Review of Lagrangian Mechanics and Reduction

Transcription

1 Review of Lagrangian Mechanics and Reduction Joel W. Burdick and Patricio Vela California Institute of Technology Mechanical Engineering, BioEngineering Pasadena, CA 91125, USA Verona Short Course, August 25-29, 2002 p.1/39

2 Variational Principles Q: How do we obtain the dynamics of a locomotive system? Verona Short Course, August 25-29, 2002 p.2/39

3 Variational Principles Q: How do we obtain the dynamics of a locomotive system? A: Use variational principles. Generic construction. Allow for constraints. Equations of motion or optimal control. Verona Short Course, August 25-29, 2002 p.2/39

4 Variational Principles Q: How do we obtain the dynamics of a locomotive system? A: Use variational principles. Generic construction. Allow for constraints. Equations of motion or optimal control. For example, the Euler-Lagrange equations, d dt L q i L q i = 0, come from a variational principle. Verona Short Course, August 25-29, 2002 p.2/39

5 The Space of Variations Define the path space, Ω(q 1, q 2, [a, b]) { q : [a, b] Q q is a C 2 curve, q(a) = q 1, q(b) = q 2 }. δq(t) q 2 q 1 q(t) Define the function C : Ω(q 1, q 2, [a, b]) R, C(c) b a L(q(t), q(t)) dt where L : T Q R. Verona Short Course, August 25-29, 2002 p.3/39

6 Hamilton s Variational Principle Theorem 1 Let L be a Lagrangian on T Q. A curve q 0 Ω(q 1, q 2, [a, b]) satisfies the Euler-Lagrange equations d dt L q L q = 0, if and only if q 0 is a critical point of the function C. Critical points satisfy: δc = δ b a L(q(t), q(t)) dt = 0, where δq(t) T q(t) Q, a t b. Verona Short Course, August 25-29, 2002 p.4/39

7 Hamilton s Variational Principle Theorem 1 Let L be a Lagrangian on T Q. A curve q 0 Ω(q 1, q 2, [a, b]) satisfies the Euler-Lagrange equations d dt L q L q = F (q, q, t), if and only if q 0 is a critical point of the function C. Critical points satisfy: δc = δ b a L(c(t), ċ(t)) dt = b a F (q, q, t)δq dt, where δq(t) T q(t) Q, a t b. Verona Short Course, August 25-29, 2002 p.4/39

8 What do we mean by δ? Let c(s, t) denote a parametrized family of curves. Let c(s, t) pass through c(t) at s 0. The infinitesimal variation of c at c 0 is Hence, u = dc(s, t) ds δc(c(t)) = d ds b a b s=s0 a q 1 L(c(t), ċ(t)) dt = 0 L(c(s, t), ċ(s, t)) dt s=0 = 0. c(t) u(t) q 2 Verona Short Course, August 25-29, 2002 p.5/39

9 Actions of Lie Groups 2 Definition 0 Let Q be a manifold and G a Lie group. A left action of G on M is a smooth mapping Φ : G Q Q with: 1. Φ(e, x) = x, x Q, and 2. Φ(g, Φ(h, x)) = Φ(gh, x), g, h G. Definition 0 The lifted action is the map T Φ g : T Q T Q (q, v) (Φ g (q), T q Φ g (v)) for all g G and q Q. Verona Short Course, August 25-29, 2002 p.6/39

10 Coordinate form of lifted Action If Φ g is left translation on q = (g, r), the lifted action has the coordinate expression: ( ) ( ) T g L T q Φ h q = h ġ hġ = ṙ ṙ Verona Short Course, August 25-29, 2002 p.7/39

11 Snakeboard Example Q = SE(2) (S 1 S 1 S 1 ); front wheels ψ back wheels l q = (x, y, θ, φ f, φ b, ψ) (x,y) φ b θ φ f Let g = (a 1, a 2, α). The (left) action and lifted action are: Φ g (q) = x cos α y sin α + a 1 x sin α + y cos α + a 2 θ + α φ f φ b ψ T q Φ g v q = v x cos α v y sin α v x sin α + v y cos α v θ v φf v φb v ψ Verona Short Course, August 25-29, 2002 p.8/39

12 Reducing the Lagrangian Definition 0 The Lagrangian is G-invariant if it is invariant under the induced action of G on T Q. L(Φ g (q), T q Φ g q) = L(q, q) Theorem 1 If the Lagrangian is G-invariant then it induces a well-defined function called the reduced Lagrangian l : T Q/G R Verona Short Course, August 25-29, 2002 p.9/39

13 The Abelian Case Let local coordinates be q = (s, r). If G is Abelian: Φ h (q) = (s + h, r) Invariance of the Lagrangian implies: L(s + g, r, ṡ, ṙ) = L(s, r, ṡ, ṙ) L(q, q) = L(r, ṡ, ṙ) The Euler-Lagrange Equations are: ( ) d L dt ṡ ( ) d L L dt ṙ r = 0 = 0 Verona Short Course, August 25-29, 2002 p.10/39

