Journey through Nonholonomic Systems: Rebels, Reductions, and Research Topics

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

2. Pre-requisites (will discuss them informally, no worry!!!) Differential geometry Vectorfields, Lie-Brackets Frobenius theorem Principal bundles, connections Mechanics: Lagrangian (TS) and Hamiltonian (T*S) Symplectic forms, Poisson brackets

3. Suggested books ( for the library ) Nonholonomic mechanics Neimark, Fufaev Dynamics of nonholonomic systems, 1972 A.BlochNonholonomic mechanics and control, 2003 prefacesupplement J. CortesGeometric, 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 partIpartII

4. Recent meetings: BanffBanff1 Surveys and historical notes Bloch, Marsden, Zenkov Borisov Geometric Mechanics Marsden Geometric Control Murray panel MurraypageOstrowski References for Symplectic Geometry: Alan WeinsteinAnaCannas + Abraham/Marsden, Arnold, …

5. NH papers (JK + friends) Kurt EhlersRichard Montgomery Reduction (ARMA,92) + BKMM96 Calgary97 pdf (Romp) TorunMoving frames (Romp) A.Graymemorialpdf Cartan connectionspdf AMSfeaturedreviewpdf Alanfestpdf Rubber rolling (RCD)

6. Pause for fun: Euler disk OReillyMoffat http://www.eulersdisk.com/index.html http://plus.maths.org/issue11/news/spin/index.html

7. 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

8. 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!)

9. 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

10. 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

11. 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

12. 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

13. 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)

14. CONSTRAINED MECHANICAL SYSTEMS: NONHOLONOMIC (D’ALEMBERT-LAGRANGE) Versus SUBRIEMANNIAN (VAKONOMIC, OPTIMAL CONTROL)

15. 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) • D’Alembert’s 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”

16. … 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”)

17. 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

18. Historical Note Poincaré on Hertz (pg 245)

19. 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

20. DICTIONARY Optimal control = “vakonomic systems” (Valery Kozlov) = subriemannian geometries = = under-actuated robotic systems = variational Nonholomic systems = = d’Alembert’s 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

21. Paradigm of NH systems: Chaplygin-Caratheodory sleigh (see NF pg 76) “Left invariant system in G = SE(2) = 3 dof with one constraint”

22. Summary of Chaplygin sleigh analysis (left invariant) Asymptotic motions: w = 0 No smooth invariant measure exists

23. 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 !

24. Spherical robots Ballbot Roball Rotundus zorbing

25. "Un viaje por los Sistemas No-Holonómicos". • Jair Koiller FGV/RJAGIMB • 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

27. Lecture II: exploring Lie group symmetries, reduction Starting point: Reduction (ARMA,92)BKMM96Additional 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?

28. Scheme for generalized Chaplygin systems Q principal bundle Group: G Base: S=Q/G E: a connection L = equivariant lagrangian in Q Reduction: get a non closed form w in T*S H = compressed look for f = time reparametrization (it does not always exists)

29. 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)

30. 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 )

31. 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.

32. Intermission: informal discussion on pre-requisites Differential geometry: Vectorfields, Lie-Brackets Frobenius theorem vs. Chow theorem Principal bundles, connections Mechanics: Lagrangian (TS) and Hamiltonian descriptions (T*S) Symplectic forms, Poisson brackets Integrable Hamiltonian Systems: action angle coords.

33. Vectorfields and Lie brackets are by now standard tools in Robotics: control of under-actuated systems Parallel parking: http://www.laas.fr/~florent/mobile_robot.html Nonholonomic motion planning: http://www.laas.fr/~jpl/book.html DARPA GRAND CHALLENGE

34. Learning about vectorfields, Lie brackets in practice: Nonholonomic motion planning Books: Li-CannyLaumondNSF Presentations (look for more in the web) Callegari Kelly-Murray Latombe Actuated rattleback Biomimetic

35. Snakeboard: a hybrid system nonholonomic system + control For beginners http://www.snakeboarder.com/main.html http://www.wikihow.com/Ride-a-Snakeboard/Streetboard http://www.snakeboard.no/Website_eng/Index_eng.htm http://streetboarding.org/shop/product_info.php?products_id=128 For other type of beginners Math of snakeboard Northwestern (look at folder for more)

36. Optimal control in Microswimming (tomorrow) www.impa.br/~jair

37. Summary of lecture II: Nonholonomic systems with symmetry Review: JK, 92: generalized Chaplygin systems Q principal bundle Group: G Base: S=Q/G E: by a connection L = equivariant lagrangian in Q Reduction: get a non closed form w in T*S Alanfest RCD f = time reparametrization

38. 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

39. 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)