Rework outline and copy some reports
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@ -113,22 +113,26 @@ Based on:
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Questions:
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- [ ] The APA model should maybe not be used here, same for the nice top and bottom plates. Here the detailed design is not yet performed
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** TODO [#A] Copy relevant parts of reports
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** DONE [#A] Copy relevant parts of reports
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CLOSED: [2025-02-06 Thu 15:27]
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- [ ] Stewart platform presentation: [[file:~/Cloud/meetings/group-meetings-me/2020-01-27-Stewart-Platform-Simscape/2020-01-27-Stewart-Platform-Simscape.org]]
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- [ ] Add some sections from here: [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/index.org]]
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- [X] Stewart platform presentation: [[file:~/Cloud/meetings/group-meetings-me/2020-01-27-Stewart-Platform-Simscape/2020-01-27-Stewart-Platform-Simscape.org]]
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- [X] Add some sections from here: [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/index.org]]
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For instance:
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- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/stewart-architecture.org][stewart architecture]]
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- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/stewart_platform.org::+TITLE: Stewart Platform - Simscape Model]]
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- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/kinematic-study.org][kinematic study]]
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- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/identification.org]]
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Effect of joints stiffnesses
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- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/cubic-configuration.org][cubic configuration]]
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- [ ] Look at the [[file:~/Cloud/work-projects/ID31-NASS/documents/state-of-thesis-2020/index.org][NASS 2020 report]]
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- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/stewart-architecture.org][stewart architecture]]
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- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/stewart_platform.org::+TITLE: Stewart Platform - Simscape Model]]
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- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/kinematic-study.org][kinematic study]]
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- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/identification.org]]
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Effect of joints stiffnesses
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- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/cubic-configuration.org][cubic configuration]]
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Not relevant here: in chapter 2
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- [X] Look at the [[file:~/Cloud/work-projects/ID31-NASS/documents/state-of-thesis-2020/index.org][NASS 2020 report]]
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Sections 5.1, 5.4
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- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/amplified_piezoelectric_stack.org][amplified_piezoelectric_stack]] (Just use 2DoF here)
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- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/nano_hexapod.org][nano_hexapod]] (it seems this report is already after the detailed design phase: yes but some parts could be interesting)
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- [ ] Should the study of effect of flexible joints be included here?
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- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/amplified_piezoelectric_stack.org][amplified_piezoelectric_stack]] (Just use 2DoF here)
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- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/nano_hexapod.org][nano_hexapod]] (it seems this report is already after the detailed design phase: yes but some parts could be interesting)
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*Will also be used in Chapter 2*
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- [X] Should the study of effect of flexible joints be included here?
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*No, considered perfect and then optimized in chapter 2*
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- [X] file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/control-vibration-isolation.org
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** DONE [#A] Make a nice outline
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@ -207,7 +211,13 @@ CLOSED: [2025-02-05 Wed 16:04]
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- control is performed
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- simulations => validation of the concept
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** TODO [#C] First time in the report that we speak about MIMO control ? Or maybe next section!
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** TODO [#A] Make sure the Simulink file for the Stewart platform is working well
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SCHEDULED: <2025-02-06 Thu>
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It should be the exact model reference that will be included in the NASS model.
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** DONE [#C] First time in the report that we speak about MIMO control ? Or maybe next section!
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CLOSED: [2025-02-06 Thu 16:01]
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Maybe should introduce:
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- "MIMO" Root locus
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@ -218,14 +228,16 @@ Or should this be in annexes?
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Maybe say that in this phd-thesis, the focus is not on the control.
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I tried multiple architectures (complementary filters, etc.), but the focus is not on that.
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** QUES [#C] Cubic architecture should be the topic here or in the detailed design?
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** ANSW [#C] Cubic architecture should be the topic here or in the detailed design?
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CLOSED: [2025-02-06 Thu 16:01]
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I suppose that it should be in the detailed design phase.
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(Review about Stewart platform design should be made in Chapter two.)
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Here, just use simple control architecture for general validation (and not optimization).
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** QUES [#C] Should I make a review of control strategies?
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** ANSW [#C] Should I make a review of control strategies?
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CLOSED: [2025-02-06 Thu 16:01]
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Yes it seems to good location for review related to control.
