Dirac Framework for Robotics Tuesday, July 8 th , (4 hours) Stefano Stramigioli

transcript

Summer School 2003, Bertinoro (I)

Dirac FrameworkDirac Frameworkforfor

RoboticsRobotics

Tuesday, July 8Tuesday, July 8thth, (4 hours), (4 hours)Stefano StramigioliStefano Stramigioli

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

1D Mechanics1D Mechanics

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

EnergEnergyy

where is the momenta the applied where is the momenta the applied force and its velocity.force and its velocity.

Co-Co-EnergyEnergy

The dynamics EquationsThe second Law of dynamics is:The second Law of dynamics is:

Integral Integral FormForm

Diff. formDiff. form

The Kernel PCH representation

Interconnection portInterconnection port

Spring

EnergEnergyy

where is the displacement the where is the displacement the applied force to the spring and its applied force to the spring and its relative velocity.relative velocity.

Co-Co-EnergyEnergy

The dynamics EquationsThe elastic force on the spring is:The elastic force on the spring is:

Integral Integral FormForm

Diff. formDiff. form

The Kernel PCH representation

Mass-Spring System

• Spring

• MassMass

Together….

Interconnection of the two subsystems (1 junc.)

Or in image representationOr in image representation

Combining…

There exists a left orthogonal There exists a left orthogonal

Finally

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

3D Mechanics3D Mechanics

Notation

Set of points in Euclidean SpaceFree Vectors in Euclidean SpaceRight handed coordinate frame I

Coordinate mapping associated to

Rotations

It can be seen that if and are purely rotated

Theorem

If is a differentiable function of time

are skew-symmetric and belong to :

Tilde operator

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 !!

SO(3) is a Group

It is a Group becauseIt is a Group because

• AssociativityAssociativity

•IdentityIdentity

•Inverse Inverse

It is a Lie Group (group AND manifold)

• •

where•

where• Lie Algebra Commutator

Lie Groups

Common Space thanks Common Space thanks to Lie group structureto Lie group structure

Dual Space

• For any finite dimensional vector space we can define the space of linear operators from that space to

The space of linear operators from The space of linear operators from to (dual space of ) is to (dual space of ) is indicated withindicated with

co-vectorco-vector

In our case we have

Configuration Independent Configuration Independent Port !Port !

General Motion

General Motions

It can be seen that in general, for right handed frames

where ,

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 ?

Theorem

If is a differentiable function of time

belong to where

Tilde operator

Elements of se(3): Twists

The following are vector and matrix The following are vector and matrix coordinatecoordinate notations for twists:notations for twists:

The following are often called twists The following are often called twists too, but they are no geometrical entities !too, but they are no geometrical entities !

99 change of coordinates ! change of coordinates !

SE(3) is a Group

It is a Group becauseIt is a Group because

• AssociativityAssociativity

•IdentityIdentity

•Inverse Inverse

SE(3) is a Lie Group (group AND manifold)

• •

where•

where• Lie Algebra Commutator

Lie Groups

Common Space thanks Common Space thanks to Lie group structureto Lie group structure

Intuition of Twists

Consider a point fixed in :Consider a point fixed in :

and consider a second referenceand consider a second reference

wherewhere

andand

Possible Choices

For the twist of with respect to For the twist of with respect to we consider and we have we consider and we have 2 possibilities2 possibilities

Left and Right Translations

Possible Choices

andand

Notation used for Twists

For the motion of body with respect For the motion of body with respect to body expressed in the reference to body expressed in the reference frameframe we use we use

The twist is an across variable ! The twist is an across variable !

Point mass geometric Point mass geometric free-vectorfree-vector

Rigid body geometric Rigid body geometric screw + screw + MagnitudeMagnitude

Chasle's Theorem and intuition of a Twist

Any twist can be written as:Any twist can be written as:

Examples of Twists

Changes of Coordinates for Twists

• It can be proven that

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:

Poinsot's Theorem and intuition of a Wrench

Any wrench can be written as:Any wrench can be written as:

Chasles vs. Poinsot

Charles Charles TheoremTheorem

Poinsot Poinsot TheoremTheorem

The inversion of the upper and lower The inversion of the upper and lower part corresponds to the use of the part corresponds to the use of the Klijn formKlijn form

Vectors, Screws as “Forces”• Forces and Wrenches are co-vectors, but:

– Euclidean metricvector interpretation of a Force

– Klein’s form screw interpretation of a Wrench

That is identification of dual spaces.

