#+TITLE: Simscape Model - Nano Hexapod
:DRAWER:
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#+EMAIL: dehaeze.thomas@gmail.com
#+AUTHOR: Dehaeze Thomas

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* Notes                                                             :noexport:
** Notes
Prefix is =nhexa=

Based on:
- [ ] Stewart platform presentation: [[file:~/Cloud/meetings/group-meetings-me/2020-01-27-Stewart-Platform-Simscape/2020-01-27-Stewart-Platform-Simscape.org]]
- [ ] Add some sections from here: [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/index.org]]
  For instance:
  - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/stewart-architecture.org][stewart architecture]]
  - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/stewart_platform.org::+TITLE: Stewart Platform - Simscape Model]]
  - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/kinematic-study.org][kinematic study]]
  - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/identification.org][stewart platform - dynamics]]
  - [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/cubic-configuration.org][cubic configuration]]
- [ ] Look at the [[file:~/Cloud/work-projects/ID31-NASS/documents/state-of-thesis-2020/index.org][NASS 2020 report]]
  Sections 5.1, 5.4
- [ ] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/amplified_piezoelectric_stack.org][amplified_piezoelectric_stack]] (Just use 2DoF here)
- [ ] [[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)
- [ ] Should the study of effect of flexible joints be included here?
- [X] file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/control-vibration-isolation.org

Questions:
- [ ] 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

** DONE [#A] Copy relevant parts of reports
CLOSED: [2025-02-06 Thu 15:27]

- [X] Stewart platform presentation: [[file:~/Cloud/meetings/group-meetings-me/2020-01-27-Stewart-Platform-Simscape/2020-01-27-Stewart-Platform-Simscape.org]]
- [X] Add some sections from here: [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/index.org]]
  For instance:
  - [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/stewart-architecture.org][stewart architecture]]
  - [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/nass-simscape/org/stewart_platform.org::+TITLE: Stewart Platform - Simscape Model]]
  - [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/kinematic-study.org][kinematic study]]
  - [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/identification.org]]
        Effect of joints stiffnesses
  - [X] [[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/cubic-configuration.org][cubic configuration]]
    Not relevant here: in chapter 2
- [X] Look at the [[file:~/Cloud/work-projects/ID31-NASS/documents/state-of-thesis-2020/index.org][NASS 2020 report]]
  Sections 5.1, 5.4
- [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*
- [X] file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/control-vibration-isolation.org

** DONE [#A] Make a nice outline
CLOSED: [2025-02-05 Wed 17:45]

*Introduction*
- Choice of architecture to do 5DoF control
- Stewart platform
- Need to model the active vibration platform
- Control

*1 - Active Vibration Platforms*:
Introduction:
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)
- Presentation of the Simscape model

*After the report* (NASS-Simscape):
- nano-hexapod on top of micro-station
- control is performed
- simulations => validation of the concept

** QUES [#A] Should I talk about APA here?

- It seems APA is not mentioned in chapter 1
- Could be nice to only talk about APA in chapter 2 as in chapter 1 we don't care too much about actual implementation/design
- [ ] Should the NASS be validated using 1 DoF actuators?
  How to deal with IFF?
  - Add parallel stiffness
  - High pass filter

** TODO [#A] Make sure the Simulink file for the Stewart platform is working well
SCHEDULED: <2025-02-10 Mon>

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 [#B] Check all notations

- [ ] Make sure they are all defined in correct order
- [ ] Make sure all vectors and matrices are bold

** DONE [#C] Better understand principle of virtual work
CLOSED: [2025-02-10 Mon 15:51]

[[*Static Forces][Static Forces]]

Better understand this: https://en.wikipedia.org/wiki/Virtual_work

Also add link or explanation for this equation.

** TODO [#C] Mention the Toolbox (maybe make a DOI for that)
** 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]

Maybe should introduce:
- "MIMO" Root locus
- "MIMO" Nyquist plot / characteristic loci

Or should this be in annexes?

Maybe say that in this phd-thesis, the focus is not on the control.
I tried multiple architectures (complementary filters, etc.), but the focus is not on that.

** ANSW [#C] Cubic architecture should be the topic here or in the detailed design?
CLOSED: [2025-02-06 Thu 16:01]

I suppose that it should be in the detailed design phase.
(Review about Stewart platform design should be made in Chapter two.)

Here, just use simple control architecture for general validation (and not optimization).

** ANSW [#C] Should I make a review of control strategies?
CLOSED: [2025-02-06 Thu 16:01]

Yes it seems to good location for review related to control.

Jacobian matrix.
Control is the frame of the struts, in the cartesian frame (CoM, CoK), modal control, etc...

[[file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org][file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org]]

** DONE [#C] Compare simscape =linearize= and analytical formula
CLOSED: [2025-02-06 Thu 16:01]

- [X] OK for $\omega=0$ (using just the Stiffness matrix)
- [ ] Should add the mass matrix and compare for all frequencies

The analytical dynamic model is taken from cite:taghirad13_paral

** DONE [#C] Output the cubic configuration with clear display of the cube and center of the cube
CLOSED: [2025-02-06 Thu 16:02]

[[file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/cubic-configuration.org][cubic configuration]]

*No, this will be in Chapter 2*

** CANC [#C] Maybe make an appendix to present the developed toolbox?
CLOSED: [2025-02-06 Thu 16:02]

- State "CANC"       from "TODO"       [2025-02-06 Thu 16:02]
* Introduction                                                        :ignore:

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).

* Active Vibration Platforms
<<sec:nhexa_platform_review>>
** Introduction                                                      :ignore:

*Goals*:
- Quick review of active vibration platforms (5 or 6DoF) similar to NASS
- Explain why Stewart platform architecture is chosen

- Wanted controlled DOF: Y, Z, Ry
- But because of continuous rotation (key specificity): X,Y,Z,Rx,Ry in the frame of the active platform

- Literature review? (*maybe more suited for chapter 2*)
  - file:~/Cloud/work-projects/ID31-NASS/matlab/stewart-simscape/org/bibliography.org
  - Talk about flexible joint? Maybe not so much as it should be topic of second chapter.
    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.
- [[cite:&taghirad13_paral]]

- 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]]

- [ ] Talk about external metrology?
  Maybe not the topic here.
- [ ] Talk about control architecture?
- [ ] Comparison with the micro-station / NASS

** Serial and Parallel Manipulators

*Goal*:
- Explain why a parallel manipulator is here preferred
- Compact, 6DoF, higher control bandwidth, linear, simpler

- Show some example of serial and parallel manipulators

- A review of Stewart platform will be given in Chapter related to the detailed design of the Nano-Hexapod


#+name: tab:nhexa_serial_vs_parallel
#+caption: Advantages and Disadvantages of both serial and parallel robots
#+attr_latex: :environment tabularx :width \linewidth :align lXX
#+attr_latex: :center t :booktabs t :float t
|                    | *Serial Robots* | *Parallel Robots* |
|--------------------+-----------------+-------------------|
| Advantages         | Large Workspace | High Stiffness    |
| Disadvantages      | Low Stiffness   | Small Workspace   |
| Kinematic Struture | Open            | Closed-loop       |

* The Stewart platform
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/nhexa_1_stewart_platform.m
:END:
<<sec:nhexa_stewart_platform>>
** Introduction                                                      :ignore:

The Stewart platform, first introduced by Stewart in 1965 [[cite:&stewart65_platf_with_six_degrees_freed]] for flight simulation applications, represents a significant milestone in parallel manipulator design.
This mechanical architecture has evolved far beyond its original purpose, finding applications across diverse fields from precision positioning systems to robotic surgery.
The fundamental design consists of two platforms connected by six adjustable struts in parallel, creating a fully parallel manipulator capable of six degrees of freedom motion.

Unlike serial manipulators where errors worsen through the kinematic chain, parallel architectures distribute loads across multiple actuators, leading to enhanced mechanical stiffness and improved positioning accuracy.
This parallel configuration also results in superior dynamic performance, as the actuators directly contribute to the platform's motion without intermediate linkages.
These characteristics of Stewart platforms have made them particularly valuable in applications requiring high precision and stiffness.

For the NASS application, the Stewart platform architecture presents three key advantages.
First, as a fully parallel manipulator, all motion errors of the micro-station can be compensated through the coordinated action of the six actuators.
Second, its compact design compared to serial manipulators makes it ideal for integration on top micro-station where only $95\,mm$ of height is available.
Third, the good dynamical properties should enable high bandwidth positioning control.

While Stewart platforms excel in precision and stiffness, they typically exhibit a relatively limited workspace compared to serial manipulators.
However, this limitation is not significant for the NASS application, as the required motion range corresponds to the positioning errors of the micro-station which are in the order of $10\,\mu m$.

This section provides a comprehensive analysis of the Stewart platform's properties, focusing on aspects crucial for precision positioning applications.
The analysis encompasses the platform's kinematic relationships (Section ref:ssec:nhexa_stewart_platform_kinematics), the use of the Jacobian matrix (Section ref:ssec:nhexa_stewart_platform_jacobian), static behavior (Section ref:ssec:nhexa_stewart_platform_static), and dynamic characteristics (Section ref:ssec:nhexa_stewart_platform_dynamics).
These theoretical foundations form the basis for subsequent design decisions and control strategies, which will be elaborated in later sections.

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** Mechanical Architecture
<<ssec:nhexa_stewart_platform_architecture>>

The Stewart platform consists of two rigid platforms connected by six struts arranged in parallel (Figure ref:fig:nhexa_stewart_architecture).
Each strut incorporates an active prismatic joint that enables controlled length variation, with its ends attached to the fixed and mobile platforms through joints.
The typical configuration consists of a universal joint at one end and a spherical joint at the other, providing the necessary degrees of freedom[fn:1].

#+name: fig:nhexa_stewart_architecture
#+caption: Schematical representation of the Stewart platform architecture.
[[file:figs/nhexa_stewart_architecture.png]]

To facilitate rigorous analysis of the Stewart platform, four reference frames are defined:
- The fixed base frame $\{F\}$, located at the center of the base platform's bottom surface, serves as the mounting reference for the support structure.
- The mobile frame $\{M\}$, situated at the center of the top platform's upper surface, provides a reference for payload mounting.
- The point-of-interest frame $\{A\}$, fixed to the base but positioned at the workspace center.
- The moving point-of-interest frame $\{B\}$, attached to the mobile platform and coincident with frame $\{A\}$ in the home position.

Frames $\{F\}$ and $\{M\}$ serve primarily to define the joint locations.
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 frame $\{B\}$ expressed in frame $\{A\}$ and the rotation matrix ${}^A\bm{R}_B$ expressing the orientation of $\{B\}$ with respect to $\{A\}$.
For the nano-hexapod, frames $\{A\}$ and $\{B\}$ are chosen to be located at the theoretical focus point of the X-ray light which is $150\,mm$ above the top platform, i.e. above $\{M\}$.

Location of the joints and orientation and length of the struts are crucial for subsequent kinematic, static, and dynamic analyses of the Stewart platform.
The center of rotation for the joint fixed to the base is noted $\bm{a}_i$, while $b_i$ is used for the top platform joints.
The struts orientation are represented 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.

#+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:

The kinematic analysis of the Stewart platform involves understanding the geometric relationships between the platform position/orientation and the actuator lengths, without considering the forces involved.

**** Loop Closure

The foundation of the kinematic analysis lies in the geometric constraints imposed by each strut, which can be expressed through loop closure equations.
For each strut $i$ (illustrated in Figure ref:fig:nhexa_stewart_loop_closure), the loop closure equation eqref:eq:nhexa_loop_closure can be written.

\begin{equation}\label{eq:nhexa_loop_closure}
  {}^A\bm{P}_B = {}^A\bm{a}_i + l_i{}^A\hat{\bm{s}}_i - \underbrace{{}^B\bm{b}_i}_{{}^A\bm{R}_B {}^B\bm{b}_i} \quad \text{for } i=1 \text{ to } 6
\end{equation}

Such equation links the 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 $l_i {}^A\hat{\bm{s}}_i$:

#+name: fig:nhexa_stewart_loop_closure
#+caption: Notations to compute the kinematic loop closure
[[file:figs/nhexa_stewart_loop_closure.png]]

**** Inverse Kinematics

The inverse kinematic problem involves determining the required strut lengths $\bm{\mathcal{L}} = \left[ l_1, l_2, \ldots, l_6 \right]^T$ for a desired platform pose $\bm{\mathcal{X}}$ (i.e. position ${}^A\bm{P}$ and orientation ${}^A\bm{R}_B$).
This problem can be solved analytically using the loop closure equations eqref:eq:nhexa_loop_closure.
The obtain strut lengths are given by eqref:eq:nhexa_inverse_kinematics.

\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

The forward kinematic problem seeks to determine the platform pose $\bm{\mathcal{X}}$ given a set of strut lengths $\bm{\mathcal{L}}$.
Unlike the inverse kinematics, this presents a significant challenge as it requires solving a system of nonlinear equations.
While various numerical methods exist for solving this problem, they can be computationally intensive and may not guarantee convergence to the correct solution.

For the nano-hexapod application, where displacements are typically small, an approximate solution based on linearization around the operating point provides a practical alternative.
This approximation, developed in subsequent sections through the Jacobian matrix analysis, proves particularly useful for real-time control applications.

# TODO - Add references

** The Jacobian Matrix
<<ssec:nhexa_stewart_platform_jacobian>>
**** Introduction                                                  :ignore:

The Jacobian matrix plays a central role in analyzing the Stewart platform's behavior, providing a linear mapping between platform and actuator velocities.
While the previously derived kinematic relationships are essential for position analysis, the Jacobian enables velocity analysis and forms the foundation for both static and dynamic studies.

**** Jacobian Computation - Velocity Loop Closure

As was shown in Section ref:ssec:nhexa_stewart_platform_kinematics, the strut lengths $\bm{\mathcal{L}}$ and the platform pose $\bm{\mathcal{X}}$ are related through a system of nonlinear algebraic equations representing the kinematic constraints imposed by the struts.

By taking the time derivative of the position loop close eqref:eq:nhexa_loop_closure, the /velocity loop closure/ is obtained eqref:eq:nhexa_loop_closure_velocity.

\begin{equation}\label{eq:nhexa_loop_closure_velocity}
  {}^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 ${}^A\hat{s}_i$, eqref:eq:nhexa_loop_closure_velocity_bis is obtained.

\begin{equation}\label{eq:nhexa_loop_closure_velocity_bis}
  {}^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}

Equation eqref:eq:nhexa_loop_closure_velocity_bis can be rearranged in a matrix form to obtain eqref:eq:nhexa_jacobian_velocities, with $\dot{\bm{\mathcal{L}}} = [ \dot{l}_1 \ \dots \ \dot{l}_6 ]^T$ the vector of strut velocities, and $\dot{\bm{\mathcal{X}}} = [{}^A\bm{v}_p ,\  {}^A\bm{\omega}]^T$ the vector of platform velocity and angular velocity.

\begin{equation}\label{eq:nhexa_jacobian_velocities}
  \boxed{\dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}}}
\end{equation}

The matrix $\bm{J}$ is called the Jacobian matrix, and is defined by eqref:eq:nhexa_jacobian, with:
- ${}^A\hat{\bm{s}}_i$ the orientation of the struts expressed in $\{A\}$
- ${}^A\bm{b}_i$ the position of the joints with respect to $O_B$ and express in $\{A\}$

\begin{equation}\label{eq:nhexa_jacobian}
  \bm{J} = \begin{bmatrix}
    {{}^A\hat{\bm{s}}_1}^T & ({}^A\bm{b}_1 \times {}^A\hat{\bm{s}}_1)^T \\
    {{}^A\hat{\bm{s}}_2}^T & ({}^A\bm{b}_2 \times {}^A\hat{\bm{s}}_2)^T \\
    {{}^A\hat{\bm{s}}_3}^T & ({}^A\bm{b}_3 \times {}^A\hat{\bm{s}}_3)^T \\
    {{}^A\hat{\bm{s}}_4}^T & ({}^A\bm{b}_4 \times {}^A\hat{\bm{s}}_4)^T \\
    {{}^A\hat{\bm{s}}_5}^T & ({}^A\bm{b}_5 \times {}^A\hat{\bm{s}}_5)^T \\
    {{}^A\hat{\bm{s}}_6}^T & ({}^A\bm{b}_6 \times {}^A\hat{\bm{s}}_6)^T
  \end{bmatrix}
\end{equation}

This Jacobian matrix $\bm{J}$ therefore links the rate of change of strut length to the velocity and angular velocity of the top platform with respect to the fixed base through a set of linear equations.
However, $\bm{J}$ needs to be recomputed for every Stewart platform pose as it depends on the actual pose of of the manipulator.

**** Approximate solution of the Forward and Inverse Kinematic problems

For small displacements $\delta \bm{\mathcal{X}} = [\delta x, \delta y, \delta z, \delta \theta_x, \delta \theta_y, \delta \theta_z ]^T$ around an operating point $\bm{\mathcal{X}}_0$ (for which the Jacobian was computed), the associated joint displacement $\delta\bm{\mathcal{L}} = [\delta l_1,\,\delta l_2,\,\delta l_3,\,\delta l_4,\,\delta l_5,\,\delta l_6]^T$ 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 $\delta\bm{\mathcal{L}}$, 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 /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

The accuracy of the Jacobian-based forward kinematics solution was estimated through a systematic error analysis.
For a series of platform positions along the $x$-axis, the exact strut lengths are computed using the analytical inverse kinematics equation eqref:eq:nhexa_inverse_kinematics.
These strut lengths are then used with the Jacobian to estimate the platform pose, from which the error between the estimated and true poses can be calculated.

The estimation errors in the $x$, $y$, and $z$ directions are shown in Figure ref:fig:nhexa_forward_kinematics_approximate_errors.
The results demonstrate that for displacements up to approximately $1\,\%$ of the hexapod's size (which corresponds to $100\,\mu m$ as the size of the Stewart platform is here $\approx 100\,mm$), the Jacobian approximation provides excellent accuracy.

This finding has particular significance for the Nano-hexapod application.
Since the maximum required stroke ($\approx 100\,\mu m$) is three orders of magnitude smaller than the stewart platform size ($\approx 100\,mm$), the Jacobian matrix can be considered constant throughout the workspace.
It can be computed once at the rest position and used for both forward and inverse kinematics with high accuracy.

#+begin_src matlab
%% Estimate the errors associated with approximate forward kinematics using the Jacobian matrix
stewart = initializeSimplifiedNanoHexapod('H', 100e-3, 'MO_B', 0);

Xrs = logspace(-6, -1, 100); % Wanted X translation of the mobile platform [m]

% Compute the strut exact length for each X-position
Ls_exact = zeros(6, length(Xrs));
for i = 1:length(Xrs)
    [~, L_exact(:, i)] = inverseKinematics(stewart, 'AP', [Xrs(i); 0; 0]);
end

% From the strut length, compute the stewart pose using the Jacobian matrix
Xrs_approx = zeros(6, length(Xrs));
for i = 1:length(Xrs)
    Xrs_approx(:, i) = inv(stewart.geometry.J)*L_exact(:, i);
end
#+end_src

#+begin_src matlab :exports none :results none
%% Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem
figure;
hold on;
plot(1e6*Xrs, 1e9*abs(Xrs - Xrs_approx(1, :)), 'DisplayName', '$\epsilon_x$');
plot(1e6*Xrs, 1e9*abs(Xrs_approx(2, :)), 'DisplayName', '$\epsilon_y$');
plot(1e6*Xrs, 1e9*abs(Xrs_approx(3, :)), 'DisplayName', '$\epsilon_z$');
plot(1e6*Xrs, 1e6*Xrs, 'k--', 'DisplayName', '$0.1\%$ error');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
leg = legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 1);
leg.ItemTokenSize(1) = 15;
xlim([1, 1e4]); ylim([1, 1e4]);
xticks([1, 10, 100, 1000, 10000]);
yticks([1, 10, 100, 1000, 10000]);
xticklabels({'$1\mu m$', '$10\mu m$', '$100\mu m$', '$1mm$', '$10mm$'});
yticklabels({'$1nm$', '$10nm$', '$100nm$', '$1\mu m$', '$10\mu m$'});
#+end_src

#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/nhexa_forward_kinematics_approximate_errors.pdf', 'width', 'wide', 'height', 'normal');
#+end_src

#+name: fig:nhexa_forward_kinematics_approximate_errors
#+caption: Errors associated with the use of the Jacobian matrix to solve the forward kinematic problem. A Stewart platform with an height of $100\,mm$ was used to perform this analysis
#+RESULTS:
[[file:figs/nhexa_forward_kinematics_approximate_errors.png]]

**** Static Forces

The static force analysis of the Stewart platform can be elegantly performed using the principle of virtual work.
This principle states that, for a system in static equilibrium, the total virtual work of all forces acting on the system must be zero for any virtual displacement compatible with the system's constraints.

