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1Introduction to SAMCEF - MECANO
Introduction to Introduction to
SAMCEF MECANOSAMCEF MECANO
2Introduction to SAMCEF - MECANO
OutlineOutline
IntroductionGeneralized coordinatesKinematic constraintsTime integrationDescription and paramerization of finite rotations
AcknowledgementsMichel Géradin (ULg)
Doan D. Bao (GoodYear)Alberto Cardona (Intec, Argentina)
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3Introduction to SAMCEF - MECANO
Fixed deformable structures
Fixed deformable structures
Finite Element Method
Finite Element Method
Articulated rigid systemsArticulated
rigid systems
MultibodyDynamics
MultibodyDynamics
?
how to analyze in a general manner flexible articulated
structures
Our GoalOur Goal
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Strategic choicesStrategic choicesKinematic descriptionKinematic description
Recursive?Absolute?Absolute
Structural behaviorStructural behavior
Rigid ?Flexible ?Rigid/flexible
Motion equationsformulation
Motion equationsformulationSymbolic ?Numerical ?Numerical Time integrationTime integration
Implicit ? Explicit ? Implicit
Software architecture
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Modelling Modelling StepStep
Finite rotation parameterization
Finite motion description
Measures of deformation and relative motion
Rigid bodieszero deformation
Flexible elementselastic strains
+ internal work
Jointsrelative motion + external work
Mechanical elements library
• rigid members• flexible members• rigid articulations• flexible articulations• active members
Control systemlibrary
FE structuralsubstructuring
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Modelling Modelling StepStep
Mechanical elements library
• rigid members• flexible members• rigid articulations• flexible articulations• active members
Control systemlibrary
FE structuralsubstructuring
DAE system equations• dynamic equilibrium• kinematic constraints
Post-processingkinematics
internal forceselement stresses
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The MECANO The MECANO softwaresoftware
Preprocessing• topological description (FE)
• mechanical properties• response specification
Preprocessing• topological description (FE)
• mechanical properties• response specification
Initial Assembly
Static analysisequilibrium configuration
static forces
Static analysisequilibrium configuration
static forces
Kinetostatic analysistrajectories
quasi-static forces
Kinetostatic analysistrajectories
quasi-static forces
Dynamic analysistrajectories
dynamic forces
Dynamic analysistrajectories
dynamic forces
Linearized analysiseigenvalues
stability
Linearized analysiseigenvalues
stability
Preprocessing• trajectories
• loads and stresses• animation
Preprocessing• trajectories
• loads and stresses• animation
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The MECANO The MECANO element libraryelement library
Some rigid joint elements
Hinge joint
Gear joint
Prismatic joint
Cylindrical joint
Spherical joint
Rack-and-pinion
Kinematic constraints
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9Introduction to SAMCEF - MECANO
The MECANO element libraryThe MECANO element library
Cam contact
Point-on-plane
Wheel + tyre model
Cable + pulley
some non-holonomic joints
Kinematic constraints
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The MECANO The MECANO element libraryelement library
BushingSpring-damper-stop
Nonlinear beam
Superelement
The basic elastic elements
Kinematic constraints
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The MECANO element libraryThe MECANO element library
Hydraulic actuator
Nonlinear constraint
Nonlinear feedback
Some nonlinear and control features
Kinematic constraints
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The MECANO element libraryThe MECANO element library
Rail-wheel contact
Kinematic constraints
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The MECANO element libraryThe MECANO element library
Flexible slider
Kinematic constraints
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The MECANO element libraryThe MECANO element library
Symbolic element representation
Kinematic constraints
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15Introduction to SAMCEF - MECANO
Domains of applicationDomains of application
• mechanical engineering• automotive industry• sport industry• aeronautics• space structures• biomechanics• ...
