Computational Geometric Mechanics:A Synthesis of Differential Geometry,Mechanics, and Numerical Analysis.

Melvin LeokMathematics, University of California, San Diego.

Math 196, Student Colloquium, UCSD

mleok@math.ucsd.eduhttp://www.math.ucsd.edu/˜mleok/

Supported by NSF DMS-0726263, DMS-100152, DMS-1010687 (CAREER),CMMI-1029445, DMS-1065972, CMMI-1334759, DMS-1411792, DMS-1345013.

2

Introduction

� Geometry and Mechanics

•Geometry is concerned with the properties of objects such ascurves, surfaces, and their higher-dimensional analogues.

• Understanding symmetry, which are the things we can do togeometric objects while keeping it the same, is important to con-servation properties.

•While the dynamics of mechanical systems is typically expressedin terms of differential equations, much can be understood fromgeometric and graphical means.

3

Equilibrium and Stability

� Equilibrium Points

• Consider a pendulum, under the influence of gravity.

• An Equilibrium Point is a state that does not change underthe dynamics.

• The fully down and fully up positions to a pendulum are ex-amples of equilibria.

4

Equilibrium and Stability

� Equilibria of Dynamical Systems

• The dynamics of the pendulum can then be visualized by plottingthe vector field, (θ, v).

• The equilibrium points correspond to the positions at which thevector field vanishes.

5

Equilibrium and Stability

� Stability of Equilibrium Points

• A point is at equilibrium if when we start the system at exactlythat point, it will stay at that point forever.

• Stability of an equilibrium point asks the question what happensif we start close to the equilibrium point, does it stay close?

• If we start near the fully down position, we will stay near it, so thefully down position is a stable equilibrium.

6

Equilibrium and Stability

� Stability of Equilibrium Points

• If we start near the fully up position, the pendulum will wanderfar away from the equilibrium, and as such, it is an unstableequilibrium.

7

Stability... always a good thing?

� Life on the brink of instability...

s

θ

m

l

g

M

l = pendulum length

m = pendulum bob mass

M = cart mass

g = acceleration due to gravity

8

Stability vs. Maneuverability

� The Northrop Grumman X-29 Forward Swept Aircraft

9

The Book Toss

(a) (b) (c)

10

Free Rigid Body Dynamics

Π3

Π2

Π1

11

Conservation Laws tell us a lot

• Spatial angular momentum is constant.

•Magnitude of body angular momentum is constant.

‖Π‖2 = Π21 + Π2

2 + Π23

A sphere in body angular momentum variables.

• Energy is constant.

K(Π) =1

2

(Π2

1

I1+

Π22

I2+

Π23

I3

)

A ellipsoid in body angular momentum variables.

12

True Polar Wander

13

The Interplanetary Superhighway

14

The Mathematics of Falling Cats

15

Connections, Curvature, and Geometric Phase

� Holonomy and Curvature

•Geometric Phase is an example of holonomy.

•Curvature can be thought of as infinitesimal holonomy.

area = A

finish

start

16

Example of Holonomy

Vatican Museum double helical staircase designed by Giuseppe Momo in 1932.

17

Falling Cats and Geometric Phases

reduced trajectory

true trajectory

horizontal lift using the

mechanical connection

dynamic phase

geometric phase

map to body variables

spherical cap with

solid angle Λ

angular mometum sphere

18

Geometry in Paramecium

19

Geometry at Home

20

Geometric Control of Spacecraft

� Geometric Phase based controllers

• Shape controlled using internal momentum wheels and gyroscopes.

• Changes in shape result in corresponding changes in orientation.

•More precise than chemical propulsion based orientation control.

rigid carrier

spinning rotors

21

Geometric Control of Spacecraft

22

Falling Cats and Astronauts

LIFE Magazine, 1968.

23

Numerically implementing geometric control algorithms

� Demand for long-time stability in control algorithms

• Trend towards autonomous space and underwater vehicle missionswith long deployment times and low energy propulsionsystems.

• Small inaccuracies in the control algorithms accumulated over longtimes will significantly diminish the operational lifespan of suchmissions.

• Need for geometric integrators to efficiently achieve good qualita-tive results for simulations over long times.

24

Geometry and Numerical Methods

� Dynamical equations preserve structure

•Many continuous systems of interest have properties that are con-served by the flow:

◦ Energy

◦ Symmetries, Reversibility, Monotonicity

◦Momentum - Angular, Linear, Kelvin Circulation Theorem.

◦ Symplectic Form

◦ Integrability

• At other times, the equations themselves are defined on a mani-fold, such as a Lie group, or more general configuration manifoldof a mechanical system, and the discrete trajectory we computeshould remain on this manifold, since the equations may not bewell-defined off the surface.

