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Start to write Stewart section

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Thomas Dehaeze 2025-02-07 18:01:00 +01:00
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3
preamble.tex
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@ -12,5 +12,8 @@
\setabbreviationstyle[acronym]{long-short}
\setglossarystyle{long-name-desc}
\usepackage{amssymb}
\usepackage{amsmath}
\makeindex
\makeglossaries

532
simscape-nano-hexapod.org
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@ -212,9 +212,63 @@ CLOSED: [2025-02-05 Wed 16:04]
- simulations => validation of the concept
** TODO [#A] Make sure the Simulink file for the Stewart platform is working well
SCHEDULED: <2025-02-06 Thu>
SCHEDULED: <2025-02-08 Sat>
It should be the exact model reference that will be included in the NASS model.
It should be the exact model reference that will be included in the NASS model (referenced subsystem).
- [X] Check what was already done for the toolbox
- [ ] Same parameters for the APA as in previous model (1N/um ?)
*kn = 1N/um*
nano hexapod mass: *15kg*
cn = 2*0.01*sqrt((ms + mn)*kn)
=> depends on the sample's mass: between 30 and 60, *cn = 50N/(m/s) seems reasonable*
*real mass of the top platform is 5kg*
- [X] Use similar strategy that for the NASS simulation (these .mat files, etc.)
- [X] Similar interfaces: {F}, {M},
- inputs: 6 actuator inputs
- output 1: 6 encoders
- output 2: 6 force sensors
- [X] joints configurable with
- [X] 2dof
- [X] 3dof
- [X] 4dof
- [X] flexible => will be added for chapter 2
- [-] actuator:
- [X] 1dof
- [X] 2dof (APA)
- [ ] FEM => will be added for chapter 2
- [X] plates: cylindrical or .STEP
Only cylindrical for now
- [X] Add payload:
- size: height, diameter/radius
- Weight
- [ ] Control configuration
- [X] Log configuration
- [ ] *Do I want to be able to change each individual parameter value of each strut => no*
** TODO [#C] Better understand principle of virtual work
[[*Static Forces][Static Forces]]
Better understand this: https://en.wikipedia.org/wiki/Virtual_work
Also add link or explanation for this equation.
** DONE [#B] Define the geometry for the simplified nano-hexapod
CLOSED: [2025-02-06 Thu 18:56]
- [X] Micro-Hexapod radius: 150mm
- [X] Top plate radius: 150mm
- [X] Total height should match height of the nano-hexapod => 95mm
- [X] Location of joints: 20mm above/below bottom/top surfaces
- [X] Joints on a radius of 120mm and 110mm at the top
- [X] Angles of the joints:
- Bottom: +/- 10 degrees (50 deg offset)
- Top: +/-15 degrees (45 deg offset)
- [X] Check order of the struts to match the (final) nano-hexapod model
- Bottom:
- 190, 290, 310, 50, 70, 170
- top: 255, 285, 15, 45, 135, 165
** DONE [#C] First time in the report that we speak about MIMO control ? Or maybe next section!
CLOSED: [2025-02-06 Thu 16:01]
@ -375,6 +429,10 @@ The main disadvantage of Stewart platforms is the small workspace when compare t
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src
@ -382,132 +440,92 @@ The main disadvantage of Stewart platforms is the small workspace when compare t
** Mechanical Architecture
<<ssec:nhexa_stewart_platform_architecture>>
- [ ] Use this file as a reference: file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/stewart-architecture.org
A Stewart manipulator consists of two platforms connected by six struts (Figure ref:fig:nhexa_stewart_architecture).
Each strut is connected to the fixed and the mobile platforms with a joint.
Typically, a universal joint is used on one side while a spherical joint is used on the other side[fn:1].
In the strut, there is an active element working as a prismatic joint.
Presentation of the typical architecture
- Explain the different frames, etc...
- explain key elements:
- two plates
- joints
- actuators
#+name: fig:nhexa_stewart_architecture
#+caption: Schematical representation of the Stewart platform architecture.
[[file:figs/nhexa_stewart_architecture.png]]
[[file:figs/nhexa_stewart_platform_conf.png]]
Such architecture allows to move the mobile platform with respect to the fixed platform in 6 degrees-of-freedom.
It is therefore a /fully/ parallel manipulator as the number of actuators is equal to the number of DoF.
It is also a symmetrical parallel manipulator as typically all the struts are identical.
The Stewart Platform:
- Has 6 degrees-of-freedom
- Is a *Fully* parallel manipulator as the number of actuators is equal to the number of dof
- Is a *Symmetrical* parallel manipulator as all the struts are the same
#+name: tab:stewart_platforms_configurations
#+attr_latex: :environment tabularx :width \linewidth :align cXXX
#+attr_latex: :center t :booktabs t :float t
| | *Base Joint* | *Actuator Joint* | *Top Joint* |
|---------+--------------+------------------+-------------|
| *6-SPS* | Spherical | Prismatic | Spherical |
| *6-UPS* | Universal | Prismatic | Spherical |
Make well defined notations.
