02/11/2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Matrix Order Number of row or columns Rank of Matrix: Order of largest non-zero determinant. A matrix

02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.

Matrix

• Order Number of row or columns

• Rank of Matrix:Order of largest non-zero determinant.A matrix whose order exceeds its rank is singular

• Matrix Operations– Addition/Subtraction

A and B must be same Order

– Multiplication • By Scalar

• By Matrix or Vector

i,ji,ji,j BAC

CBA

jii,j BAC

CBA


Inverse of a matrix

nnnn

n

n

nnnn

n

n

AAA

AA

AAA

AAA

A

aaa

aa

aaa

aaa

A

...

......

......

....

...

...

1

...

......

......

....

...

...

21

2313

22212

12111

1

21

3231

22221

11211

1

• Matrix of Cofactors divided by the determinate


Cofactors and Determinants

• Cofactor of Matrix

• Determinate of Matrix

• We could use any row or any column

jkkj

jk MA 1

n

kkjkj

n

kikik AaAaA

11


Matrices

• Inverse of a diagonal Matrix

• Inverse of a symmetrical matrix is symmetrical

• Inverse of an anti-symmetrical matrix is anti-symmetrical

• Inverse of the product of matrices is equal to the reordered product of the inverses.

• Normal Matrix

• Orthogonal Matrix

• Other Identities

II

AA

AA T

1

11

d

a

b

a

/10

0/1

0

0

111 ABBA

TAA

1 AA T


Definitions

• Actuator: A motor or transducer that converts energy (Electrical, Hydraulic, or Pneumatic Etc..) into motion.

• Transducer: A device for converting one form of energy to another. An example would be a microphone. It converts acoustic energy(Sound) to electrical energy.

• A/D: Analog to Digital converter. Converts an analog voltage to a digital value. Used to interface sensors to a computer. Also written (ATOD).

• D/A: Digital to Analog converter. Converts a digital value to an analog voltage. Often connected to the input of a control system or amp. Also written DTOA.


Definitions Continued

• Repeatability: How well a robot can return to the same point.

• Accuracy: How well a robot can move to an arbitrary point in space

• Precision: The smallest increment with which a robot can be positioned.

• Resolution: Sensor IncrementDesired Position

Actual Position

Ajacent Position

Precision Accuracy


Definitions Continued

• As upper limits the precision is equal to the resolution and the Accuracy is 1/2 the precision.

• Most robots repeatability, accuracy and precision changes throughout its workspace.

Type Hozizontal VerticalCatesian Uniform UniformCylindircal Decreases Radially UniformSpherical Decreases Radially Decreases RadiallySCARA Varies UniformArticulated Varies Varies


Type of Robot Actuation

• Direct Drive

• Geared– Belts, Gears, Harmonic Drives, Cycloidal Cam

Direct Drive Gear ReducedBacklash None Can be SignificantPower Density Poor Can be Very GoodSpeed High Suficient for Most ApplicationsFriction/Stiction Low Can be HighDisturbances Seen Directly Divided by Gear RationInetia Changes Seen Directly Divided by Gear Ration SquaredProcess Feedback Fealt Directly Is Masked by Stiction in GeartrainNoise Low Can be LoudReliability Very Good Good to PoorP:ostion Sensor Coaxial Can Take Advantage of Gear RatioWeight Heavy Low


SCARA (IBM/Sankyo)


Type of Actuation

Actuation Type

Torque/Force Density

SpeedPositional

RepeatabilityControl Type Cons

Hydraulics Very High Slow Poor Variable ValveGood Linear

Some RotationHydraulics

Leak, Control

Pneumatics Medium Fast Limited2 position or

PWMGood Linear

Some RotationLife of Seals,

Control

Electrical Low Fast Good SimpleGood Linear

Good RotationPower Density

• Electrical Most Popular

• Hydraulic used mainly in welding and underwater activities.

• Pneumatics used for gripping and detented motion


Motor

• Brush DC– Brushes may wear out. Perceived as a reliability issue. Brushes produce

dust.

