From b6a32476b54ff1a00ab9f86b69c885ace327bd0f Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 2 Apr 2025 16:52:24 +0200 Subject: [PATCH] First complete rewrite --- nass-geometry.bib | 521 +++++++++++++++--------------- nass-geometry.org | 769 ++++++++++++++++++++------------------------- nass-geometry.pdf | Bin 2550304 -> 2543680 bytes nass-geometry.tex | 783 ++++++++++++++++++---------------------------- 4 files changed, 927 insertions(+), 1146 deletions(-) diff --git a/nass-geometry.bib b/nass-geometry.bib index 66f72d6..87ad4af 100644 --- a/nass-geometry.bib +++ b/nass-geometry.bib @@ -1,3 +1,113 @@ +@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 + Stewart Platform Mechanism}, + journal = {Journal of Robotic Systems}, + volume = 10, + number = 5, + pages = {725-744}, + year = 1993, + doi = {10.1002/rob.4620100510}, + url = {https://doi.org/10.1002/rob.4620100510}, + keywords = {parallel robot}, +} + + + +@article{geng94_six_degree_of_freed_activ, + author = {Z.J. Geng and L.S. Haynes}, + title = {Six Degree-Of-Freedom Active Vibration Control Using the + Stewart Platforms}, + journal = {IEEE Transactions on Control Systems Technology}, + volume = 2, + number = 1, + pages = {45-53}, + year = 1994, + doi = {10.1109/87.273110}, + url = {https://doi.org/10.1109/87.273110}, + keywords = {parallel robot, cubic configuration}, +} + + + +@article{geng95_intel_contr_system_multip_degree, + author = {Z. Jason Geng and George G. Pan and Leonard S. Haynes and + Ben K. Wada and John A. Garba}, + title = {An Intelligent Control System for Multiple + Degree-Of-Freedom Vibration Isolation}, + journal = {Journal of Intelligent Material Systems and Structures}, + volume = 6, + number = 6, + pages = {787-800}, + year = 1995, + doi = {10.1177/1045389x9500600607}, + url = {https://doi.org/10.1177/1045389x9500600607}, + keywords = {parallel robot}, +} + + + @inproceedings{spanos95_soft_activ_vibrat_isolat, author = {J. Spanos and Z. Rahman and G. Blackwood}, title = {A Soft 6-axis Active Vibration Isolator}, @@ -100,21 +210,6 @@ -@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 @@ -148,6 +243,48 @@ +@inproceedings{defendini00_techn, + author = {Defendini, A and Vaillon, L and Trouve, F and Rouze, Th and + Sanctorum, B and Griseri, G and Spanoudakis, P and von + Alberti, M}, + title = {Technology predevelopment for active control of vibration + and very high accuracy pointing systems}, + booktitle = {Spacecraft Guidance, Navigation and Control Systems}, + year = 2000, + volume = 425, + pages = 385, +} + + + +@inproceedings{abu02_stiff_soft_stewar_platf_activ, + author = {Abu Hanieh, Ahmed and Horodinca, Mihaita and Preumont, + Andre}, + title = {Stiff and Soft Stewart Platforms for Active Damping and + Active Isolation of Vibrations}, + booktitle = {Actuator 2002, 8th International Conference on New + Actuators}, + year = 2002, + keywords = {parallel robot}, +} + + + +@inproceedings{chen03_payload_point_activ_vibrat_isolat, + author = {Hong-Jen Chen and Ronald Bishop and Brij Agrawal}, + title = {Payload Pointing and Active Vibration Isolation Using + Hexapod Platforms}, + booktitle = {44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural + Dynamics, and Materials Conference}, + year = 2003, + doi = {10.2514/6.2003-1643}, + url = {https://doi.org/10.2514/6.2003-1643}, + keywords = {parallel robot}, + month = 4, +} + + + @phdthesis{hanieh03_activ_stewar, author = {Hanieh, Ahmed Abu}, keywords = {parallel robot}, @@ -177,186 +314,6 @@ -@inproceedings{taranti01_effic_algor_vibrat_suppr, - author = {Taranti, Christian and Agrawal, Brij and Cristi, Roberto}, - title = {An Efficient Algorithm for Vibration Suppression to meet - pointing requirements of optical payloads}, - booktitle = {AIAA Guidance, Navigation, and Control Conference and - Exhibit}, - year = 2001, - pages = 4094, -} - - - -@inproceedings{chen03_payload_point_activ_vibrat_isolat, - author = {Hong-Jen Chen and Ronald Bishop and Brij Agrawal}, - title = {Payload Pointing and Active Vibration Isolation Using - Hexapod Platforms}, - booktitle = {44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural - Dynamics, and Materials Conference}, - year = 2003, - doi = {10.2514/6.2003-1643}, - url = {https://doi.org/10.2514/6.2003-1643}, - keywords = {parallel robot}, - month = 4, -} - - - -@article{chi15_desig_exper_study_vcm_based, - author = {Weichao Chi and Dengqing Cao and Dongwei Wang and Jie Tang - and Yifan Nie and Wenhu Huang}, - title = {Design and Experimental Study of a Vcm-Based Stewart - Parallel Mechanism Used for Active Vibration Isolation}, - journal = {Energies}, - volume = 8, - number = 8, - pages = {8001-8019}, - year = 2015, - doi = {10.3390/en8088001}, - url = {https://doi.org/10.3390/en8088001}, - keywords = {parallel robot}, -} - - - -@article{tang18_decen_vibrat_contr_voice_coil, - author = {Jie Tang and Dengqing Cao and Tianhu Yu}, - title = {Decentralized Vibration Control of a Voice Coil Motor-Based - Stewart Parallel Mechanism: Simulation and Experiments}, - journal = {Proceedings of the Institution of Mechanical Engineers, - Part C: Journal of Mechanical Engineering Science}, - volume = 233, - number = 1, - pages = {132-145}, - year = 2018, - doi = {10.1177/0954406218756941}, - url = {https://doi.org/10.1177/0954406218756941}, - keywords = {parallel robot}, -} - - - -@article{jiao18_dynam_model_exper_analy_stewar, - author = {Jian Jiao and Ying Wu and Kaiping Yu and Rui Zhao}, - title = {Dynamic Modeling and Experimental Analyses of Stewart - Platform With Flexible Hinges}, - journal = {Journal of Vibration and Control}, - volume = 25, - number = 1, - pages = {151-171}, - year = 2018, - doi = {10.1177/1077546318772474}, - url = {https://doi.org/10.1177/1077546318772474}, - keywords = {parallel robot, flexure}, -} - - - -@article{beijen18_self_tunin_mimo_distur_feedf, - author = {M.A. Beijen and M.F. Heertjes and J. Van Dijk and W.B.J. - Hakvoort}, - title = {Self-Tuning Mimo Disturbance Feedforward Control for Active - Hard-Mounted Vibration Isolators}, - journal = {Control Engineering Practice}, - volume = 72, - pages = {90-103}, - year = 2018, - doi = {10.1016/j.conengprac.2017.11.008}, - url = {https://doi.org/10.1016/j.conengprac.2017.11.008}, - keywords = {parallel robot, feedforward}, -} - - - -@phdthesis{tjepkema12_activ_ph, - author = {Tjepkema, D}, - title = {Active hard mount vibration isolation for precision - equipment [Ph. D. thesis]}, - university = {University of Twente, Enschede, The Netherlands}, - year = {2012}, -} - - - -@article{geng93_six_degree_of_freed_activ, - author = {Zheng Geng and Leonard S. Haynes}, - title = {Six-Degree-Of-Freedom Active Vibration Isolation Using a - Stewart Platform Mechanism}, - journal = {Journal of Robotic Systems}, - volume = 10, - number = 5, - pages = {725-744}, - year = 1993, - doi = {10.1002/rob.4620100510}, - url = {https://doi.org/10.1002/rob.4620100510}, - keywords = {parallel robot}, -} - - - -@article{geng94_six_degree_of_freed_activ, - author = {Z.J. Geng and L.S. Haynes}, - title = {Six Degree-Of-Freedom Active Vibration Control Using the - Stewart Platforms}, - journal = {IEEE Transactions on Control Systems Technology}, - volume = 2, - number = 1, - pages = {45-53}, - year = 1994, - doi = {10.1109/87.273110}, - url = {https://doi.org/10.1109/87.273110}, - keywords = {parallel robot, cubic configuration}, -} - - - -@article{geng95_intel_contr_system_multip_degree, - author = {Z. Jason Geng and George G. Pan and Leonard S. Haynes and - Ben K. Wada and John A. Garba}, - title = {An Intelligent Control System for Multiple - Degree-Of-Freedom Vibration Isolation}, - journal = {Journal of Intelligent Material Systems and Structures}, - volume = 6, - number = 6, - pages = {787-800}, - year = 1995, - doi = {10.1177/1045389x9500600607}, - url = {https://doi.org/10.1177/1045389x9500600607}, - keywords = {parallel robot}, -} - - - -@inproceedings{zhang11_six_dof, - author = {Zhen Zhang and J Liu and Jq Mao and Yx Guo and Yh Ma}, - title = {Six DOF active vibration control using stewart platform - with non-cubic configuration}, - booktitle = {2011 6th IEEE Conference on Industrial Electronics and - Applications}, - year = 2011, - doi = {10.1109/iciea.2011.5975679}, - url = {https://doi.org/10.1109/iciea.2011.5975679}, - keywords = {parallel robot}, - month = 6, -} - - - -@inproceedings{abu02_stiff_soft_stewar_platf_activ, - author = {Abu Hanieh, Ahmed and Horodinca, Mihaita and Preumont, - Andre}, - title = {Stiff and Soft Stewart Platforms for Active Damping and - Active Isolation of Vibrations}, - booktitle = {Actuator 2002, 8th International Conference on New - Actuators}, - year = 2002, - keywords = {parallel robot}, -} - - - @article{agrawal04_algor_activ_vibrat_isolat_spacec, author = {Brij N Agrawal and Hong-Jen Chen}, title = {Algorithms for Active Vibration Isolation on Spacecraft @@ -417,6 +374,21 @@ +@inproceedings{zhang11_six_dof, + author = {Zhen Zhang and J Liu and Jq Mao and Yx Guo and Yh Ma}, + title = {Six DOF active vibration control using stewart platform + with non-cubic configuration}, + booktitle = {2011 6th IEEE Conference on Industrial Electronics and + Applications}, + year = 2011, + doi = {10.1109/iciea.2011.5975679}, + url = {https://doi.org/10.1109/iciea.2011.5975679}, + keywords = {parallel robot}, + month = 6, +} + + + @article{du14_piezo_actuat_high_precis_flexib, author = {Zhijiang Du and Ruochong Shi and Wei Dong}, title = {A Piezo-Actuated High-Precision Flexible Parallel Pointing @@ -433,37 +405,52 @@ -@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{chi15_desig_exper_study_vcm_based, + author = {Weichao Chi and Dengqing Cao and Dongwei Wang and Jie Tang + and Yifan Nie and Wenhu Huang}, + title = {Design and Experimental Study of a Vcm-Based Stewart + Parallel Mechanism Used for Active Vibration Isolation}, + journal = {Energies}, + volume = 8, + number = 8, + pages = {8001-8019}, + year = 2015, + doi = {10.3390/en8088001}, + url = {https://doi.org/10.3390/en8088001}, + 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{tang18_decen_vibrat_contr_voice_coil, + author = {Jie Tang and Dengqing Cao and Tianhu Yu}, + title = {Decentralized Vibration Control of a Voice Coil Motor-Based + Stewart Parallel Mechanism: Simulation and Experiments}, + journal = {Proceedings of the Institution of Mechanical Engineers, + Part C: Journal of Mechanical Engineering Science}, + volume = 233, + number = 1, + pages = {132-145}, + year = 2018, + doi = {10.1177/0954406218756941}, + url = {https://doi.org/10.1177/0954406218756941}, + keywords = {parallel robot}, +} + + + +@article{jiao18_dynam_model_exper_analy_stewar, + author = {Jian Jiao and Ying Wu and Kaiping Yu and Rui Zhao}, + title = {Dynamic Modeling and Experimental Analyses of Stewart + Platform With Flexible Hinges}, + journal = {Journal of Vibration and Control}, + volume = 25, + number = 1, + pages = {151-171}, + year = 2018, + doi = {10.1177/1077546318772474}, + url = {https://doi.org/10.1177/1077546318772474}, + keywords = {parallel robot, flexure}, } @@ -487,32 +474,28 @@ -@inproceedings{defendini00_techn, - author = {Defendini, A and Vaillon, L and Trouve, F and Rouze, Th and - Sanctorum, B and Griseri, G and Spanoudakis, P and von - Alberti, M}, - title = {Technology predevelopment for active control of vibration - and very high accuracy pointing systems}, - booktitle = {Spacecraft Guidance, Navigation and Control Systems}, - year = 2000, - volume = 425, - pages = 385, +@article{beijen18_self_tunin_mimo_distur_feedf, + author = {M.A. Beijen and M.F. Heertjes and J. Van Dijk and W.B.J. + Hakvoort}, + title = {Self-Tuning Mimo Disturbance Feedforward Control for Active + Hard-Mounted Vibration Isolators}, + journal = {Control Engineering Practice}, + volume = 72, + pages = {90-103}, + year = 2018, + doi = {10.1016/j.conengprac.2017.11.008}, + url = {https://doi.org/10.1016/j.conengprac.2017.11.008}, + keywords = {parallel robot, feedforward}, } -@article{torii12_small_size_self_propel_stewar_platf, - author = {Akihiro Torii and Masaaki Banno and Akiteru Ueda and Kae - Doki}, - title = {A Small-Size Self-Propelled Stewart Platform}, - journal = {Electrical Engineering in Japan}, - volume = 181, - number = 2, - pages = {37-46}, - year = 2012, - doi = {10.1002/eej.21261}, - url = {https://doi.org/10.1002/eej.21261}, - keywords = {parallel robot}, +@phdthesis{tjepkema12_activ_ph, + author = {Tjepkema, D}, + title = {Active hard mount vibration isolation for precision + equipment [Ph. D. thesis]}, + university = {University of Twente, Enschede, The Netherlands}, + year = {2012}, } @@ -540,3 +523,53 @@ organization = {EUSPEN}, } + + +@inproceedings{merlet02_still, + author = {Merlet, Jean-Pierre}, + title = {Still a long way to go on the road for parallel mechanisms}, + booktitle = {Proc. ASME 2002 DETC Conf., Montreal}, + year = 2002, + keywords = {parallel robot}, +} + + + +@article{mcinroy02_model_desig_flexur_joint_stewar, + author = {J.E. McInroy}, + title = {Modeling and Design of Flexure Jointed Stewart Platforms + for Control Purposes}, + journal = {IEEE/ASME Transactions on Mechatronics}, + volume = 7, + number = 1, + pages = {95-99}, + year = 2002, + doi = {10.1109/3516.990892}, + url = {https://doi.org/10.1109/3516.990892}, + keywords = {parallel robot, flexure}, +} + + + +@book{preumont18_vibrat_contr_activ_struc_fourt_edition, + author = {Andre Preumont}, + title = {Vibration Control of Active Structures - Fourth Edition}, + year = 2018, + publisher = {Springer International Publishing}, + url = {https://doi.org/10.1007/978-3-319-72296-2}, + doi = {10.1007/978-3-319-72296-2}, + keywords = {favorite, parallel robot}, + 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, +} + diff --git a/nass-geometry.org b/nass-geometry.org index cbe1fbf..3a04041 100644 --- a/nass-geometry.org +++ b/nass-geometry.org @@ -953,48 +953,65 @@ Arguments: * Introduction :ignore: -- In the conceptual design phase, the geometry of the Stewart platform was chosen arbitrarily and not optimized -- In the detail design phase, we want to see if the geometry can be optimized to improve the overall performances -- Optimization criteria: mobility, stiffness, decoupling between the struts for decentralized control, dynamical decoupling in the cartesian frame +The performance of a Stewart platform depends on its geometric configuration, especially the orientation of its struts and the positioning of its joints. +During the conceptual design phase of the nano-hexapod, a preliminary geometry was selected based on general principles without detailed optimization. +As the project advanced to the detailed design phase, a rigorous analysis of how geometry influences system performance became essential to ensure that the final design would meet the demanding requirements of the Nano Active Stabilization System (NASS). -Outline: -- Review of Stewart platform (Section ref:sec:detail_kinematics_stewart_review) - Geometry, Actuators, Sensors, Joints -- Effect of geometry on the Stewart platform characteristics (Section ref:sec:detail_kinematics_geometry) -- Cubic configuration: special architecture that received many attention in the literature. We want to see the special properties of this architecture and if this can be applied for the nano hexapod (Section ref:sec:detail_kinematics_cubic) -- Presentation of the obtained geometry for the nano hexapod (Section ref:sec:detail_kinematics_nano_hexapod) +In this chapter, the nano-hexapod geometry is optimized through careful analysis of how design parameters influence critical performance aspects: attainable workspace, mechanical stiffness, strut-to-strut coupling for decentralized control strategies, and dynamic response in Cartesian coordinates. + +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 garnered significant attention for its purported advantages—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 <> ** Introduction :ignore: -- As was explained in the conceptual phase, Stewart platform have the following key elements: - # Section ref:sec:nhexa_stewart_platform - - Two plates connected by six struts - - Each strut is composed of: - - a flexible joint at each end - - an actuator - - 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. -- This 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 impose other challenges. - Some Stewart platforms found in the literature are listed in Table ref:tab:detail_kinematics_stewart_review -- All presented Stewart platforms are using flexible joints, as it is a prerequisites for nano-positioning capabilities. -- Most of stewart platforms are using voice coil actuators or piezoelectric actuators. - The actuators used for the Stewart platform will be chosen in the next section. - # TODO - Add reference to the section -- Depending on the application, various sensors are integrated in the struts or on the plates such as force sensors, inertial sensors or relative displacement sensors. - The choice of sensor for the nano-hexapod will be described in the next section. - # TODO - Add reference to the section -- Flexible joints can also have various implementations. This will be discussed in the next section. -- There are two main categories of Stewart platform geometry: - - Cubic architecture (Figure ref:fig:detail_kinematics_stewart_examples_cubic). - Struts are positioned along 6 sides of a cubic (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. - - Non-cubic architecture (Figure ref:fig:detail_kinematics_stewart_examples_non_cubic) - The orientation of the struts and position of the joints are chosen based on performances criteria. - Some of which are presented in Section ref:sec:detail_kinematics_geometry +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. + +#+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 ordered by appearance in the literature +#+attr_latex: :environment tabularx :width 0.8\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]] | + + +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 + +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 #+name: fig:detail_kinematics_stewart_examples_cubic #+caption: Some examples of developped Stewart platform with Cubic geometry. (\subref{fig:detail_kinematics_jpl}), (\subref{fig:detail_kinematics_uw_gsp}), (\subref{fig:detail_kinematics_ulb_pz}), (\subref{fig:detail_kinematics_uqp}) @@ -1028,6 +1045,12 @@ Outline: #+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 performance criteria. +The effect of strut orientation and position of the joints on the Stewart platform properties is discussed 