Haptics-Utah-Lect3-Physical Haptic Systems.pptmech.utah.edu/.../Haptics-Utah-Lect3-Physical_Haptic_Systems.pdf · zBasic Elements of a haptic systemBasic Elements of a haptic system

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).

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

2

Readings

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)

3

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

Physical Haptic SystemsPhysical Haptic Systems

Actuators and transmissionActuators and transmission

4



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)

5

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

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

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

7

process repeats.Source: wikipedia.org

[Slide courtesy of S. Bamberg, Univ. of Utah]

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

8

g ( pp , y gmotors to stay away from stall for efficiency. However, haptic devices are mostly standing still)


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

9

Back EMFArmatureResistance

ArmatureInductance


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)

10

good quality motors (e.g., Maxon motors)


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

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


High cable tension high dragPossibly high reflected N2 inertia

Pulley

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


capstan pulleyMotor

Capstan Pulley Cable

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


, 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

Lawrentz force magnetic l it ti d ilevitation devices

ProsPros6-DOF force feedback

ConsC l itComplexity

hardware designControllerController

Limited workspace

15

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

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

16

Festo.comwww.pneumaticsfirst.org


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

17

cylindrical shape, causing the muscle to shorten http://www.shadowrobot.com/airmuscles/

© R. Wood (Harvard ES159/259) 2008 + W. Provancher 2009

Inflated – contractedDeflated – extended

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

18

Teletaction, IEEE Int. Conf. on Robotics and Automation, April 2000.


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

19

Sarcos Dexterous Master & ArmSame basic actuator designs as pneumatics (with modifications)

www.pneumaticsfirst.org © W. Provancher 2009

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

20

Since the Austenite and undeformed Martensite phases have the same shape


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

21

( yp y ,[Scheibe et al. 2007] on next slide).


Examples of SMA T til Di lSMA Tactile Displays

[Fisher 1996]

[Kontarinis et al 1995] [Wellman et al 1997]

22

[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

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

23

after poling: createsa dipole


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.

24

Piezoelectric tactile pin array[Summers & Chanter 2002]


[Homma, et al. 2004]Winfield & Colgate (2007)http://lims.mech.northwestern.edu/projects/TPaD/index.htm


SensorsSensorsPositionSensor

ForceSensor

25



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

26

FSRsOthers…

We’ll discuss the most common of these © W. Provancher 2009

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

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

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!


S tSensor types

MagneticMagnetic Optical Acoustic Inertial Mechanical

Most important for typical haptic interfaces


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)


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

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

33

MR encoders work similarly, but with magnetic sensors and metal disk


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

34

A B C D

Disk rotation CW

Ch. 2 Encoder Sensor

iskon

Allowstransmission of light (hole)© W. Provancher 2009

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)


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


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

37

angle© A. Okamura + W. Provancher 2009

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


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 -

39© A. Okamura + W. Provancher 2009

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)

40

Almost always FILTER your velocity signal!!© W. Provancher 2009

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

41

Filtered velocity = Filtered instantaneous velocity used in controller © A. Okamura + W. Provancher 2009

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

42

Set fc near (typically above) the mechanical time constant of your systemWhy not just set filter_ratio = .999 ???



Power AmplifierPower AmplifierPower

Amplifier

43



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

44

available


Stanford haptic paddle Linear Amp

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


Duty Cycle

Physical Haptic SystemsComputer Controller

Control (DAQ) Card( )Control

Card

46



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

47

National Instruments DAC and Encoder inputs on separate cards


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


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

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

50

B B 2 = BB2

© A. Okamura 2006

Physical Haptic SystemsComputer Controller

Controller code/softwareControllerProgram

51



T i l S ft C fi tiTypical Software Configuration

52

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

The Prototypical Closed Loop Control System

Error

Command Control Driver PlantΣ+ Output

LawPlantΣ

-Feedback


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

54

e = Error = Desired Position – Actual Positionv = Velocity


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

55

course.

© A. Okamura 2006

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

56

From Craig, p. 69

© A. Okamura 2006

Ki tiKinematics

Moving between Joint Space andMoving between Joint Space and Cartesian Space

Forward kinematics: based on joint angles, calculate end-effector position

57

calculate end-effector position

© A. Okamura 2006

J i t i blJoint variables

Be careful how you define joint positionsBe careful how you define joint positions

Ab l R l i

58

Absolute Relative

© A. Okamura 2006

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

59

joint motions cause motion independently at the end effector)

© A. Okamura 2006

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 )

60

at joints (i.e., relative actuation)

© A. Okamura 2006

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


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?


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”

63

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


J i t i blJoint variables

Be careful how you define joint positionsBe careful how you define joint positions

Ab l R l i

65

Absolute Relative

© A. Okamura 2006

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

66

joint motions cause motion independently at the end effector)

© A. Okamura 2006

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 )

67

at joints (i.e., relative actuation)

© A. Okamura 2006

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


E lExample

Two possible solutionsTwo possible solutions Two approaches:

algebraic method

geometric method


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


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

71

derivatives:

© A. Okamura 2006

A bl i M t iAssemble in a Matrix


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


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


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?

75

equal zero?

© A. Okamura 2006

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?


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


Documents

Haptics-Utah-Lect3-Physical Haptic Systems.pptmech.utah.edu/.../Haptics-Utah-Lect3-Physical_Haptic_Systems.pdf · zBasic Elements of a haptic systemBasic Elements of a haptic system