Manipulator Control

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# Manipulator Control - PowerPoint PPT Presentation

Introduction to ROBOTICS. Manipulator Control. Jizhong Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu. Outline. Homework Highlights Robot Manipulator Control Control Theory Review Joint-level PD Control Computed Torque Method

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Introduction to ROBOTICS

### Manipulator Control

Jizhong Xiao

Department of Electrical Engineering

City College of New York

jxiao@ccny.cuny.edu

Outline
• Homework Highlights
• Robot Manipulator Control
• Control Theory Review
• Joint-level PD Control
• Computed Torque Method
• Non-linear Feedback Control
• Midterm Exam Scope
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

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.

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

Homework 4

Find the dynamic model of 2-link robot with mass equally distributed

• Derivation of L-E Formula

Velocity of point

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:

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
Jacobian Matrix Revisit

Forward Kinematics

(x , y)

2

l2

l1

1

Jacobian Matrix Revisit
• Example: 2-DOF planar robot arm
• Given l1, l2 ,Find: Jacobian
Robot Manipulator Control
• Robot System:
• Joint Level Controller

Find a control input (tor),

Find a control input (tor),

Robot Manipulator Control
• Control Methods
• Conventional Joint PID Control
• Widely used in industry
• Computed torque approach
• Nonlinear feedback
• Variable structure control
• ….

d e

dt

Error signal e

yd - ya

actual ya

V

desired yd

compute V using PID feedback

-

Motor

actual ya

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

Evaluating the response

overshoot

ss error -- difference from the system’s 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

rise time

Control Performance, P-type

Kp = 20

Kp = 50

Kp = 200

Kp = 500

Control Performance, PI - type

Kp = 100

Ki = 50

Ki = 200

You’ve been integrated...

Kp = 100

unstable & oscillation

Control Performance, PID-type

Kp = 100

Kd = 5

Kd = 2

Ki = 200

Kd = 10

Kd = 20

Control Theory Review (II)
• Linear Control System
• State space equation of a system
• Example: a system:
• Eigenvalue of Aare the root of characteristic equation
• Asymptotically stable all eigenvalues of A have negative real part

(Equ. 1)

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)

Control Theory Review (III)
• Feedback linearization
• Nonlinear system
• Example:

Original system:

Nonlinear feedback:

Linear system:

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
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

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
Robot Motion Control (III)
• Non-linear Feedback Control

Robot System:

Jocobian:

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:

Thank you!

HWK 5 posted on the web,

Next Class: Midterm Exam