1. Euler Parameters and Bowling ball dynamics a la Huston et al. Andrew Kickertz 5 May 2011

2. Definition Simple rotation of Θ about a unit vector 2

3. Relation to direction cosine matrix 3

4. Derivatives Derivation is in Kane, Likins, Levinson (1993). Spacecraft Dynamics. 4

5. Alternate forms of ω? From Kane, Likins, Levinson (1993). Spacecraft Dynamics. And Huston, Passerello, Winget, & Sears (1979). On the dynamics of a weighted bowling ball. Kane et al. Huston et al. 5

6. Autolev implementation % Newtonian, bodies, frames newtonian A bodies B % Variables, constants, points variables q{7}',u{7}’... % KDEs q1' = u1... % Position vectors... % Euler parameters dircos(A,B,Euler,e1,e2,e3,e4) % Velocities % Motion constraint % KDEs kindiffs(A,B,Euler,e1,e2,e3,e4) % Accelerations... % Forces... % Equations of motion, output Matlab file... 6

7. Overview CM Offset: ✓ Distinct moment of inertia: 3 Friction: Coulomb Oil pattern: uniform Newton-Euler / d’Alembert Euler Parameters 7

8. Notation A lane B ball C contact point G center of mass Q geometric center 1 2 3 8

9. Kinematics Given ω, the orientation ε can be found? 9

10. Kinematics (2) C contact point G center of mass Q geometric center Defining velocity and acceleration of geometric center Velocity and acceleration of center of mass – Q and G are on a rigid body Velocity of contact point 10

11. Slip - stick Rolling Note that ω 3 is a 3 rd degree of freedom, but does not affect the motion 11

12. Kinetics Contact force Inertia force Inertia torque N C G Q mg C 12

13. Governing equations Rolling C G Q mg C P 13

14. Parameters and initial conditions r=0.109 m mg=71.32 N (16 lbs) |ω|=10 rad/s ω 1 =ω 2 ω 3 =0 14

15. Results: hook and ω Definition of percent hook: 1 2 3 15

16. Results: hook and ω (2) Axis rotation γ and axis tilt β From King, Perkins, et al. (2010). Bowling dynamics revealed by miniature wireless MEMS IMU. 16

17. Results: hook and ω (3) Hook  with γ , for a constant |ω| Hook  with β , for a constant |ω| Hook constant with ω 3  γβ|ω|H 900100.00 750100.34 600100.64 450100.87 150101.10 00101.09 4510 0.86 4520100.82 4530100.75 451.43100.87 452.8610.010.87 455.7110.050.87 4511.310.200.87 4526.611.180.87 17

18. Results: hook and CM offset CM offsets defined in the A frame 1 2 3 18

19. Results: hook and I I defined in the A frame I 11 , H  I 22 , H  1 2 3 19