Lie Group and Homogeneous Variational Integrators: Towards a - PowerPoint PPT Presentation

Lie Group and Homogeneous Variational Integrators: Towards a Geometrically Exact Model of Elasticity Melvin Leok Mathematics, Purdue University. Joint work with Mathieu Desbrun, Anil Hirani, Taeyoung Lee, Jerrold Marsden, Harris McClamroch, and Alan Weinstein. Discrete Differential Geometry, Berlin, July, 2007. mleok@math.purdue.edu http://www.math.purdue.edu/˜mleok/ Mathematics, Purdue University NSF DMS-0504747 and DMS-0726263

2 The Mathematics of Falling Cats

3 Connections, Curvature, and Geometric Phase � Connections • Connections provide a means of comparing elements of a fiber based at different points on the manifold. � Holonomy and Curvature • Geometric Phase is an example of holonomy . area = A finish • Curvature can be thought of as infinitesimal holonomy . start

4 Example of Holonomy Vatican Museum double helical staircase designed by Giuseppe Momo in 1932.

5 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

6 Shape Dynamics of Formations of Satellites � NASA Terrestrial Planet Finder (TPF) • The NASA Terrestrial Planet Finder (TPF) and the ESA Darwin missions are examples of a potential application of the geometric formation control of satellite clusters. Small satellites in arranged in a formation to yield a large Artist’s conception of the ESA Darwin effective aperture telescope. Courtesy NASA/JPL-Caltech. flotilla. Courtesy ESA. • It is quite natural to think of controlling the shape of the cluster, and its orientation (group) separately.

7 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 manifold of a mechanical system, and the discrete trajectory we compute should remain on this manifold, since the equations may not be well-defined off the surface.

8 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 stability to computations, since the structure-preserving algorithm exactly conserves a discrete quantity that is always close to the continuous quantity we are interested in.

9 Geometric Integration: Energy Stability � Energy stability for symplectic integrators Control�on�global�error Continuous�energy Isosurface Discrete�energy Isosurface

10 Geometric Integration: Energy Stability � Energy behavior for conservative and dissipative systems 0.35 0.3 Midpoint Newmark 0.3 0.25 0.25 0.2 Variational Benchmark Explicit Newmark Variational 0.2 Energy Energy 0.15 non-variational Runge-Kutta 0.15 0.1 0.1 Runge-Kutta 0.05 0.05 Benchmark 0 0 0 100 200 300 400 500 600 700 800 900 1000 0 200 400 600 800 1000 1200 1400 1600 Time Time (a) Conservative mechanics (b) Dissipative mechanics

11 Geometric Integration: Energy Stability � Solar System Simulation • Forward Euler q k +1 = q k + h ˙ q ( q k , p k ) , p k +1 = p k + h ˙ p ( q k , p k ) . • Inverse Euler q k +1 = q k + h ˙ q ( q k +1 , p k +1 ) , p k +1 = p k + h ˙ p ( q k +1 , p k +1 ) . • Symplectic Euler q k +1 = q k + h ˙ q ( q k , p k +1 ) , p k +1 = p k + h ˙ p ( q k , p k +1 ) .

12 Geometric Integration: Energy Stability � Forward Euler 30 20 10 0 2144 −10 −20 −30 30 20 30 10 20 0 10 0 −10 −10 −20 −20 −30 −30 1 0.5 0 1980 2000 2020 2040 2060 2080 2100 2120 2140 2160 Energy error

13 Geometric Integration: Energy Stability � Inverse Euler 30 20 10 0 2077 −10 −20 −30 30 20 30 10 20 0 10 0 −10 −10 −20 −20 −30 −30 50 0 −50 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 Energy error

14 Geometric Integration: Energy Stability � Symplectic Euler 30 20 10 0 2268 −10 −20 −30 30 20 30 10 20 0 10 0 −10 −10 −20 −20 −30 −3 −30 x 10 2 0 −2 1950 2000 2050 2100 2150 2200 2250 2300 Energy error

15 Introduction to Computational Geometric Mechanics � Geometric Mechanics • Differential geometric and symmetry techniques applied to the study of Lagrangian and Hamiltonian mechanics. � Computational Geometric Mechanics • Constructing computational algorithms using ideas from geometric mechanics. • Variational integrators based on discretizing Hamilton’s principle, automatically symplectic and momentum preserving.

