ARO309 - Astronautics and Spacecraft Design

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

2. Relative Motion Chapter 7

3. 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

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

5. Co-Moving LVLH Frame • The target frame is moving at an angular rate of Ω where and • Chapter 1: Relative motion in the INERTIAL(XYZ) frame

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

7. Co-Moving LVLH Frame • Steps to find the relative state given the inertial state of A and B. Compute the angular momentum of A, hA Compute the unit vectors Compute the rotating matrix Q Compute Compute the inertial acceleration of A and B

8. Co-Moving LVLH Frame • Steps to find the relative state given the inertial state of A and B. • Compute the relative state in inertial space • Compute the relative state in the rotating coordinate system

9. Co-Moving LVLH Frame Rotating Frame

10. Linearization of the EOM (7.3) neglecting higher order terms

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

12. 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)

13. Linearization of the EOM e = 0.1 e = 0

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

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

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

17. Maneuvers in the CW Frame and

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

19. Maneuvers in the CW Frame Two-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

20. Maneuvers in the CW Frame Two-Impulse Rendezvous example:

21. Rigid Body DynamicsAttitude Dynamics Chapter 9-10

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

23. 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

24. Example 9.2 Angular Velocity of Body Angular Velocity of Panel

25. Example 9.2 (continues) 0

26. Example: Gimbal

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

28. Equations of Motion • Dynamics are divided to translational and rotational dynamics Rotational: If then where

29. Angular Momentum ?

30. Angular Momentum Since: Note:

31. Angular Momentum If has 2 planes of symmetry then therefore

32. Moments of Inertia

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

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

35. Kinetic Energy

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

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

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

39. 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

40. 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

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

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

43. Euler’s Angles (revisited)

44. Satellite Attitude Dynamics • Torque Free Motion

45. Euler’s Equation for Torque Free Motion A = B

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

47. Euler’s Equation for Torque Free Motion

48. Euler’s Equation for Torque Free Motion If A > C (prolate), γ < θ If A < C (oblate), γ > θ

49. Euler’s Equation for Torque Free Motion

50. Stability of Torque-Free S/C Assumes: