The Geometric Framework -...

The Geometric Framework

Stanislao Grazioso

Friday 6th April, 2018

Stanislao Grazioso Geometric Theory of Soft Robots Friday 6th April, 2018 1 / 43

Introduction

Screw theory

Differential geometry

Finite element method

This course will combine these three techniques/methods


Main aspects of a geometric framework

1. Rigid body transformationsCoupling of position and rotation variables

Kinematics using Euclidean transformations

Space of Euclidean transformations = Lie group structure

2. Lie derivativesDerivatives (deformations and velocities) and kinematic joint transformations expressed in local framesattached to the bodies.

Resulting EoM are invariant with respect to a superimposed Euclidean transformation, i.e. EoM do notdepend on the position and orientation of the bodies with respect to the inertial reference frame →reduced non−linearities in EoM

3. Global parametrization−free frameworkLie Group motion formalism avoids the parametrization of rotation variables → reduced non−linearitiesand no singularities


1. Rigid Body Transformations


Rigid body transformations

I

RRp

u

q = g(p)

p

reference configuration

current configuration

o

Definition

A mapping g : R3 → R3 is a rigid body transformation if:

Distance is preserved, i.e., ||g(pj)− g(pi )|| = ||pj − pi || ∀ pi ,pj ∈ R3

Cross product is preserved, i.e., u(a× b) = g(a)× g(b) ∈ R3


Rigid body transformations (cont’d)

I

RRp

u

q = g(p)

p



o

Proposition

Rigid body transformations are such that

q = u + Rp (1)

where u ∈ R3 and R ∈ SO(3) is a rotation matrix, which satisfies

RTR = I3×3 det(R) = +1



I

RRp

u

q = g(p)

p



o

Equation 1 in matrix form↓[

q1

]= H(R,u)

[p1

]

H = H(R,u) =

[R u

01×3 1

]∈ SE (3)



Change of reference frame = Euclidean transformation

The space SE (3) of Euclidean transformations is a Lie Group

H = H(R, x) =

[R x01x3 1

]∈ SE(3)

SE(3) = SO(3)× R3


Lie group

Definition

A group (G,·) is a set G of elements q together with a composition operation (·)which satisfies the four axioms of:

closure: the composition of two elements of the set yields an element of theset, i.e., ∀q1, q2 ∈ G , q1 · q2 = q3 ∈ G

associativity: q1 · (q2 · q3) = (q1 · q2) · q3

neutral element: there exists an element e of the set such thatq · e = e · q = q

inverse element: there exists an element q−1 of the set such thatq · q−1 = q−1 · q = e

DefinitionA Lie group is a continuous group for which the composition rule and the inverseare smooth


Lie group (cont’d)

Proposition

A matrix Lie group is a Lie group for which the composition rule is represented bythe matrix product

R ∈ SO(3), the special Orthogonal group

H ∈ SE (3), the special Euclidean group


2. Lie Derivatives


Lie derivatives

Derivative of a Lie group

The derivative of q ∈ G with respect to a ∈ R reads

da(q) = qaL

= aRq

where aL ∈ g and aR ∈ g are respectively called a left and right invariant vectorfield. These elements represent the Lie algebra associated to the Lie group.


Lie algebra

Definition

The Lie algebra g (se(3)) is the tangent space at the identity element of a Liegroup G (H).

da(H) = Ha


Lie algebra (cont’d)

Proposition

The Lie algebra g is isomorphic to Rk through the invertible linear map

(·) : Rk → g, a ∈ Rk 7→ a ∈ g



SO(3) so(3) R3

SE (3) se(3) R6



da(H) = Ha

Left invariant vector field on SE (3)=

Invariant under a superimposed Euclidean transformation=

Intrinsic quantity


Twist

Time derivative of a Lie group

The time derivative of H reads

H = HηL

= ηRH

where ηL ∈ se(3) and ηR ∈ se(3) are respectively called a left and right invariantvector field.

The element η ∈ se is the Lie algebra associated to the Lie group H ∈ SE (3).

In the screw theory , the Lie algebra η ∈ se is called twist.


Twist (cont’d)

se(3) and so(3)

The Lie algebra η is the 4× 4 matrix

η =

[ω v

01×3 1

]∈ se(3)

where the Lie algebra ω is the skew-symmetric matrix

ω =

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

∈ so(3)


Twist (cont’d)

Velocity vector

According to the isomorphism g ' Rk , so(3) is isomorphic to R3 with

ω = [ω1 ω2 ω3]T , while se(3) is isomorphic to R6 with

η =

[v

ω

]


Screws

s

θq

pqθ

−sθ × q

Figure: A screw axis S represented by a point q, a unit direction vector s and a pitch p.