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. (\subref{fig:detail_kinematics_pph}), (\subref{fig:detail_kinematics_zhang11}), (\subref{fig:detail_kinematics_yang19}), (\subref{fig:detail_kinematics_naves}) #+attr_latex: :options [htbp] @@ -1060,42 +1083,6 @@ Outline: #+end_subfigure #+end_figure -#+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 ordered by appearance in the literature -#+attr_latex: :environment tabularx :width \linewidth :align llllX -#+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]] | - -Conclusion: -- Various Stewart platform designs: - - geometry, sizes, orientation of struts - - Lot's have a "cubic" architecture that will be discussed in Section ref:sec:detail_kinematics_cubic - - actuator types - - various sensors - - flexible joints -- The effect of geometry on the properties of the Stewart platform is studied in section ref:sec:detail_kinematics_geometry -- It is determined what is the optimal geometry for the NASS - * Effect of geometry on Stewart platform properties :PROPERTIES: :HEADER-ARGS:matlab+: :tangle matlab/detail_kinematics_1_geometry.m @@ -1103,16 +1090,12 @@ Conclusion: <> ** Introduction :ignore: -# Section ref:sec:nhexa_stewart_platform (page pageref:sec:nhexa_stewart_platform), -- As was shown during the conceptual phase, the geometry of the Stewart platform influences: - - the stiffness and compliance properties - - the mobility or workspace - - the force authority - - the dynamics of the manipulator -- It is therefore important to understand how the geometry impact these properties, and to be able to optimize the geometry for a specific application. +# TODO - Section ref:sec:nhexa_stewart_platform (page pageref:sec:nhexa_stewart_platform), +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. -One important tool to study this is the Jacobian matrix which depends on the $\bm{b}_i$ (join position w.r.t top platform) and $\hat{\bm{s}}_i$ (orientation of struts). -The choice of frames ($\{A\}$ and $\{B\}$), independently of the physical Stewart platform geometry, impacts the obtained kinematics and stiffness matrix, as it is defined for forces and motion evaluated at the chosen frame. +An important analytical tool for this study is the Jacobian matrix, which depends on $\bm{b}_i$ (joint position with respect to the top platform) and $\hat{\bm{s}}_i$ (orientation of struts). +The choice of frames ($\{A\}$ and $\{B\}$), 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: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -1140,19 +1123,17 @@ The choice of frames ($\{A\}$ and $\{B\}$), independently of the physical Stewar #+end_src ** Platform Mobility / Workspace +<> **** Introduction :ignore: -The mobility of the Stewart platform (or any manipulator) is here defined as the range of motion that it can perform. -It corresponds to the set of possible pose (i.e. combined translation and rotation) of frame {B} with respect to frame {A}. -It is therefore a six dimensional property which is difficult to represent. -Depending on the applications, only the translation mobility (i.e. fixed orientation workspace) or the rotation mobility may be represented. -This is equivalent as to project the six dimensional value into a 3 dimensional space, easier to represent. +The mobility of the Stewart platform (or any manipulator) is defined as the range of motion that it can perform. +It corresponds to the set of possible poses (i.e., combined translation and rotation) of frame $\{B\}$ with respect to frame $\{A\}$. +This represents a six-dimensional property which is difficult to represent. +Depending on the applications, only the translation mobility (i.e., fixed orientation workspace) or the rotation mobility may be represented. +This approach is equivalent to projecting the six-dimensional value into a three-dimensional space, which is easier to represent. -Mobility of parallel manipulators are inherently difficult to study as the translational and orientation workspace are coupled [[cite:&merlet02_still]]. -Things are getting much more simpler when considering small motions as the Jacobian matrix can be considered constant and the equations are linear. - -As was shown during the conceptual phase, for small displacements, the Jacobian matrix can be used to link the strut motion to the motion of frame B with respect to A through equation eqref:eq:detail_kinematics_jacobian. -# Section ref:ssec:nhexa_stewart_platform_jacobian (page pageref:ssec:nhexa_stewart_platform_jacobian). +Mobility of parallel manipulators is inherently difficult to study as the translational and orientation workspace are coupled [[cite:&merlet02_still]]. +The analysis is significantly simplified when considering small motions, as the Jacobian matrix can be used to link the strut motion to the motion of frame $\{B\}$ with respect to $\{A\}$ through eqref:eq:detail_kinematics_jacobian, which is a linear equation. \begin{equation}\label{eq:detail_kinematics_jacobian} \begin{bmatrix} \delta l_1 \\ \delta l_2 \\ \delta l_3 \\ \delta l_4 \\ \delta l_5 \\ \delta l_6 \end{bmatrix} = \begin{bmatrix} @@ -1165,25 +1146,15 @@ As was shown during the conceptual phase, for small displacements, the Jacobian \end{bmatrix} \begin{bmatrix} \delta x \\ \delta y \\ \delta z \\ \delta \theta_x \\ \delta \theta_y \\ \delta \theta_z \end{bmatrix} \end{equation} -Therefore, the mobility of the Stewart platform (set of $[\delta x\ \delta y\ \delta z\ \delta \theta_x\ \delta \theta_y\ \delta \theta_z]$) depends on: -- the stroke of each strut -- the geometry of the Stewart platform (embodied in the Jacobian matrix) - -More specifically: -- the XYZ mobility only depends on the si (orientation of struts) -- the mobility in rotation depends on bi (position of top joints) +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 $s_i$ (orientation of struts), while the mobility in rotation also depends on $b_i$ (position of top joints). **** Mobility in translation -Here, for simplicity, only translations are first considered (i.e. fixed orientation of the Stewart platform): -- Let's consider a general Stewart platform geometry shown in Figure ref:fig:detail_kinematics_mobility_trans_arch. -- In the general case: the translational mobility can be represented by a 3D shape with 12 faces (each actuator limits the stroke along its orientation in positive and negative directions). - The faces are therefore perpendicular to the strut direction. - The obtained mobility for the considered stewart platform geometry is shown in Figure ref:fig:detail_kinematics_mobility_trans_result. - In reality, the workspace boundaries are portion of spheres, but they are well approximated by flat surfaces for short stroke hexapods -- Considering an actuator stroke of $\pm d$, the mobile platform can be translated in any direction with a stroke of $d$ - This means that a sphere with radius $d$ is contained in the general shape as illustrated in Figure ref:fig:detail_kinematics_mobility_trans_result. - The sphere will touch the shape along six lines defined by the strut axes. +For simplicity, only translations are first considered (i.e., fixed orientation of the Stewart platform). +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. +The faces are therefore perpendicular to the strut direction. +The obtained mobility for the Stewart platform geometry shown in Figure ref:fig:detail_kinematics_mobility_trans_arch is computed and represented in Figure ref:fig:detail_kinematics_mobility_trans_result. #+begin_src matlab :exports none :results none %% Example of one Stewart platform and associated translational mobility @@ -1270,15 +1241,16 @@ exportFig('figs/detail_kinematics_mobility_trans_result.pdf', 'width', 'normal', #+end_subfigure #+end_figure -To better understand how the geometry of the Stewart platform impacts the translational mobility, two configurations are compared: -- Struts oriented horizontally (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts). - This leads to having more stroke in the horizontal direction and less stroke in the vertical direction (Figure ref:fig:detail_kinematics_mobility_translation_strut_orientation). -- Struts oriented vertically (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts). - More stroke in vertical direction +With the previous interpretations of the 12 faces making the translational mobility 3D shape, it can be concluded that for a strut stroke of $\pm d$, a sphere with radius $d$ is contained in the 3D shape and touches it along the six lines defined by the strut axes, as illustrated in Figure ref:fig:detail_kinematics_mobility_trans_result. +This means that the mobile platform can be translated in any direction with a stroke of $d$. -It can be counter intuitive to have less stroke in the direction of the struts. -This is because the struts are forming a lever mechanism that amplifies the motion. -The amplification factor increases when the struts have an high angle with the direction and motion and is equal to one when it is aligned with the direction of motion. +To better understand how the geometry of the Stewart platform impacts the translational mobility, two configurations are compared with struts oriented vertically (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts) and struts oriented horizontally (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts). +The vertically oriented struts lead to greater stroke in the horizontal direction and reduced stroke in the vertical direction (Figure ref:fig:detail_kinematics_mobility_translation_strut_orientation). +Conversely, horizontal oriented struts provide more stroke in the vertical direction. + +It may seem counterintuitive that less stroke is available in the direction of the struts. +This phenomenon occurs because the struts form a lever mechanism that amplifies the motion. +The amplification factor increases when the struts have a high angle with the direction of motion and equals one when aligned with the direction of motion. #+begin_src matlab :exports none :results none %% Stewart platform with Vertically oriented struts @@ -1410,22 +1382,13 @@ exportFig('figs/detail_kinematics_mobility_translation_strut_orientation.pdf', ' **** 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. -For instance, having the struts more vertical (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts) gives less rotational stroke along the vertical direction than having the struts oriented more horizontally (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts). +For instance, having the struts more vertical (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts) provides less rotational stroke along the vertical direction than having the struts oriented more horizontally (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts). -Two cases are considered with same strut orientation but with different top joints positions: -- struts close to each other (Figure ref:fig:detail_kinematics_stewart_mobility_close_struts) -- struts further apart (Figure ref:fig:detail_kinematics_stewart_mobility_space_struts) - -The mobility for pure rotations are compared in Figure ref:fig:detail_kinematics_mobility_angle_strut_distance. -Note that the same strut stroke are considered in both cases to evaluate the mobility. -Having struts further apart decreases the "level arm" and therefore the rotational mobility is reduced. - -For rotations and translations, having more mobility also means increasing the effect of actuator noise on the considering degree of freedom. -Somehow, the level arm is increased, so any strut vibration gets amplified. -Therefore, the designed Stewart platform should just have the necessary mobility. +Two cases are considered with the same strut orientation but with different top joint positions: struts positioned close to each other (Figure ref:fig:detail_kinematics_stewart_mobility_close_struts) and struts positioned further apart (Figure ref:fig:detail_kinematics_stewart_mobility_space_struts). +The mobility for pure rotations is compared in Figure ref:fig:detail_kinematics_mobility_angle_strut_distance. +Having struts further apart decreases the "lever arm" and therefore reduces the rotational mobility. #+begin_src matlab :exports none :results none %% Stewart platform with struts close to each other @@ -1550,27 +1513,24 @@ exportFig('figs/detail_kinematics_mobility_angle_strut_distance.pdf', 'width', ' **** Combined translations and rotations -It is possible to consider combined translations and rotations. -Displaying such mobility is more complex. -It will be used for the nano-hexapod to verify that the obtained design has the necessary mobility. - -For a fixed geometry and a wanted mobility (combined translations and rotations), it is possible to estimate the required minimum actuator stroke. -It will be done in Section ref:sec:detail_kinematics_nano_hexapod to estimate the required actuator stroke for the nano-hexapod geometry. +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. +This analysis was conducted in Section ref:sec:detail_kinematics_nano_hexapod to estimate the required actuator stroke for the nano-hexapod geometry. ** Stiffness +<> **** Introduction :ignore: -Stiffness matrix: -- defines how the nano-hexapod deforms (frame $\{B\}$ with respect to frame $\{A\}$) due to static forces/torques applied on $\{B\}$. -- It depends on the Jacobian matrix (i.e. the geometry) and the strut axial stiffness eqref:eq:detail_kinematics_stiffness_matrix -- The contribution of joints stiffness is here not considered [[cite:&mcinroy00_desig_contr_flexur_joint_hexap;&mcinroy02_model_desig_flexur_joint_stewar]] +The stiffness matrix defines how the nano-hexapod deforms (frame $\{B\}$ with respect to frame $\{A\}$) due to static forces/torques applied on $\{B\}$. +It depends on the Jacobian matrix (i.e., the geometry) and the strut axial stiffness as shown in equation eqref:eq:detail_kinematics_stiffness_matrix. +The contribution of joints stiffness is not considered here, as there were optimized after the geometry was fixed, but several work were done to quantify the impact of the flexible joint stiffness [[cite:&mcinroy00_desig_contr_flexur_joint_hexap;&mcinroy02_model_desig_flexur_joint_stewar]]. \begin{equation}\label{eq:detail_kinematics_stiffness_matrix} \bm{K} = \bm{J}^T \bm{\mathcal{K}} \bm{J} \end{equation} -It is assumed that the stiffness of all strut is the same: $\bm{\mathcal{K}} = k \cdot \mathbf{I}_6$. -Obtained stiffness matrix linearly depends on the strut stiffness $k$, and is structured as shown in eqref:eq:detail_kinematics_stiffness_matrix_simplified. +It is assumed that the stiffness of all struts is the same: $\bm{\mathcal{K}} = k \cdot \mathbf{I}_6$. +In that case, the obtained stiffness matrix linearly depends on the strut stiffness $k$, and is structured as shown in equation eqref:eq:detail_kinematics_stiffness_matrix_simplified. \begin{equation}\label{eq:detail_kinematics_stiffness_matrix_simplified} \bm{K} = k \bm{J}^T \bm{J} = @@ -1585,62 +1545,51 @@ Obtained stiffness matrix linearly depends on the strut stiffness $k$, and is st **** Translation Stiffness -As shown by eqref:eq:detail_kinematics_stiffness_matrix_simplified, the translation stiffnesses (the 3x3 top left terms of the stiffness matrix): -- Only depends on the orientation of the struts and not their location: $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T$ -- Extreme case: all struts are vertical $s_i = [0,\ 0,\ 1]$ => vertical stiffness of $6 k$, but null stiffness in X and Y directions -- If two struts along X, two struts along Y, and two struts along Z => $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3$ - Stiffness is well distributed along directions. - This corresponds to the cubic architecture presented in Section ref:sec:detail_kinematics_cubic. +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 with $s_i = [0,\ 0,\ 1]$, a vertical stiffness of $6k$ is achieved, but with null stiffness in the X and Y directions. +If two struts are aligned along the X axis, two struts along the Y axis, and two struts along the Z axis, then $\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3$, resulting in well-distributed stiffness along all directions. +This configuration corresponds to the cubic architecture presented in Section ref:sec:detail_kinematics_cubic. -If struts more vertical (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts): -- increase vertical stiffness -- decrease horizontal stiffness -- increase Rx,Ry stiffness -- decrease Rz stiffness - -Opposite conclusions if struts are not horizontal (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts). +When struts are oriented more vertically (Figure ref:fig:detail_kinematics_stewart_mobility_vert_struts), vertical stiffness increases while 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 (Figure ref:fig:detail_kinematics_stewart_mobility_hori_struts). **** Rotational Stiffness -The rotational stiffnesses depends 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 B). - -Same orientation but increased distances (bi) by a factor 2 => rotational stiffness increased by factor 4. -Compact stewart platform of Figure ref:fig:detail_kinematics_stewart_mobility_close_struts as therefore less rotational stiffness than the Stewart platform of Figure ref:fig:detail_kinematics_stewart_mobility_space_struts. +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 $\{B\}$). +With the same orientation but increased distances ($b_i$) 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. **** Diagonal Stiffness Matrix -Having the stiffness matrix $\bm{K}$ diagonal can be beneficial for control purposes as it would make the plant in the cartesian frame decoupled at low frequency. -This depends on the geometry and on the chosen {B} frame. -For specific geometry and chose of B frame, it is possible to have a diagonal K matrix. - -This will be discussed in Section ref:ssec:detail_kinematics_cubic_static. +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 $\{B\}$ frame. +For specific geometry and choice of $\{B\}$ frame, it is possible to achieve a diagonal $K$ matrix. +This is discussed in Section ref:ssec:detail_kinematics_cubic_static. ** Dynamical properties -**** In the Cartesian Frame +<> -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 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. # Section ref:ssec:nhexa_stewart_platform_dynamics (page pageref:ssec:nhexa_stewart_platform_dynamics). -The dynamics depends both on the geometry (Jacobian matrix) but also on the payload being placed on top of the platform. - -Under