02/11/2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Matrix Order Number of row or columns Rank of...
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Transcript of 02/11/2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Matrix Order Number of row or columns Rank of...
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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.
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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.
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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.
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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.
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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.
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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.
02/11/2002 Robotics 1 Copyright Martin P. Aalund, Ph.D.
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