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@ -234,53 +246,58 @@ Control is the frame of the struts, in the cartesian frame (CoM, CoK), modal con
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[[file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org][file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org]]
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** TODO [#C] Compare simscape =linearize= and analytical formula
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** DONE [#C] Compare simscape =linearize= and analytical formula
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CLOSED: [2025-02-06 Thu 16:01]
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- [X] OK for $\omega=0$ (using just the Stiffness matrix)
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- [ ] Should add the mass matrix and compare for all frequencies
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The analytical dynamic model is taken from cite:taghirad13_paral
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** TODO [#C] Output the cubic configuration with clear display of the cube and center of the cube
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** DONE [#C] Output the cubic configuration with clear display of the cube and center of the cube
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CLOSED: [2025-02-06 Thu 16:02]
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[[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/cubic-configuration.org][cubic configuration]]
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** TODO [#C] Make sure the Simulink file for the Stewart platform is working well
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*No, this will be in Chapter 2*
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It should be the exact model reference that will be included in the NASS model.
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** TODO [#C] Maybe make an appendix to present the developed toolbox?
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** CANC [#C] Maybe make an appendix to present the developed toolbox?
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CLOSED: [2025-02-06 Thu 16:02]
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- State "CANC" from "TODO" [2025-02-06 Thu 16:02]
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* Introduction :ignore:
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Introduction:
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- Choice of architecture to do 5DoF control (Section ref:sec:nhexa_platform_review)
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- Stewart platform (Section ref:sec:nhexa_stewart_platform)
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Show what is an hexapod, how we can define its geometry, stiffness, etc...
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Some kinematics: stiffness matrix, mass matrix, etc...
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- Need to model the active vibration platform: multi-body model (Section ref:sec:nhexa_model)
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Explain what we want to capture with this model
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Key elements (plates, joints, struts): for now simplistic model (rigid body elements, perfect joints, ...), but in next section, FEM will be used
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- Control (Section ref:sec:nhexa_control)
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Now that the multi-body model of the micro-station has been developed and validated using dynamical measurements, a model of the active vibration platform can be integrated.
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First, the mechanical architecture of the active platform needs to be carefully chosen.
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In Section ref:sec:nhexa_platform_review, a quick review of active vibration platforms is performed.
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The chosen architecture is the Stewart platform, which is presented in Section ref:sec:nhexa_stewart_platform.
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It is a parallel manipulator that require the use of specific tools to study its kinematics.
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However, to study the dynamics of the Stewart platform, the use of analytical equations is very complex.
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Instead, a multi-body model of the Stewart platform is developed (Section ref:sec:nhexa_model), that can then be easily integrated on top of the micro-station's model.
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From a control point of view, the Stewart platform is a MIMO system with complex dynamics.
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To control such system, it requires several tools to study interaction (Section ref:sec:nhexa_control).
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* Active Vibration Platforms
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<<sec:nhexa_platform_review>>
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** Introduction :ignore:
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*Goals*:
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- Quick review of active vibration platforms (5 or 6DoF) similar to NASS
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- Explain why Stewart platform architecture is chosen
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- Explain what is a Stewart platform (quickly as it will be shown in details in the next section)
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- Quick review of active vibration platforms (5 or 6DoF)
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Active vibration platform with 5DoF or 6DoF?
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Synchrotron applications?
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- Wanted controlled DOF: Y, Z, Ry
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- But because of continuous rotation (key specificity): X,Y,Z,Rx,Ry in the frame of the active platform
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- Literature review? (*maybe more suited for chapter 2*)
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- file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org
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- Talk about flexible joint? Maybe not so much as it should be topic of second chapter.
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Just say that we must of flexible joints that can be defined as 3 to 6DoF joints, and it will be optimize in chapter 2.
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- [[cite:&taghirad13_paral]]
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- For some systems, just XYZ control (stack stages), example: holler
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- For other systems, Stewart platform (ID16a), piezo based
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- Examples of Stewart platforms for general vibration control, some with Piezo, other with Voice coil. IFF, ...
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@ -292,6 +309,7 @@ Synchrotron applications?
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[[file:~/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A0-nass-introduction/nass-introduction.org::*Review of stages with online metrology for Synchrotrons][Review of stages with online metrology for Synchrotrons]]
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- [ ] Talk about external metrology?
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Maybe not the topic here.
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- [ ] Talk about control architecture?
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- [ ] Comparison with the micro-station / NASS
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@ -305,6 +323,17 @@ Synchrotron applications?
