Math 692A: Geometric Numerical Integration

Transcription

1 Math 692A: Geometric Numerical Integration Melvin Leok Mathematics, Purdue University. mleok/ Mathematics, Purdue University NSF DMS-54747, and DMS

2 The Mathematics of Falling Cats 2

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

4 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 5 spinning rotors

6 Geometric Control of Spacecraft 6

7 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. 7 Small satellites in arranged in a formation to yield a large effective aperture telescope. Courtesy NASA/JPL-Caltech. Artist s conception of the ESA Darwin flotilla. Courtesy ESA. It is quite natural to think of controlling the shape of the cluster, and its orientation (group) separately.

8 Numerically implementing geometric control algorithms Demand for long-time stability in control algorithms Trend towards autonomous space and underwater vehicle missions with long deployment times and low energy propulsion systems. Small inaccuracies in the control algorithms accumulated over long times will significantly diminish the operational lifespan of such missions. Need for geometric integrators to efficiently achieve good qualitative results for simulations over long times. 8

9 Geometry and Numerical Methods Dynamical equations preserve structure Many continuous systems of interest have properties that are conserved 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 manifold, 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. 9

10 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. 1

11 11 Geometric Integration: Energy Stability Energy stability for symplectic integrators Control on global error Continuous energy Isosurface Discrete energy Isosurface

12 Geometric Integration: Energy Stability Energy behavior for conservative and dissipative systems Midpoint Newmark Energy.2.15 Benchmark Variational Explicit Newmark non-variational Runge-Kutta Energy.2.15 Variational.1.1 Runge-Kutta.5.5 Benchmark Time (a) Conservative mechanics Time (b) Dissipative mechanics

13 13 Geometric Integration: Energy Stability Solar System Simulation Forward Euler Inverse Euler Symplectic Euler q k+1 = q k + h q(q k, p k ), p k+1 = p k + hṗ(q k, p k ). q k+1 = q k + h q(q k+1, p k+1 ), p k+1 = p k + hṗ(q k+1, p k+1 ). q k+1 = q k + h q(q k, p k+1 ), p k+1 = p k + hṗ(q k, p k+1 ).

14 14 Geometric Integration: Energy Stability Forward Euler Energy error

15 15 Geometric Integration: Energy Stability Inverse Euler Energy error

16 16 Geometric Integration: Energy Stability Symplectic Euler x Energy error

17 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. 17

18 Introduction to Computational Geometric Mechanics Why adopt this approach? Provides a systematic framework to construct structure-preserving numerical schemes that are consistent with the underlying geometry of the problem. Structure-preservation can be important from the point of view of stability, accuracy, and efficiency. Provides insight into the discrete analogues of differential geometric structures that are preserved by numerical schemes. 18

19 Traditional Approach Discrete Mechanics Discretization is performed at the level of the equation, for example, finite-differences and finite-elements applied to the differential equations of motion. Alternative Approach Discretize underlying variational principles. 19

20 Continuous Mechanics Lagrangian Formulation of Mechanics Given a Lagrangian L : T Q R, the solution curve satisfies Hamilton s variational principle, δ t1 t L (q (t), q (t)) dt =. This is equivalent to the Euler-Lagrange equation, d L dt q L q =. When the Lagrangian has the form L = 1 2 qt M q V (q), this reduces to Newton s Second Law, d dt (M q) = V q. 2

21 Discrete Mechanics Discretization of Mechanics Approximate T Q by Q Q. Diagonal group action of G on Q Q The discrete Lagrangian L d : Q Q R is a generating function of the discrete flow. A particularly simple choice of discrete Lagrangian is given by, ( qk+1 + q L d (q k, q k+1 ) = h L k, q ) k+1 q k 2 h Consider the discrete action sum S : Q N+1 R defined by S = N 1 k= L d (q k, q k+1 ). which is a discrete analogue of the action integral. 21

22 Discrete Mechanics Discrete Variational Principle Hamilton s variational principle states that the solution curve has an action integral that is stationary. Analogously, the discrete variational principle states that, δ N 1 k= L d (q k, q k+1 ) =. 22

23 23 Discrete Mechanics Discrete Variational Principle q(t) varied curve Q q(a) δq(t) q(b) q i varied point Q q N q δq i

24 24 Discrete Mechanics Discrete Euler-Lagrange equation To extremize S over q,, q N, we set This implies, S q k =, k =,..., N. D 1 L d (q k, q k+1 ) δq k +D 2 L d (q k 1, q k ) δq k =, k = 1,..., N 1. Since δq = δq N = and the rest of the variations are arbitrary, we obtain the discrete Euler-Lagrange equation, D 1 L d Φ + D 2 L d =.

