Zac Manchester
January 13, 2017
67

# Recursive Inertia Estimation With Semidefinite Programming

A new algorithm for estimating a spacecraft's inertia matrix on orbit. Presented at AIAA SciTech 2017. Check out the paper here: http://zacinaction.github.io/docs/Inertia_Estimation.pdf

January 13, 2017

## Transcript

1. ### Zac  Manchester Harvard  University Recursive  Inertia  Estimation  with   Semidefinite

Programming Mason  Peck Cornell  University

3. ### Gyrostat  Dynamics 2 Inertia Rotor  Torque External  Torque Angular  Velocity

Rotor  Momentum J ˙ ! + !⇥(J! + ⇢) + ˙ ⇢ = ⌧
4. ### A  Least-­‐Squares  Problem 3 j = ⇥ J11 J22 J33

J12 J13 J23 ⇤T
5. ### A  Least-­‐Squares  Problem 4 J! = G(!)j j = ⇥

J11 J22 J33 J12 J13 J23 ⇤T
6. ### A  Least-­‐Squares  Problem 5 J! = G(!)j G( ˙ !)

+ !⇥G(!) j = ⌧ ˙ ⇢ !⇥⇢ j = ⇥ J11 J22 J33 J12 J13 J23 ⇤T
7. ### A  Least-­‐Squares  Problem 6 J! = G(!)j H y G(

˙ !) + !⇥G(!) j = ⌧ ˙ ⇢ !⇥⇢ j = ⇥ J11 J22 J33 J12 J13 J23 ⇤T
8. ### A  Least-­‐Squares  Problem 7 J! = G(!)j H y G(

˙ !) + !⇥G(!) j = ⌧ ˙ ⇢ !⇥⇢ H(!, ˙ !)j = y(!, ⇢, ˙ ⇢, ⌧) j = ⇥ J11 J22 J33 J12 J13 J23 ⇤T

˙ ⇢, ⌧)
10. ### Some  Least-­‐Squares  Problems… 9 H(!, ˙ !)j = y(!, ⇢,

˙ ⇢, ⌧) We  don’t  have  measurements  of  this
11. ### Some  Least-­‐Squares  Problems… 10 H(!, ˙ !)j = y(!, ⇢,

˙ ⇢, ⌧) We  don’t  have  measurements  of  this This  doesn’t  necessarily  form  a  valid  inertia  matrix
12. ### Principle  of  Least  Action 11 L(q, ˙ q) = T

V S = Z tf t0 L (q(t), ˙ q(t)) dt = 0
13. ### 12 Discrete  Mechanics = N X k=0 Z tk+1 tk

L(q, ˙ q) dt S = Z tf t0 L(q, ˙ q) dt
14. ### 13 Discrete  Mechanics = N X k=0 Z tk+1 tk

L(q, ˙ q) dt S = Z tf t0 L(q, ˙ q) dt ⇡ N X k=0 L ✓ qk, qk+1 qk h ◆ h
15. ### 14 Discrete  Mechanics = N X k=0 Z tk+1 tk

L(q, ˙ q) dt S = Z tf t0 L(q, ˙ q) dt Sd = N X k=0 Ld(qk, qk+1) = 0 ⇡ N X k=0 L ✓ qk, qk+1 qk h ◆ h
16. ### 15 Discrete  Gyrostat  Equation fk = 2 6 6 6

4 k q 1 T k k 3 7 7 7 5 H( k , k+1)j = y( k , k+1 , ⇢k , ⇢k+1 , ⌧k+1)

19. ### What  Makes  A  “Valid”  Inertia  Matrix? 18 J 2 S3

J > 0 x T Jx > 0 8 x 6= 0
20. ### 19 J 2 S3 J > 0 Jii  Jkk

+ J`` x T Jx > 0 8 x 6= 0 What  Makes  A  “Valid”  Inertia  Matrix?
21. ### Semidefinite  Programming  (SDP) 20 minimize x cT x subject to

F0 + n X i =1 xiFi 0  x z z y 0
22. ### 21 Schur Complement  A B BT C 0 =)

( C > 0 A BC 1BT 0
23. ### 22 SDP  Inertia  Estimation  Formulation minimize ⇥ · · ·

0 · · · 1 ⇤  i s subject to 8 > > > > > > > > < > > > > > > > > :  s ( Hj y ) T ( Hj y ) I 0 J > 0 J11 + J22 J33 0 J11 + J33 J22 0 J22 + J33 J11 0
24. ### 0 20 40 60 80 -0.2 -0.1 0 0.1 1

0 20 40 60 80 -0.1 -0.05 0 0.05 2 0 20 40 60 80 Time (s) -0.04 -0.02 0 0.02 3 0 20 40 60 80 -0.05 0 0.05 0.1 1 0 20 40 60 80 -0.05 0 0.05 0.1 2 0 20 40 60 80 Time (s) -0.05 0 0.05 0.1 3 23 Slewing  Spacecraft  Example
25. ### 0 20 40 60 80 10-9 10-7 10-5 10-3 10-1

J 11 SDP Variational SDP Finite Diff. Momentum Based 0 20 40 60 80 10-9 10-7 10-5 10-3 10-1 J 22 0 20 40 60 80 Time (s) 10-9 10-7 10-5 10-3 10-1 J 33 0 20 40 60 80 10-9 10-7 10-5 10-3 10-1 J 12 0 20 40 60 80 10-9 10-7 10-5 10-3 10-1 J 13 0 20 40 60 80 Time (s) 10-9 10-7 10-5 10-3 10-1 J 23 24 Slewing  Spacecraft  Example
26. ### 0 10 20 30 40 50 60 -0.2 0 0.2

1 0 10 20 30 40 50 60 -0.2 -0.1 0 0.1 2 0 10 20 30 40 50 60 Time (s) 0.76 0.77 0.78 0.79 3 0 10 20 30 40 50 60 -0.1 0 0.1 1 0 10 20 30 40 50 60 -0.1 0 0.1 2 0 10 20 30 40 50 60 Time (s) -0.1 0 0.1 3 25 Spinning  Spacecraft  Example
27. ### 0 10 20 30 40 50 60 10-8 10-6 10-4

10-2 100 J 12 0 10 20 30 40 50 60 10-8 10-6 10-4 10-2 100 J 13 0 10 20 30 40 50 60 Time (s) 10-8 10-6 10-4 10-2 100 J 23 0 10 20 30 40 50 60 10-8 10-6 10-4 10-2 100 J 11 SDP Variational SDP Finite Diff. Momentum Based 0 10 20 30 40 50 60 10-8 10-6 10-4 10-2 100 J 22 0 10 20 30 40 50 60 Time (s) 10-8 10-6 10-4 10-2 100 J 33 26 Spinning  Spacecraft  Example
28. ### 27 Conclusions • SDP  formulation  guarantees  that  a  valid  inertia

is  returned • Discrete  mechanics  formulation  eliminates  noise   amplification  problems • A  priori  knowledge  can  be  included  in  the  estimator  to   improve  convergence