Let $\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T$ represent the vector of actuator forces applied in each strut, and $\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T$ denote the external wrench (combined force $\bm{f}$ and torque $\bm{n}$) acting on the mobile platform at point $\bm{O}_B$.
The virtual work $\delta W$ consists of two contributions:
- The work performed by the actuator forces through virtual strut displacements $\delta \bm{\mathcal{L}}$: $\bm{\tau}^T \delta \bm{\mathcal{L}}$
- The work performed by the external wrench through virtual platform displacements $\delta \bm{\mathcal{X}}$: $-\bm{\mathcal{F}}^T \delta \bm{\mathcal{X}}$

The principle of virtual work can thus be expressed as:
\begin{equation}
\delta W = \bm{\tau}^T \delta \bm{\mathcal{L}} - \bm{\mathcal{F}}^T \delta \bm{\mathcal{X}} = 0
\end{equation}

Using the Jacobian relationship that links virtual displacements eqref:eq:nhexa_inverse_kinematics_approximate, this equation becomes:
\begin{equation}
\left( \bm{\tau}^T \bm{J} - \bm{\mathcal{F}}^T \right) \delta \bm{\mathcal{X}} = 0
\end{equation}

Since this equation must hold for any virtual displacement $\delta \bm{\mathcal{X}}$, the following force mapping relationships can be derived:

\begin{equation}\label{eq:nhexa_jacobian_forces}
\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}

These equations establish that the transpose of the Jacobian matrix maps actuator forces to platform forces and torques, while its inverse transpose maps platform forces and torques to required actuator forces.

** Static Analysis
<<ssec:nhexa_stewart_platform_static>>

The static stiffness characteristics of the Stewart platform play a crucial role in its performance, particularly for precision positioning applications.
These characteristics are fundamentally determined by both the actuator properties and the platform geometry.

Starting from the individual actuators, the relationship between applied force $\delta \tau_i$ and resulting displacement $\delta l_i$ for each strut $i$ is characterized by its stiffness $k_i$:
\begin{equation}
\tau_i = k_i \delta l_i, \quad i = 1,\ \dots,\ 6
\end{equation}

These individual relationships can be combined into a matrix form using the diagonal stiffness matrix $\mathcal{K}$:
\begin{equation}
\bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}}, \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right]
\end{equation}

By applying the force mapping relationships eqref:eq:nhexa_jacobian_forces derived in the previous section and the Jacobian relationship for small displacements eqref:eq:nhexa_forward_kinematics_approximate, the relationship between applied wrench $\bm{\mathcal{F}}$ and resulting platform displacement $\delta \bm{\mathcal{X}}$ is obtained eqref:eq:nhexa_stiffness_matrix.

\begin{equation}\label{eq:nhexa_stiffness_matrix}
\bm{\mathcal{F}} = \underbrace{\bm{J}^T \mathcal{K} \bm{J}}_{\bm{K}} \delta \bm{\mathcal{X}}
\end{equation}
where $\bm{K} = \bm{J}^T \mathcal{K} \bm{J}$ is identified as the platform stiffness matrix.

The inverse relationship is given by the compliance matrix $\bm{C}$:

\begin{equation}
\delta \bm{\mathcal{X}} = \underbrace{(\bm{J}^T \mathcal{K} \bm{J})^{-1}}_{\bm{C}} \bm{\mathcal{F}}
\end{equation}

These relationships reveal that the overall platform stiffness and compliance characteristics are determined by two factors:
- The individual actuator stiffnesses represented by $\mathcal{K}$
- The geometric configuration embodied in the Jacobian matrix $\bm{J}$

This geometric dependency means that the platform's stiffness varies throughout its workspace, as the Jacobian matrix changes with the platform's position and orientation.
For the NASS application, where the workspace is relatively small compared to the platform dimensions, these variations can be considered minimal.
However, the initial geometric configuration significantly impacts the overall stiffness characteristics.
The relationship between maximum stroke and stiffness presents another important design consideration.
As both parameters are influenced by the geometric configuration, their optimization involves inherent trade-offs that must be carefully balanced based on application requirements.
The optimization of this configuration to achieve desired stiffness properties while having enough stroke will be addressed during the detailed design phase.

** Dynamic Analysis
<<ssec:nhexa_stewart_platform_dynamics>>

The dynamic behavior of a Stewart platform can be analyzed through various approaches, depending on the desired level of model fidelity.
For initial analysis, we consider a simplified model with the following assumptions:
- Massless struts
- Ideal joints without friction or compliance
- Rigid platform and base

Under these assumptions, the system dynamics can be expressed in the Cartesian space as:
\begin{equation}
M s^2 \mathcal{X} = \Sigma \mathcal{F}
\end{equation}

where $M$ represents the platform mass matrix, $\mathcal{X}$ the platform pose, and $\Sigma \mathcal{F}$ the sum of forces acting on the platform.

The primary forces acting on the system are actuator forces $\bm{\tau}$, elastic forces due to strut stiffness $-\mathcal{K} \mathcal{L}$ and damping forces in the struts $\mathcal{C} \dot{\mathcal{L}}$.

\begin{equation}
\Sigma \bm{\mathcal{F}} = \bm{J}^T (\tau - \mathcal{K} \mathcal{L} - s \mathcal{C} \mathcal{L}), \quad \mathcal{K} = \text{diag}(k_1\,\dots\,k_6),\  \mathcal{C} = \text{diag}(c_1\,\dots\,c_6)
\end{equation}

Combining these forces and using eqref:eq:nhexa_forward_kinematics_approximate yields the complete dynamic equation eqref:eq:nhexa_dynamical_equations.

\begin{equation}\label{eq:nhexa_dynamical_equations}
M s^2 \mathcal{X} = \mathcal{F} - J^T \mathcal{K} J \mathcal{X} - J^T \mathcal{C} J s \mathcal{X}
\end{equation}

The transfer function in the Cartesian frame becomes eqref:eq:nhexa_transfer_function_cart.

\begin{equation}\label{eq:nhexa_transfer_function_cart}
\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}

Through coordinate transformation using the Jacobian matrix, the dynamics in the actuator space is obtained eqref:eq:nhexa_transfer_function_struts.

\begin{equation}\label{eq:nhexa_transfer_function_struts}
\frac{\mathcal{L}}{\tau}(s) = ( \bm{J}^{-T} M \bm{J}^{-1} s^2 + \mathcal{C} + \mathcal{K} )^{-1}
\end{equation}

While this simplified model provides useful insights, real Stewart platforms exhibit more complex behaviors.
Several factors significantly increase model complexity:
- Strut dynamics, including mass distribution and internal resonances
- Joint compliance and friction effects
- Supporting structure dynamics and payload dynamics, which are both very critical for NASS

# TODO - Add citations [[cite:&mcinroy00_desig_contr_flexur_joint_hexap;&mcinroy02_model_desig_flexur_joint_stewar]]

These additional effects make analytical modeling impractical for complete system analysis.

** Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:

The fundamental characteristics of the Stewart platform have been analyzed in this chapter.
Essential kinematic relationships were developed through loop closure equations, from which both exact and approximate solutions for the inverse and forward kinematic problems were derived.
The Jacobian matrix was established as a central mathematical tool, through which crucial insights into velocity relationships, static force transmission, and dynamic behavior of the platform were obtained.

For the NASS application, where displacements are typically limited to the micrometer range, the accuracy of linearized models using a constant Jacobian matrix has been demonstrated, by which both analysis and control can be significantly simplified.
However, additional complexities such as strut masses, joint compliance, and supporting structure dynamics must be considered in the full dynamic behavior.
This will be performed in the next section using a multi-body model.

All these characteristics (maneuverability, stiffness, dynamics, etc.) are fundamentally determined by the platform's geometry.
While a reasonable geometric configuration will be used to validate the NASS during this conceptual phase, the optimization of these geometric parameters will be explored during the detailed design phase.

* Multi-Body Model
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/nhexa_2_model.m
:END:
<<sec:nhexa_model>>
** Introduction                                                      :ignore:

*Goal*:
- Study the dynamics of Stewart platform
- Instead of working with complex analytical models: a multi-body model is used.
  Complex because has to model the inertia of the struts.
  Cite papers that tries to model the stewart platform analytically
  Advantage: it will be easily included in the model of the NASS

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<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

** Model Definition
<<ssec:nhexa_model_def>>
**** Geometry

#+begin_src matlab
nano_hexapod = initializeSimplifiedNanoHexapod();
#+end_src

The geometry of the Stewart platform (see Figure ref:fig:nhexa_stewart_model_def) is defined by the position of frame $\{F\}$ with respect to $\{M\}$ and by the locations of the joints ${}^Fa_i$ and ${}^Mb_i$.
The point of interest, indicated by frame $\{A\}$ is located $150\,mm$ above the top platform (i.e. above the $\{M\}$ frame).
Parameters that defines the geometry of the nano-hexapod multi-body models are summarized in Table ref:tab:nhexa_stewart_model_geometry.

From this, the orientation $\hat{s}_i$ and length $l_i$ of the struts can be computed, the Jacobian matrix $\bm{J}$ can be computed, and the kinematics of the Stewart platform can be studied.

#+attr_latex: :options [b]{0.6\linewidth}
#+begin_minipage
#+name: fig:nhexa_stewart_model_def
#+caption: Geometry of the stewart platform
#+attr_latex: :float nil :scale 1
[[file:figs/nhexa_stewart_model_def.png]]
#+end_minipage
\hfill
#+attr_latex: :options [b]{0.38\linewidth}
#+begin_minipage
#+begin_scriptsize
#+latex: \centering
#+attr_latex: :environment tabularx :width 0.75\linewidth :placement [b] :align Xrrr
#+attr_latex: :booktabs t :float nil :center nil
|           | $\bm{x}$ | $\bm{y}$ | $\bm{z}$ |
|-----------+----------+----------+----------|
| ${}^MO_B$ | $0$      | $0$      | $150$    |
| ${}^FO_M$ | $0$      | $0$      | $95$     |
| ${}^Fa_1$ | $-92$    | $-77$    | $20$     |
| ${}^Fa_2$ | $92$     | $-77$    | $20$     |
| ${}^Fa_3$ | $113$    | $-41$    | $20$     |
| ${}^Fa_4$ | $21$     | $118$    | $20$     |
| ${}^Fa_5$ | $-21$    | $118$    | $20$     |
| ${}^Fa_6$ | $-113$   | $-41$    | $20$     |
| ${}^Mb_1$ | $-28$    | $-106$   | $-20$    |
| ${}^Mb_2$ | $28$     | $-106$   | $-20$    |
| ${}^Mb_3$ | $106$    | $28$     | $-20$    |
| ${}^Mb_4$ | $78$     | $78$     | $-20$    |
| ${}^Mb_5$ | $-78$    | $78$     | $-20$    |
| ${}^Mb_6$ | $-106$   | $28$     | $-20$    |
#+latex: \captionof{table}{\label{tab:nhexa_stewart_model_geometry}Parameter values in [mm]}
#+end_scriptsize
#+end_minipage

**** Inertia of Plates

Both the fixed base and the top platform are modelled are solid bodies.
The bottom plate is a cylinder with radius of $120\,mm$ (matching the size of the micro-hexapod's top platform) and a thickness of $15\,mm$.
The top plate is also modelled as a cylinder with a radius of $110\,mm$ and a thickness of $15,mm$.
Both have a mass of $5\,kg$.

