"Un viaje por los Sistemas No-Holon

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• "Un viaje por los Sistemas No-Holonómicos".
• Jair Koiller FGV/RJ AGIMB
• Martes Rebeldes sin causa, ejemplos
  sorprendentes.
• II. Miércoles Reducción y simetrías.
• III. Jueves Temas de investigación
• GMCNetwork

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Pre-requisites (will discuss them informally,
no worry!!!) Differential geometry

Vectorfields, Lie-Brackets
Frobenius theorem
Principal bundles, connections Mechanics Lagran
gian (TS) and Hamiltonian (TS) Symplectic
forms, Poisson brackets
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Suggested books ( for the library
) Nonholonomic mechanics Neimark, Fufaev
Dynamics of nonholonomic systems, 1972 A.Bloch
Nonholonomic mechanics and control, 2003
preface supplement J. Cortes Geometric, Control
and Numerical Aspects of NH Systems,
2002 Control F.Bullo, A. Lewis, Geometric
Control of Mechanical Systems, 2005 Geometric
mechanics W.Oliva, Geometric Mechanics D.
Holm, Geometric Mechanics partI partII
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Recent meetings Banff Banff1 Surveys and
historical notes Bloch, Marsden, Zenkov
Borisov
Geometric
Mechanics Marsden Geometric Control Murray
panel Murraypage Ostrowski References
for Symplectic Geometry Alan Weinstein
AnaCannas Abraham/Marsden, Arnold,




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NH papers (JK friends)
Kurt Ehlers Richard
Montgomery Reduction (ARMA,92)
BKMM96 Calgary97 pdf (Romp)
Torun Moving frames (Romp)
A.Graymemorial pdf

Cartan connections pdf

AMSfeaturedreview
pdf Alanfest pdf
Rubber rolling (RCD)
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Pause for fun Euler disk OReilly
Moffat http//www.eulersdisk.com/index.html
http//plus.maths.org/issue11/news/spin/index.ht
ml
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Lecture I. Rebels with a cause. Surprising
examples Angular momentum is not
conserved! Celtic stone
Volume in phase space is not conserved!
Chaplygin sleigh Rattleback
Wobblestone Celtic stone Anagyre
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Prologue in holonomic (unconstrained)
mechanics conservation of angular momentum J is
a basic principle
yet, reorientation is easy for bodies that can
change shape Q SO(3) x SO(3) J 0 all
the time Astronauts, gymnasts and pool divers
do the same trick (Crocodiles too!)

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Mathematics of deforming bodies Principal
bundles with connection Q configuration
space G group acting in Q by rigid motions
(no change in shape) S Q/G shape space A
physical principle gives a natural way to
uniquelly associate to an infinitesimal unlocated
shape change an infinitesimal located change.
After a closed loop in shape space, located
shape will be in a different place
HOLONOMY the group element that
relates them
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Examples of Principal Bundles in Applied
Mathematics Chemistry Guichardet connection
for molecules Iwai Space satellite
reorientation in space without rockets Cats
and Gymnasts landing on their feet In the
above examples, the connection is given by the
physical principle
Total angular momentum 0 Microorganism
swimming (zero Reynolds number) Connection given
by total force 0 , total torque 0
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Terminology G group Q configurations
S Q/G shapes Vertical spaces Vq , q in
Q infinitesimal rigid motions of a

located shape Connection distribution of
horizontal spaces Hq , q in Q
equivariance g . Hq Hgq
Vq and Hq are complementary
in TqQ Curvature of connection measures how
much on can move
in the vertical direction using
sequence of
infinitesimal horizontal
moves Mathematical tools differential forms