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Dynamic car Dynamic car behaviorbehavior
M. Ebalard, J. Mercier (PSA Citroën)Utilisation du logiciel Mecano pour le comportement routier à PSA
Adequacy of Mecano• association of kinematicjoints with finite elements• open software through user element capability
Examples of application
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Landing gear analysisLanding gear analysis
Research conducted in framework of ELGAR consortium
Examples of application
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Landing gear analysisLanding gear analysisFunctional phases• deployment/retraction• ground impact• rolling• breaking• taxiing
Examples of application
Model components• mechanism• damper (internal hydraulic system)• tyre: contact, deformation, friction
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MEA antenna: result animationMEA antenna: result animation
Examples of application
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Simulation of Cluster satellite boom deploymentSimulation of Cluster satellite boom deployment
Examples of application
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21Introduction to SAMCEF - MECANO
Generalized coordinates for mechanism analysisGeneralized coordinates for mechanism analysis• Generalized co-ordinates
description of various typesthe 4-bar mechanism example
• The finite element method for articulated systemskinematics
elasto-dynamicsdifferential-algebraic equations of motionnumerical integration by Newmark algorithmthe double pendulum example
Generalized coordinates
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IntroductionIntroduction• Classical approach: minimal number of DOF.
• Lagrangian coordinates: best suited to robotics
• open-tree structure + associated graph
• loop closure conditions
• lack of generality, difficult for flexible systems
• Cartesian coordinates: large scale simulation packages
• inherent topology description
• large sets of differential-algebraic equations (DAE)
• still difficult for flexible systems
• Nonlinear Finite element coordinates:
• inherent topology description
• simplification of kinematic constraints
• geometrically nonlinear effects naturally included
Generalized coordinates
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The 4The 4--bar mechanism examplebar mechanism example
• Simplest planar, closed-loop mechanism• one single variable: crank angle β
Gruebler’s formula m : number of jointsN: number of bodiesp : number of DOF removed by joint i
F = 1F = 3(N − 1)−mXi=1
pi
Generalized coordinates
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Minimal coordinatesMinimal coordinates Expression in terms of minimal coordinate β
cosθ2 =d2 + s2 + 2r` cosβ
2ds− r2 − `2
sin(θ3 + φ) =c
ρ
θj = f(β) j = 2, . . . 4
From geometry
(`− r cosβ) cos θ3 + r sinβ sin θ3 = s− d cos θ2
θ1 = 2π − β − θ2 − θ3
Generalized coordinates
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Minimal coordinates: Minimal coordinates: kinematickinematic transformertransformer
Input-output relationship
Complex mechanisms as assembly of transformers
Generalized coordinates
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LagrangianLagrangian coordinatescoordinates
3 generalized coordinates
2 closure conditions
Equations of motion: 3 differential Equations of motion: 3 differential equations linked by 2 constraintsequations linked by 2 constraints(DAE)(DAE)
qT = [θ1 θ2 θ3 ]
`1 cosθ1 + `2 cos(θ1 + θ2) + `3 cos(θ1 + θ2 + θ3)− `4 = 0`1 sin θ1 + `2 sin(θ1 + θ2) + `3 sin(θ1 + θ2 + θ3) = 0
Generalized coordinates
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Cartesian coordinatesCartesian coordinates
8 kinematic constraints
9 generalized coordinates qT = [ x1 y1 θ1 x2 y2 θ2 x3 y3 θ3 ]
x1 −`12 cosθ1 = 0 y1 −
`12 sin θ1 = 0
x1 +`12 cosθ1 − x2 +
`22 cos θ2 = 0 y1 +
`12 sin θ1 − y2 +
`22 sin θ2 = 0
x2 +`22 cosθ1 − x2 +
`32 cos θ3 = 0 y2 +
`22 sin θ2 − y3 +
`32 sin θ3 = 0
x3 +`32 cosθ3 = `4 y3 +
`32 sin θ3 = 0
Equations of motion: 9 Equations of motion: 9 differential equations linked differential equations linked by 8 constraintsby 8 constraints (DAE)(DAE)
Generalized coordinates
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Finite Element coordinatesFinite Element coordinates