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- A review of Stewart platform will be given in Chapter related to the detailed design of the Nano-Hexapod
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#+name: tab:nhexa_serial_vs_parallel
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#+caption: Advantages and Disadvantages of both serial and parallel robots
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#+attr_latex: :environment tabularx :width \linewidth :align lXX
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#+attr_latex: :center t :booktabs t :float t
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| | *Serial Robots* | *Parallel Robots* |
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|--------------------+-----------------+-------------------|
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| Advantages | Large Workspace | High Stiffness |
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| Disadvantages | Low Stiffness | Small Workspace |
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| Kinematic Struture | Open | Closed-loop |
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* The Stewart platform
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:PROPERTIES:
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:HEADER-ARGS:matlab+: :tangle matlab/nhexa_1_stewart_platform.m
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@ -322,6 +351,13 @@ Synchrotron applications?
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- Presentation of tools used to analyze the properties of the Stewart platform => useful for design and control
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The Stewart Platform is very adapted for the NASS application for the following reasons:
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- it is a fully parallel manipulator, thus all the motions errors can be compensated
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- it is very compact compared to a serial manipulator
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- it has high stiffness and good dynamic performances
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The main disadvantage of Stewart platforms is the small workspace when compare the serial manipulators which is not a problem here.
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** Matlab Init :noexport:ignore:
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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@ -346,7 +382,7 @@ Synchrotron applications?
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** Mechanical Architecture
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<<ssec:nhexa_stewart_platform_architecture>>
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file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/stewart-architecture.org
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- [ ] Use this file as a reference: file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/stewart-architecture.org
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Presentation of the typical architecture
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- Explain the different frames, etc...
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@ -355,24 +391,209 @@ Presentation of the typical architecture
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- joints
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- actuators
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[[file:figs/nhexa_stewart_platform_conf.png]]
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The Stewart Platform:
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- Has 6 degrees-of-freedom
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- Is a *Fully* parallel manipulator as the number of actuators is equal to the number of dof
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- Is a *Symmetrical* parallel manipulator as all the struts are the same
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#+name: tab:stewart_platforms_configurations
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#+attr_latex: :environment tabularx :width \linewidth :align cXXX
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#+attr_latex: :center t :booktabs t :float t
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| | *Base Joint* | *Actuator Joint* | *Top Joint* |
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|---------+--------------+------------------+-------------|
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| *6-SPS* | Spherical | Prismatic | Spherical |
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| *6-UPS* | Universal | Prismatic | Spherical |
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Make well defined notations.
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- {F}, {M}
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- si, li, ai, bi, etc.
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- [ ] Make figure with defined frames, joints, etc...
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Maybe can use this figure as an example:
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[[file:/home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A0-nass-introduction/figs/introduction_stewart_du14.svg]]
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These frames are used to describe the relative motion of the two platforms through the position vector ${}^A\bm{P}_B$ of $\{B\}$ expressed in $\{A\}$ and the rotation matrix ${}^A\bm{R}_B$ expressing the orientation of $\{B\}$ with respect to $\{A\}$.
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For the nano-hexapod, these frames are chosen to be located at the theoretical focus point of the X-ray light (xxx mm above the top platform).
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#+begin_quote
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Stewart platforms are generated in multiple steps.
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We define 4 important *frames*:
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- $\{F\}$: Frame fixed to the *Fixed* base and located at the center of its bottom surface.
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This is used to fix the Stewart platform to some support.
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- $\{M\}$: Frame fixed to the *Moving* platform and located at the center of its top surface.
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This is used to place things on top of the Stewart platform.
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- $\{A\}$: Frame fixed to the fixed base.
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It defined the center of rotation of the moving platform.
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- $\{B\}$: Frame fixed to the moving platform.
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The motion of the moving platforms and forces applied to it are defined with respect to this frame $\{B\}$.
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Then, we define the *location of the spherical joints*:
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- $\bm{a}_{i}$ are the position of the spherical joints fixed to the fixed base
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- $\bm{b}_{i}$ are the position of the spherical joints fixed to the moving platform
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We define the *rest position* of the Stewart platform:
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- For simplicity, we suppose that the fixed base and the moving platform are parallel and aligned with the vertical axis at their rest position.
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- Thus, to define the rest position of the Stewart platform, we just have to defined its total height $H$.
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$H$ corresponds to the distance from the bottom of the fixed base to the top of the moving platform.
