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