Geometric Mechanics and Global Nonlinear Control for Multi-Body Dynamics

Transcription

1 Geometric Mechanics and Global Nonlinear Control for Multi-Body Dynamics Harris McClamroch Aerospace Engineering, University of Michigan Joint work with Taeyoung Lee (George Washington University) Melvin Leok (University of California, San Diego)

2 2 Anthony Bloch Tony Bloch at UM in 2010

3 3 Anthony Bloch U of Michigan: , 1994-present Respect for his scholarly and personal qualities Respect his commitment to research collaboration with faculty and students Joint research publications on Hamiltonian systems with holonomic and nonholonomic constraints His influence on my subsequent research, including topics mentioned subsequently, has been profound Thank you, Tony, for our long and valued friendship

4 4 Multi-body control system? Multi-body systems are connections of rigid and deformable bodies Lagrangian and/or Hamiltonian formulations Inclusion of control features Primary motivation for classical physics Historical contributions from most important names in mathematics and science Modeling, analysis and computational challenges remain Importance in aerospace, robotics, biomechanics, etc. Technical challenges: complexity and singularities arise from the most common use of local coordinates

5 5 Themes of this presentation Multi-body systems defined by: Lumped masses elements Rigid bodies Interconnection and interaction structures Use configuration manifold to encode constraints from the interconnection structure Number of configuration variables is not minimal No singularities or ambiguities in definition of the configuration manifold Avoid use of Lagrange multipliers

6 New Lagrangian and Hamiltonian formulations for multi-body dynamics on a configuration manifold Key ingredients in the formulation Configuration manifold Hyper-regular Lagrangian function defined on the tangent bundle of the configuration manifold Relative simplicity of equations due to their geometric structure Globally valid descriptions for control analysis and computations A comment on prior work that follows this perspective 6

7 Background This presentation Research publications of Taeyoung Lee, Melvin Leok and Harris McClamroch: 2007 to present Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds, by LeLeMc, Springer, to appear in 2016 Inspired by differential geometric concepts mechanics by Tony Bloch, Jerry Marsden and other participants in this workshop Cover the main formulations of Lagrangian and Hamiltonian dynamics on manifolds for multi-body systems Illustrate the main themes with a few examples Comment on control analysis and computational tasks at the end 7

8 8 Framework for multi-body dynamics M is a differentiable manifold, embedded in a finite dimensional vector space M may have a product structure: M = M 1 M n M encodes holonomic constraints Linear manifolds Spherical manifolds, such as S 1, S 2 Lie group manifolds, such as SO(3) and SE(3) Hyper-regular Lagrangian function L : T M R 1 is L(q, q) = T ( q, q) U(q)

9 Action integral Multi-body Lagrangian dynamics Variations δq T q M at q M can be expressed in terms of orthogonal projections P (q) : R n T q M for q M Hamilton s variational principle Assume (q, q) : [t 0, t f ] T M. The variation of the action integral is zero iff the Lagrangian variational condition holds: { ( ) } P T d L(q, q) L(q, q) (q) = 0 dt q i q Equivalent to Arnold s theorem in Mathematical Methods of Classical Mechanics Defines a smooth Lagrangian vector field on T M 9

10 10 Multi-body Hamiltonian dynamics Legendre transformation (q, q) T M (q, µ) T M and Hamiltonian function H : T M R 1 are: µ = P T L(q, q) (q) q H(q, µ) = µ q L(q, q) Assume (q, µ) : [t 0, t f ] T M. The variation of the action integral is zero iff the Hamiltonian variational condition holds:

11 11 H(q, µ) q = P (q) µ µ = P T H(q, µ) (q) + q P T (q) + ( P T (q)µ x ) ( ) P T T (q)µ P (q) q ( P (q) P T (q) Defines a smooth Hamiltonian covector field on T M ) P T T (q)µ q H(q, µ) µ

12 Comments The challenge: find the orthogonal projection operator Solvable for configuration manifolds that have a product structure in terms of the previously mentioned example manifolds It is often easier to carry out the variational analysis by direct analysis Determine expressions for the infinitesimal variation of the configuration Use special group or geometric features of the configuration manifold, as in the previously mentioned example manifolds Follow standard variational procedures to obtain Euler Lagrange equations and Hamilton s equations 12

13 Lagrangians for multi-body dynamics Special form of the configuration manifold M = M 1 M 2 M n where M i is a differentiable manifold embedded in R 3, i = 1,..., n Special form of the Lagrangian function L(q, q) = 1 n n m 2 ij q i T q j U(q) i=1 j=1 q i M i, that is q = (q 1,..., q n ) M m ij R 1, i = 1,..., n, j = 1,..., n scalar m ij = m ji and n n matrix [m ij ] is positive definite U : R 3n R 1 is differentiable potential function 13

