Modeling of Robotics Systems (2:30h)

Modeling of Robotics Systems (2:30h)

Why Geometrical Robotics?
In 3D, especially due to the topology of rotations, geometry is important. Coordinates can bring you to WRONG conclusions, tensors (geometry) NOT ! Complex systems need Powerful Tools to be manageable With the right tools 3D can be as simple as 1D ! 11/20/2018 Semester on Control, Barcelona 2005

Semester on Control, Barcelona 2005
Contents 1D Mechanics: as introduction 3D Mechanics Points, vectors, line vectors screws Rotations and Homogeneous matrices Screw Ports 11/20/2018 Semester on Control, Barcelona 2005

1D Mechanics

1D Mechanics In 1D Mechanics there is no geometry for the ports: efforts/Forces and flows/velocities are scalars Starting point to introduce the basic elements for 3D 11/20/2018 Semester on Control, Barcelona 2005

Mass Energy Co-Energy where is the momenta the applied force and its velocity. 11/20/2018 Semester on Control, Barcelona 2005

The dynamics Equations
The second Law of dynamics is: Integral Form Diff. form 11/20/2018 Semester on Control, Barcelona 2005

The Kernel PCH representation
Interconnection port 11/20/2018 Semester on Control, Barcelona 2005

Spring Energy Co-Energy where is the displacement the applied force to the spring and its relative velocity. 11/20/2018 Semester on Control, Barcelona 2005

The dynamics Equations
The elastic force on the spring is: Integral Form Diff. form 11/20/2018 Semester on Control, Barcelona 2005

The Kernel PCH representation
Interconnection port 11/20/2018 Semester on Control, Barcelona 2005

Mass-Spring System Spring Mass 11/20/2018 Semester on Control, Barcelona 2005

Together…. 11/20/2018 Semester on Control, Barcelona 2005

Interconnection of the two subsystems (1 junc.)
Or in image representation 11/20/2018 Semester on Control, Barcelona 2005

Combining… There exists a left orthogonal 11/20/2018 Semester on Control, Barcelona 2005

Finally 11/20/2018 Semester on Control, Barcelona 2005

Summary and Conclusions
All possible 1D networks of elements can be expressed in this form Dissipation can be easily included terminating a port on a dissipating element Interconnection of elements still give the same form No geometry In 3D we need a common language for interconnection: Lie groups ! 11/20/2018 Semester on Control, Barcelona 2005

The Basics on Space

Euclidian Spaces and motions
A 3-dimensional Euclidean space is characterized by a scalar product We can consider the relative position of as a vector : The set of these free-vectors will be indicateds with 11/20/2018 Semester on Control, Barcelona 2005

Notation and concepts Scalar Product Norm (distance) Orthogonality of 11/20/2018 Semester on Control, Barcelona 2005

Notation and concepts (cont.)
Angles between vectors Conclusion All concepts defined using only the scalar product. 11/20/2018 Semester on Control, Barcelona 2005

Coordinate Systems A coordinate system is composed of an origin and 3 linear independent free vectors: 11/20/2018 Semester on Control, Barcelona 2005

Coordinate Systems (cont.)
A Coordinate System is ortho-normal iff 11/20/2018 Semester on Control, Barcelona 2005

Coordinates Coordinates are real numbers: For (points) For (vectors) We will use often the following notation: 11/20/2018 Semester on Control, Barcelona 2005

Coordinate Mappings 11/20/2018 Semester on Control, Barcelona 2005

Change of coordinates A change of coordinates from to can be expressed with: Which is a mapping like 11/20/2018 Semester on Control, Barcelona 2005

Orientation The vector product operation often used in mechanics, is NOT as often thought an extra structure/operation defined, but it is only a consequence of the Lie group structure of motions: 11/20/2018 Semester on Control, Barcelona 2005

Tilde Operator (Notation)
We will often use the following notation In 20-SIM Y=Skew(X) 11/20/2018 Semester on Control, Barcelona 2005

Rotations

Approach We will first introduce change of coordinates for rotated frames 2D 3D Relate changes of coordinates to motion of objects 11/20/2018 Semester on Control, Barcelona 2005

Rotations in 2D Given What is the relation of the coordinates of in and in ? 11/20/2018 Semester on Control, Barcelona 2005

We have that Or equivalently With , Similarly, for we have 11/20/2018 Semester on Control, Barcelona 2005

and since , using the expression of in , we have that 11/20/2018 Semester on Control, Barcelona 2005

Conclusion Rotation Matrix 11/20/2018 Semester on Control, Barcelona 2005

Remarks Due to the fact we are considering right-handed orthonormal frames: The columns and row vectors of have length 1 and they are orthogonal 11/20/2018 Semester on Control, Barcelona 2005

