Variational Principle
Numerical Simulation
Introduction
A Lie Group Variational Integrator
for the Attitude dynamics of a Rigid body
Taeyoung Lee
Student Seminar in Flight Dynamics and Control
Department of Aerospace Engineering
University of Michigan
February 4, 2005
Lie Group Variational Integrator
1/28
Variational Principle
Numerical Simulation
Introduction
Motivation
Dynamics and Control of a System
The motion of a system is described by a Differential Equation.
Geometric feature is crucial for a dynamics and control problem.
The analytic solution of a differential equation is hard to find.
Numerical Integration
The numerical solution is used to study dynamics of the system,
to design/verify a controller.
There has been little attention given to computational approaches.
Generally, numerical integration methods DO NOT preserve the
characteristics of the system.
Geometric Numerical Integration
Lie Group Variational Integrator
2/28
Variational Principle
Numerical Simulation
Introduction
Motivation
Dynamics and Control of a System
The motion of a system is described by a Differential Equation.
Geometric feature is crucial for a dynamics and control problem.
The analytic solution of a differential equation is hard to find.
Numerical Integration
The numerical solution is used to study dynamics of the system,
to design/verify a controller.
There has been little attention given to computational approaches.
Generally, numerical integration methods DO NOT preserve the
characteristics of the system.
Geometric Numerical Integration
Lie Group Variational Integrator
2/28
Variational Principle
Numerical Simulation
Introduction
Motivation
Dynamics and Control of a System
The motion of a system is described by a Differential Equation.
Geometric feature is crucial for a dynamics and control problem.
The analytic solution of a differential equation is hard to find.
Numerical Integration
The numerical solution is used to study dynamics of the system,
to design/verify a controller.
There has been little attention given to computational approaches.
Generally, numerical integration methods DO NOT preserve the
characteristics of the system.
Geometric Numerical Integration
Lie Group Variational Integrator
2/28
Variational Principle
Numerical Simulation
Introduction
Motivation
Numerical Example I : Two Body Problem
Exact flow of the Two Body Problem
Normalized Equation of motion
q2
.
(q21 + q22 )3/2
!
r
1+e
q10 = 1 − e, q20 = 0, q̇10 = 0, q̇20 =
1−e
q̈1 = −
q1
,
(q21 + q22 )3/2
q̈2 = −
The trajectory is a conic section.
The total energy and the angular momentum are conserved.
H=
1 2
1
q̇ + q̇22 − 2
2 1
(q1 + q22 )1/2
h = q1 q̇2 − q2 q̇1
Lie Group Variational Integrator
3/28
Variational Principle
Numerical Simulation
Introduction
Motivation
Numerical Example I : Two Body Problem
Numerical integration result
Variational step-size Runge-Kutta method (Matlab ode45.m)
1
−0.49
Energy
−0.5
0.5
−0.51
−0.52
q2
−0.53
−0.54
0
0
1
2
3
4
5
t/T
6
7
8
9
10
1
2
3
4
5
t/T
6
7
8
9
10
Angular Momentum
0.808
−0.5
−1
−1.5
−1
−0.5
q1
0
0.5
(Exact flow
0.806
0.804
0.802
0.8
0.798
0
, Runge-Kutta
)
The trajectory is not closed orbit. (e = 0.6)
The total energy and the angular momentum are not conserved.
Lie Group Variational Integrator
4/28
Variational Principle
Numerical Simulation
Introduction
Motivation
Numerical Example II : Pendulum
Exact flow of a Pendulum
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
....
.
q
x
Equation of motion
q̈ = − sin q
The total energy is conserved.
H=
1 2
q̇ − cos q
2
The flow is symplectic.