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In all these examples Vertical spaces and
horizontal spaces are perpendicular with respect
to the natural metric of the problem In
mechanics Metric the kinetic energy In
microswimming Metric hydrodynamical power
expenditure
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How cats do it? Kane Total angular
momentum J 0 defines a connection Universal
joint links upper and lower body G SO(3) acts
on Q SO(3) x SO(3) S SO(3) shape space
base Change of inertia matrix in time (feet and
tail help) Curvature is fat (the opposite of
flat)
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CONSTRAINED MECHANICAL SYSTEMS NONHOLONOMIC
(DALEMBERT-LAGRANGE) Versus SUBRIEMANNIAN
(VAKONOMIC, OPTIMAL CONTROL)
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Getting started Nonholonomic mechanics, by the
old way (this is elementary physics, but can
solve most problems) Newton m a
external reaction forces (due to
constraints) DAlemberts principle
Reactions do not perform work when
constraints are enforced Engineers solve them by
clever tricks Quasi-velocities projections
of true velocities on moving frames. using
them the constraints are automatically
eliminated Weird thing quasi-velocities
exist, quasi-coordinates do not.
Contraints are
non-integrable
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and nonholonomic mechanics by the modern way
Mathematical framework extending Lagrangian or
Hamiltonian systems, using 2 forms and
bivectors (non closed and non-Jacobi) To
reduce the dynamics look for external (left)
lie group symmetries or for internal or
material (right) symmetries Try to get the
reduced equations in the same category as the
original ones Can the reduced dynamics be
Hamiltonized? (by a coordinate dependent change
of time). Is the reduced system integrable?
Reconstruct the full dynamics, analytically or
numerically. For the latter, use a
discrete nonholonomic mechanics (peg-leg)
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Is the modern way useful? Necessary?
Robotic engineers are using them! The
geometric language helps setting up the
equations in systematic way using the
obvious symmetries to reduce finding hidden
symmetries relating a problem to classic
ones when not solvable analytically set up
good numerical methods
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Historical Note Poincaré on Hertz (pg 245)
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What Hertz already said (with different
terminology) in his Foundations of Mechanics
Two different theories, but using the same
ingredients the ODES and properties are very
different L T V Lagrangian in TQ, Q
configurations space H distribution of
horizontal spaces in TQ (the constraint
distribution) Optimal control (subriemannian
geometry) vs. NH systems
SHORTEST
STRAIGHTEST
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DICTIONARY Optimal control vakonomic
systems (Valery Kozlov)
subriemannian geometries
under-actuated robotic
systems variational Nonholomic systems
dAlemberts
principle NOT variational principle Hertz
(1890) NH systems minimize curvature LOCALLY
subject to the
constraints theories coincide for holonomic
systems Cartan (1928) NH systems obey a
projected affine connection
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Paradigm of NH systems Chaplygin-Caratheodory
sleigh (see NF pg 76)
Left invariant system in G SE(2) 3 dof
with one constraint
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Summary of Chaplygin sleigh analysis (left
invariant)
Asymptotic motions w 0 No smooth
invariant measure exists
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Rolling without sliping nor twisting interesting
example of nonholonomic constraints, right
invariant JK-KE(RCD) Fatima Leite
GeometryofRolling FatimaGMC
Both curves have same geodesic
curvature !
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Spherical robots
Ballbot Roball Rotundus
zorbing
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• "Un viaje por los Sistemas No-Holonómicos".
• Jair Koiller FGV/RJ AGIMB
• Martes Rebeldes sin causa, ejemplos
  sorprendentes.
• II. Miércoles Reducción y simetrías.
• III. Jueves Temas de investigación
• GMCNetwork

26
Summary of Lecture I. Rebels with a cause.
1. Angular momentum is not conserved!

paradigm Celtic stone 2.
Volume in phase space is not conserved!
paradigm Chaplygin sleigh
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Lecture II exploring Lie group symmetries,
reduction Starting point Reduction (ARMA,92)
BKMM96 Additional material Left invariant
metrics in Lie groups constraints are
left invariant 1-forms (vs. right
invariant 1-forms, Fedorov) Principal
bundles with connections (generalized
Chaplygin systems) Open question hybrid cases?
More or less dimensions?
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Scheme for generalized Chaplygin systems