3D Rotation In 3D, if the coordinates we use are all right handed, we have for a general rotation around the origin: Basis of espressed in Basis of espressed in 11/20/2018 Semester on Control, Barcelona 2005

Example A rotation around the y axis would be 11/20/2018 Semester on Control, Barcelona 2005

Special Orthonormal Group
A square matrix such that is called orthonormal. The group of orthonormal matrices with determinant 1 is called Special Orthonormal group of and indicated as: 11/20/2018 Semester on Control, Barcelona 2005

Theorem If is a differentiable function of time, and are Skew-Symmetric and belonging to : 11/20/2018 Semester on Control, Barcelona 2005

Angular velocities This implies that such that: Remember: 11/20/2018 Semester on Control, Barcelona 2005

is a Lie algebra The linear combination of skew-symmetric matrices is still skew-symmetric To each matrix we can associate a vector such that … It is a vector space It is a Lie Algebra !! 11/20/2018 Semester on Control, Barcelona 2005

SO(3) is a Group It is a Group because Associativity Identity Inverse 11/20/2018 Semester on Control, Barcelona 2005

It is a Lie Group (group AND manifold)
where Lie Algebra Commutator 11/20/2018 Semester on Control, Barcelona 2005

Common Space thanks to Lie group structure
11/20/2018 Semester on Control, Barcelona 2005

Why is the Lie group structure so important?
It makes it possible to talk about motion without knowing the pose of the object! We can get rid of any dependency from configuration: ESSENTIAL to talk about interconnection throw Power Ports of bodies with different poses. 11/20/2018 Semester on Control, Barcelona 2005

Dual Space For any finite dimensional vector space we can define the space of linear operators from that space to co-vector The space of linear operators from to (dual space of ) is indicated with 11/20/2018 Semester on Control, Barcelona 2005

Configuration Independent Port !
In our case we have Configuration Independent Port ! 11/20/2018 Semester on Control, Barcelona 2005

Other Rotations Representations
Generalized angles: Euler, Bryan etc. (singularity positions and NOT geometric Quaternions: 11/20/2018 Semester on Control, Barcelona 2005

Other Rotations Representations
matrix Lie group of unitary matrices of determinant +1 Unit Quaternions and are identifiable with points of a 3-sphere 11/20/2018 Semester on Control, Barcelona 2005

Double Covering of The quaternions and double cover Quaternions and are simply connected is connected but NOT simply connected: see the Dirac Belt Trick ! is one of the few examples of a NON simply connected manifolds WITHOUT holes! 11/20/2018 Semester on Control, Barcelona 2005

About the topology of rotations

General Motions

General Motions It can be seen that in general, for right handed frames where , 11/20/2018 Semester on Control, Barcelona 2005

Homogeneous Matrices Due to the group structure of it is easy to compose changes of coordinates in rotations Can we do the same for general motions ? 11/20/2018 Semester on Control, Barcelona 2005

Projective space Improper hyperplane 11/20/2018 Semester on Control, Barcelona 2005

Motions versus Coordinate Changes
A rigit motion is a map like: 11/20/2018 Semester on Control, Barcelona 2005

Features of motions must be both: Isometry: it does not change the distance between points. Orientation preserving: it does not change the orientation: swap the z axis. 11/20/2018 Semester on Control, Barcelona 2005

Connection with coordinate changes
we want that the moved points in the moved coordinate system have the same coordinates than the original points in the original coordinate system : 11/20/2018 Semester on Control, Barcelona 2005

But is the same as: The representation of the motion from 1 to 2 in the coordinate 1 (1 o h o 1-1) is the INVERSE of the coordinate change from 1 to 2 (2 o 1-1). 11/20/2018 Semester on Control, Barcelona 2005

Theorem If is a differentiable function of time belong to where 11/20/2018 Semester on Control, Barcelona 2005

Tilde operator In 20-SIM Y=Tilde(X) 11/20/2018 Semester on Control, Barcelona 2005

Elements of se(3): Twists
The following are vector and matrix coordinate notations for twists: The following are often called twists too, but they are no geometrical entities ! 11/20/2018 9 change of coordinates ! Semester on Control, Barcelona 2005

SE(3) is a Group It is a Group because Associativity Identity Inverse 11/20/2018 Semester on Control, Barcelona 2005

SE(3) is a Lie Group (group AND manifold)
where Lie Algebra Commutator 11/20/2018 Semester on Control, Barcelona 2005

Common Space independent on H thanks to Lie group structure

Intuition of Twists Consider a point fixed in : and consider a second reference where and 11/20/2018 Semester on Control, Barcelona 2005

For each of the points in the body

Possible Choices For the twist of with respect to we consider and we have 2 possibilities 11/20/2018 Semester on Control, Barcelona 2005

