---
_id: '13188'
abstract:
- lang: eng
  text: "The Kirchhoff rod model describes the bending and twisting of slender elastic
    rods in three dimensions, and has been widely studied to enable the prediction
    of how a rod will deform, given its geometry and boundary conditions. In this
    work, we study a number of inverse problems with the goal of computing the geometry
    of a straight rod that will automatically deform to match a curved target shape
    after attaching its endpoints to a support structure. Our solution lets us finely
    control the static equilibrium state of a rod by varying the cross-sectional profiles
    along its length.\r\nWe also show that the set of physically realizable equilibrium
    states admits a concise geometric description in terms of linear line complexes,
    which leads to very efficient computational design algorithms. Implemented in
    an interactive software tool, they allow us to convert three-dimensional hand-drawn
    spline curves to elastic rods, and give feedback about the feasibility and practicality
    of a design in real time. We demonstrate the efficacy of our method by designing
    and manufacturing several physical prototypes with applications to interior design
    and soft robotics."
acknowledged_ssus:
- _id: M-Shop
acknowledgement: We thank the anonymous reviewers for their generous feedback, and
  Julian Fischer for his help in proving Proposition 1. This project has received
  funding from the European Research Council (ERC) under the European Union’s Horizon
  2020 research and innovation programme (grant agreement No. 715767).
article_number: '171'
article_processing_charge: No
article_type: original
author:
- first_name: Christian
  full_name: Hafner, Christian
  id: 400429CC-F248-11E8-B48F-1D18A9856A87
  last_name: Hafner
- first_name: Bernd
  full_name: Bickel, Bernd
  id: 49876194-F248-11E8-B48F-1D18A9856A87
  last_name: Bickel
  orcid: 0000-0001-6511-9385
citation:
  ama: Hafner C, Bickel B. The design space of Kirchhoff rods. <i>ACM Transactions
    on Graphics</i>. 2023;42(5). doi:<a href="https://doi.org/10.1145/3606033">10.1145/3606033</a>
  apa: Hafner, C., &#38; Bickel, B. (2023). The design space of Kirchhoff rods. <i>ACM
    Transactions on Graphics</i>. Association for Computing Machinery. <a href="https://doi.org/10.1145/3606033">https://doi.org/10.1145/3606033</a>
  chicago: Hafner, Christian, and Bernd Bickel. “The Design Space of Kirchhoff Rods.”
    <i>ACM Transactions on Graphics</i>. Association for Computing Machinery, 2023.
    <a href="https://doi.org/10.1145/3606033">https://doi.org/10.1145/3606033</a>.
  ieee: C. Hafner and B. Bickel, “The design space of Kirchhoff rods,” <i>ACM Transactions
    on Graphics</i>, vol. 42, no. 5. Association for Computing Machinery, 2023.
  ista: Hafner C, Bickel B. 2023. The design space of Kirchhoff rods. ACM Transactions
    on Graphics. 42(5), 171.
  mla: Hafner, Christian, and Bernd Bickel. “The Design Space of Kirchhoff Rods.”
    <i>ACM Transactions on Graphics</i>, vol. 42, no. 5, 171, Association for Computing
    Machinery, 2023, doi:<a href="https://doi.org/10.1145/3606033">10.1145/3606033</a>.
  short: C. Hafner, B. Bickel, ACM Transactions on Graphics 42 (2023).
date_created: 2023-07-04T07:41:30Z
date_published: 2023-09-20T00:00:00Z
date_updated: 2024-03-25T23:30:26Z
day: '20'
ddc:
- '516'
department:
- _id: BeBi
doi: 10.1145/3606033
ec_funded: 1
external_id:
  isi:
  - '001086833300010'
file:
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  date_created: 2023-07-04T08:11:28Z
  date_updated: 2023-07-04T08:11:28Z
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  date_created: 2023-07-04T07:46:28Z
  date_updated: 2023-07-04T07:46:28Z
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  content_type: video/mp4
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  date_created: 2023-07-04T07:47:10Z
  date_updated: 2023-07-04T07:47:10Z
  file_id: '13193'
  file_name: matlab-submission.zip
  file_size: 25790
  relation: supplementary_material
  title: Matlab Source Code with Example
file_date_updated: 2023-07-04T08:11:28Z
has_accepted_license: '1'
intvolume: '        42'
isi: 1
issue: '5'
keyword:
- Computer Graphics
- Computational Design
- Computational Geometry
- Shape Modeling
language:
- iso: eng
month: '09'
oa: 1
oa_version: Submitted Version
project:
- _id: 24F9549A-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '715767'
  name: 'MATERIALIZABLE: Intelligent fabrication-oriented Computational Design and
    Modeling'
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
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    relation: part_of_dissertation
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status: public
title: The design space of Kirchhoff rods
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 42
year: '2023'
...
---
_id: '9376'
abstract:
- lang: eng
  text: This paper presents a method for designing planar multistable compliant structures.
    Given a sequence of desired stable states and the corresponding poses of the structure,
    we identify the topology and geometric realization of a mechanism—consisting of
    bars and joints—that is able to physically reproduce the desired multistable behavior.
    In order to solve this problem efficiently, we build on insights from minimally
    rigid graph theory to identify simple but effective topologies for the mechanism.
