Control of Robots

1

References:

- M. W. Spong, S. Hutchinson, M. Vidayasagar, Robot Modeling and

Control, Wiley, 2006 (Chaps. 6,8,10).

- H. D. Taghirad, Parallel Robots: Mechanics and Control, CRC Press,

2013 (Chap. 6).

Definitions of a Robot [pronunciation: \rō-bät, -bət\ (US) \rō-bɒt\ (Br)]

� A Robot (Oxford Dictionary):

� A machine capable of carrying out a complex series of actions

automatically, especially one programmable by a computer.

� (especially in science fiction) a machine resembling a human being

and able to replicate certain human movements and functions

automatically.

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automatically.

� A Robot (Robot Institute of America, 1979):

� A re-programmable, multifunctional manipulator designed to move

material, parts, tools, or specialized devices through various

programmed motions for the performance of a variety of tasks.

� Manipulating industrial robot (ISO 8373):

� An automatically controlled, reprogrammable, multipurpose,

manipulator programmable in three or more axes, which may be

either fixed in place or mobile for use in industrial automation

applications.

Two General Category of Robots

� Manipulators:

� Motion Control (position or velocity of the end-effector)

xy

z

3

� Force Control

xy

A

B

Two General Category of Robots (Continued)

� Mobile Robots:

� Position Control

4

� Velocity Control

Trajectory Following Control System

� Reference Input:

� At each instant, the desired (reference) joint angle (θd) is given from the trajectory planner.

� Control System Output:

� The measured joint angles or displacements.

xy

z

B

5

Controller

(Compensator)Robot

Tθd

� The measured joint angles or displacements.

� Robot Input:

� Torque: controller torque + disturbance torque (load, friction, effects

of other joints).

A

A General Feedback Control System

� The Controller (Compensator):

� determines the input to the system, so that the error between the

reference value and the measured value are compensated (and

removed).

� Power Amplifier:

� Imports power to the system, so that proper actuation of the plant is

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� Imports power to the system, so that proper actuation of the plant is

possible.

� Examples: an electronic amplifier, a hydraulic steering system.

� Usually, motor or hydraulic valve driver does the amplification task.

� Sometimes, the voltage or force (torque) level is also increased by

the amplifier.

Two Types of Inputs of a Plant

� Control Input:

� The input from the controller output that is applied to eliminate the

error between the reference input and the output.

� This input is determined by the controller (by a hardware circuit or a

computer program)

� The power of the controller output is usually amplified.

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� The power of the controller output is usually amplified.

� Disturbance Input:

� The external inputs that are applied to the plant (separate from the

control input) and affect the plant variables, specially the controlled

variable (plant output). Examples: friction, load, effects of other links.

Two Critical Elements of a Feedback Control

System (Chaps. 7 & 8, Niku’s Book )

� Sensor:

� Measures the value of the controlled variable (output) and feeds it

back.

� Measurement errors and noises can affect the control quality.

� Actuator:

� Applies the controller output to the plant.

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� Applies the controller output to the plant.

� Actuators are usually added to the plant (as a built-in part) to enable

the input of proper commands to the plant.

� Examples: an electric motor used for applying torque to a manipulator, a

hydraulic or pneumatic valve and a cylinder used to apply force in a

hydraulic robot.

Electric Motor Actuators

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Hydraulic Actuators

10

Independent Joint Control

� Each joint angle is controlled separately.

� An independent control system is designed for each joint.

� Each joint has a SISO (Single-Input Single-Output) control system:

� Input: Torque or Force (applied by an actuator )

� Output: Angle or Displacement (measured by a sensor)

� Force and Displacement used for translational (prismatic) joints.

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� Force and Displacement used for translational (prismatic) joints.

� The dynamic effects of other joints (links) are treated as disturbance

inputs.

� Independent joint control strategy is used in many low speed

robots.

� For high speed robots, the dynamic effects of other joints are

significant and must be taken into account.

Independent (Rotational) Joint Control

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Multi-Input Multi-Output (MIMO) Control of a

Robot

� The input torque of all joints are controlled based on the

measurements of all angles.

� Multiple Input: Several torques to several links.

� Multiple Output: Measured angles of several links.

