Robot Kinematics

Representation of an object in 3-D

Representation of the position

=

z

y

xU

q

q

q

Q ; 3x1 matrix

Representation of the Orientation

Frame Transformations

Frame: A set of four vectors carrying position and orientation information

]ZYX[R BU

BU

BUU

B =3x3 matrix

Translation of a Frame

Rotation of a Frame

QQQB

Borg

UU+=

QRQBU

B

U=

Translation and Rotation of a Frame

Borg

UBUB

UQQRQ +=

Borg

UBUB

UQQRQ +=

QTQBU

B

U=⇒

where T: transformation

−−−−

−−−−−−−−=

−−−−⇒

−−−−

−−−−−−−−=

−−−−⇒

1

)1x3(Q

1|000

|

)1x3(Q|)3x3(R

1

)1x3(Q

)1x3(Q

|

|

)1x3(Q|)3x3(R)1x3(Q

B

Borg

UUB

U

B

Borg

UUB

U

Let [T]: Homogeneous transformation matrix

[ ]

−−−−−−−−=1|000

|

)1x3(Q|)3x3(R

TBorg

UUB

[ ]

=

1000

qrrr

qrrr

qrrr

Tz333231

y232221

x131211Say

Translation Operator

Trans : Translation of q units along x-direction

Trans =

Note: Trans operators are commutative in nature

( ) ,X q

( ) ,X q

1000

0100

0010

q001

( ) ( ) ( ) ( )Trans X q Trans Y q Trans Y q Trans X qx y y x , , , ,=

Rotational Operator

Rot : Rotation about axis by an angle θ (anticlockwise sense)( ) ,Z θ Z

OC q Cosx= θDC q Sinx= θAQ q Cosy= θAD BC q Siny= = θ

q q Cos q Sin q XX x y z= − +θ θ 0

q q Sin q Cos q XY x y z= + +θ θ 0

q q X q X q XZ x y z= + +0 0 1

θθθ−θ

=

z

y

x

Z

Y

X

q

q

q

100

0cossin

0sincos

q

q

q

( )

θθθ−θ

=θ100

0cossin

0sincos

,ZRot

In matrix form:

( )

θθθ−θ=θ

cossin0

sincos0

001

,XRot

Similarly, we get

( )

θθ−

θθ=θ

cos0sin

010

sin0cos

,YRot

Properties of Rotation Matrix

• Each row/column of a rotation matrix is a unit vector• Inner (dot) product of each row of a rotation matrix with each

other row becomes equal to 0. The same is true for each column also.

• Rotation matrices are not commutative in nature

• Inverse of a rotation matrix is nothing but its transpose

•

( ) ( ) ( ) ( )1221 ,XROT,YROT,YROT,XROT θθ≠θθ

( ) ( )θ=θ− ,XROT,XROT T1

BA

ABT T= −1

Composite Rotation MatrixComposite rotation matrix representing a rotation of α angle about , followed by a rotation of β angle about axis , followed by a rotation of γ angle about axis.

( ) ( ) ( )αβγ= ,ZROT,YROT,XROTROTcomposite

ZY

X

Representations of Position in Other Than Cartesian Coordinate SystemCylindrical Coordinate System

Steps:1.Starting from the origin O, translate by r units along axis2.Rotate in anti-clockwise sense about axis by an angle θ3.Translate along axis by z units

ˆUZ

ˆUZ

ˆUX

[ ] ( ) ( ) ( )ˆ ˆ ˆ, , ,U U UcompositeT TRANS Z z ROT Z TRANS X rθ=

θθθθθ−θ

=

1000

z100

sinr0cossin

cosr0sincos

We get qx = rcosθqy = rsinθqz = z

Spherical Coordinate System

Steps:1.Starting from the origin O, translate along axis by r units2.Rotate in anti-clockwise sense about axis by an angle α3.Rotate in anti-clockwise sense about axis by an angle β