- {F}, {M}
- si, li, ai, bi, etc.
- [ ] Make figure with defined frames, joints, etc...
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.
In order to study the Stewart platform, four important frames are typically defined:
- $\{F\}$: Frame fixed on the 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.
- $\{M\}$: 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 $\{B\}$.
- $\{A\}$: Frame fixed to the fixed base, but located at the point-of-interest.
- $\{B\}$: Frame fixed to the moving platform and located at the same point-of-interest than $\{A\}$.
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
Frames $\{F\}$ and $\{M\}$ are useful to describe the location of the joints in a meaningful frame.
On the other hand, frames $\{A\}$ and $\{B\}$ 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 ($150\,mm$ above the top platform, i.e. above $\{M\}$).
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.
Location of the joints and orientation and length of the struts are very important for the study of the Stewart platform as well.
The center of rotation for the joint fixed to the base is noted $\bm{a}_i$, while $b_i$ is used for the top joints.
The struts orientation are indicated by the unit vectors $\hat{\bm{s}}_i$ and their lengths by the scalars $l_i$.
This is summarized in Figure ref:fig:nhexa_stewart_notations.
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
#+end_quote
#+name: fig:nhexa_stewart_notations
#+caption: Frame and key notations for the Stewart platform
[[file:figs/nhexa_stewart_notations.png]]
** Kinematic Analysis
<<ssec:nhexa_stewart_platform_kinematics>>
**** Introduction :ignore:
Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces that cause the motion.
The relation between the geometry of the manipulator with the final motion of the moving platform is derived and analyzed.
*Definition of the geometry of the Stewart Platform*:
- $\bm{a}_i$: position of the attachment points on the fixed base
- $\bm{b}_i$: position of moving attachment points
- $l_i$: length of each limb
- $\hat{\bm{s}}_i$: unit vector representing the direction of each limb
#+name: fig:nhexa_stewart_schematic
#+caption: Geometry of a Stewart Platform
#+attr_latex: :scale 1
[[file:figs/nhexa_stewart_schematic.png]]
**** Loop Closure
At the displacement level, the *closure of each kinematic loop* can be express in the vector form as
\[ \vec{AB} = \vec{AA_i} + \vec{A_iB_i} - \vec{BB_i} \quad \text{for } i = 1,2,\dots,n \]
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.
At the displacement level, the /closure/ of each kinematic loop (illustrated in Figure ref:fig:nhexa_stewart_loop_closure) can be express in the vector form as
\begin{equation}
\vec{ab} = \vec{aa_i} + \vec{a_ib_i} - \vec{bb_i} \quad \text{for } i = 1 \text{ to } 6
\end{equation}
in which $\vec{aa_i}$ and $\vec{bb_i}$ can be easily obtained from the location of the joint on the base and on 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 $l_i {}^A\hat{\bm{s}}_i$:
\begin{equation*}
{}^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
\end{equation*}
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 strut vector $l_i {}^A\hat{\bm{s}}_i$:
\begin{equation}\label{eq:nhexa_loop_close}
{}^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 \text{ to } 6
\end{equation}
#+name: fig:nhexa_stewart_loop_closure
#+caption: Notations to compute the kinematic loop closure
[[file:figs/nhexa_stewart_loop_closure.png]]
**** Inverse Kinematics
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.
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 $\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]$.
This problem can be easily solved using the loop closures eqref:eq:nhexa_loop_close.
The obtain joint variables are:
\begin{equation*}
\begin{aligned}
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\\
&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}
\end{aligned}
\end{equation*}
\begin{equation}\label{eq:nhexa_inverse_kinematics}
l_i = \sqrt{{}^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 + 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}
\end{equation}
If the position and orientation of the platform lie in the feasible workspace, the solution is unique.
Otherwise, the solution gives complex numbers.
**** Forward Kinematics
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 ${}^A\bm{R}_B$.
In /forward kinematic analysis/, it is assumed that the vector of strut lengths $\bm{\mathcal{L}}$ 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.
In vector calculus, the Jacobian matrix represents the best linear approximation of a vector-valued function near a working point.
Consider a function $\bm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$ with continuous first-order partial derivatives.
For any input $\bm{x} \in \mathbb{R}^n$, this function produces an output $\bm{f}(\bm{x}) \in \mathbb{R}^m$.
The Jacobian matrix $\bm{J}$ of $\bm{f}$ at point $\bm{x}$ is the $m \times n$ matrix whose $(i,j)$ entry is:
$J_{ij} = \frac{\partial f_i}{\partial x_j}$
This matrix represents the linear transformation that best approximates $\bm{f}$ in a neighborhood of $\bm{x}$.
In other words, for points sufficiently close to $\bm{x}$, the function $\bm{f}$ behaves approximately like its Jacobian matrix.
