© Stefano Stramigioli II Geoplex-EURON Summer School Bertinoro, (I) 18-22 July 2005 Mathematical Background (1h)

© Stefano Stramigioli

II Geoplex-EURON Summer SchoolII Geoplex-EURON Summer School Bertinoro, (I) 18-22 July 2005 Bertinoro, (I) 18-22 July 2005

Mathematical BackgroundMathematical Background(1h)(1h)

04/18/23 II EURON/Geoplex Summer School 2


Goal

To give a basic and INTUITIVE background of the tools which will be used for the school and the understanding of basic concepts of physics and robotics.



General Remarks

• Stop me at ANY time if something is not clear!!

• The treatment will NOT be mathematically precise but yields to convey intuition

• Hopefully you will appreciate concepts which are very often discarded in basic courses, but that they are ESSENTIAL in engineering sciences (i.e.tensors).



• More meant as a reference, do not concentrate on the details!!!

• Most of the material is not ALL essential for understanding, but it usually HELPs in focusing the robotics concepts and understand them deeply



Contents

• Vectors and Co-Vectors• Higher Order Tensors

•Linear maps, Quadratic Forms..• Intuition of Differential Geometric Concepts

• manifolds, (co-)tangent spaces • Groups• Lie-Groups



Vectors and Co-vectorsVectors and Co-vectors



Vector Space (finite dimensional)

• A real vector space is characterized by– An origin– Elements can be scaled by any

real number and still belong to the vector space:

– You can take a linear combination of elements and get again an element in the vector space



From a Vector Space to

Change of Change of basebase

basebase11

basebase22



Co-vectors

Consider the set of LINEAR operators:

Once a base has been chosen, it can Once a base has been chosen, it can be seen that numerically they are be seen that numerically they are represented by an n-dim row vector:represented by an n-dim row vector:

==



Changing coordinates

Te vector space of this linear Te vector space of this linear operators, called dual space is operators, called dual space is indicated with .indicated with .Consider the velocity of a point Consider the velocity of a point mass as an element of a 3D vector mass as an element of a 3D vector spacespace

A force applied to this mass will A force applied to this mass will transfer power to the mass. transfer power to the mass. The The value of power is a scalar value of power is a scalar INDEPENDENT of the coordinate INDEPENDENT of the coordinate choice.choice.



Changing coordinates

Coord 1Coord 1 Coord 2Coord 2

Consider the change of base such Consider the change of base such thatthat

Different!Different!!!



Co-vectors

•Co-vectors are also represented by Co-vectors are also represented by numerical arrays once a base is chosen, numerical arrays once a base is chosen, but they are different than vectors since but they are different than vectors since they transform diferently !!they transform diferently !!

•A A velocityvelocity is a is a vectorvector

•A A forceforce is a is a co-vectorco-vector NOT a vector !! NOT a vector !!

•A force is a linear operator from A force is a linear operator from velocity to powervelocity to power



Examples

• Vectors– Velocities– Twists

• Co-vectors– Forces– Wrenches



TensorsTensors



Tensors

A tensor of typeis defined as a multi-linear operator of

the following form:

VectoVectorr

Co-vectorCo-vector

Order 1(=p+q) tensors areOrder 1(=p+q) tensors are



Matrices and 2nd order tensors

What does a matrix represent ? 4 options:

- Map (linear map)- Map (quadratic form)- Map (linear map on dual

space)- Map (quadratic form on d.s.)



Linear Map

We are used to think of a linear map We are used to think of a linear map asas

But can be seen as a l.m. mapBut can be seen as a l.m. map



Linear Map

Change of coord of vectorsChange of coord of vectors



Quadratic Form (i.e. Kinetic energy)

Change of coord of vectorsChange of coord of vectors



Makes sense to take eigen-values of a matrix ?

vectovectorr

vectovectorr

Linear operator!!Linear operator!!

If A had been a quadratic form like an If A had been a quadratic form like an inertia matrix, it would not mean inertia matrix, it would not mean anything to take eigenvalues: other anything to take eigenvalues: other coordinates would give different coordinates would give different values!!values!!



Conclusions

• Vectors transform differently than co-vectors changing coordinates

• Linear maps transform differently than quadratic forms changing coordinates

• Always think about the kind of objects you are working with (Einstein notation !!)

• Be aware of nonsense like eigenvalues of an inertia matrix!



Examples

• (1 1) Tensors– Mechanisms transformation matrix

• (0 2) Tensors– Inertia matrices, Inertial elipsoids,

point mass, stiffness matrices at equilibrium, symplectic structure, any metric (!!)

• (2 0) Tensors– Poisson structure



Remarks

• Frames are an artifact: physics is not based on coordinates

• Be aware of `user metric choices’ !!– Manipulability index– Pseudo-inverses– Hybrid Position-Force Control

•No metric in se(3) !!

….



Pure Contra-variant tensors: Type (0 q)



Skew Symmetric Tensor

Since all arguments of this linear function Since all arguments of this linear function now are the same, we can ask wethear now are the same, we can ask wethear commuting them gives the same or commuting them gives the same or different value:different value:

=?=?



Skew Symmetric tensor

IfIf

We call the tensor We call the tensor skew-symmetric of skew-symmetric of q-formq-form



Example

• q=2: Skew-simmetric matrix q=3: Skew-simmetric “cube” …..



Volume

• A skew-symmetric (0 q) tensor is used to measure the volume of the q-dim parallelepiped scanned by its arguments:

++

Volume!Volume!



Example in 2D

++



ManifoldsManifolds



Manifold

Diff. Diff. on on intersintersectionection



Tangent Spaces



Co-Tangent spaces

The vector space of co-vectors at The vector space of co-vectors at each configuration is indicated each configuration is indicated withwith



Examples

• Configuration space of a manipulator• Relative configurations of bodies• Line segment• Earth surface• Space• …



GroupsGroups



Group

A group is a set and an operation A group is a set and an operation for whichfor which

AssociativityAssociativity

•IdentityIdentity

•Inverse Inverse



Examples

• Set of nonsingular matrices with matrix product operation

• Flow of a differential equation• Object motions• Quaternions• …



Lie GroupsLie Groups



What is a Lie-Group

• A Lie group is a ``manifold group’’– It is smooth– It is a group

• The tangent space in the identity is an algebra (has a skew symmetric operation) called Lie algebra



Lie Group

Lie AlgebraLie Algebra

Lie GroupLie Group



Lie Groups

Common Space thanks Common Space thanks to Lie group structureto Lie group structure



Examples

• Space of Rotation matrices• Space of Homogeneous matrices• Space of “Abstract” rotations• Space of “Abstract” motions• Unit Quaternions• …



Intuition of Differential Intuition of Differential FormsForms



Tensor Fields

• Given a n-dimensional manifold, we can associate to each point:– A value in R (function on a manifold)– A vector in its tangent space (vector

field)– A covector in its co-tangent space

(co-vector field)– A n-form– …



Integration on manifold

• If we have a smooth n-form on a n-dim manifold, we can integrate it on the manifold after having created infinitesimal parallelograms!

© Stefano Stramigioli II Geoplex-EURON Summer School Bertinoro, (I) 18-22 July 2005 Mathematical Background (1h)

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