Review of Lagrangian Mechanics and Reduction

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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