---
_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:
- access_level: open_access
  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
- access_level: open_access
  checksum: 3b6e874e30bfa1bfc3ad3498710145a1
  content_type: video/mp4
  creator: bbickel
  date_created: 2021-05-08T17:38:22Z
  date_updated: 2021-05-08T17:38:22Z
  file_id: '9378'
  file_name: multistable-video.mp4
  file_size: 76542901
  relation: main_file
  success: 1
- access_level: open_access
  checksum: 20dc3bc42e1a912a5b0247c116772098
  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
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
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 40
year: '2021'
...
---
_id: '7389'
abstract:
- lang: eng
  text: "Recently Kloeckner described the structure of the isometry group of the quadratic
    Wasserstein space W_2(R^n). It turned out that the case of the real line is exceptional
    in the sense that there exists an exotic isometry flow. Following this line of
    investigation, we compute Isom(W_p(R)), the isometry group of the Wasserstein
    space\r\nW_p(R) for all p \\in [1,\\infty) \\setminus {2}. We show that W_2(R)
    is also exceptional regarding the\r\nparameter p: W_p(R) is isometrically rigid
    if and only if p is not equal to 2. Regarding the underlying\r\nspace, we prove
    that the exceptionality of p = 2 disappears if we replace R by the compact\r\ninterval
    [0,1]. Surprisingly, in that case, W_p([0,1]) is isometrically rigid if and only
    if\r\np is not equal to 1. Moreover, W_1([0,1]) admits isometries that split mass,
    and Isom(W_1([0,1]))\r\ncannot be embedded into Isom(W_1(R))."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Gyorgy Pal
  full_name: Geher, Gyorgy Pal
  last_name: Geher
- first_name: Tamas
  full_name: Titkos, Tamas
  last_name: Titkos
- first_name: Daniel
  full_name: Virosztek, Daniel
  id: 48DB45DA-F248-11E8-B48F-1D18A9856A87
  last_name: Virosztek
  orcid: 0000-0003-1109-5511
citation:
  ama: Geher GP, Titkos T, Virosztek D. Isometric study of Wasserstein spaces - the
    real line. <i>Transactions of the American Mathematical Society</i>. 2020;373(8):5855-5883.
    doi:<a href="https://doi.org/10.1090/tran/8113">10.1090/tran/8113</a>
  apa: Geher, G. P., Titkos, T., &#38; Virosztek, D. (2020). Isometric study of Wasserstein
    spaces - the real line. <i>Transactions of the American Mathematical Society</i>.
    American Mathematical Society. <a href="https://doi.org/10.1090/tran/8113">https://doi.org/10.1090/tran/8113</a>
  chicago: Geher, Gyorgy Pal, Tamas Titkos, and Daniel Virosztek. “Isometric Study
    of Wasserstein Spaces - the Real Line.” <i>Transactions of the American Mathematical
    Society</i>. American Mathematical Society, 2020. <a href="https://doi.org/10.1090/tran/8113">https://doi.org/10.1090/tran/8113</a>.
  ieee: G. P. Geher, T. Titkos, and D. Virosztek, “Isometric study of Wasserstein
    spaces - the real line,” <i>Transactions of the American Mathematical Society</i>,
    vol. 373, no. 8. American Mathematical Society, pp. 5855–5883, 2020.
  ista: Geher GP, Titkos T, Virosztek D. 2020. Isometric study of Wasserstein spaces
    - the real line. Transactions of the American Mathematical Society. 373(8), 5855–5883.
  mla: Geher, Gyorgy Pal, et al. “Isometric Study of Wasserstein Spaces - the Real
    Line.” <i>Transactions of the American Mathematical Society</i>, vol. 373, no.
    8, American Mathematical Society, 2020, pp. 5855–83, doi:<a href="https://doi.org/10.1090/tran/8113">10.1090/tran/8113</a>.
  short: G.P. Geher, T. Titkos, D. Virosztek, Transactions of the American Mathematical
    Society 373 (2020) 5855–5883.
date_created: 2020-01-29T10:20:46Z
date_published: 2020-08-01T00:00:00Z
date_updated: 2023-08-17T14:31:03Z
day: '01'
ddc:
- '515'
department:
- _id: LaEr
doi: 10.1090/tran/8113
ec_funded: 1
external_id:
  arxiv:
  - '2002.00859'
  isi:
  - '000551418100018'
intvolume: '       373'
isi: 1
issue: '8'
keyword:
- Wasserstein space
- isometric embeddings
- isometric rigidity
- exotic isometry flow
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2002.00859
month: '08'
oa: 1
oa_version: Preprint
page: 5855-5883
project:
- _id: 26A455A6-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '846294'
  name: Geometric study of Wasserstein spaces and free probability
publication: Transactions of the American Mathematical Society
publication_identifier:
  eissn:
  - '10886850'
  issn:
  - '00029947'
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
status: public
title: Isometric study of Wasserstein spaces - the real line
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 373
year: '2020'
...