Transcript Introduction - Electrical Engineering

Introduction to ROBOTICS
Midterm Exam Review
Prof. John (Jizhong) Xiao
Department of Electrical Engineering
City College of New York
[email protected]
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Grades Distribution
12
10
8
6
Students
4
2
0
<60 61~70 71~80 81~89 >90
37 students taking Exam
Minimum grade: 37
Maximum grade: 99
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Q1 and Q2
• Q1 (a): 11/37
• Q1 (b): 14/37
• Q2: 1/37
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Composite Rotation Matrix
• A sequence of finite rotations
– matrix multiplications do not commute
– rules:
• if rotating coordinate O-U-V-W is rotating about
principal axis of OXYZ frame, then Pre-multiply
the previous (resultant) rotation matrix with an
appropriate basic rotation matrix
• if rotating coordinate OUVW is rotating about its
own principal axes, then post-multiply the
previous (resultant) rotation matrix with an
appropriate basic rotation matrix
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Homogeneous Representation
• A frame in space (Geometric
Interpretation)
z
 R33 P31 
F 

1 
 0
 nx
n
F  y
 nz

0
sx
sy
sz
0
ax
ay
az
0
P( px , py , pz )
a
s
n
px 
p y 
pz 

1
y
x
Principal axis n w.r.t. the reference coordinate system
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Jacobian Matrix Revisit
Forward Kinematics
n s a p 
T 

0
0
0
1

 44
6
0
 h1 (q ) 
h (q )
Y61  h(q )   2 
  


h
(
q
)
 6 
Y  J q
61
6n n1
 x 
 y 
 
 z 
Y     
 
 
 
 
 x   h1 ( q ) 
p   y   h2 ( q ) 
 z   h3 ( q ) 
 (q )  h4 (q )
{n, s, a}   (q )    h5 (q ) 
 (q ) h6 (q ) 
 dh(q) 

J  
 dq  6n
 q1 
 
 dh(q )  q 2 
 dq    

 6n
 
q n  n1
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 h1
 q
 1
 h2
  q1
 
 h
 6
 q1
h1
q2
h2
q2

h6
q2
h1 
qn 

h2 

qn 

 
h6 


qn  6n

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Example
Example: 1-link robot with point mass (m)
concentrated at the end of the arm.
L
Set up coordinate frame as in the figure
l 
0 
r11   
0 
 
1
According to physical meaning:
1 2 2
l m1
2
P  9.8m  l  S1
K
m
Y0
Y1
LK P
X1
1
X0
d L
L
  (  )
 l 2 m1  9.8m  l  C1
dt 1 1
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Manipulator Dynamics
• Potential energy of link i
r0i : Center of mass
w.r.t. base frame
Pi  mi gr0i  mi g (T0i ri i )
ri i : Center of mass
w.r.t. i-th frame
g  ( g x , g y , g z ,0)
g  9.8m / sec2
g
: gravity row vector
expressed in base frame
• Potential energy of a robot arm
n
n
P   Pi  [mi g (T0i ri i )]
i 1
i 1
Function of
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qi
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Manipulator Dynamics
• Dynamics Model of n-link Arm
  D(q)q  H (q, q )  C (q)
The Acceleration-related Inertia term, Symmetric Matrix
The Coriolis and Centrifugal terms
The Gravity terms
 1  Driving torque
     applied on each link
 n 
Non-linear, highly coupled , second order differential equation
Joint torque
Robot motion
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Robot Motion Control
• Computed torque method
– Robot system:
D(q)q  H (q, q )  C (q)  

Y  h(q)

– Controller:
d  kv (q d  q)  k p (qd  q)]  H (q, q)  C(q)
tor  D(q)[q
d  q
)  kv (q d  q)  k p (qd  q)  0
(q
Error dynamics
How to chose
Kp, Kv ?
e  kv e  k p e  0
Advantage: compensated for the dynamic effects
Condition: robot dynamic model is known
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Robot Motion Control
Error dynamics
e  kv e  k p e  0
Define states:
x1  e
x2  e
x1  x2
x2  kv x2  k p x1
x
 0
In matrix form:  1   
 x 
 2
Characteristic equation:
How to chose Kp, Kv
to make the system
stable?
 k p
1   x1 
 AX



 kv   x2 
I  A 

1
kp
  kv
 2  kv   k p  0
 k v  k v  4k p
2
The eigenvalue of A matrix is:
1, 2 
Condition:  have negative real part
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One of a
selections:
kv  0
kp  0
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