Example of the use of a Wrench

Finding the contact centroid

Measured Wrench

6D sensor

Contact Point(Center of Pressure)

Transformation of Wrenches

• How do wrenches transform changing coordinate systems? We have seen that for twists:

Changes of coordinates

MTFMTF

In Dirac Kernel form

MTFMTF

Power Port

belong to vector spaces in duality:

such that there exists a bilinear such that there exists a bilinear operatoroperator

Finite dimensional case

• If is finite dimensional is uniquely defined, namely

where indicates the uniquely where indicates the uniquely defined set of linear operators from defined set of linear operators from to to

Elements of are vectorsElements of are vectors

Elements of are Elements of are co-vectorsco-vectors

In Robotics

Is the v.s. of TwistsIs the v.s. of Twists

Is the v.s. of WrenchesIs the v.s. of Wrenches

Power and Inf. Dim Case

• represents the instantaneous power flowing from A to B

• For inf.dim. systems they belong to k and (n-k) (Lie-algebra-valued) forms

DynamicsDynamics

Contents

• time derivative

• Rigid Body dynamics• Spatial Springs• Kinematic Pairs• Mechanism Topology

time derivative

• is function of time• It can be proven that

Transformations of

If we have ,how does look If we have ,how does look like? like?

It can be shown that in general

Rigid Bodies DynamicsRigid 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

Generalization of Newton’s law

In an inertial frame, for a point mass we had

This can be generalized for rigid This can be generalized for rigid bodiesbodies

Where Where NNii00 is the moment of body is the moment of body

expressed in the inertial frame expressed in the inertial frame 00 . .

That is why momenta is a co-That is why momenta is a co-vector !!vector !!

And in body coordinates ?

Using the derivative of Using the derivative of AdAdHH

multiplying on the left for we getmultiplying on the left for we get

and since we have thatand since we have that

and we eventually obtainand we eventually obtain

Momentum dynamics

which is called Lie-Poisson reduction.

NOTE: No information on configuration !

Other form

DefiningDefining

which is linear and anti-which is linear and anti-symmetricsymmetric

Port-Hamiltonian form

Storage portStorage port

ModulatioModulationn

Geometric SpringsGeometric Springs

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 TheoryMorse Theory

4 cells: 1 stable+3 unstable 4 cells: 1 stable+3 unstable pointspoints

Spatial Springs

wherewherewherewhere

For Constant Spatial Spring

It could be shown that:It could be shown that:

Storage portStorage port

to to integrate!integrate!

Parametric Changes (Scalar Case)

Variable Spatial Springs (Geometric Case)

BodBody 1y 1

BodBody 2y 2

Length Length VariationVariation

Variation RCCVariation RCC

It can be shown that varying RCC It can be shown that varying RCC does NOT exchange energy !!does NOT exchange energy !!

Kinematic PairsKinematic 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 subspaceAllowed subspace of of dimension dimension nn

Actuation subspaceActuation subspace of dimension of dimension nn

Decomposition of and

6-6-nn

Representations of subspaces

To satisfy To satisfy power power

continuitycontinuity

And in the Kernel Dirac representation

Actuators portsActuators ports

Interconnection Interconnection portport

Mechanism TopologyMechanism 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

Primary Graph

• The Primary graph is characterised by the Incedence Matrix

Lagrangian Tree

Fundamental Loop Matrix

Lagrangian Lagrangian TreeTree

Primary GraphPrimary Graph

Fundamental Cut-set Matrix

Lagrangian Lagrangian TreeTree

Primary GraphPrimary Graph

`Power Continuity

Power Power continuitcontinuit

y !y !

Mechanism Dirac Structure

Power Ports Rigid Power Ports Rigid BodiesBodies

Power Ports Nodic Power Ports Nodic ElementsElements

Further Steps…

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

Dirac Framework for Robotics Tuesday, July 8 th , (4 hours) Stefano Stramigioli

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