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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)
- [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/amplified_piezoelectric_stack.org][amplified_piezoelectric_stack]] (Just use 2DoF here)
- [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)
*Will also be used in Chapter 2*
- [X] Should the study of effect of flexible joints be included here?
*No, considered perfect and then optimized in chapter 2*
Maybe no sections, just a review discussing several aspect of the platforms.
1. Review of active vibration platforms (focused on Synchrotron applications)
2. Serial and Parallel Architecture: advantages and disadvantages of both
3. Which architecture => Parallel manipulator? Why *Stewart platform*?
*2 - The Stewart Platform*:
Introduction: some history about Stewart platform and why it is so used
1. Architecture (plates, struts, joints)
2. Kinematics and Jacobian
4. Static Analysis
5. Dynamic Analysis: very complex => multi-body model
For instance, compute the plant for massless struts and perfect joints (will be compared with Simscape model).
But say that if we want to model more complex cases, it becomes impractical (cite papers).
*3 - Multi-Body model of the Stewart platform*:
Introduction: Complex dynamics => analytical formulas can be complex => Choose to study the dynamics using a multi-body model
1. Model definition: (Matlab Toolbox), frames, inertias of parts, stiffnesses, struts, etc...
2. Joints: perfect 2dof/3dof (+ mass-less)
3. Actuators: APA + Encoder (mass-less)
4. Nano-Hexapod: definition of each part + Plant with defined inputs/outputs (force sensor, relative displacement sensor, etc...)
Compare with analytical formulas (see number of states)
*4 - Control of the Stewart Platform*:
Introduction: MIMO control => much more complex than SISO control because of interaction. Possible to ignore interaction when good decoupling (important to have tools to study interaction)
1. Centralized and Decentralized Control
2. Decoupling Control / Choice of control space file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org
Estimate coupling: RGA
- Jacobian matrices, CoK, CoM, control in the frame of the struts, ...
- Discussion of cubic architecture (quick, as it is going to be in detailed in chapter 2)
- SVD, Modal, ...
3. Active Damping: decentralized IFF
Guaranteed stability?
For decentralized control: "MIMO root locus"
How to optimize the added damping to all modes?
4. HAC-LAC
Stability of closed-loop: Nyquist (main advantage: possible to do with experimental FRF)
*Conclusion*:
- Configurable Stewart platform model
- Will be included in the multi-body model of the micro-station => nass multi body model
** DONE [#A] Location of this report in the complete thesis
CLOSED: [2025-02-05 Wed 16:04]
*Before the report* (assumptions):
- Uniaxial model: no stiff actuator, HAC-LAC strategy
- Rotating model:
Soft actuators are problematic due to gyroscopic effects
Use moderately stiff (1um/N).
IFF can be applied with APA architecture
- Model of Micro-station is ready
*In this report*:
- Goal: build a flexible (i.e. configurable) multi-body model of a Stewart platform that will be used in the next section to perform dynamical analysis and simulate experiments with the complete NASS
- Here, I propose to work with "perfect" stewart platforms:
- almost mass-less struts
- joints with zero stiffness in free DoFs (i.e. 2-DoF and 3-DoF joints)
- Presentation of Stewart platforms (Literature review about stewart platforms will be done in chapter 2)
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.
First, the mechanical architecture of the active platform needs to be carefully chosen.
In Section ref:sec:nhexa_platform_review, a quick review of active vibration platforms is performed.
The chosen architecture is the Stewart platform, which is presented in Section ref:sec:nhexa_stewart_platform.
It is a parallel manipulator that require the use of specific tools to study its kinematics.
However, to study the dynamics of the Stewart platform, the use of analytical equations is very complex.
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.
From a control point of view, the Stewart platform is a MIMO system with complex dynamics.
To control such system, it requires several tools to study interaction (Section ref:sec:nhexa_control).
- For some systems, just XYZ control (stack stages), example: holler
- For other systems, Stewart platform (ID16a), piezo based
- Examples of Stewart platforms for general vibration control, some with Piezo, other with Voice coil. IFF, ...
Show different geometry configuration
- DCM: tripod?
** Active vibration control of sample stages
[[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]]
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 $\{A\}$.
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).
