© stefano stramigioli ii geoplex-euron summer school bertinoro, (i) 18-22 july 2005 mathematical...
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© 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
II Geoplex-EURON Summer SchoolII Geoplex-EURON Summer School Bertinoro, (I) 18-22 July 2005 Bertinoro, (I) 18-22 July 2005
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.
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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).
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• 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
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Contents
• Vectors and Co-Vectors• Higher Order Tensors
•Linear maps, Quadratic Forms..• Intuition of Differential Geometric Concepts
• manifolds, (co-)tangent spaces • Groups• Lie-Groups
© Stefano Stramigioli
II Geoplex-EURON Summer SchoolII Geoplex-EURON Summer School Bertinoro, (I) 18-22 July 2005 Bertinoro, (I) 18-22 July 2005
Vectors and Co-vectorsVectors and Co-vectors
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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
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From a Vector Space to
Change of Change of basebase
basebase11
basebase22
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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:
==
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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.
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Changing coordinates
Coord 1Coord 1 Coord 2Coord 2
Consider the change of base such Consider the change of base such thatthat
Different!Different!!!
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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
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Examples
• Vectors– Velocities– Twists
• Co-vectors– Forces– Wrenches
© Stefano Stramigioli
II Geoplex-EURON Summer SchoolII Geoplex-EURON Summer School Bertinoro, (I) 18-22 July 2005 Bertinoro, (I) 18-22 July 2005
TensorsTensors
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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
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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.)
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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
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Linear Map
Change of coord of vectorsChange of coord of vectors
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Quadratic Form (i.e. Kinetic energy)
Change of coord of vectorsChange of coord of vectors
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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!!
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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!
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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
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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) !!
….
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Pure Contra-variant tensors: Type (0 q)
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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:
=?=?
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Skew Symmetric tensor
IfIf
We call the tensor We call the tensor skew-symmetric of skew-symmetric of q-formq-form
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Example
• q=2: Skew-simmetric matrix q=3: Skew-simmetric “cube” …..
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Volume
• A skew-symmetric (0 q) tensor is used to measure the volume of the q-dim parallelepiped scanned by its arguments:
++
Volume!Volume!
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Example in 2D
++
© Stefano Stramigioli
II Geoplex-EURON Summer SchoolII Geoplex-EURON Summer School Bertinoro, (I) 18-22 July 2005 Bertinoro, (I) 18-22 July 2005
ManifoldsManifolds
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Manifold
Diff. Diff. on on intersintersectionection
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Tangent Spaces
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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
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Examples
• Configuration space of a manipulator• Relative configurations of bodies• Line segment• Earth surface• Space• …
© Stefano Stramigioli
II Geoplex-EURON Summer SchoolII Geoplex-EURON Summer School Bertinoro, (I) 18-22 July 2005 Bertinoro, (I) 18-22 July 2005
GroupsGroups
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Group
A group is a set and an operation A group is a set and an operation for whichfor which
AssociativityAssociativity
•IdentityIdentity
•Inverse Inverse
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Examples
• Set of nonsingular matrices with matrix product operation
• Flow of a differential equation• Object motions• Quaternions• …
© Stefano Stramigioli
II Geoplex-EURON Summer SchoolII Geoplex-EURON Summer School Bertinoro, (I) 18-22 July 2005 Bertinoro, (I) 18-22 July 2005
Lie GroupsLie Groups
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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
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Lie Group
Lie AlgebraLie Algebra
Lie GroupLie Group
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Lie Groups
Common Space thanks Common Space thanks to Lie group structureto Lie group structure
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Examples
• Space of Rotation matrices• Space of Homogeneous matrices• Space of “Abstract” rotations• Space of “Abstract” motions• Unit Quaternions• …
© Stefano Stramigioli
II Geoplex-EURON Summer SchoolII Geoplex-EURON Summer School Bertinoro, (I) 18-22 July 2005 Bertinoro, (I) 18-22 July 2005
Intuition of Differential Intuition of Differential FormsForms
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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– …
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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!