A geometrical interpretation of twist

The twist η corresponding to an angular velocity θ about the screw axis S:

η =

[v

ω

]=

[−s θ × q + ps θ

s θ

]Stanislao Grazioso Geometric Theory of Soft Robots Friday 6th April, 2018 20 / 43

Wrenches

Force vectorSix–dimensional vector comprising the linear force and the moment as

τ =

[f

m

]∈ R6


Adjoint representation

DefinitionThe adjoint representation of a Lie algebra element is defined as

Adq : g→ g, a 7→ qaq−1

Adjoint representation of a se(3) element

AdH(a) = HaH−1

AdH(a) =

[R uR

03×3 R

]a


Lie bracket

DefinitionThe Lie bracket operator is the bilinear operator defined as

[·, ·] : g× g→ g,[a, b]7→ db(a)− da(b)

Cross derivatives

db(a)− da(b) =[a, b]

db(a)− da(b) = ab = adab a =

[aω aω03×3 aω

]

Definition

The linear operator (·) is the bilinear operator defined as

(·) : Rk → Rk×k , a 7→ a = A


Equations of motion of a rigid body


Variations

Variations of a Lie group element

δ(R) = Rδθ

δ(H) = Hδh

where δθ ∈ so(3) is an arbitrary infinitesimal rotation associated with the axialvector δθ ∈ R3 and δhu = RT δu ∈ R3 is an arbitrary infinitesimal displacement

Variations of a twist element

δ(η)− (δh)· =[η, δh

]δ(η)− (δh)· = ηδh = −δhη


Kinetic energy K

K =1

2

∫V

ρqT q dV

q = u + Rp

q = u + Rωp = R [I3×3 − p]η

⇓

K =1

2ηTMη

M =

[mI3×3 JT1 (p)J1(p) J2(p)

]

m =

∫V

ρ dV ; J1(p) =

∫V

ρp dV ; J2(p) =

∫V

ρpT p dV


Potential energy Vext

Vext =

∫V

qTge dV

ge = 3× 1 vector of applied external forces expressed in the fixed reference frame.


Hamiltonian formulation

Hamilton’s principle ∫ t1

t0

(δ(K)− δ(Vext)) dt = 0 .

δ(K) = δ(η)TMη =

= ((δhT )· + δhT ηT )Mη

δ(Vext) =

∫V

δ(q)Tge dV =

= δhTgext

δ(q) = R [I3×3 − p] δh

gext =

[gext,ugext,ω

]=

∫V

[I3×3

p

]RTge dV



Dynamic equilibrium equations

weak form[δhT (Mη)

]t1

t0−∫ t1

t0δhT (Mη − ηTMη − gext) dt = 0

strong form Mη − ηTMη = gext



Equations of motion of a rigid body

H = Hη

Mη − ηTMη = gext


3. Global parametrization–free framework


Global parametrization of rotation

Euler’s equations (free rotating rigid body)

R = Rω

Jω + ωTJω = 03x1

⇓ Global parametrization of rotation

R = R(α1, α2, α3)

ω = T(α)α

ω = T(α)α+ T(α, α)α

Discretized Euler’s equationsR = R(α)

(J(Tα+ Tα) + (αTTT )JTα) = 03x1


Global parametrization–free equations of motion

Euler’s equations (free rotating rigid body)

R = Rω

Jω + ωTJω = 03x1

⇓ Exponential map

R = Rω ⇒ R(t) = R0 expSO(3)(ωt)

Euler’s equations discretized on the Lie groupRn+1 = Rn expSO(3)(nn+1)

Jωn+1 + ωTn+1Jωn+1 = 03x1


Global parametrization–free equations of motion (cont’d)

Integration formulae (implicit generalized−α scheme)

nn+1 = hωn + (0.5− β)h2an + βh2an+1

ωn+1 = ωn + (1− γ)han + γhan+1

an+1 =1

(1− αm)((1− αf )ωn+1 + αf ωn − αman)

αm =2ρ− 1

ρ+ 1; αf =

ρ

ρ+ 1; γ =

3− ρ2(ρ+ 1)

; β =1

(ρ+ 1)2

ρ ∈ [0, 1]


Global parametrization–free equations of motion (cont’d)

Classic vs Geometric

Discrete dynamics (classic) TT (J(Tα+ Tα) + (Tα)JTα) = 03x1

Discrete dynamics (geometric) Jω + ωTJω = 03x1

non singularities singularitiesquadratic high nonlinearitiesintrinsic orientation dependent


Exponential map

DefinitionThe exponential map projects an element of the Lie algebra into an element of theLie group

exp : g→ G , a 7→ exp(a)

and it is given by

exp(a) =∞∑i=0

ai

i !