**** Jacobian Computation - Velocity Loop Closure
@ -533,17 +551,32 @@ The *general Jacobian matrix* is defined as:
\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*}
\[{}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i = l_i {}^A\hat{\bm{s}}_i + {}^A\bm{a}_i\]
By taking the time derivative of the position loop close eqref:eq:nhexa_loop_close, the velocity loop closure is obtained:
\begin{equation}
{}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{a}_i}_{=0}
\end{equation}
Moreover, we have:
- ${}^A\dot{\bm{R}}_B {}^B\bm{b}_i = {}^A\bm{\omega} \times {}^A\bm{R}_B {}^B\bm{b}_i = {}^A\bm{\omega} \times {}^A\bm{b}_i$ in which ${}^A\bm{\omega}$ denotes the angular velocity of the moving platform expressed in the fixed frame $\{\bm{A}\}$.
- $l_i {}^A\dot{\hat{\bm{s}}}_i = l_i \left( {}^A\bm{\omega}_i \times \hat{\bm{s}}_i \right)$ in which ${}^A\bm{\omega}_i$ is the angular velocity of strut $i$ express in fixed frame $\{\bm{A}\}$.
By multiplying both sides by ${}^A\hat{s}_i$:
\begin{equation}
{}^A\hat{\bm{s}}_i {}^A\bm{v}_p + \underbrace{{}^A\hat{\bm{s}}_i ({}^A\bm{\omega} \times {}^A\bm{b}_i)}_{=({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) {}^A\bm{\omega}} = \dot{l}_i + \underbrace{{}^A\hat{s}_i l_i \left( {}^A\bm{\omega}_i \times {}^A\hat{\bm{s}}_i \right)}_{=0}
\end{equation}
Finally:
\begin{equation}
\hat{\bm{s}}_i {}^A\bm{v}_p + ({}^A\bm{b}_i \times \hat{\bm{s}}_i) {}^A\bm{\omega} = \dot{l}_i
\end{equation}
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}
\begin{equation}\label{eq:nhexa_jacobian}
\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 \\
@ -555,49 +588,65 @@ We can rearrange the equations in a matrix form:
\end{equation}
$\bm{J}$ then depends only on:
- $\hat{\bm{s}}_i$ the orientation of the limbs expressed in $\{A\}$
- $\hat{\bm{s}}_i$ the orientation of the struts expressed in $\{A\}$
- $\bm{b}_i$ the position of the joints with respect to $O_B$ and express in $\{A\}$
The Jacobian matrix links the rate of change of strut length to the velocity and angular velocity of the top platform with respect to the fixed base.
This Jacobian matrix needs to be recomputed for every Stewart platform pose.
**** 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*}
\begin{equation}\label{eq:nhexa_inverse_kinematics_approximate}
\boxed{\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*}
\begin{equation}\label{eq:nhexa_forward_kinematics_approximate}
\boxed{\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
As we know how to exactly solve the Inverse kinematic problem, we can compare the exact solution with the approximate solution using the Jacobian matrix.
For small displacements, the approximate solution is expected to work well.
We would like here to determine up to what displacement this approximation can be considered as correct.
Then, we can determine the range for which the approximate inverse kinematic is valid.
This will also gives us the range for which the approximate forward kinematic is valid.
- [ ] [[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]]
Let's first compare the perfect and approximate solution of the inverse for pure $x$ translations.
The approximate and exact required strut stroke to have the wanted mobile platform $x$ displacement are computed.
The estimated error is shown in Figure etc...
For small wanted displacements (up to $\approx 1\%$ of the size of the Hexapod), the approximate inverse kinematic solution using the Jacobian matrix is quite correct.
In the case of the Nano-hexapod, the maximum stroke is estimate to the around $100\,\mu m$ while its size is around $100\,mm$, therefore the fixed Jacobian matrix is a very good approximate for the forward and 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*}
Let's note $\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T$ the vector of actuator forces applied in each strut and $\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T$ external force/torque action on the mobile platform at $\bm{O}_B$.
The /principle of virtual work/ states that the total virtual work $\delta W$, done by all actuators and external forces is equal to zero:
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 $\bm{O}_B$
\begin{equation}
\delta W = \bm{\tau}^T \delta \bm{\mathcal{L}} - \bm{\mathcal{F}}^T \delta \bm{\mathcal{X}} = 0
\end{equation}
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:
\begin{equation}
\bm{\tau}^T \bm{J} - \bm{\mathcal{F}}^T = 0 \quad \Rightarrow \quad \boxed{\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}} \quad \text{and} \quad \boxed{\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}}
\end{equation}
\[ \bm{\tau}^T \bm{J} - \bm{\mathcal{F}}^T = 0 \quad \Rightarrow \quad \tcmbox{\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}} \quad \text{and} \quad \tcmbox{\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}} \]
**** Singularities
- Briefly mention singularities, and say that for small stroke, it is not an issue, the Jacobian matrix may be considered constant
Therefore, the same Jacobian matrix can also be used to map actuator forces to forces and torques applied on the mobile platform at the defined frame $\{B\}$.
