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

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manipulator control

Introduction to ROBOTICS

Manipulator Control

Jizhong Xiao

Department of Electrical Engineering

City College of New York

  • Homework Highlights
  • Robot Manipulator Control
    • Control Theory Review
    • Joint-level PD Control
    • Computed Torque Method
    • Non-linear Feedback Control
  • Midterm Exam Scope
homework 2
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
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
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 41
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

homework 42
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
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
Jacobian Matrix Revisit

Forward Kinematics

jacobian matrix revisit1

(x , y)





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

Find a control input (tor),

  • Task Level Controller

Find a control input (tor),

robot manipulator control1
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
      • ….

d e


Error signal e

yd - ya

actual ya


desired yd

compute V using PID feedback



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


steady-state error

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

the steady-state error?

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
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 ii1
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
Control Theory Review (III)
  • Feedback linearization
    • Nonlinear system
    • Example:

Original system:

Nonlinear feedback:

Linear system:

robot motion control i
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
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 ii1
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
Robot Motion Control (III)
  • Non-linear Feedback Control

Robot System:


robot motion control iii1
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
Thank you!

HWK 5 posted on the web,

Next Class: Midterm Exam

Time: 6:30-9:00, Please on time!