SBRML Part1 Differential Geometry in Robotics · 2018. 11. 5. · A differential geometry...

Differential Geometry in Robotics

Sensor Based Manipulation and Locomotion

Differential Geometric Concepts

• Motivation

• Manifolds

• Tangent vectors and tangential space

• Cotangent vectors and cotangent space

• Transformation of vectors and covectors

• Tensors

• Coordinate transformations for tensors


Motivation

What is a vector in robotics?

Usual understanding: an element of a metric Euclidian space

Example: joint position, joint velocity, and joint accelerationjoint torques

First remove again all certainties…

Cartesian accelerationsCartesian forces/torques

Cartesian positions, Cartesian velocities


What's to object against?

1. All these "vector" quantities transform in different ways betweencoordinate systems

)(qTx T - Homogeneoustransformation

qqJx )(

FqJ T )(J - Jacobian matrix

xq

F)(qJT

)(qJ

Joint dynamics Cartesian dynamics

A differential geometry perspective enables a deeper understanding of these aspects

Obviously, there is a richer structure behind, than vector spaces

A

),( 21 xxx

x111 ,, qq

222 ,, qq

),( yx FFF



2. The six-dimensional vector spaces of Cartesian velocities, rotational velocities (twists) and of Cartesian forces and torques (wrench) are not metric spaces!

Neither the scalar product , nor the induced

vector norm

can be defined in a physically meaningful way, namely:- independent of the choice of the coordinate system- independent of the units

21 yyT

yyy T



Example:Tx ]2,2,2,1,1,1[1 Tx ]1,1,1,2,2,2[2

with velocity [m/s], angular velocity [rad/s]

021 xx

with velocity [mm/s], angular velocity [rad/s]

but

Tx ]2,2,2,101,101,101[ 3331

Tx ]1,1,1102,102,102[ 3332

021 xx

care is needed when generalizing the notion of orthogonality(familiar from translations) to twist und wrenches



3. Rotations in 3D are not described by a vector space(in contrast to translations)

- for example, rotations are not comutative

Further care is therefore needed when generalizing the concepts of stiffness,potentials, controller design etc. from translational to rotational case.


Manifolds

2R

A differential manifold is a topological space which is locally diffeomorphicto the Euclidian space .mR

A diffeomorphism is a differentiable function which is invertible and whoseinverse is differentiable

What is the space of all configurations of the robot tip?

x x x

x xx

MExample:

why only locally?:

2R

2S

x

x

x

xxx

2T

TCP

On which manifold does the TCP of the following robot move?And its joints?


Manifolds: Examples I1. The earth

Local coordinates: to each point x in M there exists one subset U in Mwhich contains x and a bijective mapping

manifold M

x in the vicinityof the south pole

local (polar) coordinates (r,)

localmaparound x

singularity at the poles

nx R

n: R UUU U mapping

map imagelocal coordinate

To describe a manifold one needs in general several, overlapping maps=> AtlasChange of coordinates: the maps and contain x),( U ),( V ))(( 1

xx

local coordinate:Geogr. longitude and latitude

local (conical)coordinates


Chicago Cloud Gate

o


Embedded ManifoldsnR

pn: RR hAn embedded manifold is a subspace of , defined through the solution set of a vector function

0),,(

0),,(

1

1

np

n1

xxh

xxh

(1)

If there exist p coordinates, such that the Jacobian in a pointis invertible, then the manifold has the dimension m=n-p in the

vicinity of the point and (1) can be locally solved. (implicit function theorem)

)(),,(),,(

01

1 xxxhh

p

p

n0 Rx

),,(

),,(

1

111

nppp

np

xxfx

xxfx

),,( 1 mpnp xx are then m local coordinates in the vicinity of x0

pn


Embedded Manifolds: Examples2R 0),( 2121 bxaxxxh1

)(112 ax

bx1x - Local coordinates

1x

2x

1x2x

3x

´3R

0),,( 321321 cxbxaxxxxh1

21, xx - Local coordinates )(1213 bxax

cx

3R2E

1x

2x

3x 0),,( 23

22

21321 rcxbxaxxxxh1

21, xx - Local coordinates(e.g. for the northern hemisphere)