**** Joints

The top and bottom joints, different number of DoF can be considered.
universal joint, spherical joint, with added axial stiffness and even with added lateral stiffnesses.
For each DoF, stiffnesses can be added.

During the conceptual design phase, bottom joints are modelled with universal joints (2-DoF) while top joints are modelled with spherical joints (3-DoF).
Both have no stiffness along their DoF and are mass-less.

**** Actuators

In its simplest form, the actuators are modelled with one prismatic joint having some internal stiffness and damping.

As was shown

- [ ] Add schematic of the 1DoF model (possibly with added parallel stiffness)
- [ ] Add model parameters for the 2DoF actuator used for the model (choose something reasonable for an APA)


- Actuators: can be modelled as wanted
- Talk about 2DoF and 1DoF


During the conceptual design: Actuators = 1DoF + Encoder (mass-less)
- k = 1N/um
- Force sensor


** Model Dynamics
<<ssec:nhexa_model_dynamics>>

- [ ] Make schematic with inputs and outputs (maybe add frames {F} and {M} to connect to outside world)
- [ ] Screenshot of the obtained multi-body model ?

- 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 parallel on IFF plant?

#+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_kp', 1e4, ...
    '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

#+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

** Nano Hexapod

#+begin_src matlab
initializeSimplifiedNanoHexapod();
#+end_src

** Conclusion
:properties:
:unnumbered: t
:end:

- Validation of multi-body model in a simple case
- Possible to increase the model complexity when required
  - If considered 6dof joint stiffness, model order increases
  - Can have an effect on IFF performances: [[cite:&preumont07_six_axis_singl_stage_activ]]
  - Conclusion: during the conceptual design, we consider a perfect, but will be taken into account later
  - Optimization of the Flexible joint will be performed in Chapter 2.2
- MIMO system: how to control? => next section

* Control of Stewart Platforms
:PROPERTIES:
:HEADER-ARGS:matlab+: :tangle matlab/nhexa_3_control.m
:END:
<<sec:nhexa_control>>
** Introduction                                                      :ignore:

MIMO control: much more complex than SISO control because of interaction.
Possible to ignore interaction when good decoupling is achieved.
Important to have tools to study interaction
Different ways to try to decouple a MIMO plant.

Reference book: [[cite:&skogestad07_multiv_feedb_contr]]

** Matlab Init                                              :noexport:ignore:
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir>>
#+end_src

#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init>>
#+end_src

#+begin_src matlab :tangle no :noweb yes
<<m-init-path>>
#+end_src

#+begin_src matlab :eval no :noweb yes
<<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

** Centralized and Decentralized Control
<<ssec:nhexa_control_centralized_decentralized>>

- Explain what is centralized and decentralized:
  - 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?
  Study of interaction: RGA

** Choice of the control space
<<ssec:nhexa_control_space>>

- [ ] file:~/Cloud/research/matlab/decoupling-strategies/svd-control.org

- 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)

*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>>

Guaranteed stability: [[cite:&preumont08_trans_zeros_struc_contr_with]]
- [ ] I think there is another paper about that

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?

** Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:


* Conclusion
:PROPERTIES:
:UNNUMBERED: t
:END:
<<sec:nhexa_conclusion>>

- Configurable Stewart platform model
- Will be included in the multi-body model of the micro-station => nass multi body model
- Control: complex problem, try to use simplest architecture

* Bibliography                                                        :ignore:
#+latex: \printbibliography[heading=bibintoc,title={Bibliography}]

* Helping Functions                                                 :noexport:
** Initialize Path
#+NAME: m-init-path
#+BEGIN_SRC matlab
addpath('./matlab/');     % Path for scripts

%% Path for functions, data and scripts
addpath('./matlab/mat/'); % Path for Computed FRF
addpath('./matlab/src/'); % Path for functions
addpath('./matlab/subsystems/'); % Path for Subsystems Simulink files

%% Data directory
data_dir = './matlab/mat/'
#+END_SRC

#+NAME: m-init-path-tangle
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./mat/'); % Path for Data
addpath('./src/'); % Path for functions
addpath('./subsystems/'); % Path for Subsystems Simulink files

%% Data directory
data_dir = './mat/';
#+END_SRC

** Initialize Simscape Model
#+NAME: m-init-simscape
#+begin_src matlab
% Simulink Model name
mdl = 'nano_hexapod_model';
#+end_src

** Initialize other elements
#+NAME: m-init-other
#+BEGIN_SRC matlab
%% Colors for the figures
colors = colororder;

%% Frequency Vector
freqs = logspace(0, 3, 1000);
#+END_SRC

* Matlab Functions                                                  :noexport:
** =initializeLoggingConfiguration=: Logging Configuration
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/initializeLoggingConfiguration.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:

*** Function description
#+begin_src matlab
  function [] = initializeLoggingConfiguration(args)
#+end_src

*** Optional Parameters
#+begin_src matlab
  arguments
    args.log      char   {mustBeMember(args.log,{'none', 'all', 'forces'})} = 'none'
    args.Ts (1,1) double {mustBeNumeric, mustBePositive} = 1e-3
  end
#+end_src

*** Structure initialization
#+begin_src matlab
  conf_log = struct();
#+end_src

*** Add Type
#+begin_src matlab
  switch args.log
    case 'none'
      conf_log.type = 0;
    case 'all'
      conf_log.type = 1;
    case 'forces'
      conf_log.type = 2;
  end
#+end_src

*** Sampling Time
#+begin_src matlab
  conf_log.Ts = args.Ts;
#+end_src

*** Save the Structure
#+begin_src matlab
if exist('./mat', 'dir')
    if exist('./mat/nano_hexapod_model_conf_log.mat', 'file')
        save('mat/nano_hexapod_model_conf_log.mat', 'conf_log', '-append');
    else
        save('mat/nano_hexapod_model_conf_log.mat', 'conf_log');
    end
elseif exist('./matlab', 'dir')
    if exist('./matlab/mat/nano_hexapod_model_conf_log.mat', 'file')
        save('matlab/mat/nano_hexapod_model_conf_log.mat', 'conf_log', '-append');
    else
        save('matlab/mat/nano_hexapod_model_conf_log.mat', 'conf_log');
    end
end
#+end_src

** =initializeController=: Initialize Controller
#+begin_src matlab :tangle matlab/src/initializeController.m :comments none :mkdirp yes :eval no
function [] = initializeController(args)

    arguments
        args.type char {mustBeMember(args.type,{'open-loop', 'iff', 'dvf', 'hac-dvf', 'ref-track-L', 'ref-track-iff-L', 'cascade-hac-lac', 'hac-iff', 'stabilizing'})} = 'open-loop'
    end

    controller = struct();

    switch args.type
      case 'open-loop'
        controller.type = 1;
        controller.name = 'Open-Loop';
      case 'dvf'
        controller.type = 2;
        controller.name = 'Decentralized Direct Velocity Feedback';
      case 'iff'
        controller.type = 3;
        controller.name = 'Decentralized Integral Force Feedback';
      case 'hac-dvf'
        controller.type = 4;
        controller.name = 'HAC-DVF';
      case 'ref-track-L'
        controller.type = 5;
        controller.name = 'Reference Tracking in the frame of the legs';
      case 'ref-track-iff-L'
        controller.type = 6;
        controller.name = 'Reference Tracking in the frame of the legs + IFF';
      case 'cascade-hac-lac'
        controller.type = 7;
        controller.name = 'Cascade Control + HAC-LAC';
      case 'hac-iff'
        controller.type = 8;
        controller.name = 'HAC-IFF';
      case 'stabilizing'
        controller.type = 9;
        controller.name = 'Stabilizing Controller';
    end

    if exist('./mat', 'dir')
        save('mat/nano_hexapod_model_controller.mat', 'controller');
    elseif exist('./matlab', 'dir')
        save('matlab/mat/nano_hexapod_model_controller.mat', 'controller');
    end

end
#+end_src

** =initializeSample=: Sample

#+begin_src matlab :tangle matlab/src/initializeSample.m :comments none :mkdirp yes :eval no
function [sample] = initializeSample(args)

    arguments
        args.type char {mustBeMember(args.type,{'none', 'cylindrical'})} = 'none'
        args.H (1,1) double {mustBeNumeric, mustBePositive} = 200e-3 % Height [m]
        args.R (1,1) double {mustBeNumeric, mustBePositive} = 110e-3 % Radius [m]
        args.m (1,1) double {mustBeNumeric, mustBePositive} = 1 % Mass [kg]
    end

    sample = struct();

    switch args.type
      case 'none'
        sample.type = 0;
        sample.m = 0;
      case 'cylindrical'
        sample.type = 1;

        sample.H = args.H;
        sample.R = args.R;
        sample.m = args.m;
    end

    if exist('./mat', 'dir')
        if exist('./mat/nano_hexapod.mat', 'file')
            save('mat/nano_hexapod.mat', 'sample', '-append');
        else
            save('mat/nano_hexapod.mat', 'sample');
        end
    elseif exist('./matlab', 'dir')
        if exist('./matlab/mat/nano_hexapod.mat', 'file')
            save('matlab/mat/nano_hexapod.mat', 'sample', '-append');
        else
            save('matlab/mat/nano_hexapod.mat', 'sample');
        end
    end

end
#+end_src

** Stewart platform
*** =initializeSimplifiedNanoHexapod=: Nano Hexapod

#+begin_src matlab :tangle matlab/src/initializeSimplifiedNanoHexapod.m :comments none :mkdirp yes :eval no
function [nano_hexapod] = initializeSimplifiedNanoHexapod(args)