Logarithmic map

DefinitionThe logarithmic map projects an element of the Lie group into an element of theLie algebra

log : G → g, q 7→ log(q) = a

and it is given by

log(q) =∞∑i=0

(e − q)i

i


Tangent map

Definition

T : Rk → Rk , u 7→ T(u)da(u) = a

with

T(u) =∞∑i=0

(−1)iui

(i + 1)!


Inverse of the tangent map

Definition

T−1 : Rk → Rk , u, a 7→ T−1(u)a = da(u)

with

T−1(u) =∞∑i=0

(−1)iBiui

(i)!

where Bi is the Bernoulli number of the first kind.


The exponential map on SO(3)

Exponential map expSO(3)(hω) = I3×3 + α(hω)hω + β(hω)2 h2

ω

Logarithmic map logSO(3)(R) = θ2sinθ (R− RT )

Tangent operator TSO(3)(hω) = I3×3 − β(hω)2 hω + 1−α(hω)

‖hω‖2 h2ωs

Inverse of the tangent operator T−1SO(3)(hω) = I3×3 + 1

2 hω + 1−γ(hω)‖hω‖2 h2

ω

α(hω) =sin(‖hω‖)‖hω‖

β(hω) = 21− cos(‖hω‖)‖hω‖2

γ(hω) =‖hω‖

2cot

(‖hω‖

2

)

θ = acos

(1

2(trace(R)− 1

), θ < π


The exponential map on SE (3)

Exponential map expSE(3)(h) =

[expSO(3)(hω) TT

SO(3)(hω)hu01×3 1

]

Logarithmic map logSE(3)(H) =

[hω T−TSO(3)(hω)hu01×3 0

]

Tangent operator TSE(3)(h) =

[TSO(3)(hω) Tuω+(hu,hω)

03×3 TSO(3)(hω)

]

Inverse of the tangent operator T−1SE(3)(h) =

[T−1

SO(3)(hω) Tuω−(hu,hω)

03×3 T−1SO(3)(hω)

]

Tuω+(hω , hu) =−β2

hω +1− α‖hω‖2

[hω , hu ] +hTu hω

‖hu‖2

((β − α)hu + (

β

2−

3(1− α‖hu‖2

)h2u

)Tuω−(hω , hu) =

1

2hω +

1− γ‖hω‖2

[hω , hu ] +hTu hω

‖hu‖4

((

1

β+ γ − 2)h2

u

)Stanislao Grazioso Geometric Theory of Soft Robots Friday 6th April, 2018 41 / 43

References (screw theory to multibody dynamics androbotics)

[Ball00] R Ball ”A treatise on the theory of screws”, 1900 (Reprinted 1998).

[Bro83] RW Brockett ”Robotic manipulators and the product of exponentials formula”,International symposium on the mathematical theory of networks and systems,pp. 120–129, 1983.

[MLS94] R M Murray, Z Li, S S Sastry, ”A mathematical introduction to roboticmanipulation”, CRC press, 1994.

[Se04] J M Selig ”Geometric fundamentals of robotics”, Springer Science andBusiness Media, 2004.

[LP17] K M Lynch and F C Park ”Modern robotics: Mechanics, Planning andControl”, Cambridge University Press, 2017.


References (geometric time integration)

[CG93] P E Crounch and R Grossman ”Numerical integration of ordinary differentialequations on manifolds”, Journal of Nonlinear Science, vol. 3, no. 1, pp. 1–33,1993.

[MK98] H Munthe–Kaas ”Runge-kutta methods on lie groups”, BIT NumericalMathematics, vol. 38, no. 1, pp. 92–111, 1998.

[MK98] J Park and W K Chung ”Geometric integration on Euclidean group withapplication to articulated multibody systems”, IEEE Transaction on Robotics,21(5), pp 850–863, 2005.

[BCA12] O Bruls, A Cardona and M Arnold ”Lie group generalized–α time integration ofconstrained flexible multibody systems”, Mechanism and Machine Theory, vol.48, no. 1, pp. 121–137, 2012.

[TMZ15] Z Terze, A Muller and D Zlatar ”Lie–group integration method for constrainedmultibody systems in state space”, Multibody System Dynamics, vol. 34, no.3, pp. 275–305, 2015.


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