14 Remarks s is said to be a "cyclic" variable. There is a constant of the motion L ṡ i.e., a conserved quantity that can be interpreted as a "momentum". Equations of motion evolve on the reduced space T Q/S (S is the set of cyclic variables). The reduced Lagrangian is simply: L(r, ṡ, ṙ) We must generalize these ideas! Verona Short Course, August 25-29, 2002 p.11/39

15 Lie Group Reduction Group invariances led to reduction for vector fields. We d like accomplish reduction more generally. T G X T L g T G X G Lg G Verona Short Course, August 25-29, 2002 p.12/39

16 Lie Group Reduction Group invariances led to reduction for vector fields. We d like accomplish reduction more generally. X g T e L g X L (G) X e Lg G Verona Short Course, August 25-29, 2002 p.12/39

17 Lie Group Reduction Group invariances led to reduction for vector fields. We d like accomplish reduction more generally. L g 1 G X X L (G) T L g 1 e Xξ g Verona Short Course, August 25-29, 2002 p.12/39

18 Lie Group Reduction Group invariances led to reduction for vector fields. We d like accomplish reduction more generally. For vector fields: In general: L g 1 G X X L (G) e Xξ T L g 1 g Q π G F Q/G f F (Q) π G (F (Q)) f(q/g), f(π G (F (Q)) where π G (W ) : W W/G. Verona Short Course, August 25-29, 2002 p.12/39

19 Reducing Spaces If the action of G on Q is free and proper, then Q may be reduced. For principal bundle, Q/G = M. Recall T G/G = g. Action on G induces an action on T Q. π G (Q) Q T T Q π G (T Q) Q/G? T Q/G Verona Short Course, August 25-29, 2002 p.13/39

20 Reducing Spaces If the action of G on Q is free and proper, then Q may be reduced. For principal bundle, Q/G = M. Recall T G/G = g. Action on G induces an action on T Q. π G (Q) Q T D π G (D) Q/G Aloc DQ/G Verona Short Course, August 25-29, 2002 p.13/39

21 Infinitesimal Generators The action of G on Q induces a vector field on Q. The Lie algebra exponential exp defines a curve on Q, Φ(exp(tˆξ), q) which after time differentiation, gives the infinitesimal generator. ξ Q (q) d dt Φ(exp(tˆξ), q) t=0 Verona Short Course, August 25-29, 2002 p.14/39

22 Infinitesimal Generator in Coordinates Let local coordinates be q = (g, r) ξ Q = d ( ) dɛ ɛ=0 Φ(exp(ɛξ), (g, r)) = d exp(ɛξ)g dɛ r ( ) ( ) ( ) = d R g exp(ɛξ) T e R g ξ ξg = = dɛ r 0 0 ɛ=0 ɛ=0 Snakeboard: Let ξ = (b 1, b 2, α). Then ξ Q (x, y, θ, φ f, φ b, ψ) = (b 1 y α, b 2 + x α, α, 0, 0, 0) Verona Short Course, August 25-29, 2002 p.15/39

23 The Adjoints Differentiation of the inner automorphism leads to the adjoint operator: Ad g : g g, Ad g η T e I g η Differentiation of the adjoint operator (with respect to g) leads to the Lie bracket, sometimes denoted by ad, ad ξ η T e (Adη) ξ = [ξ, η] Transformation of observer. Used for body/spatial transformations. Verona Short Course, August 25-29, 2002 p.16/39

24 The Momentum Map Definition 0 Given a Lagrangian L : T Q R, the momentum map is the mapping, J : T Q g, J(v q ), ξ = FL(v q ), ξ Q (q) where FL(v q ) = L q (v q). For a G-invariant Lagrangian, the momentum map is Ad-equivariant, J T q Φ g (q, q) = Ad g 1 J(q, q) Related to body/spatial references. Verona Short Course, August 25-29, 2002 p.17/39

25 Noether s Theorem Theorem 1 Let L : T Q R be a G-invariant Lagrangian. Then, for a solution of the Euler-Lagrange equations, the quantity J is a constant in time. Proof based on Hamilton s variational principle. Introduces invariance in the variations. Combination implies conservation law. G M Q Verona Short Course, August 25-29, 2002 p.18/39

26 Euler-Poincaré Reduction Theorem 2 Let l : g R be the reduced Lagrangian of a G-invariant Lagrangian L : T G R. A curve, ξ 0 (t) Ω(ξ 1, ξ 2, [a, b]) satisfies the Euler-Poincaré equations, d dt l ξ ad ξ l ξ = 0 if and only if ξ 0 (t) is a critical point of the function c. Critical points satisfy, δc = δ b a l(ξ(t)) dt = 0 with variations of the form δξ = η + [ξ, η] where η vanishes at the endpoints. Verona Short Course, August 25-29, 2002 p.19/39

