tdehaeze/phd-nass-geometry
Archived
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Christophe's review

Browse Source
This commit is contained in:
Thomas Dehaeze 2025-04-04 17:52:43 +02:00
parent ab39bbdf0a
commit de4b8bc88c
7 changed files with 455 additions and 227 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
figs/detail_geometry_gough_paper.jpg Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 139 KiB

BIN
figs/detail_geometry_stewart_flight_simulator.jpg Normal file
View File

Binary file not shown.
Side by Side

After

Width:  |  Height:  |  Size: 53 KiB

293
nass-geometry.bib
View File

@ -1,64 +1,3 @@
@phdthesis{afzali-far16_vibrat_dynam_isotr_hexap_analy_studies,
author = {Afzali-Far, Behrouz},
school = {Lund University},
title = {Vibrations and Dynamic Isotropy in Hexapods-Analytical
Studies},
year = 2016,
keywords = {parallel robot},
}
@article{mcinroy00_desig_contr_flexur_joint_hexap,
author = {J.E. McInroy and J.C. Hamann},
title = {Design and Control of Flexure Jointed Hexapods},
journal = {IEEE Transactions on Robotics and Automation},
volume = 16,
number = 4,
pages = {372-381},
year = 2000,
doi = {10.1109/70.864229},
url = {https://doi.org/10.1109/70.864229},
keywords = {parallel robot},
}
@article{yang19_dynam_model_decoup_contr_flexib,
author = {Yang, XiaoLong and Wu, HongTao and Chen, Bai and Kang,
ShengZheng and Cheng, ShiLi},
title = {Dynamic Modeling and Decoupled Control of a Flexible
Stewart Platform for Vibration Isolation},
journal = {Journal of Sound and Vibration},
volume = 439,
pages = {398-412},
year = 2019,
doi = {10.1016/j.jsv.2018.10.007},
url = {https://doi.org/10.1016/j.jsv.2018.10.007},
issn = {0022-460X},
keywords = {parallel robot, flexure, decoupled control},
month = {Jan},
publisher = {Elsevier BV},
}
@article{furutani04_nanom_cuttin_machin_using_stewar,
author = {Katsushi Furutani and Michio Suzuki and Ryusei Kudoh},
title = {Nanometre-Cutting Machine Using a Stewart-Platform Parallel
Mechanism},
journal = {Measurement Science and Technology},
volume = 15,
number = 2,
pages = {467-474},
year = 2004,
doi = {10.1088/0957-0233/15/2/022},
url = {https://doi.org/10.1088/0957-0233/15/2/022},
keywords = {parallel robot, cubic configuration},
}
@article{geng93_six_degree_of_freed_activ,
author = {Zheng Geng and Leonard S. Haynes},
title = {Six-Degree-Of-Freedom Active Vibration Isolation Using a
@ -210,6 +149,21 @@
@article{mcinroy00_desig_contr_flexur_joint_hexap,
author = {J.E. McInroy and J.C. Hamann},
title = {Design and Control of Flexure Jointed Hexapods},
journal = {IEEE Transactions on Robotics and Automation},
volume = 16,
number = 4,
pages = {372-381},
year = 2000,
doi = {10.1109/70.864229},
url = {https://doi.org/10.1109/70.864229},
keywords = {parallel robot},
}
@inproceedings{li01_simul_vibrat_isolat_point_contr,
author = {Xiaochun Li and Jerry C. Hamann and John E. McInroy},
title = {Simultaneous Vibration Isolation and Pointing Control of
@ -330,6 +284,22 @@
@article{furutani04_nanom_cuttin_machin_using_stewar,
author = {Katsushi Furutani and Michio Suzuki and Ryusei Kudoh},
title = {Nanometre-Cutting Machine Using a Stewart-Platform Parallel
Mechanism},
journal = {Measurement Science and Technology},
volume = 15,
number = 2,
pages = {467-474},
year = 2004,
doi = {10.1088/0957-0233/15/2/022},
url = {https://doi.org/10.1088/0957-0233/15/2/022},
keywords = {parallel robot, cubic configuration},
}
@inproceedings{ting06_desig_stewar_nanos_platf,
author = {Yung Ting and H.-C. Jar and Chun-Chung Li},
title = {Design of a 6DOF Stewart-type Nanoscale Platform},
@ -500,6 +470,25 @@
@article{yang19_dynam_model_decoup_contr_flexib,
author = {Yang, XiaoLong and Wu, HongTao and Chen, Bai and Kang,
ShengZheng and Cheng, ShiLi},
title = {Dynamic Modeling and Decoupled Control of a Flexible
Stewart Platform for Vibration Isolation},
journal = {Journal of Sound and Vibration},
volume = 439,
pages = {398-412},
year = 2019,
doi = {10.1016/j.jsv.2018.10.007},
url = {https://doi.org/10.1016/j.jsv.2018.10.007},
issn = {0022-460X},
keywords = {parallel robot, flexure, decoupled control},
month = 1,
publisher = {Elsevier BV},
}
@phdthesis{naves20_desig,
author = {Mark Naves},
school = {Univeristy of Twente},
@ -525,6 +514,177 @@
@phdthesis{afzali-far16_vibrat_dynam_isotr_hexap_analy_studies,
author = {Afzali-Far, Behrouz},
school = {Lund University},
title = {Vibrations and Dynamic Isotropy in Hexapods-Analytical
Studies},
year = 2016,
keywords = {parallel robot},
}
@article{gough62_univer_tyre_test_machin,
author = {Gough, V Eric},
title = {Universal Tyre Test Machine},
journal = {Proc. FISITA 9th Int. Technical Congr., London, 1962},
pages = {117--137},
year = 1962,
}
@article{stewart65_platf_with_six_degrees_freed,
author = {Stewart, Doug},
title = {A Platform With Six Degrees of Freedom},
journal = {Proceedings of the institution of mechanical engineers},
volume = 180,
number = 1,
pages = {371--386},
year = 1965,
publisher = {Sage Publications Sage UK: London, England},
}
@article{dasgupta00_stewar_platf_manip,
author = {Bhaskar Dasgupta and T.S. Mruthyunjaya},
title = {The Stewart Platform Manipulator: a Review},
journal = {Mechanism and Machine Theory},
volume = 35,
number = 1,
pages = {15-40},
year = 2000,
doi = {10.1016/s0094-114x(99)00006-3},
url = {https://doi.org/10.1016/s0094-114x(99)00006-3},
keywords = {parallel robot},
}
@article{kazezkhan14_dynam_model_stewar_platf_nansh_radio_teles,
author = {Guljaina Kazezkhan and Binbin Xiang and Na Wang and Aili
Yusup},
title = {Dynamic Modeling of the Stewart Platform for the Nanshan
Radio Telescope},
journal = {Advances in Mechanical Engineering},
volume = 12,
number = 7,
pages = {nil},
year = 2014,
doi = {10.1177/1687814020940072},
url = {http://dx.doi.org/10.1177/1687814020940072},
DATE_ADDED = {Fri Apr 4 16:01:49 2025},
}
@article{yun19_devel_isotr_stewar_platf_teles_secon_mirror,
author = {Hai Yun and Lei Liu and Qing Li and Wenbo Li and Liang
Tang},
title = {Development of an Isotropic Stewart Platform for Telescope
Secondary Mirror},
journal = {Mechanical Systems and Signal Processing},
volume = 127,
number = {nil},
pages = {328-344},
year = 2019,
doi = {10.1016/j.ymssp.2019.03.001},
url = {http://dx.doi.org/10.1016/j.ymssp.2019.03.001},
DATE_ADDED = {Fri Apr 4 16:02:00 2025},
}
@article{russo24_review_paral_kinem_machin_tools,
author = {Matteo Russo and Dan Zhang and Xin-Jun Liu and Zenghui Xie},
title = {A Review of Parallel Kinematic Machine Tools: Design,
Modeling, and Applications},
journal = {International Journal of Machine Tools and Manufacture},
volume = 196,
number = {nil},
pages = 104118,
year = 2024,
doi = {10.1016/j.ijmachtools.2024.104118},
url = {http://dx.doi.org/10.1016/j.ijmachtools.2024.104118},
DATE_ADDED = {Fri Apr 4 16:06:19 2025},
}
@inproceedings{marion04_hexap_esrf,
author = {Marion, Ph and Comin, F and Rostaining, G and others},
title = {Hexapods at the ESRF: mechanical aspects results obtained},
booktitle = {MEDSI 2004 proceedings},
year = 2004,
pages = {1--9},
keywords = {esrf},
}
@article{villar18_nanop_esrf_id16a_nano_imagin_beaml,
author = {F. Villar and L. Andre and R. Baker and S. Bohic and J. C.
da Silva and C. Guilloud and O. Hignette and J. Meyer and A.
Pacureanu and M. Perez and M. Salome and P. van der Linden and
Y. Yang and P. Cloetens},
title = {Nanopositioning for the Esrf Id16a Nano-Imaging Beamline},
journal = {Synchrotron Radiation News},
volume = 31,
number = 5,
pages = {9-14},
year = 2018,
doi = {10.1080/08940886.2018.1506234},
url = {http://dx.doi.org/10.1080/08940886.2018.1506234},
keywords = {esrf},
}
@book{merlet06_paral_robot,
author = {Merlet, J. P.},
title = {Parallel Robots},
year = 2006,
publisher = {Springer Publishing Company, Incorporated},
edition = {2nd},
isbn = 9048170532,
}
@phdthesis{li01_simul_fault_vibrat_isolat_point,
author = {Li, Xiaochun},
keywords = {parallel robot},
school = {University of Wyoming},
title = {Simultaneous, Fault-tolerant Vibration Isolation and
Pointing Control of Flexure Jointed Hexapods},
year = 2001,
}
@article{kim00_robus_track_contr_desig_dof_paral_manip,
author = {Dong Hwan Kim and Ji-Yoon Kang and Kyo-Il Lee},
title = {Robust Tracking Control Design for a 6 Dof Parallel
Manipulator},
journal = {Journal of Robotic Systems},
volume = 17,
number = 10,
pages = {527-547},
year = 2000,
doi = {10.1002/1097-4563(200010)17:10$<$527::AID-ROB2>3.0.CO;2-A},
url =
{https://doi.org/10.1002/1097-4563(200010)17:10$<$527::AID-ROB2>3.0.CO;2-A},
keywords = {parallel robot},
}
@inproceedings{merlet02_still,
author = {Merlet, Jean-Pierre},
title = {Still a long way to go on the road for parallel mechanisms},
@ -562,14 +722,3 @@
series = {Solid Mechanics and Its Applications},
}
@phdthesis{li01_simul_fault_vibrat_isolat_point,
author = {Li, Xiaochun},
keywords = {parallel robot},
school = {University of Wyoming},
title = {Simultaneous, Fault-tolerant Vibration Isolation and
Pointing Control of Flexure Jointed Hexapods},
year = 2001,
}