Strong formStrong form of of kinematickinematic constraintsconstraints• 4 boundary nodal constraints
• 4 assembly nodal constraints
13 generalized coordinates:• 12 finite element DOF q• 1 driving DOF βq = [x1 y1 . . . x6 y6 ]
T
• 3 zero strain constraints
• 1 driving constraints (implicit)
x1 = 0, y1 = 0x6 = 0, y6 = `4
x2 = x3 y2 = y3x4 = x5 y4 = y5
²1 = 0 ²2 = 0 ²3 = 0
y2 cosβ − x2 sinβ = 0ΦD(q) = 0
²i =1
2
`2i − `2i,0
`2i,0
Generalized coordinates
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Finite element description of a mechanical systemFinite element description of a mechanical system
q = DOF at structural level
qe = DOF at element level
Le = DOF localization operator(boolean)
λ = lagrangian multipliers
qe = Leq
Φ(qe, qe, t) = 0
• Boolean constraints: localization of DOF• Implicit constraints: algebraic treatment
System topology results implicitly from Boolean assembly and
constraint description
Kinematic constraints
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Classification of Classification of kinematickinematic constraintsconstraints
• Holonomic constraints
• Restrict the number of DOF of the system
Φ(q, t) = 0
• Non-holonomic constraints
• Restrict the system behavior
• Unilateral constraints
ex: prismatic joint
ex: rolling motion
Ψ(q, q, t) ≥ 0
Ψ(q, q, t) = 0
Kinematic constraints
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Classical mechanical pairsClassical mechanical pairs
revolute
cylindrical
prismatic
screw
planar
spherical
Only 3 lower pairs• surface contact• motion reversibility
• All other mechanical pairs are higher pairs
Other modes of classification:• number of DOF (1 for all lower pairs)• mode of closure
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The algebraic constrained problemThe algebraic constrained problem⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
minqF(q)
subject to
Φ(q) = 0
Possible methods of solution• constraint elimination• Lagrange multipliers• penalty function• augmented Lagrangian• perturbed Lagrangian
Formulated in terms of
Residual vector
r(q) =∂F
∂q
Jacobian matrix
Hessian matrix
Bmj =∂Φm∂qj
B such that
H such that Hij =∂2F
∂qi∂qj
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Constraint elimination methodConstraint elimination method• First variation of constraints
δΦ =Bδq = 0• Partitioning into independent and dependent variables
BIδqI +BDδqD = 0
• Elimination of dependent variables
δq =RδqIreduction
matrix R =
⎡⎣ I
−B−1D BI
⎤⎦
RT ∂F
∂q=RT r(q) = 0
• Projected residual equation
• exact verification of constraints• complex and computationally expensive
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• provides exact solution regardless of penalty and scaling factors
• quadratic term reinforces positive character of H
• scaling factor required for pivoting
Augmented Augmented LagrangianLagrangian methodmethod
F∗p (q,λ) = F(q) + (kλTΦ+
p
2ΦTΦ)
⎡⎣H + pBTB kBT
kB 0
⎤⎦⎡⎣∆q∆λ
⎤⎦ =⎡⎣−r? −BT (kλ? + pΦ)
kΦ
⎤⎦
p penalty coefficientk scaling factor
Augmented functional
Second-order solution method: by Newton-Raphson
q = q? +∆qλ = λ? +∆λ
pBTB
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Constrained dynamic problemsConstrained dynamic problems
Z t2
t1[δ(L− kλTΦ− pΦTΦ) + δW]dt = 0
Mq +BT (pΦ+ kλ) = g(q, q, t)Φ(q, t) = 0
Hamilton’s principle
Matrix form of constrained motion
equations
• Differential - Algebraic nature (DAE)• vanishing of penalty term at equilibrium → exact• scaling of constraints for numerical conditioning• multipliers λ = forces needed to close constraints• all remaining forces (internal, external, complementary inertia) in RHS
Kinematic constraints
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Constrained equations of motionConstrained equations of motion
Mq +BT (pΦ+ kλ) = g(q, q, t)Φ(q, t) = 0
Methods of solution
• Constraint reduction• Constraint regularization → m+n ODE + constraint stabilization• second-order solution → linearization
But: specific problem of numerical stability of time integration
Kinematic constraints
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Linearization of motion equationsLinearization of motion equationsM 00 0
¸∆q∆λ
¸+Ct 00 0
¸∆q∆λ
¸+Kt kBT
0 kB
¸∆q∆λ
¸=r?
