Introduction to Robotics (Fag 3480) · Fag 3480 - Introduction to Robotics. side 30. Rotation...

Introduction to Robotics (Fag 3480)Vår 2010

Robert Wood (Harward Engineeering and Applied Sciences-Basis)Ole Jakob Elle, PhD (Modified for IFI/UIO)

Førsteamanuensis II, Institutt for InformatikkUniversitetet i Oslo

Seksjonsleder Teknologi, Intervensjonssenteret, Oslo Universitetssykehus (Rikshospitalet)

Fag 3480 - Introduction to Roboticsside 2

Ch. 2: Rigid Body Motions and Homogeneous Transforms


Industrial robotsHigh precision and repetitive tasks

Pick and place, painting, etc

Hazardous environments


RepresentationsFor the majority of this class, we will consider robotic manipulators as open or closed chains of links and joints

Two types of joints: revolute (θ) and prismatic (d)


Arm configurationsThe most frequent arm configurations are :•Open kinematic chains :

•Jointed articulated or anthropomorphic (human-like arms) (RRR)•Spherical (RRP)•Scara (RRP)•Cylindrical (RPP)•Cartesian (PPP)•Multi-joined (RRRRRR.....) , Redundant configurations

•Closed kinematic chains


DefinitionsEnd-effector/Tool

Device that is in direct contact with the environment. Usually very task-specific

Configuration

Complete specification of every point on a manipulator

set of all possible configurations is the configuration space

For rigid links, it is sufficient to specify the configuration space by the joint angles,

State space

Current configuration (joint positions q) and velocities

Work space

The reachable space the tool can achieve

Reachable workspace

Dextrous workspace

[ ]Tnqqqq ...21=

q


Common configurations: wristsMany manipulators will be a sequential chain of links and joints forming the ‘arm’ with multiple DOFs concentrated at the ‘wrist’


Common configurations: elbow manipulatorAnthropomorphic arm: ABB IRB1400 or KUKA

Very similar to the lab arm NACHI (RRR)


Antropomorphic arm (RRR)


Common configurations: SCARA (RRP)


Spherical Manipulator (RRP)


Common configurations: cylindrical robot (RPP)

workspace forms a cylinder

Seiko RT3300 Robot


Common configurations: Cartesian robot (PPP)

Increased structural rigidity, higher precision

Pick and place operations

Epson Chartesian robot (EZ-modules)


Workspace comparison(a) spherical

(b) SCARA

(c) cylindrical

(d) Cartesian


Parallel manipulators

6DOF Stewart platform

some of the links will form a closed chain with ground

Advantages:

Motors can be proximal: less powerful, higher bandwidth, easier to control

Disadvantages:

Generally less motion, kinematics can be challenging

ABB IRB940 Tricept


CASPAR - Maquet

ROBODOC –Integrated Surgical Systems Inc.

PathFinder –Armstrong HealthCare Lmt.

Image-guided robots


Simple example: control of a 2DOF planar manipulator

Move from ‘home’ position and follow the path AB with a constant contact force F all using visual feedback


Coordinate frames & forward kinematicsThree coordinate frames:

Positions:

Orientation of the tool frame:0 1

2

( )( )

=

11

11

1

1

sincos

θθ

aa

yx

( ) ( )( ) ( ) ty

xaaaa

yx

≡

++++

=

21211

21211

2

2

sinsincoscos

θθθθθθ

0 1 2

+++−+

=

⋅⋅⋅⋅

=)cos()sin()sin()cos(

ˆˆˆˆˆˆˆˆ

2121

2121

0202

020202 θθθθ

θθθθyyyxxyxx

R

=

=

10ˆ

01ˆ 00 yx ,

++−

=

++

=)cos()sin(ˆ

)sin()cos(ˆ

21

212

21

212 θθ

θθθθθθ

yx ,


Ch. 2: Rigid Body Motions and Homogeneous Transforms


Definition: coordinate frame

A set n of orthonormal basis vectors spanning Rn

For example,

When representing a point p, we need to specify a coordinate frame

With respect to o0:

With respect to o1:

v1 and v2 are invariant geometric entities

But the representation is dependant

upon choice of coordinate frame

Representing position

=

650p

−=

2.48.21p

=

=

=

100

ˆ

010

ˆ

001

ˆ kji , ,

−=

−=

=

=

2.48.2

11.5

8.077.7

65 1

202

11

01 vvvv , , ,


2D rotations

Representing one coordinate frame in terms of another

Where the unit vectors are defined as:

This is a rotation matrix

Rotations

[ ]01

01

01 yxR =

−=

=

θθ

θθ

cossinˆ

sincosˆ 0

010

01 yyxx ,

−=

θθθθ

cossinsincos0

1R


Rotation matrices as projections

Projecting the axes of from o1 onto the axes of frame o0

Alternate approach

⋅⋅

=

⋅⋅

=01

0101

01

0101 ˆˆ

ˆˆˆˆˆˆ

yyxy

yyxxx

x ,

−=

−

+

=

⋅⋅⋅⋅

=

θθθθ

θθπ

πθθ

cossinsincos

cosˆˆ2

cosˆˆ

2cosˆˆcosˆˆ

ˆˆˆˆˆˆˆˆ

0101

0101

0101

010101

yyyx

xyxx

yyyxxyxx

R


Properties of rotation matricesInverse rotations:

Or, another interpretation uses odd/even properties:

( )TR

yyyx

xyxx

yyyxxyxx

R

01

1010

1010

1010

101010

cossinsincos

cosˆˆ2

cosˆˆ

2cosˆˆcosˆˆ

ˆˆˆˆˆˆˆˆ

=

−

=

+

−

=

⋅⋅⋅⋅

=

θθθθ

θπθ

θπθ

( ) ( )( ) ( )

( )TR

R

01

10

cossinsincos

cossinsincos

=

−

=

−−−−−

=

θθθθ

θθθθ


Properties of rotation matricesInverse of a rotation matrix:

The determinant of a rotation matrix is always ±1

+1 if we only use right-handed convention

( )

( ) ( )TRR

yyyx

xyxx

yyyxxyxx

R

010

1

1

1

0101

0101

1

0101

0101101

cossinsincos

det1

cossinsincos

cosˆˆ2

cosˆˆ

2cosˆˆcosˆˆ

ˆˆˆˆˆˆˆˆ

=

−

=

−=

−

−

=

⋅⋅⋅⋅

=

−

−

−−

θθθθ

θθθθ

θθπ

πθθ


Properties of rotation matricesSummary:

Columns (rows) of R are mutually orthogonal

Each column (row) of R is a unit vector

The set of all n x n matrices that have these properties are called the Special Orthogonal group of order n

( ) 1det

1

== −

RRRT

( )nSOR ∈


3D rotations

( )3ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ

010101

010101

01010101 SO

zzzyzxyzyyyxxzxyxx

R ∈

⋅⋅⋅⋅⋅⋅⋅⋅⋅

=

General 3D rotation:

Special cases

Basic rotation matrices

−=

−=

θθ

θθ

θθθθ

θ

θ

cos0sin010

sin0cos

cossin0sincos0001

,

,

y

x

R

R

−=

1000cossin0sincos

, θθθθ

θzR


Properties of rotation matrices (cont’d)

SO(3) is a group under multiplication

Closure:

Identity:

Inverse:

Associativity:

In general, members of SO(3) do not commute

( ) ( )33 2121 SORRSORR ∈⇒∈ , if

( )3100010001

SOI ∈

=

1−= RRT

( ) ( )321321 RRRRRR = Allows us to combine rotations:bcabac RRR =

1221 RRRR ≠


Now assume p is a fixed point on the rigid object with fixed coordinate frame o1

The point p can be represented in the

frame o0 (p0) again by the projection onto

the base frame

Rotational transformations

=

wvu

p1

( )( )( )

101

010101

010101

010101

010101

010101

010101

0111

0111

0111

01

01

01

0

ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ

ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ

ˆˆˆˆˆˆˆˆˆˆˆˆ

ˆˆˆ

pRwvu

zzzyzxyzyyyxxzxyxx

zzwzyvzxuyzwyyvyxuxzwxyvxxu

zzwyvxuyzwyvxuxzwyvxu

zpypxp

p

=

⋅⋅⋅⋅⋅⋅⋅⋅⋅

=

⋅+⋅+⋅⋅+⋅+⋅⋅+⋅+⋅

=

⋅++⋅++⋅++

=

⋅⋅⋅

=


Rotating a vectorAnother interpretation of a rotation matrix:

Rotating a vector about an axis in a fixed frame

Ex: rotate v0 about y0 by π/2

=

110

0v

=

−=

−=

=

=

011

110

001010100

110

cos0sin010

sin0cos

2/

02/,

1

πθ

π

θθ

θθ

vRv y


Rotation matrix summaryThree interpretations for the role of rotation matrix:

Representing the coordinates of a point in two different frames

Orientation of a transformed coordinate frame with respect to a fixed frame

Rotating vectors in the same coordinate frame


Similarity transformsAll coordinate frames are defined by a set of basis vectors

These span Rn

Ex: the unit vectors i, j, k

In linear algebra, a n x n matrix A is a mapping from Rn to Rn

y = Ax, where y is the image of x under the transformation A

Think of x as a linear combination of unit vectors (basis vectors), for example the unit vectors:

Then the columns of A are the images of these basis vectors

If we want to represent vectors with respect to a different basis, e.g.: f1, …, fn, the transformation A can be represented by:

Where the columns of T are the vectors f1, …, fn,

[ ] [ ]TnT ee 1...000...011 == ..., ,

ATTA 1−=′


Similarity transformsA’ and A have the same eigenvalues

An eigenvector x of A corresponds to an eigenvector T-1x of A’

Rotation matrices are also a change of basis

If A is a linear transformation in o0 and B is a linear transformation in o1, then they are related as follows:

Ex: the frames o0 and o1 are related as follows:

If the matrix A is also a rotation matrix Rz,θ (relative to o0) the rotation expressed in o1 is:

( ) 01

101 ARRB −

=

−=

001010100

01R

( )

−=

−

−

−==

−

θθθθθθ

θθ

cossin0sincos0

001

001010100

1000cossin0sincos

001010100

01

101 ARRB


Compositions of rotationsw/ respect to the current frame

Ex: three frames o0, o1, o2

This defines the composition law for successive rotations about the currentreference frame: post-multiplication

202

0

212

1

101

0

pRp

pRp

pRp

=

=

=21

201

0 pRRp = 12

01

02 RRR =


Compositions of rotationsEx: R represents rotation about the current y-axis by φ followed by θ about

the current z-axis

−

−=

−

−=

=

φθφθφθθ

φθφθφθθθθ

φφ

φφθφ

cossinsincossin0cossin

sinsincoscoscos

1000cossin0sincos

cos0sin010

sin0cos,, zy RRR


Compositions of rotationsw/ respect to a fixed reference frame (o0)

Let the rotation between two frames o0 and o1 be defined by R10

Let R be a desired rotation w/ respect to the fixed frame o0

Using the definition of a similarity transform, we have:

This defines the composition law for successive rotations about a fixed reference frame: pre-multiplication

( )[ ] 01

01

101

01

02 RRRRRRR ==

−


Compositions of rotationsEx: we want a rotation matrix R that is a composition of φ about y0 (Ry,φ) and then θ about z0

(Rz,θ)

the second rotation needs to be projected back to the initial fixed frame

Now the combination of the two rotations is:

( )θθθ

θθθ

,,,

,,1

,02

yzy

yzy

RRRRRRR

−

−

=

=

[ ] φθφθφφ ,,,,,, yzyzyy RRRRRRR == −


Compositions of rotationsSummary:

Consecutive rotations w/ respect to the current reference frame:

Post-multiplying by successive rotation matrices

w/ respect to a fixed reference frame (o0)

Pre-multiplying by successive rotation matrices

We can also have hybrid compositions of rotations with respect to the current and a fixed frame using these same rules


Parameterizing rotationsThere are three parameters that need to be specified to create

arbitrary rigid body rotations

We will describe three such parameterizations:

Euler angles

Roll, Pitch, Yaw angles

Axis/Angle


Parameterizing rotationsEuler angles

Rotation by φ about the z-axis, followed by θ about the current y-axis, then ψ about the current z-axis

−+−+−−−

=

−

−

−==

θψθψθ

θφψφψθφψφψθφ

θφψφψθφψφψθφ

ψψ

ψψ

θθ

θθ

φφ

φφ

ψθφ

csscsssccscsscccssccssccssccc

cssc

cs

sccssc

RRRR zyzZYZ

10000

0010

0

10000

,,,


Parameterizing rotationsRoll, Pitch, Yaw angles

Three consecutive rotations about the fixed principal axes:

Yaw (x0) ψ, pitch (y0) θ, roll (z0) φ

−+−+

++−=

−

−

−=

=

ψθψθθ

ψθφψφψθφψφθφ

ψθφψφψθφψφθφ

ψψ

ψψ

θθ

θθ

φφ

φφ

ψθφ

ccscscssscssscccs

cscssssccscc

cssc

cs

sccssc

RRRR xyzXYZ

00

001

0010

0

10000

,,,


Parameterizing rotationsAxis/Angle representation

Any rotation matrix in SO(3) can be represented as a single rotation about a suitable axis through a set angle

For example, assume that we have a unit vector:

Given θ, we want to derive Rk,θ:

Intermediate step: project the z-axis onto k:

Where the rotation R is given by:

=

z

y

x

kkk

k̂

1,,

−= RRRR zk θθ

αβθβαθ

βα

−−=⇒

=

,,,,,,

,,

zyzyzk

yz

RRRRRRRRR


Parameterizing rotationsAxis/Angle representation

This is given by:

Inverse problem:

Given arbitrary R, find k and θ

++−−+++−+

=

θθθθθθ

θθθθθθ

θθθθθθ

θ

cvkskvkkskvkkskvkkcvkskvkkskvkkskvkkcvk

R

zxzyyzx

xzyyzyx

yzxzyxx

k2

2

2

,

( )

−−−

=

−

= −

1221

3113

2332

1

sin21ˆ

21cos

rrrrrr

k

RTr

θ

θ


Alternative approachRotation matrices in SO(3) can be derived using an alternate method using the matrix

exponential

Assume that you have a unit vector, ω, that represents an axis of rotation

Take a vector q(t) and rotate it about ω with unity velocity

This gives:

Where the ‘cross-product’ matrix is given as follows:

Now integrate to find the vector q(t):

Rotating for θ units of time gives:

Thus the rotation about ω can be represented by the matrix exponential

For notation, we can call:

( ) ( )( )tq

tqtq

ωωˆ=

×=

−−

−=

00

0ˆ

12

13

23

ωωωω

ωωω

( ) ( )0ˆ qetq t ω=

( ) ( ) ( )00 ,ˆ qRqetq

θω

θω ==

( )3ˆ so∈ω


Rigid motionsRigid motion is a combination of rotation and translation

Defined by a rotation matrix (R) and a displacement vector (d)

the group of all rigid motions (d,R) is known as the Special Euclidean group, SE(3)

Consider three frames, o0, o1, and o2 and corresponding rotation matrices R21, and

R10

Let d21 be the vector from the origin o1 to o2, d1

0 from o0 to o1

For a point p2 attached to o2, we can represent this vector in frames o0 and o1:

( )3

3R∈

∈

dSOR

( ) ( )33 3 SOSE ×= R

( )01

12

01

212

01

01

12

212

01

01

101

0

12

212

1

ddRpRR

ddpRR

dpRp

dpRp

++=

++=

+=

+=


Homogeneous transformsWe can represent rigid motions (rotations and translations) as matrix

multiplication

Define:

Now the point p2 can be represented in frame o0:

Where the P0 and P2 are:

=

=

10

1012

121

2

01

010

1

dRH

dRH

212

01

0 PHHP =

=

=

1 ,

1

22

00 p

Pp

P


Homogeneous transformsThe matrix multiplication H is known as a homogeneous

transform and we note that

Inverse transforms:

( )3SEH ∈

−=−

101 dRRH

TT


Homogeneous transformsBasic transforms:

Three pure translation, three pure rotation

=

=

=

1000100

00100001

10000100

0100001

100001000010

001

,

,

,

c

b

a

cz

by

ax

Trans

Trans

Trans

−

=

−=

−

=

100001000000

100000001000

100000000001

,

,

,

γγ

γγ

γ

ββ

ββ

β

αα

ααα

cssc

cs

sc

cssc

z

y

x

Rot

Rot

Rot

Introduction to Robotics (Fag 3480) · Fag 3480 - Introduction to Robotics. side 30. Rotation...

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