Direct Kinematics

Link Description

The concept of Direct Kinematics

• Choosing wisely the coordinate systems on the links

• If the wise choice was made, each link can be represented with 4 parameters

• When the parameters are found, the transformation matrices between the links can be found from a closed formula

DK Algorithm

• 1) Draw sketch• 2) Identify and number robot links. Base = 0, Last = n• 3) Draw axis Zi for joint i. For rotating joint, Zi is the

rotation axis. For prismatic (translating) joint, Zi can merge with the DOF axis or be perpendicular to it.

• 5) Determine joint length ai-1 between Zi-1 and Zi

• 6) Draw axis Xi-1 along the shortest distance between Zi-1 and Zi. If the distance is 0, choose the direction of Xi-1 to be a normal to the plane that they create.

DK Algorithm (2)

• 7) Determine joint twist i-1 measured around Xi-1 (between Zi-1 and Zi)

• 8) Determine the joint offset di

• 9) Determine joint angle i around Zi

• 10) Write DH table• 11+12) Write link transformations and calculate

the common transformation

Kinematics Parameters of a link

1i

Link length  

Link twist

1ia

What are the kinematics parameters of this link?

• a = 7 = 450

Kinematics Parameters of a link

• Link offset d• Joint angle  

Summary of the link parameters in terms of link frames

• ai = the distance from Zi to Zi+1 measured along Xi i = the angle between Zi and Zi+1 measured about Xi• di = the distance from Xi-1 to Xi measured along Zi i = the angle between Xi-1 and Xi measured about Zi

• We usually choose ai > 0 since it corresponds to a distance;

• However, i , di , i are signed quantities.

There is no unique attachment of frames to links:

• 1. When we align Zi axis with joint axis i, two choices of the Zi direction.

• 2. When we have intersecting joint axes (ai=0), two choices of the Xi direction, corresponding to choice of signs for the normal to the plane containing Zi and Zi+1.

• 3. When axes i and i+1 are parallel, the choice of origin location for {i} is arbitrary (generally chosen in order to cause di to be zero).

PTTTTPTP iPi

QP

RQ

iR

iii

i 111

TTTTT Ri

PR

RQ

iR

ii

11

iZiZiXiXi

i dDRaDRT 111

iiZiiXi

i dScrewaScrewT ,, 111

1000

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

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

0)sin()cos(

1111

1111

1

1

iiiiiii

iiiiiii

iii

ii d

d

a

T

Three link Arm : RPR mechanism

• “Cylindrical” robot – 2 joints analogous to polar coordinates when viewed from above.

• Schematic: point – axes intersection; prismatic joint at minimal extension

• Find coordinate systems and a, , d, (i=3)

i ai i di i

0 0 0

1 0 90 0 1

2 0 0 d2 0

3 L2 3

DH table:

1000

0100

00)cos()sin(

00)sin()cos(

11

11

01

T

1000

0010

100

0001

212

dT

1000

100

00)cos()sin(

00)sin()cos(

2

33

33

23 LT

TTTTT NNN12

312

01

0 ...