ˆUZ

ˆUZUY

[ ] ( ) ( ) ( )ˆ ˆ ˆ, , ,U U UcompositeT ROT Z ROT Y TRANS Z rβ α=

ααα−βαβαββαβαβαβ−βα

=

1000

cosrcos0sin

sinsinrsinsincossincos

cossinrcossinsincoscos

We get qx = rsinαcosβqy = rsinαsinβqz = rcosα

Representations of Orientation in Other Than Cartesian Coordinate SystemRoll, Pitch and Yaw Angles

Steps:1.Rotate {B} about by an angle α rolling

2.Rotate {B|} about by an angle β pitching

3.Rotate {B||} about by an angle γ yawing

ˆUX

ˆUZUY

We compare with

( ) ( ) ( )αβγ= ,XROT,YROT,ZROTR UUUrpy:compositeUB

βααββ−γβα+γα−γβα+γαγβ

γβα+γαγβα+γα−γβ=

ccscs

ssccssssccsc

cscsscsssccc

=

333231

232221

131211UB

rrr

rrr

rrr

R

We get

α

β

γ

=

=−

+

=

−

−

−

tan

tan

tan

1 32

33

1 31

112

212

1 21

11

r

r

r

r r

r

r

Using Euler Angles

BU

UBR R= −1

Steps:1. Rotate {B} about by an angle α in anti-clockwise sense2. Rotate {B} about by an angle β in anti-clockwise sense3. Rotate {B} about by an angle γ in anti-clockwise sense

BZ

'BY

"BX

1U BB UR R−=

( ) ( ) ( )" 'ˆ ˆ ˆ, , ,B

U Eulerangles B B BR ROT X ROT Y ROT Zγ β α= − − −

γβγββ−αγ−αγβγα+αγββαγα+αγβγα−αγββα

=ccscs

csscsccssscs

ssccscscsscc

=

333231

232221

131211UB

rrr

rrr

rrr

R

We compare with

UBR

1 21

11

1 31

2 211 21

1 32

33

tan

tan

tan

r

r

r

r r

r

r

α

β

γ

−

−

−

= ÷

− ÷= ÷+

= ÷

Denavit-Hartenberg Notations•Proposed in the year 1955

Link and Joint Parameters

•Offset of linki (di): It is the distance measured from a point where ai-1 intersects the Axisi-1 to the point where ai intersects the Axisi-1 measured along the said axis• Joint Angle (θ_i): It is defined as the angle between the extension

of ai-1 and ai measured about the Axisi-1

Notes:•Revolute joint: θi is variable

• Prismatic joint: di is variable

• Length of linki (ai): It is the mutual perpendicular distance between Axisi-1 and Axisi

•Angle of twist of link_i (αi): It is defined as the angle between Axisi-1 and Axisi

Rules for Coordinate Assignment

• Zi is an axis about which the rotation is considered or along which the translation takes place• If Zi-1 and Zi axes are parallel to each other, X axis will be

directed from Zi-1 to Zi along their common normal

• If Zi-1 and Zi axes intersect each other, X axis can be selected along either of two remaining directions• If Zi-1 and Zi axes act along a straight line, X axis can be selected

anywhere in a plane perpendicular to them•Y axis is decided as Y = ZxX

We have

ii

Ai

BA

CB

iCT T T T T− −=1 1

( ) ( ) ( ) ( )= ROT Z, TRANS Z,d ROT X TRANS X ai i i iθ α, ,

= Screw ScrewZ X

Now,

1

0

0 0 0 1

i i i i i i i

i i i i i i iii

i i i

c s c s s a c

s c c c s a sT

s c d

θ θ α θ α θθ θ α θ α θ

α α−

− − =

[ ]ii

iiT T−− −

=11 1

α−ααθ−αθα−ααθαθ−

−θθ

=

1000

cdcscss

sdscccs

a0sc

iiiiiii

iiiiiii

iii