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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