(Area conservation in phase plane)
Lie Group Variational Integrator
5/28
Variational Principle
Numerical Simulation
Introduction
Motivation
Numerical Example II : Pendulum
Numerical integration result
2
−0.45
1
−0.46
0
−0.47
−1
−0.48
−2
0
5
10
15
20
25
30
time (sec)
35
40
45
50
Velocity
1
−0.49
Energy
Position
Variational step-size Runge-Kutta method (Matlab ode45.m)
−0.5
−0.51
0.5
−0.52
0
−0.53
−0.5
−0.54
−1
0
5
10
15
20
25
30
time (sec)
35
40
45
(Exact flow
50
−0.55
0
5
10
15
20
, Runge-Kutta
25
time
30
35
40
45
50
)
The position and velocity errors increase as t.
The total energy is not conserved.
Lie Group Variational Integrator
6/28
Variational Principle
Numerical Simulation
Introduction
Motivation
Numerical Example II : Pendulum
Numerical integration result
The numerical flow is not symplectic.
Exact Flow
Lie Group Variational Integrator
Numerical Flow
7/28
Variational Principle
Numerical Simulation
Introduction
Motivation
Numerical Example II : Pendulum
Numerical integration result
The numerical flow is not symplectic.
2
1
0
−1
t = 50.0 (sec)
−2
−3
−2
−1
Exact Flow
Lie Group Variational Integrator
0
1
2
3
Numerical Flow
7/28
Variational Principle
Numerical Simulation
Introduction
Geometric Numerical Integration
Energy Momentum method [LaBudde 1976],[Simo 1992]
The parameters of numerical integration is chosen such that
energy and momentum are conserved.
The states are projected to the constant energy momentum
surface.
Variational Integrator [Moser 1991], [Marsden 2001]
A systematic method of constructing structure-preserving
integrator
It approximates the continuous dynamics by discretizing
Hamilton’s principle.
It inherits the properties of the continuous dynamics.
Lie group method [Iserles 2000]
Many practical differential equations evolve on a Lie group.
It preserves the geometry of the configuration space.
Lie Group Variational Integrator
8/28
Variational Principle
Numerical Simulation
Introduction
Geometric Numerical Integration
Energy Momentum method [LaBudde 1976],[Simo 1992]
The parameters of numerical integration is chosen such that
energy and momentum are conserved.
The states are projected to the constant energy momentum
surface.
Variational Integrator [Moser 1991], [Marsden 2001]
A systematic method of constructing structure-preserving
integrator
It approximates the continuous dynamics by discretizing
Hamilton’s principle.
It inherits the properties of the continuous dynamics.
Lie group method [Iserles 2000]
Many practical differential equations evolve on a Lie group.
It preserves the geometry of the configuration space.
Lie Group Variational Integrator
8/28
Variational Principle
Numerical Simulation
Introduction
Geometric Numerical Integration
Energy Momentum method [LaBudde 1976],[Simo 1992]
The parameters of numerical integration is chosen such that
energy and momentum are conserved.
The states are projected to the constant energy momentum
surface.
Variational Integrator [Moser 1991], [Marsden 2001]
A systematic method of constructing structure-preserving
integrator
It approximates the continuous dynamics by discretizing
Hamilton’s principle.
It inherits the properties of the continuous dynamics.
Lie group method [Iserles 2000]
Many practical differential equations evolve on a Lie group.
It preserves the geometry of the configuration space.
Lie Group Variational Integrator
8/28
Variational Principle
Numerical Simulation
Introduction
Problem Definition
Attitude Dynamics of a Rigid body under a Potential depends on the Attitude
Assumption
V
.........
..... ..........
...
...
...O .r... ........
.
.
....
... ....
....
.. ....
..
...... ......... .......... .
..... ........ρ .........
....
...
...
...
......
...
...
...
...
...
...
q
...
e2
..
...
CM
..
?....
...
.
.
.
.
.
e3 ........
.
.
.............................
Lie Group Variational Integrator
e1
The O − e1 e2 e3 is an inertial frame.
V : SO(3) 7→ R
Geometric Feature
The configuration space is SO(3).
The Hamiltonian is conserved.
Other first integral may exist according
to the symmetries of V.
Goal
A numerical integrator that respects the
geometric features.
The integrator updates ω, R.
9/28
Variational Principle
Numerical Simulation
Introduction
Problem Definition
Attitude Dynamics of a Rigid body under a Potential depends on the Attitude
Assumption
V
.........