208
nass-geometry.org
View File

@ -117,6 +117,34 @@ Optimal geometry?
- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/kinematic-study.org::*Estimation of the Joint required Stroke][Estimation of the Joint required Stroke]]
** Not used
*** Review table
#+name: tab:detail_kinematics_stewart_review
#+caption: Examples of Stewart platform developed. When not specifically indicated, sensors are included in the struts. All presented Stewart platforms are using flexible joints. The table is sorted by order of appearance in the literature
#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccc
#+attr_latex: :center t :booktabs t :font \scriptsize
| | *Geometry* | *Actuators* | *Sensors* | *Reference* |
|------------------------------------------+--------------+------------------------------+------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| | Cubic | Magnetostrictive | Force, Accelerometers | [[cite:&geng93_six_degree_of_freed_activ;&geng94_six_degree_of_freed_activ;&geng95_intel_contr_system_multip_degree]] |
| Figure ref:fig:detail_kinematics_jpl | Cubic | Voice Coil (0.5 mm) | Force | [[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax]] |
| | Cubic | Voice Coil (10 mm) | Force, LVDT, Geophones | [[cite:&thayer98_stewar;&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]] |
| Figure ref:fig:detail_kinematics_uw_gsp | Cubic | Voice Coil | Force | [[cite:&mcinroy99_dynam;&mcinroy99_precis_fault_toler_point_using_stewar_platf;&mcinroy00_desig_contr_flexur_joint_hexap;&li01_simul_vibrat_isolat_point_contr;&jafari03_orthog_gough_stewar_platf_microm]] |
| | Cubic | Piezoelectric ($25\,\mu m$) | Force | [[cite:&defendini00_techn]] |
| Figure ref:fig:detail_kinematics_ulb_pz | Cubic | APA ($50\,\mu m$) | Force | [[cite:&abu02_stiff_soft_stewar_platf_activ]] |
| Figure ref:fig:detail_kinematics_pph | Non-Cubic | Voice Coil | Accelerometers | [[cite:&chen03_payload_point_activ_vibrat_isolat]] |
| | Cubic | Voice Coil | Force | [[cite:&hanieh03_activ_stewar;&preumont07_six_axis_singl_stage_activ]] |
| Figure ref:fig:detail_kinematics_uqp | Cubic | Piezoelectric ($50\,\mu m$) | Geophone | [[cite:&agrawal04_algor_activ_vibrat_isolat_spacec]] |
| | Non-Cubic | Piezoelectric ($16\,\mu m$) | Eddy Current | [[cite:&furutani04_nanom_cuttin_machin_using_stewar]] |
| | Cubic | Piezoelectric ($120\,\mu m$) | (External) Capacitive | [[cite:&ting06_desig_stewar_nanos_platf;&ting13_compos_contr_desig_stewar_nanos_platf]] |
| | Non-Cubic | Piezoelectric ($160\,\mu m$) | (External) Capacitive | [[cite:&ting07_measur_calib_stewar_microm_system]] |
| Figure ref:fig:detail_kinematics_zhang11 | Non-cubic | Magnetostrictive | Accelerometer | [[cite:&zhang11_six_dof]] |
| | Non-Cubic | Piezoelectric | Strain Gauge | [[cite:&du14_piezo_actuat_high_precis_flexib]] |
| | Cubic | Voice Coil | Accelerometer | [[cite:&chi15_desig_exper_study_vcm_based;&tang18_decen_vibrat_contr_voice_coil;&jiao18_dynam_model_exper_analy_stewar]] |
| | Cubic | Piezoelectric | Force | [[cite:&wang16_inves_activ_vibrat_isolat_stewar]] |
| | Almost cubic | Voice Coil | Force, Accelerometer | [[cite:&beijen18_self_tunin_mimo_distur_feedf;&tjepkema12_activ_ph]] |
| Figure ref:fig:detail_kinematics_yang19 | Almost cubic | Piezoelectric | Force, Strain gauge | [[cite:&yang19_dynam_model_decoup_contr_flexib]] |
| Figure ref:fig:detail_kinematics_naves | Non-Cubic | 3-phase rotary motor | Rotary Encoder | [[cite:&naves20_desig;&naves20_t_flex]] |
*** Dynamic isotropy
[[cite:&afzali-far16_vibrat_dynam_isotr_hexap_analy_studies]]:
@ -336,6 +364,54 @@ ylim([1e-10, 2e-3])
#+end_src
** TODO [#B] Change review based on christophe's comments
SCHEDULED: <2025-04-04 Fri>
- [-] make sure that all papers are cited
- [X] geng93_six_degree_of_freed_activ
- [X] geng94_six_degree_of_freed_activ
- [X] geng95_intel_contr_system_multip_degree
- [X] spanos95_soft_activ_vibrat_isolat
- [X] rahman98_multiax
- [X] thayer98_stewar
- [X] thayer02_six_axis_vibrat_isolat_system
- [X] hauge04_sensor_contr_space_based_six
- [X] mcinroy99_dynam
- [ ] mcinroy99_precis_fault_toler_point_using_stewar_platf
- [X] mcinroy00_desig_contr_flexur_joint_hexap
- [X] li01_simul_vibrat_isolat_point_contr
- [X] jafari03_orthog_gough_stewar_platf_microm
- [X] defendini00_techn
- [X] abu02_stiff_soft_stewar_platf_activ
- [X] chen03_payload_point_activ_vibrat_isolat
- [X] hanieh03_activ_stewar
- [X] preumont07_six_axis_singl_stage_activ
- [X] agrawal04_algor_activ_vibrat_isolat_spacec
- [X] furutani04_nanom_cuttin_machin_using_stewar
- [ ] ting06_desig_stewar_nanos_platf
- [X] ting13_compos_contr_desig_stewar_nanos_platf
- [X] ting07_measur_calib_stewar_microm_system
- [X] zhang11_six_dof
- [X] du14_piezo_actuat_high_precis_flexib
- [X] chi15_desig_exper_study_vcm_based
- [X] tang18_decen_vibrat_contr_voice_coil
- [X] jiao18_dynam_model_exper_analy_stewar
- [X] wang16_inves_activ_vibrat_isolat_stewar
- [ ] beijen18_self_tunin_mimo_distur_feedf
- [ ] tjepkema12_activ_ph
- [X] yang19_dynam_model_decoup_contr_flexib
- [ ] naves20_desig
- [ ] naves20_t_flex
- [-] make sure that all figures are cited
- [X] Figure ref:fig:detail_kinematics_jpl
- [X] Figure ref:fig:detail_kinematics_uw_gsp
- [X] Figure ref:fig:detail_kinematics_ulb_pz
- [X] Figure ref:fig:detail_kinematics_pph
- [X] Figure ref:fig:detail_kinematics_uqp
- [X] Figure ref:fig:detail_kinematics_zhang11
- [X] Figure ref:fig:detail_kinematics_yang19
- [ ] Figure ref:fig:detail_kinematics_naves
** DONE [#B] Read everything again and check figures
CLOSED: [2025-04-01 Tue 18:17] SCHEDULED: <2025-04-01 Tue>
@ -961,54 +1037,48 @@ In this chapter, the nano-hexapod geometry is optimized through careful analysis
The chapter begins with a comprehensive review of existing Stewart platform designs in Section ref:sec:detail_kinematics_stewart_review, surveying various approaches to geometry, actuation, sensing, and joint design from the literature.
Section ref:sec:detail_kinematics_geometry develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process.
Section ref:sec:detail_kinematics_cubic examines the cubic configuration, a specialized architecture that has gathered significant attention, to evaluate its suitability for the nano-hexapod application.
Section ref:sec:detail_kinematics_cubic examines the cubic configuration, a specific architecture that has gathered significant attention, to evaluate its suitability for the nano-hexapod application.
Finally, Section ref:sec:detail_kinematics_nano_hexapod presents the optimized nano-hexapod geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS.
* Review of Stewart platforms
<<sec:detail_kinematics_stewart_review>>
As was explained in the conceptual phase, Stewart platforms have the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors.
# Section ref:sec:nhexa_stewart_platform
The exact geometry (i.e., position of joints and orientation of the struts) can be chosen freely depending on the application, which results in many different designs found in the literature.
The focus is here made on Stewart platforms for nano-positioning and vibration control.
Long stroke Stewart platforms are not considered here as their design imposes other challenges.
Some Stewart platforms found in the literature are listed in Table ref:tab:detail_kinematics_stewart_review.
The first parallel platform similar to the Stewart platform was built in 1954 by Gough [[cite:&gough62_univer_tyre_test_machin]], for a tyre test machine (shown in Figure ref:fig:detail_geometry_gough_paper).
Subsequently, Stewart proposed a similar design for a flight simulator (shown in Figure ref:fig:detail_geometry_stewart_flight_simulator) in a 1965 publication [[cite:&stewart65_platf_with_six_degrees_freed]].
Since then, the Stewart platform (sometimes referred to as the Stewart-Gough platform) has been utilized across diverse applications [[cite:&dasgupta00_stewar_platf_manip]], including large telescopes [[cite:&kazezkhan14_dynam_model_stewar_platf_nansh_radio_teles;&yun19_devel_isotr_stewar_platf_teles_secon_mirror]], machine tools [[cite:&russo24_review_paral_kinem_machin_tools]], and Synchrotron instrumentation [[cite:&marion04_hexap_esrf;&villar18_nanop_esrf_id16a_nano_imagin_beaml]].
#+name: tab:detail_kinematics_stewart_review
#+caption: Examples of Stewart platform developed. When not specifically indicated, sensors are included in the struts. All presented Stewart platforms are using flexible joints. The table is sorted by order of appearance in the literature
#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcccc
#+attr_latex: :center t :booktabs t :font \scriptsize
| | *Geometry* | *Actuators* | *Sensors* | *Reference* |
|------------------------------------------+--------------+------------------------------+------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| | Cubic | Magnetostrictive | Force, Accelerometers | [[cite:&geng93_six_degree_of_freed_activ;&geng94_six_degree_of_freed_activ;&geng95_intel_contr_system_multip_degree]] |
| Figure ref:fig:detail_kinematics_jpl | Cubic | Voice Coil (0.5 mm) | Force | [[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax]] |
| | Cubic | Voice Coil (10 mm) | Force, LVDT, Geophones | [[cite:&thayer98_stewar;&thayer02_six_axis_vibrat_isolat_system;&hauge04_sensor_contr_space_based_six]] |
| Figure ref:fig:detail_kinematics_uw_gsp | Cubic | Voice Coil | Force | [[cite:&mcinroy99_dynam;&mcinroy99_precis_fault_toler_point_using_stewar_platf;&mcinroy00_desig_contr_flexur_joint_hexap;&li01_simul_vibrat_isolat_point_contr;&jafari03_orthog_gough_stewar_platf_microm]] |
| | Cubic | Piezoelectric ($25\,\mu m$) | Force | [[cite:&defendini00_techn]] |
| Figure ref:fig:detail_kinematics_ulb_pz | Cubic | APA ($50\,\mu m$) | Force | [[cite:&abu02_stiff_soft_stewar_platf_activ]] |
| Figure ref:fig:detail_kinematics_pph | Non-Cubic | Voice Coil | Accelerometers | [[cite:&chen03_payload_point_activ_vibrat_isolat]] |
| | Cubic | Voice Coil | Force | [[cite:&hanieh03_activ_stewar;&preumont07_six_axis_singl_stage_activ]] |
| Figure ref:fig:detail_kinematics_uqp | Cubic | Piezoelectric ($50\,\mu m$) | Geophone | [[cite:&agrawal04_algor_activ_vibrat_isolat_spacec]] |
| | Non-Cubic | Piezoelectric ($16\,\mu m$) | Eddy Current | [[cite:&furutani04_nanom_cuttin_machin_using_stewar]] |
| | Cubic | Piezoelectric ($120\,\mu m$) | (External) Capacitive | [[cite:&ting06_desig_stewar_nanos_platf;&ting13_compos_contr_desig_stewar_nanos_platf]] |
| | Non-Cubic | Piezoelectric ($160\,\mu m$) | (External) Capacitive | [[cite:&ting07_measur_calib_stewar_microm_system]] |