** Static Analysis
<<ssec:nhexa_stewart_platform_static>>
@ -616,32 +665,114 @@ If we combine these 6 relations:
\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*}
\begin{equation}
\bm{\mathcal{F}} = \bm{J}^T \mathcal{K} \bm{J} \cdot \delta \bm{\mathcal{X}}
\end{equation*}
\end{equation}
And then we identify the stiffness matrix $\bm{K}$:
\begin{equation*}
\begin{equation}
\bm{K} = \bm{J}^T \mathcal{K} \bm{J}
\end{equation*}
\end{equation}
If the stiffness matrix $\bm{K}$ is inversible, the *compliance matrix* of the manipulator is defined as
\begin{equation*}
\begin{equation}
\bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1}
\end{equation*}
\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*}
\begin{equation}
\delta \bm{\mathcal{X}} = \bm{C} \cdot \bm{\mathcal{F}}
\end{equation*}
\end{equation}
Conclusion: stiffness/compliance of the Stewart platform depends on the Jacobian matrix, therefore on the position and orientation of the struts.
** Dynamic Analysis
<<ssec:nhexa_stewart_platform_dynamics>>
If one wants to study the dynamics of the Stewart platform, ...
Let's suppose that the struts are mass-less, that the joints are perfect.
Suppose the
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).
\begin{equation}
M s^2 \mathcal{X} = \Sigma \mathcal{F}
\end{equation}
Forces are:
- Actuator forces: $\mathcal{F} = \bm{J}^T \tau$
- Stiffness of the struts: $-J^T \mathcal{K} J \mathcal{X}$
- Damping of the struts: $-J^T \mathcal{C} J \dot{\mathcal{X}}$
\begin{equation}
M s^2 \mathcal{X} = \mathcal{F} - J^T \mathcal{K} J \mathcal{X} - J^T \mathcal{C} J s \mathcal{X}
\end{equation}
Equation in the cartesian frame:
\begin{equation}
\frac{\mathcal{X}}{\mathcal{F}}(s) = ( M s^2 + \bm{J}^{T} \mathcal{C} J s + \bm{J}^{T} \mathcal{K} J )^{-1}
\end{equation}
Using the Jacobian, equation in the strut frame:
\begin{equation}
\frac{\mathcal{L}}{\tau}(s) = ( \bm{J}^{-T} M \bm{J}^{-1} s^2 + \mathcal{C} + \mathcal{K} )^{-1}
\end{equation}
It becomes much more complex when:
- model the mass of the struts, or more complex strut dynamics
- take into account flexible joint stiffnesses
- would not be practical to combine with the dynamical equations of the micro-station
#+begin_src matlab
%% Plant using Analytical Equations
% Stewart platform definition
k = 1e6; % Actuator stiffness [N/m]
c = 1e1; % Actuator damping [N/(m/s)]
stewart = initializeSimplifiedNanoHexapod('Mpm', 1e-3, 'actuator_type', '1dof', 'actuator_k', k, 'actuator_c', c);
% Payload: Cylinder
h = 300e-3; % Height of the cylinder [m]
r = 110e-3; % Radius of the cylinder [m]
m = 10; % Mass of the payload [kg]
initializeSample('type', 'cylindrical', 'm', m, 'H', h, 'R', r);
% Mass Matrix
M = zeros(6,6);
M(1,1) = m;
M(2,2) = m;
M(3,3) = m;
M(4,4) = 1/12*m*(3*r^2 + h^2);
M(5,5) = 1/12*m*(3*r^2 + h^2);
M(6,6) = 1/2*m*r^2;
% Stiffness and Damping matrices
K = k*eye(6);
C = c*eye(6);
% Compute plant in the frame of the struts
G_analytical = inv(ss(inv(stewart.geometry.J')*M*inv(stewart.geometry.J)*s^2 + C*s + K));
% Compare with Simscape model
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'dL'); io_i = io_i + 1; % Encoders [m]
G_simscape = linearize(mdl, io);
G_simscape.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G_simscape.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};
#+end_src
#+begin_src matlab
bodeFig({G_analytical(1,1), G_simscape(1,1), G_analytical(1,2), G_simscape(1,2)})
#+end_src
** Conclusion
:PROPERTIES:
:UNNUMBERED: t
@ -653,6 +784,10 @@ All depends on the geometry.
Reasonable choice of geometry is made in chapter 1.
Optimization of the geometry will be made in chapter 2.
The static analysis supposed that joints are perfect.
It gets more complex if flexible joints are used with stiffnesses that are not negligible.
[[cite:&mcinroy00_desig_contr_flexur_joint_hexap;&mcinroy02_model_desig_flexur_joint_stewar]]
* Multi-Body Model
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/nhexa_2_model.m
@ -690,6 +825,10 @@ Optimization of the geometry will be made in chapter 2.
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src
@ -730,7 +869,113 @@ Definition of each part + Plant with defined inputs/outputs (force sensor, relat
- If all is perfect (mass-less struts, perfect joints, etc...), maybe compare analytical model with simscape model?
- Say something about the model order
Model order is 12, and that we can compute modes from matrices M and K, compare with the Simscape model
- 4 observed modes (due to symmetry, in reality 6 modes)
- Compare with analytical formulas (see number of states)
- Effect of 2DoF APA on IFF plant?