25

Motivation: Geometric Integration

� Main Goal of Geometric Integration:

Structure preservation in order to reproduce long time behavior.

� Role of Discrete Structure-Preservation:Discrete conservation laws impart long time numerical stabilityto computations, since the structure-preserving algorithm exactlyconserves a discrete quantity that is always close to the continuousquantity we are interested in.

26

Geometric Integration: Energy Stability

� Energy stability for symplectic integrators

Continuous energyIsosurface

Discrete energyIsosurface

Control on global error

27

Geometric Integration: Energy Stability

� Energy behavior for conservative and dissipative systems

0 200 400 600 800 1000 1200 1400 16000

0.05

0.1

0.15

0.2

0.25

0.3

Time

En

erg

y

Variational

Runge-Kutta

Benchmark

0 100 200 300 400 500 600 700 800 900 10000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time

En

erg

y

Midpoint Newmark

Explicit NewmarkVariational

non-variational Runge-Kutta

Benchmark

(a) Conservative mechanics (b) Dissipative mechanics

28

Geometric Integration: Energy Stability

� Solar System Simulation

• Forward Euler

qk+1 = qk + hq(qk,pk),

pk+1 = pk + hp(qk,pk).

• Inverse Euler

qk+1 = qk + hq(qk+1,pk+1),

pk+1 = pk + hp(qk+1,pk+1).

• Symplectic Euler

qk+1 = qk + hq(qk,pk+1),

pk+1 = pk + hp(qk,pk+1).

29

Geometric Integration: Energy Stability

� Forward Euler

−30−20

−100

1020

30

−30

−20

−10

0

10

20

30−30

−20

−10

0

10

20

30

2144

1980 2000 2020 2040 2060 2080 2100 2120 2140 21600

0.5

1

Energy error

30

Geometric Integration: Energy Stability

� Inverse Euler

−30−20

−100

1020

30

−30

−20

−10

0

10

20

30−30

−20

−10

0

10

20

30

2077

1990 2000 2010 2020 2030 2040 2050 2060 2070 2080−50

0

50

Energy error

31

Geometric Integration: Energy Stability

� Symplectic Euler

−30−20

−100

1020

30

−30

−20

−10

0

10

20

30−30

−20

−10

0

10

20

30

2268

1950 2000 2050 2100 2150 2200 2250 2300−2

0

2x 10

−3

Energy error

32

Introduction to Computational Geometric Mechanics

� Geometric Mechanics

• Differential geometric and symmetry techniques applied to thestudy of Lagrangian and Hamiltonian mechanics.

� Computational Geometric Mechanics

• Constructing computational algorithms using ideas from geometricmechanics.

• Variational integrators based on discretizing Hamilton’s principle,automatically symplectic and momentum preserving.

33

Continuous Mechanics

� Lagrangian Formulation of Mechanics

• Given a Lagrangian L : TQ → R, the solution curve satisfiesHamilton’s variational principle,

δ

∫ t1

t0

L (q (t) , q (t)) dt = 0.

• This is equivalent to the Euler-Lagrange equation,

d

dt

∂L

∂q− ∂L

∂q= 0.

•When the Lagrangian has the form L = 12qTMq − V (q), this

reduces to Newton’s Second Law,

d

dt(Mq) = −∂V

∂q.

34

Discrete Mechanics

� Discrete Variational Principle

q a( )

q b( )

dq t( )

Q

q t( ) varied curve

q0

qN

dqi

Q

qi varied point

•Discrete Lagrangian

Ld ≈∫ h

0L (q(t), q(t)) dt

•Discrete Euler-Lagrange equation

D2Ld(q0, q1) + D1Ld(q1, q2) = 0

35

Symplecticity in the Planar Pendulum

I.1 First Problems and Methods 5

degrees of freedom; Hp and Hq are the vectors of partial derivatives. One verifieseasily by differentiation (see Sect. IV.1) that, along the solution curves of (1.10),

H!p(t), q(t)

"= Const , (1.11)

i.e., the Hamiltonian is an invariant or a first integral. More details about Hamil-tonian systems and their derivation from Lagrangian mechanics will be given inSect. VI.1.

m

cos q 1q

Pendulum. The mathematical pendulum (mass m = 1,massless rod of length ! = 1, gravitational accelerationg = 1) is a system with one degree of freedom having theHamiltonian

H(p, q) =1

2p2 ! cos q, (1.12)

so that the equations of motion (1.10) become

p = ! sin q, q = p. (1.13)