)(1 22

213 bxaxr

cx


Manifold Examples: Examples II2. Orientation of a rigid body

33xRdescribed by a rotation matrix

The configuration space is the set of all rotation matrices and isdenoted as SO(3) – Special Orthogonal Group

properties: IRRT 1)det( R

- orthogonal- right-handed coordinate system

Exercise: check that SO(3) fulfills the group definition(closure, associativity, identity element, inverse element)however not the additional abelian group requirement (comutativity)

9R

Local coordinates: - Euler angles () (in all alternatives)- Axis-angle representation (r, )- Roll-Pitch-Yaw angles ()

SO(3) is a 3 – dimensional manifold, embedded for example in which is isomorph to the set of all 3x3 matrices

Smallest singularity-free representation: , e.g. quaternions4R


Addition: Rotational Representations

1221

3113

2332

3322111

sin21

21cos

rrrrrr

k

rrr

Axis-angle Representation

Roll-Pitch-Yaw

1kk

(Repetition from "Grundlagen Intelligenter Roboter")

Quaternions (Euler Parameter) not minimal, singularity-free, global

2sin,

2cos3,2,1,0

k

Singular for! n


Manifolds: Examples III3. Configuration of a rigid body in three-dimensional space

44xTdescribed by a homogeneous transformation

10 31x

pRT

Local coordinates: x = (px,py,pz,) (with all alternatives)

The configuration space is the set of all homogeneous transformations and is denoted as SE(3) –Special Euclidian Group (= SO(3)x ) 6 – dimensional manifold

3R

p position

6R

4. configuration of a (6-dimensional) robot Q

joint angles q: - global map

Locally, in the vicinity of the point q0, the forward kinematics x=f(q) can be regarded as a mapping between the coordinates of two manifolds.

5. The configuration space and the workspaceof a non- planar, 2-joint robot with endless turning joints is each a torus


Tangent Vectors and Tangential Space

1v 2v

xGiven an m - dimensional manifold M,we define at each point x the tangent spaceTxM as the m-dimensional vector space of allvelocities passing through x (along the manifold).

Only these quantities are called "vectors" in differential geometry. They are represented as column vectors.

Examples:

z

y

x

z

y

x

ppp

x

n

i

q

q

q

q

1

joint velocities in Cartesian velocities in SE3nR

(contravariant vectors or simply vectors)

vector on Q vector on SE3

vector on SO3


Cotangent Vectors and Cotangent Space

The cotangent Space at a point x of the manifold M isthe linear vector space of all linear functionals

MTx*

RMTx:

The elements of the cotangent space are called covectors

(covectors, covector space)

If the manifold is m-dimensional, the cotanget space has alsodimension m.

Examples:

z

y

x

z

y

x

mmmfff

f

n

i

1

Cartesian wrenchtorque in joint space

Pqq T )( Pxfxf T )(power


Transformation of Vectors and CovectorsWhy is it important to distinguish between vectors and covectors?

Vectors and covectors transform in a different way between coordinate systems or between manifolds.

Example:

The forward kinematics x=f(q) is a local mapping between Q und SE3

Velocity transformation:

qqJqqqf

tqfx )()(

d)(d

JacobianSince the power P is independent of representation :

x

f

qP xP

xq PP

qqJfxfP TTx )(

qP Tq => fqJ T )(

Remark: the product between vector and covector is thus well defined, i.e.invariant with respect to the change of coordinates or to representation manifolds for a given physical system


Transformation of Vectors and Covectors

Generally, if is a mapping between two manifolds and with

x - coordinates ofy - coordinates of

mM nN

mMnN

)(xy

then the vectors transform in a contravariant manner:

while the covectors transform covariantly:

xxJxxyy )(

yT

y

T

x xJxy )(

Why is it important to distinguish between vectors and covectors?


TensorsWhat are the remaining quantities in robotics? How do they transform?

