Reference Coordinate Systems For Missile Motion

Synopsis The earth is a spherical rotating mass. Hence, if a Cartesian system of co-ordinate axes is chosen as reference, the reference axis orientation on ground will always change with respect to an object in flight unless certain assumptions are made or a different system of axes is followed for the aerial object. This reference system will also help in understanding the forces acting on the missile and to consider its motion in space.

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Chapter 1 Introduction 1.1

Overview

The different systems of co-ordinate axes [1]are as given below:(a) Earth-based System. (b) Conditionally Stationary System. (c) Body Axes System. (d) Wind System of Co-ordinate Axes. (e) Conditionally Moving System.

1.2

Earth-based System

The earth based system of co-ordinate axes considers the motion of center of gravity of the missile. The same is depicted in Fig.1.1. The system determines the position of the earth’s gravity force vector G with respect to the earth based system of axes and also computes the altitude of the flight H of the guided missile. (a) Point O on the earth’s surface is considered as the origin of the system. 1

Figure 1.1: Earth Based System Of Co-ordinate Axes

(b) This point coincides with the centre of gravity of the guided missile when the missile is placed on the launcher. (c) Axis OYe is directed verically upwards while the axes OXe and OZe are located horizontally. All three axes are mutually perpendicular to each other forming a right handed co-ordinate axis system. (d) The assumptions made are that the duration of the missile is short and hence rotation of earth is not taken into account making the system stationary in space.

1.3

Body Axes System

In case of body axes system, the origin is the centre of gravity,S, of the missile during its entire flight. This is as depicted in Fig.1.2. (a) The axis SXba coincides with longitudinal axis of the missile. The axes SYba and SZba are the lateral axes of the missile. They are located in the main planes of symmetry, forming a right-handed system. (b) The body axes system moves with respect the earth-based system of axes or the conditionally stationary system of axes as the missile moves in space. 2

Figure 1.2: Body System Of Co-ordinate Axes

(c) Since the flight control and stabilisation of the missile is with respect to the planes of its wings i.e., taking into consideration roll, pitch and yaw, the body axes system ∗ is mostly preferred with the axes Yba∗ and Zba now directed along the upper left

hand wing and upper right hand wing respectively with no change in longitudinal axis. (d) Thus the engine thrust T , and ruddervator deflection angles are normally acting along or calculated with respect to the body axes.

1.4

Conditionally Statonary System of Axes

The conditionally stationary system of axes is used when the guided missile is rollstabilised. It is similar to the earth-based system as far as its origin is concerned i.e., here also it is the centre of gravity of the missile. However the difference is that here the X axis (OXcs ) coincides with the direction of the longitudinal axis of the missile at the launching moment. Z axis (OZcs ) is perpendicular to X axis in the horizontal plane while Y axis (OYcs ) is oriented vertically upwards thus forming a right hand system. 3

Figure 1.3: Conditionally Stationary System Of Co-ordinate Axes

The same is depicted in Fig.1.3. The conditional system of axes is considered stationary with regard to the earth-based system of axes throughout the flight period of the missile as shown in Fig.1.3.

1.5

Wind System of Axes

The wind system of axes is similar to the body axes system where the origin coincides with the centre of gravity of the missile throughout its entire flight. However, the Xaxis (SXwnd ) is directed along the velocity vector of the guided missile as shown in Fig.1.4. The Y -axis (SYwnd ) is perpendicular to the plane of symmetry containing Xaxis, while Z-axis (SZwnd ) is on the same plane of X-axis but perpendicular to the X-axis thus forming a right handed system. The wind system axes is used for calculating the aerodynamic forces on the missile and to compute the angle of attack, α.

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Figure 1.4: Wind System Of Co-ordinate Axes

1.6

Conditionally Moving System of Axes

This system is used to determing the position of the guided missile with repect to the guidance system in the process of guiding the missile to the target. This is as depicted in Fig.5. (a) The origin of the system is stationary and coincides with the centre of rotation of the missile guidance radar antenna. (b) The axis O0 Xcm coincides with the direction of the target line of sight (LOS). (c) The axis O0 Ycm lies in the plane of rotation of the position finding radar antenna. (d) The axis O0 Zcm lies in the plane of rotation of the target tracking antenna in azimuth. (e) Thus the system O0 Xcm Ycm Zcm is always moving with respect to the target motion. (f) The conditionally moving system of axis is used for computing the distance,r, of the missile from the missile guidance radar, the displacement angle, ξ, of the 5

Figure 1.5: Conditionally Moving System of Co-ordinate Axes

missile with respect to LOS and the translational displacement, h, of the missile with respect to its ideal trajectory.

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References [1] Siouris, G. M., Missile Guidance and Control Systems, Springer, New York, 2003.

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