On the Problem of On the Problem of Meas ing Va iable Meas ing Va iable Measuring Variable Measuring Variable Impedance Impedance Gi i G i li* Giorgio Grioli* Antonio Bicchi*^ *Center for Robotics and Bioengineering “E. Piaggio” l d l ^ IIT - Istituto Italiano di T ecnologia
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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
– 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