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Geometric Dynamics Overview

Zac Manchester
October 19, 2011

Geometric Dynamics Overview

A very brief introduction to the differential-geometric view of classical dynamics for engineers. This was a talk I presented to my research group at Cornell.

Zac Manchester

October 19, 2011
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  1. The  Geometric  View  of   Dynamics  

  2. What  is  a  Manifold?   •  N-­‐dimensional  generalization  of  the

     concept  of  a  surface   •  2D  surfaces  embedded  in  3D  Euclidean  space  are  2D   manifolds   •  All  manifolds  can  be  embedded  in  a  higher  dimensional   Euclidean  space  
  3. Configuration  Manifold   •  Single  Pendulum:  S1   •  Double

     Pendulum:  S1  X  S1  =  T2  
  4. Tangent  Bundle  =  Phase  Space   •  Velocity  vectors  do

     not  live  in  the  same  space  as  the   configuration!   •  Set  of  all  possible  velocity  vectors  at  every  point  on  the   configuration  manifold  makes  up  tangent  bundle   S1  X  R  
  5. What  is  a  Lie  Group?   •  Mathematical  definition  of

     a  group   –  A  set  that  is  closed  under  an  associative  operation   –  Examples:   •  Integers  with  addition   •  Positive  real  numbers  with  multiplication   •  Matrices  with  matrix  multiplication   •  A  Lie  group  is  a  group  with  a    continuous  parameter   –  All  the  useful  ones  can  be  represented  as  matrices   –  Examples:   •  2D  and  3D  rotations  –  SO(2)  and  SO(3)   •  2D  and  3D  rigid  body  motion  –  SE(2)  and  SE(3)  
  6. What  is  a  Lie  Algebra?   •  The  vector  space

     tangent  to  the  Lie  group  at  the  origin  (think  tangent   plane  to  the  manifold)   •  The  exponential  map  takes  elements  of  the  Lie  algebra  and  maps  them   onto  the  Lie  group  (i.e.  the  standard  matrix  exponential)   –  For  the  unit  circle  in  the  complex  plane  S1,  the  Lie  algebra  is  the  complex   numbers   –  For  the  rotation  group  SO(3),  the  associated  Lie  algebra  so(3)  is  the  set  of   angular  velocity  vectors  written  as  skew-­‐symmetric  matrices  
  7. What  can  we  do  with  this  stuff?   •  Write

     numerical  integrators  that  conserve  things  like   energy  and  momentum  and  don’t  require  normalization   tricks   –  By  only  using  the  group  operation  to  propagate  the  state,  we   guarantee  that  the  result  is  also  a  member  of  the  group   •  These  benefits  extend  to  estimators  and  controllers  as   well,  and  allow  a  more  explicit  handling  of  nonlinearity