14 14 Multi-body dynamics on linear manifolds Configuration submanifolds M i = { q R 3 } : A i q i = b i ) Orthogonal projection operator P i = (I 3 3 A T i (A ia T i ) 1 A i Euler Lagrange equations on T M m ii q i + ( ) m ij I 3 3 A T i (A ia T i ) 1 A i q j j i ( ) + I 3 3 A T i (A ia T U(q) i ) 1 A i = 0, i = 1,..., n q i Defines a smooth Lagrangian vector field on T M Naturally extended to smooth Lagrangian vector field on T R 3n

15 Legendre transformation ) µ i = m ii q i + (I 3 3 A T i (A ia T i ) 1 A i m ij q j, i = 1,..., n. Invertibility Hamilton s equations on T M n q i = m I ij µ j, i = 1,..., n j=1 j i ( ) µ i = I 3 3 A T i (A ia T U(q) i ) 1 A i, i = 1,..., n q i Defines a smooth Hamiltonian covector field on T M Naturally extended to smooth Hamiltonian covector field on T R 3n 15

16 Examples: Reduces to classical case if M i = R 3, i = 1,..., n n mass particles moving on straight lines and/or flat surfaces in R 3 under a mutual potential 16

17 17 Multi-body dynamics on (S 1 ) n { } Configuration submanifolds M i = q R 2 : q i 2 = 1 ( ) Orthogonal projection operator P i (q i ) = I 2 2 q i qi T Euler Lagrange equations on T M ( ) m ii q i + I 2 2 q i qi T m ij q j + m ii q i 2 q i + j i ( I 2 2 q i q T i ) U(q) q i Defines a smooth Lagrangian vector field on T M = 0, i = 1,..., n Naturally extended to smooth Lagrangian vector field on T R 2n

18 Legendre transformation ( µ i = m ii q i + I 2 2 q i qi T Invertibility Hamilton s equations on T M n q i = m I ij µ j, i = 1,..., n j=1 j=1 ) m ij q j, i = 1,..., n n µ i =(µ i qi T q i µ T i ) ( ) m I ij µ j I 2 2 q i qi T U(x) ), i = 1,..., n x i j i Defines a smooth Hamiltonian covector field on T M Naturally extended to smooth Hamiltonian covector field on T R 2n 18

19 Examples based on lumped mass approximations: Planar pendulum Double planar pendulum Mass particles on a torus Planar geometrically exact rod (approximated with finite elements) Planar tether (approximated with finite elements) Planar robot manipulator Planar robot locomotion, e.g. SLIP 19

20 20 Multi-body dynamics on (S 2 ) n { } Configuration submanifolds M i = q R 3 : q i 2 = 1 ( ) Orthogonal projection operator P i (q i ) = I 3 3 q i qi T Euler Lagrange equations on T M ( ) m ii q i + I 3 3 q i qi T m ij q j + m ii q i 2 q i + j i ( I 3 3 q i q T i ) U(q) q i Defines a smooth Lagrangian vector field on T M = 0, i = 1,..., n Naturally extended to smooth Lagrangian vector field on T R 3n

21 Legendre transformation ( µ i = m ii q i + I 3 3 q i qi T Invertibility assumption Hamilton s equations on T M n q i = m I ij µ j, i = 1,..., n µ i =( j=1 j=1 ) m ij q j, i = 1,..., n j i n ( ) m I ij µ j) (µ i q i ) I 3 3 q i qi T U(x) ), i = 1,..., n x i Defines a smooth Hamiltonian covector field on T M Naturally extended to smooth Hamiltonian covector field on T R 3n 21

22 Examples based on lumped mass approximations: Spherical pendulum Double spherical pendulum 3D geometrically exact rod (approximated with finite elements) 3D tether (approximated with finite elements) 3D robot manipulator 3D robot locomotion 22

23 Other lumped mass multi-body examples we have studied Slider and crank mechanism: M = R 1 S 1 Planar pendulum on a cart: M = R 1 S 1 Double planar pendulum on a cart: M = R 1 (S 1 ) 2 Spherical pendulum on a cart: M = R 2 S 2 Double spherical pendulum on a cart: M = R 2 (S 2 ) 2 3D geometrically exact rod in space: M = R 3 (S 2 ) n 3D tether in space: M = R 3 (S 2 ) n 23

24 24 Full body dynamics Two (irregular) rigid bodies acting under mutual Newtonian gravity Important coupling between attitude and orbit dynamics Configuration manifold M = (SE(3)) 2 Twelve degrees of freedom Rigid body kinematics Ṙ i = R i S(ω i ), i = 1, 2 Notation: R = (R 1, R 2 ), x = (x 1, x 2 ), ω = (ω 1, ω 2 )

