areas interest in robotics
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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 - PowerPoint PPT PresentationTRANSCRIPT
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Areas Interest in Robotics
Industrial Engineering Department
Binghamton University
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Outline• Introduction
– Historical Example– Mechanical Engineering and Robotics
• Review of Basic Kinematics and Dynamics• Transformation Matrices/Denavit-
Hartenberg• Dynamics and Controls• Example: Surgical Robot
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Robot Configurations
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Cartesian Cylindrical
SphericalSCARA
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Phillip John McKerrow, Introduction to Robotics (1991)
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Review of Basic Kinematics and Dynamics
• Case Study: Dynamic Analysis
• Software for Dynamic Analysis: ADAMS
• Rigid Body Kinematics
• Rigid Body Dynamics
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Kinematics of Rigid Bodies
General Plane Motion: Translation plus Rotation
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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
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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.
Kinematics of Rigid Bodies (cont.)
Motion
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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.
Kinematics of Rigid Bodies (cont.)
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General Plane Motion: drA drB
Kinematics of Rigid Bodies (cont.)
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General Plane Motion; (1) Translation
measured from originalPoint A
Kinematics of Rigid Bodies (cont.)
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General Plane Motion:(2) Rotation about axis through Point A
Kinematics of Rigid Bodies (cont.)
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General Plane Motion = Translation + Rotation
Kinematics of Rigid Bodies (cont.)
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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.)
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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
Kinematics of Rigid Bodies (cont.)
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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
Kinematics of Rigid Bodies (cont.)
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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
Kinematics of Rigid Bodies (cont.)
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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
Kinematics of Rigid Bodies (cont.)
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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
'''
Kinematics of Rigid Bodies (cont.)
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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
'''
Kinematics of Rigid Bodies (cont.)
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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
Kinematics of Rigid Bodies (cont.)
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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
Kinematics of Rigid Bodies (cont.)
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R
X Y
Z
O
A
X'
Y'
Z'
r
'Z
AAdt
Ada
XYZ
2
2
Kinematics of Rigid Bodies (cont.)
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Summary of Equations: Kinematics of Rigid Bodies
A
BA
B rraa AB
ABrvv AB
Kinematics of Rigid Bodies (cont.)
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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.
Kinematics of Rigid Bodies (cont.)
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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
Kinematics of Rigid Bodies (cont.)
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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
Kinematics of Rigid Bodies (cont.)
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Links, Joints, and Kinematic Chains
Link = df. A rigid body which
possesses at least two nodes which are points for attachment to other links.
Kinematics of Rigid Bodies (cont.)
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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.”
Kinematics of Rigid Bodies (cont.)
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Type of contact between links
Lower pair: surface contact
Higher pair: line or point contactSix Lower Pairs
Kinematics of Rigid Bodies (cont.)
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“Constrained Pin” “Screw”
“Slide” “Sliding Pin”
Kinematics of Rigid Bodies (cont.) CS 480A-34
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Planar (F) Joint – 3 DOF
“Ball and Socket”
Kinematics of Rigid Bodies (cont.)
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Kinematics of Rigid Bodies (cont.)
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Open/Closed Kinematic Chain (Mechanism)Closed Kinematic Chain = df. A kinematic chain in
which there are no open attachment points or nodes.
Kinematics of Rigid Bodies (cont.)
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Open Kinematic Chain = df. A kinematic chain in
which there is at least one open attachment point or node.
Kinematics of Rigid Bodies (cont.)
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Dynamics of Rigid Bodies
Dynamic Equivalence
Lumped Parameter Dynamic Model
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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
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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.)
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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
Dynamics of Rigid Bodies (cont.)
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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
Dynamics of Rigid Bodies (cont.)
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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.)
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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
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Position Kinematics
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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
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Mark W. Spong and M. Vidyasagar, Robot Dynamics and Control (1989)
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
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Example
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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
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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
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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
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Similarly for any robot configuration:
1000333231
232221
131211
60
z
y
x
drrr
drrr
drrr
T
Stanford manipulator configuration:
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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
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Velocity Kinematics
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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
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00
0cos
0sin
11
11
1ql
ql
J c
c
vc
00
coscoscos
sinsinsin
21221111
21221211
2qqlqqlql
qqlqqlql
J cc
cc
vc
Linear Velocities
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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
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In general the problem can be stated:Given the 4x4 D-H homogeneous transformation
Mark W. Spong and M. Vidyasagar, Robot Dynamics and Control (1989)
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
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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
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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.
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• Dynamics – Determine the forces and torques required for or resulting from the given kinematic motions.
Robot Dynamics
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• 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
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Feedback Control System
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DC Motor
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Surgical Instrument
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Good software cannot fix the problems caused by poor mechanical design. – Phillip John McKerrow, Introduction to Robotics
(1991)