very specific conditions, the equations of motion in the Cartesian frame eqref:eq:nhexa_transfer_function_cart can be decoupled. -These are studied in Section ref:ssec:detail_kinematics_cubic_dynamic. +Under very specific conditions, the equations of motion in the Cartesian frame, given by equation eqref:eq:nhexa_transfer_function_cart, can be decoupled. +These conditions are studied in Section ref:ssec:detail_kinematics_cubic_dynamic. \begin{equation}\label{eq:nhexa_transfer_function_cart} \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{T} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{T} \bm{\mathcal{K}} \bm{J} )^{-1} \end{equation} -**** In the frame of the Struts - -In the frame of the struts, the equations of motion eqref:eq:nhexa_transfer_function_struts are well decoupled at low frequency. -This is why most of Stewart platforms are controlled in the frame of the struts: bellow the resonance frequency, the system is decoupled and SISO control may be applied for each strut, independently of the payload being used. +In the frame of the struts, the equations of motion given by equation eqref:eq:nhexa_transfer_function_struts are well decoupled at low frequency. +This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is decoupled and SISO control may be applied for each strut, independently of the payload being used. \begin{equation}\label{eq:nhexa_transfer_function_struts} \frac{\bm{\mathcal{L}}}{\bm{f}}(s) = ( \bm{J}^{-T} \bm{M} \bm{J}^{-1} s^2 + \bm{\mathcal{C}} + \bm{\mathcal{K}} )^{-1} \end{equation} -Coupling between sensors (force sensors, relative position sensor, inertial sensors) in different struts may also be important for decentralized control. -In section ref:ssec:detail_kinematics_decentralized_control, it will be study if the Stewart platform geometry can be optimized to have lower coupling between the struts. +Coupling between sensors (force sensors, relative position sensors, inertial sensors) in different struts may also be important for decentralized control. +In section ref:ssec:detail_kinematics_decentralized_control, it will be studied whether the Stewart platform geometry can be optimized to have lower coupling between the struts. ** Conclusion :PROPERTIES: @@ -1648,12 +1597,11 @@ In section ref:ssec:detail_kinematics_decentralized_control, it will be study if :END: The effects of two changes in the manipulator's geometry, namely the position and orientation of the legs, are summarized in Table ref:tab:detail_kinematics_geometry. -These results could have been easily deduced based on some mechanical principles, but thanks to the kinematic analysis, they can be quantified. - -These trade-offs give some guidelines when choosing the Stewart platform geometry. +These results could have been easily deduced based on mechanical principles, but thanks to the kinematic analysis, they can be quantified. +These trade-offs provide important guidelines when choosing the Stewart platform geometry. #+name: tab:detail_kinematics_geometry -#+attr_latex: :environment tabularx :width 0.9\linewidth :align Xcc +#+attr_latex: :environment tabularx :width 0.8\linewidth :align Xcc #+attr_latex: :center t :booktabs t :float t :font \small #+caption: Effect of a change in geometry on the manipulator's stiffness, force authority and stroke | *Struts* | *Vertically Oriented* | *Increased separation* | @@ -1676,13 +1624,11 @@ These trade-offs give some guidelines when choosing the Stewart platform geometr ** Introduction :ignore: The Cubic configuration for the Stewart platform was first proposed in [[cite:&geng94_six_degree_of_freed_activ]]. -This configuration is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure ref:fig:detail_kinematics_cubic_architecture_examples. +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 size than the cube's edge, as shown in Figure ref:fig:detail_kinematics_cubic_architecture_example. -Practical implementations of such configuration are shown in Figures ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_uqp. - -It is also possible to have the struts length smaller than the cube's edge (Figure ref:fig:detail_kinematics_cubic_architecture_example_small). -An example of such Stewart platform is shown in Figure ref:fig:detail_kinematics_ulb_pz. +Typically, the struts have similar length to the cube's edges, as illustrated in Figure ref:fig:detail_kinematics_cubic_architecture_example. +Practical implementations of such configurations can be observed in Figures ref:fig:detail_kinematics_jpl, ref:fig:detail_kinematics_uw_gsp and ref:fig:detail_kinematics_uqp. +It is also possible to implement designs with strut lengths smaller than the cube's edges (Figure ref:fig:detail_kinematics_cubic_architecture_example_small), as exemplified in Figure ref:fig:detail_kinematics_ulb_pz. #+begin_src matlab :exports none :results none %% Example of a typical "cubic" architecture @@ -1760,23 +1706,14 @@ exportFig('figs/detail_kinematics_cubic_architecture_example_small.pdf', 'width' #+end_figure -A number of properties are attributed to the cubic configuration, which have made this configuration widely popular ([[cite:&geng94_six_degree_of_freed_activ;&preumont07_six_axis_singl_stage_activ;&jafari03_orthog_gough_stewar_platf_microm]]): -- Simple kinematics relationships and dynamical analysis [[cite:&geng94_six_degree_of_freed_activ]] -- Uniform stiffness in all directions [[cite:&hanieh03_activ_stewar]] -- Uniform mobility [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt.8.5.2]] -- Minimization of the cross coupling between actuators and sensors in other struts [[cite:&preumont07_six_axis_singl_stage_activ]]. - This is attributed to the fact that the struts are orthogonal to each other. - This is said to facilitate collocated sensor-actuator control system design, i.e. the implementation of decentralized control [[cite:&geng94_six_degree_of_freed_activ;&thayer02_six_axis_vibrat_isolat_system]]. +Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption [[cite:&geng94_six_degree_of_freed_activ;&preumont07_six_axis_singl_stage_activ;&jafari03_orthog_gough_stewar_platf_microm]]: simplified kinematics relationships and dynamical analysis [[cite:&geng94_six_degree_of_freed_activ]]; uniform stiffness in all directions [[cite:&hanieh03_activ_stewar]]; uniform mobility [[cite:&preumont18_vibrat_contr_activ_struc_fourt_edition, chapt.8.5.2]]; and minimization of the cross coupling between actuators and sensors in different struts [[cite:&preumont07_six_axis_singl_stage_activ]]. +This minimization is attributed to the fact that the struts are orthogonal to each other, and is said to facilitate collocated sensor-actuator control system design, i.e., the implementation of decentralized control [[cite:&geng94_six_degree_of_freed_activ;&thayer02_six_axis_vibrat_isolat_system]]. - -Such properties are studied to see if they are useful for the nano-hexapod and the associated conditions: -- The mobility and stiffness properties of the cubic configuration are studied in Section ref:ssec:detail_kinematics_cubic_static. -- Dynamical decoupling is studied in Section ref:ssec:detail_kinematics_cubic_dynamic -- Decentralized control, important for the NASS, is studied in Section ref:ssec:detail_kinematics_decentralized_control - -As the cubic architecture has some restrictions on the geometry, alternative designs are proposed in Section ref:ssec:detail_kinematics_cubic_design. - -The goal is to determine if the cubic architecture is interesting for the nano-hexapod. +These properties are examined in this section to assess their relevance for the nano-hexapod. +The mobility and stiffness properties of the cubic configuration are analyzed in Section ref:ssec:detail_kinematics_cubic_static. +Dynamical decoupling is investigated in Section ref:ssec:detail_kinematics_cubic_dynamic, while decentralized control, crucial for the NASS, is examined in Section ref:ssec:detail_kinematics_decentralized_control. +Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section ref:ssec:detail_kinematics_cubic_design. +The ultimate objective is to determine the suitability of the cubic architecture for the nano-hexapod. ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -1806,8 +1743,10 @@ The goal is to determine if the cubic architecture is interesting for the nano-h ** Static Properties <> **** Stiffness matrix for the Cubic architecture + +Consider the cubic architecture depicted in Figure ref:fig:detail_kinematics_cubic_schematic_full. 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 eqref:eq:detail_kinematics_cubic_s. +The unit vectors corresponding to the edges of the cube are described by equation eqref:eq:detail_kinematics_cubic_s. \begin{equation}\label{eq:detail_kinematics_cubic_s} \hat{\bm{s}}_1 = \begin{bmatrix} \sqrt{2}/\sqrt{3} \\ 0 \\ 1/\sqrt{3} \end{bmatrix} \quad @@ -1818,14 +1757,6 @@ The unit vectors corresponding to the edges of the cube are described by eqref:e \hat{\bm{s}}_6 = \begin{bmatrix} -1/\sqrt{6} \\ 1/\sqrt{2} \\ 1/\sqrt{3} \end{bmatrix} \end{equation} -Coordinates of the cube's vertices relevant for the top joints, expressed with respect to the cube's center eqref:eq:detail_kinematics_cubic_vertices. - -\begin{equation}\label{eq:detail_kinematics_cubic_vertices} - \tilde{\bm{b}}_1 = \tilde{\bm{b}}_2 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-\sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad - \tilde{\bm{b}}_3 = \tilde{\bm{b}}_4 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{ \sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad - \tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix} -\end{equation} - #+begin_src latex :file detail_kinematics_cubic_schematic_full.pdf :results file \begin{tikzpicture} \begin{scope}[rotate={45}, shift={(0, 0, -4)}] @@ -1990,6 +1921,40 @@ Coordinates of the cube's vertices relevant for the top joints, expressed with r #+end_subfigure #+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. + +\begin{equation}\label{eq:detail_kinematics_cubic_vertices} + \tilde{\bm{b}}_1 = \tilde{\bm{b}}_2 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-\sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad + \tilde{\bm{b}}_3 = \tilde{\bm{b}}_4 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{ \sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad + \tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix} +\end{equation} + +In the case where top joints are positioned at the cube's vertices, a diagonal stiffness matrix is obtained as shown in equation eqref:eq:detail_kinematics_cubic_stiffness. +Translation stiffness is twice the stiffness of the struts, and rotational stiffness is proportional to the square of the cube's size $H_c$. + +\begin{equation}\label{eq:detail_kinematics_cubic_stiffness} + \bm{K}_{\{B\} = \{C\}} = k \begin{bmatrix} + 2 & 0 & 0 & 0 & 0 & 0 \\ + 0 & 2 & 0 & 0 & 0 & 0 \\ + 0 & 0 & 2 & 0 & 0 & 0 \\ + 0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 & 0 \\ + 0 & 0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 \\ + 0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\ + \end{bmatrix} +\end{equation} + +However, typically, the top joints are not placed at the cube's vertices but at positions along the cube's edges (Figure ref:fig:detail_kinematics_cubic_schematic). +In that case, the location of the top joints can be expressed by equation eqref:eq:detail_kinematics_cubic_edges, yet the computed stiffness matrix remains identical to Equation eqref:eq:detail_kinematics_cubic_stiffness. + +\begin{equation}\label{eq:detail_kinematics_cubic_edges} + \bm{b}_i = \tilde{\bm{b}}_i + \alpha \hat{\bm{s}}_i +\end{equation} + + +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 (or torques) applied at the cube's center will induce pure translations (or 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). + #+begin_src matlab %% Analytical formula for Stiffness matrix of the Cubic Stewart platform % Define symbolic variables @@ -2056,39 +2021,12 @@ disp('Analytical Stiffness Matrix:'); pretty(K); #+end_src -In that case (top joints at the cube's vertices), a diagonal stiffness matrix is obtained eqref:eq:detail_kinematics_cubic_stiffness. -Translation stiffness is twice the stiffness of the struts, and rotational stiffness is proportional to the square of the cube's size $H_c$. - -\begin{equation}\label{eq:detail_kinematics_cubic_stiffness} - \bm{K}_{\{B\} = \{C\}} = k \begin{bmatrix} - 2 & 0 & 0 & 0 & 0 & 0 \\ - 0 & 2 & 0 & 0 & 0 & 0 \\ - 0 & 0 & 2 & 0 & 0 & 0 \\ - 0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 & 0 \\ - 0 & 0 & 0 & 0 & \frac{3}{2} H_c^2 & 0 \\ - 0 & 0 & 0 & 0 & 0 & 6 H_c^2 \\ - \end{bmatrix} -\end{equation} - -But typically, the top joints are not placed at the cube's vertices but anywhere along the cube's edges (Figure ref:fig:detail_kinematics_cubic_schematic). -In that case, the location of the top joints can be expressed by eqref:eq:detail_kinematics_cubic_edges. -But the computed stiffness matrix is the same eqref:eq:detail_kinematics_cubic_stiffness. - -\begin{equation}\label{eq:detail_kinematics_cubic_edges} - \bm{b}_i = \tilde{\bm{b}}_i + \alpha \hat{\bm{s}}_i -\end{equation} - -The Stiffness matrix is therefore diagonal when the considered {B} frame is located at the center of the cube. -This corresponds to forces and torques applied on the top platform, but expressed at the center of the cube, and for translations and rotations of the top platform expressed with respect to the cube's center. -We may call this specific location (where the Stiffness matrix is diagonal) the "Center of Stiffness" (in analogy with the "Center of Mass" where the mass matrix is diagonal). - **** Effect of having frame $\{B\}$ off-centered -However, as soon as the location of the A and B frames are shifted from the cube's center, off diagonal elements in the stiffness matrix appear. +When the reference frames $\{A\}$ and $\{B\}$ are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix. -Let's consider here a vertical shift as shown in Figure ref:fig:detail_kinematics_cubic_schematic. -In that case, the stiffness matrix is eqref:eq:detail_kinematics_cubic_stiffness_off_centered. -Off diagonal elements are increasing with the height difference between the cube's center and the considered B frame. +Considering a vertical shift as shown in Figure ref:fig:detail_kinematics_cubic_schematic, the stiffness matrix transforms into that shown in Equation eqref:eq:detail_kinematics_cubic_stiffness_off_centered. +Off-diagonal elements increase proportionally with the height difference between the cube's center and the considered $\{B\}$ frame. \begin{equation}\label{eq:detail_kinematics_cubic_stiffness_off_centered} \bm{K}_{\{B\} \neq \{C\}} = k \begin{bmatrix} @@ -2101,29 +2039,30 @@ Off diagonal elements are increasing with the height difference between the cube \end{bmatrix} \end{equation} +This stiffness matrix structure is characteristic of Stewart platforms exhibiting symmetry, and is not an exclusive property of cubic architectures. +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. -Such structure of the stiffness matrix is very typical with Stewart platform that have some symmetry, but not necessary only for cubic architectures. -Therefore, the stiffness of the cubic architecture is special only when considering a frame located at the center of the cube. -This is not very convenient, as in the vast majority of cases, the interesting frame (where motion are relevant and forces are applied) is located about the top platform. - -Note that the cube's center needs not to be at the "center" of the Stewart platform. +It should be noted that the cube's center need not be at the "center" of the Stewart platform. This can lead to interesting architectures shown in Section ref:ssec:detail_kinematics_cubic_design. +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. +This observation leads to the interesting alternative architectures presented in Section ref:ssec:detail_kinematics_cubic_design. + **** Uniform Mobility -The mobility in translation of the Stewart platform is studied with constant orientation. -Considering limited actuator stroke (i.e. elongation of each strut), the maximum XYZ position that can be reached can be estimated. -The obtained mobility in X,Y,Z directions for the Cubic architecture is shown in Figure ref:fig:detail_kinematics_cubic_mobility_translations. -- It corresponds to a cube, whose axis are aligned with the struts, and the length of the cube's edge is equal to the strut axial stroke. -- We can say that the mobility in not uniform in the XYZ directions, but is uniform in the directions aligned with the cube's edges. - Claims of the cubic architecture having the property of having a translational mobility of a sphere is wrong [[cite:&mcinroy00_desig_contr_flexur_joint_hexap]]. -- Nevertheless, it can be said that the obtained mobility is somehow more uniform than other architecture, as the ones shown in Figure ref:fig:detail_kinematics_mobility_trans. -- Note that the mobility in translation does not depend on the cube's size. +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. -Also show mobility in Rx,Ry,Rz (Figure ref:fig:detail_kinematics_cubic_mobility_rotations): -- More mobility in Rx and Ry than in Rz -- Mobility decreases with the size of the cube +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. +This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure ref:fig:detail_kinematics_mobility_trans. +It is worth noting that the translational mobility properties remain independent of the cube's size. + +The rotational mobility, illustrated in Figure ref:fig:detail_kinematics_cubic_mobility_rotations, exhibit greater achievable angular displacements 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 %% Cubic configuration @@ -2254,9 +2193,11 @@ exportFig('figs/detail_kinematics_cubic_mobility_rotations.pdf', 'width', 'norma <> **** Introduction :ignore: -In this section, the dynamics of the platform in the cartesian frame is studied. +This section examines the dynamics of the cubic architecture in the Cartesian frame. This corresponds to the transfer function from forces and torques $\bm{\mathcal{F}}$ to translations and rotations $\bm{\mathcal{X}}$ of the top platform. -If relative motion sensor are located in each strut ($\bm{\mathcal{L}}$ is measured), the pose $\bm{\mathcal{X}}$ is computed using the Jacobian matrix as shown in Figure ref:fig:detail_kinematics_centralized_control. +When relative motion sensors are integrated in each strut (measuring $\bm{\mathcal{L}}$), the pose $\bm{\mathcal{X}}$ is computed using the Jacobian matrix as shown in Figure ref:fig:detail_kinematics_centralized_control. + +The analysis aims to identify whether the cubic configuration exhibits special properties for control in the Cartesian frame. #+begin_src latex :file detail_kinematics_centralized_control.pdf \begin{tikzpicture} @@ -2283,45 +2224,25 @@ If relative motion sensor are located in each strut ($\bm{\mathcal{L}}$ is measu #+RESULTS: [[file:figs/detail_kinematics_centralized_control.png]] -We want to see if the Stewart platform has some special properties for control in the cartesian frame. - **** Low frequency and High frequency coupling -As was derived during the conceptual design phase, the dynamics from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ is described by eqref:eq:detail_kinematics_transfer_function_cart - -\begin{equation}\label{eq:detail_kinematics_transfer_function_cart} - \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{T} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{T} \bm{\mathcal{K}} \bm{J} )^{-1} -\end{equation} - - -At low frequency: the static behavior of the platform depends on the stiffness matrix eqref:eq:detail_kinematics_transfer_function_cart_low_freq. -In section ref:ssec:detail_kinematics_cubic_static, it was shown that for the cubic configuration, the stiffness matrix is diagonal if frame $\{B\}$ is taken at the cube's center. -In that case, the "cartesian" plant is decoupled at low frequency. - -\begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq} - \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1} -\end{equation} - - -At high frequency, the behavior depends on the mass matrix (evaluated at frame B) eqref:eq:detail_kinematics_transfer_function_high_freq. -To have the mass matrix diagonal, the center of mass of the mobile parts needs to coincide with the B frame and the principal axes of inertia of the body also needs to coincide with the axis of the B frame. - -\begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq} - \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1} -\end{equation} - -To verify that, -- A cubic stewart platform is used with a cylindrical payload on top (Figure ref:fig:detail_kinematics_cubic_payload) -- The transfer functions from F to X are computed for two specific locations of the B frames: - - center of mass: coupled at low frequency due to non diagonal stiffness matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com) - - center of stiffness: coupled at high frequency due to non diagonal mass matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_cok) -- In both cases, similar dynamics for a non-cubic stewart platform would be obtained and the cubic architecture does not show any clear advantage. +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 static behavior of the platform depends on the stiffness matrix eqref:eq:detail_kinematics_transfer_function_cart_low_freq. +In Section ref:ssec:detail_kinematics_cubic_static, it was demonstrated that for the cubic configuration, the stiffness matrix is diagonal if frame $\{B\}$ is positioned at the cube's center. +In this case, the "Cartesian" plant is decoupled at low frequency. +At high frequency, the behavior is governed by the mass matrix (evaluated at frame $\{B\}$) eqref:eq:detail_kinematics_transfer_function_high_freq. +To achieve a diagonal mass matrix, the center of mass of the mobile components must coincide with the $\{B\}$ frame, and the principal axes of inertia must align with the axes of the $\{B\}$ frame. #+name: fig:detail_kinematics_cubic_payload #+caption: Cubic stewart platform with top cylindrical payload #+attr_latex: :width 0.6\linewidth [[file:figs/detail_kinematics_cubic_payload.png]] +To verify these properties, a cubic Stewart platform with a cylindrical payload on top (Figure ref:fig:detail_kinematics_cubic_payload) was analyzed. +Transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ were computed for two specific locations of the $\{B\}$ frames. +When the $\{B\}$ frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_com). +Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure ref:fig:detail_kinematics_cubic_cart_coupling_cok). + #+begin_src matlab %% Input/Output definition of the Simscape model clear io; io_i = 1; @@ -2505,15 +2426,12 @@ exportFig('figs/detail_kinematics_cubic_cart_coupling_cok.pdf', 'width', 'half', **** Payload's CoM at the cube's center -It is natural to try to have the cube's center (center of stiffness) and the center of mass of the moving part coincide at the same location [[cite:&li01_simul_fault_vibrat_isolat_point]]. -To do so, the payload is located below the top platform, such that the center of mass of the moving body is at the cube's center (Figure ref:fig:detail_kinematics_cubic_centered_payload). - -This is what is physically done in [[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]] (Figure ref:fig:detail_kinematics_uw_gsp). - -The obtained dynamics is indeed well decoupled, thanks to the diagonal stiffness matrix and mass matrix at the same time. - -The main issue with this is that usually we want the payload to be located above the top platform, as it is the case for the nano-hexapod. -Indeed, if a similar design than the one shown in Figure ref:fig:detail_kinematics_cubic_centered_payload was used, the x-ray beam will hit the different struts during the rotation of the spindle. +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 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. #+begin_src matlab %% Cubic Stewart platform with payload above the top platform @@ -2610,24 +2528,22 @@ exportFig('figs/detail_kinematics_cubic_cart_coupling_com_cok.pdf', 'width', 'ha **** Conclusion -Some conclusions can be drawn from the above analysis: -- Static Decoupling <=> Diagonal Stiffness matrix <=> {A} and {B} at the cube's center - Can also have static decoupling with non-cubic architecture, if there is some symmetry between the struts. -- Dynamic Decoupling <=> Static Decoupling + CoM of mobile platform coincident with {A} and {B}. - This is very powerful, but requires to have the payload at the cube's center which is very restrictive and often not possible. - This is also not specific to the cubic architecture. -- Same stiffness in XYZ, which can be interesting for some applications. +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. +This property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements. +Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame $\{B\}$. +While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical. +Additionally, the cubic architecture provides uniform stiffness in XYZ directions, which may be advantageous for certain applications. ** Decentralized Control <> **** Introduction :ignore: -This is reasonable to think that as the struts are orthogonal to each other for the cubic architecture, the coupling between the struts should be minimized and it should therefore be especially interesting for decentralized control. +The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control. +This section examines whether the cubic architecture actually demonstrates advantageous properties for decentralized control in the frame of the struts. -In this section, we wish to see if the cubic architecture has indeed some interesting properties related to decentralized control in the frame of the struts. - -Here two sensors integrated in the struts are considered: a displacement sensor and a force sensor. -The control architecture is shown in Figure ref:fig:detail_kinematics_decentralized_control where $\bm{K}_{\mathcal{L}}$ is a diagonal transfer function matrix. +Two sensor types integrated in the struts are considered: displacement sensors and force sensors. +The control architecture is illustrated in Figure ref:fig:detail_kinematics_decentralized_control, where $\bm{K}_{\mathcal{L}}$ represents a diagonal transfer function matrix. #+begin_src latex :file detail_kinematics_decentralized_control.pdf \begin{tikzpicture} @@ -2651,10 +2567,9 @@ The control architecture is shown in Figure ref:fig:detail_kinematics_decentrali #+RESULTS: [[file:figs/detail_kinematics_decentralized_control.png]] -The "strut plant" are compared for two Stewart platforms: -- with cubic architecture shown in Figure ref:fig:detail_kinematics_cubic_payload (page pageref:fig:detail_kinematics_cubic_payload) -- with a Stewart platform shown in Figure ref:fig:detail_kinematics_non_cubic_payload. It has the same payload and strut dynamics than for the cubic architecture. - The struts are oriented more vertically to be far away from the cubic architecture +The obtained plant dynamics in the frame of the struts are compared for two Stewart platforms. +The first employs a cubic architecture shown in Figure ref:fig:detail_kinematics_cubic_payload. +The second uses a non-cubic Stewart platform shown in Figure ref:fig:detail_kinematics_non_cubic_payload, featuring identical payload and strut dynamics but with struts oriented more vertically to differentiate it from the cubic architecture. #+name: fig:detail_kinematics_non_cubic_payload #+caption: Stewart platform with non-cubic architecture @@ -2739,14 +2654,12 @@ G_non_cubic.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ... **** Relative Displacement Sensors -The transfer functions from actuator force included in each strut to the relative motion of the struts are shown in Figure ref:fig:detail_kinematics_decentralized_dL. -As expected from the equations of motion from $\bm{f}$ to $\bm{\mathcal{L}}$ eqref:eq:nhexa_transfer_function_struts, the $6 \times 6$ plants are decoupled at low frequency. +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:nhexa_transfer_function_struts, the $6 \times 6$ plant is decoupled at low frequency. +At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal. -At high frequency, the plant is coupled as the mass matrix projected in the frame of the struts is not diagonal. - -No clear advantage can be seen for the cubic architecture (figure ref:fig:detail_kinematics_cubic_decentralized_dL) as compared to the non-cubic architecture (Figure ref:fig:detail_kinematics_non_cubic_decentralized_dL). - -Note that the resonance frequencies are not the same in both cases as having the struts oriented more vertically changed the stiffness properties of the Stewart platform and hence the frequency of different modes. +No significant advantage is evident for the cubic architecture (Figure ref:fig:detail_kinematics_cubic_decentralized_dL) compared to the non-cubic architecture (Figure ref:fig:detail_kinematics_non_cubic_decentralized_dL). +The resonance frequencies differ between the two cases because the more vertical strut orientation in the non-cubic architecture alters the stiffness properties of the Stewart platform, consequently shifting the frequencies of various modes. #+begin_src matlab :exports none :results none %% Decentralized plant - Actuator force to Strut displacement - Cubic Architecture @@ -2828,11 +2741,9 @@ exportFig('figs/detail_kinematics_cubic_decentralized_dL.pdf', 'width', 'half', **** Force Sensors -Similarly, the transfer functions from actuator force to force sensors included in each strut are extracted both for the cubic and non-cubic Stewart platforms. -The results are shown in Figure ref:fig:detail_kinematics_decentralized_fn. - -The system is well decoupled at high frequency in both cases. -There are no evidence of an advantage of the cubic architecture. +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. #+begin_src matlab :exports none :results none %% Decentralized plant - Actuator force to strut force sensor - Cubic Architecture @@ -2912,30 +2823,23 @@ exportFig('figs/detail_kinematics_cubic_decentralized_fn.pdf', 'width', 'half', **** Conclusion -The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut and thus provides no advantages for decentralized control. -No evidence of specific advantages of the cubic architecture for decentralized control has been found in the literature, despite many claims. +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 in our modeling scenarios, suggesting that the benefits of orthogonal strut arrangement may be more nuanced than commonly described for decentralized control. ** Cubic architecture with Cube's center above the top platform <> **** Introduction :ignore: -As was shown in Section ref:ssec:detail_kinematics_cubic_dynamic, the cubic architecture can have very interesting dynamical properties when the center of mass of the moving body is at the cube's center. +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. +As shown in Section ref:ssec:detail_kinematics_cubic_static, the stiffness matrix is diagonal when the considered $\{B\}$ frame is located at the cube's center. +However, the $\{B\}$ frame is typically positioned above the top platform where forces are applied and displacements are measured. -This is because, both the mass and stiffness matrices are diagonal. -As shown in in section ref:ssec:detail_kinematics_cubic_static, the stiffness matrix is diagonal when the considered B frame is located at the cube's center. +This section proposes modifications to the cubic architecture to enable positioning the payload above the top platform while still leveraging the advantageous dynamical properties of the cubic configuration. -Or, typically the $\{B\}$ frame is taken above the top platform where forces are applied and where displacements are expressed. +Three key parameters define the geometry of the cubic Stewart platform: $H$, the height of the Stewart platform (distance from fixed base to mobile platform); $H_c$, the height of the cube, as shown in Figure ref:fig:detail_kinematics_cubic_schematic_full; and $H_{CoM}$, the height of the center of mass relative to the mobile platform (coincident with the cube's center). -In this section, modifications of the Cubic architectures are proposed in order to be able to have the payload above the top platform while still benefiting from interesting dynamical properties of the cubic architecture. - -There are three key parameters for the geometry of the Cubic Stewart platform: -- $H$ height of the Stewart platform (distance from fix base to mobile platform) -- $H_c$ height of the cube, as shown in Figure ref:fig:detail_kinematics_cubic_schematic_full -- $H_{CoM}$ height of the center of mass with respect to the mobile platform. It is also the cube's center. - -Depending on the considered cube's size $H_c$ compared to $H$ and $H_{CoM}$, different designs are obtained. - -In the three examples shows bellow, $H = 100\,mm$ and $H_{CoM} = 20\,mm$. +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 $H_{CoM} = 20\,mm$. #+begin_src matlab %% Cubic configurations with center of the cube above the top platform @@ -2946,18 +2850,17 @@ FOc = H + MO_B; % Center of the cube with respect to {F} **** Small cube -When the considered cube size $H_c$ is smaller than twice the height of the CoM $H_{CoM}$, the obtained design looks like Figure ref:fig:detail_kinematics_cubic_above_small. +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. \begin{equation}\label{eq:detail_kinematics_cube_small} H_c < 2 H_{CoM} \end{equation} -This is similar to [[cite:&furutani04_nanom_cuttin_machin_using_stewar]], even though it is not mentioned that the system has a cubic configuration. # TODO - Add link to Figure ref:fig:nhexa_stewart_piezo_furutani (page pageref:fig:nhexa_stewart_piezo_furutani) +This configuration is similar to that described in [[cite:&furutani04_nanom_cuttin_machin_using_stewar]], although they do not explicitly identify it as a cubic configuration. +Adjacent struts are parallel to each other, differing from the typical architecture where parallel struts are positioned opposite to each other. -Adjacent struts are parallel to each other, which is quite different from the typical architecture in which parallel struts are opposite to each other. - -This lead to a compact architecture, but as the cube's size is small, the rotational stiffness may be too low. +This approach yields a compact architecture, but the small cube size may result in insufficient rotational stiffness. #+begin_src matlab %% Small cube @@ -3033,13 +2936,13 @@ exportFig('figs/detail_kinematics_cubic_above_small_top.pdf', 'width', 'half', ' **** Medium sized cube -Increasing the cube size with an height close to the stewart platform height leads to an architecture in which the struts are crossing. +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). \begin{equation}\label{eq:detail_kinematics_cube_medium} 2 H_{CoM} < H_c < 2 (H_{CoM} + H) \end{equation} -This is similar to cite:yang19_dynam_model_decoup_contr_flexib (Figure ref:fig:detail_kinematics_yang19 in page pageref:fig:detail_kinematics_yang19), even though it is not cubic (but the struts are crossing). +This configuration resembles the design proposed in [[cite:&yang19_dynam_model_decoup_contr_flexib]] (Figure ref:fig:detail_kinematics_yang19), although their design is not strictly cubic. #+begin_src matlab :exports none :results none %% Example of a cubic architecture with cube's center above the top platform - Medium cube size @@ -3115,7 +3018,7 @@ exportFig('figs/detail_kinematics_cubic_above_medium_top.pdf', 'width', 'half', **** Large cube -When the cube's height is more than twice the platform height added to the CoM height, the architecture shown in Figure ref:fig:detail_kinematics_cubic_above_large is obtained. +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. \begin{equation}\label{eq:detail_kinematics_cube_large} 2 (H_{CoM} + H) < H_c @@ -3258,8 +3161,8 @@ simplify(sqrt(bi_z_H(:,1).^2 + bi_z_H(:,2).^2)) simplify(sqrt(bi_z_0(:,1).^2 + bi_z_0(:,2).