| Figure ref:fig:detail_kinematics_zhang11 | Non-cubic | Magnetostrictive | Accelerometer | [[cite:&zhang11_six_dof]] |
| | Non-Cubic | Piezoelectric | Strain Gauge | [[cite:&du14_piezo_actuat_high_precis_flexib]] |
| | Cubic | Voice Coil | Accelerometer | [[cite:&chi15_desig_exper_study_vcm_based;&tang18_decen_vibrat_contr_voice_coil;&jiao18_dynam_model_exper_analy_stewar]] |
| | Cubic | Piezoelectric | Force | [[cite:&wang16_inves_activ_vibrat_isolat_stewar]] |
| | Almost cubic | Voice Coil | Force, Accelerometer | [[cite:&beijen18_self_tunin_mimo_distur_feedf;&tjepkema12_activ_ph]] |
| Figure ref:fig:detail_kinematics_yang19 | Almost cubic | Piezoelectric | Force, Strain gauge | [[cite:&yang19_dynam_model_decoup_contr_flexib]] |
| Figure ref:fig:detail_kinematics_naves | Non-Cubic | 3-phase rotary motor | Rotary Encoder | [[cite:&naves20_desig;&naves20_t_flex]] |
#+name: fig:detail_geometry_stewart_origins
#+caption: Two of the earliest developments of Stewart platforms
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_geometry_gough_paper}Tyre test machine proposed by Gough \cite{gough62_univer_tyre_test_machin}}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :height 5.5cm
[[file:figs/detail_geometry_gough_paper.jpg]]
#+end_subfigure
#+attr_latex: :caption \subcaption{\label{fig:detail_geometry_stewart_flight_simulator}Flight simulator proposed by Stewart \cite{stewart65_platf_with_six_degrees_freed}}
#+attr_latex: :options {0.48\textwidth}
#+begin_subfigure
#+attr_latex: :height 5.5cm
[[file:figs/detail_geometry_stewart_flight_simulator.jpg]]
#+end_subfigure
#+end_figure
# TODO - Section ref:sec:nhexa_stewart_platform
As explained in the conceptual phase, Stewart platforms comprise the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors.
All presented Stewart platforms utilize flexible joints, as this is a prerequisite for nano-positioning capabilities.
Flexible joints can have various implementations, which will be discussed when designing the nano-hexapod flexible joints.
# TODO - ref:sec:detail_fem_joint
The specific geometry (i.e., position of joints and orientation of the struts) can be selected based on the application requirements, resulting in numerous designs throughout the literature.
This discussion focuses primarily on Stewart platforms designed for nano-positioning and vibration control, which necessitates the use of flexible joints.
The implementation of these flexible joints, will be discussed when designing the nano-hexapod flexible joints.
Long stroke Stewart platforms are not addressed here as their design presents different challenges, such as singularity-free workspace and complex kinematics [[cite:&merlet06_paral_robot]].
In terms of actuation, most Stewart platforms employ either voice coil actuators (such as the ones shown in Figures ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_pph) or piezoelectric actuators (such as the ones shown in Figures ref:fig:detail_kinematics_ulb_pz, ref:fig:detail_kinematics_uqp and ref:fig:detail_kinematics_yang19).
Various sensors are integrated in the struts or on the plates depending on the application requirements.
These include force sensors, inertial sensors, or relative displacement sensors.
The actuator and sensor selection for the nano-hexapod will also be described in the next section.
# TODO - Add reference to the section
In terms of actuation, mainly two types are used: voice coil actuators and piezoelectric actuators.
Voice coil actuators, providing stroke ranges from $0.5\,mm$ to $10\,mm$, are commonly implemented in cubic architectures (as illustrated in Figures ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_pph) and are mainly used for vibration isolation [[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax;&thayer98_stewar;&mcinroy99_dynam;&preumont07_six_axis_singl_stage_activ]].
For applications requiring short stroke (typically smaller than $500\,\mu m$), piezoelectric actuators present an interesting alternative, as shown in [[cite:&agrawal04_algor_activ_vibrat_isolat_spacec;&furutani04_nanom_cuttin_machin_using_stewar;&yang19_dynam_model_decoup_contr_flexib]].
Examples of piezoelectric-actuated Stewart platforms are presented in Figures ref:fig:detail_kinematics_ulb_pz, ref:fig:detail_kinematics_uqp and ref:fig:detail_kinematics_yang19.
Although less frequently encountered, magnetostrictive actuators have been successfully implemented in [[cite:&zhang11_six_dof]] (Figure ref:fig:detail_kinematics_zhang11).
#+name: fig:detail_kinematics_stewart_examples_cubic
#+caption: Some examples of developped Stewart platform with Cubic geometry
@ -1042,11 +1112,23 @@ The actuator and sensor selection for the nano-hexapod will also be described in
#+end_subfigure
#+end_figure
There are two main categories of Stewart platform geometry.
The first is cubic architecture (some exampled are presented in Figure ref:fig:detail_kinematics_stewart_examples_cubic), where struts are positioned along six sides of a cube (and are therefore orthogonal to each other).
Such specific architecture has some special properties that will be studied in Section ref:sec:detail_kinematics_cubic.
The second is non-cubic architecture (Figure ref:fig:detail_kinematics_stewart_examples_non_cubic), where the orientation of the struts and position of the joints can be optimized based on defined performance criteria.
The effect of strut orientation and position of the joints on the Stewart platform properties is discussed in Section ref:sec:detail_kinematics_geometry.
The sensors integrated in these platforms are selected based on specific control requirements, as different sensors offer distinct advantages and limitations [[cite:&hauge04_sensor_contr_space_based_six]].
Force sensors are typically integrated within the struts in a collocated arrangement with actuators to enhance control robustness.
Stewart platforms incorporating force sensors are frequently utilized for vibration isolation [[cite:&spanos95_soft_activ_vibrat_isolat;&rahman98_multiax]] and active damping applications [[cite:&geng95_intel_contr_system_multip_degree;&abu02_stiff_soft_stewar_platf_activ]], as exemplified in Figure ref:fig:detail_kinematics_ulb_pz.
Inertial sensors (accelerometers and geophones) are commonly employed in vibration isolation applications [[cite:&chen03_payload_point_activ_vibrat_isolat;&chi15_desig_exper_study_vcm_based]].
These sensors are predominantly aligned with the struts [[cite:&hauge04_sensor_contr_space_based_six;&li01_simul_fault_vibrat_isolat_point;&thayer02_six_axis_vibrat_isolat_system;&zhang11_six_dof;&jiao18_dynam_model_exper_analy_stewar;&tang18_decen_vibrat_contr_voice_coil]], although they may also be fixed to the top platform [[cite:&wang16_inves_activ_vibrat_isolat_stewar]].
For high-precision positioning applications, various displacement sensors are implemented, including LVDTs [[cite:&thayer02_six_axis_vibrat_isolat_system;&kim00_robus_track_contr_desig_dof_paral_manip;&li01_simul_fault_vibrat_isolat_point;&thayer98_stewar]], capacitive sensors [[cite:&ting07_measur_calib_stewar_microm_system;&ting13_compos_contr_desig_stewar_nanos_platf]], eddy current sensors [[cite:&chen03_payload_point_activ_vibrat_isolat;&furutani04_nanom_cuttin_machin_using_stewar]], and strain gauges [[cite:&du14_piezo_actuat_high_precis_flexib]].
Notably, some designs incorporate external sensing methodologies rather than integrating sensors within the struts [[cite:&li01_simul_fault_vibrat_isolat_point;&chen03_payload_point_activ_vibrat_isolat;&ting13_compos_contr_desig_stewar_nanos_platf]].
A recent design [[cite:&naves20_desig]], although not strictly speaking a Stewart platform, has demonstrated the use of 3-phase rotary motors with rotary encoders for achieving long-stroke and highly repeatable positioning, as illustrated in Figure ref:fig:detail_kinematics_naves.
Two primary categories of Stewart platform geometry can be identified.
The first is cubic architecture (examples presented in Figure ref:fig:detail_kinematics_stewart_examples_cubic), wherein struts are positioned along six sides of a cube (and therefore oriented orthogonally to each other).
This architecture represents the most prevalent configuration for vibration isolation applications in the literature.
Its distinctive properties will be examined in Section ref:sec:detail_kinematics_cubic.
The second category comprises non-cubic architectures (Figure ref:fig:detail_kinematics_stewart_examples_non_cubic), where strut orientation and joint positioning can be optimized according to defined performance criteria.
The influence of strut orientation and joint positioning on Stewart platform properties is analyzed in Section ref:sec:detail_kinematics_geometry.
#+name: fig:detail_kinematics_stewart_examples_non_cubic
#+caption: Some examples of developped Stewart platform with non-cubic geometry
@ -1091,7 +1173,7 @@ The effect of strut orientation and position of the joints on the Stewart platfo
As was demonstrated during the conceptual phase, the geometry of the Stewart platform impacts the stiffness and compliance characteristics, the mobility (or workspace), the force authority, and the dynamics of the manipulator.
It is therefore essential to understand how the geometry impacts these properties, and to develop methodologies for optimizing the geometry for specific applications.
An important analytical tool for this study is the Jacobian matrix, which depends on $\bm{b}_i$ (joints' position with respect to the top platform) and $\hat{\bm{s}}_i$ (struts' orientation).
A useful analytical tool for this study is the Jacobian matrix, which depends on $\bm{b}_i$ (joints' position with respect to the top platform) and $\hat{\bm{s}}_i$ (struts' orientation).
The choice of $\{A\}$ and $\{B\}$ frames, independently of the physical Stewart platform geometry, impacts the obtained kinematics and stiffness matrix, as these are defined for forces and motion evaluated at the chosen frame.
** Matlab Init :noexport:ignore:
@ -1569,7 +1651,7 @@ This is discussed in Section ref:ssec:detail_kinematics_cubic_static.
<<ssec:detail_kinematics_geometry_dynamics>>
The dynamical equations (both in the Cartesian frame and in the frame of the struts) for the Stewart platform were derived during the conceptual phase with simplifying assumptions (massless struts and perfect joints).
The dynamics depend both on the geometry (Jacobian matrix) and on the payload being placed on top of the platform.