..... ..........
...
...
...O .r... ........
.
.
....
... ....
....
.. ....
..
...... ......... .......... .
..... ........ρ .........
....
...
...
...
......
...
...
...
...
...
...
q
...
e2
..
...
CM
..
?....
...
.
.
.
.
.
e3 ........
.
.
.............................
Lie Group Variational Integrator
e1
The O − e1 e2 e3 is an inertial frame.
V : SO(3) 7→ R
Geometric Feature
The configuration space is SO(3).
The Hamiltonian is conserved.
Other first integral may exist according
to the symmetries of V.
Goal
A numerical integrator that respects the
geometric features.
The integrator updates ω, R.
9/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Variational Principle
The basic idea of the variational integrator
Continuous time
Discrete time
Configuration space
Configuration space
(q, q̇) ∈ TQ
(qk , qk+1 ) ∈ Q × Q
Lagrangian
Lagrangian
L(q, q̇)
Ld (qk , qk+1 )
Action Integral
G=
Rt
f
t0
L(q, q̇) dt
Variation
δG =
d
G
d
=0
Action Sum
Gd =
PN−1
k=0
Ld (qk , qk+1 )
Variation
δGd =
d
G
d d
=0
Equation of motion
Equation of motion
q̇ = f (q, q̇)
qk+2 = fd (qk , qk+1 )
Lie Group Variational Integrator
10/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Variational Principle
The basic idea of the variational integrator
Continuous time
Discrete time
Configuration space
Configuration space
(q, q̇) ∈ TQ
(qk , qk+1 ) ∈ Q × Q
Lagrangian
Lagrangian
L(q, q̇)
Ld (qk , qk+1 )
Action Integral
G=
Rt
f
t0
L(q, q̇) dt
Variation
δG =
d
G
d
=0
Action Sum
Gd =
PN−1
k=0
Ld (qk , qk+1 )
Variation
δGd =
d
G
d d
=0
Equation of motion
Equation of motion
q̇ = f (q, q̇)
qk+2 = fd (qk , qk+1 )
Lie Group Variational Integrator
10/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Lagrangian
Two equivalent forms of Lagrangian
L(R, ω) =
.................
... .q .....
- e1
... ....... .....
... ....... ......
... ....... .....
.... .......... ........
... ρ̃ ...... ....
.... ..
c ...
.....
...
e2 ...
.
.
?.
...
.
.
e3 ........
.
....................
1
2
Z
kω × ρ̃k2 dm − V(R).
Body
Using kω × ρ̃k2 = kS(ρ̃)ωk2 ,
L(R, ω) =
1 T
ω
2
Z
S(ρ̃)T S(ρ̃)dm ω − V(R),
Body
1
, ω T Jω − V(R).
2
(1)
Using kω × ρ̃k2 = kS(ω)ρ̃k2 ,
L(R, ω) =
1
2
Z
tr S(ω)ρ̃ρ̃T S(ω)T dm − V(R),
Body
1
= tr S(ω)
2
Z
Body
1 , tr S(ω)Jd S(ω)T − V(R),
2
Lie Group Variational Integrator
ρ̃ρ̃T dm S(ω)T − V(R),
(2)
11/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Lagrangian
Two equivalent forms of Lagrangian
L(R, ω) =
.................
... .q .....
- e1
... ....... .....
... ....... ......
... ....... .....
.... .......... ........
... ρ̃ ...... ....
.... ..
c ...
.....
...
e2 ...
.
.
?.
...
.
.
e3 ........
.
....................
1
2
Z
kω × ρ̃k2 dm − V(R).
Body
Using kω × ρ̃k2 = kS(ρ̃)ωk2 ,
L(R, ω) =
1 T
ω
2
Z
S(ρ̃)T S(ρ̃)dm ω − V(R),
Body
1
, ω T Jω − V(R).