Q principal bundle Group G Base SQ/G
E a connection L equivariant
lagrangian in Q Reduction get a non closed
form w in TS H compressed
look for f time reparametrization
(it does not always exists)
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Javelin http//mae.ucdavis.edu/biosport/
Eng.-in-sports1 Hubbard Hubbard1 In an
international javelin competition a selection of
sanctioned javelins is available for competitors
to use personal javelins are banned. This means
that each competitor has access to the latest and
greatest equipment, thus leveling the playing
field (excuse the pun) for all In the ideal
world, engineering developments would be
inexpensively available to all competitors (as in
a javelin competition), thus advancing the sport
and ensuring the best athlete wins - surely the
ideal of most sports. R.Smith, Symscape
Eng.-in-sports
Homework Solve a toy NH model for the
javelin (a Chaplygin system)
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Homework. apply the reduction procedure in the
following example of a Chaplygin system
Homogeneous ball on a vertical cylinder (NF, pg
95)
5 Degrees of freedom R x S1 x SO(3) 2
rolling constraints 2 transversal
symmetries can eliminate two angles
(which ones?) Dynamics reduces to R x S2
(why?) It is integrable since fully
symmetric (3 equal inertias I1 I2 I3
. Open,doable I1 I2 ? I3 )
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Homogeneous ball on a vertical cylinder (NF, pg
95) Calculations show z is a sinusoidal
function (gravity g does not
appear) Conclusion the ball does not fall !
Tokieda Dynamics of basketball-rim
interactions, H. Okubo and M. Hubbard, Sports
Engineering, 7, 15-29.
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Intermission informal discussion on
pre-requisites
Differential geometry Vectorfie
lds, Lie-Brackets Frobenius
theorem vs. Chow theorem
Principal bundles, connections Mechanics Lagr
angian (TS) and Hamiltonian descriptions (TS)
Symplectic forms, Poisson brackets Integrable
Hamiltonian Systems action angle coords.
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Vectorfields and Lie brackets are by now standard
tools in Robotics control of under-actuated
systems Parallel parking http//www.laas.fr/f
lorent/mobile_robot.html Nonholonomic motion
planning http//www.laas.fr/jpl/book.html
DARPA GRAND CHALLENGE
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Learning about vectorfields, Lie brackets in
practice Nonholonomic motion planning Books
Li-Canny Laumond NSF Presentations (look
for more in the web) Callegari Kelly-Murray Latomb
e Actuated rattleback Biomimetic

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Snakeboard a hybrid system nonholonomic system
control For beginners http//www.snakeboarder.co
m/main.html http//www.wikihow.com/Ride-a-Snakebo
ard/Streetboard http//www.snakeboard.no/Website_e
ng/Index_eng.htm http//streetboarding.org/shop/p
roduct_info.php?products_id128

For other type of beginners

Math of snakeboard

Northwestern
(look
at folder for more)
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Optimal control in Microswimming
(tomorrow) www.impa.br/jair
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Summary of lecture II Nonholonomic systems
with symmetry Review JK, 92 generalized
Chaplygin systems

Q principal bundle Group G Base SQ/G
E by a connection L equivariant
lagrangian in Q Reduction get a non closed
form w in TS Alanfest
RCD
f time reparametrization
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Bottomline if d( f w ) 0 then .
the reduced system can be
Hamiltonized SO
WHAT? Recent advances (look at folders)
Examples of integrable high dimensional
generalized Chaplygin systems (Fedorov,
Jovanovic) Adhoc examples by Borisov
group Systematic study by Luis Garcia Naranjo
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Lecture III. Recent advances in NH mechanics,
research topics Singular reduction/dimension
jumps (Cushman, Sniatycki) Existence of
invariant measures (Blackall , Bloch,
Zenkov, de Leon) Almost Poisson description
(Mashke/van der Shaft, Marle, Koon-Marsden,
Spain group) Lie-Algebroid description (Spain
group) Affine-connection description (A.
Lewis, Kurt Ehlers/JK)