16 Discrete Mechanics � Discrete Variational Principle q i varied�curve varied�point q t ( ) Q Q (��) q b q N � q t ( ) � q i q 0 (��) q a • Discrete Lagrangian � h L d ≈ L ( q ( t ) , ˙ q ( t )) dt 0 • Discrete Euler-Lagrange equation D 2 L d ( q 0 , q 1 ) + D 1 L d ( q 1 , q 2 ) = 0

17 Comparing representations of the rotation group � Euler Angles • Local coordinate chart, exhibits singularities. • Requires change of charts to simulate large attitude maneuvers. � Unit Quaternions • Reprojection used to stay on unit 3-sphere. • The 3-sphere is a double-cover of SO (3) which causes topological problems for optimization. � Rotation Matrices • 9 dimensional space (3 × 3 matrices) with a 6 dimensional constraint (orthogonality), but the exponential map saves the day.

18 Variational Lie Group Techniques � Basic Idea • To stay on the Lie group, we parametrize the curve by the initial point g 0 , and elements of the Lie algebra ξ i , such that, �� � ξ s ˜ g d ( t ) = exp l κ,s ( t ) g 0 • This involves standard interpolatory methods on the Lie algebra that are lifted to the group using the exponential map. • Automatically stays on SO ( n ) without the need for reprojection, constraints, or local coordinates. • Order of accuracy of method is independent of the retraction, as the variational principle is at the level of the Lie group, and the retraction is simply used to locally parametrize the group.

19 Model Problem � 3D Pendulum • A rigid body, with a pivot point and a center of mass that are not collocated, under gravita- tional forces. • Three degrees of freedom. • Exhibits surprisingly rich and complex dynamics.

20 Physical Realization of a 3D Pendulum � Triaxial Attitude Control Testbed Attitude Dynamics and Control Laboratory, University of Michigan, Ann Arbor.

21 Example of a Lie Group Variational Integrator � 3D Pendulum • Lagrangian � L ( R, ω ) = 1 2 dm − V ( R ) , � � (˜ ρ ) ω � 2 Body · : R 3 → R 3 × 3 is a skew mapping such that � where � xy = x × y . • Equations of motion J ˙ ω + ω × Jω = M, T R − R T ∂V where � M = ∂V ∂R . ∂R ˙ R = R � ω.

22 Example of a Lie Group Variational Integrator � 3D Pendulum • Discrete Lagrangian L d ( R k , F k ) = 1 h tr [( I 3 × 3 − F k ) J d ] − h 2 V ( R k ) − h 2 V ( R k +1 ) . • Discrete Equations of Motion Jω k +1 = F T k Jω k + hM k +1 , � � S ( Jω k ) = 1 F k J d − J d F T , k h R k +1 = R k F k .

23 Example of a Lie Group Variational Integrator � Automatically staying on the rotation group • The magic begins with the ansatz, F k = exp( � f k ) , and the Rodrigues’ formula, which converts the equation, � � Jω k = 1 � F k J d − J d F T , k h into hJω k = sin � f k � Jf k + 1 − cos � f k � f k × Jf k . � f k � 2 � f k � • Since F k is the exponential of a skew matrix, it is a rotation matrix, and by matrix multiplication R k +1 = R k F k is a rotation matrix.

24 Numerical Simulation � Chaotic Motion of a 3D Pendulum

25 Numerical Simulations � Flyby of two dumbbells

26 Numerical Simulations � Effect of representations (Runge-Kutta) 4 4 T ro t T ro t T tran T tran 3 3 T T 2 2 1 1 0 0 0 5 10 15 0 5 10 15 4 4 T U E T U E 2 2 E E 0 0 −2 −2 −4 −4 0 5 10 15 0 5 10 15 t t Runge-Kutta with quaternions Runge-Kutta with Euler angles

27 Numerical Simulations � Effect of representations (Runge-Kutta) −3 4 3 x 10 T ro t T tran 3 2 T 2 ∆ E 1 1 0 0 −1 0 5 10 15 0 5 10 15 −3 4 6 x 10 T U E R T R � R T � I − � I − 2 R 2 � 2 SO(3) error 4 E 0 2 −2 −4 0 0 5 10 15 0 5 10 15 t t Runge-Kutta with SO(3) Runge-Kutta with SO(3) (Integration error)

28 Numerical Simulations � Lie group variational integrator on SO(3) 4 T ro t T tran 3 T 2 1 0 0 5 10 15 4 T U E 2 E 0 −2 −4 0 5 10 15 t Trajectory in inertial frame Transfer of energy