Jizhong Xiao

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Manipulator Control
Introduction to ROBOTICS

• Jizhong Xiao
• Department of Electrical Engineering
• City College of New York
• jxiao_at_ccny.cuny.edu

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Outline

• Homework Highlights
• Robot Manipulator Control
• Control Theory Review
• Joint-level PD Control
• Computed Torque Method
• Non-linear Feedback Control
• 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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Manipulator Control
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Manipulator Dynamics Revisit

• Dynamics Model of n-link Arm

The Acceleration-related Inertia term, Symmetric
Matrix
The Coriolis and Centrifugal terms
The Gravity terms
Driving torque applied on each link
Non-linear, highly coupled , second order
differential equation
Joint torque Robot motion
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Jacobian Matrix Revisit
Forward Kinematics
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Jacobian Matrix Revisit

• Example 2-DOF planar robot arm
• Given l1, l2 , Find Jacobian

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Robot Manipulator Control

• Robot System

• Joint Level Controller

Find a control input (tor),

• Task Level Controller

Find a control input (tor),
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Robot Manipulator Control

• Control Methods
• Conventional Joint PID Control
• Widely used in industry
• Advanced Control Approaches
• Computed torque approach
• Nonlinear feedback
• Adaptive control
• Variable structure control
• .

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Control Theory Review (I)
PID controller Proportional / Integral /
Derivative control
e yd - ya V Kp e Ki ? e dt Kd
)
Closed Loop Feedback Control
Reference book Modern Control Engineering,
Katsuhiko Ogata, ISBN0-13-060907-2
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Evaluating the response
overshoot
steady-state error
ss error -- difference from the systems desired
value
settling time
overshoot -- of final value exceeded at first
oscillation
rise time -- time to span from 10 to 90 of the
final value
settling time -- time to reach within 2 of the
final value
How can we eliminate the steady-state error?
rise time
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Control Performance, P-type
Kp 20
Kp 50
Kp 200
Kp 500
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Control Performance, PI - type
Kp 100
Ki 50
Ki 200
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Youve been integrated...
Kp 100
unstable oscillation
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Control Performance, PID-type
Kp 100
Kd 5
Ki 200
Kd 2
Kd 10
Kd 20
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PID final control
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Control Theory Review (II)

• Linear Control System
• State space equation of a system
• Example a system
• Eigenvalue of A are the root of characteristic
  equation
• Asymptotically stable all eigenvalues of A
  have negative real part

(Equ. 1)
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Control Theory Review (II)

• Find a state feedback control
  such that the closed loop system is
  asymptotically stable
• Closed loop system becomes
• Chose K, such that all eigenvalues of A(A-BK)
  have negative real parts

(Equ. 2)
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Control Theory Review (III)

• Feedback linearization
• Nonlinear system
• Example

Original system
Nonlinear feedback
Linear system
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Robot Motion Control (I)

• 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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Robot Motion Control (II)

• 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
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Robot Motion Control (II)
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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Robot Motion Control (III)

• Non-linear Feedback Control

Robot System
Jocobian
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Robot Motion Control (III)

• Non-linear Feedback Control

Design the nonlinear feedback controller as
Then the linearized dynamic model
Design the linear controller
Error dynamic equation
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Project

• Simulation study of Non-linear Feedback Control

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Thank you!
HWK 5 posted on the web, Due Oct. 23 before
class Next Class (Oct. 23) Midterm Exam Time
630-900, Please on time!