Dr. John (Jizhong) Xiao

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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

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Outline

• Homework Highlights
• Course Review
• Midterm Exam Scope

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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
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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

Erroneous answer
Velocity of point
Kinetic energy of link i
point at link i
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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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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

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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
Forward
Jacobian Matrix
Kinematics
Inverse
Jacobian Matrix Relationship between joint
space velocity with task space velocity
Joint Space
Task Space
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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 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

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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

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Link Parameters
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
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Example Puma 560
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Jacobian Matrix
Kinematics
Jacobian is a function of q, it is not a constant!
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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
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

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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
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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

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
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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

How to chose Kp, Kv ?
Error dynamics
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
One of a selections
Condition have negative real part
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Task Level Controller

• Non-linear Feedback Control

Robot System
Jocobian
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Task Level Controller

• Non-linear Feedback Control

Nonlinear feedback controller
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
630-900