    arguments
        %% initializeFramesPositions
        args.H    (1,1) double {mustBeNumeric, mustBePositive} = 95e-3 % Height of the nano-hexapod [m]
        args.MO_B (1,1) double {mustBeNumeric} = 150e-3  % Height of {B} w.r.t. {M} [m]
        %% generateGeneralConfiguration
        args.FH  (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 % Height of fixed joints [m]
        args.FR  (1,1) double {mustBeNumeric, mustBePositive} = 120e-3 % Radius of fixed joints [m]
        args.FTh (6,1) double {mustBeNumeric} = [220, 320, 340, 80, 100, 200]*(pi/180) % Angles of fixed joints [rad]
        args.MH  (1,1) double {mustBeNumeric, mustBePositive} = 15e-3 % Height of mobile joints [m]
        args.MR  (1,1) double {mustBeNumeric, mustBePositive} = 110e-3 % Radius of mobile joints [m]
        args.MTh (6,1) double {mustBeNumeric} = [255, 285, 15, 45, 135, 165]*(pi/180) % Angles of fixed joints [rad]
        %% Actuators
        args.actuator_type char {mustBeMember(args.actuator_type,{'1dof', '2dof', 'flexible'})} = '1dof'
        args.actuator_k   (1,1) double {mustBeNumeric, mustBePositive} = 380000
        args.actuator_kp  (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.actuator_ke  (1,1) double {mustBeNumeric, mustBePositive} = 4952605
        args.actuator_ka  (1,1) double {mustBeNumeric, mustBePositive} = 2476302
        args.actuator_c   (1,1) double {mustBeNumeric, mustBePositive} = 5
        args.actuator_cp  (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.actuator_ce  (1,1) double {mustBeNumeric, mustBePositive} = 100
        args.actuator_ca  (1,1) double {mustBeNumeric, mustBePositive} = 50
        %% initializeCylindricalPlatforms
        args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 5 % Mass of the fixed plate [kg]
        args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3 % Thickness of the fixed plate [m]
        args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 150e-3 % Radius of the fixed plate [m]
        args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 5 % Mass of the mobile plate [kg]
        args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3 % Thickness of the mobile plate [m]
        args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 150e-3 % Radius of the mobile plate [m]
        %% initializeCylindricalStruts
        args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % Mass of the fixed part of the strut [kg]
        args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 60e-3 % Length of the fixed part of the struts [m]
        args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3 % Radius of the fixed part of the struts [m]
        args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 1e-3 % Mass of the mobile part of the strut [kg]
        args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 60e-3 % Length of the mobile part of the struts [m]
        args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3 % Radius of the fixed part of the struts [m]
        %% Bottom and Top Flexible Joints
        args.flex_type_F char {mustBeMember(args.flex_type_F,{'2dof', '3dof', '4dof', '6dof', 'flexible'})} = '2dof'
        args.flex_type_M char {mustBeMember(args.flex_type_M,{'2dof', '3dof', '4dof', '6dof', 'flexible'})} = '3dof'
        args.Kf_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Cf_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Kt_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ct_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Kf_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Cf_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Kt_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ct_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ka_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ca_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Kr_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Cr_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ka_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ca_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Kr_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Cr_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        %% inverseKinematics
        args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
        args.ARB (3,3) double {mustBeNumeric} = eye(3)
    end

    stewart = initializeStewartPlatform();

    stewart = initializeFramesPositions(stewart, ...
                                        'H', args.H, ...
                                        'MO_B', args.MO_B);

    stewart = generateGeneralConfiguration(stewart, ...
                                           'FH', args.FH, ...
                                           'FR', args.FR, ...
                                           'FTh', args.FTh, ...
                                           'MH', args.MH, ...
                                           'MR', args.MR, ...
                                           'MTh', args.MTh);

    stewart = computeJointsPose(stewart);

    stewart = initializeStrutDynamics(stewart, ...
                                      'type', args.actuator_type, ...
                                      'k',  args.actuator_k, ...
                                      'kp',  args.actuator_kp, ...
                                      'ke', args.actuator_ke, ...
                                      'ka', args.actuator_ka, ...
                                      'c',  args.actuator_c, ...
                                      'cp',  args.actuator_cp, ...
                                      'ce', args.actuator_ce, ...
                                      'ca', args.actuator_ca);

    stewart = initializeJointDynamics(stewart, ...
                                      'type_F', args.flex_type_F, ...
                                      'type_M', args.flex_type_M, ...
                                      'Kf_M', args.Kf_M, ...
                                      'Cf_M', args.Cf_M, ...
                                      'Kt_M', args.Kt_M, ...
                                      'Ct_M', args.Ct_M, ...
                                      'Kf_F', args.Kf_F, ...
                                      'Cf_F', args.Cf_F, ...
                                      'Kt_F', args.Kt_F, ...
                                      'Ct_F', args.Ct_F, ...
                                      'Ka_F', args.Ka_F, ...
                                      'Ca_F', args.Ca_F, ...
                                      'Kr_F', args.Kr_F, ...
                                      'Cr_F', args.Cr_F, ...
                                      'Ka_M', args.Ka_M, ...
                                      'Ca_M', args.Ca_M, ...
                                      'Kr_M', args.Kr_M, ...
                                      'Cr_M', args.Cr_M);

    stewart = initializeCylindricalPlatforms(stewart, ...
                                             'Fpm', args.Fpm, ...
                                             'Fph', args.Fph, ...
                                             'Fpr', args.Fpr, ...
                                             'Mpm', args.Mpm, ...
                                             'Mph', args.Mph, ...
                                             'Mpr', args.Mpr);

    stewart = initializeCylindricalStruts(stewart, ...
                                          'Fsm', args.Fsm, ...
                                          'Fsh', args.Fsh, ...
                                          'Fsr', args.Fsr, ...
                                          'Msm', args.Msm, ...
                                          'Msh', args.Msh, ...
                                          'Msr', args.Msr);

    stewart = computeJacobian(stewart);

    stewart = initializeStewartPose(stewart, ...
                                    'AP', args.AP, ...
                                    'ARB', args.ARB);

    nano_hexapod = stewart;
    if exist('./mat', 'dir')
        if exist('./mat/nano_hexapod.mat', 'file')
            save('mat/nano_hexapod.mat', 'nano_hexapod', '-append');
        else
            save('mat/nano_hexapod.mat', 'nano_hexapod');
        end
    elseif exist('./matlab', 'dir')
        if exist('./matlab/mat/nano_hexapod.mat', 'file')
            save('matlab/mat/nano_hexapod.mat', 'nano_hexapod', '-append');
        else
            save('matlab/mat/nano_hexapod.mat', 'nano_hexapod');
        end
    end
end
#+end_src

*** =initializeStewartPlatform=: Initialize the Stewart Platform structure

#+begin_src matlab :tangle matlab/src/initializeStewartPlatform.m :comments none :mkdirp yes :eval no
function [stewart] = initializeStewartPlatform()
% initializeStewartPlatform - Initialize the stewart structure
%
% Syntax: [stewart] = initializeStewartPlatform(args)
%
% Outputs:
%    - stewart - A structure with the following sub-structures:
%      - platform_F -
%      - platform_M -
%      - joints_F   -
%      - joints_M   -
%      - struts_F   -
%      - struts_M   -
%      - actuators  -
%      - geometry   -
%      - properties -

    stewart = struct();
    stewart.platform_F = struct();
    stewart.platform_M = struct();
    stewart.joints_F   = struct();
    stewart.joints_M   = struct();
    stewart.struts_F   = struct();
    stewart.struts_M   = struct();
    stewart.actuators  = struct();
    stewart.sensors    = struct();
    stewart.sensors.inertial = struct();
    stewart.sensors.force    = struct();
    stewart.sensors.relative = struct();
    stewart.geometry   = struct();
    stewart.kinematics = struct();

end
#+end_src

*** =initializeFramesPositions=: Initialize the positions of frames {A}, {B}, {F} and {M}

#+begin_src matlab :tangle matlab/src/initializeFramesPositions.m :comments none :mkdirp yes :eval no
function [stewart] = initializeFramesPositions(stewart, args)
% initializeFramesPositions - Initialize the positions of frames {A}, {B}, {F} and {M}
%
% Syntax: [stewart] = initializeFramesPositions(stewart, args)
%
% Inputs:
%    - args - Can have the following fields:
%        - H    [1x1] - Total Height of the Stewart Platform (height from {F} to {M}) [m]
%        - MO_B [1x1] - Height of the frame {B} with respect to {M} [m]
%
% Outputs:
%    - stewart - A structure with the following fields:
%        - geometry.H      [1x1] - Total Height of the Stewart Platform [m]
%        - geometry.FO_M   [3x1] - Position of {M} with respect to {F} [m]
%        - platform_M.MO_B [3x1] - Position of {B} with respect to {M} [m]
%        - platform_F.FO_A [3x1] - Position of {A} with respect to {F} [m]

    arguments
        stewart
        args.H    (1,1) double {mustBeNumeric, mustBePositive} = 90e-3
        args.MO_B (1,1) double {mustBeNumeric} = 50e-3
    end

    H = args.H; % Total Height of the Stewart Platform [m]

    FO_M = [0; 0; H]; % Position of {M} with respect to {F} [m]

    MO_B = [0; 0; args.MO_B]; % Position of {B} with respect to {M} [m]

    FO_A = MO_B + FO_M; % Position of {A} with respect to {F} [m]

    stewart.geometry.H      = H;
    stewart.geometry.FO_M   = FO_M;
    stewart.platform_M.MO_B = MO_B;
    stewart.platform_F.FO_A = FO_A;

end
#+end_src

*** =generateGeneralConfiguration=: Generate a Very General Configuration

#+begin_src matlab :tangle matlab/src/generateGeneralConfiguration.m :comments none :mkdirp yes :eval no
function [stewart] = generateGeneralConfiguration(stewart, args)
% generateGeneralConfiguration - Generate a Very General Configuration
%
% Syntax: [stewart] = generateGeneralConfiguration(stewart, args)
%
% Inputs:
%    - args - Can have the following fields:
%        - FH  [1x1] - Height of the position of the fixed joints with respect to the frame {F} [m]
%        - FR  [1x1] - Radius of the position of the fixed joints in the X-Y [m]
%        - FTh [6x1] - Angles of the fixed joints in the X-Y plane with respect to the X axis [rad]
%        - MH  [1x1] - Height of the position of the mobile joints with respect to the frame {M} [m]
%        - FR  [1x1] - Radius of the position of the mobile joints in the X-Y [m]
%        - MTh [6x1] - Angles of the mobile joints in the X-Y plane with respect to the X axis [rad]
%
% Outputs:
%    - stewart - updated Stewart structure with the added fields:
%        - platform_F.Fa  [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
%        - platform_M.Mb  [3x6] - Its i'th column is the position vector of joint bi with respect to {M}

    arguments
        stewart
        args.FH  (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
        args.FR  (1,1) double {mustBeNumeric, mustBePositive} = 115e-3;
        args.FTh (6,1) double {mustBeNumeric} = [-10, 10, 120-10, 120+10, 240-10, 240+10]*(pi/180);
        args.MH  (1,1) double {mustBeNumeric, mustBePositive} = 15e-3
        args.MR  (1,1) double {mustBeNumeric, mustBePositive} = 90e-3;
        args.MTh (6,1) double {mustBeNumeric} = [-60+10, 60-10, 60+10, 180-10, 180+10, -60-10]*(pi/180);
    end