The dynamics depends both on the geometry (Jacobian matrix) and on the payload being placed on top of the platform.
# Section ref:ssec:nhexa_stewart_platform_dynamics (page pageref:ssec:nhexa_stewart_platform_dynamics).
Under very specific conditions, the equations of motion in the Cartesian frame, given by equation eqref:eq:detail_kinematics_transfer_function_cart, can be decoupled.
@ -1646,7 +1728,7 @@ These trade-offs provide important guidelines when choosing the Stewart platform
** Introduction :ignore:
The Cubic configuration for the Stewart platform was first proposed in [[cite:&geng94_six_degree_of_freed_activ]].
The Cubic configuration for the Stewart platform was first proposed by Dr. Gough in a comment to the original paper by Dr. Stewart [[cite:&stewart65_platf_with_six_degrees_freed]].
This configuration is characterized by active struts arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure ref:fig:detail_kinematics_cubic_architecture_example.
Typically, the struts have similar length to the cube's edges, as illustrated in Figure ref:fig:detail_kinematics_cubic_architecture_example.
@ -1903,7 +1985,7 @@ The unit vectors corresponding to the edges of the cube are described by equatio
#+end_src
#+name: fig:detail_kinematics_cubic_schematic_cases
#+caption: Cubic architecture. Struts are represented un blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic})
#+caption: Cubic architecture. Struts are represented in blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic})
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:detail_kinematics_cubic_schematic_full}Full cube}
@ -2042,7 +2124,7 @@ This stiffness matrix structure is characteristic of Stewart platforms exhibitin
Therefore, the stiffness characteristics of the cubic architecture are only distinctive when considering a reference frame located at the cube's center.
This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform.
It should be noted that for the stiffness matrix to be diagonal, the cube's center need not coincide with the geometric center of the Stewart platform.
It should be noted that for the stiffness matrix to be diagonal, the cube's center doesn't need to coincide with the geometric center of the Stewart platform.
This observation leads to the interesting alternative architectures presented in Section ref:ssec:detail_kinematics_cubic_design.
**** Uniform Mobility
@ -2052,10 +2134,10 @@ Considering limited actuator stroke (elongation of each strut), the maximum achi
The resulting mobility in X, Y, and Z directions for the cubic architecture is illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_translations.
The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke.
This findings suggest that the mobility pattern is more nuanced than sometimes described in the literature [[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions.
These findings suggest that the mobility pattern is more subtle than sometimes described in the literature [[cite:&mcinroy00_desig_contr_flexur_joint_hexap]], exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions.
This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure ref:fig:detail_kinematics_mobility_trans.
The rotational mobility, illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_rotations, exhibit greater achievable angular stroke in the $R_x$ and $R_y$ directions compared to the $R_z$ direction.
The rotational mobility, illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_rotations, exhibits greater achievable angular stroke in the $R_x$ and $R_y$ directions compared to the $R_z$ direction.
Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities.
#+begin_src matlab
@ -2430,7 +2512,7 @@ exportFig('figs/detail_kinematics_cubic_cart_coupling_cok.pdf', 'width', 'half',
An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components [[cite:&li01_simul_fault_vibrat_isolat_point]].
This can be achieved by positioning the payload below the top platform, such that the center of mass of the moving body coincides with the cube's center (Figure ref:fig:detail_kinematics_cubic_centered_payload).
This approach was physically implemented in several studies [[cite:&mcinroy99_dynam;&jafari03_orthog_gough_stewar_platf_microm]], as shown in Figure ref:fig:detail_kinematics_uw_gsp.
The resulting dynamics are indeed well-decoupled (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com_cok), benefiting from simultaneously diagonal stiffness and mass matrices.
The resulting dynamics are indeed well-decoupled (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com_cok), taking advantage from diagonal stiffness and mass matrices.
The primary limitation of this approach is that, for many applications including the nano-hexapod, the payload must be positioned above the top platform.
If a design similar to Figure ref:fig:detail_kinematics_cubic_centered_payload were employed for the nano-hexapod, the X-ray beam would intersect with the struts during spindle rotation.
@ -2741,7 +2823,7 @@ exportFig('figs/detail_kinematics_cubic_decentralized_dL.pdf', 'width', 'half',
Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms.
The results are presented in Figure ref:fig:detail_kinematics_decentralized_fn.
The system demonstrates good decoupling at high frequency in both cases, with no evidence suggesting any advantage for the cubic architecture.
The system demonstrates good decoupling at high frequency in both cases, with no clear advantage for the cubic architecture.
#+begin_src matlab :exports none :results none
%% Decentralized plant - Actuator force to strut force sensor - Cubic Architecture
@ -2822,7 +2904,7 @@ exportFig('figs/detail_kinematics_cubic_decentralized_fn.pdf', 'width', 'half',
**** Conclusion
The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature.
Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement may be more nuanced than commonly described for decentralized control.
Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement for decentralized control is less obvious than often reported in the literature.
** Cubic architecture with Cube's center above the top platform
<<ssec:detail_kinematics_cubic_design>>
@ -3167,18 +3249,12 @@ For the proposed configuration, the top joints $\bm{b}_i$ (resp. the bottom join
Since the rotational stiffness for the cubic architecture scales with the square of the cube's height eqref:eq:detail_kinematics_cubic_stiffness, the cube's size can be determined based on rotational stiffness requirements.
Subsequently, using Equation eqref:eq:detail_kinematics_cube_joints, the dimensions of the top and bottom platforms can be calculated.
**** Conclusion
The configurations proposed in this analysis represent derivations from the classical cubic architecture, wherein the cube's center is typically located at the Stewart platform's center.
Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform.
This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame.
** Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:
The analysis of the cubic architecture for Stewart platforms has yielded several important findings.
The analysis of the cubic architecture for Stewart platforms yielded several important findings.
While the cubic configuration provides uniform stiffness in the XYZ directions, it stiffness property becomes particularly advantageous when forces and torques are applied at the cube's center.
Under these conditions, the stiffness matrix becomes diagonal, resulting in a decoupled Cartesian plant at low frequencies.
@ -3193,7 +3269,8 @@ Fully decoupled dynamics in the Cartesian frame can be achieved when the center
However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications.
To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform.
These configurations maintain the fundamental advantages of the cubic architecture while enabling practical payload placement.
Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform.
This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame.
* Nano Hexapod
:PROPERTIES:
@ -3527,7 +3604,7 @@ exportFig('figs/detail_kinematics_nano_hexapod_mobility.pdf', 'width', 'wide', '
#+end_src
#+name: fig:detail_kinematics_nano_hexapod_mobility
#+caption: Wanted translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume).
#+caption: Specified translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume).
#+RESULTS:
[[file:figs/detail_kinematics_nano_hexapod_mobility.png]]
@ -3537,8 +3614,7 @@ exportFig('figs/detail_kinematics_nano_hexapod_mobility.pdf', 'width', 'wide', '
With the nano-hexapod geometry and mobility requirements established, the flexible joint angular stroke necessary to avoid limiting the achievable workspace can be determined.
This analysis focuses solely on bending stroke, as the torsional stroke of the flexible joints is expected to be minimal given the absence of vertical rotation requirements.
The required angular stroke for both fixed and mobile joints is calculated to be $1\,\text{mrad}$.
The required angular stroke for both fixed and mobile joints is estimated to be equal to $1\,\text{mrad}$.
This specification will guide the design of the flexible joints.
# TODO - Add link to section
@ -3596,7 +3672,7 @@ sprintf('Mobile joint stroke should be %.1f mrad', 1e3*max(max_angles_M))
This chapter has explored the optimization of the nano-hexapod geometry for the Nano Active Stabilization System (NASS).
First, a review of existing Stewart platforms revealed two main geometric categories: cubic architectures, characterized by mutually orthogonal struts arranged along the edges of a cube, and non-cubic architectures with varied strut orientations.
While cubic architectures are prevalent in the literature and attributed with beneficial properties such as simplified kinematics, uniform stiffness, and reduced cross-coupling, the performed analysis revealed that some of these advantages may be more nuanced or context-dependent than commonly described.
While cubic architectures are prevalent in the literature and attributed with beneficial properties such as simplified kinematics, uniform stiffness, and reduced cross-coupling, the performed analysis revealed that some of these advantages should be more nuanced or context-dependent than commonly described.
The analytical relationships between Stewart platform geometry and its mechanical properties were established, enabling a better understanding of the trade-offs between competing requirements such as mobility and stiffness along different axes.
These insights were useful during the nano-hexapod geometry optimization.

