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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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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
Mass
Energy
where is the momenta
and
its velocity.
Co-Energy
the applied force
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Summer School 2003, Bertinoro (I)
The dynamics Equations
The second Law of dynamics is:
Integral Form
Diff. form
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Summer School 2003, Bertinoro (I)
The Kernel PCH representation
Interconnection port
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Summer School 2003, Bertinoro (I)
Spring
Energy
Co-Energy
where is the displacement
the applied
force to the spring and
its relative
velocity.
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Summer School 2003, Bertinoro (I)
The dynamics Equations
The elastic force on the spring is:
Integral Form
Diff. form
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Summer School 2003, Bertinoro (I)
The Kernel PCH representation
Interconnection port
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Summer School 2003, Bertinoro (I)
Mass-Spring System
• Spring
• Mass
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Summer School 2003, Bertinoro (I)
Together….
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Summer School 2003, Bertinoro (I)
Interconnection of the two subsystems (1 junc.)
Or in image representation
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Summer School 2003, Bertinoro (I)
Combining…
There exists a left orthogonal
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Summer School 2003, Bertinoro (I)
Finally
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
Rotations
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Summer School 2003, Bertinoro (I)
Rotations
It can be seen that if
purely rotated
and
are
where
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Summer School 2003, Bertinoro (I)
Theorem
If
of time
is a differentiable function
are skew-symmetric and belong to
:
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Summer School 2003, Bertinoro (I)
Tilde operator
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
SO(3) is a Group
It is a Group because
• Associativity
•Identity
•Inverse
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Summer School 2003, Bertinoro (I)
It 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)
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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Summer School 2003, Bertinoro (I)
In our case we have
Configuration Independent Port !
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Summer School 2003, Bertinoro (I)
General Motion
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Summer School 2003, Bertinoro (I)
General Motions
It can be seen that in general, for right
handed frames
where
,
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
SE(3)
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Summer School 2003, Bertinoro (I)
Theorem
If
of time
belong to
is a differentiable function
where
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
Left and Right Translations
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
Chasle's Theorem and intuition of a Twist
Any twist can be written as:
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Summer School 2003, Bertinoro (I)
Examples of Twists
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Summer School 2003, Bertinoro (I)
Examples of Twists
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Summer School 2003, Bertinoro (I)
Changes of Coordinates for Twists
• It can be proven that
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
Poinsot's Theorem and intuition of a Wrench
Any wrench can be written as:
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
Example of the use of a Wrench
Finding the contact centroid
Contact Point
(Center of Pressure)
Measured
Wrench
6D sensor
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Summer School 2003, Bertinoro (I)
Transformation of Wrenches
• How do wrenches transform changing
coordinate systems? We have seen that for
twists:
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Summer School 2003, Bertinoro (I)
Changes of coordinates
MTF
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Summer School 2003, Bertinoro (I)
In Dirac Kernel form
MTF
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
And in body coordinates ?
Using the derivative of AdH
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Summer School 2003, Bertinoro (I)
…..
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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Summer School 2003, Bertinoro (I)
Momentum dynamics
which is called Lie-Poisson reduction.
NOTE: No information on configuration !
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Summer School 2003, Bertinoro (I)
Other form
Defining
which is linear and anti-symmetric
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Summer School 2003, Bertinoro (I)
Port-Hamiltonian form
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Summer School 2003, Bertinoro (I)
Port-Hamiltonian form
Modulation
Storage port
Interconnection port
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
For Constant Spatial Spring
It could be shown that:
Storage port
to integrate!
Interconnection port
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Summer School 2003, Bertinoro (I)
Parametric Changes (Scalar Case)
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
Decomposition of
and
!
n
n
6-n
6-n
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Summer School 2003, Bertinoro (I)
Representations of subspaces
To satisfy power
continuity
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Summer School 2003, Bertinoro (I)
And in the Kernel Dirac representation
Interconnection port
Actuators ports
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Summer School 2003, Bertinoro (I)
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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Summer School 2003, Bertinoro (I)
Lagrangian Tree
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Summer School 2003, Bertinoro (I)
Fundamental Loop Matrix
Lagrangian Tree
Primary Graph
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Summer School 2003, Bertinoro (I)
Fundamental Cut-set Matrix
Lagrangian Tree
Primary Graph
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Summer School 2003, Bertinoro (I)
`Power Continuity
Power
continuity !
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Summer School 2003, Bertinoro (I)
Mechanism Dirac Structure
Power Ports Rigid Bodies
Power Ports Nodic Elements
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Summer School 2003, Bertinoro (I)
Further Steps…
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Summer School 2003, Bertinoro (I)
•
•
•
•
•
•
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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