    Fa = zeros(3,6);
    Mb = zeros(3,6);

    for i = 1:6
        Fa(:,i) = [args.FR*cos(args.FTh(i)); args.FR*sin(args.FTh(i));  args.FH];
        Mb(:,i) = [args.MR*cos(args.MTh(i)); args.MR*sin(args.MTh(i)); -args.MH];
    end

    stewart.platform_F.Fa = Fa;
    stewart.platform_M.Mb = Mb;

end
#+end_src

*** =computeJointsPose=: Compute the Pose of the Joints

#+begin_src matlab :tangle matlab/src/computeJointsPose.m :comments none :mkdirp yes :eval no
function [stewart] = computeJointsPose(stewart)
% computeJointsPose -
%
% Syntax: [stewart] = computeJointsPose(stewart)
%
% Inputs:
%    - stewart - A structure with the following fields
%        - platform_F.Fa   [3x6] - Its i'th column is the position vector of joint ai with respect to {F}
%        - platform_M.Mb   [3x6] - Its i'th column is the position vector of joint bi with respect to {M}
%        - platform_F.FO_A [3x1] - Position of {A} with respect to {F}
%        - platform_M.MO_B [3x1] - Position of {B} with respect to {M}
%        - geometry.FO_M   [3x1] - Position of {M} with respect to {F}
%
% Outputs:
%    - stewart - A structure with the following added fields
%        - geometry.Aa    [3x6]   - The i'th column is the position of ai with respect to {A}
%        - geometry.Ab    [3x6]   - The i'th column is the position of bi with respect to {A}
%        - geometry.Ba    [3x6]   - The i'th column is the position of ai with respect to {B}
%        - geometry.Bb    [3x6]   - The i'th column is the position of bi with respect to {B}
%        - geometry.l     [6x1]   - The i'th element is the initial length of strut i
%        - geometry.As    [3x6]   - The i'th column is the unit vector of strut i expressed in {A}
%        - geometry.Bs    [3x6]   - The i'th column is the unit vector of strut i expressed in {B}
%        - struts_F.l     [6x1]   - Length of the Fixed part of the i'th strut
%        - struts_M.l     [6x1]   - Length of the Mobile part of the i'th strut
%        - platform_F.FRa [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the bottom of the i'th strut from {F}
%        - platform_M.MRb [3x3x6] - The i'th 3x3 array is the rotation matrix to orientate the top of the i'th strut from {M}

    assert(isfield(stewart.platform_F, 'Fa'),   'stewart.platform_F should have attribute Fa')
    Fa = stewart.platform_F.Fa;

    assert(isfield(stewart.platform_M, 'Mb'),   'stewart.platform_M should have attribute Mb')
    Mb = stewart.platform_M.Mb;

    assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
    FO_A = stewart.platform_F.FO_A;

    assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
    MO_B = stewart.platform_M.MO_B;

    assert(isfield(stewart.geometry,   'FO_M'), 'stewart.geometry should have attribute FO_M')
    FO_M = stewart.geometry.FO_M;

    Aa = Fa - repmat(FO_A, [1, 6]);
    Bb = Mb - repmat(MO_B, [1, 6]);

    Ab = Bb - repmat(-MO_B-FO_M+FO_A, [1, 6]);
    Ba = Aa - repmat( MO_B+FO_M-FO_A, [1, 6]);

    As = (Ab - Aa)./vecnorm(Ab - Aa); % As_i is the i'th vector of As

    l = vecnorm(Ab - Aa)';

    Bs = (Bb - Ba)./vecnorm(Bb - Ba);

    FRa = zeros(3,3,6);
    MRb = zeros(3,3,6);

    for i = 1:6
        FRa(:,:,i) = [cross([0;1;0], As(:,i)) , cross(As(:,i), cross([0;1;0], As(:,i))) , As(:,i)];
        FRa(:,:,i) = FRa(:,:,i)./vecnorm(FRa(:,:,i));

        MRb(:,:,i) = [cross([0;1;0], Bs(:,i)) , cross(Bs(:,i), cross([0;1;0], Bs(:,i))) , Bs(:,i)];
        MRb(:,:,i) = MRb(:,:,i)./vecnorm(MRb(:,:,i));
    end

    stewart.geometry.Aa = Aa;
    stewart.geometry.Ab = Ab;
    stewart.geometry.Ba = Ba;
    stewart.geometry.Bb = Bb;
    stewart.geometry.As = As;
    stewart.geometry.Bs = Bs;
    stewart.geometry.l  = l;

    stewart.struts_F.l  = l/2;
    stewart.struts_M.l  = l/2;

    stewart.platform_F.FRa = FRa;
    stewart.platform_M.MRb = MRb;

end
#+end_src

*** =initializeCylindricalPlatforms=: Initialize the geometry of the Fixed and Mobile Platforms

#+begin_src matlab :tangle matlab/src/initializeCylindricalPlatforms.m :comments none :mkdirp yes :eval no
function [stewart] = initializeCylindricalPlatforms(stewart, args)
% initializeCylindricalPlatforms - Initialize the geometry of the Fixed and Mobile Platforms
%
% Syntax: [stewart] = initializeCylindricalPlatforms(args)
%
% Inputs:
%    - args - Structure with the following fields:
%        - Fpm [1x1] - Fixed Platform Mass [kg]
%        - Fph [1x1] - Fixed Platform Height [m]
%        - Fpr [1x1] - Fixed Platform Radius [m]
%        - Mpm [1x1] - Mobile Platform Mass [kg]
%        - Mph [1x1] - Mobile Platform Height [m]
%        - Mpr [1x1] - Mobile Platform Radius [m]
%
% Outputs:
%    - stewart - updated Stewart structure with the added fields:
%      - platform_F [struct] - structure with the following fields:
%        - type = 1
%        - M [1x1] - Fixed Platform Mass [kg]
%        - I [3x3] - Fixed Platform Inertia matrix [kg*m^2]
%        - H [1x1] - Fixed Platform Height [m]
%        - R [1x1] - Fixed Platform Radius [m]
%      - platform_M [struct] - structure with the following fields:
%        - M [1x1] - Mobile Platform Mass [kg]
%        - I [3x3] - Mobile Platform Inertia matrix [kg*m^2]
%        - H [1x1] - Mobile Platform Height [m]
%        - R [1x1] - Mobile Platform Radius [m]

    arguments
        stewart
        args.Fpm (1,1) double {mustBeNumeric, mustBePositive} = 1
        args.Fph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
        args.Fpr (1,1) double {mustBeNumeric, mustBePositive} = 125e-3
        args.Mpm (1,1) double {mustBeNumeric, mustBePositive} = 1
        args.Mph (1,1) double {mustBeNumeric, mustBePositive} = 10e-3
        args.Mpr (1,1) double {mustBeNumeric, mustBePositive} = 100e-3
    end

    I_F = diag([1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
                1/12*args.Fpm * (3*args.Fpr^2 + args.Fph^2), ...
                1/2 *args.Fpm * args.Fpr^2]);

    I_M = diag([1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
                1/12*args.Mpm * (3*args.Mpr^2 + args.Mph^2), ...
                1/2 *args.Mpm * args.Mpr^2]);

    stewart.platform_F.type = 1;

    stewart.platform_F.I = I_F;
    stewart.platform_F.M = args.Fpm;
    stewart.platform_F.R = args.Fpr;
    stewart.platform_F.H = args.Fph;

    stewart.platform_M.type = 1;

    stewart.platform_M.I = I_M;
    stewart.platform_M.M = args.Mpm;
    stewart.platform_M.R = args.Mpr;
    stewart.platform_M.H = args.Mph;

end
#+end_src

*** =initializeCylindricalStruts=: Define the inertia of cylindrical struts

#+begin_src matlab :tangle matlab/src/initializeCylindricalStruts.m :comments none :mkdirp yes :eval no
function [stewart] = initializeCylindricalStruts(stewart, args)
% initializeCylindricalStruts - Define the mass and moment of inertia of cylindrical struts
%
% Syntax: [stewart] = initializeCylindricalStruts(args)
%
% Inputs:
%    - args - Structure with the following fields:
%        - Fsm [1x1] - Mass of the Fixed part of the struts [kg]
%        - Fsh [1x1] - Height of cylinder for the Fixed part of the struts [m]
%        - Fsr [1x1] - Radius of cylinder for the Fixed part of the struts [m]
%        - Msm [1x1] - Mass of the Mobile part of the struts [kg]
%        - Msh [1x1] - Height of cylinder for the Mobile part of the struts [m]
%        - Msr [1x1] - Radius of cylinder for the Mobile part of the struts [m]
%
% Outputs:
%    - stewart - updated Stewart structure with the added fields:
%      - struts_F [struct] - structure with the following fields:
%        - M [6x1]   - Mass of the Fixed part of the struts [kg]
%        - I [3x3x6] - Moment of Inertia for the Fixed part of the struts [kg*m^2]
%        - H [6x1]   - Height of cylinder for the Fixed part of the struts [m]
%        - R [6x1]   - Radius of cylinder for the Fixed part of the struts [m]
%      - struts_M [struct] - structure with the following fields:
%        - M [6x1]   - Mass of the Mobile part of the struts [kg]
%        - I [3x3x6] - Moment of Inertia for the Mobile part of the struts [kg*m^2]
%        - H [6x1]   - Height of cylinder for the Mobile part of the struts [m]
%        - R [6x1]   - Radius of cylinder for the Mobile part of the struts [m]

