haptics-utah-lect3-physical haptic...
TRANSCRIPT
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HapticsME7960, Sect. 007
Lect. 3: Physical HapticLect. 3: Physical Haptic Systems
Spring 2009Prof William ProvancherProf. William Provancher
Prof. Jake AbbottUniversity of Utah
Salt Lake City, UT USAy,We would like to acknowledge the many colleagues whose course materials were borrowed and adapted in putting together this course, namely: Drs. Allison Okamura (JHU), Katherine Kuchenbecker (U Penn), Francois Conti, Federico Barbagli, and Kenneth Salisbury (Stanford), Ed Colgate (Northwestern), Hong Tan (Purdue), Blake Hannaford and Ganesh Sankaranarayanan (U. Washington), and Karon MacLean (UBC).
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T d ’ ClToday’s ClassPhysical Haptic SystemsPhysical Haptic Systems
Actuators for haptic/tactile systemsActuator types Actuator examples
SensorsAmplifierAmplifierControl CardOverview of Control Architecture
Overall system viewKinematics
Readings
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Readings
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Ph i l H ti S tPhysical Haptic SystemsBasic Elements of a haptic systemBasic Elements of a haptic system
Physical device (the plant of a control system) Actuators and transmissionS
Discussed today
SensorsHigh power (current) amplifier + power supply (what magnitude of currents?)Controller
Computer ControllerControl (DAQ) Card( )Servo control loop (typicallyat ≥1kHz) Why???
Analog Controller (quite rare)
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Analog Controller (quite rare)
Illustration from K. J. Kuchenbecker and G. Niemeyer, “Induced Master Motion in Force-Reflecting Teleoperation.” ASME Journal of Dynamic Systems, Measurement, and Control. Volume 128(4):800-810, December 2006.
© W. Provancher 2009
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Physical Haptic SystemsPhysical Haptic Systems
Actuators and transmissionActuators and transmission
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Illustration from K. J. Kuchenbecker and G. Niemeyer, “Induced Master Motion in Force-Reflecting Teleoperation.” ASME Journal of Dynamic Systems, Measurement, and Control. Volume 128(4):800-810, December 2006.
© W. Provancher 2009
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H ti /T til A t t TElectromagnetic Actuators
Haptic/Tactile Actuator Types Not typically used in haptic Electromagnetic Actuators
Electric motors DC Brushed PM Motors (most commonly used in haptics)DC B hl
systems. Why??
DC BrushlessStepper Motors
Voice coils and linear motorsLawrence force W
/Kg)
Lawrence forcePneumatic Actuators HydraulicSMA r D
ensi
ty (W
SMAPiezoelectricElectro-rheological
Pow
erWeight (Kg)
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Magneto-rheological (and magnetic particle brakes)… An interesting exception can be found in
Lawrence, D. A., Pao, L. Y., and Aphanuphong, S. 2006. Unwarping Encoder Ripple in Low Cost Haptic Interfaces. In Proceedings of Haptics Symposium (March 25 - 29, 2006), Washington, DC, 12.
© A. Okamura 2006 + W. Provancher 2009
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Brushed DC (direct current) PM ( t t) t(permanent magnet) motors
Most commonly usedBrushRotor
(armature)Most commonly used actuator in haptic devicesHow do they work? Brush
Commutator( )
Rotating armature with coil windings is caused to rotate relative to a permanent magnet
current is transmitted through brushes to armature, Stator
(Permanent
Housing
and is constantly switched so that the armature magnetic field remains fixed
(Permanent Magnets)
6© A. Okamura 2006 + W. Provancher 2009
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DC Motors: Two-pole ExamplePermanent
3-pole most common (Maxon motors have more)
PermanentMagnet
( )
Power supplied through brushes generates a magnetic field around the armaturearmature.
The north pole of the armature is pushed away from the north magnet andfrom the north magnet and pulled toward the south, causing rotation.
