analytical and applied kinematics - school of engineeringmoreno/kinematics/me 5150 kinematics...

Analytical and Applied Kinematics

Vito Moreno

UTEB 388

moreno@engr.uconn.edu

860-486-5342 office

860-614-2365 (cell)

http://www.engr.uconn.edu/~moreno

Office hours: Tuesdays 2:00 to 4:30 pm

1

Kinematics Challenge

• ~45 students

– MS, MENG, PhD, UG

– Storrs, Avery Pt, CCAT, Virginia..

• Notes:

– Text +http://www.engr.uconn.edu/~moreno

• Homework:

– Text, notes or http://www.engr.uconn.edu/~moreno

– Onsite – pass in to me

– Offsite – email

• Quizzes

2

Kinematics Introduction

3

This course introduces a unified and analytical approach to two (2) and

three (3) dimensional kinematics and planar and spatial geometry and

constraint motion.

Applications to: mechanisms, robotics, biomechanics…

Some topics covered:

Coordinate transformation operators

Displacement operators

Motion invariants

Velocity and acceleration operators

Link and joint constraints

Analytical methods of mechanism synthesis and analysis

Geometric error modeling

Computational methods in kinematics and geometry

4

Syllabus Rev A

Revised 16-Jan

Week Date Chapter/Topics Reading + Notes HW HW Due

1 21-Jan Introduction 1.1 -1.7 1.36,1.39,1.52 4-Feb

2 28-Jan Kinematic Analysis, Displ, Vel, Acc 3.1-3.3 TBD

3 4-Feb Kinematic Analysis, Displ, Vel, Acc Oscillating Slider 11-Feb

4 11-Feb Matrix Methods TBD

5 18-Feb Matrix Methods/HD Notation Salute 25-Feb

6 25-Feb Test #1

7 4-Mar HD Notation Robot Manipulator Puma

8 11-Mar Forward Kinematics Manipulator

9 18-Mar Spring Break

10 25-Mar Test #2

11 1-Apr Design Synthesis

12 8-Apr Position Synthesis Linkage synthesis

13 15-Apr Path Synthesis

14 22-Apr Function Synthesis

15 29-Apr Test #3 - last class

ME 5150 Solid Kinematics

Reference texts:

1. Mechanism Design, Erdman, A.G., Sandor, G.N., Kota, S., Prentice

Hall, 4th ed. 2001. (E&S)

2. Kinematics and Mechanisms Design, Suh,C.H. and Radcliffe, C.W.,

John Wiley and Sons, 1978. (S&R)

3. Mechanism and Dynamics of Machinery, Mabie, H.H., Reinholtz,

C.F., John Wiley and Sons, 4th ed. 1987.

4. Theoretical Kinematics, Bottema, O., Roth, B., Dover Publications,

1979.

5. Introduction to Theoretical Kinematics, McCarthy, J.M. The MIT

Press, 1990.

5

Kinematics Introduction

6

6. http://highered.mcgraw-

hill.com/sites/dl/free/0073121584/366642/Norton_Ch02.pdf

Kinematics Introduction

Basic definitions:

Kinematics is part of Solid Mechanics

Statics – study of forces and moments apart from motion

Kinetics – study of the action of forces and moments on

the motion of bodies

Dynamics

Kinematics – Study of the relative motion apart from

forces

7

Kinematics Introduction

Mechanism – combination of several rigid bodies which are connected

is such a way that relative motion between them is

allowed.

No relative motion = Structure

Function of a mechanism – to transmit or transform motion from one

rigid body to another (source to output).

Types of mechanisms-

Planar and spatial linkages

Gear systems

Cam systems

8

Kinematics Introduction

9

Kinematics Introduction

Examples of Mechanisms

10

Kinematics Introduction

Examples of Mechanisms

11

Kinematics Introduction

Examples of Mechanisms

Cams Gears

12

Gears

Cams

Planar and spatial linkages

Kinematics Introduction

13

Kinematics Introduction

Machine – a mechanism or combination of mechanisms for

the purpose of transferring force or motion.

Motion

Plane (2D) motion – translation, rotation

Spatial (3D) Motion

Helical – pitch rotation and translation

Spherical – all points at a fixed distance from a given point

Cylindrical – free rotation and translation along an axis

• Mechanism Categories

– Motion (rigid body guidance)