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From $\bm{a}_{i}$ and $\bm{b}_{i}$, we can determine the *length and orientation of each strut*:
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- $l_{i}$ is the length of the strut
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- ${}^{A}\hat{\bm{s}}_{i}$ is the unit vector align with the strut
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#+end_quote
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** Kinematic Analysis
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<<ssec:nhexa_stewart_platform_kinematics>>
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**** Introduction :ignore:
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Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces that cause the motion.
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The relation between the geometry of the manipulator with the final motion of the moving platform is derived and analyzed.
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*Definition of the geometry of the Stewart Platform*:
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- $\bm{a}_i$: position of the attachment points on the fixed base
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- $\bm{b}_i$: position of moving attachment points
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- $l_i$: length of each limb
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- $\hat{\bm{s}}_i$: unit vector representing the direction of each limb
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#+name: fig:nhexa_stewart_schematic
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#+caption: Geometry of a Stewart Platform
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#+attr_latex: :scale 1
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[[file:figs/nhexa_stewart_schematic.png]]
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**** Loop Closure
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At the displacement level, the *closure of each kinematic loop* can be express in the vector form as
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\[ \vec{AB} = \vec{AA_i} + \vec{A_iB_i} - \vec{BB_i} \quad \text{for } i = 1,2,\dots,n \]
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in which $\vec{AA_i}$ and $\vec{BB_i}$ can be easily obtained from the geometry of the attachment points in the base and in the moving platform.
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The *loop closure* can be written as the unknown pose variables ${}^A\bm{P}$ and ${}^A\bm{R}_B$, the position vectors describing the known geometry of the base and of the moving platform, $\bm{a}_i$ and $\bm{b}_i$, and the limb vector $l_i {}^A\hat{\bm{s}}_i$:
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\begin{equation*}
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{}^A\bm{P} = {}^A\bm{a}_i + l_i{}^A\hat{\bm{s}}_i - {}^A\bm{R}_B {}^B\bm{b}_i \quad \text{for } i=1,2,\dots,n
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\end{equation*}
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**** Inverse Kinematics
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For /inverse kinematic analysis/, it is assumed that the position ${}^A\bm{P}$ and orientation of the moving platform ${}^A\bm{R}_B$ are given and the problem is to obtain the joint variables $\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]^T$.
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This problem can be easily solved using the loop closures.
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The obtain joint variables are:
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\begin{equation*}
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\begin{aligned}
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l_i = &\Big[ {}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + \dots\\
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&2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i \Big]^{1/2}
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\end{aligned}
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\end{equation*}
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If the position and orientation of the platform lie in the feasible workspace, the solution is unique.
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Otherwise, the solution gives complex numbers.
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**** Forward Kinematics
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**** Jacobian Matrix
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In /forward kinematic analysis/, it is assumed that the vector of limb lengths $\bm{L}$ is given and the problem is to find the position ${}^A\bm{P}$ and the orientation ${}^A\bm{R}_B$.
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- Velocity Loop Closure
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- Static Forces
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This is a difficult problem that requires to solve nonlinear equations.
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In a next section, an approximate solution of the forward kinematics problem is proposed for small displacements.
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** The Jacobian Matrix
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**** Introduction :ignore:
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In vector calculus, the Jacobian matrix of a vector-valued function in several variables is the matrix of all its first-order partial derivatives.
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Suppose $\bm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a function such that each of its first-order partial derivatives exist on $\mathbb{R}^n$.
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This function takes a point $\bm{x} \in \mathbb{R}^n$ as input and produces the vector $\bm{f}(\bm{x}) \in \mathbb{R}^m$ as output.
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Then the Jacobian matrix $\bm{J}$ of $\bm{f}$ is defined to be an $n \times m$ matrix, whose its (i,j)'th entry is $J_{ij} = \frac{\partial f_i}{\partial x_j}$.
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The Jacobian matrix is the *linear transformation* that best approximates $\bm{f}$ for points close to $\bm{x}$.
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*Summary*: Linear approximation of a function with several inputs and outputs around a working point.