#+begin_quote
Stewart platforms are generated in multiple steps.
We define 4 important *frames*:
- $\{F\}$: Frame fixed to the *Fixed* base and located at the center of its bottom surface.
This is used to fix the Stewart platform to some support.
- $\{M\}$: Frame fixed to the *Moving* platform and located at the center of its top surface.
This is used to place things on top of the Stewart platform.
- $\{A\}$: Frame fixed to the fixed base.
It defined the center of rotation of the moving platform.
- $\{B\}$: Frame fixed to the moving platform.
The motion of the moving platforms and forces applied to it are defined with respect to this frame $\{B\}$.
Then, we define the *location of the spherical joints*:
- $\bm{a}_{i}$ are the position of the spherical joints fixed to the fixed base
- $\bm{b}_{i}$ are the position of the spherical joints fixed to the moving platform
We define the *rest position* of the Stewart platform:
- For simplicity, we suppose that the fixed base and the moving platform are parallel and aligned with the vertical axis at their rest position.
- Thus, to define the rest position of the Stewart platform, we just have to defined its total height $H$.
$H$ corresponds to the distance from the bottom of the fixed base to the top of the moving platform.
From $\bm{a}_{i}$ and $\bm{b}_{i}$, we can determine the *length and orientation of each strut*:
- $l_{i}$ is the length of the strut
- ${}^{A}\hat{\bm{s}}_{i}$ is the unit vector align with the strut
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.
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 $l_i {}^A\hat{\bm{s}}_i$:
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 $\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]^T$.
This problem can be easily solved using the loop closures.
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 ${}^A\bm{R}_B$.
This is a difficult problem that requires to solve nonlinear equations.
In a next section, an approximate solution of the forward kinematics problem is proposed for small displacements.
** The Jacobian Matrix
**** Introduction :ignore:
In vector calculus, the Jacobian matrix of a vector-valued function in several variables is the matrix of all its first-order partial derivatives.
Suppose $\bm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is a function such that each of its first-order partial derivatives exist on $\mathbb{R}^n$.
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.
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 $J_{ij} = \frac{\partial f_i}{\partial x_j}$.
The Jacobian matrix is the *linear transformation* that best approximates $\bm{f}$ for points close to $\bm{x}$.
*Summary*: Linear approximation of a function with several inputs and outputs around a working point.
$\bm{\mathcal{L}}$ 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:
- $\hat{\bm{s}}_i$ 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):
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):
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:
- linked to the sensor position relative to the actuators
- linked to the fact that sensors and actuators pairs are "independent" or each other (related to the control architecture, not because there is no coupling)
- When can decentralized control be used and when centralized control is necessary?
- Jacobian matrices, CoK, CoM, control in the frame of the struts, SVD, Modal, ...
- Combined CoM and CoK => Discussion of cubic architecture ? (quick, as it is going to be in detailed in chapter 2)
- Explain also the link with the setpoint: it is interesting to have the controller in the frame of the performance variables
Also speak about disturbances? (and how disturbances can be mixed to different outputs due to control and interaction)
- Table that summarizes the trade-off for each strategy
- Say that in this study, we will do the control in the frame of the struts for simplicity (even though control in the cartesian frame was also tested)
For decentralized control: "MIMO root locus" can be used to estimate the damping / optimal gain
Poles and converging towards /transmission zeros/
How to optimize the added damping to all modes?
- [ ] Add some papers citations
Compute:
- [ ] Plant dynamics
- [ ] Root Locus
** MIMO High-Authority Control - Low-Authority Control
<<ssec:nhexa_control_hac_lac>>
Compute:
- [ ] compare open-loop and damped plant (outputs are the encoders)
- [ ] Implement decentralized control?
- [ ] Check stability:
- Characteristic Loci: Eigenvalues of $G(j\omega)$ plotted in the complex plane
- Generalized Nyquist Criterion: If $G(s)$ has $p_0$ unstable poles, then the closed-loop system with return ratio $kG(s)$ is stable if and only if the characteristic loci of $kG(s)$, taken together, encircle the point $-1$, $p_0$ times anti-clockwise, assuming there are no hidden modes
- [ ] Show some performance metric? For instance compliance?