• Brushless DC– Require a full H-Bridge and a sensor for comutation

• AC Induction– Requi

• Stepper

• Reluctance


Electric Motor Types

Motor Type CommutationPower Density

Fields Rotor

Field Stator

Torque Ripple

Thermal Amp Type Reliabilty Speed

Brush DC Mechanical HighDC

WindingsMagnets Low

Poor Windings on Rotor

4 Transistors

Brushes Medium

Brushless DC 6 Step

Electrical Halls High Magnets3

Windings Medium

Good Windings on Stator

6 Transistors

None High

Brushless DC Sinusoidal

Sensor Based High Magnets3

Windings Very Low

Good Windings on Stator

6 Transistors

None High

Stepper None MediumMagnets or Iron

N Windings

High FairN

TransistorsNone Low

Reluctance Sensor BasedMedium

LowIron

3 Windings

Low Fair6

TransistorsNone Medium

Inductance Sensor BasedMedium

LowInduced

3 Windings

Very Low Fair6

TransistorsNone Medium


Position Sensing

• Sense at Joint– Don’t Worry about Deflection or Backlash

• Sense at Motor– Low Cost Sensor

• Sense at End-Effector– Limited View

– Cost


Sensor Comparison

• Incremental Encoders and Resolvers are Most Popular

Type

Typical Resolution

BitsIncremental or Absolute

Signal Types Signals

Homing Required Cost

Electrical Imunity

Number of Wires

Electrical Interface

Encoder Incremental

12 Incremental Digital A, B, Index Yes $ Very Good 4-8 None

Pseudo Ambsolute

20+ Apsolute DigutalA, B, or Sin

Cos Coded RefSmall Motion

$$ Good 6-10 Yes

Encoder Absolute

16 Absolute Digital Binary No $$$ Very Good 20+ None

Sinusoidal Encoders

20+ AbsoluteDigital and

AnalogSin, Cos, 485 No $$ Good 6-10 Interpolator

Resolver14 Absolute

Analog Volts RMS

Sin,Cos No $$$ Good 6 R/D Converter

Inductosyn24 Incremental

Analog millivolts

Sin, Cos Yes $$$$ Fair 6 Amp +Converter

Capacitive24 Either

Analog millivolts

Sin, Cos Yes/No $$ Fair 6-10 Converter

Inductive24 Either

Analog millivolts

Sin, Cos Yes/No $$ Fair 6-10 Converter


Encoder

• Generally Have two picks up that are 90 degrees out of Phase (A and B) This allows you to determine the direction of rotation and thus count up or down

• By using the rising and falling edges of both A and B we can get 4 times the number of slots.

• May have one or more index marks for homing.


Absolute Encoder

• Gray Code Vs Binary

• Gray Code only changes by one bit per transition.

• At least one sensor per track.

1011

01 00

1011

00 01


Resolver

Sin

Cosine

Reference

• Uses an AC signal to excite the rotor winding.

• Stator has two windings at 90 degrees to each other.

• As the rotor turns the coupling to the two windings will change

• Can have multiple poles, but lose absolute capability.

• Converters usually are analog and can be expensive, $200 for 14-16 bits.


Inductosyn

• Similar to a resolver but made in two planes.

• The Inductosyn has many pole pairs, 50 +

• The output will repeat ones for each pole pair.

• Each cycle can be decoded to 14+ bits

• Require very precise alignment, and high quality amplifiers.

• Expensive

• Analog Encoders offer similar solution at a lower cost.


Analog Encoder.

• Uses a pattern and a matched diffraction grate to transmit light at different amounts as a function of rotation

• Optical sensors generate voltages proportional to the light hitting them.

• These voltages are digitize and used to produce absolute position values for a cycle.

• Encoders can be designed to produce multiple cycles per revolution.For example a disk can have 2048 cycles and each cycle can be decoded to 10 bits to result in 22 bits of position information.

• Multiple tracks can be place on a diskOne track with many cycles can be used to obtain fine resolutionOne tack can be used to determine which cycle of the fine track the encoder is in. Similar to an Hour, Minute and second hand on a clock.

• Requires additional Electronics to decode.

• LEDs require relatively high power.


Sin/Cosine


Reflective Sin/Cosine


Capacitive and Inductive Sensors

• Operate similar to analog encoders.

• Patterns are placed on the rotor and stator.

• Rotor and stator can be made of low cost materials

• Technology similar to printed circuit board fabrication.

• Capable of very low power operation. This would allow for battery baked operation.

• Resolution similar to analog encoders and Inductosyns

• Electronics utilize Digital to Analog converters and DSPs or PLDs.


Potentiometer

• Apply Voltage across Resistive Element

• Uses a Brush Sliding on a Resistive Element

• Brush Acts as a Voltage Divider

• Low Cost

• Noisy

• Varies with Temperature and Time

• Contact will Wear

• May produce particles

• New Laser Trimmed Films Show Promise for Linear Applications

Documents

02/11/2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Matrix Order Number of row or columns Rank of Matrix: Order of largest non-zero determinant. A matrix