Parallel parking, falling cats, spherical robots and differential geometry€¦ · A Comprehensive Introduction to Di erential Geometry - M. Spivak, Publish or Perish Inc., 1979

Parallel parking Vector spaces Smooth manifolds Spherical robots References

Parallel parking, falling cats, spherical robots anddifferential geometry

Ravi N [email protected] 1

1Systems and Control Engineering,IIT Bombay, India

( Visiting Professor, IIT-Gandhinagar)A popular talk

February 1, 2017.

February 1, 2017

A popular talk IIT-Gandhinagar


Outline

1 Parallel parking

2 Vector spaces and beyond

3 Smooth manifolds

4 Satellites and spherical robots

5 Recommended texts



Outline

1 Parallel parking


3 Smooth manifolds


5 Recommended texts



Parallel parking

• To begin with lateral motion of the vehicle was not permissible

• Yet, a series of two actions (steering and driving) produced netdisplacement in the lateral direction

AB

Figure: Parallel parking

• Essential ideas: vector fields and non-commutativity of vector fieldsdefined over manifolds



Parallel parking: some math

• Configuration variables evolve on a smooth manifold

qqq = (x, y, θ, φ) ∈M = R1 × R1 × S1× S1

x

y

Figure: Rolling coin

• Constraints - No slip and no sliding

x sin θ − y cos θ = 0 No lateral motion

x cos θ + y sin θ = rφ Pure rolling



Equations of motion

• Equations of motion with us = θ and ud = φxyθ

=

r cos θr sin θ

0

ud +

001

us (1)

• Vector fields

g1(x)4=

r cos θr sin θ

0

, g2(x)4=

001

(2)

• Lie bracket of g1 and g2 is denoted as [g1, g2].



Parallel parking: the sequence of actions

Parallel parking sequence



Vector field

• We have encountered differential equations of the form

x1 = f1(x1, . . . , xn+1)

...

xn = fn+1(x1, . . . , xn+1) (3)

OR (in compact notation)

x = f(x); x(t) ∈ Rn (4)

• f(x) is a vector field. Determines evolution in the space where thesystem lives

• Example: a spring-mass system. The equations of motion are:

x1 = x; x2 = x(x1x2

)=

(x2− k

mx1

)(5)



High school physics: vectors in R3 and matrices

• Vectors in R3: binary operation: the sum of two vectors is defined andyields a vector

• Other operations in R3: 1. the dot product of two vectors yields ascalar, 2. the cross product of two vectors yields another vector.

• Rn×n (n× n matrices of real numbers): the sum of two matrices isdefined and yields a matrix

• Other operations in Rn×n: the multiplication of two matrices yields amatrix.

• Matrix multiplication does not commute: AB 6= BA. We define ameasure of non-commutativity as [A,B] = AB −BA.



Outline

1 Parallel parking


3 Smooth manifolds


5 Recommended texts



Vector spaces

• Objects called vectors: a set with a binary operation, an associationwith scalars,

• Examples: Rn, Cn, set of continuous real-valued functions on aninterval [a, b], polynomials of degree ≤ n.

• More structure could be introduced: norms (notion of size) and innerproducts (a notion of othrogonality)

• Bedrock of much of engineering applications - Fourier series, leastsquares optimization, linear estimation, linear filtering, modal analysis



Outline

1 Parallel parking


3 Smooth manifolds


5 Recommended texts



Smooth manifolds

• Objects that are “locally Euclidean” (but not globally) - like the earth,which is a sphere (geometric notation S2), a circle (S1), a torus(S1× S1)

• Evolution of a dynamical system on smooth manifolds

• Questions: What is a metric on such objects ? What is the shortestpath ? What are zero acceleration curves ?



2 × 2 real, orthogonal matrices

Characterized asO(2) = {A ∈ R2×2|ATA = I}

• Parameterize A as

A =

(a bc d

)(6)

with the constraints a2 + c2 = 1, ab+ cd,= 0, b2 + d2 = 1.