When the armaturecommutator
When the armature becomes horizontal, the connections between the commutator and the brushes reverse, and the process repeats
PermanentMagnetBrush
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process repeats.Source: wikipedia.org
[Slide courtesy of S. Bamberg, Univ. of Utah]
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DC t tDC motor terms Cogging / Torque RippleCogging / Torque Ripple
Tendency for torque output to ripple as the brushes transfer power (on bad motors this can be felt)
Friction/damping Caused by bearings and eddy currents (mostly, it’s coulomb friction that you feel but at high speedscoulomb friction that you feel, but at high speeds, especially through a geared transmission, viscous friction can begin to dominate)
St ll tStall torque Max torque delivered by motor when operated continuously without cooling (In most applications, you design
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g ( pp , y gmotors to stay away from stall for efficiency. However, haptic devices are mostly standing still)
© A. Okamura 2006 + W. Provancher 2009
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M t E tiMotor Equations
Torque constant KTorque constant, KT
Kt = Ke → (for Kt ≡ N*m/A, Ke ≡ V/(Rad/s))Kt = 9 55 x 10-3 Ke N*m/A → (for Kt ≡ N*m/A Ke ≡ V/KRPM)
Dynamic equation
Kt = 9.55 x 10 Ke N m/A → (for Kt ≡ N m/A, Ke ≡ V/KRPM)Kt = 1.3524 Ke oz*in/A → (for Kt ≡ oz*in/A, Ke ≡ V/KRPM)
Dynamic equation
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Back EMFArmatureResistance
ArmatureInductance
© A. Okamura 2006 + W. Provancher 2009
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B h d DC tBrushed DC motorsPros BrushRotor
(armature)ProsHigh bandwidth linear actuatorEasy to obtain with a wide
Brush
Commutator( )
Easy to obtain with a wide variety of sizesEasy to control
ConsConsTo get large torques, the armature inertia gets too high to make a good Stator
(Permanent
Housing
impedance deviceGenerates lots of heatRelatively expensive for
d lit t (
(Permanent Magnets)
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good quality motors (e.g., Maxon motors)
© A. Okamura 2006 + W. Provancher 2009
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DC Motors with leadscrews or b ll f li tiballscrews for linear motion
ProsProsHigh mechanical advantage and high forcesg
Can render stiff interactionsCons
Haptic Master
Most not backdrivableRequires (expensive) force sensor Haptic Master
(Moog Inc.)sensorDifficult controller designPossible eccentric vibration
11© W. Provancher 2009
http://www.engineersedge.com/hydraulic/images/linear-actuator-animation.gif
Possible eccentric vibration
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DC Motors with capstan drives for rotary motionfor rotary motion
MotorCapstanPulley
ProsCog-less “gear-ratio” transmission
SensAble (Premium) Phantom
MotorEduHaptics.org
ratio” transmissionCons
Issues with frictionTMotor
Capstan Pulley
CableIssues with frictiontransmission of cable on capstan pulley Sector
P lle
Cable
12© W. Provancher 2009
High cable tension high dragPossibly high reflected N2 inertia
Pulley
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DC Motors with capstan drives for linear motion
ProsCog-less linear transmission
Clark, J. E., Provancher, W. R., Mitiguy, P. “Theory, Simulation, And Hardware: Lab Design For An Integrated System Dynamics Education,” In Proc. of ASME IMECE, Orlando, FL, Nov. 5–11, 2005, p 147–153.