BIN
nass-geometry.pdf
View File

Binary file not shown.

174
nass-geometry.tex
View File

@ -1,4 +1,4 @@
% Created 2025-04-03 Thu 10:35
% Created 2025-04-04 Fri 17:48
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -31,53 +31,44 @@ In this chapter, the nano-hexapod geometry is optimized through careful analysis
The chapter begins with a comprehensive review of existing Stewart platform designs in Section \ref{sec:detail_kinematics_stewart_review}, surveying various approaches to geometry, actuation, sensing, and joint design from the literature.
Section \ref{sec:detail_kinematics_geometry} develops the analytical framework that connects geometric parameters to performance characteristics, establishing quantitative relationships that guide the optimization process.
Section \ref{sec:detail_kinematics_cubic} examines the cubic configuration, a specialized architecture that has gathered significant attention, to evaluate its suitability for the nano-hexapod application.
Section \ref{sec:detail_kinematics_cubic} examines the cubic configuration, a specific architecture that has gathered significant attention, to evaluate its suitability for the nano-hexapod application.
Finally, Section \ref{sec:detail_kinematics_nano_hexapod} presents the optimized nano-hexapod geometry derived from these analyses and demonstrates how it addresses the specific requirements of the NASS.
\chapter{Review of Stewart platforms}
\label{sec:detail_kinematics_stewart_review}
As was explained in the conceptual phase, Stewart platforms have the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors.
The exact geometry (i.e., position of joints and orientation of the struts) can be chosen freely depending on the application, which results in many different designs found in the literature.
The focus is here made on Stewart platforms for nano-positioning and vibration control.
Long stroke Stewart platforms are not considered here as their design imposes other challenges.
Some Stewart platforms found in the literature are listed in Table \ref{tab:detail_kinematics_stewart_review}.
\begin{table}[htbp]
\caption{\label{tab:detail_kinematics_stewart_review}Examples of Stewart platform developed. When not specifically indicated, sensors are included in the struts. All presented Stewart platforms are using flexible joints. The table is sorted by order of appearance in the literature}
\centering
\scriptsize
\begin{tabularx}{0.9\linewidth}{Xcccc}
\toprule
& \textbf{Geometry} & \textbf{Actuators} & \textbf{Sensors} & \textbf{Reference}\\
\midrule
& Cubic & Magnetostrictive & Force, Accelerometers & \cite{geng93_six_degree_of_freed_activ,geng94_six_degree_of_freed_activ,geng95_intel_contr_system_multip_degree}\\
Figure \ref{fig:detail_kinematics_jpl} & Cubic & Voice Coil (0.5 mm) & Force & \cite{spanos95_soft_activ_vibrat_isolat,rahman98_multiax}\\
& Cubic & Voice Coil (10 mm) & Force, LVDT, Geophones & \cite{thayer98_stewar,thayer02_six_axis_vibrat_isolat_system,hauge04_sensor_contr_space_based_six}\\
Figure \ref{fig:detail_kinematics_uw_gsp} & Cubic & Voice Coil & Force & \cite{mcinroy99_dynam,mcinroy99_precis_fault_toler_point_using_stewar_platf,mcinroy00_desig_contr_flexur_joint_hexap,li01_simul_vibrat_isolat_point_contr,jafari03_orthog_gough_stewar_platf_microm}\\
& Cubic & Piezoelectric (\(25\,\mu m\)) & Force & \cite{defendini00_techn}\\
Figure \ref{fig:detail_kinematics_ulb_pz} & Cubic & APA (\(50\,\mu m\)) & Force & \cite{abu02_stiff_soft_stewar_platf_activ}\\
Figure \ref{fig:detail_kinematics_pph} & Non-Cubic & Voice Coil & Accelerometers & \cite{chen03_payload_point_activ_vibrat_isolat}\\
& Cubic & Voice Coil & Force & \cite{hanieh03_activ_stewar,preumont07_six_axis_singl_stage_activ}\\
Figure \ref{fig:detail_kinematics_uqp} & Cubic & Piezoelectric (\(50\,\mu m\)) & Geophone & \cite{agrawal04_algor_activ_vibrat_isolat_spacec}\\
& Non-Cubic & Piezoelectric (\(16\,\mu m\)) & Eddy Current & \cite{furutani04_nanom_cuttin_machin_using_stewar}\\
& Cubic & Piezoelectric (\(120\,\mu m\)) & (External) Capacitive & \cite{ting06_desig_stewar_nanos_platf,ting13_compos_contr_desig_stewar_nanos_platf}\\
& Non-Cubic & Piezoelectric (\(160\,\mu m\)) & (External) Capacitive & \cite{ting07_measur_calib_stewar_microm_system}\\
Figure \ref{fig:detail_kinematics_zhang11} & Non-cubic & Magnetostrictive & Accelerometer & \cite{zhang11_six_dof}\\
& Non-Cubic & Piezoelectric & Strain Gauge & \cite{du14_piezo_actuat_high_precis_flexib}\\
& Cubic & Voice Coil & Accelerometer & \cite{chi15_desig_exper_study_vcm_based,tang18_decen_vibrat_contr_voice_coil,jiao18_dynam_model_exper_analy_stewar}\\
& Cubic & Piezoelectric & Force & \cite{wang16_inves_activ_vibrat_isolat_stewar}\\
& Almost cubic & Voice Coil & Force, Accelerometer & \cite{beijen18_self_tunin_mimo_distur_feedf,tjepkema12_activ_ph}\\
Figure \ref{fig:detail_kinematics_yang19} & Almost cubic & Piezoelectric & Force, Strain gauge & \cite{yang19_dynam_model_decoup_contr_flexib}\\
Figure \ref{fig:detail_kinematics_naves} & Non-Cubic & 3-phase rotary motor & Rotary Encoder & \cite{naves20_desig,naves20_t_flex}\\
\bottomrule
\end{tabularx}
\end{table}
The first parallel platform similar to the Stewart platform was built in 1954 by Gough \cite{gough62_univer_tyre_test_machin}, for a tyre test machine (shown in Figure \ref{fig:detail_geometry_gough_paper}).
Subsequently, Stewart proposed a similar design in a 1965 publication \cite{stewart65_platf_with_six_degrees_freed}, for a flight simulator (shown in Figure \ref{fig:detail_geometry_stewart_flight_simulator}).
Since then, the Stewart platform (sometimes referred to as the Stewart-Gough platform) has been utilized across diverse applications \cite{dasgupta00_stewar_platf_manip}, including large telescopes \cite{kazezkhan14_dynam_model_stewar_platf_nansh_radio_teles,yun19_devel_isotr_stewar_platf_teles_secon_mirror}, machine tools \cite{russo24_review_paral_kinem_machin_tools}, and Synchrotron instrumentation \cite{marion04_hexap_esrf,villar18_nanop_esrf_id16a_nano_imagin_beaml}.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,height=5.5cm]{figs/detail_geometry_gough_paper.jpg}
\end{center}
\subcaption{\label{fig:detail_geometry_gough_paper}Tyre test machine proposed by Gough \cite{gough62_univer_tyre_test_machin}}
\end{subfigure}
\begin{subfigure}{0.48\textwidth}
\begin{center}
\includegraphics[scale=1,height=5.5cm]{figs/detail_geometry_stewart_flight_simulator.jpg}
\end{center}
\subcaption{\label{fig:detail_geometry_stewart_flight_simulator}Flight simulator proposed by Stewart \cite{stewart65_platf_with_six_degrees_freed}}
\end{subfigure}
\caption{\label{fig:detail_geometry_stewart_origins}Two of the earliest developments of Stewart platforms}
\end{figure}
As explained in the conceptual phase, Stewart platforms comprise the following key elements: two plates connected by six struts, with each strut composed of a joint at each end, an actuator, and one or several sensors.
The specific geometry (i.e., position of joints and orientation of the struts) can be selected based on the application requirements, resulting in numerous designs throughout the literature.
This discussion focuses primarily on Stewart platforms designed for nano-positioning and vibration control, which necessitates the use of flexible joints.
The implementation of these flexible joints, will be discussed when designing the nano-hexapod flexible joints.
Long stroke Stewart platforms are not addressed here as their design presents different challenges, such as singularity-free workspace and complex kinematics \cite{merlet06_paral_robot}.
In terms of actuation, mainly two types are used: voice coil actuators and piezoelectric actuators.
Voice coil actuators, providing stroke ranges from \(0.5\,mm\) to \(10\,mm\), are commonly implemented in cubic architectures (as illustrated in Figures \ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_pph}) and are mainly used for vibration isolation \cite{spanos95_soft_activ_vibrat_isolat,rahman98_multiax,thayer98_stewar,mcinroy99_dynam,preumont07_six_axis_singl_stage_activ}.
For applications requiring smaller stroke (typically smaller than \(500\,\mu m\)), piezoelectric actuators present an interesting alternative, as shown in \cite{agrawal04_algor_activ_vibrat_isolat_spacec,furutani04_nanom_cuttin_machin_using_stewar,yang19_dynam_model_decoup_contr_flexib}.
Examples of piezoelectric-actuated Stewart platforms are presented in Figures \ref{fig:detail_kinematics_ulb_pz}, \ref{fig:detail_kinematics_uqp} and \ref{fig:detail_kinematics_yang19}.
Although less frequently encountered, magnetostrictive actuators have been successfully implemented in \cite{zhang11_six_dof} (Figure \ref{fig:detail_kinematics_zhang11}).
All presented Stewart platforms utilize flexible joints, as this is a prerequisite for nano-positioning capabilities.
Flexible joints can have various implementations, which will be discussed when designing the nano-hexapod flexible joints.
In terms of actuation, most Stewart platforms employ either voice coil actuators (such as the ones shown in Figures \ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_pph}) or piezoelectric actuators (such as the ones shown in Figures \ref{fig:detail_kinematics_ulb_pz}, \ref{fig:detail_kinematics_uqp} and \ref{fig:detail_kinematics_yang19}).
Various sensors are integrated in the struts or on the plates depending on the application requirements.
These include force sensors, inertial sensors, or relative displacement sensors.
The actuator and sensor selection for the nano-hexapod will also be described in the next section.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
\begin{center}
@ -108,11 +99,23 @@ The actuator and sensor selection for the nano-hexapod will also be described in
\caption{\label{fig:detail_kinematics_stewart_examples_cubic}Some examples of developped Stewart platform with Cubic geometry}
\end{figure}
There are two main categories of Stewart platform geometry.
The first is cubic architecture (some exampled are presented in Figure \ref{fig:detail_kinematics_stewart_examples_cubic}), where struts are positioned along six sides of a cube (and are therefore orthogonal to each other).
Such specific architecture has some special properties that will be studied in Section \ref{sec:detail_kinematics_cubic}.
