Transcript Summer School 2003, Bertinoro

Summer School 2003, Bertinoro (I)
Dirac Framework
for
Robotics
Tuesday, July 8th, (4 hours)
Stefano Stramigioli
© Stefano Stramigioli
Summer School 2003, Bertinoro (I)
Contents
• 1D Mechanics: as introduction
• 3D Mechanics
– Points, vectors, line vectors screws
– Rotations and Homogeneous matrices
– Screw Ports
– Rigid Body Kinematics and Dynamics
– Springs
– Interconnection and Mechanisms
Dynamics
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1D Mechanics
© Stefano Stramigioli
Summer School 2003, Bertinoro (I)
1D Mechanics
• In 1D Mechanics there is no geometry for
the ports: efforts/Forces and
flows/velocities are scalar
• Starting point to introduce the basic
elements for 3D
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Mass
Energy
where is the momenta
and
its velocity.
Co-Energy
the applied force
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The dynamics Equations
The second Law of dynamics is:
Integral Form
Diff. form
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The Kernel PCH representation
Interconnection port
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Spring
Energy
Co-Energy
where is the displacement
the applied
force to the spring and
its relative
velocity.
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The dynamics Equations
The elastic force on the spring is:
Integral Form
Diff. form
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The Kernel PCH representation
Interconnection port
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Mass-Spring System
• Spring
• Mass
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Together….
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Interconnection of the two subsystems (1 junc.)
Or in image representation
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Combining…
There exists a left orthogonal
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Finally
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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
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3D Mechanics
© Stefano Stramigioli
Summer School 2003, Bertinoro (I)
Notation
Set of points in Euclidean Space
Free Vectors in Euclidean Space
Right handed coordinate frame I
Coordinate mapping associated to
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Rotations
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Rotations
It can be seen that if
purely rotated
and
are
where
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Theorem
If
of time
is a differentiable function
are skew-symmetric and belong to
:
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Tilde operator
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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 !!
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SO(3) is a Group
It is a Group because
• Associativity
•Identity
•Inverse
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It is a Lie Group (group AND manifold)
•
•
where
•
where
• Lie Algebra Commutator
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Lie Groups
Common Space thanks to
Lie group structure
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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
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In our case we have
Configuration Independent Port !
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General Motion
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General Motions
It can be seen that in general, for right
handed frames
where
,
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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 ?
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SE(3)
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Theorem
If
of time
belong to
is a differentiable function
where
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Tilde operator
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Summer School 2003, Bertinoro (I)
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 !
9 change of coordinates !
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Summer School 2003, Bertinoro (I)
SE(3) is a Group
It is a Group because
• Associativity
•Identity
•Inverse
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Summer School 2003, Bertinoro (I)
SE(3) is a Lie Group (group AND manifold)
•
•
where
•
where
• Lie Algebra Commutator
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Summer School 2003, Bertinoro (I)
Lie Groups
Common Space thanks to
Lie group structure
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Summer School 2003, Bertinoro (I)
Intuition of Twists
Consider a point
fixed in
:
and consider a second reference
where
and
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Summer School 2003, Bertinoro (I)
Possible Choices
For the twist of
we consider
possibilities
with respect to
and we have 2
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Left and Right Translations
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Possible Choices
and
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Summer School 2003, Bertinoro (I)
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
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Chasle's Theorem and intuition of a Twist
Any twist can be written as:
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Examples of Twists
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Examples of Twists
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Changes of Coordinates for Twists
• It can be proven that
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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:
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Poinsot's Theorem and intuition of a Wrench
Any wrench can be written as:
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Chasles vs. Poinsot
Charles Theorem
Poinsot Theorem
The inversion of the upper and lower part
corresponds to the use of the Klijn form
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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.
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Example of the use of a Wrench
Finding the contact centroid
Contact Point
(Center of Pressure)
Measured
Wrench
6D sensor
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Transformation of Wrenches
• How do wrenches transform changing
coordinate systems? We have seen that for
twists:
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Changes of coordinates
MTF
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In Dirac Kernel form
MTF
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Power Port
A
B
belong to vector spaces in duality:
such that there exists a bilinear operator
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Summer School 2003, Bertinoro (I)
Finite dimensional case
• If
is finite dimensional
defined, namely
is uniquely
where
indicates the uniquely defined
set of linear operators from
to
Elements of
are vectors
Elements of
are co-vectors
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Summer School 2003, Bertinoro (I)
In Robotics
Is the v.s. of Twists
Is the v.s. of Wrenches
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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
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Dynamics
© Stefano Stramigioli
Summer School 2003, Bertinoro (I)
Contents
•
time derivative
•
•
•
•
Rigid Body dynamics
Spatial Springs
Kinematic Pairs
Mechanism Topology
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Summer School 2003, Bertinoro (I)
time derivative
•
is function of time
• It can be proven that
where
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Transformations of
If we have
like?
,how does
look
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Summer School 2003, Bertinoro (I)
It can be shown that in general
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Rigid Bodies Dynamics
© Stefano Stramigioli
Summer School 2003, Bertinoro (I)
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
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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
expressed in the inertial frame 0
.
That is why momenta is a co-vector !!
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And in body coordinates ?
Using the derivative of AdH
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…..
multiplying on the left for
we get
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Summer School 2003, Bertinoro (I)
and since
we have that
and we eventually obtain
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Momentum dynamics
which is called Lie-Poisson reduction.
NOTE: No information on configuration !
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Other form
Defining
which is linear and anti-symmetric
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Port-Hamiltonian form
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Port-Hamiltonian form
Modulation
Storage port
Interconnection port
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Geometric Springs
© Stefano Stramigioli
Summer School 2003, Bertinoro (I)
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
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Summer School 2003, Bertinoro (I)
Spatial Springs
where
where
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For Constant Spatial Spring
It could be shown that:
Storage port
to integrate!
Interconnection port
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Parametric Changes (Scalar Case)
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Variable Spatial Springs (Geometric Case)
Length Variation
Body
1
Variation RCC
Body
2
It can be shown that varying RCC does
NOT exchange energy !!
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Kinematic Pairs
© Stefano Stramigioli
Summer School 2003, Bertinoro (I)
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
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Decomposition of
and
!
n
n
6-n
6-n
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Representations of subspaces
To satisfy power
continuity
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And in the Kernel Dirac representation
Interconnection port
Actuators ports
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Mechanism Topology
© Stefano Stramigioli
Summer School 2003, Bertinoro (I)
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
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Summer School 2003, Bertinoro (I)
Primary Graph
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Summer School 2003, Bertinoro (I)
Primary Graph
• The Primary graph is characterised by the
Incedence Matrix
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Lagrangian Tree
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Fundamental Loop Matrix
Lagrangian Tree
Primary Graph
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Fundamental Cut-set Matrix
Lagrangian Tree
Primary Graph
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`Power Continuity
Power
continuity !
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Mechanism Dirac Structure
Power Ports Rigid Bodies
Power Ports Nodic Elements
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Further Steps…
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•
•
•
•
•
•
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
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