City College of New York
1
Dr. John (Jizhong) Xiao
Department of Electrical Engineering
City College of New York
jxiao@ccny.cuny.edu
Review for Midterm Exam
Introduction to ROBOTICS
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Outline
•
Homework Highlights
•
Course Review
•
Midterm Exam Scope
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Homework 2
Joint variables ?
Find the forward kinematics, Roll

Pitch

Yaw
representation of orientation
Why use atan2 function?
Inverse trigonometric functions have multiple solutions:
Limit x to [

180, 180] degree
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Homework 3
Find kinematics model of 2

link robot, Find the inverse kinematics solution
Inverse: know position (Px,Py,Pz) and
orientation (n, s, a), solve joint variables.
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Homework 4
Find the dynamic model of 2

link
robot with mass equally distributed
•
Calculate D, H, C terms directly
Physical meaning?
Interaction effects of motion of joints j & k on link i
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Homework 4
Find the dynamic model of 2

link
robot with mass equally distributed
•
Derivation of L

E Formula
point at link
i
Velocity of point
Kinetic energy of link i
Erroneous answer
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Homework 4
Example: 1

link robot with point mass (m)
concentrated at the end of the arm.
Set up coordinate frame as in the figure
According to physical meaning:
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Course Review
•
What are Robots?
–
Machines with sensing, intelligence and
mobility (NSF)
•
Why use Robots?
–
Perform 4A tasks in 4D environments
4A: Automation, Augmentation, Assistance, Autonomous
4D: Dangerous, Dirty, Dull, Difficult
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Course Coverage
•
Robot
Manipulator
–
Kinematics
–
Dynamics
–
Control
•
Mobile Robot
–
Kinematics/Control
–
Sensing and Sensors
–
Motion planning
–
Mapping/Localization
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Robot Manipulator
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Position
vector
Rotation
matrix
Scaling
Homogeneous Transformation Matrix
•
Composite Homogeneous Transformation Matrix
•
Rules:
–
Transformation (rotation/translation) w.r.t. (X,Y,Z) (OLD
FRAME), using
pre

multiplication
–
Transformation (rotation/translation) w.r.t. (U,V,W) (NEW
FRAME), using
post

multiplication
Homogeneous Transformation
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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)
Principal axis
n
w.r.t. the reference coordinate system
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Manipulator Kinematics
Joint Space
Task Space
Forward
Inverse
Kinematics
Jacobian Matrix: Relationship between joint
space velocity with task space velocity
Jacobian
Matrix
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Manipulator Kinematics
•
Steps to derive kinematics model:
–
Assign D

H coordinates frames
–
Find link parameters
–
Transformation matrices of adjacent joints
–
Calculate kinematics model
•
chain product of successive coordinate transformation
matrices
–
When necessary, Euler angle representation
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Denavit

Hartenberg Convention
•
Number the joints from 1 to n starting with the base and ending with
the end

effector.
•
Establish the base coordinate system.
Establish a right

handed
orthonormal coordinate system at the supporting base
with axis lying along the axis of motion of joint 1.
•
Establish joint axis.
Align the Z
i
with the axis of motion (rotary or
sliding) of joint i+1.
•
Establish the origin of the ith coordinate system.
Locate the origin of
the ith coordinate at the intersection of the Z
i
& Z
i

1
or at the
intersection of common normal between the Z
i
& Z
i

1
axes and the Z
i
axis.
•
Establish X
i
axis.
Establish or along the
common normal between the Z
i

1
& Z
i
axes when they are parallel.
•
Establish Y
i
axis.
Assign to complete the
right

handed coordinate system.
•
Find the link and joint parameters
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Denavit

Hartenberg Convention
1.
Number the joints
2.
Establish base frame
3.
Establish joint axis Z
i
4.
Locate origin, (intersect.
of Z
i
& Z
i

1
) OR (intersect
of common normal & Z
i
)
5.
Establish X
i
,Y
i
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Link Parameters
: angle from Z
i

1
to Z
i
about X
i
: distance from intersection
of Z
i

1
& X
i
to O
i
along X
i
Joint distance : distance from O
i

1
to intersection of Z
i

1
& X
i
along Z
i

1
: angle from X
i

1
to X
i
about Z
i

1
t
0
0
6
0
0
90
5
8
0

90
4
0
0
90
3
8
0
2
13
0

90
1
J

l
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Example:
Puma 560
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Jacobian Matrix
Jacobian is a function of
q, it is not a constant!
Kinematics:
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Jacobian Matrix Revisit
Forward Kinematics
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Trajectory Planning
•
Motion Planning:
–
Path planning
•
Geometric path
•
Issues: obstacle avoidance, shortest
path
–
Trajectory planning,
•
“interpolate” or “approximate” the
desired path by a class of
polynomial functions and generates
a sequence of time

based “control
set points” for the control of
manipulator from the initial
configuration to its destination.
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Trajectory planning
•
Path Profile
•
Velocity Profile
•
Acceleration Profile
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Trajectory Planning
•
n

th order polynomial, must satisfy 14 conditions,
•
13

th order polynomial
•
4

3

4 trajectory
•
3

5

3 trajectory
t0
t1, 5 unknow
t1
t2, 4 unknow
t2
tf, 5 unknow
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Manipulator Dynamics
•
Lagrange

Euler Formulation
–
Lagrange function is defined
•
K
: Total kinetic energy of robot
•
P
: Total potential energy of robot
•
: Joint variable of i

th joint
•
: first time derivative of
•
: Generalized force (torque) at i

th joint
Joint torques Robot motion, i.e. position velocity,
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Manipulator Dynamics
•
Dynamics Model of n

link Arm
The Acceleration

related Inertia
matrix term, Symmetric
The Coriolis and Centrifugal terms
The Gravity terms
Driving torque
applied on each link
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Example
Example: 1

link robot with point mass (m)
concentrated at the end of the arm.
Set up coordinate frame as in the figure
According to physical meaning:
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Manipulator Dynamics
•
Potential energy of link i
: gravity row vector
expressed in base frame
: Center of mass
w.r.t. i

th frame
: Center of mass
w.r.t. base frame
•
Potential energy of a robot arm
Function of
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Robot Motion Control
•
Joint level PID control
–
each joint is a servo

mechanism
–
adopted widely in industrial robot
–
neglect dynamic behavior of whole arm
–
degraded control performance especially in
high speed
–
performance depends on configuration
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Joint Level Controller
•
Computed torque method
–
Robot system:
–
Controller:
Error dynamics
How to chose
Kp, Kv ?
Advantage: compensated for the dynamic effects
Condition: robot dynamic model is known exactly
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Robot Motion Control
How to chose Kp, Kv
to make the system
stable?
Error dynamics
Define states:
In matrix form:
Characteristic equation:
The eigenvalue of A matrix is:
Condition: have negative real part
One of a
selections:
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•
Non

linear Feedback Control
Jocobian:
Robot System:
Task Level Controller
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Task Level Controller
Nonlinear feedback controller:
•
Non

linear Feedback Control
Then the linearized dynamic model:
Linear Controller:
Error dynamic equation:
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Midterm Exam
Scope
•
Study lecture notes
•
Understand homework and examples
•
Have clear concept
•
2.5 hour exam
•
close book, close notes
•
But you can bring one

page cheat sheet
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Thank you!
Next class:
Oct. 23
(Tue): Midterm Exam
Time: 6:30

9:00
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