Since the vector field (1.13) is 2"-periodic in q, it is natural to consider q as a vari-able on the circle S1. Hence, the phase space of points (p, q) becomes the cylinderR " S1. Fig. 1.3 shows some level curves of H(p, q). By (1.11), the solution curvesof the problem (1.13) lie on such level curves.

exact flowexact flow explicit Eulerexplicit Euler symplectic Eulersymplectic Euler

Fig. 1.3. Exact and numerical flow for the pendulum problem (1.13); step sizes h = t = 1

Area Preservation. Figure 1.3 (first picture) illustrates that the exact flow of aHamiltonian system (1.10) is area preserving. This can be explained as follows: thederivative of the flow #t with respect to initial values (p, q),

#!t(p, q) =

$!p(t), q(t)

"

$(p, q),

36

Symplecticity in the Planar PendulumVI.2 Symplectic Transformations 185

!1 0 1 2 3 4 5 6 7 8 9

!2

!1

1

2

A

!!/2(A)

!!(A)

B

!!/2(B)

!!(B)

!3!/2(B)

Fig. 2.2. Area preservation of the flow of Hamiltonian systems

is symmetric). We therefore obtain

d

dt

!"!"t

!y0

#T

J"!"t

!y0

#$=

" d

dt

!"t

!y0

#T

J"!"t

!y0

#+

"!"t

!y0

#T

J" d

dt

!"t

!y0

#

="!"t

!y0

#T

!2H%"t(y0)

&J!T J

"!"t

!y0

#+

"!"t

!y0

#T

!2H%"t(y0)

&"!"t

!y0

#= 0,

because JT = "J and J!T J = "I . Since the relation"!"t

!y0

#T

J"!"t

!y0

#= J (2.6)

is satisfied for t = 0 ("0 is the identity map), it is satisfied for all t and all (p0, q0),as long as the solution remains in the domain of definition of H . #$Example 2.5. We illustrate this theorem with the pendulum problem (Example 1.2)using the normalization m = # = g = 1. We have q = $, p = $, and the Hamil-tonian is given by

H(p, q) = p2/2 " cos q.

Fig. 2.2 shows level curves of this function, and it also illustrates the area preser-vation of the flow "t. Indeed, by Theorem 2.4 and Lemma 2.3, the areas of A and"t(A) as well as those of B and "t(B) are the same, although their appearance iscompletely different.

We next show that symplecticity of the flow is a characteristic property forHamiltonian systems. We call a differential equation y = f(y) locally Hamiltonian,if for every y0 % U there exists a neighbourhood where f(y) = J!1!H(y) forsome function H .

Theorem 2.6. Let f : U & R2d be continuously differentiable. Then, y = f(y) islocally Hamiltonian if and only if its flow "t(y) is symplectic for all y % U and forall sufficiently small t.

37

Symplecticity in the Planar Pendulum188 VI. Symplectic Integration of Hamiltonian Systems

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

0 2 4 6 8

!2

2

explicit Euler Runge, order 2

symplectic Euler Verlet

implicit Euler midpoint rule

Fig. 3.1. Area preservation of numerical methods for the pendulum; same initial sets as inFig. 2.2; first order methods (left column): h = !/4; second order methods (right column):h = !/3; dashed: exact flow

Example 3.2. We consider the pendulum problem of Example 2.5 with the sameinitial sets as in Fig. 2.2. We apply six different numerical methods to this problem:the explicit Euler method (I.1.5), the symplectic Euler method (I.1.9), and the im-plicit Euler method (I.1.6), as well as the second order method of Runge (II.1.3)(the right one), the Stormer–Verlet scheme (I.1.17), and the implicit midpoint rule(I.1.7). For two sets of initial values (p0, q0) we compute several steps with step sizeh = !/4 for the first order methods, and h = !/3 for the second order methods.One clearly observes in Fig. 3.1 that the explicit Euler, the implicit Euler and thesecond order explicit method of Runge are not symplectic (not area preserving). Weshall prove below that the other methods are symplectic. A different proof of theirsymplecticity (using generating functions) will be given in Sect. VI.5.

In the following we show the symplecticity of various numerical methods fromChapters I and II when they are applied to the Hamiltonian system in the vari-ables y = (p, q),

p = !Hq(p, q)

q = Hp(p, q)or equivalently y = J!1"H(y),

where Hp and Hq denote the column vectors of partial derivatives of the Hamil-tonian H(p, q) with respect to p and q, respectively.

38

Variational Lie Group Techniques

� Basic Idea

• To stay on the Lie group, we parametrize the curve by the initialpoint g0, and elements of the Lie algebra ξi, such that,

gd(t) = exp(∑

ξslκ,s(t))g0

• This involves standard interpolatory methods on the Lie algebrathat are lifted to the group using the exponential map.