ConsRelatively low forces
TCableIssues with friction transmission of cable on
t ll
TMotorCable
13© W. Provancher 2009
capstan pulleyMotor
Capstan Pulley Cable
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V i C il A t tVoice Coil Actuators
ProsProsVery high bandwidth linear actuatoractuatorRelatively high forces
ConsConsLarge relative sizeShort strokeShort strokeGenerates lots of heat
Illustrations from Dimitrios A. Kontarinis and Robert D. Howe, “Tactile Display of Vibratory Information in
14© W. Provancher 2009
, p y yTeleoperation and Virtual Environments.” PRESENCE, 4(4):387-402, 1995Murray, A.M. and Klatzky, R.L. and Khosla, P.K, Psychophysical Characterization and Testbed Validation of a Wearable Vibrotactile Glove for Telemanipulation, Presence: Teleoperators & Virtual Environments, MIT Press 12:2, pp. 156-182, 2003
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Lawrentz force magnetic l it ti d ilevitation devices
ProsPros6-DOF force feedback
ConsC l itComplexity
hardware designControllerController
Limited workspace
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Illustrations from butterflyhaptics.com/ andBerkelman, P. “A Novel Coil Configuration to Extend the Motion Range of Lorentz Force Magnetic Levitation Devices for Haptic Interaction”, Proceedings of the 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems San Diego, CA, USA, Oct 29 - Nov 2, 2007 © W. Provancher 2009
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P ti t tPneumatic actuatorsHow do they work?How do they work?
Compressed air pressure is used to transfer energy from the power source to haptic interface. p p
Many different types (piston, McKibben, …) Concerns are friction and bandwidthConcerns are friction and bandwidth
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Festo.comwww.pneumaticsfirst.org
© A. Okamura 2006 + W. Provancher 2009
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Pneumatic actuators,cont.
McKibben Artificial MuscleMcKibben Artificial Muscle, pneumatically actuated with similar properties to biological musclesmuscles
Spring-like, flexible, for use within robotic arms and for prosthesis Shadowrobot handHigh force to weight ratioConstructed from internal air bladder and outer braided weave
Shadowrobot McKibben ActuatorshellAs bladder inflates, shell allows for expansion while maintaining cylindrical shape causing the
Shadowrobot. McKibben Actuator
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cylindrical shape, causing the muscle to shorten http://www.shadowrobot.com/airmuscles/
© R. Wood (Harvard ES159/259) 2008 + W. Provancher 2009
Inflated – contractedDeflated – extended
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Examples of pneumatic actuators f t til di lfor tactile displays
4x6 Pneumatic Array4x6 Pneumatic Array
M.B. Cohn, M. Lam, and R.S. Fearing, ``Tactile Feedback for Teleoperation'' SPIE Conf. 1833, Telemanipulator Technology, Boston, MA, Nov. 15-16, 1992.
G. Moy, C. Wagner, and R.S. Fearing, A Compliant Tactile Display for Teletaction IEEE Int Conf on Robotics and Automation April 2000
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Teletaction, IEEE Int. Conf. on Robotics and Automation, April 2000.
© W. Provancher 2009
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H d li t tHydraulic actuatorsProsPros
High forcesSmall actuator size on arm
d ff tor end effectorCons
Dangerous: Can kill you!!g yNeed big hydraulic pumpHydraulic seals need maintenancemaintenance
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Sarcos Dexterous Master & ArmSame basic actuator designs as pneumatics (with modifications)
www.pneumaticsfirst.org © W. Provancher 2009
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Shape Memory Alloy (SMA)(Li A t t )(Linear Actuators)
Most commonly in wire form: contracts when heatedMost commonly in wire form: contracts when heatedRepeated strains can be up to 3%
Based upon a solid state phase change (molecular realignment)
Two phases: Martensite and Austenite At low temperatures the material is MartensiteAt low temperatures, the material is MartensiteIf the Martensite material is loaded, this creates a deformation (typically seen as a shortening wire)‘Shape memory’ occurs when heated: the material willShape memory occurs when heated: the material will ‘remember’ its undeformed shape by heating to the Austenite phase
Since the Austenite and undeformed Martensite phases have
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Since the Austenite and undeformed Martensite phases have the same shape
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Sh M All (SMA)Shape Memory Alloy (SMA)
ProsProsRelatively robustHigh force / High power densityHigh force / High power densitySmall sizeInexpensive and readily available (especially inInexpensive and readily available (especially in wire form)
ConsConsLow efficiency (bad for portable applications)Gets hot (bad for direct contact with skin typically… however, see
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( yp y ,[Scheibe et al. 2007] on next slide).
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Examples of SMA T til Di lSMA Tactile Displays
[Fisher 1996]
[Kontarinis et al 1995] [Wellman et al 1997]
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[Kontarinis, et al. 1995] [Wellman, et al. 1997]
Scheibe, R. and Moehring, M. and Froehlich, B., Tactile Feedback at the Finger Tips for Improved Direct Interaction in Immersive Environments, Proc. IEEE Symposium on 3D User Interfaces 2007 pp. 10-14, 2007 © W. Provancher 2009
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Pi l t i A t tPiezoelectric ActuatorsPiezoelectric crystal strains when voltage is appliedPiezoelectric crystal strains when voltage is applied
Strains can be up to 0.1%Details
Direct piezoelectric effect: a piezoelectric material will generate charges (electric field) upon the application of stress along a properly poled directionConverse piezoelectric effect: a piezoelectric material will change its shape upon the application of an electric field (along a poled direction)
charge separation
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after poling: createsa dipole
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Examples of Pi l t i T til Di lPiezoelectric Tactile Displays
Glassmire thesis (2006)
TPaD: Tactile Pattern Display.
Piezoelectric tactile pin array
TPaD: Tactile Pattern Display.
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Piezoelectric tactile pin array[Summers & Chanter 2002]
© W. Provancher 2009
[Homma, et al. 2004]Winfield & Colgate (2007)http://lims.mech.northwestern.edu/projects/TPaD/index.htm
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Physical Haptic SystemsPhysical Haptic Systems
SensorsSensorsPositionSensor
ForceSensor
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Illustration from K. J. Kuchenbecker and G. Niemeyer, “Induced Master Motion in Force-Reflecting Teleoperation.” ASME Journal of Dynamic Systems, Measurement, and Control. Volume 128(4):800-810, December 2006.
© W. Provancher 2009
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S i h ti tSensors in haptic systemsMotion sensors
Rotary motionIncremental and absolute Encoders (also resolvers, but less common)Hall effect sensorPotentiometer (not common in haptics – noisyTachometer (pulse or generator)Tachometer (pulse or generator)Optical gray-scale analog sensors
Linear (translational) motion (less common)Encoders, potentiometer, optical emitter-detector,
AccelerationMEMS (most common now)MEMS (most common now)Piezoelectric and others
3D, 6-DOF motionVision based (Vicon), ultrasonic (Logitech), magnetic (Polhemus, Ascension Flock of Birds), others
( ) ( )Pressure (Tactile) (Also shape, but less common)Capacitive, piezoelectric…
Force sensorsLoad cells (flexure + strain/displacement sensor)FSRs
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FSRsOthers…
We’ll discuss the most common of these © W. Provancher 2009
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Force Sensors6-Axis JR3 ForceTorque Load Cell
How do they work? Typically a flexure + a strain gage (sometimes alsogage (sometimes also piezoelectric sensors, but these tend to drift)
Good quality 6-axis force-torque JR3 comGood quality 6 axis force torque sensor is ~$6-7K
Force Sensing Resistor (FSR)
JR3.com
ATI Nano17 Transducer
(FSR)Piezoresistive inkTons of sensor driftCh $10
Interlink FSR
Cheap ~$10
www.ati-ia.comTekscan Flexiforce FSR
27© W. Provancher 2009www.tekscan.com/flexiforce
www.interlinkelectronics.com
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T ki S A li tiTracking Sensor Applications
Eye trackingEye tracking Head tracking Body tracking Hand tracking
Most important for typical haptic interfaces
28© A. Okamura 2006
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D f F dDegrees of Freedom
Number of independent position variablesNumber of independent position variables needed to in order to locate all parts of a mechanismmechanism DOF of motion DOF f iDOF of sensing DOF of actuation DOF does not always correspond to number of joints!
29© A. Okamura 2006
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S tSensor types
MagneticMagnetic Optical Acoustic Inertial Mechanical
Most important for typical haptic interfaces
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M h i l T kMechanical Trackers
Ground based linkages most commonly usedGround-based linkages most commonly used Position Sensors
Di i l d ( i l MR)Digital: encoders (optical or MR)Analog: Hall-effect (magnetic)
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O ti l E dOptical Encoders How do they work? EmitterDetectorHow do they work?
Focused beam of light aimed at a paired photodetector is interrupted by a coded patterned disk
Incremental Encoder
a coded patterned disk Produces a number of pulses per revolution (↑ pulses = ↑ cost)
Quantization problems at lowQuantization problems at low speeds Absolute vs. differential encoders
Wikipedia.org
Absolute Encoder
Why choose one vs. other?Extra considerations if choosing incremental?
32© A. Okamura 2006 + W. Provancher 2009
saba.kntu.ac.irAbsolute Encoder Disk
Wikipedia.org
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O ti l E dOptical Encoders Phase-quadrature encoderPhase quadrature encoder
2 channels, 90° out of phase allows sensing of direction of rotation g4-fold increase in resolution
Schmitt trigger electronics create square waveform from analog detector signal
MR d k i il l b t ith ti d
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MR encoders work similarly, but with magnetic sensors and metal disk
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Quadrature Encoder States & DecodingDisk rotation CCW
1 1
00
Ch. 1 Encoder Signal
1
00
1
00
Ch. 2 Encoder Signal
Encoder States
00
A B C DSimplified Encoder Disk
EmitterDetectorEncoder States
A B C D
Disk rotation CCW
Simplified Encoder Disk(4 CPR, 16 PPR) (shown in state A)
Ch. 1 Encoder SensorBlockstransmission
Ch. 1
Ch. 2 0
0
0
1
1
1
1
0 45 deg.
CC
W D
Rotatio
of light
2
1
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A B C D
Disk rotation CW
Ch. 2 Encoder Sensor
iskon
Allowstransmission of light (hole)© W. Provancher 2009
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H ll Eff t SHall-Effect Sensors
How do they work?How do they work? A small transverse voltage is generated across a current-carrying conductor in the presence of acurrent-carrying conductor in the presence of a magnetic field
(Discovery made in 1879, but not useful until the advent ofuntil the advent of semiconductor technology)
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H ll Eff t SHall-Effect Sensors
Amount of voltage output related to the strength of magnetic field passing through. Linear over small range of motion
Need to be calibrated Affected by temperature, other magnetic objects in the environments
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H ll Eff t SHall-Effect SensorsFrom StanfordH ti P ddl
θ
Haptic Paddle
magnet
Hall-effectHall effectSensor with
3 leads
The voltage varies sinusoidally with rotation
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M i V l itMeasuring Velocity
Differentiate positionDifferentiate position advantage: use same sensor as position sensor disadvantage: get noise signaldisadvantage: get noise signal
Alternative f d ti b t ti kfor encoders, measure time between ticks
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Digital differentiationDigital differentiation Many different methods ySimple Example:
Position reading at time 1 = P1 V = P2 – P1
tPosition reading at time 2 = P2 t is the period of the servo loop (in sec. or counts)
Th iti i t i ll l d fi d i t lThe position is typically sampled on a fixed intervalDifferentiation increases noise -
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Noisy Velocity readings
Noise on velocity signal can create jitter, especially at high K gains (many haptic controllers don’t even includeat high Kv gains (many haptic controllers don t even include damping)Common solutions