2
(1)
Using kω × ρ̃k2 = kS(ω)ρ̃k2 ,
L(R, ω) =
1
2
Z
tr S(ω)ρ̃ρ̃T S(ω)T dm − V(R),
Body
1
= tr S(ω)
2
Z
Body
1 , tr S(ω)Jd S(ω)T − V(R),
2
Lie Group Variational Integrator
ρ̃ρ̃T dm S(ω)T − V(R),
(2)
11/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Lagrangian
Relationship between J and Jd
Using the relationship, S(ρ̃)T S(ρ̃) = ρ̃T ρ̃I3×3 − ρ̃ρ̃T , J and Jd are related as
Z
ρ̃T ρ̃dm
J=
I3×3 − Jd
Body
= tr[Jd ] I3×3 − Jd .
If we express ρ̃ in coordinates (x, y, z), J and Jd are expressed as
"R
J=
y2R+ z2 dm
− R xy dm
− xz dm
R−
R
xy dm
2
R+ x dm
− yz dm
z2
R
"R
#
− R xz dm
R − yz dm ,
x2 + y2 dm
Jd =
2
R x dm
xy
dm
R
xz dm
R
R xy dm
2
R y dm
yz dm
R
#
R xz dm
R yz dm .
z2 dm
Furthermore, the following equation is satisfied for ω ∈ R3 .
S(Jω) = S(ω)Jd + Jd S(ω).
Lie Group Variational Integrator
(3)
12/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Variation
Action Integral
Z t
f
G=
Z t
f
L(R, ω)dt, =
t0
t0
i
1 h
tr S(ω)Jd S(ω)T − V(R) dt.
2
(4)
Variation of R, ω
The variation should respect the geometry of the configuration space.
The varied rotation matrix R ∈ SO(3) can be expressed as
R = Reη ,
(5)
where ∈ R, η ∈ so(3)
Using the kinematic relationship Ṙ = RS(ω), the variation of ω is induced
from R .
S(ω ) = RT Ṙ = e−η S(ω)eη + η̇,
= S(ω) + [η̇ + S(ω)η − ηS(ω)] + O(2 ).
Lie Group Variational Integrator
(6)
13/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Equation of Motion
Variation of Action Integral
d
δG = G =
d
=0
Z t
f
t0
1
∂V
tr η S(J ω̇ + ω × Jω) + 2RT
2
∂R
dt.
Continuous Equation of motion
From Hamilton’s principle, δG = 0. Then, we obtain
S(J ω̇ + ω × Jω) =
∂V T
∂V
R − RT
,
∂R
∂R
or equivalently,
J ω̇ + ω × Jω = M,
3
where M ∈ R is determined by S(M) =
∂V T
R
∂R
−
(7)
RT ∂V
.
∂R
More explicitly, it can be shown that
M = r1 × vr1 + r2 × vr2 + r3 × vr3 ,
where ri , vri ∈ R1×3 are the ith row vectors of R and
Lie Group Variational Integrator
∂V
,
∂R
(8)
respectively.
14/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Variational Principle
The basic idea of the variational integrator
Continuous time
Discrete time
Configuration space
Configuration space
(q, q̇)
(qk , qk+1 )
Lagrangian
Lagrangian
L(q, q̇)
Ld (qk , qk+1 )
Action Integral
G=
Rt
f
t0
L(q, q̇) dt
Variation
δG =
d
G
d
=0
Action Sum
Gd =
PN−1
k=0
Ld (qk , qk+1 )
Variation
δGd =
d
G
d d
=0
Equation of motion
Equation of motion
q̇ = f (q, q̇)
qk+2 = fd (qk , qk+1 )
Lie Group Variational Integrator
15/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Discrete Lagrangian
Discrete Lagrangian
Define new variable Fk ∈ SO(3) such that Rk+1 = Rk Fk .