^2)) #+end_src -The top joints $\bm{b}_i$ are located on a circle with radius $R_{b_i}$ eqref:eq:detail_kinematics_cube_top_joints. -The bottom joints $\bm{a}_i$ are located on a circle with radius $R_{a_i}$ eqref:eq:detail_kinematics_cube_bot_joints. +In order to determine the approximate size of the platform as a function of +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. \begin{subequations}\label{eq:detail_kinematics_cube_joints} \begin{align} @@ -3268,33 +3171,39 @@ The bottom joints $\bm{a}_i$ are located on a circle with radius $R_{a_i}$ eqref \end{align} \end{subequations} -The size of the platforms increase with the cube's size and the height of the location of the center of mass (also coincident with the cube's center). -The size of the bottom platform also increases with the height of the Stewart platform. - -As the rotational stiffness for the cubic architecture is scaled as the square of the cube's height eqref:eq:detail_kinematics_cubic_stiffness, the cube's size can be determined from the requirements in terms of rotational stiffness. -Then, using eqref:eq:detail_kinematics_cube_joints, the size of the top and bottom platforms can be determined. +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 -For each of the proposed configuration, the Stiffness matrix is diagonal with $k_x = k_y = k_y = 2k$ with $k$ is the stiffness of each strut. -However, the rotational stiffnesses are increasing with the cube's size but the required size of the platform is also increasing, so there is a trade-off here. +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. -We found that we can have a diagonal stiffness matrix using the cubic architecture when $\{A\}$ and $\{B\}$ are located above the top platform. -Depending on the cube's size, 3 different configurations were obtained. +These proposed architectures maintain the fundamental advantages inherent to the cubic configuration, such as uniform stiffness and uniform mobility, while providing favorable dynamical properties when payloads are placed on top of the mobile platform. +This approach allows for practical payload positioning while preserving the desirable control characteristics associated with the cubic architecture, making these configurations potentially useful for applications requiring both specific payload placement and good dynamic performance. ** Conclusion :PROPERTIES: :UNNUMBERED: t :END: -Cubic architecture can be interesting when specific payloads are being used. -In that case, the center of mass of the payload should be placed at the center of the cube. -For the classical cubic architecture, it is often not possible. +The analysis of the cubic architecture for Stewart platforms has 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. -Architectures with the center of the cube about the top platform are proposed to overcome this issue. +Regarding mobility, the translational capabilities of the cubic configuration exhibit uniformity along the directions of the orthogonal struts, rather than complete uniformity in the Cartesian space. +This understanding refines the characterization of cubic architecture mobility commonly presented in literature. -This study was necessary to determine if the Cubic configuration has specific properties that would be interesting for the nano-hexapod. -During this study, it was found that some properties attributed to the cubic configuration (such as uniform mobility and natural decoupling between the struts) were not verified or require more nuances than typically done. +The analysis of decentralized control in the frame of the struts revealed more nuanced results than expected. +While cubic architectures are frequently associated with reduced coupling between actuators and sensors, our comparative study showed that these benefits may be more subtle or context-dependent than commonly described. +Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations appeared similar. + +Fully decoupled dynamics can be achieved when the center of mass of the moving body coincides with the cube's 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. * Nano Hexapod :PROPERTIES: @@ -3303,28 +3212,10 @@ During this study, it was found that some properties attributed to the cubic con <> ** Introduction :ignore: -For the NASS, the chosen frame $\{A\}$ and $\{B\}$ coincide with the sample's point of interest, which is $150\,mm$ above the top platform. -This is where we want to control the sample's position. +Based on previous analysis, this section aims to determine the nano-hexapod geometry. -Requirements: -- The nano-hexapod should fit within a cylinder with radius of $120\,mm$ and with a height of $95\,mm$. -- Based on the measured errors of all the stages of the micro-stations, the required mobility of the nano-hexapod should be (with some safety margins): - It should be able to perform combined translation in any direction of +/-50um. - At any position, it should be able to perform Rx and Ry rotations of +/-50urad -- In terms of stiffness: - Having the resonance frequencies well above the maximum rotational velocity of $2\pi\,\text{rad/s}$ to limit the gyroscopic effects. - Having the resonance below the problematic modes of the micro-station to decouple from the micro-station complex dynamics. -- In terms of dynamics: - - Be able to apply IFF in a decentralized way with good robustness and performances (good damping of modes) - - Having good decoupling for the High authority controller - -The main difficulty for the design optimization of the nano-hexapod, is that the payloads will have various inertia, with masses ranging from 1 to 50kg. -It is therefore not possible to have one geometry that gives good dynamical properties for all the payloads. - -It could have been an option to have a cubic architecture as proposed in section ref:ssec:detail_kinematics_cubic_design, but having the cube's center 150mm above the top platform would have lead to platforms well exceeding the maximum available size. -In that case, each payload would have to be calibrated in inertia before placing on top of the nano-hexapod, which would require a lot of work from the future users. - -Considering the fact that it would not be possible to have the center of mass at the cube's center, the cubic architecture was considered not interesting for the nano-hexapod. +For the NASS, the chosen reference frames $\{A\}$ and $\{B\}$ coincide with the sample's point of interest, which is positioned $150\,mm$ above the top platform. +This is the location where precise control of the sample's position is required, as it is where the x-ray beam is focused. ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -3351,28 +3242,36 @@ Considering the fact that it would not be possible to have the center of mass at <> #+end_src +** Requirements +<> + +The design of the nano-hexapod must satisfy several constraints. +The device should fit within a cylinder with radius of $120\,mm$ and height of $95\,mm$. +Based on the measured errors of all stages of the micro-stations, and incorporating safety margins, the required mobility should enable combined translations in any direction of $\pm 50\,\mu m$. +At any position, the system should be capable of performing $R_x$ and $R_y$ rotations of $\pm 50\,\mu \text{rad}$. +Regarding stiffness, the resonance frequencies should be well above the maximum rotational velocity of $2\pi\,\text{rad/s}$ to minimize gyroscopic effects, while remaining below the problematic modes of the micro-station to ensure decoupling from its complex dynamics. +In terms of dynamics, the design should facilitate implementation of Integral Force Feedback (IFF) in a decentralized manner, and provide good decoupling for the high authority controller in the frame of the struts. + +A significant challenge in optimizing the nano-hexapod design arises from the variety of payloads that will be used, with masses ranging from 1 to 50kg. +This variation in payload characteristics makes it impossible to develop a single geometry that provides optimal dynamical properties for all possible configurations. + ** Obtained Geometry <> -Based on previous analysis: -- The geometry can be optimized to have the wanted trade-off between stiffness in different directions and mobility in different directions -- But as the payloads will be so different in terms of inertia, it was found difficult to optimize the geometry so that the wanted dynamical properties of the nano-hexapod are obtained for all the payloads. +Based on the previous analysis of Stewart platform configurations, while the geometry can be optimized to achieve the desired trade-off between stiffness and mobility in different directions, the wide range of potential payloads complicates the optimization process for obtaining consistent dynamical properties across all usage scenarios. -Therefore, the geometry was chosen by: -- Height between the two plates is 95mm -- Taking both platforms with the maximum size available: - Joints are offset by 15mm from the plate surfaces, and are positioned along a circle with radius 120mm for the fixed joints and 110mm for the mobile joints. -- Make reasonable choice of the angles of the struts. - The positioning angles (Figure ref:fig:detail_kinematics_nano_hexapod_top) are $[255, 285, 15, 45, 135, 165]$ degrees for the top joints and $[220, 320, 340, 80, 100, 200]$ degrees for the bottom joints. +For the nano-hexapod design, the struts were oriented more vertically compared to a cubic architecture due to several important considerations. +First, the requirements in the vertical direction are more stringent than in the horizontal direction. +This vertical strut orientation decreases the amplification factor in the vertical direction, providing greater resolution and reducing the effects of actuator noise. +Second, the micro-station's vertical modes exhibit higher frequencies than its lateral modes. +Therefore, higher resonance frequencies of the nano-hexapod in the vertical direction compared to the horizontal direction enhance the decoupling properties between the micro-station and the nano-hexapod. -Obtained geometry is shown in Figure ref:fig:detail_kinematics_nano_hexapod. -The geometry will be slightly refined during the detailed mechanical design for several reason: easy of mount, manufacturability, ... but will stay close to the defined geometry. +Regarding dynamic properties, particularly for control in the frame of the struts, no specific optimization was implemented since the analysis revealed that the particular geometry has minimal impact on the resulting coupling characteristics. -This geometry will be used for: -- Estimate required actuator stroke (Section ref:ssec:detail_kinematics_nano_hexapod_actuator_stroke) -- Estimate flexible joint stroke (Section ref:ssec:detail_kinematics_nano_hexapod_joint_stroke) -- When performing noise budgeting for the choice of instrumentation -- For control purposes +Consequently, the geometry was selected according to practical constraints. +The height between the two plates is set at $95\,mm$. +Both platforms utilize the maximum available size, with joints offset by $15\,mm$ from the plate surfaces and positioned along circles with radii of $120\,mm$ for the fixed joints and $110\,mm$ for the mobile joints. +The positioning angles, as shown in Figure ref:fig:detail_kinematics_nano_hexapod_top, are $[255,\ 285,\ 15,\ 45,\ 135,\ 165]$ degrees for the top joints and $[220,\ 320,\ 340,\ 80,\ 100,\ 200]$ degrees for the bottom joints. #+begin_src matlab %% Obtained Nano Hexapod Design @@ -3449,20 +3348,30 @@ exportFig('figs/detail_kinematics_nano_hexapod_top.pdf', 'width', 500, 'height', #+end_subfigure #+end_figure +The resulting geometry is illustrated in Figure ref:fig:detail_kinematics_nano_hexapod. +While minor refinements may occur during detailed mechanical design to address manufacturing and assembly considerations, the fundamental geometry will remain consistent with this configuration. +This geometry serves as the foundation for estimating required actuator stroke (Section ref:ssec:detail_kinematics_nano_hexapod_actuator_stroke), determining flexible joint stroke requirements (Section ref:ssec:detail_kinematics_nano_hexapod_joint_stroke), performing noise budgeting for instrumentation selection, and developing control strategies. +# TODO - Add link to sections + +Implementing a cubic architecture as proposed in Section ref:ssec:detail_kinematics_cubic_design was considered. +However, positioning the cube's center $150\,mm$ above the top platform would have resulted in platform dimensions exceeding the maximum available size. +Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the nano-hexapod, ensuring that its center of mass coincides with the cube's center. +Given the impracticality of consistently aligning the center of mass with the cube's center, the cubic architecture was deemed unsuitable for the nano-hexapod application. + ** Required Actuator stroke <> -Now that the geometry is fixed, the required actuator stroke to have the wanted mobility can be computed. +With the geometry established, the actuator stroke necessary to achieve the desired mobility can be determined. -Wanted mobility: -- Combined translations in the xyz directions of +/-50um (basically "cube") -- At any point of the cube, be able to do combined Rx and Ry rotations of +/-50urad -- Rz is always at 0 -- Say that it is frame B with respect to frame A, but it is motion expressed at the point of interest (at the focus point of the light) +The required mobility parameters include combined translations in the XYZ directions of $\pm 50\,\mu m$ (essentially a cubic workspace). +Additionally, at any point within this workspace, combined $R_x$ and $R_y$ rotations of $\pm 50\,\mu \text{rad}$, with $R_z$ maintained at 0, should be possible. -First the minimum actuator stroke to have the wanted mobility is computed. -With the chosen geometry, an actuator stroke of +/-94um is found. -This stroke will be used when choosing the actuator. +Calculations based on the selected geometry indicate that an actuator stroke of $\pm 94\,\mu m$ is required to achieve the desired mobility. +This specification will be used during the actuator selection process. +# TODO - Add link to section + +Figure ref:fig:detail_kinematics_nano_hexapod_mobility illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the nano-hexapod with an actuator stroke of $\pm 94\,\mu m$. +The diagram confirms that the required workspace fits within the system's capabilities. #+begin_src matlab %% Estimate required actuator stroke for the wanted mobility @@ -3507,11 +3416,6 @@ end sprintf('Actuator stroke should be from %.0f um to %.0f um', 1e6*L_min, 1e6*L_max) #+end_src -Considering combined rotations and translations, the wanted mobility and the obtained mobility of the Nano hexapod are shown in Figure ref:fig:detail_kinematics_nano_hexapod_mobility. - -It can be seen that just wanted mobility (displayed as a cube), just fits inside the obtained mobility. -Here the worst case scenario is considered, meaning that whatever the angular position in Rx and Ry (in the range +/-50urad), the top platform can be positioned anywhere inside the cube. - #+begin_src matlab %% Compute mobility in translation with combined angular motion % Direction of motion (spherical coordinates) @@ -3611,20 +3515,20 @@ 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 and computed Mobility +#+caption: Wanted translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume). #+RESULTS: [[file:figs/detail_kinematics_nano_hexapod_mobility.png]] ** Required Joint angular stroke <> -Now that the geometry of the nano-hexapod is fixed and the wanted mobility is know, the flexible joint angular stroke to not limit the achievable workspace is determined. +With the nano-hexapod geometry and mobility requirements established, the flexible joint angular stroke necessary to avoid limiting the achievable workspace can be determined. -Only the bending stroke is considered here. -The torsional stroke of the flexible joints is estimated to be very small, considering that no vertical rotation is expected. +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 fixed and mobile joints are required angular stroke are found to be 1mrad. -This be used to design flexible joints. +The required angular stroke for both fixed and mobile joints is calculated to be $1\,\text{mrad}$. +This specification will guide the design of the flexible joints. +# TODO - Add link to section #+begin_src matlab %% Estimate required actuator stroke for the wanted mobility @@ -3677,12 +3581,19 @@ sprintf('Mobile joint stroke should be %.1f mrad', 1e3*max(max_angles_M)) * Conclusion <> -- quick review of the literature about stewart platform. - lots of different architectures was found -- link was made between the geometry of the stewart platform and its static (stiffness, mobility) and dynamical properties -- most of stewart platforms have a payload with fixed inertia properties, which also for fine optimization -- for the NASS, inertia used for experiments will be very broad, which makes the optimization impossible -- Specific geometry is not found to have a huge impact on performances, especially when considering control in the frame of the struts +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. + +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. + +For the cubic configuration, complete dynamical decoupling in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center, but this arrangement is often impractical for real-world applications. +Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the nano-hexapod due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center. + +For the nano-hexapod design, a key challenge was addressing the wide range of potential payloads (1 to 50kg), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios. +This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors, better matching the micro-station's modal characteristics with higher vertical resonance frequencies, and accommodating the stringent vertical positioning requirements. * Bibliography :ignore: #+latex: \printbibliography[heading=bibintoc,title={Bibliography}] diff --git a/nass-geometry.pdf b/nass-geometry.pdf index 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architecture presented in Section \ref{sec:detail_kinematics_cubic}. -\end{itemize} +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 with \(s_i = [0,\ 0,\ 1]\), a vertical stiffness of \(6k\) is achieved, but with null stiffness in the X and Y directions. +If two struts are aligned along the X axis, two struts along the Y axis, and two struts along the Z axis, then \(\hat{\bm{s}}_i \cdot \hat{\bm{s}}_i^T = 2 \bm{I}_3\), resulting in well-distributed stiffness along all directions. +This configuration corresponds to the cubic architecture presented in Section \ref{sec:detail_kinematics_cubic}. -If struts more vertical (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}): -\begin{itemize} -\item increase vertical stiffness -\item decrease horizontal stiffness -\item increase Rx,Ry stiffness -\item decrease Rz stiffness -\end{itemize} - -Opposite conclusions if struts are not horizontal (Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}). +When struts are oriented more vertically (Figure \ref{fig:detail_kinematics_stewart_mobility_vert_struts}), vertical stiffness increases while 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 (Figure \ref{fig:detail_kinematics_stewart_mobility_hori_struts}). \paragraph{Rotational Stiffness} -The rotational stiffnesses depends 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 B). - -Same orientation but increased distances (bi) by a factor 2 => rotational stiffness increased by factor 4. -Compact stewart platform of Figure \ref{fig:detail_kinematics_stewart_mobility_close_struts} as therefore less rotational stiffness than the Stewart platform of Figure \ref{fig:detail_kinematics_stewart_mobility_space_struts}. +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 \(\{B\}\)). +With the same orientation but increased distances (\(b_i\)) 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} -Having the stiffness matrix \(\bm{K}\) diagonal can be beneficial for control purposes as it would make the plant in the cartesian frame decoupled at low frequency. -This depends on the geometry and on the chosen \{B\} frame. -For specific geometry and chose of B frame, it is possible to have a diagonal K matrix. - -This will be discussed in Section \ref{ssec:detail_kinematics_cubic_static}. +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 \(\{B\}\) frame. +For specific geometry and choice of \(\{B\}\) frame, it is possible to achieve a diagonal \(K\) matrix. +This is discussed in Section \ref{ssec:detail_kinematics_cubic_static}. \section{Dynamical properties} -\paragraph{In the Cartesian Frame} +\label{ssec:detail_kinematics_geometry_dynamics} -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 depends both on the geometry (Jacobian matrix) but also on the payload being placed on top of the platform. - -Under very specific conditions, the equations of motion in the Cartesian frame \eqref{eq:nhexa_transfer_function_cart} can be decoupled. -These are studied in Section \ref{ssec:detail_kinematics_cubic_dynamic}. +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. +Under very specific conditions, the equations of motion in the Cartesian frame, given by equation \eqref{eq:nhexa_transfer_function_cart}, can be decoupled. +These conditions are studied in Section \ref{ssec:detail_kinematics_cubic_dynamic}. \begin{equation}\label{eq:nhexa_transfer_function_cart} \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{T} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{T} \bm{\mathcal{K}} \bm{J} )^{-1} \end{equation} -\paragraph{In the frame of the Struts} - -In the frame of the struts, the equations of motion \eqref{eq:nhexa_transfer_function_struts} are well decoupled at low frequency. -This is why most of Stewart platforms are controlled in the frame of the struts: bellow the resonance frequency, the system is decoupled and SISO control may be applied for each strut, independently of the payload being used. +In the frame of the struts, the equations of motion given by equation \eqref{eq:nhexa_transfer_function_struts} are well decoupled at low frequency. +This is why most Stewart platforms are controlled in the frame of the struts: below the resonance frequency, the system is decoupled and SISO control may be applied for each strut, independently of the payload being used. \begin{equation}\label{eq:nhexa_transfer_function_struts} \frac{\bm{\mathcal{L}}}{\bm{f}}(s) = ( \bm{J}^{-T} \bm{M} \bm{J}^{-1} s^2 + \bm{\mathcal{C}} + \bm{\mathcal{K}} )^{-1} \end{equation} -Coupling between sensors (force sensors, relative position sensor, inertial sensors) in different struts may also be important for decentralized control. -In section \ref{ssec:detail_kinematics_decentralized_control}, it will be study if the Stewart platform geometry can be optimized to have lower coupling between the struts. +Coupling between sensors (force sensors, relative position sensors, inertial sensors) in different struts may also be important for decentralized control. +In section \ref{ssec:detail_kinematics_decentralized_control}, it will be studied whether the Stewart platform geometry can be optimized to have lower coupling between the struts. \section*{Conclusion} The effects of two changes in the manipulator's geometry, namely the position and orientation of the legs, are summarized in Table \ref{tab:detail_kinematics_geometry}. -These results could have been easily deduced based on some mechanical principles, but thanks to the kinematic analysis, they can be quantified. - -These trade-offs give some guidelines when choosing the Stewart platform geometry. +These results could have been easily deduced based on mechanical principles, but thanks to the kinematic analysis, they can be quantified. +These trade-offs provide important guidelines when choosing the Stewart platform geometry. \begin{table}[htbp] \caption{\label{tab:detail_kinematics_geometry}Effect of a change in geometry on the manipulator's stiffness, force authority and stroke} \centering \small -\begin{tabularx}{0.9\linewidth}{Xcc} +\begin{tabularx}{0.8\linewidth}{Xcc} \toprule \textbf{Struts} & \textbf{Vertically Oriented} & \textbf{Increased separation}\\ \midrule @@ -462,13 +371,11 @@ Horizontal rotation stroke & \(\searrow\) & \(\searrow\)\\ \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}. -This configuration is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube, as shown in Figure \ref{fig:detail_kinematics_cubic_architecture_examples}. +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 size than the cube's edge, as shown in Figure \ref{fig:detail_kinematics_cubic_architecture_example}. -Practical implementations of such configuration are shown in Figures \ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_uqp}. - -It is also possible to have the struts length smaller than the cube's edge (Figure \ref{fig:detail_kinematics_cubic_architecture_example_small}). -An example of such Stewart platform is shown in Figure \ref{fig:detail_kinematics_ulb_pz}. +Typically, the struts have similar length to the cube's edges, as illustrated in Figure \ref{fig:detail_kinematics_cubic_architecture_example}. +Practical implementations of such configurations can be observed in Figures \ref{fig:detail_kinematics_jpl}, \ref{fig:detail_kinematics_uw_gsp} and \ref{fig:detail_kinematics_uqp}. +It is also possible to implement designs with strut lengths smaller than the cube's edges (Figure \ref{fig:detail_kinematics_cubic_architecture_example_small}), as exemplified in Figure \ref{fig:detail_kinematics_ulb_pz}. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -487,32 +394,21 @@ An example of such Stewart platform is shown in Figure \ref{fig:detail_kinematic \end{figure} -A number of properties are attributed to the cubic configuration, which have made this configuration widely popular (\cite{geng94_six_degree_of_freed_activ,preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm}): -\begin{itemize} -\item Simple kinematics relationships and dynamical analysis \cite{geng94_six_degree_of_freed_activ} -\item Uniform stiffness in all directions \cite{hanieh03_activ_stewar} -\item Uniform mobility \cite[, chapt.8.5.2]{preumont18_vibrat_contr_activ_struc_fourt_edition} -\item Minimization of the cross coupling between actuators and sensors in other struts \cite{preumont07_six_axis_singl_stage_activ}. -This is attributed to the fact that the struts are orthogonal to each other. -This is said to facilitate collocated sensor-actuator control system design, i.e. the implementation of decentralized control \cite{geng94_six_degree_of_freed_activ,thayer02_six_axis_vibrat_isolat_system}. -\end{itemize} +Several advantageous properties attributed to the cubic configuration have contributed to its widespread adoption \cite{geng94_six_degree_of_freed_activ,preumont07_six_axis_singl_stage_activ,jafari03_orthog_gough_stewar_platf_microm}: simplified kinematics relationships and dynamical analysis \cite{geng94_six_degree_of_freed_activ}; uniform stiffness in all directions \cite{hanieh03_activ_stewar}; uniform mobility \cite[, chapt.8.5.2]{preumont18_vibrat_contr_activ_struc_fourt_edition}; and minimization of the cross coupling between actuators and sensors in different struts \cite{preumont07_six_axis_singl_stage_activ}. +This minimization is attributed to the fact that the struts are orthogonal to each other, and is said to facilitate collocated sensor-actuator control system design, i.e., the implementation of decentralized control \cite{geng94_six_degree_of_freed_activ,thayer02_six_axis_vibrat_isolat_system}. - -Such properties are studied to see if they are useful for the nano-hexapod and the associated conditions: -\begin{itemize} -\item The mobility and stiffness properties of the cubic configuration are studied in Section \ref{ssec:detail_kinematics_cubic_static}. -\item Dynamical decoupling is studied in Section \ref{ssec:detail_kinematics_cubic_dynamic} -\item Decentralized control, important for the NASS, is studied in Section \ref{ssec:detail_kinematics_decentralized_control} -\end{itemize} - -As the cubic architecture has some restrictions on the geometry, alternative designs are proposed in Section \ref{ssec:detail_kinematics_cubic_design}. - -The goal is to determine if the cubic architecture is interesting for the nano-hexapod. +These properties are examined in this section to assess their relevance for the nano-hexapod. +The mobility and stiffness properties of the cubic configuration are analyzed in Section \ref{ssec:detail_kinematics_cubic_static}. +Dynamical decoupling is investigated in Section \ref{ssec:detail_kinematics_cubic_dynamic}, while decentralized control, crucial for the NASS, is examined in Section \ref{ssec:detail_kinematics_decentralized_control}. +Given that the cubic architecture imposes strict geometric constraints, alternative designs are proposed in Section \ref{ssec:detail_kinematics_cubic_design}. +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} + +Consider the cubic architecture depicted in Figure \ref{fig:detail_kinematics_cubic_schematic_full}. 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 \eqref{eq:detail_kinematics_cubic_s}. +The unit vectors corresponding to the edges of the cube are described by equation \eqref{eq:detail_kinematics_cubic_s}. \begin{equation}\label{eq:detail_kinematics_cubic_s} \hat{\bm{s}}_1 = \begin{bmatrix} \sqrt{2}/\sqrt{3} \\ 0 \\ 1/\sqrt{3} \end{bmatrix} \quad @@ -523,14 +419,6 @@ The unit vectors corresponding to the edges of the cube are described by \eqref{ \hat{\bm{s}}_6 = \begin{bmatrix} -1/\sqrt{6} \\ 1/\sqrt{2} \\ 1/\sqrt{3} \end{bmatrix} \end{equation} -Coordinates of the cube's vertices relevant for the top joints, expressed with respect to the cube's center \eqref{eq:detail_kinematics_cubic_vertices}. - -\begin{equation}\label{eq:detail_kinematics_cubic_vertices} - \tilde{\bm{b}}_1 = \tilde{\bm{b}}_2 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-\sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad - \tilde{\bm{b}}_3 = \tilde{\bm{b}}_4 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{ \sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad - \tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix} -\end{equation} - \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} \begin{center} @@ -547,7 +435,15 @@ Coordinates of the cube's vertices relevant for the top joints, expressed with r \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})} \end{figure} -In that case (top joints at the cube's vertices), a diagonal stiffness matrix is obtained \eqref{eq:detail_kinematics_cubic_stiffness}. +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}. + +\begin{equation}\label{eq:detail_kinematics_cubic_vertices} + \tilde{\bm{b}}_1 = \tilde{\bm{b}}_2 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{-\sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad + \tilde{\bm{b}}_3 = \tilde{\bm{b}}_4 = H_c \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{ \sqrt{3}}{\sqrt{2}} \\ \frac{1}{2} \end{bmatrix}, \quad + \tilde{\bm{b}}_5 = \tilde{\bm{b}}_6 = H_c \begin{bmatrix} \frac{-2}{\sqrt{2}} \\ 0 \\ \frac{1}{2} \end{bmatrix} +\end{equation} + +In the case where top joints are positioned at the cube's vertices, a diagonal stiffness matrix is obtained as shown in equation \eqref{eq:detail_kinematics_cubic_stiffness}. Translation stiffness is twice the stiffness of the struts, and rotational stiffness is proportional to the square of the cube's size \(H_c\). \begin{equation}\label{eq:detail_kinematics_cubic_stiffness} @@ -561,25 +457,24 @@ Translation stiffness is twice the stiffness of the struts, and rotational stiff \end{bmatrix} \end{equation} -But typically, the top joints are not placed at the cube's vertices but anywhere along the cube's edges (Figure \ref{fig:detail_kinematics_cubic_schematic}). -In that case, the location of the top joints can be expressed by \eqref{eq:detail_kinematics_cubic_edges}. -But the computed stiffness matrix is the same \eqref{eq:detail_kinematics_cubic_stiffness}. +However, typically, the top joints are not placed at the cube's vertices but at positions along the cube's edges (Figure \ref{fig:detail_kinematics_cubic_schematic}). +In that case, the location of the top joints can be expressed by equation \eqref{eq:detail_kinematics_cubic_edges}, yet the computed stiffness matrix remains identical to Equation \eqref{eq:detail_kinematics_cubic_stiffness}. \begin{equation}\label{eq:detail_kinematics_cubic_edges} \bm{b}_i = \tilde{\bm{b}}_i + \alpha \hat{\bm{s}}_i \end{equation} -The Stiffness matrix is therefore diagonal when the considered \{B\} frame is located at the center of the cube. -This corresponds to forces and torques applied on the top platform, but expressed at the center of the cube, and for translations and rotations of the top platform expressed with respect to the cube's center. -We may call this specific location (where the Stiffness matrix is diagonal) the ``Center of Stiffness'' (in analogy with the ``Center of Mass'' where the mass matrix is diagonal). + +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 (or torques) applied at the cube's center will induce pure translations (or 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} -However, as soon as the location of the A and B frames are shifted from the cube's center, off diagonal elements in the stiffness matrix appear. +When the reference frames \(\{A\}\) and \(\{B\}\) are shifted from the cube's center, off-diagonal elements emerge in the stiffness matrix. -Let's consider here a vertical shift as shown in Figure \ref{fig:detail_kinematics_cubic_schematic}. -In that case, the stiffness matrix is \eqref{eq:detail_kinematics_cubic_stiffness_off_centered}. -Off diagonal elements are increasing with the height difference between the cube's center and the considered B frame. +Considering a vertical shift as shown in Figure \ref{fig:detail_kinematics_cubic_schematic}, the stiffness matrix transforms into that shown in Equation \eqref{eq:detail_kinematics_cubic_stiffness_off_centered}. +Off-diagonal elements increase proportionally with the height difference between the cube's center and the considered \(\{B\}\) frame. \begin{equation}\label{eq:detail_kinematics_cubic_stiffness_off_centered} \bm{K}_{\{B\} \neq \{C\}} = k \begin{bmatrix} @@ -592,33 +487,30 @@ Off diagonal elements are increasing with the height difference between the cube \end{bmatrix} \end{equation} +This stiffness matrix structure is characteristic of Stewart platforms exhibiting symmetry, and is not an exclusive property of cubic architectures. +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. -Such structure of the stiffness matrix is very typical with Stewart platform that have some symmetry, but not necessary only for cubic architectures. -Therefore, the stiffness of the cubic architecture is special only when considering a frame located at the center of the cube. -This is not very convenient, as in the vast majority of cases, the interesting frame (where motion are relevant and forces are applied) is located about the top platform. - -Note that the cube's center needs not to be at the ``center'' of the Stewart platform. +It should be noted that the cube's center need not be at the ``center'' of the Stewart platform. This can lead to interesting architectures shown in Section \ref{ssec:detail_kinematics_cubic_design}. +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. +This observation leads to the interesting alternative architectures presented in Section \ref{ssec:detail_kinematics_cubic_design}. + \paragraph{Uniform Mobility} -The mobility in translation of the Stewart platform is studied with constant orientation. -Considering limited actuator stroke (i.e. elongation of each strut), the maximum XYZ position that can be reached can be estimated. -The obtained mobility in X,Y,Z directions for the Cubic architecture is shown in Figure \ref{fig:detail_kinematics_cubic_mobility_translations}. -\begin{itemize} -\item It corresponds to a cube, whose axis are aligned with the struts, and the length of the cube's edge is equal to the strut axial stroke. -\item We can say that the mobility in not uniform in the XYZ directions, but is uniform in the directions aligned with the cube's edges. -Claims of the cubic architecture having the property of having a translational mobility of a sphere is wrong \cite{mcinroy00_desig_contr_flexur_joint_hexap}. -\item Nevertheless, it can be said that the obtained mobility is somehow more uniform than other architecture, as the ones shown in Figure \ref{fig:detail_kinematics_mobility_trans}. -\item Note that the mobility in translation does not depend on the cube's size. -\end{itemize} +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}. -Also show mobility in Rx,Ry,Rz (Figure \ref{fig:detail_kinematics_cubic_mobility_rotations}): -\begin{itemize} -\item More mobility in Rx and Ry than in Rz -\item Mobility decreases with the size of the cube -\end{itemize} +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. +This configuration still offers more consistent mobility characteristics compared to alternative architectures illustrated in Figure \ref{fig:detail_kinematics_mobility_trans}. +It is worth noting that the translational mobility properties remain independent of the cube's size. + +The rotational mobility, illustrated in Figure \ref{fig:detail_kinematics_cubic_mobility_rotations}, exhibit greater achievable angular displacements 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] \begin{subfigure}{0.48\textwidth} @@ -638,52 +530,25 @@ Also show mobility in Rx,Ry,Rz (Figure \ref{fig:detail_kinematics_cubic_mobility \section{Dynamical Decoupling} \label{ssec:detail_kinematics_cubic_dynamic} -In