Introduction to Robotics

Tutorial II

Alfred BrucksteinYaniv Altshuler

Denavit-Hartenberg

•Specialized description of articulated figures

•Each joint has only one degree of freedom

•rotate around its z-axis

•translate along its z-axis

Reminder

Denavit-Hartenberg• Link length ai

• The perpendicular distance between the axes of jointi and jointi+1

• Link twist αi

• The angle between the axes of jointi and jointi+1

• Angle around xi-axis

Denavit-Hartenberg

• Link offset di

• The distance between the origins of the coordinate frames attached to jointi and jointi+1

• Measured along the axis of jointi

Denavit-Hartenberg

• Link rotation (joint angle) φi

• The angle between the link lenghts αi-1 and αi

• Angle around zi-axis

Denavit-Hartenberg

Denavit-Hartenberg

1.Compute the link vector ai and the link length

2.Attach coordinate frames to the joint axes

3.Compute the link twist αi

4.Compute the link offset di

5.Compute the joint angle φi

6.Compute the transformation (i-1)Ti which transforms entities from linki to linki-1

Denavit-Hartenberg

This transformation is done in several steps :

•Rotate the link twist angle αi-1 around the axis xi

•Translate the link length ai-1 along the axis xi

•Translate the link offset di along the axis zi

•Rotate the joint angle φi around the axis zi

iiii xxzzi

i RotTransTransRotT 1

Denavit-Hartenberg

1000

0cossin0

0sincos0

0001

ii

iixi

Rot

Denavit-Hartenberg

1000

0100

0010

001 i

x

a

Transi

Denavit-Hartenberg

1000

100

0010

0001

iz d

Transi

Denavit-Hartenberg

1000

0100

00cossin

00sincos

ii

ii

ziRot

Denavit-Hartenberg

Multiplying the matrices :

iiii xxzzi

i RotTransTransRotT 1

1000

cossin0

sinsincoscoscossin

cossinsincossincos

1

iii

iiiiiii

iiiiiii

ii

d

a

a

T

DH Example3 revolute joints

Shown in home position

Link 1 Link 3

Link 2

joint 1

joint 2 joint 3

R

L1 L2

Shown with joints in non-zero positions

Z1

Z0

Z21

2 3

x0

x1

x2

z3

x3

Observe that frame i moves with link i

DH Example

Z1

Z0

Z2

1

23

x0

x1 x2 x3

Z3

Link Var d a

1 1 1 0 90o R

2 2 2 0 0 L1

3 3 3 0 0 L2

R

L1 L2

1

DH Example

1 = 90o (rotate by 90o around x0 to

align Z0 and Z1)

Link Var d a

1 1 1 0 90o R

2 2 2 0 0 L1

3 3 3 0 0 L2

DH Example

DH Example

z1

z0

z21

2 3

x0

x1

x2

z3

x3

Origin of {1} w.r.t. {0}

x1 axis expressed wrt {0}

y1 axis expressed wrt {0}

z1 axis expressed wrt {0}

DH Example

z1

z0

z21

2 3

x0

x1

x2

z3

x3

Origin of {2} w.r.t. {1}

x2 axis expressed wrt {1}

y2 axis expressed wrt {1}

z2 axis expressed wrt {1}

DH Example

z1

z0

z21

2 3

x0

x1

x2

z3

x3

Origin of {3} w.r.t. {2}

x3 axis expressed wrt {2}

y3 axis expressed wrt {2}

z3 axis expressed wrt {2}

DH Example

where

DH Example

Example – the Stanford Arm

Example – the Stanford Arm

X1Y1

Z1

X2

Z2

X3

Z3

X4

X5

X6

Z4

Z5

Z6

X7

Z7

Example – the Stanford Arm

X1Y1

Z1

X2

Z2

X3

Z3

X4

X5

X6

Z4

Z5

Z6

X7

Z7

i ai di i i

1 0 d1 90° 1

2

3

4

5

6

Example – the Stanford Arm

X1Y1

Z1

X2

Z2

X3

Z3

X4

X5

X6

Z4

Z5

Z6

X7

Z7

i ai di i i

1 0 d1 90° 1

2 0 d2 90° 2

3

4

5

6

Example – the Stanford Arm

X1Y1

Z1

X2

Z2

X3

Z3

X4

X5

X6

Z4

Z5

Z6

X7

Z7

i ai di i i

1 0 d1 90° 1

2 0 d2 90° 2

3 0 d3

(var)90° 90°

4

5

6

Example – the Stanford Arm

X1Y1

Z1

X2

Z2

X3

Z3

X4

X5

X6

Z4

Z5

Z6

X7

Z7

i ai di i i

1 0 d1 90° 1

2 0 d2 90° 2

3 0 d3

(var)90° 90°

4 0 d4 90° 4

5

6

Example – the Stanford Arm

X1Y1

Z1

X2

Z2

X3

Z3

X4

X5

X6

Z4

Z5

Z6

X7

Z7

i ai di i i

1 0 d1 90° 1

2 0 d2 90° 2

3 0 d3

(var)90° 90°

4 0 d4 90° 4

5 0 d5 0° 5

6

Example – the Stanford Arm

X1Y1

Z1

X2

Z2

X3

Z3

X4

X5

X6

Z4

Z5

Z6

X7

Z7

i ai di i i

1 0 d1 90° 1

2 0 d2 90° 2

3 0 d3

(var)90° 90°

4 0 d4 90° 4

5 0 d5 0° 5

6 0 d6 0° 6