Tach/GeneratorVoltage a speed (same source as back EMF)Resolution only as great as your A/D converterNot great at low speeds
Integrate the signal from an accelerometerIntegrate the signal from an accelerometerTime per tick rather than ticks per time
use a special chip that measures time between ticks Especially good to do at slow speedsp y g pFares poorly at high velocities
Filtering (conventional to smooth or Kalman filtering to combine sensor signals)
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Almost always FILTER your velocity signal!!© W. Provancher 2009
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Digital differentiationD li ith i b filt iDealing with noise by filtering
How do you deal with noiseHow do you deal with noiseFiltering
Example 1: Pre-filter positionsV = P2 – P1
tAverage 20 readings = P1 Average next 20 readings = P2
Example 2: Post-filter Velocities (more common)Example 2: Post filter Velocities (more common)Position reading at time 1 = P1 Position reading at time 2 = P2C l l t i t t l itCalculate instantaneous velocitySet velocity equal to filtered version of instantaneous velocity
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Digital FilteringCommon filter types
FIR filters Median (with defined sample window)Moving average (with defined sample window)
IIR filtersFading memory filterg y
Filtered value = old_value * filter_ratio+ current value * (1 – filter_ratio)
Typical filter ratio 0.8 to 0.9ypCan simulate in Matlab with a step input to figure out equivalent cutoff frequency (function of filter_ratio and sample frequency)
Filter output = 0.667 at τ sec and 10%-90% rise time ~2.2 τTi t t RC 1/ C f f 1/(2 )Time constant τ ≡ RC = 1/ω, Corner frequency ≡ fc = 1/(2 π τ) E.g., 90:10 fading memory filter (i.e., filter ratio = 0.9) has fc = 15.2 Hz when running a 1 KHz servo loop.
Set fc near (typically above) the mechanical time constant
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Set fc near (typically above) the mechanical time constant of your systemWhy not just set filter_ratio = .999 ???
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Physical Haptic SystemsPhysical Haptic Systems
Power AmplifierPower AmplifierPower
Amplifier
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Illustration from K. J. Kuchenbecker and G. Niemeyer, “Induced Master Motion in Force-Reflecting Teleoperation.” ASME Journal of Dynamic Systems, Measurement, and Control. Volume 128(4):800-810, December 2006.
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Power (Current) AmplifiersLMD18200 H-Bridge PWM Amp
Courtesy of Stanford SPDL
PWM (Pulse Width Modulation)More efficientCopley, AMC, etc. make p yPWM amps with closed-loop current control capability (~$350)
Voltage in proportional current outOther low end H-Bridge PWM amps
l il bl ( LMD18200 $10)
AMC PWM Amplifier
also available (e.g., LMD18200, ~$10)Linear
Higher bandwidth than PWM, but not typically neededyp yHaptic paddle uses a LM675 high current op. ampMany commercial linear amps also available
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available
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Stanford haptic paddle Linear Amp
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Example of PWM Control SignalExample of PWM Control Signal
Varying the duty cycle of the PWM signal is analogous to changing the voltage Equivalent
Voltage
50%CDuty Cycle
~90%Duty Cycle
45© W. Provancher 2009
Duty Cycle
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Physical Haptic SystemsComputer Controller
Control (DAQ) Card( )Control
Card
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Illustration from K. J. Kuchenbecker and G. Niemeyer, “Induced Master Motion in Force-Reflecting Teleoperation.” ASME Journal of Dynamic Systems, Measurement, and Control. Volume 128(4):800-810, December 2006.