this section, the dynamics of the platform in the cartesian frame is studied. +This section examines the dynamics of the cubic architecture in the Cartesian frame. This corresponds to the transfer function from forces and torques \(\bm{\mathcal{F}}\) to translations and rotations \(\bm{\mathcal{X}}\) of the top platform. -If relative motion sensor are located in each strut (\(\bm{\mathcal{L}}\) is measured), the pose \(\bm{\mathcal{X}}\) is computed using the Jacobian matrix as shown in Figure \ref{fig:detail_kinematics_centralized_control}. +When relative motion sensors are integrated in each strut (measuring \(\bm{\mathcal{L}}\)), the pose \(\bm{\mathcal{X}}\) is computed using the Jacobian matrix as shown in Figure \ref{fig:detail_kinematics_centralized_control}. + +The analysis aims to identify whether the cubic configuration exhibits special properties for control in the Cartesian frame. \begin{figure}[htbp] \centering \includegraphics[scale=1]{figs/detail_kinematics_centralized_control.png} \caption{\label{fig:detail_kinematics_centralized_control}From Strut coordinate to Cartesian coordinate using the Jacobian matrix} \end{figure} - -We want to see if the Stewart platform has some special properties for control in the cartesian frame. \paragraph{Low frequency and High frequency coupling} -As was derived during the conceptual design phase, the dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) is described by \eqref{eq:detail_kinematics_transfer_function_cart} - -\begin{equation}\label{eq:detail_kinematics_transfer_function_cart} - \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{T} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{T} \bm{\mathcal{K}} \bm{J} )^{-1} -\end{equation} - - -At low frequency: the static behavior of the platform depends on the stiffness matrix \eqref{eq:detail_kinematics_transfer_function_cart_low_freq}. -In section \ref{ssec:detail_kinematics_cubic_static}, it was shown that for the cubic configuration, the stiffness matrix is diagonal if frame \(\{B\}\) is taken at the cube's center. -In that case, the ``cartesian'' plant is decoupled at low frequency. - -\begin{equation}\label{eq:detail_kinematics_transfer_function_cart_low_freq} - \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to 0]{} \bm{K}^{-1} -\end{equation} - - -At high frequency, the behavior depends on the mass matrix (evaluated at frame B) \eqref{eq:detail_kinematics_transfer_function_high_freq}. -To have the mass matrix diagonal, the center of mass of the mobile parts needs to coincide with the B frame and the principal axes of inertia of the body also needs to coincide with the axis of the B frame. - -\begin{equation}\label{eq:detail_kinematics_transfer_function_high_freq} - \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(j \omega) \xrightarrow[\omega \to \infty]{} - \omega^2 \bm{M}^{-1} -\end{equation} - -To verify that, -\begin{itemize} -\item A cubic stewart platform is used with a cylindrical payload on top (Figure \ref{fig:detail_kinematics_cubic_payload}) -\item The transfer functions from F to X are computed for two specific locations of the B frames: -\begin{itemize} -\item center of mass: coupled at low frequency due to non diagonal stiffness matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_com}) -\item center of stiffness: coupled at high frequency due to non diagonal mass matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_cok}) -\end{itemize} -\item In both cases, similar dynamics for a non-cubic stewart platform would be obtained and the cubic architecture does not show any clear advantage. -\end{itemize} +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 static behavior of the platform depends on the stiffness matrix \eqref{eq:detail_kinematics_transfer_function_cart_low_freq}. +In Section \ref{ssec:detail_kinematics_cubic_static}, it was demonstrated that for the cubic configuration, the stiffness matrix is diagonal if frame \(\{B\}\) is positioned at the cube's center. +In this case, the ``Cartesian'' plant is decoupled at low frequency. +At high frequency, the behavior is governed by the mass matrix (evaluated at frame \(\{B\}\)) \eqref{eq:detail_kinematics_transfer_function_high_freq}. +To achieve a diagonal mass matrix, the center of mass of the mobile components must coincide with the \(\{B\}\) frame, and the principal axes of inertia must align with the axes of the \(\{B\}\) frame. \begin{figure}[htbp] \centering @@ -691,6 +556,11 @@ To verify that, \caption{\label{fig:detail_kinematics_cubic_payload}Cubic stewart platform with top cylindrical payload} \end{figure} +To verify these properties, a cubic Stewart platform with a cylindrical payload on top (Figure \ref{fig:detail_kinematics_cubic_payload}) was analyzed. +Transfer functions from \(\bm{\mathcal{F}}\) to \(\bm{\mathcal{X}}\) were computed for two specific locations of the \(\{B\}\) frames. +When the \(\{B\}\) frame was positioned at the center of mass, coupling at low frequency was observed due to the non-diagonal stiffness matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_com}). +Conversely, when positioned at the center of stiffness, coupling occurred at high frequency due to the non-diagonal mass matrix (Figure \ref{fig:detail_kinematics_cubic_cart_coupling_cok}). + \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} \begin{center} @@ -709,15 +579,12 @@ To verify that, \paragraph{Payload's CoM at the cube's center} -It is natural to try to have the cube's center (center of stiffness) and the center of mass of the moving part coincide at the same location \cite{li01_simul_fault_vibrat_isolat_point}. -To do so, the payload is located below the top platform, such that the center of mass of the moving body is at the cube's center (Figure \ref{fig:detail_kinematics_cubic_centered_payload}). - -This is what is physically done in \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} (Figure \ref{fig:detail_kinematics_uw_gsp}). - -The obtained dynamics is indeed well decoupled, thanks to the diagonal stiffness matrix and mass matrix at the same time. - -The main issue with this is that usually we want the payload to be located above the top platform, as it is the case for the nano-hexapod. -Indeed, if a similar design than the one shown in Figure \ref{fig:detail_kinematics_cubic_centered_payload} was used, the x-ray beam will hit the different struts during the rotation of the spindle. +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 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. \begin{figure}[htbp] \begin{subfigure}{0.49\textwidth} @@ -737,24 +604,20 @@ Indeed, if a similar design than the one shown in Figure \ref{fig:detail_kinemat \paragraph{Conclusion} -Some conclusions can be drawn from the above analysis: -\begin{itemize} -\item Static Decoupling <=> Diagonal Stiffness matrix <=> \{A\} and \{B\} at the cube's center -Can also have static decoupling with non-cubic architecture, if there is some symmetry between the struts. -\item Dynamic Decoupling <=> Static Decoupling + CoM of mobile platform coincident with \{A\} and \{B\}. -This is very powerful, but requires to have the payload at the cube's center which is very restrictive and often not possible. -This is also not specific to the cubic architecture. -\item Same stiffness in XYZ, which can be interesting for some applications. -\end{itemize} +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. +This property can also be obtained with non-cubic architectures that exhibit symmetrical strut arrangements. +Dynamic decoupling requires both static decoupling and coincidence of the mobile platform's center of mass with reference frame \(\{B\}\). +While this configuration offers powerful control advantages, it requires positioning the payload at the cube's center, which is highly restrictive and often impractical. +Additionally, the cubic architecture provides uniform stiffness in XYZ directions, which may be advantageous for certain applications. \section{Decentralized Control} \label{ssec:detail_kinematics_decentralized_control} -This is reasonable to think that as the struts are orthogonal to each other for the cubic architecture, the coupling between the struts should be minimized and it should therefore be especially interesting for decentralized control. +The orthogonal arrangement of struts in the cubic architecture suggests a potential minimization of inter-strut coupling, which could theoretically create favorable conditions for decentralized control. +This section examines whether the cubic architecture actually demonstrates advantageous properties for decentralized control in the frame of the struts. -In this section, we wish to see if the cubic architecture has indeed some interesting properties related to decentralized control in the frame of the struts. - -Here two sensors integrated in the struts are considered: a displacement sensor and a force sensor. -The control architecture is shown in Figure \ref{fig:detail_kinematics_decentralized_control} where \(\bm{K}_{\mathcal{L}}\) is a diagonal transfer function matrix. +Two sensor types integrated in the struts are considered: displacement sensors and force sensors. +The control architecture is illustrated in Figure \ref{fig:detail_kinematics_decentralized_control}, where \(\bm{K}_{\mathcal{L}}\) represents a diagonal transfer function matrix. \begin{figure}[htbp] \centering @@ -762,12 +625,9 @@ The control architecture is shown in Figure \ref{fig:detail_kinematics_decentral \caption{\label{fig:detail_kinematics_decentralized_control}From Strut coordinate to Cartesian coordinate using the Jacobian matrix} \end{figure} -The ``strut plant'' are compared for two Stewart platforms: -\begin{itemize} -\item with cubic architecture shown in Figure \ref{fig:detail_kinematics_cubic_payload} (page \pageref{fig:detail_kinematics_cubic_payload}) -\item with a Stewart platform shown in Figure \ref{fig:detail_kinematics_non_cubic_payload}. It has the same payload and strut dynamics than for the cubic architecture. -The struts are oriented more vertically to be far away from the cubic architecture -\end{itemize} +The obtained plant dynamics in the frame of the struts are compared for two Stewart platforms. +The first employs a cubic architecture shown in Figure \ref{fig:detail_kinematics_cubic_payload}. +The second uses a non-cubic Stewart platform shown in Figure \ref{fig:detail_kinematics_non_cubic_payload}, featuring identical payload and strut dynamics but with struts oriented more vertically to differentiate it from the cubic architecture. \begin{figure}[htbp] \centering @@ -776,14 +636,12 @@ The struts are oriented more vertically to be far away from the cubic architectu \end{figure} \paragraph{Relative Displacement Sensors} -The transfer functions from actuator force included in each strut to the relative motion of the struts are shown in Figure \ref{fig:detail_kinematics_decentralized_dL}. -As expected from the equations of motion from \(\bm{f}\) to \(\bm{\mathcal{L}}\) \eqref{eq:nhexa_transfer_function_struts}, the \(6 \times 6\) plants are decoupled at low frequency. +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:nhexa_transfer_function_struts}, the \(6 \times 6\) plant is decoupled at low frequency. +At high frequency, coupling is observed as the mass matrix projected in the strut frame is not diagonal. -At high frequency, the plant is coupled as the mass matrix projected in the frame of the struts is not diagonal. - -No clear advantage can be seen for the cubic architecture (figure \ref{fig:detail_kinematics_cubic_decentralized_dL}) as compared to the non-cubic architecture (Figure \ref{fig:detail_kinematics_non_cubic_decentralized_dL}). - -Note that the resonance frequencies are not the same in both cases as having the struts oriented more vertically changed the stiffness properties of the Stewart platform and hence the frequency of different modes. +No significant advantage is evident for the cubic architecture (Figure \ref{fig:detail_kinematics_cubic_decentralized_dL}) compared to the non-cubic architecture (Figure \ref{fig:detail_kinematics_non_cubic_decentralized_dL}). +The resonance frequencies differ between the two cases because the more vertical strut orientation in the non-cubic architecture alters the stiffness properties of the Stewart platform, consequently shifting the frequencies of various modes. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} @@ -803,11 +661,9 @@ Note that the resonance frequencies are not the same in both cases as having the \paragraph{Force Sensors} -Similarly, the transfer functions from actuator force to force sensors included in each strut are extracted both for the cubic and non-cubic Stewart platforms. -The results are shown in Figure \ref{fig:detail_kinematics_decentralized_fn}. - -The system is well decoupled at high frequency in both cases. -There are no evidence of an advantage of the cubic architecture. +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. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} @@ -827,42 +683,33 @@ There are no evidence of an advantage of the cubic architecture. \paragraph{Conclusion} -The Cubic architecture seems to not have any significant effect on the coupling between actuator and sensors of each strut and thus provides no advantages for decentralized control. -No evidence of specific advantages of the cubic architecture for decentralized control has been found in the literature, despite many claims. +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 in our modeling scenarios, suggesting that the benefits of orthogonal strut arrangement may be more nuanced than commonly described for decentralized control. \section{Cubic architecture with Cube's center above the top platform} \label{ssec:detail_kinematics_cubic_design} -As was shown in Section \ref{ssec:detail_kinematics_cubic_dynamic}, the cubic architecture can have very interesting dynamical properties when the center of mass of the moving body is at the cube's center. +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. +As shown in Section \ref{ssec:detail_kinematics_cubic_static}, the stiffness matrix is diagonal when the considered \(\{B\}\) frame is located at the cube's center. +However, the \(\{B\}\) frame is typically positioned above the top platform where forces are applied and displacements are measured. -This is because, both the mass and stiffness matrices are diagonal. -As shown in in section \ref{ssec:detail_kinematics_cubic_static}, the stiffness matrix is diagonal when the considered B frame is located at the cube's center. +This section proposes modifications to the cubic architecture to enable positioning the payload above the top platform while still leveraging the advantageous dynamical properties of the cubic configuration. -Or, typically the \(\{B\}\) frame is taken above the top platform where forces are applied and where displacements are expressed. +Three key parameters define the geometry of the cubic Stewart platform: \(H\), the height of the Stewart platform (distance from fixed base to mobile platform); \(H_c\), the height of the cube, as shown in Figure \ref{fig:detail_kinematics_cubic_schematic_full}; and \(H_{CoM}\), the height of the center of mass relative to the mobile platform (coincident with the cube's center). -In this section, modifications of the Cubic architectures are proposed in order to be able to have the payload above the top platform while still benefiting from interesting dynamical properties of the cubic architecture. - -There are three key parameters for the geometry of the Cubic Stewart platform: -\begin{itemize} -\item \(H\) height of the Stewart platform (distance from fix base to mobile platform) -\item \(H_c\) height of the cube, as shown in Figure \ref{fig:detail_kinematics_cubic_schematic_full} -\item \(H_{CoM}\) height of the center of mass with respect to the mobile platform. It is also the cube's center. -\end{itemize} - -Depending on the considered cube's size \(H_c\) compared to \(H\) and \(H_{CoM}\), different designs are obtained. - -In the three examples shows bellow, \(H = 100\,mm\) and \(H_{CoM} = 20\,mm\). +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} -When the considered cube size \(H_c\) is smaller than twice the height of the CoM \(H_{CoM}\), the obtained design looks like Figure \ref{fig:detail_kinematics_cubic_above_small}. +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}. \begin{equation}\label{eq:detail_kinematics_cube_small} H_c < 2 H_{CoM} \end{equation} -This is similar to \cite{furutani04_nanom_cuttin_machin_using_stewar}, even though it is not mentioned that the system has a cubic configuration. -Adjacent struts are parallel to each other, which is quite different from the typical architecture in which parallel struts are opposite to each other. +This configuration is similar to that described in \cite{furutani04_nanom_cuttin_machin_using_stewar}, although they do not explicitly identify it as a cubic configuration. +Adjacent struts are parallel to each other, differing from the typical architecture where parallel struts are positioned opposite to each other. -This lead to a compact architecture, but as the cube's size is small, the rotational stiffness may be too low. +This approach yields a compact architecture, but the small cube size may result in insufficient rotational stiffness. \begin{figure}[htbp] \begin{subfigure}{0.36\textwidth} @@ -888,13 +735,13 @@ This lead to a compact architecture, but as the cube's size is small, the rotati \paragraph{Medium sized cube} -Increasing the cube size with an height close to the stewart platform height leads to an architecture in which the struts are crossing. +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}). \begin{equation}\label{eq:detail_kinematics_cube_medium} 2 H_{CoM} < H_c < 2 (H_{CoM} + H) \end{equation} -This is similar to \cite{yang19_dynam_model_decoup_contr_flexib} (Figure \ref{fig:detail_kinematics_yang19} in page \pageref{fig:detail_kinematics_yang19}), even though it is not cubic (but the struts are crossing). +This configuration resembles the design proposed in \cite{yang19_dynam_model_decoup_contr_flexib} (Figure \ref{fig:detail_kinematics_yang19}), although their design is not strictly cubic. \begin{figure}[htbp] \begin{subfigure}{0.36\textwidth} @@ -920,7 +767,7 @@ This is similar to \cite{yang19_dynam_model_decoup_contr_flexib} (Figure \ref{fi \paragraph{Large cube} -When the cube's height is more than twice the platform height added to the CoM height, the architecture shown in Figure \ref{fig:detail_kinematics_cubic_above_large} is obtained. +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. \begin{equation}\label{eq:detail_kinematics_cube_large} 2 (H_{CoM} + H) < H_c @@ -950,8 +797,8 @@ When the cube's height is more than twice the platform height added to the CoM h \paragraph{Platform size} -The top joints \(\bm{b}_i\) are located on a circle with radius \(R_{b_i}\) \eqref{eq:detail_kinematics_cube_top_joints}. -The bottom joints \(\bm{a}_i\) are located on a circle with radius \(R_{a_i}\) \eqref{eq:detail_kinematics_cube_bot_joints}. +In order to determine the approximate size of the platform as a function of +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}. \begin{subequations}\label{eq:detail_kinematics_cube_joints} \begin{align} @@ -960,86 +807,72 @@ The bottom joints \(\bm{a}_i\) are located on a circle with radius \(R_{a_i}\) \ \end{align} \end{subequations} -The size of the platforms increase with the cube's size and the height of the location of the center of mass (also coincident with the cube's center). -The size of the bottom platform also increases with the height of the Stewart platform. - -As the rotational stiffness for the cubic architecture is scaled as the square of the cube's height \eqref{eq:detail_kinematics_cubic_stiffness}, the cube's size can be determined from the requirements in terms of rotational stiffness. -Then, using \eqref{eq:detail_kinematics_cube_joints}, the size of the top and bottom platforms can be determined. +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} -For each of the proposed configuration, the Stiffness matrix is diagonal with \(k_x = k_y = k_y = 2k\) with \(k\) is the stiffness of each strut. -However, the rotational stiffnesses are increasing with the cube's size but the required size of the platform is also increasing, so there is a trade-off here. +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. -We found that we can have a diagonal stiffness matrix using the cubic architecture when \(\{A\}\) and \(\{B\}\) are located above the top platform. -Depending on the cube's size, 3 different configurations were obtained. +These proposed architectures maintain the fundamental advantages inherent to the cubic configuration, such as uniform stiffness and uniform mobility, while providing favorable dynamical properties when payloads are placed on top of the mobile platform. +This approach allows for practical payload positioning while preserving the desirable control characteristics associated with the cubic architecture, making these configurations potentially useful for applications requiring both specific payload placement and good dynamic performance. \section*{Conclusion} -Cubic architecture can be interesting when specific payloads are being used. -In that case, the center of mass of the payload should be placed at the center of the cube. -For the classical cubic architecture, it is often not possible. +The analysis of the cubic architecture for Stewart platforms has 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. -Architectures with the center of the cube about the top platform are proposed to overcome this issue. +Regarding mobility, the translational capabilities of the cubic configuration exhibit uniformity along the directions of the orthogonal struts, rather than complete uniformity in the Cartesian space. +This understanding refines the characterization of cubic architecture mobility commonly presented in literature. -This study was necessary to determine if the Cubic configuration has specific properties that would be interesting for the nano-hexapod. -During this study, it was found that some properties attributed to the cubic configuration (such as uniform mobility and natural decoupling between the struts) were not verified or require more nuances than typically done. +The analysis of decentralized control in the frame of the struts revealed more nuanced results than expected. +While cubic architectures are frequently associated with reduced coupling between actuators and sensors, our comparative study showed that these benefits may be more subtle or context-dependent than commonly described. +Under the conditions analyzed, the coupling characteristics of cubic and non-cubic configurations appeared similar. + +Fully decoupled dynamics can be achieved when the center of mass of the moving body coincides with the cube's 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. \chapter{Nano Hexapod} \label{sec:detail_kinematics_nano_hexapod} -For the NASS, the chosen frame \(\{A\}\) and \(\{B\}\) coincide with the sample's point of interest, which is \(150\,mm\) above the top platform. -This is where we want to control the sample's position. +Based on previous analysis, this section aims to determine the nano-hexapod geometry. -Requirements: -\begin{itemize} -\item The nano-hexapod should fit within a cylinder with radius of \(120\,mm\) and with a height of \(95\,mm\). -\item Based on the measured errors of all the stages of the micro-stations, the required mobility of the nano-hexapod should be (with some safety margins): -It should be able to perform combined translation in any direction of +/-50um. -At any position, it should be able to perform Rx and Ry rotations of +/-50urad -\item In terms of stiffness: -Having the resonance frequencies well above the maximum rotational velocity of \(2\pi\,\text{rad/s}\) to limit the gyroscopic effects. -Having the resonance below the problematic modes of the micro-station to decouple from the micro-station complex dynamics. -\item In terms of dynamics: -\begin{itemize} -\item Be able to apply IFF in a decentralized way with good robustness and performances (good damping of modes) -\item Having good decoupling for the High authority controller -\end{itemize} -\end{itemize} +For the NASS, the chosen reference frames \(\{A\}\) and \(\{B\}\) coincide with the sample's point of interest, which is positioned \(150\,mm\) above the top platform. +This is the location where precise control of the sample's position is required, as it is where the x-ray beam is focused. +\section{Requirements} +\label{ssec:detail_kinematics_nano_hexapod_requirements} -The main difficulty for the design optimization of the nano-hexapod, is that the payloads will have various inertia, with masses ranging from 1 to 50kg. -It is therefore not possible to have one geometry that gives good dynamical properties for all the payloads. +The design of the nano-hexapod must satisfy several constraints. +The device should fit within a cylinder with radius of \(120\,mm\) and height of \(95\,mm\). +Based on the measured errors of all stages of the micro-stations, and incorporating safety margins, the required mobility should enable combined translations in any direction of \(\pm 50\,\mu m\). +At any position, the system should be capable of performing \(R_x\) and \(R_y\) rotations of \(\pm 50\,\mu \text{rad}\). +Regarding stiffness, the resonance frequencies should be well above the maximum rotational velocity of \(2\pi\,\text{rad/s}\) to minimize gyroscopic effects, while remaining below the problematic modes of the micro-station to ensure decoupling from its complex dynamics. +In terms of dynamics, the design should facilitate implementation of Integral Force Feedback (IFF) in a decentralized manner, and provide good decoupling for the high authority controller in the frame of the struts. -It could have been an option to have a cubic architecture as proposed in section \ref{ssec:detail_kinematics_cubic_design}, but having the cube's center 150mm above the top platform would have lead to platforms well exceeding the maximum available size. -In that case, each payload would have to be calibrated in inertia before placing on top of the nano-hexapod, which would require a lot of work from the future users. +A significant challenge in optimizing the nano-hexapod design arises from the variety of payloads that will be used, with masses ranging from 1 to 50kg. +This variation in payload characteristics makes it impossible to develop a single geometry that provides optimal dynamical properties for all possible configurations. -Considering the fact that it would not be possible to have the center of mass at the cube's center, the cubic architecture was considered not interesting for the nano-hexapod. \section{Obtained Geometry} \label{ssec:detail_kinematics_nano_hexapod_geometry} -Based on previous analysis: -\begin{itemize} -\item The geometry can be optimized to have the wanted trade-off between stiffness in different directions and mobility in different directions -\item But as the payloads will be so different in terms of inertia, it was found difficult to optimize the geometry so that the wanted dynamical properties of the nano-hexapod are obtained for all the payloads. -\end{itemize} +Based on the previous analysis of Stewart platform configurations, while the geometry can be optimized to achieve the desired trade-off between stiffness and mobility in different directions, the wide range of potential payloads complicates the optimization process for obtaining consistent dynamical properties across all usage scenarios. -Therefore, the geometry was chosen by: -\begin{itemize} -\item Height between the two plates is 95mm -\item Taking both platforms with the maximum size available: -Joints are offset by 15mm from the plate surfaces, and are positioned along a circle with radius 120mm for the fixed joints and 110mm for the mobile joints. -\item Make reasonable choice of the angles of the struts. -The positioning angles (Figure \ref{fig:detail_kinematics_nano_hexapod_top}) are \([255, 285, 15, 45, 135, 165]\) degrees for the top joints and \([220, 320, 340, 80, 100, 200]\) degrees for the bottom joints. -\end{itemize} +For the nano-hexapod design, the struts were oriented more vertically compared to a cubic architecture due to several important considerations. +First, the requirements in the vertical direction are more stringent than in the horizontal direction. +This vertical strut orientation decreases the amplification factor in the vertical direction, providing greater resolution and reducing the effects of actuator noise. +Second, the micro-station's vertical modes exhibit higher frequencies than its lateral modes. +Therefore, higher resonance frequencies of the nano-hexapod in the vertical direction compared to the horizontal direction enhance the decoupling properties between the micro-station and the nano-hexapod. -Obtained geometry is shown in Figure \ref{fig:detail_kinematics_nano_hexapod}. -The geometry will be slightly refined during the detailed mechanical design for several reason: easy of mount, manufacturability, \ldots{} but will stay close to the defined geometry. +Regarding dynamic properties, particularly for control in the frame of the struts, no specific optimization was implemented since the analysis revealed that the particular geometry has minimal impact on the resulting coupling characteristics. -This geometry will be used for: -\begin{itemize} -\item Estimate required actuator stroke (Section \ref{ssec:detail_kinematics_nano_hexapod_actuator_stroke}) -\item Estimate flexible joint stroke (Section \ref{ssec:detail_kinematics_nano_hexapod_joint_stroke}) -\item When performing noise budgeting for the choice of instrumentation -\item For control purposes -\end{itemize} +Consequently, the geometry was selected according to practical constraints. +The height between the two plates is set at \(95\,mm\). +Both platforms utilize the maximum available size, with joints offset by \(15\,mm\) from the plate surfaces and positioned along circles with radii of \(120\,mm\) for the fixed joints and \(110\,mm\) for the mobile joints. +The positioning angles, as shown in Figure \ref{fig:detail_kinematics_nano_hexapod_top}, are \([255,\ 285,\ 15,\ 45,\ 135,\ 165]\) degrees for the top joints and \([220,\ 320,\ 340,\ 80,\ 100,\ 200]\) degrees for the bottom joints. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} @@ -1057,54 +890,58 @@ This geometry will be used for: \caption{\label{fig:detail_kinematics_nano_hexapod}Obtained architecture for the Nano Hexapod} \end{figure} +The resulting geometry is illustrated in Figure \ref{fig:detail_kinematics_nano_hexapod}. +While minor refinements may occur during detailed mechanical design to address manufacturing and assembly considerations, the fundamental geometry will remain consistent with this configuration. +This geometry serves as the foundation for estimating required actuator stroke (Section \ref{ssec:detail_kinematics_nano_hexapod_actuator_stroke}), determining flexible joint stroke requirements (Section \ref{ssec:detail_kinematics_nano_hexapod_joint_stroke}), performing noise budgeting for instrumentation selection, and developing control strategies. +Implementing a cubic architecture as proposed in Section \ref{ssec:detail_kinematics_cubic_design} was considered. +However, positioning the cube's center \(150\,mm\) above the top platform would have resulted in platform dimensions exceeding the maximum available size. +Additionally, to benefit from the cubic configuration's dynamical properties, each payload would require careful calibration of inertia before placement on the nano-hexapod, ensuring that its center of mass coincides with the cube's center. +Given the impracticality of consistently aligning the center of mass with the cube's center, the cubic architecture was deemed unsuitable for the nano-hexapod application. + \section{Required Actuator stroke} \label{ssec:detail_kinematics_nano_hexapod_actuator_stroke} -Now that the geometry is fixed, the required actuator stroke to have the wanted mobility can be computed. +With the geometry established, the actuator stroke necessary to achieve the desired mobility can be determined. -Wanted mobility: -\begin{itemize} -\item Combined translations in the xyz directions of +/-50um (basically ``cube'') -\item At any point of the cube, be able to do combined Rx and Ry rotations of +/-50urad -\item Rz is always at 0 -\item Say that it is frame B with respect to frame A, but it is motion expressed at the point of interest (at the focus point of the light) -\end{itemize} +The required mobility parameters include combined translations in the XYZ directions of \(\pm 50\,\mu m\) (essentially a cubic workspace). +Additionally, at any point within this workspace, combined \(R_x\) and \(R_y\) rotations of \(\pm 50\,\mu \text{rad}\), with \(R_z\) maintained at 0, should be possible. -First the minimum actuator stroke to have the wanted mobility is computed. -With the chosen geometry, an actuator stroke of +/-94um is found. - -Considering combined rotations and translations, the wanted mobility and the obtained mobility of the Nano hexapod are shown in Figure \ldots{} - -It can be seen that just wanted mobility (displayed as a cube), just fits inside the obtained mobility. -Here the worst case scenario is considered, meaning that whatever the angular position in Rx and Ry (in the range +/-50urad), the top platform can be positioned anywhere inside the cube. +Calculations based on the selected geometry indicate that an actuator stroke of \(\pm 94\,\mu m\) is required to achieve the desired mobility. +This specification will be used during the actuator selection process. +Figure \ref{fig:detail_kinematics_nano_hexapod_mobility} illustrates both the desired mobility (represented as a cube) and the calculated mobility envelope of the nano-hexapod with an actuator stroke of \(\pm 94\,\mu m\). +The diagram confirms that the required workspace fits within the system's capabilities. \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 and computed Mobility} +\caption{\label{fig:detail_kinematics_nano_hexapod_mobility}Wanted translation mobility of the Nano-Hexapod (grey cube) and computed Mobility (red volume).} \end{figure} -Therefore, in Section \ldots{}, the specification for actuator stroke is +/-100um - \section{Required Joint angular stroke} \label{ssec:detail_kinematics_nano_hexapod_joint_stroke} -Now that the mobility of the Stewart platform is know, the corresponding flexible joint stroke can be estimated. +With the nano-hexapod geometry and mobility requirements established, the flexible joint angular stroke necessary to avoid limiting the achievable workspace can be determined. -\begin{itemize} -\item conclude on the required joint angle: 1mrad? -Will be used to design flexible joints. -\end{itemize} +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 \(1\,\text{mrad}\). +This specification will guide the design of the flexible joints. \chapter{Conclusion} \label{sec:detail_kinematics_conclusion} -Inertia used for experiments will be very broad => difficult to optimize the dynamics -Specific geometry is not found to have a huge impact on performances. -Practical implementation is important. +This chapter has explored the optimization of the nano-hexapod geometry for the Nano Active Stabilization System (NASS). -Geometry impacts the static and dynamical characteristics of the Stewart platform. -Considering the design constrains, the slight change of geometry will not significantly impact the obtained results. +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. + +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. + +For the cubic configuration, complete dynamical decoupling in the Cartesian frame can be achieved when the center of mass of the moving body coincides with the cube's center, but this arrangement is often impractical for real-world applications. +Modified cubic architectures with the cube's center positioned above the top platform were proposed as a potential solution, but proved unsuitable for the nano-hexapod due to size constraints and the impracticality of ensuring that different payloads' centers of mass would consistently align with the cube's center. + +For the nano-hexapod design, a key challenge was addressing the wide range of potential payloads (1 to 50kg), which made it impossible to optimize the geometry for consistent dynamic performance across all usage scenarios. +This led to a practical design approach where struts were oriented more vertically than in cubic configurations to address several application-specific needs: achieving higher resolution in the vertical direction by reducing amplification factors, better matching the micro-station's modal characteristics with higher vertical resonance frequencies, and accommodating the stringent vertical positioning requirements. \printbibliography[heading=bibintoc,title={Bibliography}] \end{document}