• Automatically stays on SO(n) without the need for reprojection,constraints, or local coordinates.

• Cayley transform based methods perform 5-6 times faster, withoutloss of geometric conservation properties.

39

Example of a Lie Group Variational Integrator

� 3D Pendulum

•Discrete Lagrangian

Ld(Rk, Fk) =1

htr [(I3×3 − Fk)Jd]−

h

2V (Rk)− h

2V (Rk+1).

•Discrete Equations of Motion

Jωk+1 = FTk Jωk + hMk+1,

S(Jωk) =1

h

(FkJd − JdFTk

),

Rk+1 = RkFk.

40

Example of a Lie Group Variational Integrator

� Automatically staying on the rotation group

• The magic begins with the ansatz,

Fk = eS(fk),

and the Rodrigues’ formula, which converts the equation,

S(Jωk) =1

h

(FkJd − JdFTk

),

into

hJωk =sin ‖fk‖‖fk‖

Jfk +1− cos ‖fk‖‖fk‖2

fk × Jfk.

• Since Fk is the exponential of a skew matrix, it is a rotation matrix,and by matrix multiplication Rk+1 = RkFk is a rotation matrix.

41

Numerical Simulation

� Chaotic Motion of a 3D Pendulum

42

Numerical Simulation

� 3D Pendulum with an Internal Mass

43

Numerical Simulations

� Connected Rigid Bodies in a Perfect Fluid

Nonlinear Dynamics and Control Lab, University of Washington

•Most models are limited to carangiform locomotion, wherethe motion is concentrated in the rear of the body and the tail.

• Lie group and homogeneous space methods allow for the modelingof non-carangiform locomotion.

44

Numerical Simulations

� Falling Cat Models

Robot Locomotion GroupCSAIL, MIT

45

Applications to Asteroid Simulations

46

Under-actuated control of a 3D pendulum

47

Under-actuation in Everyday Life

� Parallel Parking and Lie Brackets

• Consider a vector field X which generates rotations, and a vectorfield Y which generates translations, then the Lie bracket of thetwo vector fields is,

φh2

[X,Y ] ≈ φ−hY ◦ φ−hX ◦ φ

hY ◦ φhX ,

and this new control direction allows us to parallel park a car.

48

Under-actuated control with symmetry of a 3D pendulum

−2

−1.5

−1

−0.5

−2

−1.5

−1

−0.5

49

Under-actuated control with symmetry of a 3D pendulum

−12

−10

−8

−6

−4

−2

−12

−10

−8

−6

−4

−2

50

Under-actuated control of a 3D pendulum on a cart

51

Variational Integrators for S2

� Spherical Pendulum

• Lagrangian

L(q, q) =1

2ml2q · q + mgle3 · q,

where e3 is the gravity direction.

•Variation of qδq = ξ × q,

where ξ ∈ R3 is orthogonal to the unit vector, i.e. ξ · q = 0.

• Equation of motion

q + (q · q)q +g

l(q × (q × e3)) = 0.

52

Variational Integrators for S2

� Spherical Pendulum

• First order equationsSince q = ω × q for some angular velocity ω ∈ R3 satisfyingω · q = 0, this can also be equivalently written as

ω − g

lq × e3 = 0,

q = ω × q.• These are global, singularity free equations of motion.

53

Comparison with Spherical Polar Coordinates

� Spherical Pendulum

• Lagrangian

L =ml2

2[θ2 + (ϕ sin θ)2]−mgl cos θ

• Equations of motion

θ =pθml2

, pθ =cos θ

ml2 sin3 θp2ϕ + mgl sin θ,

ϕ =pϕ

ml2 sin2 θ, pϕ = 0.

• Equations are singular when θ = 0, or θ = π, corresponding to thenorth and south poles of the sphere.

• These are purely due to coordinate singularities.

54

Variational Integrators for S2

� Double Spherical Pendulum Simulation

55

Variational Integrators for S2

� 3-body problem on the sphere

56

Variational Integrators for S2

� Geometrically Exact Elastic Rod

57

Variational Integrators for S2

� Magnetic Arrays

58

Conclusion

� Discrete Geometry, Mechanics, and Control

• Geometry has an important role to play in nonlinear control, andnumerical methods.

• Geometric control theory takes into account the interaction be-tween shape and group variables.

• Discrete geometry and mechanics is important for developing ac-curate and efficient computational schemes.

59

Questions?

http://www.math.ucsd.edu/˜mleok/