#+begin_src matlab
initializeSimplifiedNanoHexapod('flex_type_F', '2dof', 'flex_type_M', '3dof', 'actuator_type', '1dof');
initializeSample('type', 'cylindrical', 'm', 50, 'H', 300e-3);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'dL'); io_i = io_i + 1; % Encoders [m]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'fn'); io_i = io_i + 1; % Force Sensors [N]
% With no payload
G = linearize(mdl, io);
G.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6', ...
'fn1', 'fn2', 'fn3', 'fn4', 'fn5', 'fn6'};
size(G)
#+end_src
#+begin_src matlab :exports none
%% Diagonal elements of the FRF matrix from u to de
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G(1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$d_{ei}/u_i$ - Model')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-9, 1e-4]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.5]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :exports none
%% Diagonal elements of the FRF matrix from u to de
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G(6+1,1), freqs, 'Hz'))), 'color', colors(2,:), ...
'DisplayName', '$d_{ei}/u_i$ - Model')
for i = 2:6
plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', colors(2,:), ...
'HandleVisibility', 'off');
end
for i = 1:5
for j = i+1:6
plot(freqs, abs(squeeze(freqresp(G(6+i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]);
ylim([1e-5, 1e2]);
leg = legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
ax2 = nexttile;
hold on;
for i = 1:6
plot(freqs, 180/pi*angle(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', [colors(2,:),0.5]);
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([-180, 180]);
yticks([-180, -90, 0, 90, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
** Conclusion
:PROPERTIES:
@ -776,6 +1021,10 @@ Reference book: [[cite:&skogestad07_multiv_feedb_contr]]
<<m-init-path-tangle>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape>>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other>>
#+end_src
@ -801,6 +1050,31 @@ Reference book: [[cite:&skogestad07_multiv_feedb_contr]]
- 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)
*Maybe all details about control should be in chapter 2, dedicated to control*
*Here, just say that using kinematics, we control in the frame of the struts*
#+begin_src matlab
%% Control at the CoM
stewart = initializeSimplifiedNanoHexapod('Mpm', 1e-3); % Massless top platform
initializeSample('type', 'cylindrical', 'm', 10, 'H', 300e-3);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'open-loop');
% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N]
io(io_i) = linio([mdl, '/plant'], 2, 'openoutput', [], 'dL'); io_i = io_i + 1; % Encoders [m]
% With no payload
G = linearize(mdl, io);
G.InputName = {'f1', 'f2', 'f3', 'f4', 'f5', 'f6'};
G.OutputName = {'dL1', 'dL2', 'dL3', 'dL4', 'dL5', 'dL6'};
J = stewart.geometry.J;
Gm = inv(J)*G*inv(J');
#+end_src
** Active Damping with Decentralized IFF
<<ssec:nhexa_control_iff>>

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@ -1,4 +1,4 @@
% Created 2025-02-05 Wed 17:49
% Created 2025-02-07 Fri 16:42
% Intended LaTeX compiler: pdflatex
\documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt}
@ -24,33 +24,30 @@
\clearpage
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.
Introduction:
\begin{itemize}
\item Choice of architecture to do 5DoF control (Section \ref{sec:nhexa_platform_review})
\item Stewart platform (Section \ref{sec:nhexa_stewart_platform})
Show what is an hexapod, how we can define its geometry, stiffness, etc\ldots{}
Some kinematics: stiffness matrix, mass matrix, etc\ldots{}
\item Need to model the active vibration platform: multi-body model (Section \ref{sec:nhexa_model})
Explain what we want to capture with this model
Key elements (plates, joints, struts): for now simplistic model (rigid body elements, perfect joints, \ldots{}), but in next section, FEM will be used
\item Control (Section \ref{sec:nhexa_control})
\end{itemize}
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}).
\chapter{Active Vibration Platforms}
\label{sec:nhexa_platform_review}
\textbf{Goals}:
\begin{itemize}
\item Quick review of active vibration platforms (5 or 6DoF) similar to NASS
\item Explain why Stewart platform architecture is chosen
\item Explain what is a Stewart platform (quickly as it will be shown in details in the next section)
\item Quick review of active vibration platforms (5 or 6DoF)
\end{itemize}
Active vibration platform with 5DoF or 6DoF?
Synchrotron applications?
\item Wanted controlled DOF: Y, Z, Ry
\item But because of continuous rotation (key specificity): X,Y,Z,Rx,Ry in the frame of the active platform
\begin{itemize}
\item Literature review? (\textbf{maybe more suited for chapter 2})
\begin{itemize}
\item \url{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org}
@ -58,6 +55,7 @@ Synchrotron applications?
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.
\end{itemize}
\item \cite{taghirad13_paral}
\item For some systems, just XYZ control (stack stages), example: holler
\item For other systems, Stewart platform (ID16a), piezo based
\item Examples of Stewart platforms for general vibration control, some with Piezo, other with Voice coil. IFF, \ldots{}
@ -70,6 +68,7 @@ Show different geometry configuration
\begin{itemize}
\item[{$\square$}] Talk about external metrology?