27 Euler-Poincaré Reduction Theorem 2 Let l : g R be the reduced Lagrangian of a G-invariant Lagrangian L : T G R. A curve, ξ 0 (t) Ω(ξ 1, ξ 2, [a, b]) satisfies the Euler-Poincaré equations, d dt l ξ ad ξ l ξ = F(ξ) if and only if ξ 0 (t) is a critical point of the function c. Critical points satisfy, δc = δ b a l(ξ(t)) dt = b a F(ξ(t))δξ dt with variations of the form δξ = η + [ξ, η] where η vanishes at the endpoints. Verona Short Course, August 25-29, 2002 p.19/39

28 Equivalence The Euler-Lagrange equations of motion, are equivalent to: d dt L q L q = 0, The Euler-Poincaré equations of motion d dt l ξ ad ξ with the reconstruction equation, ġ = gξ(t). l ξ = 0, Verona Short Course, August 25-29, 2002 p.20/39

29 Momentum Form of the Equations Interpret ξ l as a "body momentum", p. The Euler-Poincaré equation and reconstruction equations become: ṗ ad ξp = 0 g 1 ġ = ξ = I 1 p where the locked inertia tensor, I : g g, is defined by, Iξ, η FL(ξ Q ), η Q Note: the "spatial" momentum is constant. µ = (Ad g) 1 p Verona Short Course, August 25-29, 2002 p.21/39

30 Systems with Constraints There are many ways to describe constraints, If they just involve the configuration states: ϕ(q) = 0 If they involve the velocities: ω(q, q) = 0 or as a distribution: D = { (q, v) T Q v is an admissable velocity at q Q } Verona Short Course, August 25-29, 2002 p.22/39

31 Systems with Constraints There are many ways to describe constraints, If they just involve the configuration states: ϕ(q) = 0 If they involve the velocities: φ q q = 0 ω(q, q) = 0 or as a distribution: D = { (q, v) T Q v is an admissable velocity at q Q } Verona Short Course, August 25-29, 2002 p.22/39

32 Systems with Constraints There are many ways to describe constraints, If they just involve the configuration states: ϕ(q) = 0 If they involve the velocities: ω(q, q) = 0 or as a distribution: { ω 1,..., ω m } T Q D = { (q, v) T Q v is an admissable velocity at q Q } Verona Short Course, August 25-29, 2002 p.22/39

33 Lagrange Multiplier Theorem Suppose that N M is a submanifold. Theorem 2 (Lagrange Multiplier Theorem) The following are equivalent for x 0 N and h : M R smooth: 1. x 0 is a critical point of h N ; and 2. there exists λ 0 (x) such that λ 0 (x 0 ) is a critical point of the function h : E R defined by h(λ(x)) = h(x) λ(x), ϕ(x) Q Verona Short Course, August 25-29, 2002 p.23/39

34 Lagrange Multipliers Consider the vector bundle E over M, π : E M. The fiber is a vector space V. Constraints are a section of the vector bundle transverse to the fibers, ω E. The equations of motion with Lagrange Multipliers are: d dt L q L q + λω(q, q) = 0, Lagrange multipliers are a section of the covector bundle, λ E. Finding the λ can be challenging. Can it be avoided? Verona Short Course, August 25-29, 2002 p.24/39

35 The Constrained Lagrangian For holonomic systems N Q is a submanifold, and by extension T N T Q. Therefore, δ b a is equivalent to (L(q(t), q(t)) λ(q(t), t)ϕ(q(t))) dt = 0 δ b a L c (q(t), q(t)) dt = 0 where L c is the constrained Lagrangian. Explains why generalized coordinates work. Q: What about nonholonomic constraints? Verona Short Course, August 25-29, 2002 p.25/39

36 The d Alembert Principle A: Use a different variational principle. Theorem 2 The Lagrange-d Alembert equations of motion for a system with constraints are determined by δ b a L(q(t), q(t)) dt = 0, where δq(t) D q(t), a t b. Variations lie within the constraint distribution D T Q. The variations cannot do work. Verona Short Course, August 25-29, 2002 p.26/39

37 Bundle Structure Revisited Configuration space, Q, is a bundled space, with projection, π : Q R, model fiber S, a manifold, and where the fiber is vertical. S r q V q R Q There is a natural notion of verticality. V Q = ker(t π) r π R Verona Short Course, August 25-29, 2002 p.27/39

38 Bundle Structure Revisited Configuration space, Q, is a bundled space, with projection, π : Q R, model fiber S, a manifold, and where the fiber is vertical. S r q V q R Q There is a natural notion of verticality. V Q = ker(t π) r π R Q: What about the constraints? Verona Short Course, August 25-29, 2002 p.27/39