    We then optimize its geometric parameters, such as joint positions and bar lengths,
    to obtain correct transitions between the given poses. Simultaneously, we ensure
    adequate stability of each pose based on an effective approximate error metric
    related to the elastic energy Hessian of the bars in the mechanism. As demonstrated
    by our results, we obtain functional multistable mechanisms of manageable complexity
    that can be fabricated using 3D printing. Further, we evaluated the effectiveness
    of our method on a large number of examples in the simulation and fabricated several
    physical prototypes.
acknowledged_ssus:
- _id: M-Shop
acknowledgement: 'We would like to thank everyone who contributed to this paper, the
  authors of artworks for all the examples, including @macrovec-tor_official and Wikimedia
  for the FLAG semaphore, and @pikisuper-star for the FIGURINE. The photos of iconic
  poses in the teaser were supplied by (from left to right): Mike Hewitt/Olympics
  Day 8 - Athletics/Gettty Images, Oneinchpunch/Basketball player training on acourt
  in New york city/Shutterstock, and Andrew Redington/TigerWoods/Getty Images. We
  also want to express our gratitude to Christian Hafner for insightful discussions,
  the IST Austria machine shop SSU, all proof-readers, and anonymous reviewers. This
  project has received funding from the European Union’s Horizon 2020 research and
  innovation programme, under the Marie Skłodowska-Curie grant agreement No 642841
  (DISTRO), and under the European Research Council grant agreement No 715767 (MATERIALIZABLE).'
article_number: '186'
article_processing_charge: No
article_type: original
author:
- first_name: Ran
  full_name: Zhang, Ran
  id: 4DDBCEB0-F248-11E8-B48F-1D18A9856A87
  last_name: Zhang
  orcid: 0000-0002-3808-281X
- first_name: Thomas
  full_name: Auzinger, Thomas
  id: 4718F954-F248-11E8-B48F-1D18A9856A87
  last_name: Auzinger
  orcid: 0000-0002-1546-3265
- first_name: Bernd
  full_name: Bickel, Bernd
  id: 49876194-F248-11E8-B48F-1D18A9856A87
  last_name: Bickel
  orcid: 0000-0001-6511-9385
citation:
  ama: Zhang R, Auzinger T, Bickel B. Computational design of planar multistable compliant
    structures. <i>ACM Transactions on Graphics</i>. 2021;40(5). doi:<a href="https://doi.org/10.1145/3453477">10.1145/3453477</a>
  apa: Zhang, R., Auzinger, T., &#38; Bickel, B. (2021). Computational design of planar
    multistable compliant structures. <i>ACM Transactions on Graphics</i>. Association
    for Computing Machinery. <a href="https://doi.org/10.1145/3453477">https://doi.org/10.1145/3453477</a>
  chicago: Zhang, Ran, Thomas Auzinger, and Bernd Bickel. “Computational Design of
    Planar Multistable Compliant Structures.” <i>ACM Transactions on Graphics</i>.
    Association for Computing Machinery, 2021. <a href="https://doi.org/10.1145/3453477">https://doi.org/10.1145/3453477</a>.
  ieee: R. Zhang, T. Auzinger, and B. Bickel, “Computational design of planar multistable
    compliant structures,” <i>ACM Transactions on Graphics</i>, vol. 40, no. 5. Association
    for Computing Machinery, 2021.
  ista: Zhang R, Auzinger T, Bickel B. 2021. Computational design of planar multistable
    compliant structures. ACM Transactions on Graphics. 40(5), 186.
  mla: Zhang, Ran, et al. “Computational Design of Planar Multistable Compliant Structures.”
    <i>ACM Transactions on Graphics</i>, vol. 40, no. 5, 186, Association for Computing
    Machinery, 2021, doi:<a href="https://doi.org/10.1145/3453477">10.1145/3453477</a>.
  short: R. Zhang, T. Auzinger, B. Bickel, ACM Transactions on Graphics 40 (2021).
date_created: 2021-05-08T17:37:08Z
date_published: 2021-10-08T00:00:00Z
date_updated: 2023-08-08T13:31:38Z
day: '08'
ddc:
- '000'
department:
- _id: BeBi
doi: 10.1145/3453477
ec_funded: 1
external_id:
  isi:
  - '000752079300003'
file:
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  checksum: 8564b3118457d4c8939a8ef2b1a2f16c
  content_type: application/pdf
  creator: bbickel
  date_created: 2021-05-08T17:36:59Z
  date_updated: 2021-05-08T17:36:59Z
  file_id: '9377'
  file_name: Multistable-authorversion.pdf
  file_size: 18926557
  relation: main_file
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  checksum: 3b6e874e30bfa1bfc3ad3498710145a1
  content_type: video/mp4
  creator: bbickel
  date_created: 2021-05-08T17:38:22Z
  date_updated: 2021-05-08T17:38:22Z
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  file_name: multistable-video.mp4
  file_size: 76542901
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  content_type: application/pdf
  creator: bbickel
  date_created: 2021-12-17T08:13:51Z
  date_updated: 2021-12-17T08:13:51Z
  description: This document provides additional results and analyzes the robustness
    and limitations of our approach.
  file_id: '10562'
  file_name: multistable-supplementary material.pdf
  file_size: 3367072
  relation: supplementary_material
  title: Supplementary Material for “Computational Design of Planar Multistable Compliant
    Structures”
file_date_updated: 2021-12-17T08:13:51Z
has_accepted_license: '1'
intvolume: '        40'
isi: 1
issue: '5'
keyword:
- multistability
- mechanism
- computational design
- rigidity
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: 2508E324-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '642841'
  name: Distributed 3D Object Design
- _id: 24F9549A-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '715767'
  name: 'MATERIALIZABLE: Intelligent fabrication-oriented Computational Design and
    Modeling'
publication: ACM Transactions on Graphics
publication_identifier:
  eissn:
  - 1557-7368
  issn:
  - 0730-0301
publication_status: published
publisher: Association for Computing Machinery
quality_controlled: '1'
status: public
title: Computational design of planar multistable compliant structures
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type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 40
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...