Left and Right Translations

Remarks The twists are a description independent of H and have a pure geometrical interpretation. Being independent on configuration, they will be the key to interconnection between parts with different configurations 11/20/2018 Semester on Control, Barcelona 2005

Possible Choices and 11/20/2018 Semester on Control, Barcelona 2005

Notation used for Twists
For the motion of body with respect to body expressed in the reference frame we use or The twist is an across variable ! Point mass geometric free-vector Rigid body geometric screw + Magnitude 11/20/2018 Semester on Control, Barcelona 2005

Chasle's Theorem and intuition of a Twist
Any twist can be written as: 11/20/2018 Semester on Control, Barcelona 2005

Calculation of geometrical screw from (,v)
Supposing r perpendicular to omega , it is furthermore possible to see that: and 11/20/2018 Semester on Control, Barcelona 2005

Examples of Twists 11/20/2018 Semester on Control, Barcelona 2005

Changes of Coordinates for Twists
It can be proven that In 20-SIM S=Adjoint(H) 11/20/2018 Semester on Control, Barcelona 2005

… in Matrix form 11/20/2018 Semester on Control, Barcelona 2005

Wrenches Twists belong geometrically to Wrenches are DUAL of twist: Wrenches are co-vectors and NOT vectors: linear operators from Twists to Power Using coordinates: 11/20/2018 Semester on Control, Barcelona 2005

Poinsot's Theorem and intuition of a Wrench
Any wrench can be written as: 11/20/2018 Semester on Control, Barcelona 2005

Chasles vs. Poinsot Charles Theorem Poinsot Theorem The inversion of the upper and lower part corresponds to the use of the Klijn form 11/20/2018 Semester on Control, Barcelona 2005

Vectors, Screws as “Forces”
Forces and Wrenches are co-vectors, but: Euclidean metric vector interpretation of a Force Klein’s form screw interpretation of a Wrench That is identification of dual spaces. 11/20/2018 Semester on Control, Barcelona 2005

Example of the use of a Wrench
Finding the contact centroid 11/20/2018 Semester on Control, Barcelona 2005

Transformation of Wrenches
How do wrenches transform changing coordinate systems? We have seen that for twists: 11/20/2018 Semester on Control, Barcelona 2005

Changes of coordinates
MTF 11/20/2018 Semester on Control, Barcelona 2005

Implicit relation Dirac Structure! MTF 11/20/2018 Semester on Control, Barcelona 2005

Conclusions Coordinate free interpretation of motion can be related to twists, element of the Lie albebra se(3) Dual elements are called wrenches and they also have a coordinate free interpretation thanks to the Hyperbolic form. The pairing of this two items will be the kern for interconnection 11/20/2018 Semester on Control, Barcelona 2005

Power Port A B belong to vector spaces in duality: such that there exists a bilinear operator 11/20/2018 Semester on Control, Barcelona 2005

Finite dimensional case
If is finite dimensional is uniquely defined, namely where indicates the uniquely defined set of linear operators from to Elements of are vectors Elements of are co-vectors 11/20/2018 Semester on Control, Barcelona 2005

In Robotics Is the v.s. of Twists Is the v.s. of Wrenches 11/20/2018 Semester on Control, Barcelona 2005

Remarks A geometrical (coordinate free) way to talk about instantaneous motion is using a Twist and about a system of forces is a Wrench Twists and Wrenches are geometrically a screw (motor): line-bounded-vector (rotor)+pitch. Twists and Wrenches are dual and can be used to define a power-port 11/20/2018 Semester on Control, Barcelona 2005

Power and Inf. Dim Case A B represents the instantaneous power flowing from A to B For inf.dim. systems they belong to k and (n-k) (Lie-algebra-valued) forms 11/20/2018 Semester on Control, Barcelona 2005

Dynamics

Contents time derivative Rigid Body dynamics Spatial Springs Kinematic Pairs Mechanism Topology 11/20/2018 Semester on Control, Barcelona 2005

time derivative is function of time It can be proven that where 11/20/2018 Semester on Control, Barcelona 2005

Transformations of If we have ,how does look like? 11/20/2018 Semester on Control, Barcelona 2005

It can be shown that in general

Rigid Bodies Dynamics

Rigid bodies A rigid Body is characterised by a (0,2) tensor called Inertia Tensor: and we can then define the momentum screw: where the Kinetic energy is 11/20/2018 Semester on Control, Barcelona 2005

Generalization of Newton’s law
In an inertial frame, for a point mass we had This can be generalized for rigid bodies Where Ni0 is the moment of body i expressed in the inertial frame 0 . That is why momenta is a co-vector !! 11/20/2018 Semester on Control, Barcelona 2005

And in body coordinates ?
Using the derivative of AdH 11/20/2018 Semester on Control, Barcelona 2005