• The circle had two variables and one constraint equation. We now have4 variables and 3 constraint equations.



Dynamical systems and geometry

θ

S1

θ1

θ2

S1 × S1

1

Figure: Single pendulum (circle) and double pendulum (torus)



A glimpse of differential geometry

• Familiar with calculus (differentiation, integration) on the real line (R)or Rn. Extend this calculus to surfaces - say on a circle, a sphere, atorus and more complicated surfaces

• But why ? From a dynamical systems and control theory point ofview, systems need not always evolve on the real line (R) or multiplereal lines (Rn). They may evolve on non-Euclidean surfaces. On suchsurfaces, we wish to talk of ”rate of change (velocity) ⇒ differentiation,to talk of ”rate of rate of change (acceleration), to talk of accumulation⇒ integration

• Can these surfaces which are not Euclidean be locally represented asEuclidean ? This would allow us to employ the calculus that we arefamiliar with for these surfaces



Motion on a sphere (S2)

Start

Finish

Figure: Motion on a sphere (Picture taken from J. Marsden’s, Lectures onMechanics)



The falling cat

• Watch: https://www.youtube.com/watch?v=RtWbpyjJqrU

• Starts with zero angular momentum with its back facing the ground

• Lands on its paws after a complete rotation

• 3 interconnected bodies each evolving on a manifold and obeying a netconservation of angular momentum.



Outline

1 Parallel parking


3 Smooth manifolds


5 Recommended texts



A prototype

Figure: A spherical robot fabricated in the Systems and Control group, IITBombay



Figure: A spherical robot with internal rotors for locomotion



A satellite with an internal rotor

Inertial frame I

Body frame B

Gimbal frame G

β

γ

Satellite

Wheel

Gimbal frame

Figure: A satellite actuated by an internal rotor



On “abstract” mathematics: An address by V. I. Arnold

Excerpt from the address at the discussion on mathematics in Palaisde Decouverte in Paris on 7 March 1997.

‘In the middle of the twentieth century it was attempted to divide physicsand mathematics. The consequences turned out to be catastrophic. Wholegenerations of mathematicians grew up without knowing half of their scienceand, of course, in total ignorance of any other sciences. They first beganteaching their ugly scholastic pseudo-mathematics to their students, then toschoolchildren (forgetting Hardy’s warning that ugly mathematics has nopermanent place under the Sun). Since scholastic mathematics that is cutoff from physics is fit neither for teaching nor for application in any otherscience, the result was the universal hate towards mathematicians - both onthe part of the poor schoolchildren (some of whom in the meantime becameministers) and of the users. ’



Outline

1 Parallel parking


3 Smooth manifolds


5 Recommended texts



References

A Comprehensive Introduction to Differential Geometry - M. Spivak,Publish or Perish Inc., 1979.An Introduction to Differentiable Manifolds and Riemannian Geometry - W.M. Boothby, Academic Press, 1986.Introduction to Mechanics and Symmetry - J. Marsden and T. Ratiu,Springer-Verlag, 1994.Geometric Mechanics and Symmetry - D .D. Holm, T. Schmah and C.Stoica, Oxford University Press, 2009.An Introduction to Manifolds - L. W. Tu, Springer 2008.Ordinary Differential Equations - V. I. Arnold, Springer, 92Elementary Topics in Differential Geometry - J. A. Thorpe, Springer 2004.Nonholonomic Mechanics and Control - A. M. Bloch, Springer, 2003A Mathematical Introduction to Robot Manipulation and Control - R.Murray, Z. Li and S. Sastry, CRC Press, 1992Control Theory from the Geometric Viewpoint - A. Agrachev and Y.Sachkov, Springer, 2004.


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Parallel parking, falling cats, spherical robots and differential geometry€¦ · A Comprehensive Introduction to Di erential Geometry - M. Spivak, Publish or Perish Inc., 1979