Zac Manchester
September 16, 2016
64

# Lyapunov Control for Flat Spin Recovery and Spin Inversion

I developed new control laws for stabilizing a tumbling spacecraft using reaction wheels. This presentation is from the 2016 AIAA Astrodynamics Specialist Conference.

## Zac Manchester

September 16, 2016

## Transcript

1. Zac  Manchester
Harvard  Agile  Robotics  Lab
(Formerly  Cornell  Space  Systems  Design  Studio)
Lyapunov-­‐Based  Control  Laws  for
Flat  Spin  Recovery  and  Spin  Inversion

2. 2
Motivation

3. 3
Problem:
• Achieve  spin  about  minor  axis  of  inertia  from  arbitrary  initial  conditions
• Align  + face  of  the  spacecraft  with  the  angular  momentum  vector
Constraints:
• Limited  reaction  wheel  torque
• Limited  angular  momentum  storage  in  reaction  wheels
Motivation

4. 4
Gyrostat  Dynamics:
J = I 1
˙
h = h ⇥ J ·(h ⇢)
h = I·! + ⇢
I· ˙
! + ! ⇥ (I·! + ⇢) + ˙
⇢ = ⌧
Background

5. 0
0.5
1
1.5
2
5
Lyapunov Stability:
˙
x
=
f
(
x
)
V
(
x
) 0
˙
V
(
x
) = @V
@x
d
x
d
t
 0
f
(
x
⇤) = 0
Background

6. 6
Some  Candidates:
VQ =
1
2
hd
·J ·hd
1
2
h·J ·h
Lyapunov  Functions

7. 7
Some  Candidates:
VL = hd
·J ·hd hd
·J ·h
Lyapunov  Functions

8. 8
Periodic  Averaging:
V 0
L
= hd
·J ·hd
1
T
Z T
0
hd
·J ·h dt
Lyapunov  Functions

9. 9
Periodic  Averaging:
V =
3
2
hd
·J ·hd
1
T
Z T
0
hd
·J ·h +
1
2
h·J ·h dt
Lyapunov  Functions

10. 10
Discretization:
hk = h(t0 + k t) ⇢k = ⇢(t0 + k t)
xk =
2
6
4
hk
.
.
.
hk+N 1
3
7
5 uk =
2
6
4
⇢k
.
.
.
⇢k+N 1
3
7
5
xd =
2
6
4
hd
.
.
.
hd
3
7
5
Calculating  ̇

11. 11
Discretization:
1
T
Z T
0
hd
·
J
· h +
1
2
h ·
J
· h d
t

1
N
x
|
d
¯
J xk +
1
2
N
x
|
k
¯
J xk
¯
J =
2
6
6
6
4
J 0 · · · 0
0 J · · · 0
.
.
.
.
.
.
...
.
.
.
0 0 · · · J
3
7
7
7
5
Calculating  ̇

12. 12
Periodic  System:
A =
2
6
6
6
6
6
6
4
0 13⇥3 0 · · · 0
0 0 13⇥3
.
.
. 0
.
.
.
.
.
.
...
...
.
.
.
0 0 · · · 0 13⇥3
13⇥3 0 · · · 0 0
3
7
7
7
7
7
7
5
Bk = t
2
6
6
6
6
4
h⇥
k
J 0 · · · 0
0 h⇥
k+1
J 0
.
.
.
.
.
.
...
... 0
0 · · · 0 h⇥
k+N 1
J
3
7
7
7
7
5
Calculating  ̇
˙
h = h ⇥
J
·(h ⇢) !
xk+1 =
Axk +
Bkuk

13. 13
Continuum  Limit:
V
=
1
N
x
|
d
¯
J xk +
1
2
N
x
|
k
¯
J xk
1
N
x
|
d
¯
J xk+1
1
2
N
x
|
k+1
¯
J xk+1
=
1
N
x
|
d
¯
J Bk uk
1
N
x
|
k A
| ¯
J Bk uk
1
2
N
u
|
k B
|
k
¯
J Bk uk
˙
V = lim
N!1
V
t
=
1
T
Z t0+T
t0
(h + hd) · J · h ⇥ J · ⇢ dt
Calculating  ̇

14. 14
Almost-­‐Global  Asymptotic  Stability:
˙
V =
1
T
Z T
0
(h + hd)·J ·h ⇥ J ·⇢ dt
b(h)
⇢ ⇡ ⇢
max
sign(b)
Control  Laws

15. 15
Smooth  Control  Law:
⇢ = ⇢
max
tanh(↵b)
x
-5 0 5
tanh(x)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Control  Laws

16. 16
Momentum  Vector  Trajectory:
0.5
0
-0.5
-1
1
0.5
0
-0.5
0.5
0
-0.5
1
Flat-­‐Spin  Recovery

17. 0 2 4 6 8
;
1
-0.01
0
0.01
0 2 4 6 8
;
2
-0.01
0
0.01
Time (min)
0 2 4 6 8
;
3
-0.01
0
0.01
17
Control  Inputs:
Flat-­‐Spin  Recovery

18. 18
Momentum  Vector  Trajectory:
0.5
0
-0.5
-0.5
0
0.5
-1
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
Spin  Inversion

19. 0 1 2 3 4 5 6
;
1
-0.01
0
0.01
0 1 2 3 4 5 6
;
2
-0.01
0
0.01
Time (min)
0 1 2 3 4 5 6
;
3
-0.01
0
0.01
19
Control  Inputs:
Spin  Inversion

20. 20
[email protected]
Questions?