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## Dr. John (Jizhong) Xiao

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### Introduction to ROBOTICS Review for Midterm Exam Dr. John (Jizhong) Xiao Department of Electrical Engineering City College of New York jxiao_at_ccny.cuny.edu – PowerPoint PPT presentation

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Title: Dr. John (Jizhong) Xiao

1
Review for Midterm Exam
Introduction to ROBOTICS
• Dr. John (Jizhong) Xiao
• Department of Electrical Engineering
• City College of New York
• jxiao_at_ccny.cuny.edu

2
Outline
• Homework Highlights
• Course Review
• Midterm Exam Scope

3
Homework 2
Find the forward kinematics, Roll-Pitch-Yaw
representation of orientation
Joint variables ?
Why use atan2 function?
Inverse trigonometric functions have multiple
solutions
Limit x to -180, 180 degree
4
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.
5
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
6
Homework 4
Find the dynamic model of 2-link robot with mass
equally distributed
• Derivation of L-E Formula

Velocity of point
7
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
8
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
9
Course Coverage
• Robot Manipulator
• Kinematics
• Dynamics
• Control
• Mobile Robot
• Kinematics/Control
• Sensing and Sensors
• Motion planning
• Mapping/Localization

10
Robot Manipulator
11
Homogeneous Transformation
Homogeneous Transformation Matrix
Rotation matrix
Position vector
Scaling
• 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

12
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

13
Homogeneous Representation
• A frame in space (Geometric Interpretation)

Principal axis n w.r.t. the reference coordinate
system
14
Manipulator Kinematics
Forward
Jacobian Matrix
Kinematics
Inverse
Jacobian Matrix Relationship between joint
space velocity with task space velocity
Joint Space
15
Manipulator Kinematics
• Steps to derive kinematics model
• Assign D-H coordinates frames
• Transformation matrices of adjacent joints
• Calculate kinematics model
• chain product of successive coordinate
transformation matrices
• When necessary, Euler angle representation

16
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 Zi with the axis
of motion (rotary or sliding) of joint i1.
• Establish the origin of the ith coordinate
system. Locate the origin of the ith coordinate
at the intersection of the Zi Zi-1 or at the
intersection of common normal between the Zi
Zi-1 axes and the Zi axis.
• Establish Xi axis. Establish
or along the common normal
between the Zi-1 Zi axes when they are
parallel.
• Establish Yi axis. Assign
to complete the right-handed
coordinate system.
• Find the link and joint parameters

17
Denavit-Hartenberg Convention
1. Number the joints
2. Establish base frame
3. Establish joint axis Zi
4. Locate origin, (intersect. of Zi Zi-1) OR
(intersect of common normal Zi )
5. Establish Xi,Yi

18
angle from Xi-1 to Xi about Zi-1
angle from Zi-1 to Zi about Xi
distance from intersection of Zi-1 Xi to
Oi along Xi
Joint distance distance from Oi-1 to
intersection of Zi-1 Xi along Zi-1
19
Example Puma 560
20
Jacobian Matrix
Kinematics
Jacobian is a function of q, it is not a constant!
21
Jacobian Matrix Revisit
Forward Kinematics
22
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.

23
Trajectory planning
• Path Profile
• Velocity Profile
• Acceleration Profile

24
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
25
Manipulator Dynamics
Joint torques Robot motion, i.e.
position velocity,
• 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

26
Manipulator Dynamics
• Dynamics Model of n-link Arm

The Acceleration-related Inertia matrix term,
Symmetric
The Coriolis and Centrifugal terms
Driving torque applied on each link
The Gravity terms
27
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
28
Manipulator Dynamics
• Potential energy of link i

Center of mass w.r.t. base frame
Center of mass w.r.t. i-th frame
gravity row vector expressed in base frame
• Potential energy of a robot arm

Function of
29
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

30
Joint Level Controller
• Computed torque method
• Robot system
• Controller

How to chose Kp, Kv ?
Error dynamics
Advantage compensated for the dynamic effects
Condition robot dynamic model is known exactly
31
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
One of a selections
Condition have negative real part
32
• Non-linear Feedback Control

Robot System
Jocobian
33
• Non-linear Feedback Control

Nonlinear feedback controller
Then the linearized dynamic model
Linear Controller
Error dynamic equation
34
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

35
Thank you!
Next class Oct. 23 (Tue) Midterm Exam Time
630-900