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DAQ/Controller Computer CardUsually a PCI bus card
Common control cardsCommon control cardsSensoray 626 (~$700)
Four 14-bit D/A outputs, up to 6 encoder inputs,…Four 14 bit D/A outputs, up to 6 encoder inputs,…dSpace 1104 (ACE1104CLP, ~$5K)Quanser Q4 and Q8 (≥ $4K)IOTech (e.g., DaqBoard/3001 ~$1K))Servo2go (ISA bus only, but 8 ch. for ~$1K)
Sensoray 626
Computer BoardsDAC and Encoder inputs on separate cards
National Instruments
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National Instruments DAC and Encoder inputs on separate cards
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S TSome Terms
AD/DAAD/DA analog to digital digital to analogdigital to analog
Interrupt service routine S LServo Loop Servo rate
Usually needs to be >500 Hz, and most often is 1,000 Hz
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D/A d A/DD/A and A/D
Converts between He adecimaBinarDecimalConverts between voltages and counts C t t
Hexadecimal
Binary Decimal
0 0000 0
1 0001 1
2 0010 2
Computer stores information digitally, and communicates with the
3 0011 3
4 0100 4
5 0101 5
6 0110 6 communicates with the outside world using +/-10V signals
7 0111 7
8 1000 8
9 1001 9
A10101010V signals E.g., for 8-bit 0-5V ADC 2.5V = 10000000
A 1010 10
B 1011 11
C 1100 12
D 1101 13
49MSB
E 1110 14
F 1111 15 LSB© A. Okamura 2006
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D/A d A/DD/A and A/D
Converts voltages to counts and vice versaConverts voltages to counts and vice versa A 12-bit card:
212 d i l b (4096)212 decimal numbers (4096)
Decimal (base 10): 2994
Binary (base 2):
1 1 0 0 1 0 0 1 0 1
Hexadecimal (base 16): 1 1 0 0 1 0 0 1 0 1
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B B 2 = BB2
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Physical Haptic SystemsComputer Controller
Controller code/softwareControllerProgram
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Illustration from K. J. Kuchenbecker and G. Niemeyer, “Induced Master Motion in Force-Reflecting Teleoperation.” ASME Journal of Dynamic Systems, Measurement, and Control. Volume 128(4):800-810, December 2006.
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T i l S ft C fi tiTypical Software Configuration
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Illustration from V. Hayward and K. MacLean. “Do It Yourself Haptics: Part 1.” IEEE Robotics and Automation Magazine, December, 2007, pp. 88-104.
© K. Kuchenbecker 2008
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The Prototypical Closed Loop Control System
Error
Command Control Driver PlantΣ+ Output
LawPlantΣ
-Feedback
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Example controllerExample controllerImplementing the simplestpossible PID Controllerpossible PID Controller
Control Effort = K * e + K * Σe K * vControl Effort = KP e + KI Σe – KD v
KP = Proportional GainP
KI = Integral Gain
KD = Derivative Gain
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e = Error = Desired Position – Actual Positionv = Velocity
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Other things your controller ill dwill needGrounded interfaces (e g force feedbackGrounded interfaces (e.g., force feedback devices)
Very similar to robots Need Kinematics
Determine endpoint position Calculate velocitiesCalculate velocities Calculate force-torque relationships
Sometimes need Dynamics If you are weak on these topics, you may want to check out Introduction to robotics: mechanics and control by Craig (1989) or notes from your robotics
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course.
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Ki tiKinematics
Think of aThink of a manipulator/ interface as a set of bodiesas a set of bodies connected by a chain of jointsof joints Revolute most common for hapticcommon for haptic interfaces
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From Craig, p. 69
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Ki tiKinematics
Moving between Joint Space andMoving between Joint Space and Cartesian Space
Forward kinematics: based on joint angles, calculate end-effector position
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calculate end-effector position
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J i t i blJoint variables
Be careful how you define joint positionsBe careful how you define joint positions
Ab l R l i
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Absolute Relative
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Ab l t f d ki tiAbsolute forward kinematics
x = L1cos(θ1) + L2cos(θ2) y L sin(θ ) + L sin(θ )y = L1sin(θ1) + L2sin(θ2)
Sometimes done this way with haptic devices (e g can be usedhaptic devices (e.g., can be used when controlling elbow and shoulder joints of a Phantom Premium since its joint motions cause motion
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joint motions cause motion independently at the end effector)
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R l ti f d ki tiRelative forward kinematics
x = L1cos(θ1) + L2cos(θ1+θ2)y = L sin(θ ) + L sin(θ + θ )y = L1sin(θ1) + L2sin(θ1+ θ2)
Usually done this way with robots, sometimes with hapticrobots, sometimes with haptic devicesTypically used if you have actuators t j i t (i l ti t ti )
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at joints (i.e., relative actuation)
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V l itVelocity
Recall that the forward kinematics tells us theRecall that the forward kinematics tells us the end-effector position based on joint positions H d l l t l it ?How do we calculate velocity? Use a matrix called the Jacobian
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F l ti j i t tFormulating joint torques
The Jacobian can be used to relate jointThe Jacobian can be used to relate joint torques to end-effector forces:
TT
Why is this important for haptic displays?