The second is non-cubic architecture (Figure \ref{fig:detail_kinematics_stewart_examples_non_cubic}), where the orientation of the struts and position of the joints can be optimized based on defined performance criteria.
The effect of strut orientation and position of the joints on the Stewart platform properties is discussed in Section \ref{sec:detail_kinematics_geometry}.
The sensors integrated in these platforms are selected based on specific control requirements, as different sensors offer distinct advantages and limitations \cite{hauge04_sensor_contr_space_based_six}.
Force sensors are typically integrated within the struts in a collocated arrangement with actuators to enhance control robustness.
Stewart platforms incorporating force sensors are frequently utilized for vibration isolation \cite{spanos95_soft_activ_vibrat_isolat,rahman98_multiax} and active damping applications \cite{geng95_intel_contr_system_multip_degree,abu02_stiff_soft_stewar_platf_activ}, as exemplified in Figure \ref{fig:detail_kinematics_ulb_pz}.
Inertial sensors (accelerometers and geophones) are commonly employed in vibration isolation applications \cite{chen03_payload_point_activ_vibrat_isolat,chi15_desig_exper_study_vcm_based}.
These sensors are predominantly aligned with the struts \cite{hauge04_sensor_contr_space_based_six,li01_simul_fault_vibrat_isolat_point,thayer02_six_axis_vibrat_isolat_system,zhang11_six_dof,jiao18_dynam_model_exper_analy_stewar,tang18_decen_vibrat_contr_voice_coil}, although they may also be fixed to the top platform \cite{wang16_inves_activ_vibrat_isolat_stewar}.
For high-precision positioning applications, various displacement sensors are implemented, including LVDTs \cite{thayer02_six_axis_vibrat_isolat_system,kim00_robus_track_contr_desig_dof_paral_manip,li01_simul_fault_vibrat_isolat_point,thayer98_stewar}, capacitive sensors \cite{ting07_measur_calib_stewar_microm_system,ting13_compos_contr_desig_stewar_nanos_platf}, eddy current sensors \cite{chen03_payload_point_activ_vibrat_isolat,furutani04_nanom_cuttin_machin_using_stewar}, and strain gauges \cite{du14_piezo_actuat_high_precis_flexib}.
Notably, some designs incorporate external sensing methodologies rather than integrating sensors within the struts \cite{li01_simul_fault_vibrat_isolat_point,chen03_payload_point_activ_vibrat_isolat,ting13_compos_contr_desig_stewar_nanos_platf}.
A recent design \cite{naves20_desig}, although not strictly speaking a Stewart platform, has demonstrated the use of 3-phase rotary motors with rotary encoders for achieving long-stroke and highly repeatable positioning, as illustrated in Figure \ref{fig:detail_kinematics_naves}.
Two primary categories of Stewart platform geometry can be identified.
The first is cubic architecture (examples presented in Figure \ref{fig:detail_kinematics_stewart_examples_cubic}), wherein struts are positioned along six sides of a cube (and therefore oriented orthogonally to each other).
This architecture represents the most prevalent configuration for vibration isolation applications in the literature.
Its distinctive properties will be examined in Section \ref{sec:detail_kinematics_cubic}.
The second category comprises non-cubic architectures (Figure \ref{fig:detail_kinematics_stewart_examples_non_cubic}), where strut orientation and joint positioning can be optimized according to defined performance criteria.
The influence of strut orientation and joint positioning on Stewart platform properties is analyzed in Section \ref{sec:detail_kinematics_geometry}.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@ -148,7 +151,7 @@ The effect of strut orientation and position of the joints on the Stewart platfo
As was demonstrated during the conceptual phase, the geometry of the Stewart platform impacts the stiffness and compliance characteristics, the mobility (or workspace), the force authority, and the dynamics of the manipulator.
It is therefore essential to understand how the geometry impacts these properties, and to develop methodologies for optimizing the geometry for specific applications.
An important analytical tool for this study is the Jacobian matrix, which depends on \(\bm{b}_i\) (joints' position with respect to the top platform) and \(\hat{\bm{s}}_i\) (struts' orientation).
A useful analytical tool for this study is the Jacobian matrix, which depends on \(\bm{b}_i\) (joints' position with respect to the top platform) and \(\hat{\bm{s}}_i\) (struts' orientation).
The choice of \(\{A\}\) and \(\{B\}\) frames, independently of the physical Stewart platform geometry, impacts the obtained kinematics and stiffness matrix, as these are defined for forces and motion evaluated at the chosen frame.
\section{Platform Mobility / Workspace}
\label{ssec:detail_kinematics_geometry_mobility}
@ -174,7 +177,7 @@ The analysis is significantly simplified when considering small motions, as the
Therefore, the mobility of the Stewart platform (defined as the set of achievable \([\delta x\ \delta y\ \delta z\ \delta \theta_x\ \delta \theta_y\ \delta \theta_z]\)) depends on two key factors: the stroke of each strut and the geometry of the Stewart platform (embodied in the Jacobian matrix).
More specifically, the XYZ mobility only depends on the \(\hat{\bm{s}}_i\) (orientation of struts), while the mobility in rotation also depends on \(\bm{b}_i\) (position of top joints).
\paragraph{Mobility in translation}
\subsubsection{Mobility in translation}
For simplicity, only translations are first considered (i.e., the Stewart platform is considered to have fixed orientation).
In the general case, the translational mobility can be represented by a 3D shape having 12 faces, where each actuator limits the stroke along its axis in positive and negative directions.
@ -229,7 +232,7 @@ The amplification factor increases when the struts have a high angle with the di
\end{subfigure}
\caption{\label{fig:detail_kinematics_stewart_mobility_translation_examples}Effect of strut orientation on the obtained mobility in translation. Two Stewart platform geometry are considered: struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_vert_struts}) and struts oriented vertically (\subref{fig:detail_kinematics_stewart_mobility_hori_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_translation_strut_orientation}).}
\end{figure}
\paragraph{Mobility in rotation}
\subsubsection{Mobility in rotation}
As shown by equation \eqref{eq:detail_kinematics_jacobian}, the rotational mobility depends both on the orientation of the struts and on the location of the top joints.
Similarly to the translational case, to increase the rotational mobility in one direction, it is advantageous to have the struts more perpendicular to the rotational direction.
@ -261,7 +264,7 @@ Having struts further apart decreases the ``lever arm'' and therefore reduces th
\end{subfigure}
\caption{\label{fig:detail_kinematics_stewart_mobility_rotation_examples}Effect of strut position on the obtained mobility in rotation. Two Stewart platform geometry are considered: struts close to each other (\subref{fig:detail_kinematics_stewart_mobility_close_struts}) and struts further appart (\subref{fig:detail_kinematics_stewart_mobility_space_struts}). Obtained mobility for both geometry are shown in (\subref{fig:detail_kinematics_mobility_angle_strut_distance}).}
\end{figure}
\paragraph{Combined translations and rotations}
\subsubsection{Combined translations and rotations}
It is possible to consider combined translations and rotations, although displaying such mobility becomes more complex.
For a fixed geometry and a desired mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke.
@ -290,7 +293,7 @@ In that case, the obtained stiffness matrix linearly depends on the strut stiffn
\end{array}
\right]
\end{equation}
\paragraph{Translation Stiffness}
\subsubsection{Translation Stiffness}
As shown by equation \eqref{eq:detail_kinematics_stiffness_matrix_simplified}, the translation stiffnesses (the \(3 \times 3\) top left terms of the stiffness matrix) only depend on the orientation of the struts and not their location: \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T\).
In the extreme case where all struts are vertical (\(s_i = [0\ 0\ 1]\)), a vertical stiffness of \(6k\) is achieved, but with null stiffness in the horizontal directions.
@ -300,12 +303,12 @@ This configuration corresponds to the cubic architecture presented in Section \r
When the struts are oriented more vertically, as shown in Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}, the vertical stiffness increases while the horizontal stiffness decreases.
Additionally, \(R_x\) and \(R_y\) stiffness increases while \(R_z\) stiffness decreases.
The opposite conclusions apply if struts are oriented more horizontally, illustrated in Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}.
\paragraph{Rotational Stiffness}
\subsubsection{Rotational Stiffness}
The rotational stiffnesses depend both on the orientation of the struts and on the location of the top joints with respect to the considered center of rotation (i.e., the location of frame \(\{A\}\)).
With the same orientation but increased distances to the frame \(\{A\}\) by a factor of 2, the rotational stiffness is increased by a factor of 4.
Therefore, the compact Stewart platform depicted in Figure \ref{fig:detail_kinematics_stewart_mobility_close_struts} has less rotational stiffness than the Stewart platform shown in Figure \ref{fig:detail_kinematics_stewart_mobility_space_struts}.
\paragraph{Diagonal Stiffness Matrix}
\subsubsection{Diagonal Stiffness Matrix}
Having a diagonal stiffness matrix \(\bm{K}\) can be beneficial for control purposes as it would make the plant in the Cartesian frame decoupled at low frequency.
This property depends on both the geometry and the chosen \(\{A\}\) frame.
@ -315,7 +318,7 @@ This is discussed in Section \ref{ssec:detail_kinematics_cubic_static}.
\label{ssec:detail_kinematics_geometry_dynamics}
The dynamical equations (both in the Cartesian frame and in the frame of the struts) for the Stewart platform were derived during the conceptual phase with simplifying assumptions (massless struts and perfect joints).
The dynamics depend both on the geometry (Jacobian matrix) and on the payload being placed on top of the platform.