� MIMO Control Strategies:

� Feedback Linearization (Inverse Dynamic)

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� Feedback Linearization (Inverse Dynamic)

� Robust Control

� Adaptive Control

MIMO Controller Robot

nd

d

d

θ

θθ

M

2

1

nT

T

T

M

2

1

nθ

θ

θ

M

2

1

d

Typical Electrical Actuator for a Joint (a Link)

� Electric systems often require gear reduction:

� Torque is increased.

� Speed is reduced.

� The gear ratio: r (range:20-200)

Gear

load torque

(disturbance)

14

motor

Gear

reductionload

inertia

Modeling An Electric Actuator (DC motor)

� Force applied to a wire (conductor):

� Resulting motor torque:� Torque: τm (N-m),� Magnetic Flux: φ (Webers),� Armature Current: ia (amp),

� A propertionility constant: K1Torque constant: K (N-m/amp)

φ×= iFam iK φτ 1= am iK=

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� Torque constant: Km (N-m/amp)

� Induced back emf voltage:

� Armature Voltage: Vb (Volts),

� Magnetic Flux: φ (Webers),� Rotor angular velocity: ω (rad/sec),� A proportionality constant: K2� Back emf constant:

Kb Volts/(rad/sec)

� Numerical values of

Km and Kb are the same!

mb KV φω2= mbK ω=

Modeling the Electric Circuit of the Motor

� The circuit differential equation:

� Taking Laplace Transform:

mbbaa KVVViRdt

diL ω−=−=+

)()()()( sKsVsIRLs ω−=+

16

)()(

)()()()(

ssKsV

sKsVsIRLs

mb

mba

θ

ω

−=

−=+

Evaluating the Torque Constant

� The torque constant can be evaluated from the Torque-Speed

curves of a motor.

� At stall (zero speed) situation, when the voltage Vr is applied to the

motor and the transients have settled down, the current is ia :

diamam iKiK /00 ττ =⇒= Rτ

=⇒

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mbaa KViRdt

diL ω−=+ 00 bra KViR −=+⇒

amam 00

RVi ra /=⇒ rm

V

RK 0

τ=⇒

Modeling the Mechanical Part of the Motor

� Euler’s equation of rotational motion:

� If the inertias of gear and load are not negligible, Jm must

be replaced by an equivalent inertia:

riKrdt

dB

dt

dJ lamlm

mm

mm //2

2

τττθθ

−=−=+

2++=

18

� Taking Laplace Transform:

rssIKssBsJ lammmm /)()()()(2 τθ −=+

2/ rJJJJ lgmeq ++=

The Block Diagram of the Motor

� Electric part:

� Mechanical part:

� Transfer functions from the two inputs to the output:

rssIKssBsJ lammmm /)()()()(2 τθ −=+

)()()()( ssKsVsIRLs mba θ−=+

,]))([()(

)( mm

KKBsJRLss

K

sV

s

+++=

θ]))([(

/)(

)(

)(m

KKBsJRLss

rRLs

s

s

++++−

=τθ

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Electrical part

]))([()( mbmm KKBsJRLsssV +++ ]))([()( mbmml KKBsJRLsss +++τ

)(ss mθ

Simplifying the Transfer Functions

� Rewriting the denominators in the first-order form of “τs+1”, the electrical time constant “L/R” is much smaller than the mechanical

time constant “Jm/ Bm” (previous slide).

� The dominant time response is the mechanical response, and the

electrical transients can be neglected:

� L/R=0 ⇒ L=0

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� L/R=0 ⇒ L=0

� Then, the transfer functions are simplified as:

� If both inputs are applied simultaneously:

� Taking inverse Laplace transform:

,]/[)(

)(

RKKBsJsR

K

sV

s

mbmm

mm

++=

θ

])([

/

)(

)(

mbmml

m

KKBsJRs

rR

s

s

++−

=τθ

)(]/[

/1)(

]/[

/)( s

RKKBsJs

rsV

RKKBsJs

RKs l

mbmmmbmm

mm τθ ++

−++

=

rttVRKtRKKBtJ lmmmbmmm /)()()/()()/()( τθθ −=++ &&&

Simplified Block Diagram

� Simplified notations for the equation of the system:

� The simplified block diagram of the motor, gearbox

rttVRKtRKKBtJ lmmmbmmm /)()()/()()/()( τθθ −=++ &&&

)()()()( tdtutBtJ mm −=+ θθ &&&

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� The simplified block diagram of the motor, gearbox

and load: