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Analytical and Applied Kinematics
Vito Moreno
UTEB 388
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
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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.
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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
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
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Kinematics Introduction
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
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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
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N=12
F1=15 (12+3)
F=3(12-1)-2(15)=+3
At Q, 3 links,2 joints
Kinematics Introduction
(E&S)
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)
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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
51
Kinematics Introduction – Homework #1
11
Honda
Problem 1.52
Determine the DOF for the
Mechanisms shown