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

February 2004 Center for the Foundations of Robotics 2 Motivation: Two Mass System Two masses on the real line: Lagrangian: Set A: Two 2 nd order differential equations Set B: One 1 st order and one 2 nd 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?

February 2004 Center for the Foundations of Robotics 3 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

February 2004 Center for the Foundations of Robotics 4 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

February 2004 Center for the Foundations of Robotics 5 Lifted Bundle Projection Map Two-mass System – Choose as fiber, as base space.

February 2004 Center for the Foundations of Robotics 6 Velocity Decomposition Vertical Space Horizontal Space Two mass example Why do velocity decomposition? – Understand how fiber velocities and base space velocities interact.

February 2004 Center for the Foundations of Robotics 7 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

February 2004 Center for the Foundations of Robotics 8 Principal Connections Definition: A principal connection on the principal bundle is a map that is linear on each tangent space such that 1) 2)

February 2004 Center for the Foundations of Robotics 9 Connection Property 1

February 2004 Center for the Foundations of Robotics 10 Connection Property 2

February 2004 Center for the Foundations of Robotics 11 Principal Connection on Choose as the fiber, as base space. – Projection and Lifted Projection

February 2004 Center for the Foundations of Robotics 12 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.

February 2004 Center for the Foundations of Robotics 13 Group Actions on Q Group Action of G on Q: Translation along fiber Lifted Action of G on Q

February 2004 Center for the Foundations of Robotics 14 A connection on Choose a connection of the form Need to verify if satisfies the connection properties.

February 2004 Center for the Foundations of Robotics 15 Connection: Property 1 Exponential map on : Generator on Q satisfies Property 1.

February 2004 Center for the Foundations of Robotics 16 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.

February 2004 Center for the Foundations of Robotics 17 Velocity Decomposition satisfies Lemma: Definition: Horizontal Space Thus, and decompose into components. Vertical and Horizontal Spaces in

February 2004 Center for the Foundations of Robotics 18 Velocity Decomposition: Illustration

February 2004 Center for the Foundations of Robotics 19 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

February 2004 Center for the Foundations of Robotics 20  define connection based on conservation of momentum  Momentum Map  Locked Inertia Tensor  Mechanical Connection  Reconstruction Equation Outline: Mechanical Systems Connections for mechanical systems:

February 2004 Center for the Foundations of Robotics 21 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 :

February 2004 Center for the Foundations of Robotics 22 Two-mass example Two masses on the real line: Lagrangian: Mass Matrix: Generator for Lifted map

February 2004 Center for the Foundations of Robotics 23 Example: momentum map Note: is indeed the momentum of the system in spatial coordinates

February 2004 Center for the Foundations of Robotics 24 Body Momentum Map Definition: with, and Physical Intuition: is  momentum of the system measured in the instantaneous body frame  representation in body coordinates

February 2004 Center for the Foundations of Robotics 25 Example: body momentum map and Note: is the momentum of the system measured in the body and represented in body coordinates with

February 2004 Center for the Foundations of Robotics 26 Locked Inertia Tensor Definition: with and Physical Intuition: is  inertia of the locked system  all base variables are fixed  representation in spatial coordinates

February 2004 Center for the Foundations of Robotics 27 Example: locked inertia tensor Note: is indeed the locked inertia of the system (for fixed)

February 2004 Center for the Foundations of Robotics 28 Intuition: Mechanical Connection Compute Lie-Algebra velocity such that the locked system has the momentum with

February 2004 Center for the Foundations of Robotics 29 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

February 2004 Center for the Foundations of Robotics 30 Example: mechanical connection We are ready to compute and

February 2004 Center for the Foundations of Robotics 31 Example: velocities Vertical velocities: Horizontal velocities: and Recall

February 2004 Center for the Foundations of Robotics 32 Example: velocities  movement along the fiber without movement in the base Movement in the base induces movement along the fiber

February 2004 Center for the Foundations of Robotics 33 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:

February 2004 Center for the Foundations of Robotics 34 Example: local form

February 2004 Center for the Foundations of Robotics 35 Reconstruction Reconstruction: General Case: Zero Momentum:

February 2004 Center for the Foundations of Robotics 36 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 1 st order and one 2 nd order differential equation

February 2004 Center for the Foundations of Robotics 37 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

February 2004 Center for the Foundations of Robotics 38 Choose some Let Note that Define Thus, In general, Note that Thus, Define Proof: Horizontal Space satisfies Lemma: Definition: Horizontal Space