April 12, 2023

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Theory and Design of

Statically Balanced Tensegrity Mechanisms

Graduation Colloquium

Mark Schenk

Department of BioMechanical EngineeringFaculty of Mechanical, Maritime and Materials Engineering

Exam Committee

prof. dr. ir. Peter Wieringa

dr. ir. Just Herder

dr. Simon Guest

dr. ir. Arend Schwab

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Contents

Introduction• tensegrity structures• static balancingMSc. thesis • objectivesZero Stiffness Tensegrity Structures• (zero) stiffnessPrototypeConclusions

statically balanced tensegrity mechanisms

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Needle Tower II, 1969Kröller-Müller Museum

Tensegrity Structures– introduction & examples –

Special class of bar framework• pin-jointed• prestressed

pure compression, pure tension -> cables and bars

pretension required for stiffness -> tensile integrity = tensegrity-> delicate balance

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Tensegrity Structures– engineering applications –

Engineering applications:• obvious architectural / design appeal• light-weight• stowable / deployable• biologically inspired

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Static Balancing– introduction & applications –

Static Balancing= continuous equilibrium= neutral stability= zero stiffness (Herder 2001)

Energy-free / energy-efficient design• e.g. rehabilitation, robotics

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Static Balancing– zero-free-length springs –

Zero-free-length springs• tension is proportional to length

normal pretensioned zero-free-length

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Static Balancing– design methodology & limitations –

Current design methodology• basic designs are extended

by means of modification rules• complex results possible

Limitations• ad hoc solutions, no real generic understanding• bottom-up vs. top-down• “limited” to 2D

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Statically Balanced Tensegrity Mechanisms– a combination of two fields –

tensegrity structures + static balancing

= zero stiffness tensegrity structures

= statically balanced tensegrity mechanisms

Previous examples:

(Herder, 2001)

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Statically Balanced Tensegrity Mechanisms– applications & relevance –

Totally new class of structures / mechanisms• robotics, rehabilitation, deployable structures

Academic interest• new insights for static balancing -> generic theory?• tensegrities <-> 3D pin-jointed structures

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Graduation project– title & objective –

Title“Theory and Design of Statically Balanced Tensegrity

Mechanisms.”

Objectives1. fundamental understanding zero stiffness

tensegrities-> generic description of static balancing-> develop design guidelines

2. build demonstration model

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Graduation project– research approach –

Theoretical research : zero stiffness

Stiffness of tensegrity structures• engineers

structural engineering• mathematicians

rigidity theory

combination

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Stiffness of Structures– tangent stiffness matrix –

Stiffness of a structure• counting rules (e.g. Maxwell’s rule)• tensegrity is prestressed• tangent stiffness matrix

Tangent stiffness matrix• displacements d and forces F

• formulations -> zero-free-length springs

FdK t

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Tangent Stiffness Matrix- zero stiffness -

Zero stiffness

Two components:

• both components are zero• components cancel out

t material geometrical K K K ΩKK t

~ˆ

0FdK t

0dΩdKdK t ~ˆ

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Zero Stiffness Tensegrity– conventional structures –

Conventional interpretation• internal mechanisms (undesirable)

• stabilized by self-stress

P

δ

L/2 L/2

3

3

8L

AEP

0dK ˆ 0dΩ ~

0dK t

Lto

4

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Zero Stiffness Tensegrity– structures with zero-free-length springs –

Zero stiffness tensegrity structures• affine transformations (scaling/shear)• preserve length of conventional members

0dΩ ~

0dK ˆ 0dK t

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Zero Stiffness Tensegrity- example structure -

top side

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Zero Stiffness Tensegrity– length-preserving affine & conic –

Length-preserving affine transformation• iff all bar directions lie on conic• infinitesimal -> finite• only zero-stiffness modes

Number of zero-stiffness modes• no. of unique bar directions k on conic

k ≥ 6 : zero-stiffness modes = 1k < 6 : zero-stiffness modes = 6 - k

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Zero Stiffness Babytoy

Babytoy • 6 bars• 3 bar directions• direction lie on conic

k ≥ 6 : zero-stiffness modes = 1k < 6 : zero-stiffness modes = 6 - k

• 3 zero-stiffness modes

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Prototype Structure- description -

Prototype structure• demonstrate properties• from structure to mechanism

Prototype design• classic tensegrity structure• no zero-free-length springs• 3 bars, 9 springs

-> 3 springs on each bar

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Prototype Structure- final model -

… but, there is a lot of friction

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Conclusions

Theory and Design of Statically Balanced Tensegrity Mechanisms

• theory of statically balanced tensegrity mechanisms• length-preserving affine transformations• conventional member directions on conic• finite mechanisms

• design of a statically balanced tensegrity mechanism• working prototype

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Introduction – MSc. thesis – Zero Stiffness – Prototype – Conclusions

Questions