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**** Jacobian Computation - Velocity Loop Closure
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Let's note:
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- $\bm{\mathcal{L}} = \left[ l_1, l_2, \ldots, l_6 \right]^T$: vector of actuated joint coordinates
|
||||
- $\bm{\mathcal{X}} = \left[ {}^A\bm{P}, \bm{}^A\hat{\bm{s}} \right]^T$: vector of platform motion variables
|
||||
|
||||
$\bm{\mathcal{L}}$ and $\bm{\mathcal{X}}$ are related through a system of /nonlinear algebraic equations/ representing the /kinematic constraints imposed by the struts/, which can be generally written as $f(\bm{\mathcal{L}}, \bm{\mathcal{X}}) = 0$.
|
||||
|
||||
We can differentiate this equation with respect to time and obtain:
|
||||
\begin{equation*}
|
||||
\bm{J}_x \dot{\bm{\mathcal{X}}} = \bm{J}_l \dot{\bm{\mathcal{L}}} \quad \text{where} \quad
|
||||
\bm{J}_x = \frac{\partial f}{\partial \bm{\mathcal{X}}} \quad \text{and} \quad \bm{J}_l = -\frac{\partial f}{\partial \bm{\mathcal{L}}}
|
||||
\end{equation*}
|
||||
|
||||
With:
|
||||
- $\dot{\bm{\mathcal{L}}} = [ \dot{l}_1, \dot{l}_2, \dot{l}_3, \dot{l}_4, \dot{l}_5, \dot{l}_6 ]^T$
|
||||
- $\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T$:
|
||||
|
||||
The *general Jacobian matrix* is defined as:
|
||||
\begin{equation*}
|
||||
\dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}} \quad \text{with} \quad \bm{J} = {\bm{J}_l}^{-1} \bm{J}_x
|
||||
\end{equation*}
|
||||
|
||||
The *velocity loop closures* are used for *obtaining the Jacobian matrices* in a straightforward manner:
|
||||
\begin{align*}
|
||||
{}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i = l_i {}^A\hat{\bm{s}}_i + {}^A\bm{a}_i
|
||||
& \underset{\frac{\partial}{\partial t}}{\rightarrow}
|
||||
{}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i \\
|
||||
& \Leftrightarrow\hat{\bm{s}}_i {}^A\bm{v}_p + ({}^A\bm{b}_i \times \hat{\bm{s}}_i) {}^A\bm{\omega} = \dot{l}_i
|
||||
\end{align*}
|
||||
|
||||
We can rearrange the equations in a matrix form:
|
||||
\[ \dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}} \quad \text{with} \ \dot{\bm{\mathcal{L}}} = [ \dot{l}_1 \ \dots \ \dot{l}_6 ]^T \ \text{and} \ \dot{\bm{\mathcal{X}}} = [{}^A\bm{v}_p ,\ {}^A\bm{\omega}]^T \]
|
||||
|
||||
\begin{equation}
|
||||
\bm{J} = \begin{bmatrix}
|
||||
{\hat{\bm{s}}_1}^T & (\bm{b}_1 \times \hat{\bm{s}}_1)^T \\
|
||||
{\hat{\bm{s}}_2}^T & (\bm{b}_2 \times \hat{\bm{s}}_2)^T \\
|
||||
{\hat{\bm{s}}_3}^T & (\bm{b}_3 \times \hat{\bm{s}}_3)^T \\
|
||||
{\hat{\bm{s}}_4}^T & (\bm{b}_4 \times \hat{\bm{s}}_4)^T \\
|
||||
{\hat{\bm{s}}_5}^T & (\bm{b}_5 \times \hat{\bm{s}}_5)^T \\
|
||||
{\hat{\bm{s}}_6}^T & (\bm{b}_6 \times \hat{\bm{s}}_6)^T
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
$\bm{J}$ then depends only on:
|
||||
- $\hat{\bm{s}}_i$ the orientation of the limbs expressed in $\{A\}$
|
||||
- $\bm{b}_i$ the position of the joints with respect to $O_B$ and express in $\{A\}$
|
||||
|
||||
**** Approximate solution of the Forward and Inverse Kinematic problems
|
||||
|
||||
For small displacements mobile platform displacement $\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T$ around $\bm{\mathcal{X}}_0$, the associated joint displacement can be computed using the Jacobian (approximate solution of the inverse kinematic problem):
|
||||
\begin{equation*}
|
||||
\delta\bm{\mathcal{L}} = \bm{J} \delta\bm{\mathcal{X}}
|
||||
\end{equation*}
|
||||
|
||||
Similarly, for small joint displacements $\delta\bm{\mathcal{L}} = [ \delta l_1,\ \dots,\ \delta l_6 ]^T$ around $\bm{\mathcal{L}}_0$, it is possible to find the induced small displacement of the mobile platform (approximate solution of the forward kinematic problem):
|
||||
\begin{equation*}
|
||||
\delta\bm{\mathcal{X}} = \bm{J}^{-1} \delta\bm{\mathcal{L}}
|
||||
\end{equation*}
|
||||
|
||||
These two relations solve the forward and inverse kinematic problems for small displacement in a /approximate/ way.