    arguments
        stewart
        args.Fsm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
        args.Fsh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
        args.Fsr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
        args.Msm (1,1) double {mustBeNumeric, mustBePositive} = 0.1
        args.Msh (1,1) double {mustBeNumeric, mustBePositive} = 50e-3
        args.Msr (1,1) double {mustBeNumeric, mustBePositive} = 5e-3
    end

    stewart.struts_M.type = 1;

    stewart.struts_M.M = args.Msm;
    stewart.struts_M.R = args.Msr;
    stewart.struts_M.H = args.Msh;

    stewart.struts_F.type = 1;

    stewart.struts_F.M = args.Fsm;
    stewart.struts_F.R = args.Fsr;
    stewart.struts_F.H = args.Fsh;

end
#+end_src

*** =initializeStrutDynamics=: Add Stiffness and Damping properties of each strut

#+begin_src matlab :tangle matlab/src/initializeStrutDynamics.m :comments none :mkdirp yes :eval no
function [stewart] = initializeStrutDynamics(stewart, args)
% initializeStrutDynamics - Add Stiffness and Damping properties of each strut
%
% Syntax: [stewart] = initializeStrutDynamics(args)
%
% Inputs:
%    - args - Structure with the following fields:
%        - K [6x1] - Stiffness of each strut [N/m]
%        - C [6x1] - Damping of each strut [N/(m/s)]
%
% Outputs:
%    - stewart - updated Stewart structure with the added fields:
%      - actuators.type = 1
%      - actuators.K [6x1] - Stiffness of each strut [N/m]
%      - actuators.C [6x1] - Damping of each strut [N/(m/s)]

    arguments
        stewart
        args.type char {mustBeMember(args.type,{'1dof', '2dof', 'flexible'})} = '1dof'
        args.k  (1,1) double {mustBeNumeric, mustBeNonnegative} = 20e6
        args.kp (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.ke (1,1) double {mustBeNumeric, mustBeNonnegative} = 5e6
        args.ka (1,1) double {mustBeNumeric, mustBeNonnegative} = 60e6
        args.c  (1,1) double {mustBeNumeric, mustBeNonnegative} = 2e1
        args.cp (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.ce (1,1) double {mustBeNumeric, mustBeNonnegative} = 1e6
        args.ca (1,1) double {mustBeNumeric, mustBeNonnegative} = 10

        args.F_gain (1,1) double {mustBeNumeric} = 1
        args.me (1,1) double {mustBeNumeric} = 0.01
        args.ma (1,1) double {mustBeNumeric} = 0.01
    end

    if strcmp(args.type, '1dof')
        stewart.actuators.type = 1;
    elseif strcmp(args.type, '2dof')
        stewart.actuators.type = 2;
    elseif strcmp(args.type, 'flexible')
        stewart.actuators.type = 3;
    end

    stewart.actuators.k = args.k;
    stewart.actuators.c = args.c;

    % Parallel stiffness
    stewart.actuators.kp = args.kp;
    stewart.actuators.cp = args.cp;

    stewart.actuators.ka = args.ka;
    stewart.actuators.ca = args.ca;

    stewart.actuators.ke = args.ke;
    stewart.actuators.ce = args.ce;

    stewart.actuators.F_gain = args.F_gain;

    stewart.actuators.ma = args.ma;
    stewart.actuators.me = args.me;

end
#+end_src

*** =initializeJointDynamics=: Add Stiffness and Damping properties for spherical joints

#+begin_src matlab :tangle matlab/src/initializeJointDynamics.m :comments none :mkdirp yes :eval no
function [stewart] = initializeJointDynamics(stewart, args)
% initializeJointDynamics - Add Stiffness and Damping properties for the spherical joints
%
% Syntax: [stewart] = initializeJointDynamics(args)
%
% Inputs:
%    - args - Structure with the following fields:
%        - type_F - 'universal', 'spherical', 'universal_p', 'spherical_p'
%        - type_M - 'universal', 'spherical', 'universal_p', 'spherical_p'
%        - Kf_M [6x1] - Bending (Rx, Ry) Stiffness for each top joints [(N.m)/rad]
%        - Kt_M [6x1] - Torsion (Rz) Stiffness for each top joints [(N.m)/rad]
%        - Cf_M [6x1] - Bending (Rx, Ry) Damping of each top joint [(N.m)/(rad/s)]
%        - Ct_M [6x1] - Torsion (Rz) Damping of each top joint [(N.m)/(rad/s)]
%        - Kf_F [6x1] - Bending (Rx, Ry) Stiffness for each bottom joints [(N.m)/rad]
%        - Kt_F [6x1] - Torsion (Rz) Stiffness for each bottom joints [(N.m)/rad]
%        - Cf_F [6x1] - Bending (Rx, Ry) Damping of each bottom joint [(N.m)/(rad/s)]
%        - Cf_F [6x1] - Torsion (Rz) Damping of each bottom joint [(N.m)/(rad/s)]
%
% Outputs:
%    - stewart - updated Stewart structure with the added fields:
%      - stewart.joints_F and stewart.joints_M:
%        - type - 1 (universal), 2 (spherical), 3 (universal perfect), 4 (spherical perfect)
%        - Kx, Ky, Kz [6x1] - Translation (Tx, Ty, Tz) Stiffness [N/m]
%        - Kf [6x1] - Flexion (Rx, Ry) Stiffness [(N.m)/rad]
%        - Kt [6x1] - Torsion (Rz) Stiffness [(N.m)/rad]
%        - Cx, Cy, Cz [6x1] - Translation (Rx, Ry) Damping [N/(m/s)]
%        - Cf [6x1] - Flexion (Rx, Ry) Damping [(N.m)/(rad/s)]
%        - Cb [6x1] - Torsion (Rz) Damping [(N.m)/(rad/s)]

    arguments
        stewart
        args.type_F     char   {mustBeMember(args.type_F,{'2dof', '3dof', '4dof', '6dof', 'flexible'})} = '2dof'
        args.type_M     char   {mustBeMember(args.type_M,{'2dof', '3dof', '4dof', '6dof', 'flexible'})} = '3dof'
        args.Kf_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Cf_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Kt_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ct_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Kf_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Cf_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Kt_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ct_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ka_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ca_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Kr_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Cr_F (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ka_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Ca_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Kr_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.Cr_M (1,1) double {mustBeNumeric, mustBeNonnegative} = 0
        args.K_M        double {mustBeNumeric} = zeros(6,6)
        args.M_M        double {mustBeNumeric} = zeros(6,6)
        args.n_xyz_M    double {mustBeNumeric} = zeros(2,3)
        args.xi_M       double {mustBeNumeric} = 0.1
        args.step_file_M char {} = ''
        args.K_F        double {mustBeNumeric} = zeros(6,6)
        args.M_F        double {mustBeNumeric} = zeros(6,6)
        args.n_xyz_F    double {mustBeNumeric} = zeros(2,3)
        args.xi_F       double {mustBeNumeric} = 0.1
        args.step_file_F char {} = ''
    end

    switch args.type_F
      case '2dof'
        stewart.joints_F.type = 1;
      case '3dof'
        stewart.joints_F.type = 2;
      case '4dof'
        stewart.joints_F.type = 3;
      case '6dof'
        stewart.joints_F.type = 4;
      case 'flexible'
        stewart.joints_F.type = 5;
      otherwise
        error("joints_F are not correctly defined")
    end

    switch args.type_M
      case '2dof'
        stewart.joints_M.type = 1;
      case '3dof'
        stewart.joints_M.type = 2;
      case '4dof'
        stewart.joints_M.type = 3;
      case '6dof'
        stewart.joints_M.type = 4;
      case 'flexible'
        stewart.joints_M.type = 5;
      otherwise
        error("joints_M are not correctly defined")
    end

    stewart.joints_M.Ka = args.Ka_M;
    stewart.joints_M.Kr = args.Kr_M;

    stewart.joints_F.Ka = args.Ka_F;
    stewart.joints_F.Kr = args.Kr_F;

    stewart.joints_M.Ca = args.Ca_M;
    stewart.joints_M.Cr = args.Cr_M;

    stewart.joints_F.Ca = args.Ca_F;
    stewart.joints_F.Cr = args.Cr_F;

    stewart.joints_M.Kf = args.Kf_M;
    stewart.joints_M.Kt = args.Kt_M;

    stewart.joints_F.Kf = args.Kf_F;
    stewart.joints_F.Kt = args.Kt_F;

    stewart.joints_M.Cf = args.Cf_M;
    stewart.joints_M.Ct = args.Ct_M;

    stewart.joints_F.Cf = args.Cf_F;
    stewart.joints_F.Ct = args.Ct_F;

    stewart.joints_F.M = args.M_F;
    stewart.joints_F.K = args.K_F;
    stewart.joints_F.n_xyz = args.n_xyz_F;
    stewart.joints_F.xi = args.xi_F;
    stewart.joints_F.xi = args.xi_F;
    stewart.joints_F.step_file = args.step_file_F;

    stewart.joints_M.M = args.M_M;
    stewart.joints_M.K = args.K_M;
    stewart.joints_M.n_xyz = args.n_xyz_M;
    stewart.joints_M.xi = args.xi_M;
    stewart.joints_M.step_file = args.step_file_M;

end
#+end_src

*** =initializeStewartPose=: Determine the initial stroke in each leg to have the wanted pose

#+begin_src matlab :tangle matlab/src/initializeStewartPose.m :comments none :mkdirp yes :eval no
function [stewart] = initializeStewartPose(stewart, args)
% initializeStewartPose - Determine the initial stroke in each leg to have the wanted pose
%                         It uses the inverse kinematic
%
% Syntax: [stewart] = initializeStewartPose(stewart, args)
%
% Inputs:
%    - stewart - A structure with the following fields
%        - Aa   [3x6] - The positions ai expressed in {A}
%        - Bb   [3x6] - The positions bi expressed in {B}
%    - args - Can have the following fields:
%        - AP   [3x1] - The wanted position of {B} with respect to {A}
%        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
%    - stewart - updated Stewart structure with the added fields:
%      - actuators.Leq [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}

    arguments
        stewart
        args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
        args.ARB (3,3) double {mustBeNumeric} = eye(3)
    end