….. multiplying on the left for we get 11/20/2018 Semester on Control, Barcelona 2005

and since we have that and we eventually obtain 11/20/2018 Semester on Control, Barcelona 2005

Momentum dynamics which is called Lie-Poisson reduction. NOTE: No information on configuration ! 11/20/2018 Semester on Control, Barcelona 2005

Screw Port representation (Bondgraph)
=0 ! stored input 11/20/2018 Semester on Control, Barcelona 2005

Relation to what known Expressing the previous equation in the principal inertial frame k we obtain: 11/20/2018 Semester on Control, Barcelona 2005

Other form Defining which is linear and anti-symmetric 11/20/2018 Semester on Control, Barcelona 2005

Port-Hamiltonian form

Port-Hamiltonian form
Modulation Storage port Interconnection port 11/20/2018 Semester on Control, Barcelona 2005

Rotational Dynamics in more details

Example: Ellipsoid Dynamics
Semi-axis lengths= 11/20/2018 Semester on Control, Barcelona 2005

Example: Elipsoid Dynamics (cont.)
In SI Units: 11/20/2018 Semester on Control, Barcelona 2005

If no external torque is applied
Note: in what follows P indicates angular momenta, NOT homogeneous position If no torques are applied: Square of angular momentum in body coordinate is also conserved!! 11/20/2018 Semester on Control, Barcelona 2005

Momentum conservation
These are the equation of a sphere in the body coordinate momentum frame 11/20/2018 Semester on Control, Barcelona 2005

Energy conservation Ellipsoid! 11/20/2018 Semester on Control, Barcelona 2005

Orbits of the momentum Body coords!! Conservation of momentum Casimir ! sphere Conservation of energy ellipsoid 11/20/2018 Semester on Control, Barcelona 2005

Orbit Sphere With the intersection of the sphere and ellipsoid we obtain the famous plot: 11/20/2018 Semester on Control, Barcelona 2005

Rotation around the biggest inertia

Rotation around middle inertia

Rotation around smallest inertia

Geometric Springs

4 cells: 1 stable+3 unstable points
Spatial Springs If, by means of control, we define a 3D spring using a parameterization like Euler angles, we do not have a geometric description of the spring: no information about the center of compliance, instead: Morse Theory 4 cells: 1 stable+3 unstable points 11/20/2018 Semester on Control, Barcelona 2005

Spatial Springs where where 11/20/2018 Semester on Control, Barcelona 2005

For Constant Spatial Spring
It could be shown that: Storage port Interconnection port to integrate! 11/20/2018 Semester on Control, Barcelona 2005

Kinematic Pairs

Kinematic Pair A n-dof K.P. is an ideal constraint between 2 rigid bodies which allows n independent motions For each relative configuration of the bodies we can define Allowed subspace of dimension n Actuation subspace of dimension n 11/20/2018 Semester on Control, Barcelona 2005

Decomposition of and ! n n 6-n 6-n 11/20/2018 Semester on Control, Barcelona 2005

Representations of subspaces
To satisfy power continuity 11/20/2018 Semester on Control, Barcelona 2005

And in the Kernel Dirac representation
Interconnection port Actuators ports 11/20/2018 Semester on Control, Barcelona 2005

Mechanism Topology

Network Topology Interconnection of q rigid bodies by n nodic elements (kinematic pairs, springs or dampers). We can deﬁne the Primary Graph describing the mechanism and than: Port connection graph= Lagrangian tree + Primary Graph 11/20/2018 Semester on Control, Barcelona 2005

Primary Graph 11/20/2018 Semester on Control, Barcelona 2005

Primary Graph The Primary graph is characterised by the Incedence Matrix 11/20/2018 Semester on Control, Barcelona 2005

Lagrangian Tree 11/20/2018 Semester on Control, Barcelona 2005

Fundamental Loop Matrix
Lagrangian Tree Primary Graph 11/20/2018 Semester on Control, Barcelona 2005

Fundamental Cut-set Matrix
Lagrangian Tree Primary Graph 11/20/2018 Semester on Control, Barcelona 2005

`Power Continuity Power continuity ! 11/20/2018 Semester on Control, Barcelona 2005

Mechanism Dirac Structure
Power Ports Rigid Bodies Power Ports Nodic Elements 11/20/2018 Semester on Control, Barcelona 2005

Further Steps… 11/20/2018 Semester on Control, Barcelona 2005

Conclusions Any 3D part can be modeled in the Dirac framework Any interconnection also ! In this case the ports have a geometrical structures: no scalars ! Some steps still to go to bring the system in explicit form A lot of extensions are possible Not trivial to bring everything in simplified explicit form 11/20/2018 Semester on Control, Barcelona 2005

Modeling of Robotics Systems (2:30h)

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Modeling of Robotics Systems (2:30h)

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