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

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School2 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.

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School3 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).

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School4 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

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School5 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 School Bertinoro, (I) July 2005 Vectors and Co-vectors

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School7 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

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School8 From a Vector Space to Change of base base1 base2

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School9 Co-vectors Consider the set of LINEAR operators: Once a base has been chosen, it can be seen that numerically they are represented by an n-dim row vector: =

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School10 Changing coordinates Te vector space of this linear operators, called dual space is indicated with. Consider the velocity of a point mass as an element of a 3D vector space A force applied to this mass will transfer power to the mass. The value of power is a scalar INDEPENDENT of the coordinate choice.

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School11 Changing coordinates Coord 1 Coord 2 Consider the change of base such that Different!!

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School12 Co-vectors Co-vectors are also represented by numerical arrays once a base is chosen, but they are different than vectors since they transform diferently !!Co-vectors are also represented by numerical arrays once a base is chosen, but they are different than vectors since they transform diferently !! A velocity is a vectorA velocity is a vector A force is a co-vector NOT a vector !!A force is a co-vector NOT a vector !! A force is a linear operator from velocity to powerA force is a linear operator from velocity to power

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School13 Examples Vectors –Velocities –Twists Co-vectors –Forces –Wrenches

© Stefano Stramigioli II Geoplex-EURON Summer School Bertinoro, (I) July 2005 Tensors

6/10/2015II EURON/Geoplex Summer School15 Tensors A tensor of type is defined as a multi-linear operator of the following form: Vector Co-vector Order 1(=p+q) tensors are

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School16 Matrices and 2 nd 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.)

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School17 Linear Map We are used to think of a linear map as But can be seen as a l.m. map

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School18 Linear Map Change of coord of vectors

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School19 Quadratic Form (i.e. Kinetic energy) Change of coord of vectors

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School20 Makes sense to take eigen-values of a matrix ? vector vector Linear operator!! If A had been a quadratic form like an inertia matrix, it would not mean anything to take eigenvalues: other coordinates would give different values!!

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School21 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!

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School22 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

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School23 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) !! ….

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School24 Pure Contra-variant tensors: Type (0 q)

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School25 Skew Symmetric Tensor Since all arguments of this linear function now are the same, we can ask wethear commuting them gives the same or different value: =?

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School26 Skew Symmetric tensor If We call the tensor skew-symmetric of q- form

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School27 Example q=2: Skew-simmetric matrix q=3: Skew-simmetric “cube” …..

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School28 Volume A skew-symmetric (0 q) tensor is used to measure the volume of the q-dim parallelepiped scanned by its arguments: + Volume!

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School29 Example in 2D +

© Stefano Stramigioli II Geoplex-EURON Summer School Bertinoro, (I) July 2005 Manifolds

6/10/2015II EURON/Geoplex Summer School31 Manifold Diff. on intersec tion

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School32 Tangent Spaces

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School33 Co-Tangent spaces The vector space of co-vectors at each configuration is indicated with

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School34 Examples Configuration space of a manipulator Relative configurations of bodies Line segment Earth surface Space …

© Stefano Stramigioli II Geoplex-EURON Summer School Bertinoro, (I) July 2005 Groups

6/10/2015II EURON/Geoplex Summer School36 Group A group is a set and an operation for which Associativity IdentityIdentity InverseInverse

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School37 Examples Set of nonsingular matrices with matrix product operation Flow of a differential equation Object motions Quaternions …

© Stefano Stramigioli II Geoplex-EURON Summer School Bertinoro, (I) July 2005 Lie Groups

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School39 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

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School40 Lie Group Lie Algebra Lie Group

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School41 Lie Groups Common Space thanks to Lie group structure

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School42 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 School Bertinoro, (I) July 2005 Intuition of Differential Forms

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School44 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 –…

II Geoplex-EURON Summer School Bertinoro, (I) July /10/2015II EURON/Geoplex Summer School45 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!