Maybe not the topic here.
\item[{$\square$}] Talk about control architecture?
\item[{$\square$}] Comparison with the micro-station / NASS
\end{itemize}
@ -86,6 +85,22 @@ Show different geometry configuration
\item A review of Stewart platform will be given in Chapter related to the detailed design of the Nano-Hexapod
\end{itemize}
\begin{table}[htbp]
\centering
\begin{tabularx}{\linewidth}{lXX}
\toprule
& \textbf{Serial Robots} & \textbf{Parallel Robots}\\
\midrule
Advantages & Large Workspace & High Stiffness\\
Disadvantages & Low Stiffness & Small Workspace\\
Kinematic Struture & Open & Closed-loop\\
\bottomrule
\end{tabularx}
\caption{\label{tab:nhexa_serial_vs_parallel}Advantages and Disadvantages of both serial and parallel robots}
\end{table}
\chapter{The Stewart platform}
\label{sec:nhexa_stewart_platform}
\begin{itemize}
@ -96,53 +111,241 @@ Explain advantages compared to serial architecture
Complete review of Stewart platforms will be made in Chapter 2
\item Presentation of tools used to analyze the properties of the Stewart platform => useful for design and control
\end{itemize}
The Stewart Platform is very adapted for the NASS application for the following reasons:
\begin{itemize}
\item it is a fully parallel manipulator, thus all the motions errors can be compensated
\item it is very compact compared to a serial manipulator
\item it has high stiffness and good dynamic performances
\end{itemize}
The main disadvantage of Stewart platforms is the small workspace when compare the serial manipulators which is not a problem here.
\section{Mechanical Architecture}
\label{ssec:nhexa_stewart_platform_architecture}
\url{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/stewart-architecture.org}
A Stewart manipulator consists of two platforms connected by six struts (Figure \ref{fig:nhexa_stewart_architecture}).
Each strut is connected to the fixed and the mobile platforms with a joint.
Typically, a universal joint is used on one side while a spherical joint is used on the other side\footnote{Different architecture exists, typically referred as ``6-SPS'' (Spherical, Prismatic, Spherical) or ``6-UPS'' (Universal, Prismatic, Spherical)}.
In the strut, there is an active element working as a prismatic joint.
Presentation of the typical architecture
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_stewart_architecture.png}
\caption{\label{fig:nhexa_stewart_architecture}Schematical representation of the Stewart platform architecture.}
\end{figure}
Such architecture allows to move the mobile platform with respect to the fixed platform in 6 degrees-of-freedom.
It is therefore a \emph{fully} parallel manipulator as the number of actuators is equal to the number of DoF.
It is also a symmetrical parallel manipulator as typically all the struts are identical.
In order to study the Stewart platform, four important frames are typically defined:
\begin{itemize}
\item Explain the different frames, etc\ldots{}
\item explain key elements:
\begin{itemize}
\item two plates
\item joints
\item actuators
\end{itemize}
\item \(\{F\}\): Frame fixed on the base and located at the center of its bottom surface.
This is used to fix the Stewart platform to some support.
\item \(\{M\}\): 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.
\item \(\{A\}\): Frame fixed to the fixed base, but located at the point-of-interest.
\item \(\{B\}\): Frame fixed to the moving platform and located at the same point-of-interest than \(\{A\}\).
\end{itemize}
Make well defined notations.
\begin{itemize}
\item \{F\}, \{M\}
\item si, li, ai, bi, etc.
Frames \(\{F\}\) and \(\{M\}\) are useful to describe the location of the joints in a meaningful frame.
On the other hand, frames \(\{A\}\) and \(\{B\}\) 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 (\(150\,mm\) above the top platform, i.e. above \(\{M\}\)).
\item[{$\square$}] Make figure with defined frames, joints, etc\ldots{}
Maybe can use this figure as an example:
\begin{center}
\includesvg[scale=1]{/home/thomas/Cloud/work-projects/ID31-NASS/phd-thesis-chapters/A0-nass-introduction/figs/introduction_stewart_du14}
\end{center}
\end{itemize}
Location of the joints and orientation and length of the struts are very important for the study of the Stewart platform as well.
The center of rotation for the joint fixed to the base is noted \(\bm{a}_i\), while \(b_i\) is used for the top joints.
The struts orientation are indicated by the unit vectors \(\hat{\bm{s}}_i\) and their lengths by the scalars \(l_i\).
This is summarized in Figure \ref{fig:nhexa_stewart_notations}.
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_stewart_notations.png}
\caption{\label{fig:nhexa_stewart_notations}Frame and key notations for the Stewart platform}
\end{figure}
\section{Kinematic Analysis}
\label{ssec:nhexa_stewart_platform_kinematics}
Kinematic analysis refers to the study of the geometry of motion of a robot, without considering the forces that cause the motion.