25 Correspondance between discrete and continuous mechanics Consider the exact discrete Lagrangian given by, L d (q k, q k+1 ) = tk+1 t k L (q (t), q (t)) dt where q : [t k, t k+1 ] Q is a solution of the Euler-Lagrange equations for L which satisfies the boundary conditions q (t k ) = q k and q (t k+1 ) = q k+1. This choice is motived by the Jacobi solution of Hamilton- Jacobi Theory, where the action integral is a generating function of the Hamiltonian flow. 25

26 Correspondance between discrete and continuous mechanics. Can show that with the exact discrete Lagrangian, the discrete curve {q k } n k= satisfying the Discrete Euler-Lagange equations discretely sample the continuous solution curve q(t) of the Euler- Lagrange equations for the corresponding Lagrangian. q k = q (t k ) In practice, the exact discrete Lagrangian, like Jacobi s solution, cannot be computed exactly. As such, numerical quadrature methods are used to approximate the exact discrete Lagrangian. Can be shown that a discrete Lagrangian given by an n-th order accurate quadrature method yields an n-th order accurate symplectic integrator. This recovers a class of previously discovered Symplectic Partitioned Runger-Kutta methods. 26

27 Canonical Symplectic Structure Hamilton s Canonical Equations q i = H p i ṗ i = H q i If we adopt the coordinate system, z = (q, p), the Canonical Symplectic Form, Ω = dp i dq i has the matrix representation given by: ( ) 1 J = 1 Hamilton s equations, ż = X H = J H can be written as: Ω (X H (z), v) = dh.z 27

28 28 Fiber Derivative Discrete Symplectic Structure FL d : Q Q T Q (q, q 1 ) (q, D 1 L d (q, q 1 )) Discrete Symplectic Form ω L = (FL d ) (Ω L ) = d ((FL ( d ) p i dq i)) ( ) = d L d q i k = 2 L d q i k qj k+1 (q k, q k+1 ) dq i k (q k, q k+1 ) dq i k dqj k+1

29 Momentum Maps Continuous Momentum Map A momentum map J : T Q g is a generalization of the familar conjugate momentum in Hamiltonian mechanics. Given a mechanical system that is invariant under the action of the Lie group G on the configuration manifold Q, we define a momentum map by, J ( αq ), ξ αq, ξ Q (q) 29 Discrete Momentum Map Discrete Momentum Map J d : Q Q g, J d (q k, q k+1 ), ξ D 1 L d (q k, q k+1 ), ξ Q (q k )

30 3 Momentum Maps Example: Angular Momentum Let SO(3) act on R 3 by matrix multiplication. A q = Aq The infinitesimal generator is given by ˆω R 3(q) = ˆωq = ω q, where ω R 3. The momentum map J : T R 3 so(3) = R 3 is given by, that is, J(q, p), ω = p ˆωq = ω (q p), J(q, p) = q p, which is the familiar expression for angular momentum.

31 31 Discrete Mechanics What does this way of thinking buy you? Systematic method of constructing symplectic-momentum integrators of arbitrarily high-order. In contrast to non-constructive condition on the coefficients of the partitioned Runge Kutta scheme: b i ã ij + b j a ji = b i bj, i, j = 1,..., s, b i = b i, i = 1,... s. Naturally leads to generalized schemes such as Lie group, multiscale, and pseudospectral variational integrators.

32 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. 32

33 Basic Idea Variational Lie Group Techniques To stay on the Lie group, we parametrize the curve by the initial point g, and elements of the Lie algebra ξ i, such that, ( ) g d (t) = exp ξ s lκ,s (t) 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. g 33

34 34 Model Problem 3D Pendulum A rigid body, with a pivot point and a center of mass that are not collocated, under gravitational forces. Three degrees of freedom. Exhibits surprisingly rich and complex dynamics.

