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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
AB C
Ci,j Ai,j Bi,j
– Multiplication
• By Scalar
• By Matrix or Vector
02/11/2002
AB C
Ci,j Ai B j
Robotics 1
Copyright Martin P. Aalund, Ph.D.
Inverse of a matrix
• Matrix of Cofactors divided by the determinate
a11
a
21
a
A1 31
.
.
an1
02/11/2002
a12
a22
a32
.
.
an 2
.
.
.
.
.
.
.
.
.
.
.
.
1
. a1n
A11
A
. a2 n
12
. .
1 A13
. .
A .
.
. .
. ann
A1n
A21
A22
A23
.
.
A2 n
Robotics 1
Copyright Martin P. Aalund, Ph.D.
.
.
.
.
.
.
.
.
.
.
.
.
. An1
. An 2
. .
. .
. .
. Ann
Cofactors and Determinants
• Cofactor of Matrix
Ajk 1
j k
M jk
• Determinate of Matrix
n
n
k 1
k 1
A aik Aik akj Akj
• We could use any row or any column
02/11/2002
Robotics 1
Copyright Martin P. Aalund, Ph.D.
Matrices
• Inverse of a diagonal Matrix
a 0 1 / a 0
0 b 0 1 / d
• 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. AB1 B1A1
• Normal Matrix A A
• Orthogonal Matrix A A
• Other Identities A AT
T
1
T
A
1 1
A
I 1 I
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 Increment
Desired
Position
Precision Accuracy
Ajacent
Position
02/11/2002
Actual
Position
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
Catesian
Cylindircal
Spherical
SCARA
Articulated
02/11/2002
Hozizontal
Uniform
Decreases Radially
Decreases Radially
Varies
Varies
Vertical
Uniform
Uniform
Decreases Radially
Uniform
Varies
Robotics 1
Copyright Martin P. Aalund, Ph.D.
Type of Robot Actuation
• Direct Drive
• Geared
– Belts, Gears, Harmonic Drives, Cycloidal Cam
Backlash
Power Density
Speed
Friction/Stiction
Disturbances
Inetia Changes
Process Feedback
Noise
Reliability
P:ostion Sensor
Weight
02/11/2002
Direct Drive
None
Poor
High
Low
Seen Directly
Seen Directly
Fealt Directly
Low
Very Good
Coaxial
Heavy
Gear Reduced
Can be Significant
Can be Very Good
Suficient for Most Applications
Can be High
Divided by Gear Ration
Divided by Gear Ration Squared
Is Masked by Stiction in Geartrain
Can be Loud
Good to Poor
Can Take Advantage of Gear Ratio
Low
Robotics 1
Copyright Martin P. Aalund, Ph.D.
SCARA (IBM/Sankyo)
02/11/2002
Robotics 1
Copyright Martin P. Aalund, Ph.D.
Type of Actuation
Actuation
Type
Hydraulics
Pneumatics
Electrical
Torque/Force
Density
Speed
Positional
Repeatability
Control
Very High
Slow
Poor
Variable Valve
Medium
Fast
Limited
2 position or
PWM
Low
Fast
Good
Simple
Type
Good Linear
Hydraulics
Some Rotation Leak, Control
Good Linear
Life of Seals,
Some Rotation
Control
Good Linear
Power Density
Good Rotation
• 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.
Cons
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
Commutation
Power
Density
Mechanical
High
Brushless DC 6
Step
Electrical Halls
High
Brushless DC
Sinusoidal
Sensor Based
None
Motor Type
Brush DC
Stepper
Reluctance
Sensor Based
Inductance
Sensor Based
02/11/2002
Field
Stator
Torque
Ripple
Magnets
Low
Magnets
3
Windings
Medium
High
Magnets
3
Windings
Very Low
Medium
Magnets
or Iron
Medium
Low
Medium
Low
Fields
Rotor
DC
Windings
Iron
Induced
N
Windings
3
Windings
3
Windings
Thermal
Poor Windings
4
on Rotor
Transistors
Good Windings
6
on Stator
Transistors
Good Windings
6
on Stator
Transistors
High
Fair
Low
Fair
Very Low
Fair
Robotics 1
Copyright Martin P. Aalund, Ph.D.
Amp Type Reliabilty
N
Transistors
6
Transistors
6
Transistors
Speed
Brushes
Medium
None
High
None
High
None
Low
None
Medium
None
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
02/11/2002
Robotics 1
Copyright Martin P. Aalund, Ph.D.
Sensor Comparison
• Incremental Encoders and Resolvers are Most Popular
Type
Encoder
Incremental
Pseudo
Ambsolute
Encoder
Absolute
Sinusoidal
Encoders
Typical
Resolution Incremental
Bits
or Absolute
Resolver
Inductosyn
Capacitive
Inductive
02/11/2002
Signal
Types
Signals
Homing
Required
Cost
Electrical
Imunity
Number
of Wires
Electrical
Interface
12
Incremental
Digital
A, B, Index
Yes
$
Very Good
4-8
None
20+
Apsolute
Digutal
A, B, or Sin
Cos Coded Ref
Small
Motion
$$
Good
6-10
Yes
16
Absolute
Digital
Binary
No
$$$
Very Good
20+
None
20+
Absolute
Sin, Cos, 485
No
$$
Good
6-10
Interpolator
14
Absolute
Sin,Cos
No
$$$
Good
6
R/D Converter
24
Incremental
Sin, Cos
Yes
$$$$
Fair
6
Amp +Converter
24
Either
Sin, Cos
Yes/No
$$
Fair
6-10
Converter
24
Either
Sin, Cos
Yes/No
$$
Fair
6-10
Converter
Digital and
Analog
Analog
Volts RMS
Analog
millivolts
Analog
millivolts
Analog
millivolts
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.
11
10
11
10
01
00
00
01
02/11/2002
Robotics 1
Copyright Martin P. Aalund, Ph.D.
Resolver
•
•
•
•
•
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.
Cosine
Reference
Sin
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 disk
One track with many cycles can be used to obtain fine resolution
One 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.
Sin/Cosine
02/11/2002
Robotics 1
Copyright Martin P. Aalund, Ph.D.
Reflective Sin/Cosine
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
02/11/2002
Robotics 1
Copyright Martin P. Aalund, Ph.D.