Using the kinematic relationship Ṙ = RS(ω), S(ωk ) can be approximated as
S(ωk ) = RTk Ṙk ≈ RTk
Rk+1 − Rk
h
=
1
(Fk − I3×3 ) .
h
(9)
Consider the discrete Lagrangian Ld ,
h
h
L(Rk , ωk ) + L(Rk+1 , ωk+1 ),
2
2
h
h
1
= tr[(I3×3 − Fk ) Jd ] − V(Rk ) − V(Rk+1 ).
h
2
2
Ld (Rk , Fk ) '
Action Sum
Since the discrete Lagrangian Ld approximates
N−1
X
Gd =
N−1
X
Ld (Rk , Fk ) =
k=0
k=0
Rt
k+1
tk
(10)
L(R, ω) dt,
1
h
h
tr[(I3×3 − Fk ) Jd ] − V(Rk ) − V(Rk+1 ).
h
2
2
(11)
Lie Group Variational Integrator
16/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Variation
Variation of Rk , Fk
The varied rotation matrix Rk ∈ SO(3) can be expressed as
Rk = Rk eηk ,
(12)
where ∈ R, ηk ∈ so(3).
Using the definition of Fk , the variation of Fk is induced as
−ηk T
Fk = RT
Rk Rk+1 eηk+1 = e−ηk Fk eηk+1 .
k Rk+1 = e
(13)
Variation of Action Sum
N−1
X 1
d h
∂V
∂V
=
Gd
tr[(ηk Fk − Fk ηk+1 ) Jd ] + tr ηk RTk
+ ηk+1 RTk+1
,
d =0
h
2
∂R
∂R
k
k+1
k=0
N−1 X
=
tr ηk
k=1
Lie Group Variational Integrator
∂V
1
(Fk Jd − Jd Fk−1 ) + hRTk
h
∂Rk
.
(14)
17/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Discrete Equation of Motion
Discrete Equation of Motion in Rk , Fk
From Hamilton’s principle, δGd = 0. Then, we obtain
1
T
Fk+1 Jd − Jd Fk − Jd Fk+1
+ FkT Jd = hS(Mk+1 ),
h
Rk+1 = Rk Fk .
(15)
(16)
This yields a discrete map (R0 , F0 ) 7→ (R1 , F1 ) and the process can be repeated.
Discrete Equation of Motion in Rk , ωk
Inertial and body-fixed angular momentum are related as
S(Jω) = S(ω)Jd + Jd S(ω),
S(RJω) = RS(Jω)RT = ṘJd RT − RJd ṘT .
If we approximate Ṙk =
Rk+1 −Rk
,
h
we obtain
1
Rk+1 Jd RTk − Rk Jd RTk+1 ,
h
1
T
S(Jωk ) = Rk S(Rk Jωk )Rk =
Fk Jd − Jd FkT .
h
S(Rk Jωk ) =
Lie Group Variational Integrator
(17)
18/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Discrete Equation of Motion
Variational Integrator for the Attitude dynamics of a Rigid Body
Discrete Equation of Motion in Rk , ωk
Using the above equations, the discrete equations of motion can be expressed as
Jωk+1 = FkT Jωk + hMk+1 ,
1
Fk Jd − Jd FkT ,
S(Jωk ) =
h
Rk+1 = Rk Fk ,
where Mk ∈ R3 is determined by S(Mk ) =
∂V T
Rk
∂Rk
(18)
(19)
(20)
∂V
− RTk ∂R
. More explicitly,
k
Mk = r1 × vr1 + r2 × vr2 + r3 × vr3 ,
where ri , vri ∈ R
1×3
are the ith row vectors of Rk and
∂V
,
∂Rk
(21)
respectively.
Given R0 , and ω0 , we can obtain F0 implicitly by solving (19), R1 is updated by
(20), and the angular velocity ω1 is updated by (18). This yields a map
(R0 , ω0 ) 7→ (R1 , ω1 ) and the process is repeated.
Lie Group Variational Integrator
19/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Discrete Equation of Motion
Properties of the Variational Integrator
Preservation of conserved quantities
Noether’s theorem
Symmetries in the Lagrangian result in the conservation of the associated
momentum map.
Variational integrators exhibit a discrete analogue of Noether’s
theorem.
All the conserved momenta in the continuous dynamics are preserved in
the discrete dynamics.