35 35 Physical Realization of a 3D Pendulum Triaxial Attitude Control Testbed Attitude Dynamics and Control Laboratory, University of Michigan, Ann Arbor.

36 36 Example of a Lie Group Variational Integrator 3D Pendulum Lagrangian L(R, ω) = 1 2 Body ( ρ)ω 2 dm V (R), where : R 3 R 3 3 is a skew mapping such that xy = x y. Equations of motion where M = V T R R R T V R. J ω + ω Jω = M, Ṙ = R ω.

37 37 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 = Fk T Jω k + hm k+1, S(Jω k ) = 1 ( ) F h k J d J d Fk T R k+1 = R k F k.,

38 38 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, Ĵω k = 1 ( ) F h k J d J d Fk T, into hjω k = sin f k f k Jf k + 1 cos f k f k 2 f k Jf 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.

39 39 Numerical Simulation Chaotic Motion of a 3D Pendulum

40 4 Flyby of two dumbbells Numerical Simulations

41 41 Numerical Simulations Effect of representations (Runge-Kutta) 4 3 T ro t T tran 4 3 T ro t T tran T 2 T T U E 4 2 T U E E E t Runge-Kutta with quaternions t Runge-Kutta with Euler angles

42 42 Numerical Simulations Effect of representations (Runge-Kutta) 4 3 T 2 T ro t T tran ΔE 3 x E 2 T U E SO(3) error 6 x I R T R I R T 2 R t Runge-Kutta with SO(3) t Runge-Kutta with SO(3) (Integration error)

43 43 Numerical Simulations Lie group variational integrator on SO(3) 4 3 T ro t T tran T T U E E 2 Trajectory in inertial frame t Transfer of energy

44 44 Numerical Simulations Conservation Properties: Lie Group Integrator ΔĒ 6 14 x t deviation in total energy norm(i R T R) norm(i R 2 T R2 ) x t error in the rotation matrix

45 45 Numerical Simulations Conservation Properties: Lie Group Integrator 5 x Δ γ T 5 x x t deviation in total linear momentum 1 1 x Δ π T 5 x x t deviation in total angular momentum

46 Numerical Simulations Comparison with other methods Our Lie group variational integrator (LGVI) is a Lie Störmer Verlet method, so it is a second-order symplectic Lie group method. We compare it to other second-order accurate methods: Explicit Midpoint Rule (RK): Preserves neither symplectic nor Lie group properties. Implicit Midpoint Rule (SRK): Symplectic but does not preserve Lie group properties. Crouch-Grossman (LGM): Lie group method but not symplectic. 46

47 47 Numerical Simulations Comparison with other methods E.159 RK SRK LGM LGVI mean Δ E RK SRK LGM LGVI time Computed total energy for 3 seconds Step size Mean total energy error E E vs. step size 1 5 mean I R T R CPU time (sec) Step size Mean orthogonality error I R T R vs. step size Step size CPU time vs. step size

48 Applications to Asteroid Simulations Computational Considerations Asteroids are approximated by rubble piles (hard sphere models) or simplicial complexes. Force evaluations using fast multipole methods or polyhedral potential techniques. Computational cost is dominated by cost of force evaluation. Lie Störmer Verlet variational integrators use only one force evaluation per timestep, and are only implicit in the attitude, and converge in 2-3 Newton steps. Used by Scheeres, et al. to accurately simulate binary near-earth Asteroid (1999 KW4). 48

49 Applications to Asteroid Simulations 49

50 Numerically implementing geometric control algorithms Traditional approach Local analysis of the connection near the desired shape position. Gives a closed form expression for the geometric phase associated with infinitesimally small loops in shape space. Resulting shape trajectories are often suboptimal and slow. Alternative approach Homotopy-based optimal control algorithm using geometrically exact numerical schemes. Allows for large-amplitude trajectories that are global in nature, and more efficient than infinitesimal loops. 5

51 51 Essential Ideas Discrete Optimal Control Use the discrete Lagrange d Alembert principle, δ L d (q k, q k+1 ) + F d (q k, q k+1 ) (δq k, δq k+1 ) =, to derive the discrete forced Euler Lagrange equations, and impose these as constraints at every time-step. This yields greater fidelity to the equations of motion that imposing the dynamical constraints using the method of collocation. The resulting numerical solutions are group equivariant, which implies that the numerical solutions are independent of the choice of coordinate frame.