)(),()( qgqqcqqM

)()()( qqDqqKqg ddd

Robot dynamics:

Simple PD controller with gravity compensation:

A tensor of type (p,q) is a multilinear mapping (or operator)

R

qxx

pxx MTMTMTMTA **: p times covariant

q times contravariant

Example:1. A covector is a tensor of type (1,0), thus covariant

A MTv x, R ii

iT vvv )( (=P for ) qv ,

2. A vector is a tensor of type (0,1), thus contravariant

vA MTx*, R

iii

T vvv )( (=P for ) qv ,


TensorsExample:

3. M, K, D are tensors of type (2,0), thus twice covariant

ijaA MTvv x2,1 R jji

iijT vvaAvvvvA 2

,12121 ),(

- For qvvqMA 21,),(21

R ),()(21),( 21 qqTqqMqvvA T

Kinetic energy

mass matrix

- For qvvDA 21,,

R DT PqDqvvA ),( 21

Heat power dissipatedby the viscous damper

damping matrix

- For qvvKA 21,,21

Stiffness matrix

R pT EqKqvvA

21),( 21

Potential energy of the (m-dim.) spring

Remark: q is only approximately (for very "small increments") in tangent space. A more precise definition of Ep will follow later.

What are the remaining quantities in robotics? How do they transform?


Tensors

Remark: from the definition it follows that, when applying a covariant tensor on a vector, the result will be another covariant tensor, whose order is reduced by 1

Example

qDD (2,0)(1,0) (0,1)

qE DT

D (1,0)(0,0) (0,1)

Remark: we see that formally, a scalaris a tensor of order zero

4. Tensors of type (1,1)

- linear dynamical system: xAx

(1,1)(0,1) (0,1)

Why can x be regardedas a vector for linear systems?

The system matrix in this case is of type (1,1) - one time covariant, one time contravariant



Tensors

Examples:

5. are tensors of type (0,2), thus two times contravariant

ijaA MTx*

2,1 R jji

iijT aAA 2

,12121 ),(

11, KM

- For ))((,),(21

211 qqMppqMA

R ),()(21),( 1

21 pqTpqMpA Tkinetic energy

mobility matrix:momentums

- For eKA 21

1 ,,21

compliance matrix

R pe

Te EKA 1

21 21),(

potential energyof the (m-dim.) spring

Elastic torque



Supplemental Slide Tensors

6. Coriolis and centrifugal forces:

k

ij

j

ki

i

kjijk q

mqm

qm

c 21

jji

iijkk qqcqqc ,

),(

tensor of type (3,0)

m-elements of the mass matrix

Example:

tensor of type (1,0)(covector)


Tensor Coordinate TransformationsWhy do we need to differentiate between tensor types?

Generally, given a mapping between the manifolds and with x coordinates ofy coordinates of

mM nNmMnN )(xy

Basic idea: the scalar quantities from the tensor definition are always conserved when performing coordinate transformations (energy, power, ... ).


xyTxyy

Tyy vxJAxJvAE )()(1

xxTxx vAE )()(1 xJAxJA yx

=>

similarity transformation


xyTT

xyyTyy vxJAxJvvAvE )()(

xxTxx vAvE )()( xJAxJA y

Tx =>

congruence-transformation


Scalar Products and Metrics

Vector product and vector norm in in Euclidian space

21

,, uuuvuvuvui

iiT

Therein, M is a (positive definite - p.d.) tensor (also called metric tensor) of type (2,0) and must be transformed accordingly when changing coordinates!

Example: For velocity vectors , when choosing the mass matrix as a metric, the norm represents the kinetic energy

(this also solves the problem of different units for the vectors)

q2Mq

Remark: remember that a vector cannot be applied directly on another vector, but only on a covariant tensor.

21

,

,,MM

jiiiji

TM

uuuvMuMvuvu Generalization on manifolds

)(21 qMM


Reciprocal Twists and Wrenches

Orthogonality between vectors

is therefore not well defined in differential geometry,

We can talk instead in a consistent way only about a reciprocal relation ofvectors and a covectors:

0 i

iivuvu

0))()(( i

iiT vvvv

A vector v and a covector are called reciprocal, if:

Example: For : a wrench and a twist are reciprocal, if the powertransfered by the pair is zero.

Hybrid force/position control: One can control m linearly independent twistsand (6-m) wrenches, which are reciprocal to the m twists.

xvf ,


Tangent Bundle

1v

2v

(what is the state space of a mechanical system?)

2q

1q

the state w of the mechanical systemcan be regarded as local coordinatesof a 2n manifold, the tangent bundle,denoted by So the tangent bundle is the union of the manifold M andall tangent spaces at all points of the manifold

q coordinates of , the configuration space ofa mechanical system

nM

is a tangent vector at the point w on

Coordinate transformation: ,

,

0,

SBRML Part1 Differential Geometry in Robotics · 2018. 11. 5. · A differential geometry...

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