on the problem of meas ing va iable measuring variable

On the Problem of On the Problem of Meas ing Va iable Meas ing Va iable Measuring Variable Measuring Variable

ImpedanceImpedancepp

Gi i G i li*Giorgio Grioli*Antonio Bicchi*^

*Center for Robotics and Bioengineering“E. Piaggio”

l d l^ IIT - Istituto Italiano di Tecnologia

Variable Stiffness Actuators have limitations: Variable Stiffness Actuators have limitations:◦ two motors per joint◦ more complexity, weight

less max stiffness than conventional joints◦ less max stiffness than conventional joints

VSA and Safety◦ may not be useful if heavy links are used◦ may not be economically justified by added performance

Common sense on Robots and VSA:◦ It’s hard to think of a stiff future for Robotics

V i bl i d i i t t i t◦ Variable impedance is important in nature◦ Dynamic adaptability to tasks is a major advantage

Conclusions

Variable Stiffness Actuators(aka Robotic Muscles) @ Pisa(aka Robotic Muscles) @ Pisa

The VSA HD

H iDrumming

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Hammering

The VSA Cube

The VSA Cube

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The VSA Cube

The VSA Cube

CastingCrushing

Impedance for Non-Linear Mechanical Systems

• Simplest notion of mechanical impedance: 

Mechanical Systems

p pLinear stiffness (Hooke’s Law)

• Generalization to Non Linear Springs:• Generalization to Non‐Linear Springs:– Partial derivative

• Generalization to Dynamic Systems:– Laplace Transform: Impedance

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Impedance for Non-Linear Mechanical SystemsMechanical Systems

• Generalizing Impedance:Generalizing Impedance:– Graph

• Analytical DescriptionAnalytical  Description

– At a Regular point :At a Regular point      :Locally there exists

•

Fréchet differential

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Admittance ViewAdmittance View

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An example

• Antagonist “muscle” system– Dynamics:

where

– Gen. Stiffness:

– Gen. Damping:

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An examplep

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Measuring Impedance

“Misura ciò che e misurabile e rendi• Measurements are at the basis of science

Misura ciò che e misurabile, e rendi misurabile ciò che non lo è” (Measure what is measurable, and make measurable what is not )

Galileo

• Feedback needs measuring

Galileo

– Measuring impedance is needed for control of VIA actuators

• Impedance is a differential operator “Physical Quantity: a property of adifferential operatornot a physical quantity in a strict sense

Physical Quantity: a property of a phenomenon, body, or substance, where the property has a magnitude that can be expressed as a number and a reference”

International Vocabulary of Metrology (VIM)

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in a strict sense International Vocabulary of Metrology (VIM).Basic and General Concepts and Associated Terms.

Measuring ImpedanceImpedance Measurements State of the Art 

• In ME

• In Biomechanics

• In Robotics, etc.

Common CharacteristicsCommon Characteristics– Typically: repeated experiments with probing perturbationsperturbations

– Mostly: not applicable in real time

Always*: linear time invariant impedance

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– Always*: linear, time invariant impedance

Measuring Linear Impedance• Simple case

Impedance• Simple case

– Build a non‐linearequivalent system

Observability– ObservabilityCo‐distribution

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Measuring Linear Impedance

Build a regression, or non‐linear observer e g an Extended Kalman Filtere.g. an Extended Kalman Filter

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Measuring Nonlinear ImpedanceImpedance

The same approachThe same approachis no longer possible(at least not trivially)(at least, not trivially)

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Measuring Nonlinear

Using EKF with a nonlinear impedanceImpedanceUsing EKF with a nonlinear impedance…

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The Variable Stiffness Observers

– given

– differentiation yields

– build an estimate

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The Variable Stiffness Observer

Th.: The update law

converges to within an Uniformly Ultimatelyconverges to within an Uniformly Ultimately Bounded error region around the real stiffness value

“A Non-Invasive, Real-Time Method forMeasuring Variable Stiffness”

G. Grioli, A. Bicchiy u uf

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Robotic Science and Systems 2010, Zaragoza, Spain. Submitted paperuf y

The Variable Stiffness Observer

y u uf

uf y

•The steeper stiffness changes with position gand input, the larger is the error

•Large co-contraction velocity with slow limb

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Large co contraction velocity with slow limb displacement may cause large errors

VSO - Simulations

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More interesting Simulations

– When the limb stops– When the limb stops…

proportional 

E i b

error

– Errors in m,b

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The Variable Stiffness Observer

y u uf

uf y

•The steeper stiffness changes with position gand input, the larger is the error

•Large co-contraction velocity with slow limb

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Large co contraction velocity with slow limb displacement may cause large errors

VSO – Mass and Damping

Can we observe stiffness without knowing Damping

m and b ?

NO, if we measure only the applied torquethe applied torque– e.g. human measurements

YES if we measure the elastic force(“inside” the joint)– e.g. robots

Experimental Resultsp

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Experimental Results

Position and Force

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Position and Force

Experimental Results

Relative Error

Observed Stiffness

e a e o

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Conclusions

• A discussion of nonlinear impedance definitionsA discussion of nonlinear impedance definitions

• A real‐time, non‐invasive algorithm to estimate stiffness

• Use of “dirty” derivatives increases error, do not pose threats to filter stability (but it might if closed loop)y g p

• Open issues– Extend to n‐dof’s

– Observe (nl, tv) generalized mass and damping

– Control impedance in closed loop

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– Apply beyond robotics 

Conclusions• Variable Stiffness Actuators have limitations:

– two motors per joint

l it i ht– more complexity, weight

– less max stiffness than conventional joints

• VSA and Safety• VSA and Safety– may not be useful if heavy links are used

– may not be economically justified by added performancemay not be economically justified by added performance

• Common sense on Robots and VSA:– It’s hard to think of a stiff future for RoboticsIt s hard to think of a stiff future for Robotics

– Variable impedance is important in nature

– Dynamic adaptability to tasks is a major advantage 

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QuestionsQuestions• When will soft robots an industrial reality?When will soft robots an industrial reality?

• Aren’t  “Variable Impedance Actuators” simply  “Robot Muscles”?

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Safety and Compliance on the market soon?- on the market soon?

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