aro309 - astronautics and spacecraft design winter 2014 try lam calpoly pomona aerospace engineering

ARO309 - Astronautics and Spacecraft Design

Winter 2014

Try LamCalPoly Pomona Aerospace Engineering

Relative Motion

Chapter 7

Relative Motion and Rendezvous

• In this chapter we will look at the relative dynamics between 2 objects or 2 moving coordinate frames, especially in close proximity

• We will also look at the linearized motion, which leads to the Clohessy-Wiltshire (CW) equations

Co-Moving LVLH Frame (7.2)

Local Vertical Local Horizontal (LVLH) Frame

TARGET

CHASER(or observer)

• The target frame is moving at an angular rate of Ω

where and

• Chapter 1: Relative motion in the INERTIAL (XYZ) frame

Co-Moving LVLH Frame

• We need to find the motion in the non-inertial rotating frame

where Q is the rotating matrix from

Co-Moving LVLH Frame

• Steps to find the relative state given the inertial state of A and B.

Co-Moving LVLH Frame

1. Compute the angular momentum of A, hA

2. Compute the unit vectors

3. Compute the rotating matrix Q

4. Compute

5. Compute the inertial acceleration of A and B

• Steps to find the relative state given the inertial state of A and B.

Co-Moving LVLH Frame

6. Compute the relative state in inertial space

7. Compute the relative state in the rotating coordinate system

Co-Moving LVLH Frame

Rotating Frame

Linearization of the EOM (7.3)

neglecting higher order terms

Linearization of the EOM

Assuming

Acceleration of B relative to A in the inertial frame

Linearization of the EOM

After further simplification we get the following EOM

Thus, given some initial state R0 and V0 we can integrate the above EOM (makes no assumption on the orbit type)

Linearization of the EOM

e = 0.1

e = 0

Clohessy-Whiltshire (CW) Equations (7.4)

Assuming circular orbits:

Then EOM becomes

where

Clohessy-Whiltshire (CW) Equations

Where the solution to the CW Equations are:

Maneuvers in the CW Frame (7.5)The position and velocity can be written as

where

Maneuvers in the CW Frame

and

Maneuvers in the CW Frame

Two-Impulse Rendezvous: fromPoint B to Point A

Maneuvers in the CW FrameTwo-Impulse Rendezvous: from Point B to Point A

where

where is the relative velocity in the Rotating frame, i.e.,

If the target and s/c are in the same circular orbits then

Maneuvers in the CW FrameTwo-Impulse Rendezvous example:

Rigid Body DynamicsAttitude Dynamics

Chapter 9-10

Rigid Body Motion

Note:

Position, Velocity, and Acceleration of points on a rigid body, measure in the same inertial frame of reference.

Angular Velocity/Acceleration

• When the rigid body is connected to and moving relative to another rigid body, (example: solar panels on a rotating s/c) computation of its inertial angular velocity (ω) and the angular acceleration (α) must be done with care.

• Let Ω be the inertial angular velocity of the rigid body

Note: if

Example 9.2

Angular Velocity of Body

Angular Velocity of Panel

Example 9.2 (continues)0

Example: Gimbal

Equations of Motion

• Dynamics are divided to translational and rotational dynamics

Translational:

Equations of Motion

• Dynamics are divided to translational and rotational dynamics

Rotational:

If then where

Angular Momentum

?

Angular Momentum

Since:

Note:

Angular Momentum

If has 2 planes of symmetry then

therefore

Moments of Inertia

Euler’s Equations

• Relating M and for pure rotation. Assuming body fixed coordinate is along principal axis of inertia

• Therefore

Euler’s Equations

• Assuming that moving frame is the body frame, then this leads to Euler’s Equations:

Kinetic Energy

Spinning Top• Simple axisymmetric top spinning at point 0

Introduces the topic of

1.Precession2.Nutation3.Spin

Assumes:

Notes: If A < C (oblate)If C < A (prolate)

Spinning TopFrom the diagram we note 3 rotations:

where

therefore:

Spinning TopFrom the diagram we note the coordinate frame rotation

therefore:

Spinning Top• Some results for a spinning top

– Precession and spin rate are constant– For precession two values exist (in general) for

– If spin rate is zero then

• If A > C, then top’s axis sweeps a cone below the horizontal plane• If A < C, then top’s axis sweeps a cone above the horizontal plane

Spinning Top• Some results for a spinning top

– If then

• If , then precession occurs regardless of title angle• If , then precession occurs title angle 90 deg

– If then a minimum spin rate is required for steady precession at a constant tilt

– If then

Axisymmetric Rotor on Rotating Platform

Thus, if one applies a torque or moment (x-axis) it will precess, rotating spin axis toward moment axis

Euler’s Angles (revisited)• Rotation between body fixed x,y,z to rotation angles

using Euler’s angles (313 rotation)

Euler’s Angles (revisited)

Satellite Attitude Dynamics

• Torque Free Motion

Euler’s Equation for Torque Free Motion

A = B

Euler’s Equation for Torque Free Motion

For

Then:

If A > C (prolate), ωp > 0If A < C (oblate), ωp < 0

Euler’s Equation for Torque Free Motion

Euler’s Equation for Torque Free Motion

If A > C (prolate), γ < θIf A < C (oblate), γ > θ

Euler’s Equation for Torque Free Motion

Stability of Torque-Free S/CAssumes:

Stability of Torque-Free S/C

• If k > 0, then solution is bounded• A > C and B > C or A < C and B < C• Therefore, spin is the major axis (oblate) or minor

axis (prolate)

• If k < 0, then solution is unstable• A > C > B or A < C < B• Therefore, spin is the intermediate axis

Stability of Torque-Free S/C• With energy dissipation ( )

Stability of Torque-Free S/C• Kinetic Energy relations

Conning Maneuvers• Maneuver of a purely

spinning S/C with fixed angular momentum magnitude

Conning ManeuversBefore the Maneuver

During the Maneuver

Another maneuver is required ΔHG2 after precession 180 deg

Conning ManeuversAnother maneuver is required ΔHG2 after precession 180 deg.

At the 2nd maneuver we want to stop the precession (normal to the spin axis):

Required deflection angle to precess 180 deg for a single coning mnvr

Gyroscopic Attitude Control

• Momentum exchange gyros or reaction wheels can be used to control S/C attitude without thrusters.

• The wheels can be fixed axis (reaction wheels) or gimbal 2-axis (cmg)

Gyroscopic Attitude Control

Example:

If external torque free then

therfore

Gyroscopic Attitude ControlExample II: S/C with three identical wheels with their axis along the principal axis of the S/C bus, where the wheels spin axis moment of inertial is I and other axis are J. Also, the bus moment of inertia are diagonal elements (A, B, C).