\paragraph{Loop Closure}
At the displacement level, the \emph{closure} of each kinematic loop (illustrated in Figure \ref{fig:nhexa_stewart_loop_closure}) can be express in the vector form as
\begin{equation}
\vec{ab} = \vec{aa_i} + \vec{a_ib_i} - \vec{bb_i} \quad \text{for } i = 1 \text{ to } 6
\end{equation}
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 \(l_i {}^A\hat{\bm{s}}_i\):
\begin{equation}\label{eq:nhexa_loop_close}
{}^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 \text{ to } 6
\end{equation}
\begin{figure}[htbp]
\centering
\includegraphics[scale=1]{figs/nhexa_stewart_loop_closure.png}
\caption{\label{fig:nhexa_stewart_loop_closure}Notations to compute the kinematic loop closure}
\end{figure}
\paragraph{Inverse Kinematics}
For \emph{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 \(\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]\).
This problem can be easily solved using the loop closures \eqref{eq:nhexa_loop_close}.
The obtain joint variables are:
\begin{equation}\label{eq:nhexa_inverse_kinematics}
l_i = \sqrt{{}^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 + 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}
\end{equation}
If the position and orientation of the platform lie in the feasible workspace, the solution is unique.
Otherwise, the solution gives complex numbers.
\paragraph{Forward Kinematics}
\paragraph{Jacobian Matrix}
In \emph{forward kinematic analysis}, it is assumed that the vector of limb lengths \(\bm{\mathcal{L}}\) 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.
\section{The Jacobian Matrix}
In vector calculus, the Jacobian matrix represents the best linear approximation of a vector-valued function near a working point.
Consider a function \(\bm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m\) with continuous first-order partial derivatives.
For any input \(\bm{x} \in \mathbb{R}^n\), this function produces an output \(\bm{f}(\bm{x}) \in \mathbb{R}^m\).
The Jacobian matrix \(\bm{J}\) of \(\bm{f}\) at point \(\bm{x}\) is the \(m \times n\) matrix whose \((i,j)\) entry is:
\(J_{ij} = \frac{\partial f_i}{\partial x_j}\)
This matrix represents the linear transformation that best approximates \(\bm{f}\) in a neighborhood of \(\bm{x}\).
In other words, for points sufficiently close to \(\bm{x}\), the function \(\bm{f}\) behaves approximately like its Jacobian matrix.
\paragraph{Jacobian Computation - Velocity Loop Closure}
Let's note:
\begin{itemize}
\item Velocity Loop Closure
\item Static Forces
\item \(\bm{\mathcal{L}} = \left[ l_1, l_2, \ldots, l_6 \right]^T\): vector of actuated joint coordinates
\item \(\bm{\mathcal{X}} = \left[ {}^A\bm{P}, \bm{}^A\hat{\bm{s}} \right]^T\): vector of platform motion variables
\end{itemize}
\paragraph{Singularities}
\(\bm{\mathcal{L}}\) and \(\bm{\mathcal{X}}\) are related through a system of \emph{nonlinear algebraic equations} representing the \emph{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:
\begin{itemize}
\item \(\dot{\bm{\mathcal{L}}} = [ \dot{l}_1, \dot{l}_2, \dot{l}_3, \dot{l}_4, \dot{l}_5, \dot{l}_6 ]^T\)
\item \(\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T\):
\end{itemize}
The \textbf{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 \textbf{velocity loop closures} are used for \textbf{obtaining the Jacobian matrices} in a straightforward manner:
\[{}^A\bm{P} + {}^A\bm{R}_B {}^B\bm{b}_i = l_i {}^A\hat{\bm{s}}_i + {}^A\bm{a}_i\]
By taking the time derivative of the position loop close \eqref{eq:nhexa_loop_close}, the velocity loop closure is obtained:
\begin{equation}
{}^A\bm{v}_p + {}^A \dot{\bm{R}}_B {}^B\bm{b}_i + {}^A\bm{R}_B \underbrace{{}^B\dot{\bm{b}_i}}_{=0} = \dot{l}_i {}^A\hat{\bm{s}}_i + l_i {}^A\dot{\hat{\bm{s}}}_i + \underbrace{{}^A\dot{a}_i}_{=0}
\end{equation}
Moreover, we have:
\begin{itemize}
\item \({}^A\dot{\bm{R}}_B {}^B\bm{b}_i = {}^A\bm{\omega} \times {}^A\bm{R}_B {}^B\bm{b}_i = {}^A\bm{\omega} \times {}^A\bm{b}_i\) in which \({}^A\bm{\omega}\) denotes the angular velocity of the moving platform expressed in the fixed frame \(\{\bm{A}\}\).
\item \(l_i {}^A\dot{\hat{\bm{s}}}_i = l_i \left( {}^A\bm{\omega}_i \times \hat{\bm{s}}_i \right)\) in which \({}^A\bm{\omega}_i\) is the angular velocity of limb \(i\) express in fixed frame \(\{\bm{A}\}\).
