Full Movement of AD, Quaternion Equations

Block appearance in REPEAT VISION

  The movement of missiles will be considered in 0XeYeZe coordinate system rigidly connected to the Earth. Moreover, to simplify the task, the influence of the Earth will be taken into account by the introduction of a constant acceleration in magnitude and direction based on gravity, neglecting Coriolis acceleration; the curvature of the Earth will also not be taken into account. When writing the right-hand sides of the equations, the components of the thrust force, gravity, aerodynamic and control forces will be taken into account.

  A system of equations describing the motion of the rocket center of mass in the half-speed coordinate system 0XYZ (1):

(1)\[\begin{split} \begin{cases} \dot{\nu} =\frac{1}{m}\cdot (P_{x_1}-\sum X_{p}-X+Q_x) \\ \nu \cdot\dot{\theta} =\frac{1}{m}\cdot (P_{y^*}-\sum Y^*_{p}+Y^*+Q_y^*) \\ \nu \cdot\dot{\Psi}\cdot cos\theta =-\frac{1}{m}\cdot (P_{z^*}+\sum Z^*_{p}+Z^*) \end{cases} \end{split}\]

where:

 \(P_x, P_{y^*}, P_{z^*}\) are projections of thrust force on half-speed coordinate axes;

 \(Q_x, Q_{y^*}\) are projections of gravity on half-speed coordinate axes;

 \(X_p, Y_{p^*}, Z_{p^*}\) is a projection of aerodynamic control forces on half-speed coordinate axes;

 \(X, Y^*, Z^*\) is a projection of aerodynamic forces on half-speed coordinate axes.

 The equations of rotational motion of rockets and aircraft are usually written in projections on the associated coordinate axes (2), (3) и (4):

(2)\[J_{x_1}\cdot \dot\omega_{x_1}+(J_{z_1}-J_{y_1})\cdot \omega_{y_1} \cdot \omega_{z_1}=\sum M_{x_1}+\sum M_{px_1}\]

(3)\[J_{y_1}\cdot \dot\omega_{y_1}+(J_{x_1}-J_{z_1})\cdot \omega_{x_1} \cdot \omega_{z_1}=\sum M_{y_1}+\sum M_{py_1}\]

(4)\[J_{z_1}\cdot \dot\omega_{z_1}+(J_{y_1}-J_{x_1})\cdot \omega_{x_1} \cdot \omega_{y_1}=\sum M_{z_1}+\sum M_{pz_1}\]

where:

 \(\sum M_{x_1}, \sum M_{y_1}, \sum M_{z_1}\) is the sum of the projections of the moments of external forces and thrust force to the associated coordinate axes (excluding control forces);

 \(\sum M_{px_1}, \sum M_{py_1}, \sum M_{pz_1}\) are the sums of the projections of control forces moments on the associated coordinate axes;

 \(J_{x_1}, J_{y_1}, J_{z_1}\) are the moments of inertia along the main axes of AD.

  To establish links between velocity derivative and angular rate \(w_{x_1}, w_{y_1}, w_{z_1}\), use (5):

(5)\[\begin{split} \begin{cases} \omega_{x_1} =\dot{\psi}\cdot sin\upsilon + \dot{\gamma} \\ \omega_{y_1} =\psi\cdot cos\upsilon \cdot sin\gamma + \upsilon \cdot sin \gamma\\ \omega_{z_1} =-\dot{\psi}\cdot cos\upsilon \cdot sin \gamma + \upsilon \cdot cos\gamma \end{cases} \end{split}\]

  When determining the values of aerodynamic forces in the process of solving the spatial problem of missile motion, it is necessary to know the values of α, β and \(γ_s\) angles.