Preservation of the configuration space
The state is automatically evolves on the Rotation group.
Since SO(3) is closed under multiplication, the attitude matrix Rk
remains on SO(3). (Rk+1 = Rk Fk )
The attitude is globally defined without any projection or
constrains.
Lie Group Variational Integrator
20/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Discrete Equation of Motion
Properties of the Variational Integrator
Preservation of conserved quantities
Noether’s theorem
Symmetries in the Lagrangian result in the conservation of the associated
momentum map.
Variational integrators exhibit a discrete analogue of Noether’s
theorem.
All the conserved momenta in the continuous dynamics are preserved in
the discrete dynamics.
Preservation of the configuration space
The state is automatically evolves on the Rotation group.
Since SO(3) is closed under multiplication, the attitude matrix Rk
remains on SO(3). (Rk+1 = Rk Fk )
The attitude is globally defined without any projection or
constrains.
Lie Group Variational Integrator
20/28
Variational Principle
Numerical Simulation
Continuous time model
Variational Integrator
Discrete Equation of Motion
Computational Approach
Solution of hS(Jω) = FJd − Jd F T .
Using the Rodrigues’ formula,
F = eS(f ) = I3×3 +
sin kf k
1 − cos kf k
S(f )2 ,
S(f ) +
kf k
kf k2
where f is a vector in R3 . Then the matrix equation can be transformed into
hJω =
sin kf k
1 − cos kf k
Jf +
f × Jf , G(f ).
kf k
kf k2
(22)
Newton iteration method gives
fi+1 = fi + ∇G(fi )−1 (hJω − G(fi )) ,
where ∇G(f ) can be expressed as
∇G(f ) =
Lie Group Variational Integrator
cos kf k kf k − sin kf k T sin kf k kf k − 2(1 − cos kf k)
Jff +
(f × Jf ) f T
kf k3
kf k4
sin kf k
1 − cos kf k
+
J+
{−S(Jf ) + S(f )J} .
kf k
kf k2
21/28
Variational Principle
Numerical Simulation
3D Pendulum
Simulation Result
3D Pendulum Model
Uniform gravity potential
O.....r..............
..... ......... ............
...... ......... ..........
e2
g
?
e1
... ....
....
....
... .....
... ρ .....
...
....
...
...
...
....
...
....
...
...
....
...
...
....
...
...
? ....
...
..q
...
.
e3 .....
...
C.M.
...
.
..
...
.
...
..
.
.
.
....
................................
V = −mgeT3 Rρ,
∂V
= −mge3 ρT .
∂R
Equation of motion
Since M = r3 × vr3 = −eT3 R × mgρT ,
Jωk+1 = FkT Jωk + hmgρ × RTk+1 e3 .
Conserved quantities
Since the Lagrangian is independent of time, and
is invariant under the inertial rotation about e3 .
H=
1 T
ω Jω − mgeT3 Rρ,
2
π3 = eT3 RJω.
are conserved. The equation of motion can be
written as Rk+1 Jωk+1 − Rk Jωk = hmgRk+1 ρ × e3 ,
which shows the conservation of π3 directly.
Lie Group Variational Integrator
22/28
Variational Principle
Numerical Simulation
3D Pendulum
Simulation Result
Simulation Parameters
Mass properties
J = diag [1, 2.8, 2] kg m2 , m = 1kg, ρ = [0, 0, 1] m.
Initial conditions
1
Small perturbation from the hanging equilibrium
ω0 = [0.5, −0.5, 0.4] rad/s, R0 = I3×3 .
2
Small perturbation from the inverted equilibrium
ω0 = [0.5, −0.5, 0.4] rad/s, R0 = diag [−1, 1, −1] .
3
Perturbation from an initially horizontal position


0 0 1
ω0 = [2.5, 2.5, 0] rad/s, R0 =  0 1 0 .
−1 0 0
Lie Group Variational Integrator
23/28
Variational Principle
Numerical Simulation
3D Pendulum
Simulation Result
Simulation Result
(i) Small perturbation from the hanging equilibrium
−9.174
0
−1
0
H (J)
ω1 (rad/s)
1
5
10
15
20
25
30
−9.178
0
0
5
10
15
20
25
30
0.7995
0
4
0.4
|| I−RTR ||
ω3 (rad/s)
10
15
20
25
30
5
10
15
20
25
30
5
10
15
times (s)
20
25
30
0.8
0.5
0.3
0.2
0
5
0.8005
π3 (Nms)
ω2 (rad/s)
1
−1
0
−9.176
5
10
15
time (s)
20
25
30
Angular velocity
(h = 10−3 (sec), Variational integrator
Lie Group Variational Integrator
x 10
−4
2
0
0
Conserved Quantities
, Runge-Kutta
)
24/28
Variational Principle
Numerical Simulation
3D Pendulum
Simulation Result
Simulation Result
(ii) Small perturbation from the inverted equilibrium
ω1 (rad/s)
10
10.5
−10
0
H (J)
0
5
10
15
20
25
10.4
0
30
0
5
10
15
20
25
30
15
20
25
30
5
10
15
20
25
30
5
10
15
times (s)
20
25
30
−0.8
−0.82
0
10
|| I−RTR ||
ω3 (rad/s)
10
−0.78
5
0
−5
0
5
−0.76
π3 (Nms)
ω2 (rad/s)
5
−5
0
10.45
5
10
15
time (s)
20
25
30
Angular velocity
(h = 10−5 (sec), Variational integrator
Lie Group Variational Integrator
x 10
−3
5
0
0
Conserved Quantities
, Runge-Kutta
)
25/28
Variational Principle
Numerical Simulation
3D Pendulum
Simulation Result
Simulation Result
(iii) Perturbation from an initially horizontal position
ω1 (rad/s)
10
11.89
−10
0
H (J)
0
5
10
15
20
25
0
5
10
15
20
25
30
−2.51
0
4
|| I−RTR ||
ω3 (rad/s)
10
15
20
25
30
5
10
15
20
25
30
5
10
15
times (s)
20
25
30
−2.5
5
0
−5
0
5
−2.49
π3 (Nms)
ω2 (rad/s)
11.87
11.86
0
30
5
−5
0
11.88
5
10
15
time (s)
20
25
30
Angular velocity
(h = 10−5 (sec), Variational integrator
Lie Group Variational Integrator
x 10
−3
2
0
0
Conserved Quantities
, Runge-Kutta
)
26/28
Variational Principle
Numerical Simulation
3D Pendulum
Simulation Result
Simulation Result
Chaotic Motion : (iii) Perturbation from an initially horizontal position
Lie Group Variational Integrator
27/28
Variational Principle
Numerical Simulation
3D Pendulum
Simulation Result
Conclusion
Variational Integrator for the Attitude dynamics of a Rigid body
with the Potential depends on the Attitude
The integrator is obtained from a discrete variational principle.
It exhibits the symplectic properties and preserves conserved
quantities.
It adopts the approach of Lie group integrators.
The discrete state evolves on a rotation group automatically
without any reprojection and constraints.
The general idea can be applied to other Lie groups.
Numerical Simulation for the 3D Pendulum
The Lie group variational integrator is used to study the dynamics
of the 3D Pendulum.
Some initial condition may lead to irregular or chaotic solutions.
Lie Group Variational Integrator
28/28
Variational Principle
Numerical Simulation
3D Pendulum
Simulation Result
Conclusion
Variational Integrator for the Attitude dynamics of a Rigid body
with the Potential depends on the Attitude
The integrator is obtained from a discrete variational principle.
It exhibits the symplectic properties and preserves conserved
quantities.
It adopts the approach of Lie group integrators.
The discrete state evolves on a rotation group automatically
without any reprojection and constraints.
The general idea can be applied to other Lie groups.
Numerical Simulation for the 3D Pendulum
The Lie group variational integrator is used to study the dynamics
of the 3D Pendulum.
Some initial condition may lead to irregular or chaotic solutions.
Lie Group Variational Integrator
28/28