52 Shooting based optimization using variational integrators General approach Relax terminal boundary conditions, and guess initial control torque. Evolve attitude and control torques using equations of motion and optimality conditions. Compute sensitivity of terminal boundary conditions on initial control torques, and iterate to convergence. Robust computation of sensitivities Structure preserving properties of the variational integrator allow the robust computation of sensitivities. Problems considered do not appear to require multiple shooting to achieve convergence. 52

53 53 Under-actuated control of a 3D pendulum 2 u (Nm) Attitude Maneuver t (sec) Control input u Ω (rad/sec) Error t (sec) Angular velocity Ω Iteration Convergence rate

54 Under-actuated control of a 3D pendulum 54

55 Under-actuated control with symmetry of a 3D pendulum

56 Under-actuated control with symmetry of a 3D pendulum

57 Constrained variations Variational Integrators on S 2 The condition that q S 2 = {q R 3 q q = 1} yields a constrained variation of q in the variational principle. δq = ξ q, where ξ R 3 is constrained to be orthogonal to q, i.e., ξ q =. Lagrangian The configuration space is a cartesian product of two-spheres, Q = (S 2 ) n, and the Lagrangian has the form, L(q 1,..., q n, q 1,..., q n ) = 1 n q T 2 j M ij q i V (q 1,..., q n ), i,j=1 57

58 Variational Integrators on S 2 Spherical Pendulum [ ( r k+1 = hω k + h2 g ) 2l r k e 3 r k + r k 1 hω k + h2 g 2l r 2 k r 3 ω k+1 =ω k + hg 2l r k e 3 + hg 2l r k+1 e 3 Double Spherical Pendulum 2 1+f 1 f 1 I 3 3 2α 2β 2 1+f 1 f 1 ˆr 2ˆr 1 1+f 2 f 2 I 3 3 ] [f1 ] 1+f 2 f 2 ˆr 1ˆr 2 = where α = m 2 m 1 +m 2 l 2 l1 and β = l 1 l2. f 2 [ ] hω1k hα(r 1k (r 2k ω 2k )) + h2 g 2l 1 (r 1k e 3 ) + 2αf 2 f 2 1+f 2 f 2 ˆr 1 r 2 hω 2k hβ(r 2k (r 1k ω 1k )) + h2 g 2l 2 (r 2k e 3 ) + 2βf 1 f 1 1+f 1 f 1 ˆr 2 r 1 r 1k+1 = (I ˆf 1 )(I 3 3 ˆf 1 ) 1 r 1k r 2k+1 = (I ˆf 2 )(I 3 3 ˆf 2 ) 1 r 2k Implicit system of equations can be solved using fixed-point iteration, and requires about 5-6 iterations to reach machine precision. 58

59 59 Variational Integrators for S 2 Double Spherical Pendulum Simulation

60 6 Elastic Rod Variational Integrators for S 2

61 61 Variational Integrators for S 2 Coupled Spherical Pendula

62 62 Variational Integrators for S 2 3-body problem on the sphere

63 63 Magnetic Arrays Variational Integrators for S 2

64 Towards Larger-Scale Scientific Computing Problems Hierarchical Multi-Resolution Lie Group VIs General framework motivated by multiscale multiphysics methods. Based on the use of empirical shape functions obtained from localized simulations. Decompose a strand of DNA into its constituent nucleotides, and simulate the free nucleotide dynamics to obtain shape functions. Incorporate empirical shape functions into larger scale simulation using rigid body deformations of lower level solutions. Rigid-body motions form a closed category under recursion. Allows for computational compression by reusing redundant computations. 64

65 65 Towards Larger-Scale Scientific Computing Problems Hierarchical Multi-Resolution Lie Group VIs Schematic of hierarchical method Strand of DNA composed of nucleotides