UJWALHARODE 1

Introduction to ROBOTICS

Kinematics of Robot Manipulator

PROF.UJWAL HARODE

UJWALHARODE 2

OutlineReview

Robot Manipulators-Robot Configuration-Robot Specification

• Number of Axes, DOF• Precision, Repeatability

Kinematics-Preliminary

• World frame, joint frame, end-effector frame• Rotation Matrix, composite rotation matrix• Homogeneous Matrix

-Direct kinematics• Denavit-Hartenberg Representation• Examples

-Inverse kinematics

UJWALHARODE 3

ReviewWhat is a robot?

By general agreement a robot is:

A programmable machine that imitates the actions or appearance of an intelligent creature–usually a human.

To qualify as a robot, a machine must be able to: 1) Sensing and perception: get information from its surroundings 2) Carry out different tasks: Locomotion or manipulation, do something physical–such as move or manipulate objects3) Re-programmable: can do different things4) Function autonomously and/or interact with human beings

Why use robots? – Perform 4A tasks in 4D environments

4A: Automation, Augmentation, Assistance, Autonomous

4D: Dangerous, Dirty, Dull, Difficult

UJWALHARODE 4

Manipulators

• Robot arms, industrial robot• Rigid bodies (links)

connected by joints• Joints: revolute or

prismatic• Drive: electric or

hydraulic • End-effector (tool)

mounted on a flange or plate secured to the wrist joint of robot

5

Manipulators

Robot Configuration

SCARA: RRP

(Selective Compliance Assembly Robot Arm)

Cartesian: PPP Cylindrical: RPP Spherical: RRP

Articulated: RRR

Hand coordinate:

n: normal vector; s: sliding vector;

a: approach vector, normal to the

tool mounting plate

UJWALHARODE 6

Manipulators• Motion Control Methods

• Point to point control• a sequence of discrete points• spot welding, pick-and-place, loading

& unloading

• Continuous path control• follow a prescribed path, controlled-

path motion• Spray painting, Arc welding, Gluing

UJWALHARODE 7

Manipulators

• Robot Specifications• Number of Axes

• Major axes, (1-3) => Position the wrist• Minor axes, (4-6) => Orient the tool• Redundant, (7-n) => reaching around

obstacles, avoiding undesirable configuration

• Degree of Freedom (DOF)• Workspace• Payload (load capacity)• Precision v.s. Repeatability

UJWALHARODE 8

What is Kinematics

Forward kinematics

Given joint variables

End-effector position and orientation, -

Formula?

),,,,,,( 654321 nqqqqqqqq

),,,,,( TAOzyxY

x

y

z

UJWALHARODE 9

What is Kinematics

Inverse kinematicsEnd effector position and orientation

Joint variables -Formula?

),,,,,( TAOzyx

),,,,,,( 654321 nqqqqqqqq

UJWALHARODE 10

Example 1

)/(cos

kinematics Inverse

sin

cos

kinematics Forward

01

0

0

lx

ly

lx

l

0y

1x1y

0x

UJWALHARODE 11

Preliminary

• Robot Reference Frames1. World frame2. Joint frame3. Tool frame

W

z x

yz

x

y

T

T

R

UJWALHARODE 12

Coordinate Transformation

x

y

z

P

u

vw

O, O’

Reference coordinate frame –XYZBody-attached frame-ijk

Point represented in XYZT

zyxxyz pppP ],,[zyx kji zyxxyz pppP

Point represented in uvw:

wvu kji wvuuvw pppP

Two frames coincide

UJWALHARODE 13

Properties: Dot Product

Let and be arbitrary vectors in and be the angle from to , then

cosyxyx

0

0

0

jk

ki

ji

1||

1||

1||

k

j

i

Properties of orthonormal co-ordinate frame

Unit vectors are mutually perpendicular

UJWALHARODE 14

Coordinate TransformationReference coordinate frame OXYZBody-attached frame O’uvw

x

y

zP

u

vw

O, O’

uvwxyz RPP

How to relate the coordinate in these two frames?

UJWALHARODE 15

• Basic Rotation– , , and represent the projections of onto

OX, OY, OZ axes, respectively

– Since

xp yp zp P

wvu kji wvu pppP

wvux pppPp wxvxuxx kijiiii

wvuy pppPp wyvyuyy kjjjijj

wvuz pppPp wzvzuzz kkjkikk

UJWALHARODE 16

Basic Rotation Matrix

Rotation about x-axis with

w

v

u

z

y

x

p

p

p

p

p

p

wzvzuz

wyvyuy

wxvxux

kkjkik

kjjjij

kijiii

x

z

y

v

wP

u

CS

SCxRot

0

0

001

),(

UJWALHARODE 17

0

010

0

),(

CS

SC

yRot

100

0

0

),(

CS

SC

zRot

Rotation about y-axis with :

Rotation about z-axis with :

UJWALHARODE 18

Example 2

• A point is attached to a rotating frame, the frame rotates 60 degree about the OZ axis of the reference frame. Find the coordinates of the point relative to the reference frame after the rotation.

)2,3,4(uvwa

2

964.4

598.0

2

3

4

100

05.0866.0

0866.05.0

)60,( uvwxyz azRota

UJWALHARODE 19

Composite Rotation Matrix

• A sequence of finite rotations – matrix multiplications do not commute– rules:

• if rotating coordinate O-U-V-W is rotating about principal axis of OXYZ frame, then Pre-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix

• if rotating coordinate OUVW is rotating about its own principal axes, then post-multiply the previous (resultant) rotation matrix with an appropriate basic rotation matrix

UJWALHARODE 20

• Find the rotation matrix for the following operations:

...

axis OUabout Rotation

axisOW about Rotation

axis OYabout Rotation

Answer

SSSCCSCCSSCS

SCCCS

CSSSCCSCSSCC

CS

SCCS

SC

uRotwRotIyRotR

0

0

001

100

0

0

C0S-

010

S0C

),(),(),( 3

Pre-multiply if rotate about the OXYZ axes

Post-multiply if rotate about the OUVW axes

UJWALHARODE 21

Homogeneous Transformation• Special cases

1. Translation

2. Rotation

10 31

'33

oA

BA rIT

10

0

31

13BA

BA RT

101333 PR

F

Homogeneous transformation matrix

UJWALHARODE 22

Homogeneous Transformation

• Composite Homogeneous Transformation Matrix

• Rules:– Transformation (rotation/translation) w.r.t (X,Y,Z)

(OLD FRAME), using pre-multiplication– Transformation (rotation/translation) w.r.t (U,V,W)

(NEW FRAME), using post-multiplication

UJWALHARODE 23

Orientation Representation

• Description of Roll Pitch Yaw

X

Y

Z

YAW

PITCH

ROLL