−Φ?
¸+O(∆2)
Residual vector of equilibrium r = g(q, q, t)−Mq −BT (pΦ+ kλ)
Ct = −∂g
∂qKt = −
∂g
∂q+∂(Mq)
∂q+∂(BT (pΦ+ kλ))
∂q
Kt ' −∂g
∂q+ pBTB + k
∂(BTλ)
∂q
Tangent damping matrix Tangent stiffness matrix
approx.• plays central role for convergence in iteration• quality of approximation needed depends on penalty factor
Kinematic constraints
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Integration of motion equations: Integration of motion equations: Implicit MethodImplicit Method
Motivationsfiltering of high frequencies (elasticity)• independence of algorithm stability on time step size• one-step method simplicity of software• experience in structural dynamics.
Newmark algorithm
Application to multibody systems• appropriate integration of rotation motion• stability in presence of constraints (DAE)
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Implicit integration Implicit integration Differential - algebraic equations
Correction of solution at time t
q = q∗ +∆qq = q∗ +∆qq = q∗ +∆qλ = λ∗ +∆λ
Linearized equations of motion
Linearized matrices
r(q, q, q,λ) =Mq+BTλ− g(q, q, t) = 0Φ(q, t) = 0
(q∗, q∗, q∗, λ∗)
KT =∂r
∂q= −
∂g
∂q+
∂
∂q(Mq∗ +BTλ∗)
CT =∂r
∂q= −
∂g
∂q
Tangent damping
Tangent stiffness
M 00 0
¸∆q∆λ
¸+
CT 00 0
¸∆q∆λ
¸+
KT BT
B 0
¸∆q∆λ
¸=
r∗
−Φ∗
¸
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NewmarkNewmark formula formula
• Interpolation of displacements and velocities
qn+1 = qn + (1− γ)hqn + γhqn+1qn+1 = qn + hqn + (
12− β)h2qn ++βh
2qn+1
γ =1
2and β =
1
4
• Free parameters: average constant acceleration
γ =1
2+ α and β =
1
4(γ +
1
2)2 α > 0
• Introduction of numerical damping
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Integration procedure Integration procedure q0n+1 = 0q0n+1 = qn + (1− γ)hqnq0n+1 = qn + hqn + (
12 − β)h2qn
qk+1n+1 = qkn+1 +∆q = q
kn+1 +
1βh2∆q
qk+1n+1 = qkn+1 +∆q = q
kn+1 +
γβh∆q
qkn+1 = qkn+1 +∆q
ST BT
B 0
¸∆q∆λ
¸=
r∗
−Φ∗
¸
ST =KT +γ
βhCT +
1
βh2M
krk < ² and kΦk < η
• Prediction at time t
• Correction
• Correction equation
Iteration matrix
• Convergence check
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The double pendulum The double pendulum
qT = [x1 y1 θ1 x2 y2 θ2 ]
• Description by Cartesian coordinates (redundant)
K = 12m1(x
21 + y
21) +
12m2(x
22 + y
22)
P = m1gy1 +m2gy2
• Kinetic and potential energies
Φ =
⎡⎢⎢⎣x1 − `1 sinθ1y1 + `1 cos θ1
x2 − x1 − `2 sin θ2y2 − y1 + `2 cos θ2
⎤⎥⎥⎦• Kinematic constraints
B =
⎡⎢⎢⎣1 0 −`1 cos θ1 0 0 00 1 −`1 sinθ1 0 0 0−1 0 0 1 0 −`2 cos θ20 −1 0 0 1 −`2 sin θ2
⎤⎥⎥⎦• Constraint matrix
• External force vector and mass matrix are constant• Tangent stiffness contains only geometric terms
KT = diag [ 0 0 `1(λ1 sin θ1 − λ2 cos θ1) 0 0 `2(λ3 sin θ2 − λ4 cos θ2) ]
m1 = m2 = 1.l1 = l2 = 1.,(θ1 = π/2, θ2 = 0
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(γ = 12 , β =
14 )NewmarkNewmark integration : integration : undampedundamped
• normal evolution of angular displacements and velocities• but: numerical instability detected after 2 sec.