\end{itemize}
By multiplying both sides by \({}^A\hat{s}_i\):
\begin{equation}
{}^A\hat{\bm{s}}_i {}^A\bm{v}_p + \underbrace{{}^A\hat{\bm{s}}_i ({}^A\bm{\omega} \times {}^A\bm{b}_i)}_{=({}^A\bm{b}_i \times {}^A\hat{\bm{s}}_i) {}^A\bm{\omega}} = \dot{l}_i + \underbrace{{}^A\hat{s}_i l_i \left( {}^A\bm{\omega}_i \times {}^A\hat{\bm{s}}_i \right)}_{=0}
\end{equation}
Finally:
\begin{equation}
\hat{\bm{s}}_i {}^A\bm{v}_p + ({}^A\bm{b}_i \times \hat{\bm{s}}_i) {}^A\bm{\omega} = \dot{l}_i
\end{equation}
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}\label{eq:nhexa_jacobian}
\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:
\begin{itemize}
\item \(\hat{\bm{s}}_i\) the orientation of the limbs expressed in \(\{A\}\)
\item \(\bm{b}_i\) the position of the joints with respect to \(O_B\) and express in \(\{A\}\)
\end{itemize}
The Jacobian matrix links the rate of change of strut length to the velocity and angular velocity of the top platform with respect to the fixed base.
This Jacobian matrix needs to be recomputed for every Stewart platform pose.
\paragraph{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}\label{eq:nhexa_inverse_kinematics_approximate}
\boxed{\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}\label{eq:nhexa_forward_kinematics_approximate}
\boxed{\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 \emph{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.
\paragraph{Range validity of the approximate inverse kinematics}
As we know how to exactly solve the Inverse kinematic problem, we can compare the exact solution with the approximate solution using the Jacobian matrix.
For small displacements, the approximate solution is expected to work well.
We would like here to determine up to what displacement this approximation can be considered as correct.
Then, we can determine the range for which the approximate inverse kinematic is valid.
This will also gives us the range for which the approximate forward kinematic is valid.
\begin{itemize}
\item Briefly mention singularities, and say that for small stroke, it is not an issue, the Jacobian matrix may be considered constant
\item[{$\square$}] \href{file:///home/thomas/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/kinematic-study.org}{Estimation of the range validity of the approximate inverse kinematics}
\end{itemize}
Let's first compare the perfect and approximate solution of the inverse for pure \(x\) translations.
The approximate and exact required strut stroke to have the wanted mobile platform \(x\) displacement are computed.
The estimated error is shown in Figure etc\ldots{}
For small wanted displacements (up to \(\approx 1\%\) of the size of the Hexapod), the approximate inverse kinematic solution using the Jacobian matrix is quite correct.
In the case of the Nano-hexapod, the maximum stroke is estimate to the around \(100\,\mu m\) while its size is around \(100\,mm\), therefore the fixed Jacobian matrix is a very good approximate for the forward and inverse kinematics.
\paragraph{Static Forces}
Let's note \(\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T\) the vector of actuator forces applied in each strut and \(\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T\) external force/torque action on the mobile platform at \(\bm{O}_B\).
The \emph{principle of virtual work} states that the total virtual work \(\delta W\), done by all actuators and external forces is equal to zero:
\begin{equation}
\delta W = \bm{\tau}^T \delta \bm{\mathcal{L}} - \bm{\mathcal{F}}^T \delta \bm{\mathcal{X}} = 0
\end{equation}
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:
\begin{equation}
\bm{\tau}^T \bm{J} - \bm{\mathcal{F}}^T = 0 \quad \Rightarrow \quad \boxed{\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}} \quad \text{and} \quad \boxed{\bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}}}
\end{equation}
Therefore, the same Jacobian matrix can also be used to map actuator forces to forces and torques applied on the mobile platform at the defined frame \(\{B\}\).
\section{Static Analysis}
\label{ssec:nhexa_stewart_platform_static}
@ -150,6 +353,34 @@ 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 \textbf{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*}
\section{Dynamic Analysis}
\label{ssec:nhexa_stewart_platform_dynamics}
@ -158,6 +389,8 @@ For instance, compute the plant for massless struts and perfect joints (will be
But say that if we want to model more complex cases, it becomes impractical (cite papers).
\section*{Conclusion}
Dynamic analysis of parallel manipulators presents an \textbf{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.
@ -229,7 +462,9 @@ Definition of each part + Plant with defined inputs/outputs (force sensor, relat
\item If all is perfect (mass-less struts, perfect joints, etc\ldots{}), maybe compare analytical model with simscape model?
\item Say something about the model order
Model order is 12, and that we can compute modes from matrices M and K, compare with the Simscape model
\item 4 observed modes (due to symmetry, in reality 6 modes)
\item Compare with analytical formulas (see number of states)
\item Effect of 2DoF APA on IFF plant?
\end{itemize}
\section*{Conclusion}
@ -280,6 +515,9 @@ Also speak about disturbances? (and how disturbances can be mixed to different o
\item 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)
\end{itemize}
\textbf{Maybe all details about control should be in chapter 2, dedicated to control}
\textbf{Here, just say that using kinematics, we control in the frame of the struts}
\section{Active Damping with Decentralized IFF}
\label{ssec:nhexa_control_iff}