Forward Kinematics

University of Bridgeport

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Introduction to ROBOTICS

Kinematic• Forward (direct) Kinematics• Given: The values of the joint variables.• Required: The position and the orientation of the end

effector.

• Inverse Kinematics• Given : The position and the orientation of the

end effector.• Required : The values of the joint variables.

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Why DH notation

• Find the homogeneous transformation H relating the tool frame to the fixed base frame

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Why DH notation

• A very simple way of modeling robot links and joints that can be used for any kind of robot configuration.

• This technique has became the standard way of representing robots and modeling their motions.

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DH Techniques

1. Assign a reference frame to each joint (x-axis and z-axis). The D-H representation does not use the y-axis at all.

2. Each homogeneous transformation Ai is represented as a product of four basic transformations

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DH Techniques

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• Matrix Ai representing the four movements is found by: four movements

, , , ,i i i ii z z d x a xA Rot Trans Trans Rot

1. Rotation of about current Z axis

2. Translation of d along current Z axis

3. Translation of a along current X axis

4. Rotation of about current X axis

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i i i i i i i

i i i i i i i

i i i

c -c s s s a c

s c c -s c a s

0 s c d

0 0 0 1

iA

CS

SCxRotRx0

0

001

),(,

100

0

0

),( ,

CS

SC

zRotRz

1000

00

00

0001

1000

0100

0010

001

1000

100

0010

0001

1000

0100

00

00

ii

ii

i

i

ii

ii

i CS

SC

a

d

CS

SC

A

DH Techniques

• The link and joint parameters :

• Link length ai : the offset distance between the Zi-1 and Zi axes along the Xi axis.

• Link offset di the distance from the origin of frame i−1 to the Xi axis along the Zi-1 axis.

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DH Techniques

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•Link twist αi :the angle from the Zi-1 axis to the Zi axis about the Xi axis. The positive sense for α is determined from zi-1 and zi by the right-hand rule.

•Joint angle θi the angle between the Xi-1 and Xi axes about the Zi-1 axis.

DH Techniques

• The four parameters: ai: link length, αi: Link twist , di : Link offset and

θi : joint angle.

• The matrix Ai is a function of only a single variable qi , it turns out that three of the above four quantities are constant for a given link, while the fourth parameter is the joint variable.

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DH Techniques

• With the ith joint, a joint variable is qi associated where

All joints are represented by the z-axis.• If the joint is revolute, the z-axis is in the direction of

rotation as followed by the right hand rule.• If the joint is prismatic, the z-axis for the joint is along

the direction of the liner movement.

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DH Techniques

3. Combine all transformations, from the first joint (base) to the next until we get to the last joint, to get the robot’s total transformation matrix.

4. From , the position and orientation of the tool frame are calculated.

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01 2. .......n nT A A A

0nT

DH Techniques

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DH Techniques

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DH Techniques

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DH Techniques

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i i i i i i i

i i i i i i i

i i i

c -c s s s a c

s c c -s c a s

0 s c d

0 0 0 1

iA

Example I The two links arm

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• Base frame O0

•All Z ‘s are normal to the page

Example I The two links arm

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Where (θ1 + θ2 ) denoted by θ12 and

Example 2

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Example 2

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Example 3 The three links cylindrical

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Example 3 The three links cylindrical

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Example 3 The three links cylindrical

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Example 3 The three links cylindrical

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Example 4 Spherical wrist

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Example 4 Spherical wrist

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Example 4 Spherical wrist

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Example 4 Spherical wrist

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Example 5The three links cylindrical with Spherical wrist

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Example 5The three links cylindrical with Spherical wrist

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03T

36T• given by example 2, and given by

example 3.

Example 5The three links cylindrical with Spherical wrist

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Example 5The three links cylindrical

with Spherical wrist

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Example 5The three links cylindrical

with Spherical wrist

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• Forward kinematics:1. The position of the end-effector: (dx ,dy ,dz )

2. The orientation {Roll, Pitch, Yaw }Rotation about X axis{ROLL}

Rotation about fixed Y axis{PITCH}

Rotation about fixed Z axis{YAW}

Roll Pitch Yaw• The rotation matrix for the following

operations:

X

Y

Z

axis{YAW} Zfixedabout Rotation

}axis{PITCH Y fixedabout Rotation

axis{ROLL} Xabout Rotation

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CCSCS

CSSSCSCSSCCS

SSCSCSSCSSCC

CS

SCCS

SC

xRotyRotzRotR

0

0

001

C0S-

010

S0C

100

0

0

),(),(),(

Example 4The three links cylindrical with Spherical wrist

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• How to calculate

• Compare the matrix R with Of the matrix

, ,and

06T

C C S S C S S C S C S S

R S C C S S C S C S S S C

S C S C C

31S r 32C S r 21S C r

131( )Sin r 1 32( )

rSin

C

1 21sin ( )

r

C

11 12 13

21 22 23

31 32 33

r r r

r r r

r r r

Module 1RRR:RRR

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Links α a θ d

1 90 0 * 10

2 0 10 * 0

3 -90 0 * 0

4 90 0 * 10

5 -90 0 * 0

6 0 0 * 0

HW

• From Spong book: page 112• 3.2, 3.3, 3.4, 3.6 , 3.7, 3.8, 3.9, 3.11

• No Class on next Tuesday

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Module 1

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1 2 3, ,and • Where are Roll, Pitch, and Yaw

Representing forward kinematics

• Forward kinematics

1

2

3

4

5

6

x

y

z

p

p

p

11 12 13

21 22 23

31 32 33

0 0 0 1

x

y

z

r r r d

r r r dT

r r r d

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• Transformation Matrix

Remember

• For n joints: we have n+1 links. Link 0 is the base• Joints are numbered from 1 to n• Joint i connects link i − 1 to link i. • Frame i {Xi Yi Zi} is attached to joint i+1.

• So, frame {O0 X0 Y0 Z0}, which is attached to the robot base (inertial frame) “joint 1”.

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