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ARO309 - Astronautics and Spacecraft Design Winter 2014 Try Lam CalPoly 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 whereand 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, h A 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 R 0 and V 0 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: from Point B to Point A

Maneuvers in the CW Frame Two-Impulse Rendezvous: from Point B to Point A where whereis 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 Frame Two-Impulse Rendezvous example:

Rigid Body Dynamics Attitude 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 thenwhere

Angular Momentum ?

Since: Note:

Angular Momentum Ifhas 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, thenthis leads to Euler’s Equations:

Kinetic Energy

Spinning Top Simple axisymmetric top spinning at point 0 Introduces the topic of 1.Precession 2.Nutation 3.Spin Assumes: Notes: If A < C (oblate) If C < A (prolate)

Spinning Top From the diagram we note 3 rotations: where therefore:

Spinning Top From 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), ω p > 0 If A < C (oblate), ω p < 0

Euler’s Equation for Torque Free Motion

If A > C (prolate), γ < θ If A θ

Euler’s Equation for Torque Free Motion

Stability of Torque-Free S/C Assumes:

Stability of Torque-Free S/C If k > 0, thensolution 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 Maneuvers Before the Maneuver During the Maneuver Another maneuver is required ΔH G2 after precession 180 deg

Conning Maneuvers Another maneuver is required ΔH G2 after precession 180 deg. At the 2 nd 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 Control Example 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).

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