– Function generation

– Path generation

14

Kinematics Introduction

15

Kinematics Introduction

E&S Fig 1.2c

16

Kinematics Introduction

E&S Fig 1.6

17

Kinematics Introduction

E&S Fig 1.2b

18

Kinematics Introduction

E&S Fig 1.5

19

Kinematics Introduction

E&S Fig 1.2a

20

Kinematics Introduction

Path Generation

21

Four Bar Linkage - Terminology

Kinematics Introduction

E&S Fig 1.1a

Kinematic Diagrams

22 E&S Table 1.1 Planar Link Types

The link – a solid (rigid) body which is connected to n other links

Linkage – links connected by joints

Kinematics Introduction

23

Kinematics Introduction

Joints

Kinematic Pair (joint) = connection between two links which allows

certain relative motion

Lower pair – relative motion described by single (1) coordinate

e.g. revolute, prismatic, rolling pairs

Higher pair – relative motion >1 degree of freedom

roll/slip, spherical ball and socket

Kinematic Chain – a set of links connected by joints

24

Kinematics Introduction

Suh & Radcliffe

Fig 1.1 Kinematic Pairs

Lower

Lower

25

Kinematics Introduction

Degree of Freedom –no of independent parameters

(input coordinates) to completely

define the position of a rigid body

2D – 3 dof, 3D – 6 dof

A

X

Y

AX

B

AY

Unconstrained rigid link

Three independent variables

AX AY

26

Kinematics Introduction

A

X

Y

AX

B

AY

Before joining, multiple links will have 3n DOF

AX AY

C D

CXCY

CX

CY

ground

27

Kinematics Introduction

A

X

Y

B

Connections between links result in loss of DOF

AX AY

C D

CXCY

Pin joints loose 2 DOF, have only 1 DOF called f1 joint

Degree of Constraint = number of freedoms a free body looses after

it is connected to a fixed link

DOC+DOF=3

Kinematic pair

Linkage

28

Kinematics Introduction

A

B

Four Bar Linkage Notation

D

Mobility analysis by Gruebler’s equation

oA

oB

Input

link

Coupler

link

Follower

link

fixed

fixed 1

1

2

3

4 freedomrelativeDOwithsjof

membersn

fnFDOF

1int#

#

2)1(3

1

1

1

)4(2)14(3

4

4

1

F

F

f

n

Four Bar Linkage is a Single DOF system -1 input coordinate required to define

position of all members

Gruebler, M. (1917). Getriebelehre. Springer-Verlag: Berlin.

29

Kinematics Introduction

A

B

Slider-Crank Linkage Notation

D

Sliding connection reduces DOF

Include in # f1

oA

oB

Input

link

Coupler

link

Output

link

fixed

fixed

1

1

2 3

4

freedomrelativeDOwithsjof

membersn

fnFDOF

1int#

#

2)1(3

1

1

1

)4(2)14(3

4

4

'

1

F

F

f

n

equationsGruebler

X

Y

0, zy

30

Kinematics Introduction

Closed linkage Open linkage (Dyad)

31

N=12

F1=15 (12+3)

F=3(12-1)-2(15)=+3

At Q, 3 links,2 joints

Kinematics Introduction

(E&S)

32

Erdman and Sandor, Table 1.2 Dimensional Joints

33

Erdman and Sandor, Table 1.2 Dimensional Joints

34

DOF= 3(n-1)-2f1-1f2

n=7

f1=7

f2=1

F=3(7-1)-2(7)-1(1)=+3

(roll/slide)

Velocity equivalent linkage

(Kinematic diagram) 8

DOF= 3(n-1)-2f1-1f2

n=10

f1=12

f2=0

F=3(10-1)-2(12)-0(1)=+3

Kinematics Introduction

More complicated linkage

E&S

Fig 1.24

35

Kinematics Introduction

Gruebler’s equation: paradox

E&S Fig 1.26

Over constrained linkage

n=5

f1=6

DOF 3(5-1)-2(6)=0

But motion is allowed

3rd link is redundant

Mfg. errors could cause binding

E&S Fig 1.27

n=3

f1=3

DOF 3(3-1)-2(3)=0

But motion is allowed

Sum of radii = dist between pivot points

36

Kinematics Introduction

Gruebler’s equation paradox

E&S Fig 1.26

Overconstrained linkage

n=4

f1=3, f2=1roll/slide

DOF 3(4-1)-2(3)-1(1)=+2

Pure rolling between roller and cam

n=4, f1=4, f2=0

DOF=3(4-1)-2(4)=+1

Welded roller to arm(3)

n=3, f1=2, f2=1

DOF=3(3-1)-2(2)-1(1)=+1

E&S Fig 1.28

Passive or redundant DOF

Rotation of 4 does not affect arm 3

37

Grashof, F. (1883). Theoretische Maschinenlehre. Vol. 2. Voss: Hamburg.

Kinematics Introduction

Grashof condition addresses mobility

s=shortest link

l=longest link

p=one remaining link

q=other remaining ling

If s+l ≤ p+q one link capable of full rotation (Class I kinematic chain)

If s+l> p+q no link capable of complete revolution (Class II kinematic chain)

38

Kinematics Introduction

39

Kinematics Introduction

40

Kinematics Introduction

James Watt (1784) – 4 bar linkage

41

6 Bar Linkages

DOF =1

n=6

f1=7

Watt Linkage Stephenson Linkage

E&S Fig 1.13 a-d

Ternary links

Tracer Points

(binary links)

Kinematics Introduction

42

Kinematics Introduction

E&S

Fig 1.16

Trajectory

of L1 wrt

L6

Duplicate movement of relative

center of rotation between thigh (femur) and

leg (tibia & fibula)

n=6

f1=7

43

Kinematic Inversion Changing the fixed Link

Basic Slider -Crank

Pump Mechanism Oscillating Cylinder Pump

Whitworth Quick Return Link

input

input

input input

output

output

output output

(S&R

Fig 1.2)

Relative motion is same

Absolute motion is different

44

Kinematics Introduction – Force and Transmission of Motion

n-n transmission of force

and motion

(S&R)

cam

linkage

rolling

contact

45

Kinematics Introduction – Force and Transmission of Motion

90

Pr

angleessure

angleonTransmissi For maximum mechanical advantage

30,

0

90

ypracticall

(S&R)

46

Kinematics Introduction – Force and Transmission of Motion

90

)()(Pr

)(

angleessureDeviation

angleonTransmissiE&S Fig 3.18-3.21

47

Kinematics Introduction – Homework #1

Problem 1.36

Determine the DOF for the

Mechanisms shown

1

2

3

48

Kinematics Introduction – Homework #1

4

5

Problem 1.36

Determine the DOF for the

Mechanisms shown

49

Kinematics Introduction – Homework #1

6 7 8 9

Problem 1.36

Determine the DOF for the

Mechanisms shown

50

10

Problem 1.39

Determine the DOF for the

Mechanisms shown

51

Kinematics Introduction – Homework #1

11

Honda

Problem 1.52

Determine the DOF for the

Mechanisms shown