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R di f t tiReading for next timeV Hayward and K MacLean “Do It Yourself Haptics:V. Hayward and K. MacLean. Do It Yourself Haptics: Part 1.” IEEE Robotics and Automation Magazine, December, 2007, pp. 88-104.
Next lecture will be on “Overview of dynamics/controls & yterminology”
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Supplemental Information on Ki ti d J biKinematics and Jacobians
Although “Introduction to Robotics” is aAlthough Introduction to Robotics is a prerequisite for this class, we recognize that some of you may be a little rusty withsome of you may be a little rusty with kinematics, hence the pages that follow will provide a refresher as a reference to look atprovide a refresher as a reference to look at on your own
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J i t i blJoint variables
Be careful how you define joint positionsBe careful how you define joint positions
Ab l R l i
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Absolute Relative
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Ab l t f d ki tiAbsolute forward kinematics
x = L1cos(θ1) + L2cos(θ2) y L sin(θ ) + L sin(θ )y = L1sin(θ1) + L2sin(θ2)
Sometimes done this way with haptic devices (e g can be usedhaptic devices (e.g., can be used when controlling elbow and shoulder joints of a Phantom Premium since its joint motions cause motion
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joint motions cause motion independently at the end effector)
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R l ti f d ki tiRelative forward kinematics
x = L1cos(θ1) + L2cos(θ1+θ2)y = L sin(θ ) + L sin(θ + θ )y = L1sin(θ1) + L2sin(θ1+ θ2)
Usually done this way with robots, sometimes with hapticrobots, sometimes with haptic devicesTypically used if you have actuators t j i t (i l ti t ti )
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at joints (i.e., relative actuation)
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I Ki tiInverse Kinematics
Using the end effector position calculate theUsing the end-effector position, calculate the joint angles N t d ft i h tiNot used often in haptics
But useful for calibration Th bThere can be:
No solution (workspace issue) 1 l i1 solution More than 1 solution
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E lExample
Two possible solutionsTwo possible solutions Two approaches:
algebraic method
geometric method
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V l itVelocity
Recall that the forward kinematics tells us theRecall that the forward kinematics tells us the end-effector position based on joint positions H d l l t l it ?How do we calculate velocity? Use a matrix called the Jacobian
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F l ti th J biFormulating the Jacobian Use the chainUse the chain rule:
Substitute inSubstitute in expressions for x & y from ybefore and take partial derivatives
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derivatives:
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A bl i M t iAssemble in a Matrix
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Si l itiSingularities
Most devices will have values for which theMost devices will have values for which the Jacobian is singular Thi th t th d i h l tThis means that the device has lost one or more degrees of freedom in Cartesian Space T ki dTwo kinds:
Workspace boundary W k i t iWorkspace interior
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Si l it M thSingularity Math If the matrix is invertible then it is nonIf the matrix is invertible, then it is non-singular
Can check invertibility if J by taking theCan check invertibility if J by taking the determinant of J . If it is equal to 0, then it is singular gCan use this method to check which values of θ will cause singularities
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C l l ti Si l itiCalculating Singularities
Simplify nomenclature: i (θ +θ )Simplify nomenclature: sin(θ1+θ2)=s12
For what values of θ1 and θ2 does this equal zero?
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equal zero?
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E f lEven more useful….
The Jacobian can be used to relate jointThe Jacobian can be used to relate joint torques to end-effector forces:
TWhy is this important for haptic displays?
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H d t thi ti ?How do you get this equation?
P i i l f i t l kPrinciple of virtual work States that changing the coordinate frame does not change the total work of a systemsystem
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