|
||||
As the inverse kinematic can be easily solved exactly this is not much useful, however, as the forward kinematic problem is difficult to solve, this approximation can be very useful for small displacements.
|
||||
|
||||
**** Range validity of the approximate inverse kinematics
|
||||
|
||||
- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/kinematic-study.org::*Estimation of the range validity of the approximate inverse kinematics][Estimation of the range validity of the approximate inverse kinematics]]
|
||||
|
||||
**** Static Forces
|
||||
The *principle of virtual work* states that the total virtual work, $\delta W$, done by all actuators and external forces is equal to zero:
|
||||
|
||||
\begin{align*}
|
||||
\delta W &= \bm{\tau}^T \delta \bm{\mathcal{L}} - \bm{\mathcal{F}}^T \delta \bm{\mathcal{X}}\\
|
||||
& = 0
|
||||
\end{align*}
|
||||
|
||||
|
||||
If we note:
|
||||
- $\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T$: vector of actuator forces applied in each strut
|
||||
- $\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T$: external force/torque action on the mobile platform at $\bm{O}_B$
|
||||
|
||||
From the definition of the Jacobian ($\delta \bm{\mathcal{L}} = \bm{J} \cdot \delta \bm{\mathcal{X}}$), we have $\left( \bm{\tau}^T \bm{J} - \bm{\mathcal{F}}^T \right) \delta \bm{\mathcal{X}} = 0$ that holds for any $\delta \bm{\mathcal{X}}$, hence:
|
||||
|
||||
|
||||
\[ \bm{\tau}^T \bm{J} - \bm{\mathcal{F}}^T = 0 \quad \Rightarrow \quad \tcmbox{\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}} \quad \text{and} \quad \tcmbox{\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}} \]
|
||||
|
||||
**** Singularities
|
||||
|
||||
@ -385,6 +606,35 @@ How stiffness varies with orientation of struts.
|
||||
Same with stroke?
|
||||
Or maybe in the detailed chapter?
|
||||
|
||||
The stiffness of the actuator $k_i$ links the applied actuator force $\delta \tau_i$ and the corresponding small deflection $\delta l_i$:
|
||||
\begin{equation*}
|
||||
\tau_i = k_i \delta l_i, \quad i = 1,\ \dots,\ 6
|
||||
\end{equation*}
|
||||
If we combine these 6 relations:
|
||||
\begin{equation*}
|
||||
\bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}} \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right]
|
||||
\end{equation*}
|
||||
|
||||
Substituting $\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}$ and $\delta \bm{\mathcal{L}} = \bm{J} \cdot \delta \bm{\mathcal{X}}$ gives
|
||||
\begin{equation*}
|
||||
\bm{\mathcal{F}} = \bm{J}^T \mathcal{K} \bm{J} \cdot \delta \bm{\mathcal{X}}
|
||||
\end{equation*}
|
||||
And then we identify the stiffness matrix $\bm{K}$:
|
||||
\begin{equation*}
|
||||
\bm{K} = \bm{J}^T \mathcal{K} \bm{J}
|
||||
\end{equation*}
|
||||
|
||||
If the stiffness matrix $\bm{K}$ is inversible, the *compliance matrix* of the manipulator is defined as
|
||||
\begin{equation*}
|
||||
\bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1}
|
||||
\end{equation*}
|
||||
|
||||
The compliance matrix of a manipulator shows the mapping of the moving platform wrench applied at $\bm{O}_B$ to its small deflection by
|
||||
\begin{equation*}
|
||||
\delta \bm{\mathcal{X}} = \bm{C} \cdot \bm{\mathcal{F}}
|
||||
\end{equation*}
|
||||
|
||||
|
||||
** Dynamic Analysis
|
||||
<<ssec:nhexa_stewart_platform_dynamics>>
|
||||
|
||||
@ -397,6 +647,8 @@ But say that if we want to model more complex cases, it becomes impractical (cit
|
||||
:UNNUMBERED: t
|
||||
:END:
|
||||
|
||||
Dynamic analysis of parallel manipulators presents an *inherent complexity due to their closed-loop structure and kinematic constraints*.
|
||||
|
||||
All depends on the geometry.
|
||||
Reasonable choice of geometry is made in chapter 1.
|
||||
Optimization of the geometry will be made in chapter 2.
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user