The dynamics depends both on the geometry (Jacobian matrix) and on the payload being placed on top of the platform.
Under very specific conditions, the equations of motion in the Cartesian frame, given by equation \eqref{eq:detail_kinematics_transfer_function_cart}, can be decoupled.
These conditions are studied in Section \ref{ssec:detail_kinematics_cubic_dynamic}.
@ -359,7 +362,7 @@ Horizontal rotation mobility & \(\searrow\) & \(\searrow\)\\
\end{table}
\chapter{The Cubic Architecture}
\label{sec:detail_kinematics_cubic}
The Cubic configuration for the Stewart platform was first proposed in \cite{geng94_six_degree_of_freed_activ}.
The Cubic configuration for the Stewart platform was first proposed by Dr. Gough in a comment to the original paper by Dr. Stewart \cite{stewart65_platf_with_six_degrees_freed}.
This configuration is characterized by active struts arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure \ref{fig:detail_kinematics_cubic_architecture_example}.
Typically, the struts have similar length to the cube's edges, as illustrated in Figure \ref{fig:detail_kinematics_cubic_architecture_example}.
@ -393,7 +396,7 @@ Given that the cubic architecture imposes strict geometric constraints, alternat
The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod.
\section{Static Properties}
\label{ssec:detail_kinematics_cubic_static}
\paragraph{Stiffness matrix for the Cubic architecture}
\subsubsection{Stiffness matrix for the Cubic architecture}
Consider the cubic architecture shown in Figure \ref{fig:detail_kinematics_cubic_schematic_full}.
The unit vectors corresponding to the edges of the cube are described by equation \eqref{eq:detail_kinematics_cubic_s}.
@ -420,7 +423,7 @@ The unit vectors corresponding to the edges of the cube are described by equatio
\end{center}
\subcaption{\label{fig:detail_kinematics_cubic_schematic}Cube's portion}
\end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_schematic_cases}Cubic architecture. Struts are represented un blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic})}
\caption{\label{fig:detail_kinematics_cubic_schematic_cases}Cubic architecture. Struts are represented in blue. The cube's center by a black dot. The Struts can match the cube's edges (\subref{fig:detail_kinematics_cubic_schematic_full}) or just take a portion of the edge (\subref{fig:detail_kinematics_cubic_schematic})}
\end{figure}
Coordinates of the cube's vertices relevant for the top joints, expressed with respect to the cube's center, are shown in equation \eqref{eq:detail_kinematics_cubic_vertices}.
@ -456,7 +459,7 @@ In that case, the location of the top joints can be expressed by equation \eqref
The stiffness matrix is therefore diagonal when the considered \(\{B\}\) frame is located at the center of the cube (shown by frame \(\{C\}\)).
This means that static forces (resp torques) applied at the cube's center will induce pure translations (resp rotations around the cube's center).
This specific location where the stiffness matrix is diagonal is referred to as the ``Center of Stiffness'' (analogous to the ``Center of Mass'' where the mass matrix is diagonal).
\paragraph{Effect of having frame \(\{B\}\) off-centered}
\subsubsection{Effect of having frame \(\{B\}\) off-centered}
When the reference frames \(\{A\}\) and \(\{B\}\) are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix.
@ -478,19 +481,19 @@ This stiffness matrix structure is characteristic of Stewart platforms exhibitin
Therefore, the stiffness characteristics of the cubic architecture are only distinctive when considering a reference frame located at the cube's center.
This poses a practical limitation, as in most applications, the relevant frame (where motion is of interest and forces are applied) is located above the top platform.
It should be noted that for the stiffness matrix to be diagonal, the cube's center need not coincide with the geometric center of the Stewart platform.
It should be noted that for the stiffness matrix to be diagonal, the cube's center doesn't need to coincide with the geometric center of the Stewart platform.
This observation leads to the interesting alternative architectures presented in Section \ref{ssec:detail_kinematics_cubic_design}.
\paragraph{Uniform Mobility}
\subsubsection{Uniform Mobility}
The translational mobility of the Stewart platform with constant orientation was analyzed.
Considering limited actuator stroke (elongation of each strut), the maximum achievable positions in XYZ space were estimated.
The resulting mobility in X, Y, and Z directions for the cubic architecture is illustrated in Figure \ref{fig:detail_kinematics_cubic_mobility_translations}.
The translational workspace analysis reveals that for the cubic architecture, the achievable positions form a cube whose axes align with the struts, with the cube's edge length corresponding to the strut axial stroke.
This findings suggest that the mobility pattern is more nuanced than sometimes described in the literature \cite{mcinroy00_desig_contr_flexur_joint_hexap}, exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions.
These findings suggest that the mobility pattern is more subtle than sometimes described in the literature \cite{mcinroy00_desig_contr_flexur_joint_hexap}, exhibiting uniformity primarily along directions aligned with the cube's edges rather than uniform spherical distribution in all XYZ directions.
This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure \ref{fig:detail_kinematics_mobility_trans}.
The rotational mobility, illustrated in Figure \ref{fig:detail_kinematics_cubic_mobility_rotations}, exhibit greater achievable angular stroke in the \(R_x\) and \(R_y\) directions compared to the \(R_z\) direction.
The rotational mobility, illustrated in Figure \ref{fig:detail_kinematics_cubic_mobility_rotations}, exhibits greater achievable angular stroke in the \(R_x\) and \(R_y\) directions compared to the \(R_z\) direction.
Furthermore, an inverse relationship exists between the cube's dimension and rotational mobility, with larger cube sizes corresponding to more limited angular displacement capabilities.
\begin{figure}[htbp]
@ -518,7 +521,7 @@ When relative motion sensors are integrated in each strut (measuring \(\bm{\math
\includegraphics[scale=1]{figs/detail_kinematics_centralized_control.png}
\caption{\label{fig:detail_kinematics_centralized_control}Typical control architecture in the cartesian frame}
\end{figure}
\paragraph{Low frequency and High frequency coupling}
\subsubsection{Low frequency and High frequency coupling}
As derived during the conceptual design phase, the dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) is described by Equation \eqref{eq:detail_kinematics_transfer_function_cart}.
At low frequency, the behavior of the platform depends on the stiffness matrix \eqref{eq:detail_kinematics_transfer_function_cart_low_freq}.
@ -563,12 +566,12 @@ Conversely, when positioned at the center of stiffness, coupling occurred at hig
\end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_cart_coupling}Transfer functions for a Cubic Stewart platform expressed in the Cartesian frame. Two locations of the \(\{B\}\) frame are considered: at the center of mass of the moving body (\subref{fig:detail_kinematics_cubic_cart_coupling_com}) and at the cube's center (\subref{fig:detail_kinematics_cubic_cart_coupling_cok}).}
\end{figure}
\paragraph{Payload's CoM at the cube's center}
\subsubsection{Payload's CoM at the cube's center}
An effective strategy for improving dynamical performances involves aligning the cube's center (center of stiffness) with the center of mass of the moving components \cite{li01_simul_fault_vibrat_isolat_point}.
This can be achieved by positioning the payload below the top platform, such that the center of mass of the moving body coincides with the cube's center (Figure \ref{fig:detail_kinematics_cubic_centered_payload}).
This approach was physically implemented in several studies \cite{mcinroy99_dynam,jafari03_orthog_gough_stewar_platf_microm}, as shown in Figure \ref{fig:detail_kinematics_uw_gsp}.
The resulting dynamics are indeed well-decoupled (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_com_cok}), benefiting from simultaneously diagonal stiffness and mass matrices.
The resulting dynamics are indeed well-decoupled (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_com_cok}), taking advantage from diagonal stiffness and mass matrices.
The primary limitation of this approach is that, for many applications including the nano-hexapod, the payload must be positioned above the top platform.
If a design similar to Figure \ref{fig:detail_kinematics_cubic_centered_payload} were employed for the nano-hexapod, the X-ray beam would intersect with the struts during spindle rotation.
@ -587,7 +590,7 @@ If a design similar to Figure \ref{fig:detail_kinematics_cubic_centered_payload}
\end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_com_cok}Cubic Stewart platform with payload at the cube's center (\subref{fig:detail_kinematics_cubic_centered_payload}). Obtained cartesian plant is fully decoupled (\subref{fig:detail_kinematics_cubic_cart_coupling_com_cok})}
\end{figure}
\paragraph{Conclusion}
\subsubsection{Conclusion}
The analysis of dynamical properties of the cubic architecture yields several important conclusions.
Static decoupling, characterized by a diagonal stiffness matrix, is achieved when reference frames \(\{A\}\) and \(\{B\}\) are positioned at the cube's center.
@ -615,7 +618,7 @@ The second uses a non-cubic Stewart platform shown in Figure \ref{fig:detail_kin
\includegraphics[scale=1,width=0.6\linewidth]{figs/detail_kinematics_non_cubic_payload.png}
\caption{\label{fig:detail_kinematics_non_cubic_payload}Stewart platform with non-cubic architecture}
\end{figure}
\paragraph{Relative Displacement Sensors}
\subsubsection{Relative Displacement Sensors}
The transfer functions from actuator force in each strut to the relative motion of the struts are presented in Figure \ref{fig:detail_kinematics_decentralized_dL}.
As anticipated from the equations of motion from \(\bm{f}\) to \(\bm{\mathcal{L}}\) \eqref{eq:detail_kinematics_transfer_function_struts}, the \(6 \times 6\) plant is decoupled at low frequency.
@ -639,11 +642,11 @@ The resonance frequencies differ between the two cases because the more vertical
