Mechatronics 1

Week 3 & 4

Learning Outcomes

• By the end of week 3-4 session, students will understand kinematics of industrial robots.

Course Outline

• Forward Kinematics

• Homogeneous Transformation Matrix.

• Denavit Hartenberg (D-H) parameters.

• D-H Transformation Matrix

• Interpretation and application of homogeneous tranformation matrix

• Inverse kinematics.

Homogenous Transformation Matrix

• It provides information on position and orientation in a 4x4 matrix.

1000333231

232221

131211

RRRp

RRRp

RRRp

Tz

y

x

Link i-1 Link i

Joint i-1 axisJoint i axis Joint i+1 axis

a i-1

ai

i

Zi'

Xi'

Yi'

iZi

Yi

Xi

Yi-1Zi-1

Xi-1

di

Ref. Lee, Fu, Gonzalez, 1987

Coordinate Frame Placement(Denavit Hartenberg Concept)

ai : length of link i di : offset distance at joint i

i

i

: joint angle of link i

: twist angle of link i

Rotation Matrix with D-H Parameters

ii

iiiii

iiiii

ii

cossin0

cossincoscossin

sinsinsincoscos1R

ii

iiiii

iiiii

ii

cossin0

cossincoscossin

sinsinsincoscos

1R

Homogenous Transformation Matrix with D-H Parameters

1000

cossin0

sincossincoscossin

cossinsinsincoscos

1

iii

iiiiiii

iiiiiii

ii

d

a

a

A

1000

cossin0

sincossincoscossin

cossinsinsincoscos

1iii

iiiiiii

iiiiiii

ii

d

a

a

A

Coordinate Frame Placement (1)

d2

X0

Z0 Y0

X1

Y1

Z1

Z3

Z2

Y3

X2Y2

X3

Gravity

a2

Coordinate Frame Placement (2)

X0

Z0 Y0

Gravity

a2

Z1

X1

Y1

X2

Z2

Y2

Z3Y3

X3

2

1

3

Homogenous Transf Matrix{W }

{B}

{S}

{T }

{G }

Ref. Craig, 1987

Physical Intepretation ?

Kinematics & Inverse Kinematics

Direct Kinematics

Inverse Kinematics

Joint Coordinates

Joint Coordinates

Position & Orientation of End Effector

Link Parameters

Link Parameters

Inverse Kinematics

• Given a desired position (P) & orientation (R) of the end-effector.

• Find the joint variables which can bring the robot to the desired posture.

),,( 21 nqqqq

x

y

z

Inverse Kinematics

• More difficult

• Solution not unique• Redundant robot• Elbow-up/elbow-down

configuration A

B

1

2

A shorter path is generally desirable

Solving The Problems

• Geometric approach

• Algebraic approach

• Numeric approach

Upper & Lower Configuration(Condition arises in Inverse Kinematics problems)

A

B

1

2

Kinematics(Mitsubishi RV-M1)

L2 L3

Assumed position of endeffector

Base

J2 J3

J1

z0

y0 z1

x0 x1

y1

z2

y2

x2

z3

x3

y3

Kinematics (RV-M1)

Limitation on

Movement of Robot RV-M1

Total Span Min Max

Waist 300o -150o 150o

Shoulder 130o -100o 30o

Elbow 110o 0o 110o

i ai αi di θi

1 0 -90º 0 θ1

2 a2 = L2 0 0 θ2

3 a3 = L3 0 0 θ3

Work Space

Direction of Joint Movements

z0

x0 x1

y0 z1

y1

z2

y2

x2

Arah gerakpositif joint 1

Arah gerakpositif joint 2

Arah gerak positifjoint 3

x3

z3

y3

Joint Direction (+) Direction (–)

1 150o 150o

2 30o 100o

3 110o -