39 Bundle Structure Revisited Configuration space, Q, is a bundled space, with projection, π : Q R, model fiber S, a manifold, and where the fiber is vertical. S r q V q R Q There is a natural notion of verticality. V Q = ker(t π) r π R Q: What about the constraints? A: They give notion of horizontality. Verona Short Course, August 25-29, 2002 p.27/39

40 The Ehresmann Connection Definition 0 An Ehresmann connection, A, is a vertical valued one-form on Q that satisfies, 1. A q : T q Q V q Q is a linear map for each point q Q. 2. A is a projection, A(v q ) = v q v q V q Q. This results in a natural decomposition of the tangent bundle. The horizontal space is, HQ = ker(a) S r2 q 2 S r 1 q 1 R Q Note: HQ = D π r 2 π r 1 R Verona Short Course, August 25-29, 2002 p.28/39

41 Local Form The Ehresmman connection can be written in local coordinates q = (s, r) as A(q) q = ṡ + A loc (r, s)ṙ I.e., if A(q) q = 0, then ṡ = A loc (r, s)ṙ A vector in T q Q decomposes according to Ehresmman connection into, and hor v q = v q A(q)v q ver v q = A(q)v q S S R Q Verona Short Course, August 25-29, 2002 p.29/39

42 Principal Bundles Definition 0 A principal bundle is a fiber bundle such that the model fiber is a Lie group, G. For mechanical systems the base space, M, is sometimes called the shape space. Many control systems decompose this way. Q Gr Shape Directly controlled. q T Orb(q) q M Group What we want to control (locomote within). r π M Inherits all structures discussed. Verona Short Course, August 25-29, 2002 p.30/39

43 Examples Snakeboard front wheels ψ φ f Planar Fish ϕ 1 back wheels φ b (x,y) l θ l p ϕ 2 b T 3 SE(2) T 2 SE(2) Hilare Robot Planar Amoeba T 2 SE(2) R 3 SE(2) Verona Short Course, August 25-29, 2002 p.31/39

44 Principal Bundle Structure Suppose that, the fiber is a Lie group, S = G, i.e., Q is a principal bundle, There is a natural notion of verticality. V Q = ker(t π) = { (q, v) v = ξ Q (q), ξ g } = T q Orb (q) q Q Q G r q r T Orb(q) q M π M Verona Short Course, August 25-29, 2002 p.32/39

45 Group Orbits Definition 0 Given an action of G on Q and q Q, the orbit of q is defined by Orb (q) { Φ g (q) g G } Q The tangent space at q to the group orbit through q 0 is given by, T q Orb (q 0 ) = { ξ Q (q) ξ g } Q Orb(q) T Orb(q) q q 0 Verona Short Course, August 25-29, 2002 p.33/39

46 Reducing the Connection G-invariant constraints implied, T q Φ g (H q Q) = H Φg (q)q This implies that the Ehresmman connection is G-invariant in the following sense, G G M M Q T q Φ g A(q, q) = A T q Φ g (q, q) meaning that the connection can be reduced. Verona Short Course, August 25-29, 2002 p.34/39

47 The Principal Connection Definition 0 A principal connection, A, on the principal bundle Q is a g valued one-form such that, 1. A(ξ Q (q)) = ξ, ξ g, and q Q. 2. A is Ad-equivariant, (T q Φ g ) A(q, q) = Ad g A(q, q) Once again, the horizontal space is given by, G r 2 q 2 G r 1 q 1 M Q HQ = ker(a) The vertical space can be thought of as, V Q = (A) Q π r 2 π r 1 M Verona Short Course, August 25-29, 2002 p.35/39

48 The Local Connection Form In local coordinates q = (g, r), the connection form always takes the structure: A q = Ad g (A loc ṙ + g 1 ġ) Verona Short Course, August 25-29, 2002 p.36/39

49 The Mechanical Connection Observation 0 The function A I 1 J defines a principal connection form (the mechanical connection ) that respects the conservation law. A q = I 1 J q = ξ The conserved momentum is in spatial coordinates. We d like body coordinates. Moving to body coordinates is equivalent to finding the local form, J loc = Ad gj, Verona Short Course, August 25-29, 2002 p.37/39

50 Starting with A q = ξ and using the local form of the locked inertial, one obtains: Ad gi(g(q))ad g = I loc (r) A loc (r)ṙ + g 1 ġ = I 1 loc J loc q = I 1 loc p Verona Short Course, August 25-29, 2002 p.38/39

51 Equations of Motion In summary, using the mechanical connection, we arrive at M(r) r + B(r, ṙ) + G(r) = 0 g 1 ġ = A loc (r)ṙ + I 1 loc p ṗ = ad ξ p Verona Short Course, August 25-29, 2002 p.39/39