areas interest in robotics

Industrial Automation

Areas Interest in Robotics

Industrial Engineering Department

Binghamton University


Outline• Introduction

– Historical Example– Mechanical Engineering and Robotics

• Review of Basic Kinematics and Dynamics• Transformation Matrices/Denavit-

Hartenberg• Dynamics and Controls• Example: Surgical Robot


Robot Configurations


Cartesian Cylindrical

SphericalSCARA


Phillip John McKerrow, Introduction to Robotics (1991)


Review of Basic Kinematics and Dynamics

• Case Study: Dynamic Analysis

• Software for Dynamic Analysis: ADAMS

• Rigid Body Kinematics

• Rigid Body Dynamics


Kinematics of Rigid Bodies

General Plane Motion: Translation plus Rotation


Kinematics of Rigid Bodies (cont.)

Translation

If a body moves so that all the particles have at time t the same velocity relative to some reference, the body is said to be in translation relative to this reference.

Rectilinear Translation Curvilinear Translation


RotationIf a rigid body moves so that along some straight line all the particles of the body, or a hypothetical extension of the body, have zero velocity relative to some reference, the body is said to be in rotation relative to this reference.

The line of stationary particles is called the axis of rotation.


Motion


General Plane Motion can be analyzed as: A translation plus a rotation.

Chasle’s Theorem:1. Select any point A in the body. Assume that

all particles of the body have at the same time t a velocity equal to vA, the actual velocity of the point A.

2. Superpose a pure rotational velocity about an axis going through point A.



General Plane Motion: drA drB



General Plane Motion; (1) Translation

measured from originalPoint A



General Plane Motion:(2) Rotation about axis through Point A



General Plane Motion = Translation + Rotation



R

Derivative of a Vector Fixed in a Moving Reference O

yz

x

X Y

Z

O

Two Reference Frames:XYZx'y'z'

Let R be the vector that establishes the relative position between XYZ and x'y'z'.

P

A

Let A be the fixed vector that establishes the position between A and P.

AKinematics of Rigid Bodies (cont.)


AThe time rate of change of A as seen from x'y'z' is zero: R

Oy

z

x

X Y

Z

O

0'''

zyxdt

Ad



AAs seen from XYZ, the time rate of change of A will not necessarily be zero.

Determine the time derivative by applying Chasles’ Theorem.

1. Translation. Translational motion of R will not alter the magnitude or direction of A. (The line of action will change but the direction will not.)

R O

yz

x

X Y

Z

O



R O

yz

x

X Y

Z

O

2. Rotation. Rotation about an axis passing through O':

Establish a second stationary reference frame, X'Y'Z', such that the Z' axis coincides with the axis of rotation.

Oy

z

x

A

X'

Y'

Z'

Oy

z

xO

yz

x



R

X Y

Z

O

A

X'

Y'

Z'

Locate a set of cylindrical coordinates at the end of A.

r

'Z

'' ZZrr AAAA

Because A is a fixed vector, the magnitudes Ar, A, and AZ' are constant. Therefore:

0' Zr AAA

Also, Z' is unchanging, therefore:

0' Z



The time derivative as seen from the X'Y'Z' reference frame is: R

X Y

Z

O

A

X'

Y'

Z'

r

'Z

''''''''' ZYXZYX

rr

ZYXdt

dA

dt

dA

dt

Ad

Recall: rr

dt

d

dt

d

and Note:

rr

ZYX

AAdt

Ad

'''



The result for the time derivative as seen from the X'Y'Z' reference frame is: R

X Y

Z

O

A

X'

Y'

Z'

r

'Z

rr

ZYX

AAdt

Ad

'''

XYZZYX dt

d

dt

d

'''

Both the X'Y'Z' reference frame and the XYZ reference frame are stationary reference frames, therefore

rr

ZYXXYZ

AAdt

Ad

dt

Ad

'''



R

X Y

Z

O

A

X'

Y'

Z'

r

'Z

rr

XYZ

AAdt

Ad

For: A

AAAAAAAA

A

r

r

Zr

Zr

Zr

Zr

000000

''

'Z

'' ZZrr AAAA

Ar 0 00 0 rA

rr AA



R

X Y

Z

O

A

X'

Y'

Z'

r

'Z

For acceleration, differentiate:

Adt

dA

XYZ

By the product rule:

XYZXYZXYZdt

AdA

dt

d

dt

Ad

2

2

XYZdt

d Adt

Ad

XYZ

XYZdt

Ada

2

2



R

X Y

Z

O

A

X'

Y'

Z'

r

'Z

AAdt

Ada

XYZ

2

2



Summary of Equations: Kinematics of Rigid Bodies

A

BA

B rraa AB

ABrvv AB



Degrees of Freedom

Degrees of Freedom (DOF) = df. The

number of independent parameters (measurements, coordinates) which are needed to uniquely define a system’s position in space at any point of time.



A rigid body in plane motion has three DOF.

Note: The three parameters are not unique.x, y, – is one set of three coordinates

O

r

r, , – is also a set of three coordinates



O

r

X

A rigid body in 3-D space has six DOF.

For example,x, y, z – three linear coordinates and – three angular coordinates



Links, Joints, and Kinematic Chains

Link = df. A rigid body which

possesses at least two nodes which are points for attachment to other links.



Joint = df. A connection between two or more links (at their nodes) which allows some motion, or potential motion, between the connected links.

Also called “kinematic pairs.”



Type of contact between links

Lower pair: surface contact

Higher pair: line or point contactSix Lower Pairs



“Constrained Pin” “Screw”

“Slide” “Sliding Pin”

Kinematics of Rigid Bodies (cont.) CS 480A-34


Planar (F) Joint – 3 DOF

“Ball and Socket”



Open/Closed Kinematic Chain (Mechanism)Closed Kinematic Chain = df. A kinematic chain in

which there are no open attachment points or nodes.



Open Kinematic Chain = df. A kinematic chain in

which there is at least one open attachment point or node.



Dynamics of Rigid Bodies

Dynamic Equivalence

Lumped Parameter Dynamic Model


Dynamic System Model

For a model to be dynamically equivalent to the original body, three conditions must be satisfied:

1. The mass (m) used in the model must equal the mass of the original body.

2. The Center of Gravity (CG) in the model must be in the same location as on the original body.

3. The mass moment of inertia (I) used in the model must equal the mass moment of inertia of the original body.

m, CG, I


First Moment of Mass and Center of Gravity (CG)

The first moment of mass, or mass moment (M), about an axis is the product of the mass and the distance from the axis of interest.

m

rdmM

where: r is the radius from the axis of interest to the increment of mass

Dynamics of Rigid Bodies (cont.)


Second Moment of Mass, Mass Moment of Inertia (I)

The second moment of mass, or mass moment of inertia (I), about an axis is the product of the mass and the distance squared from the axis of interest.

m

m dmrI 2

where: r is the radius from the axis of interest to the increment of mass



Lumped Parameter Dynamic Models

The dynamic model of a mechanical system involves “lumping” the dynamic properties into three basic elements:

Mass (m or I)

Spring

Damper

m



Manipulator Dynamics and Control• Forward Kinematics – Given the angles and/or

extensions of the arm, determine the position of the end of the manipulator

• Inverse Kinematics – Given the position of the end of the manipulator, determine the angles and/or extensions of the arm needed to get there

• Dynamics – Determine the forces and torques required for or resulting from the given kinematic motions.

• Control – Given the block diagram model of the dynamic system, determine the feedback loops and gains needed to accomplish the desired performance (overshoot, settling time, etc.)


Forward Kinematics:Denavit-Hartenberg (D-H)

Transformation Matrix

• Forward Kinematics – Given the angles and/or extensions of the arm, determine the position of the end of the manipulator


Position Kinematics


While the kinematic analysis of a robot manipulator can be carried out using any arbitrary reference frame, a systematic approach using a convention known as the Denavit-Hartenberg (D-H) convention is commonly used.Any homogeneous transformation is represented as the product of four 'basic" transformations:

Mark W. Spong and M. Vidyasagar, Robot Dynamics and Control (1989)

iiii xaxdzzi RotTransTransRotA ,,,,

Start hereStart here

End hereEnd here



iiii xaxdzzi RotTransTransRotA ,,,,

1000

00

00

0001

1000

0100

0010

001

1000

100

0010

0001

1000

0100

00

00

ii

iiii

ii

cs

sc

a

d

cs

sc

A

i

ii

1000

0 i

i

i

i dcs

sascccs

casscsc

Aii

iiiiii

iiiiii


Example


1000

1110

0

0

111

111

1

1

1

sacs

casc

A

1000

1110

0

0

222

222

2

2

2

sacs

casc

A

212

0 AAT

11

0 AT


1000

0100

0

0

1211221212121

1211221212121

12

12

sascscaccsssccs

cassccacsscsscc

1000

0100

0

0

1000

0100

0

0

222

222

111

111

2

2

1

1

212

0

sacs

casc

sacs

casc

AAT

1000

1110

0

0

21

12121 12

c

cacac sinsincoscoscos

sincoscossinsin 21 s

21 s

121 12 sasa


Given the angles, 1 and 2, along with

the link lengths, a1 and a2, the position of

the end point of the two-link planar manipulator with respect to the base of the manipulator can be found using the D-H transformation matrix:

1000

1110

0

0

2112121

2112121

21

21

20

sasacs

cacasc

T


Similarly for any robot configuration:

1000333231

232221

131211

60

z

y

x

drrr

drrr

drrr

T

Stanford manipulator configuration:


52452632

2155142541621321

5412515421621321

5254233

54152542123

54152542113

65264654232

646541652646542122

646541652646542112

65264654231

646541652646542121

646541652646542111

sscccddcd

sscssccsscddcdssd

ssssccscccddsdscd

ccscsr

ssccssccsr

ssscsscccr

ssccssccsr

ccscscsssssscccsr

ccscssssssssccccr

cscsscccsr

scccsccssssccccsr

scccsscsssscccccr

z

y

x

where:

d3

d6

d2


Velocity Kinematics


JacobianThe Jacobian is a matrix valued function of derivatives.

n

nnn

n

n

v

x

f

x

f

x

f

x

f

x

f

x

fx

f

x

f

x

f

J

JJ

21

2

2

2

1

2

1

2

1

1

1


00

0cos

0sin

11

11

1ql

ql

J c

c

vc

00

coscoscos

sinsinsin

21221111

21221211

2qqlqqlql

qqlqqlql

J cc

cc

vc

Linear Velocities


Inverse Kinematics

• Inverse Kinematics – Given the position of the end of the manipulator, determine the angles and/or extensions of the arm needed to get there


In general the problem can be stated:Given the 4x4 D-H homogeneous transformation


10

dRH

Find one (or all) of the solutions of the equation

nnn

nn

AAAqqqT

HqqqT

...,...,,

:where

,...,,

21210

210

In other words, solve the system of equations:

nji

hqqqT ijnij

,...,3 ,2 ,13 ,2 ,1

:where

,...,, 21


52452632

2155142541621321

5412515421621321

5254233

54152542123

54152542113

65264654232

646541652646542122

646541652646542112

65264654231

646541652646542121

646541652646542111

sscccddcd

sscssccsscddcdssd

ssssccscccddsdscd

ccscsr

ssccssccsr

ssscsscccr

ssccssccsr

ccscscsssssscccsr

ccscssssssssccccr

cscsscccsr

scccsccssssccccsr

scccsscsssscccccr

z

y

x

For example, the system of nonlinear trigonometric equations for the Stanford manipulator is:

Solve for: 1, 2, 4, 5, 6, d3


There is no simple, universal method to solve inverse kinematic problems.A common technique used for a 6 DOF robot with a 3 DOF end-effector (roll, pitch, yaw) is "kinematic decoupling:" find a location for the robot wrist and then determine the orientation of the end-effector.

Also, in general, there is no unique solution to the inverse kinematic problem.


• Dynamics – Determine the forces and torques required for or resulting from the given kinematic motions.

Robot Dynamics


• Control – Given the block diagram model of the dynamic system, determine the feedback loops and gains needed to accomplish the desired performance (overshoot, settling time, etc.)

Robot Controls


Feedback Control System


DC Motor


Surgical Instrument


Good software cannot fix the problems caused by poor mechanical design. – Phillip John McKerrow, Introduction to Robotics

(1991)

areas interest in robotics

Documents

xyz reference frame

axis of rotation

rotation relative

stationary reference

time derivative

translation relative

velocity relative

time t