Robotics

Robot Motion Analysis

Presented ByDeepam Goyal

Department of Mechanical EngineeringNational Institute of Technical Teachers Training & Research

Chandigarh – 160 019July, 2015

Historical Perspective Robot Definitions Basic Fundamentals of Robots Types of Kinematics Transformations Geometric Interpretation of Rotation Matrix Inverse transformations Homogeneous transformations Inverse Homogeneous transformations Inverse Kinematics of a Two Link Manipulator Important Terminology References

Contents

Historical Perspective• The word robot was first used in 1921 by Czech playwright

Karel Capek in his satirical drama titled Rossum’s Universal Robots– Derived from Czech word robota which literally means

‘forced labour’.

• The term robotics was coined by American author and professor of biochemistry at Boston University, Isaac Asimov in his short story titled Runaround.

Robot Definitions• Webster dictionary

– An automatic apparatus or device that performs functions ordinarily ascribed to humans or operates with what appears to be almost human intelligence

• Robotic Institute of America (RIA)– A robot is a reprogrammable,

multifunctional manipulator designed to move material, parts, tools or specialized devices through variable programmed motions for the performance of a variety of tasks.

Basic Components of a Robot System• Manipulator

– Series of rigid members, called links, connected by joints.• Actuators

– Provide power to the manipulator.• Sensory Devices

– To monitor position, speed, acceleration, torque etc.• Controller

– Provides the intelligence to make the manipulator perform in a certain manner.

• Power Conversion Unit– Takes signal from controller and converts it into meaningful

power level so that actuators can move.

Robot Configurations• Cartesian (3P)

– Three Degrees of freedom (DOF) are linear & at right angles to each other.

– Rectangular Workspace.• Cylindrical (R2P)

– Two DOFs are linear and one DOF is rotational.– Cylindrical Workspace.

• Spherical (2RP)– One DOF is linear and two DOFs are rotational.– Spherical Workspace.

• Articulated (3R)– All three DOFs are rotational.– Irregular Workspace.

Contd..

Types of Joints

Joints Motions Degree of Freedom

Revolute joint Rotary motion One

Prismatic joint Sliding motion One

Cylindrical joint One sliding & one rotary motion Two

Planar joint Two sliding & one rotary motion Three

Screw pair One translatory & one rotary motion Two

Spherical joint Three rotary motion three

Types of Kinematics

The transformation matrix is 4X4 matrix which consists of sub-matrices :

Rotation Matrix Translation or Position Vector Perspective Transformation Scaling/ Stretching

Transformation

Geometric Interpretation of Rotation Matrix• Rotation about X-axis

• Rotation about Y-axis

• Rotation about Z-axis

Inverse transformations

Problem

Example : A mobile body reference frame OABC is rotated 60 about OY-

axis of the fixed base reference system OXYZ. If and are

the coordinates with respect to OXYZ plane, what are the

corresponding coordinates of p and q with respect to OABC frame.

Txyzp )6,4,2(

Txyzq )7,5,3(

Homogeneous Transformations

Components of Homogeneous Transformations

Contd..

ProblemExample: Determine the homogeneous transformation

matrix to represent the following sequence of operations.

a) Rotation of about OX-axisb) Translation of 4 units along OX-axisc) Translation of -6 units along OC-axisd) Rotation of about OB-axis

3

6

Inverse Homogeneous Transformation

1000100

)cossin(0cossin)sincos(0sincos

),(1

z

yx

yx

ppppp

zH

Revolute and Prismatic Joints Combined

1

X

Y

S

(x , y) )xy(tanθ 1

)y(xS 22

(i)

(ii)

Inverse Kinematics of a One Link Manipulator

(x , y)

2

1

l2

l1

Given: l1, l2 , x , yFind: 1, 2

Redundancy:A unique solution to this problem does not exist. Notice,

that using the “givens” two solutions are possible. Sometimes no solution is possible.

l2

l1

(x , y)

l2l1

Inverse Kinematics of a Two Link Manipulator

21

22

21

221

2

2212

22

1

211211212

22

1

211212

212

22

12

1211212

212

22

12

1

2222

2yxcosθ

c2

)(sins)(cc2

)(sins2)(sins)(cc2)(cc

yx)2((1)

llll

llll

llll

llllllll

The Algebraic Solution

21

21211

21211

1221

11

θθθ)(sinsy)(ccx)(

)θcos(θccosθc

iiilliilli

Only Unknown

)c(s)s(c cscss

sinsy

)()c(c ccc

ccx

2211221

12221211

21211

2212211

21221211

21211

llllll

ll

slsllsslll

ll

We know what 2 is from the previous slide. We need to solve for 1 . Now we have two equations and two unknowns (sin 1 and cos 1 )

2222221

1

2212

22

1122221

221122221

221

221

2211

yxx)c(ys

)c2(sx)c(

1

)c(s)s()c()(xy

)c()(xc

slll

llllslll

lllllsls

llsls

(Substituting for c1 and simplifying many times)

(This is the law of cosines and can be replaced by x2+ y2)

22222211

1 yxx)c(ysinθ slll

The Geometric Solution

l2

l1

2

1

(x , y)

Using the Law of Cosines:

21

22

21

221

21

22

21

22

212

22

122

222

2cosθ

2)cos(θ

)cos(θ)θ180cos()θ180cos(2)(

cos2

llllyx

llllyx

llllyx

Cabbac

2

2

22

2

Using the Law of Sines:

xy2tanα

αθθ

yx)sin(θ

yx)θsin(180θsin

sinsin

1

11

222

222

2

1

l

cC

bB

xy2tan

yx

)sin(θsinθ 1

22221

1l

Redundant since 2 could be in the first or fourth quadrant.

Redundancy caused since 2 has two possible values

Degeneracy : The robot looses a degree of freedom and thus cannot

perform as desired. When the robot’s joints reach their physical limits, and as a ٭result, cannot move any further.

-In the middle point of its workspace if the z-axes of two simi ٭lar joints becomes collinear.

Dexterity : The volume of points where one can position the robot

as desired, but not orientate it.

Important Terminology

Problems

Example 1: Determine the homogeneous transformation matrix to represent the following sequence of operations.

a) Translation of 4 units along OX-axisb) Rotation of OX-axisc) Translation of -6 units along OC-axisd) Rotation of about OB-axis

3

6

References

• Groover, M.P., Emory W. Zimmers JR. “CAD/CAM:Computer-Aided Design and Manufacturing”. 25. New Delhi: Prentice Hall of India Private Limited, 2002. 324-332. Print.

• Hegde, Ganesh S. "Robot Motion Analysis." A Textbook on Industrial Robotics. Second ed. New Delhi: U Science, 2009. 25-114. Print.

• Niku, Saeed B. "Robotic Kinematics : Position Analysis." Introduction to Robotics: Analysis, Systems, Applications. 1st ed. New Delhi: PHI Learning Private Limited, 2009. 29-90. Print.

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