[0], f[1], f[2]\) are the input force components in the \(x, y, z\) inertial directions, respectively

• \(m\) is the mass of the body.

These equations describe the dynamics of a datapoint in three-dimensional space (3DOF). Which is the rate of change of position \((x, y, z)\) with respect to time equals to the velocity, and the rate of change of velocity \((v_x, v_y, v_z)\) with respect to time equals to the force divided by the mass \((m)\).

Rotational Equations of Motion

As mentioned earlier, the rotational analog of Newton’s law describes the relationship between torque, moment of inertia, and angular acceleration. We also saw that a double integration on the translational acceleration produces the change of the body in position.

However, the angular accelerations are typically expressed with respect to a body frame and must be adjusted in order to produce the attitude of the body. For that purpose we introduced the euler angles (see Nomenclature and Conventions) which describe the body attitude with respect to an inertial frame of reference.

The orientation of the body reference frame is specified by the three Euler angles, \(\psi, \theta, \varphi\).

As a rigid body changes its orientation in space, the Euler angles change. The rates of change of the Euler angles are related to the angular rates \((p, q, r)\) of the body frame.

The rate of change of the Euler angles together with the rotational analog of Newton’s law provide the set of differential equations that describe the equations governing the motion of a rigid body:

\[ \begin{align}\begin{aligned}d\varphi = p + (q \cdot sin(\varphi) + r \cdot cos(\varphi)) \cdot tan(\theta)\\d\theta = q \cdot cos(\varphi) - r \cdot sin(\varphi)\\d\psi = {q \cdot sin(\varphi) + r \cdot cos(\phi) \over cos(\theta)}\\dp = {M[0] - q \cdot r \cdot (I_{zz} - I_{yy}) \over I_{xx}}\\dq = {M[1] - p \cdot r \cdot (I_{xx} - I_{zz}) \over I_{yy}}\\dr = {M[2] - p \cdot q \cdot (I_{yy} - I_{xx}) \over I_{zz}}\end{aligned}\end{align} \]

Where:

• \(d\varphi, d\theta, d\psi\) are the changes in Euler roll, Euler pitch, and Euler yaw angles, respectively

• \(dp, dq, dr\) are the changes in body roll rate, pitch rate, and yaw rate, respectively

• \(\varphi, \theta, \psi\) are the Euler roll, Euler pitch, and uler yaw angles, respectively

• \(p, q, r\) are the body roll rate, pitch rate, and yaw rate, respcetively

• \(M[0], M[1], M[2]\) are the input moment of force components about the \(x, y, z\) in the body direction, respectively

• \(I_{xx}, I_{yy}, I_{zz}\) are the moments of inertia about the \(x, y,\) and \(z\) in body direction, respectively

These equations describe the angular dynamics of a rigid body. Together with the equations that describe the translational motion of the body they form the six-dimensional motion in space (6DOF).

References

[MI]

17 July 1995, “Missile Flight Simulation, Part One, Surface-to-Air Missiles”, Ch 4 In: Military Handbook. 1995, MIL-HDBK-1211(MI)

Examples

For examples, see the various functions.

Functions

derivs.eqm3(dp, F)	Translational motion derivatives.
derivs.eqm6(rb, F, M)	Translational and angular motion derivatives.
integrate.int3(dp, forces, dt[, derivs_out])	A step integration of the equations of translational motion.
integrate.int6(rb, forces, moments, dt[, ...])	A step integration of the equations of motion.