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(γ = 12 , β =
14 )NewmarkNewmark integration : integration : undampedundamped
Weak instability: generated by kinematic constraints
Numerical damping
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((γ = 1
2 + α, β = 14 (12 + γ)2, α = 0.015)
NewmarkNewmark integration with damping integration with damping
Normal aspect of response over 5 sec. period, but ...
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NewmarkNewmark integration with damping integration with damping ((γ = 1
2 + α, β = 14 (12 + γ)2, α = 0.015)
Unacceptable amount of energy loss
Search for better compromise between accuracy and stability
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Kinematics of finite motionKinematics of finite motion
• Spherical motionEigenvalue analysis of rotation operatorEuler theorem Explicit expressions of rotation operator Expression in terms of linear invariantsExponential map
• General motion of rigid body• Velocity analysis of spherical motion• Explicit expression of angular velocities
Rotation parameterizationCartesian rotation vector
Finite Motion
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Spherical motionSpherical motion
Initial position: material configuration X
Current position: material configuration x
Spatial coordinates
x = [x1 x2 x3]T
X = [X1 X2 X3]T
material coordinates
[~E1 ~E2 ~E3]
Base vectors of initial configuration
Base vectors of current configuration [~e1 ~e2 ~e3]
Spherical motion such that:• length of position vector unaffected by pure rotation• relative angle between 2 directions remain constant
Linear transformation
x = RX RT =R−1 det(R) = 1withProper orthogonal
Finite Motion
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Euler theorem Euler theorem
If a rigid body undergoes a motion leaving fixed one of its points, then a set of points of the body lying on a line that passes through that point, remains fixed as well.
Finite Motion
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Explicit expressions of rotation operator Explicit expressions of rotation operator
Outer product expression
Orthonomality implies 6 constraints
rTi rj = δijR = [ r1 r2 r3 ]
R =R(α1,α2,α3)3 independent rotation parameters
R =Xi
eiETi
Inner product expression
R =
⎡⎣ET1 e1 ET1 e2 ET1 e3
ET2 e1 ET2 e2 ET2 e3
ET3 e1 ET3 e2 ET3 e3
⎤⎦
Finite Motion
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Expression in terms of linear invariantsExpression in terms of linear invariants
R=Icosφ+(1−cosφ)nnT +nsinφ
Rotation operator
Invariants
tr(R) =λ1 +λ2 +λ3 =1+2cosφvect(R) =nsinφ
Finite Motion
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Exponential mapExponential map
• Differentiate with respect to φ
x =RX
• verify thatdR
dφRT = n
dx
dφ− nx = 0 with x(0) =X
• differential motion
• integration
x = exp(nφ) X R = exp(nφ)
Finite Motion
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General motion of rigid bodyGeneral motion of rigid body
xP = x0 +RXP
Combination of translation and rotation
spatial motion of origin
material coordinates of
point P
Finite Motion
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Velocity analysis of spherical motionVelocity analysis of spherical motion
xP =RXP RT =R−1
ω = RRTvP = ωxP
ω = vect(RRT )ω = vect(RRT )
V P = ΩXP Ω =RT R
Ω = vect(RT R)Ω = vect(RT R)
vP = xP = RXP V P =RT vP
Spatial description Material descriptionDifferentiate
with
= RRT + (RRT )T = 0
with with
spatial angular velocities material angular velocities
spatial velocities material velocities
skew-symmetry
Finite Motion
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Cartesian rotation vectorCartesian rotation vector
Ψ = nφdefined as
R = I +sin kΨ k
kΨ keΨ +
1− cos kΨ k
kΨ k2eΨ eΨ
Rotation operator
Trigonometric form
R = exp( eΨ)Exponential map
Ω = T (Ψ)Ψ ω = T T (Ψ)Ψ
T (Ψ) = I +
µcos kΨk − 1
kΨk2
¶ eΨ+ µ1 − sin kΨkkΨk
¶ eΨ eΨkΨk2
Angular velocities
Tangent operator limkΨk→0
T (Ψ) = limkΨk→0
R(Ψ) = I
Limit properties
Finite Motion