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ARO309 - Astronautics and Spacecraft Design Winter 2014 Try Lam CalPoly Pomona Aerospace Engineering

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Relative Motion Chapter 7

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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

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Co-Moving LVLH Frame (7.2) Local Vertical Local Horizontal (LVLH) Frame TARGET CHASER (or observer)

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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

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We need to find the motion in the non-inertial rotating frame where Q is the rotating matrix from Co-Moving LVLH Frame

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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

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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

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Co-Moving LVLH Frame Rotating Frame

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Linearization of the EOM (7.3) neglecting higher order terms

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Linearization of the EOM Assuming Acceleration of B relative to A in the inertial frame

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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)

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Linearization of the EOM e = 0.1 e = 0

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Clohessy-Whiltshire (CW) Equations (7.4) Assuming circular orbits: Then EOM becomes where

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Clohessy-Whiltshire (CW) Equations Where the solution to the CW Equations are:

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

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Maneuvers in the CW Frame and

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

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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

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Maneuvers in the CW Frame Two-Impulse Rendezvous example:

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Rigid Body Dynamics Attitude Dynamics Chapter 9-10

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Rigid Body Motion Note: Position, Velocity, and Acceleration of points on a rigid body, measure in the same inertial frame of reference.

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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

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Example 9.2 Angular Velocity of Body Angular Velocity of Panel

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Example 9.2 (continues) 0

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Example: Gimbal

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Equations of Motion Dynamics are divided to translational and rotational dynamics Translational:

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Equations of Motion Dynamics are divided to translational and rotational dynamics Rotational: If thenwhere

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Angular Momentum ?

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Since: Note:

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Angular Momentum Ifhas 2 planes of symmetry then therefore

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Moments of Inertia

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Euler’s Equations Relating M and for pure rotation. Assuming body fixed coordinate is along principal axis of inertia Therefore

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Euler’s Equations Assuming that moving frame is the body frame, thenthis leads to Euler’s Equations:

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Kinetic Energy

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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)

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

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Spinning Top From the diagram we note the coordinate frame rotation therefore:

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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

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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

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

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Euler’s Angles (revisited) Rotation between body fixed x,y,z to rotation angles using Euler’s angles (313 rotation)

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Euler’s Angles (revisited)

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Satellite Attitude Dynamics Torque Free Motion

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Euler’s Equation for Torque Free Motion A = B

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Euler’s Equation for Torque Free Motion For Then: If A > C (prolate), ω p > 0 If A < C (oblate), ω p < 0

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Euler’s Equation for Torque Free Motion

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If A > C (prolate), γ < θ If A θ

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Euler’s Equation for Torque Free Motion

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Stability of Torque-Free S/C Assumes:

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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

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

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Stability of Torque-Free S/C Kinetic Energy relations

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Conning Maneuvers Maneuver of a purely spinning S/C with fixed angular momentum magnitude

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Conning Maneuvers Before the Maneuver During the Maneuver Another maneuver is required ΔH G2 after precession 180 deg

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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

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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)

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Gyroscopic Attitude Control Example: If external torque free then therfore

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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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