  By defining the directional cosines of the sequential transition from the related axes to the wind axes, from the wind axes to the half-speed axes and from the half-speed axes to the earth’s axes and equating them to the directional cosines of the direct transition from the related axes to the earth’s axes, the following relations are obtained between (6), (7) and (8) angles:

(6)\[sin\Theta=cos\alpha \cdot cos\beta \cdot sin\upsilon-(sin\alpha \cdot cos\beta \cdot cos\gamma+sin\beta\cdot sin\gamma)\cdot cos\upsilon\]

(7)\[sin\Psi\cdot cos\Theta=cos\alpha \cdot cos\beta \cdot sin\psi \cdot cos\upsilon-sin\alpha \cdot cos\beta \cdot (cos\psi \cdot sin\gamma +sin\psi \cdot sin\upsilon \cdot cos\gamma) - sin\beta \cdot(cos\psi \cdot cos\gamma -sin\psi \cdot sin\upsilon \cdot sin\gamma)\]

(8)\[cos\upsilon \cdot sin \gamma=sin\gamma_c \cdot cos\beta \cdot cos\theta-sin\beta \cdot sin\theta\]

 Hence (9):

(9)\[\gamma_c=arcsin \cdot \frac{cos\upsilon\cdot sin\gamma +sin\beta \cdot sin\theta}{cos\beta \cdot cos\theta}\]

  If expressions for projections of the velocity vector of the rocket center of mass on the axis of the Earth’s coordinate system are used, (10), (11) and (12) are obtained:

(10)\[\frac{dx_з}{dt}=\nu \cdot cos\theta \cdot cos\Psi\]

(11)\[\frac{dy_з}{dt}=\nu \cdot sin\theta \]

(12)\[\frac{dz_з}{dt}=-\nu \cdot cos\theta \cdot sin\Psi\]

1.  Properties

Property	Default Value	Description	Units of measurement
Initial yaw angle \(\psi_0\)	0	-	rad
Initial pitch angle tang0	0	-	rad
Initial roll angle \(\gamma_0\)	0	-	rad
Initial angle of trajectory inclination tang0	3*0.0174533	-	rad
Initial angle of trajectory inclination \(\Psi_0\)	5*0.0174533	-	rad
Initial angular speed relative to the connected axis X \(\omega_{0X}\)	0	-	rad/s
Initial angular speed relative to the connected axis Y \(\omega_{0Y}\)	0	-	rad/s
Initial angular speed relative to the connected axis Z \(\omega_{0Z}\)	0	-	rad/s
AD initial speed	200	-	m/s
Initial coordinate X	0	-	m
Initial coordinate Y	300	-	m
Initial coordinate Z	0	-	m

2.  Inputs and outputs

Parameters	Input/Output	Description	Units of measurement	Port type
{Px,Py,Pz}	Input	Engine thrust vector in the half-speed coordinate system	N	Automation
{Rx,Ry,Rz}	Input	Vector of aerodynamic forces in half-speed coordinate system	N	Automation
{Mx,My,Mz}	Input	Vector of aerodynamic moments in a related coordinate system	N\(\cdot\)m	Automation
{Ix,Iy,Iz}	Input	AD inertia moments	kg\(\cdot m^2\)	Automation
mass	Input	-	kg	Automation
accelerated free fall g	Input	-	m/s\(^2\)	Automation
integration step dt	Input	-	s	Automation
Velocity \(v\)	Output	AD speed module	m/s	Automation
Angle of trajectory inclination theta (\( heta\))	Output	-	rad	Automation
PSI trajectory rotation angle (\(\Psi\))	Output	-	rad	Automation
Coordinate X	Output	Coordinate on X axis of earth coordinate system	m	Automation
Coordinate Y	Output	Coordinate on Y axis of earth coordinate system	m	Automation
Coordinate Z	Output	Coordinate on Z axis of earth coordinate system	m	Automation
Angular velocity relative X axis omegaX (\(\omega_X\))	Output	-	rad/s	Automation
Angular velocity relative Y axis omegaY (\(\omega_Y\))	Output	-	rad/s	Automation
Angular velocity relative Z axis omegaZ (\(\omega_Z\))	Output	-	rad/s	Automation
Yaw angle psi (\(\psi\))	Output	-	rad	Automation
Pitch angle tang (\(\upsilon\))	Output	-	rad	Automation
Roll angle gamma (\(\gamma\))	Output	-	rad	Automation
Angle alpha (\(\alpha\))	Output	Attack angle	rad	Automation
Angle beta (\(\beta \))	Output	Gliding angle	rad	Automation
Angle gammaC (\(\gamma_C\))	Output	Roll angle (angle between planes 0X1Y1 and 0XY*)	rad	Automation

In order to see the connection capabilities of the block, you need to go to page Description of block ports