Snap Crackle Pop. Peter M. Thompson, Ph.D. Systems Technology, Inc. Hawthorne, CA

Transcription

1 Snap Crackle Pop Peter M. Thompson, Ph.D. Systems Technology, Inc. Hawthorne, CA Presented to AIAA Southern California Aerospace Technology and Systems Conference Santa Ana, CA May 20, 2011

2 Minimum Time Trajectories with Limits on: 2

3 Why? Sudden changes in acceleration cause: Loud bangs Damage to equipment Passenger discomfort 3

4 Outline Problem statement Limits on: Acceleration & Velocity Jerk Snap, Crackle, & Pop Linear Jerk Filter More analysis: Low Pass Approximation Feedfoward Telescope example Conclusions 4

5 Problem Statement Find the min time trajectory with limits on selected derivatives Two versions: Between stationary points Command following Implementation using feedback & feedfoward: 5

6 Limits on Acceleration & Velocity The classic minimum time problem Acceleration switches between Max positive Zero Max negative This (and all of the following) problems are solved by finding the switching times 6

7 Limits on Acceleration & Velocity: Between Stationary Points 7

8 Limits on Acceleration & Velocity: Command Following Acceleration based on phase plane location Switch when lines are crossed Digital implementation: Line crossing occur between samples Compute intermediate line crossing times Split trajectory into pieces 8

9 Limits on Acceleration & Velocity: Command Following 9

10 Limits on Acceleration & Velocity: Command Following 10

11 Limit on Jerk Jerk switches between max, min, & zero For trajectories between stationary points Four cases, depending on whether Amax & Vmax reached or not Command following: Requires 3D phase space Possible, but very complicated to implement 11

12 Limit on Jerk 12

13 Limits on Snap, Crackle, & Pop Solutions found for short move case where just the highest derivative has a limit 13

14 Limits on Snap, Crackle, & Pop 14

15 Limits on Snap, Crackle, & Pop: How is this done? Guess the switch times (exact up to jerk but not beyond) 15

16 Limits on Snap, Crackle, & Pop: How is this done? Solve a minimization problem where the endpoint derivatives other than the highest are zero, then: 16

17 Limits on Snap, Crackle, & Pop: Not done yet How is this done? Exact switch times will likely occur between sample times, resulting in large (~1%) errors when derivatives are integrated (typically need << 1% error) Therefore: Normalize time between 0 and 1 Numerically integrate piecewise polynomials Sample the piecewise polynomials Re-normalize 17

18 Limits on Snap, Crackle, & Pop: Longer Moves Limit just three derivatives Highest (jerk, snap, crackle, pop, etc) Acceleration Velocity Stretch trajectory as needed between: Time with max and min acceleration Time with max velocity Further details not included 18

19 Linear Jerk Filter Implement exact limits on acceleration & velocity Follow by a linear filter called a jerk filter Simple, works like a charm Works either point-to-point or command following 19

20 Linear Jerk Filter vs. Pop Limiter Close up to jerk, then starts to differ 20

21 Approximating Command Limiter as a Low Pass Filter A rate limit can be approximated as a first order lag 21

22 Approximating Command Limiter as a Low Pass Filter The same is true when higher order derivatives are limited, details of how to this are in the paper 22

23 Feedforward Network 23

24 Telescope Example Thirty Meter Telescope currently being designed Elevation Axis: J = 1.4e8 kg-m 2 24

25 Telescope Example: Ideal response Example requirements El-Axis mount control bandwidth > 1 Hz Short move of 60 arcsec must settle to within 50 milli-arcsec in < 5 seconds Ideal response (green line) easily meets requirement 25

26 Telescope Example: Non-Ideal response Structural resonances result in non-ideal response Lowest significant structural frequency ~ 4 Hz Causes overshoots in slewing responses Jerk filter keeps the overshoot acceptable 26

27 Conclusions For min time moves between stationary points Snap, Crackle, Pop limits are feasible Method derived for determining switch times For command following Use Accel + Velocity + Linear Jerk Filter Also limits snap, crackle & pop Also works well stationary moves 27