Tools for Reduction of Mechanical Systems Ravi Balasubramanian, Klaus Schmidt, and Elie Shammas Carnegie Mellon University

Tools for Reductionof Mechanical Systems

Ravi Balasubramanian, Klaus Schmidt, and Elie Shammas

Carnegie Mellon University

February 2004 Center for the Foundations of Robotics2

Motivation: Two Mass System

Two masses on the real line:

Lagrangian:

Set A: Two 2nd order

differential equations

Set B: One 1st order and one 2nd order

differential equations

Equations of Motion:

– Two sets of equations, but same solution.– When can we do such a simplification (reduction)?– What tools to use?


Overview

Fiber Bundle– Projection map– Lifted projection map

Decomposition of Velocity Spaces– Vertical and Horizontal Spaces

Principal Connections– Example on

Mechanical Connections– Momentum map– Locked Inertia Tensor– Local Form


Fiber Bundle

A manifold with a base space and a map is a fiber bundle if:

A fiber Y is the pre-image of b under

Property of : for every point b2B 9 U 3 b such that:

is homeomorphic to

Or locally

If locally Y is a group, Q is a principal fiber bundle. If Y is a group everywhere, Q is a trivial principal fiber bundle.

for


Lifted Bundle Projection Map

Two-mass System– Choose as fiber, as base space.


Velocity Decomposition

Vertical Space

Horizontal Space

Two mass example

Why do velocity decomposition?– Understand how fiber velocities and base space velocities interact.


Overview

Fiber Bundle– Projection map– Lifted projection map

Vertical and Horizontal Spaces Principal Connections

– Example on

Mechanical Connections– Momentum map– Locked Inertia Tensor– Local Form


Principal Connections

Definition: A principal connection on the

principal bundle

is a map that is linear on each tangent space such that

1)

2)


Connection Property 1


Connection Property 2


Principal Connection on

Choose as the fiber, as base space.– Projection and Lifted Projection


Left Action on Fiber

Action (Translation)

Trivial to show group properties for .– Thus, the fiber is a group.– Q is a trivial principal fiber bundle.


Group Actions on Q

Group Action of G on Q: Translation along fiber

Lifted Action of G on Q


A connection on

Choose a connection of the form

Need to verify if satisfies the connection properties.


Connection: Property 1

Exponential map on :

Generator on Q

satisfies Property 1.


Connection: Property 2

In , is the Identity map.(1)

LHS(1) = RHS(1) =

In , is the Identity map.– Why? No rotations, and Body velocity = Spatial Velocity.

satisfies Property 2.

Thus, is a connection.


Velocity Decomposition

satisfiesLemma:

Definition: Horizontal Space

Thus, and decompose into components.

Vertical and Horizontal Spaces in


Velocity Decomposition: Illustration


For this example, the connection is arbitrary; Mechanical systems use a specific connection.

If , then:

Motion only in base space.

Motion only along fiber.

Motion in base space and

induced motion tangent to fiber.

Motion only along fiber

If , then:

Velocity Decomposition: Interpretation


define connection based on conservation of momentum

Momentum Map

Locked Inertia Tensor

Mechanical Connection

Reconstruction Equation

Outline:

Mechanical Systems

Connections for mechanical systems:


Momentum Map

Definition:

with , and

Physical Intuition: is

momentum of the system

representation in spatial coordinates

: natural pairing between covectors and vectors

: for mass matrix and :


Two-mass example

Two masses on the real line:

Lagrangian:

Mass Matrix:

Generator for Lifted map


Example: momentum map

Note: is indeed the momentum of the system

in spatial coordinates


Body Momentum Map

Definition:

with , and


momentum of the system measured in the

instantaneous body frame

representation in body coordinates


Example: body momentum map

and

Note: is the momentum of the system measured

in the body and represented in body coordinates

with


Locked Inertia Tensor

Definition:

with and


inertia of the locked system

all base variables are fixed

representation in spatial coordinates


Example: locked inertia tensor

Note: is indeed the locked inertia of the system

(for fixed)


Intuition: Mechanical Connection

Compute Lie-Algebra velocity such that the locked system has the momentum

with


is the map that assigns to each the spatial Lie-Algebra velocity of the locked system such that the momentum in spatial coordinates is conserved:

Mechanical Connection

Definition:

: locked inertia tensor

, : (Body) Momentum Map

Definition: Body Connection


Example: mechanical connection

We are ready to compute

and


Example: velocities

Vertical velocities:

Horizontal velocities:

and Recall


Example: velocities

movement along the fiberwithout movement in the base

Movement in the base induces movement along the fiber


Local form of the connection

Proposition: Let be a principal connection on .

Then can be written as

and

is called the local form of the connection

only depends on and

is the group velocity at the origin

Note:


Example: local form


Reconstruction

Reconstruction:

General Case:

Zero Momentum:


Take home message

1) Connection explores system from momentum viewpoint.

2) Decomposition of Velocities using and

- Can compute induced motion in fiber from base

velocities.

Set B: One 1st order and one 2nd order differential equation


Conclusions

Principal Connections

Mechanical Connections

Reconstruction Equation

- zero-momentum case

Next talk:

- symmetries: reduced lagrangian

- evaluate general reconstruction equation

- introduce constraints (holonomic and nonholonomic)

- define reduced equations of motion


Choose someLet

Note thatDefineThus,

In general,

Note that

Thus, Define

Proof:

Horizontal Space

satisfiesLemma:

Definition: Horizontal Space

Documents

Tools for Reduction of Mechanical Systems Ravi Balasubramanian, Klaus Schmidt, and Elie Shammas Carnegie Mellon University