25 25 Lagrangian function L(R, x, ω, ẋ) = 1 2 m 1 ẋ ωt 1 J 1ω m 2 ẋ ωt 2 J 2ω 2 U(R, x) Newtonian gravitational potential U(R, x) = 1 2 Gdm j (ρ j )dm i (ρ i ) 2 B j xi + R i ρ i x j R j ρ j Variational analysis i=1 j i B i δr i = R i S(η i ), i = 1, 2 ω i = η i + S(ω i )η i, i = 1, 2

26 Procedure: Substitute into Hamilton s principle and simplify Integrate by parts and simplify Use fundamental lemma of calculus of variations Use matrix identities to obtain Euler Lagrange equations Euler Lagrange equations on T (SE(3)) 2 : Ṙ i = R i S(ω i ), i = 1, 2 J i ω i +ω i J i ω i M i = 0, i = 1, 2 m i ẍ i F i = 0, i = 1, 2 Gravitational moments and forces: GS(ρ i M i = )RT i (x i + R i ρ i x j R j ρ j )dm j (ρ j )dm i (ρ i ) B j i xi + R i ρ i x j R j ρ j 3 B i 26

27 F i = B i G(x i + R i ρ i x j R j ρ j )dm j (ρ j )dm i (ρ i ) B j i x i + R i ρ i x j R j ρ j 3 27 Defines a smooth Lagrangian vector field on T (SE(3)) 2 Extension to smooth Lagrangian vector field on T (R 3 3 R 3 ) 2 Legendre transformation L(R, x, ω, ẋ) Π i = = J ω i ω i, i = 1, 2 i L(R, x, ω, ẋ) p i = = m ẋ i ω i, i = 1, 2 i

28 28 Hamilton s equations on T (SE(3)) 2 : Ṙ i = R i S(J 1 i Π i ), i = 1, 2 ẋ i = p i m i, i = 1, 2 Π i = S(J 1 i Π i )Π i + M i, i = 1, 2 ṗ i = F i, i = 1, 2 Defines a smooth Hamiltonian covector field on T (SE(3)) 2 Extension to smooth Hamiltonian covector field on T (R 3 3 R 3 ) 2

29 29 Other multi-body examples we have studied 3D pendulum: M = SO(3) Full body dynamics of rigid bodies in space: M = (SE(3)) n Quad rotor flight vehicle carrying a load: M = SE(3) S 2 Connected rigid bodies in space: M = SE(3) SO(3) Rigid body in space with internal ball in slot: M = SE(3) R 1 Rigid body in space with attached elastic rod: M = SE(3) (S 2 ) n Rigid body spacecraft with attached tether: M = SE(3) (S 2 ) n

30 30 Geometric mechanics: conclusions Incorporate constraints into configuration manifold Importance of of geometry of configuration manifold Geometric formulations of Lagrangian and Hamiltonian dynamics for multi-body systems Globally valid Dynamics on T M or T M Dynamics can be naturally extended to T R n or T R n Formulations are relatively concise reflecting geometric structure

31 31 Comments on nonlinear control Make use of the proposed geometric formulations of Lagrangian and Hamiltonian dynamics on a manifold for control purposes Global control analysis: differential equations on a manifold Equilibrium solutions Linearization of flow in neighborhood of equilibrium Liapunov analysis Periodic solutions Analysis of complex dynamics Stabilization and tracking control problems Local and nonlocal effects of feedback

32 Control of complex global dynamics Impediments to global smooth stabilization Optimization and optimal control References: Chaturvedi, McClamroch, Bernstein, Asymptotic smooth stabilization of the inverted 3D pendulum, 2009 Lee, Leok, McClamroch, Optimal Attitude Control of a Rigid Body using Geometrically Exact Computations on SO(3),

33 33 Comments on dynamics computations Proposed dynamics formulations are suitable for computations Computations of dynamics on the extended vector field Computations of dynamics on the embedded manifold Geometric integrators Tailored to the geometry of the configuration manifold, e.g. if it has group related properties Implicit equations define geometric integrators Preservation of flow on the embedded manifold References: Hairer, Lubich, Wanner, Geometric Numerical Integration, 2000; Iserles, Munthe-Kaas, Norsett, Zanna, Lie-group methods, 2000

34 Variational integrators Convert to discrete time variational problem: discretize action integral as action sum Use standard variational methods to obtain discrete time Euler Lagrange and Hamilton s equations Variational integrators are implicit Preservation of flow properties good local numerical stability determined by action sum approximation good long term energy conservation preservation of conserved quantities Reference: Marsden and West, Discrete mechanics and variational integrators,

35 Combine features of geometric and variational integrators Lie group variational integrators Can obtain good features of both geometric integrators and variational integrators Computational studies of many of the multi-body examples previously mentioned verify the excellent numerical properties obtained by combining geometric and variational integrators References: Lee, Leok, McClamroch: Lie group variational integrators for the full body problem, 2007 Lagrangian mechanics and variational integrators on two-spheres,