Learning Lagrangian Dynamics: A New Approach to Understanding Mechanical Systems, Exams of Dynamics

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Learning Lagrangian Dynamics

Clancy Rowley

Joint work with Mohamadreza Ahmadi and Ufuk Topcu (UT Austin)

Overview

I (^) Many mechanical systems are described by Lagrangian dynamics I (^) Properties of the Lagrangian (e.g., symmetries) lead to physical properties of the system (e.g., conservation laws)

Lagrangian dynamics

I (^) Consider a 2n dimensional state space with variables q = (q 1 ,... , qn) (“positions”) and q˙ = ( ˙q 1 ,... , q˙n) (“velocities”). I (^) Define a real-valued function L(q, q˙). I (^) The corresponding dynamics are given by the Euler-Lagrange equations: d dt

∂L

∂ q˙i

∂L

∂qi

= 0 , i = 1 ,... , n.

Lagrangian dynamics

I (^) Consider a 2n dimensional state space with variables q = (q 1 ,... , qn) (“positions”) and q˙ = ( ˙q 1 ,... , q˙n) (“velocities”). I (^) Define a real-valued function L(q, q˙). I (^) The corresponding dynamics are given by the Euler-Lagrange equations: d dt

∂L

∂ q˙i

∂L

∂qi

= 0 , i = 1 ,... , n.

I (^) For mechanical systems, the Lagrangian is kinetic energy minus potential energy:

L(q, q˙) =

q˙T^ M q˙ − V (q)

Learning Lagrangians versus learning dynamics

Suppose the dynamics of a system are given by

x ˙ = f (x),

where x = (q, q˙) is the state. I (^) One approach to learning is to use data (e.g., samples of x(ti )) to try to find the function f : R^2 n^ → R^2 n.

Learning Lagrangians versus learning dynamics

Suppose the dynamics of a system are given by

x ˙ = f (x),

where x = (q, q˙) is the state. I (^) One approach to learning is to use data (e.g., samples of x(ti )) to try to find the function f : R^2 n^ → R^2 n. I (^) If the dynamics come from a Lagrangian L(q, q˙), then a different approach is to use data to try to find the Lagrangian L : R^2 n^ → R.

Why?

Why would we want to do this?

Symmetries

I (^) Symmetries are a type of side information. For instance, the Lagrangian may be independent of a particular variable qj , or invariant to the action of some group (e.g., rotations). I (^) For instance, suppose L(q, q˙) is independent of position qj (for some j). I (^) Then the Euler-Lagrange equation for that variable is d dt

∂L

∂ q˙j

so the “generalized momentum” ∂L/∂ q˙j is conserved.

Symmetries

I (^) Symmetries are a type of side information. For instance, the Lagrangian may be independent of a particular variable qj , or invariant to the action of some group (e.g., rotations). I (^) For instance, suppose L(q, q˙) is independent of position qj (for some j). I (^) Then the Euler-Lagrange equation for that variable is d dt

∂L

∂ q˙j

so the “generalized momentum” ∂L/∂ q˙j is conserved. I (^) Emmy Noether showed that a similar result holds for symmetries described by arbitrary Lie groups.

Idea

I (^) If there are known symmetries, incorporate these as “side information”. I (^) The resulting models will then automatically satisfy the corresponding conservation laws. (^) 6 / 16

Problem setup for learning Lagrangians

We will assume the dynamics satisfy Euler-Lagrange equations of the form d dt

∂L

∂ q˙

∂L

∂q

= B(q)u.

I (^) u(t) is a control input, which we get to choose I (^) B(q) is known I (^) The function L(q, q˙) is unknown, and we will try to determine it from data.

Learning Lagrangians from data

t

q(t), q˙(t)

q(0)

q ˙(0)

q ˙(tN )

q(tN )

I (^) The goal is to learn L(q, q˙) given data {ti , u(ti ), q(ti ), q˙(ti ), q¨(ti )}Ni= 1. I (^) Parameterize the Lagrangian:

L(q, q˙) =

∑^ d

i= 1

αi ϕi (q, q˙),

for given basis functions ϕi (e.g., monomials)

Learning Lagrangians from data

t

q(t), q˙(t)

q(0)

q ˙(0)

q ˙(tN )

q(tN )

I (^) The goal is to learn L(q, q˙) given data {ti , u(ti ), q(ti ), q˙(ti ), q¨(ti )}Ni= 1. I (^) Parameterize the Lagrangian:

L(q, q˙) =

∑^ d

i= 1

αi ϕi (q, q˙),

for given basis functions ϕi (e.g., monomials) I (^) Find constants αi that best fit the data. I (^) One can then predict the future evolution from E-L equations. (^) 8 / 16

Two warnings

1. Different Lagrangians can lead to the same dynamics
2. If L(q, q˙) = 0, the Euler-Lagrange equations are trivially satisfied for any data.

We take steps to find a particular (non-trivial) Lagrangian whose corresponding dynamics agree with the data. For details, see paper (M. Ahmadi, U. Topcu, C. Rowley, ACC 2018).

Example: forced Duffing equation

q

A cos(ωt)

N/S N/S

beam

magnet magnet

rigid frame