\end{subfigure}
\caption{\label{fig:detail_kinematics_decentralized_dL}Bode plot of the transfer functions from actuator force to relative displacement sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_dL}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_dL})}
\end{figure}
\paragraph{Force Sensors}
\subsubsection{Force Sensors}
Similarly, the transfer functions from actuator force to force sensors in each strut were analyzed for both cubic and non-cubic Stewart platforms.
The results are presented in Figure \ref{fig:detail_kinematics_decentralized_fn}.
The system demonstrates good decoupling at high frequency in both cases, with no evidence suggesting any advantage for the cubic architecture.
The system demonstrates good decoupling at high frequency in both cases, with no clear advantage for the cubic architecture.
\begin{figure}[htbp]
\begin{subfigure}{0.48\textwidth}
@ -660,10 +663,10 @@ The system demonstrates good decoupling at high frequency in both cases, with no
\end{subfigure}
\caption{\label{fig:detail_kinematics_decentralized_fn}Bode plot of the transfer functions from actuator force to force sensor in each strut. Both for a non-cubic architecture (\subref{fig:detail_kinematics_non_cubic_decentralized_fn}) and for a cubic architecture (\subref{fig:detail_kinematics_cubic_decentralized_fn})}
\end{figure}
\paragraph{Conclusion}
\subsubsection{Conclusion}
The presented results do not demonstrate the pronounced decoupling advantages often associated with cubic architectures in the literature.
Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement may be more nuanced than commonly described for decentralized control.
Both the cubic and non-cubic configurations exhibited similar coupling characteristics, suggesting that the benefits of orthogonal strut arrangement for decentralized control is less obvious than often reported in the literature.
\section{Cubic architecture with Cube's center above the top platform}
\label{ssec:detail_kinematics_cubic_design}
As demonstrated in Section \ref{ssec:detail_kinematics_cubic_dynamic}, the cubic architecture can exhibit advantageous dynamical properties when the center of mass of the moving body coincides with the cube's center, resulting in diagonal mass and stiffness matrices.
@ -676,7 +679,7 @@ Three key parameters define the geometry of the cubic Stewart platform: \(H\), t
Depending on the cube's size \(H_c\) in relation to \(H\) and \(H_{CoM}\), different designs emerge.
In the following examples, \(H = 100\,mm\) and \(H_{CoM} = 20\,mm\).
\paragraph{Small cube}
\subsubsection{Small cube}
When the cube size \(H_c\) is smaller than twice the height of the CoM \(H_{CoM}\) \eqref{eq:detail_kinematics_cube_small}, the resulting design is shown in Figure \ref{fig:detail_kinematics_cubic_above_small}.
@ -710,7 +713,7 @@ This approach yields a compact architecture, but the small cube size may result
\end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_above_small}Cubic architecture with cube's center above the top platform. A cube height of 40mm is used.}
\end{figure}
\paragraph{Medium sized cube}
\subsubsection{Medium sized cube}
Increasing the cube's size such that \eqref{eq:detail_kinematics_cube_medium} is verified produces an architecture with intersecting struts (Figure \ref{fig:detail_kinematics_cubic_above_medium}).
@ -741,7 +744,7 @@ This configuration resembles the design proposed in \cite{yang19_dynam_model_dec
\end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_above_medium}Cubic architecture with cube's center above the top platform. A cube height of 140mm is used.}
\end{figure}
\paragraph{Large cube}
\subsubsection{Large cube}
When the cube's height exceeds twice the sum of the platform height and CoM height \eqref{eq:detail_kinematics_cube_large}, the architecture shown in Figure \ref{fig:detail_kinematics_cubic_above_large} is obtained.
@ -770,7 +773,7 @@ When the cube's height exceeds twice the sum of the platform height and CoM heig
\end{subfigure}
\caption{\label{fig:detail_kinematics_cubic_above_large}Cubic architecture with cube's center above the top platform. A cube height of 240mm is used.}
\end{figure}
\paragraph{Platform size}
\subsubsection{Platform size}
For the proposed configuration, the top joints \(\bm{b}_i\) (resp. the bottom joints \(\bm{a}_i\)) and are positioned on a circle with radius \(R_{b_i}\) (resp. \(R_{a_i}\)) described by Equation \eqref{eq:detail_kinematics_cube_joints}.
@ -783,13 +786,8 @@ For the proposed configuration, the top joints \(\bm{b}_i\) (resp. the bottom jo
Since the rotational stiffness for the cubic architecture scales with the square of the cube's height \eqref{eq:detail_kinematics_cubic_stiffness}, the cube's size can be determined based on rotational stiffness requirements.
Subsequently, using Equation \eqref{eq:detail_kinematics_cube_joints}, the dimensions of the top and bottom platforms can be calculated.
\paragraph{Conclusion}
The configurations proposed in this analysis represent derivations from the classical cubic architecture, wherein the cube's center is typically located at the Stewart platform's center.
Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform.
This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame.
\section*{Conclusion}
The analysis of the cubic architecture for Stewart platforms has yielded several important findings.
The analysis of the cubic architecture for Stewart platforms yielded several important findings.
While the cubic configuration provides uniform stiffness in the XYZ directions, it stiffness property becomes particularly advantageous when forces and torques are applied at the cube's center.
Under these conditions, the stiffness matrix becomes diagonal, resulting in a decoupled Cartesian plant at low frequencies.
@ -804,7 +802,8 @@ Fully decoupled dynamics in the Cartesian frame can be achieved when the center
However, this arrangement presents practical challenges, as the cube's center is traditionally located between the top and bottom platforms, making payload placement problematic for many applications.
To address this limitation, modified cubic architectures have been proposed with the cube's center positioned above the top platform.
These configurations maintain the fundamental advantages of the cubic architecture while enabling practical payload placement.
Three distinct configurations have been identified, each with different geometric arrangements but sharing the common characteristic that the cube's center is positioned above the top platform.
This structural modification enables the alignment of the moving body's center of mass with the center of stiffness, resulting in beneficial decoupling properties in the Cartesian frame.
\chapter{Nano Hexapod}
\label{sec:detail_kinematics_nano_hexapod}
Based on previous analysis, this section aims to determine the nano-hexapod optimal geometry.
@ -876,7 +875,7 @@ The diagram confirms that the required workspace fits within the system's capabi
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/detail_kinematics_nano_hexapod_mobility.png}
\caption{\label{fig:detail_kinematics_nano_hexapod_mobility}Wanted translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume).}
\caption{\label{fig:detail_kinematics_nano_hexapod_mobility}Specified translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume).}
\end{figure}
\section{Required Joint angular stroke}
\label{ssec:detail_kinematics_nano_hexapod_joint_stroke}
@ -884,8 +883,7 @@ The diagram confirms that the required workspace fits within the system's capabi
With the nano-hexapod geometry and mobility requirements established, the flexible joint angular stroke necessary to avoid limiting the achievable workspace can be determined.
This analysis focuses solely on bending stroke, as the torsional stroke of the flexible joints is expected to be minimal given the absence of vertical rotation requirements.
The required angular stroke for both fixed and mobile joints is calculated to be \(1\,\text{mrad}\).
The required angular stroke for both fixed and mobile joints is estimated to be equal to \(1\,\text{mrad}\).
This specification will guide the design of the flexible joints.
\chapter{Conclusion}
\label{sec:detail_kinematics_conclusion}
@ -893,7 +891,7 @@ This specification will guide the design of the flexible joints.
This chapter has explored the optimization of the nano-hexapod geometry for the Nano Active Stabilization System (NASS).
First, a review of existing Stewart platforms revealed two main geometric categories: cubic architectures, characterized by mutually orthogonal struts arranged along the edges of a cube, and non-cubic architectures with varied strut orientations.
While cubic architectures are prevalent in the literature and attributed with beneficial properties such as simplified kinematics, uniform stiffness, and reduced cross-coupling, the performed analysis revealed that some of these advantages may be more nuanced or context-dependent than commonly described.
While cubic architectures are prevalent in the literature and attributed with beneficial properties such as simplified kinematics, uniform stiffness, and reduced cross-coupling, the performed analysis revealed that some of these advantages should be more nuanced or context-dependent than commonly described.
The analytical relationships between Stewart platform geometry and its mechanical properties were established, enabling a better understanding of the trade-offs between competing requirements such as mobility and stiffness along different axes.
These insights were useful during the nano-hexapod geometry optimization.

7
preamble_extra.tex
View File

@ -7,13 +7,15 @@
\usepackage{xpatch} % Recommanded for biblatex
\usepackage[ % use biblatex for bibliography
backend=biber, % use biber backend (bibtex replacement) or bibtex
style=ieee, % bib style
style=numeric-comp, % bib style
hyperref=true, % activate hyperref support
backref=true, % activate backrefs
isbn=false, % don't show isbn tags
url=false, % don't show url tags
doi=false, % don't show doi tags
urldate=long, % display type for dates
autocite=inline,%
sortcites=true, %
maxnames=3, %
minnames=1, %
maxbibnames=5, %
@ -132,3 +134,6 @@
}
\usepackage{hypcap}
\renewcommand{\topfraction}{.8}
\renewcommand{\floatpagefraction}{.8}