    [Li, dLi] = inverseKinematics(stewart, 'AP', args.AP, 'ARB', args.ARB);

    stewart.actuators.Leq = dLi;

end
#+end_src

*** =computeJacobian=: Compute the Jacobian Matrix

#+begin_src matlab :tangle matlab/src/computeJacobian.m :comments none :mkdirp yes :eval no
function [stewart] = computeJacobian(stewart)
% computeJacobian -
%
% Syntax: [stewart] = computeJacobian(stewart)
%
% Inputs:
%    - stewart - With at least the following fields:
%      - geometry.As [3x6] - The 6 unit vectors for each strut expressed in {A}
%      - geometry.Ab [3x6] - The 6 position of the joints bi expressed in {A}
%      - actuators.K [6x1] - Total stiffness of the actuators
%
% Outputs:
%    - stewart - With the 3 added field:
%        - geometry.J [6x6] - The Jacobian Matrix
%        - geometry.K [6x6] - The Stiffness Matrix
%        - geometry.C [6x6] - The Compliance Matrix

    assert(isfield(stewart.geometry, 'As'),   'stewart.geometry should have attribute As')
    As = stewart.geometry.As;

    assert(isfield(stewart.geometry, 'Ab'),   'stewart.geometry should have attribute Ab')
    Ab = stewart.geometry.Ab;

    assert(isfield(stewart.actuators, 'k'),   'stewart.actuators should have attribute k')
    Ki = stewart.actuators.k;

    J = [As' , cross(Ab, As)'];

    K = J'*diag(Ki)*J;

    C = inv(K);

    stewart.geometry.J = J;
    stewart.geometry.K = K;
    stewart.geometry.C = C;

end
#+end_src

*** =inverseKinematics=: Compute Inverse Kinematics

#+begin_src matlab :tangle matlab/src/inverseKinematics.m :comments none :mkdirp yes :eval no
function [Li, dLi] = inverseKinematics(stewart, args)
% inverseKinematics - Compute the needed length of each strut to have the wanted position and orientation of {B} with respect to {A}
%
% Syntax: [stewart] = inverseKinematics(stewart)
%
% Inputs:
%    - stewart - A structure with the following fields
%        - geometry.Aa   [3x6] - The positions ai expressed in {A}
%        - geometry.Bb   [3x6] - The positions bi expressed in {B}
%        - geometry.l    [6x1] - Length of each strut
%    - args - Can have the following fields:
%        - AP   [3x1] - The wanted position of {B} with respect to {A}
%        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%
% Outputs:
%    - Li   [6x1] - The 6 needed length of the struts in [m] to have the wanted pose of {B} w.r.t. {A}
%    - dLi  [6x1] - The 6 needed displacement of the struts from the initial position in [m] to have the wanted pose of {B} w.r.t. {A}

    arguments
        stewart
        args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
        args.ARB (3,3) double {mustBeNumeric} = eye(3)
    end

    assert(isfield(stewart.geometry, 'Aa'),   'stewart.geometry should have attribute Aa')
    Aa = stewart.geometry.Aa;

    assert(isfield(stewart.geometry, 'Bb'),   'stewart.geometry should have attribute Bb')
    Bb = stewart.geometry.Bb;

    assert(isfield(stewart.geometry, 'l'),   'stewart.geometry should have attribute l')
    l = stewart.geometry.l;

    Li = sqrt(args.AP'*args.AP + diag(Bb'*Bb) + diag(Aa'*Aa) - (2*args.AP'*Aa)' + (2*args.AP'*(args.ARB*Bb))' - diag(2*(args.ARB*Bb)'*Aa));

    dLi = Li-l;

end
#+end_src

*** =displayArchitecture=: 3D plot of the Stewart platform architecture
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/displayArchitecture.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:displayArchitecture>>

This Matlab function is accessible [[file:../src/displayArchitecture.m][here]].

**** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [] = displayArchitecture(stewart, args)
% displayArchitecture - 3D plot of the Stewart platform architecture
%
% Syntax: [] = displayArchitecture(args)
%
% Inputs:
%    - stewart
%    - args - Structure with the following fields:
%        - AP   [3x1] - The wanted position of {B} with respect to {A}
%        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%        - ARB  [3x3] - The rotation matrix that gives the wanted orientation of {B} with respect to {A}
%        - F_color [color] - Color used for the Fixed elements
%        - M_color [color] - Color used for the Mobile elements
%        - L_color [color] - Color used for the Legs elements
%        - frames    [true/false] - Display the Frames
%        - legs      [true/false] - Display the Legs
%        - joints    [true/false] - Display the Joints
%        - labels    [true/false] - Display the Labels
%        - platforms [true/false] - Display the Platforms
%        - views     ['all', 'xy', 'yz', 'xz', 'default'] -
%
% Outputs:
#+end_src

**** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
    stewart
    args.AP  (3,1) double {mustBeNumeric} = zeros(3,1)
    args.ARB (3,3) double {mustBeNumeric} = eye(3)
    args.F_color = [0 0.4470 0.7410]
    args.M_color = [0.8500 0.3250 0.0980]
    args.L_color = [0 0 0]
    args.frames    logical {mustBeNumericOrLogical} = true
    args.legs      logical {mustBeNumericOrLogical} = true
    args.joints    logical {mustBeNumericOrLogical} = true
    args.labels    logical {mustBeNumericOrLogical} = true
    args.platforms logical {mustBeNumericOrLogical} = true
    args.views     char    {mustBeMember(args.views,{'all', 'xy', 'xz', 'yz', 'default'})} = 'default'
end
#+end_src

**** Check the =stewart= structure elements
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
assert(isfield(stewart.platform_F, 'FO_A'), 'stewart.platform_F should have attribute FO_A')
FO_A = stewart.platform_F.FO_A;

assert(isfield(stewart.platform_M, 'MO_B'), 'stewart.platform_M should have attribute MO_B')
MO_B = stewart.platform_M.MO_B;

assert(isfield(stewart.geometry, 'H'),   'stewart.geometry should have attribute H')
H = stewart.geometry.H;

assert(isfield(stewart.platform_F, 'Fa'),   'stewart.platform_F should have attribute Fa')
Fa = stewart.platform_F.Fa;

assert(isfield(stewart.platform_M, 'Mb'),   'stewart.platform_M should have attribute Mb')
Mb = stewart.platform_M.Mb;
#+end_src


**** Figure Creation, Frames and Homogeneous transformations
:PROPERTIES:
:UNNUMBERED: t
:END:

The reference frame of the 3d plot corresponds to the frame $\{F\}$.
#+begin_src matlab
if ~strcmp(args.views, 'all')
    figure;
else
    f = figure('visible', 'off');
end

hold on;
#+end_src

We first compute homogeneous matrices that will be useful to position elements on the figure where the reference frame is $\{F\}$.
#+begin_src matlab
FTa = [eye(3), FO_A; ...
       zeros(1,3), 1];
ATb = [args.ARB, args.AP; ...
       zeros(1,3), 1];
BTm = [eye(3), -MO_B; ...
       zeros(1,3), 1];

FTm = FTa*ATb*BTm;
#+end_src

Let's define a parameter that define the length of the unit vectors used to display the frames.
#+begin_src matlab
d_unit_vector = H/4;
#+end_src

Let's define a parameter used to position the labels with respect to the center of the element.
#+begin_src matlab
d_label = H/20;
#+end_src

**** Fixed Base elements
:PROPERTIES:
:UNNUMBERED: t
:END:
Let's first plot the frame $\{F\}$.
#+begin_src matlab
Ff = [0, 0, 0];
if args.frames
    quiver3(Ff(1)*ones(1,3), Ff(2)*ones(1,3), Ff(3)*ones(1,3), ...
            [d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)

    if args.labels
        text(Ff(1) + d_label, ...
             Ff(2) + d_label, ...
             Ff(3) + d_label, '$\{F\}$', 'Color', args.F_color);
    end
end
#+end_src

Now plot the frame $\{A\}$ fixed to the Base.
#+begin_src matlab
if args.frames
    quiver3(FO_A(1)*ones(1,3), FO_A(2)*ones(1,3), FO_A(3)*ones(1,3), ...
            [d_unit_vector 0 0], [0 d_unit_vector 0], [0 0 d_unit_vector], '-', 'Color', args.F_color)

    if args.labels
        text(FO_A(1) + d_label, ...
             FO_A(2) + d_label, ...
             FO_A(3) + d_label, '$\{A\}$', 'Color', args.F_color);
    end
end
#+end_src

Let's then plot the circle corresponding to the shape of the Fixed base.
#+begin_src matlab
if args.platforms && stewart.platform_F.type == 1
    theta = [0:0.01:2*pi+0.01]; % Angles [rad]
    v = null([0; 0; 1]'); % Two vectors that are perpendicular to the circle normal
    center = [0; 0; 0]; % Center of the circle
    radius = stewart.platform_F.R; % Radius of the circle [m]

    points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));

    plot3(points(1,:), ...
          points(2,:), ...
          points(3,:), '-', 'Color', args.F_color);
end
#+end_src

Let's now plot the position and labels of the Fixed Joints
#+begin_src matlab
if args.joints
    scatter3(Fa(1,:), ...
             Fa(2,:), ...
             Fa(3,:), 'MarkerEdgeColor', args.F_color);
    if args.labels
        for i = 1:size(Fa,2)
            text(Fa(1,i) + d_label, ...
                 Fa(2,i), ...
                 Fa(3,i), sprintf('$a_{%i}$', i), 'Color', args.F_color);
        end
    end
end
#+end_src

**** Mobile Platform elements
:PROPERTIES:
:UNNUMBERED: t
:END:

Plot the frame $\{M\}$.
#+begin_src matlab
Fm = FTm*[0; 0; 0; 1]; % Get the position of frame {M} w.r.t. {F}

if args.frames
    FM_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
    quiver3(Fm(1)*ones(1,3), Fm(2)*ones(1,3), Fm(3)*ones(1,3), ...
            FM_uv(1,1:3), FM_uv(2,1:3), FM_uv(3,1:3), '-', 'Color', args.M_color)

    if args.labels
        text(Fm(1) + d_label, ...
             Fm(2) + d_label, ...
             Fm(3) + d_label, '$\{M\}$', 'Color', args.M_color);
    end
end
#+end_src

Plot the frame $\{B\}$.
#+begin_src matlab
FB = FO_A + args.AP;

if args.frames
    FB_uv = FTm*[d_unit_vector*eye(3); zeros(1,3)]; % Rotated Unit vectors
    quiver3(FB(1)*ones(1,3), FB(2)*ones(1,3), FB(3)*ones(1,3), ...
            FB_uv(1,1:3), FB_uv(2,1:3), FB_uv(3,1:3), '-', 'Color', args.M_color)

    if args.labels
        text(FB(1) - d_label, ...
             FB(2) + d_label, ...
             FB(3) + d_label, '$\{B\}$', 'Color', args.M_color);
    end
end
#+end_src

Let's then plot the circle corresponding to the shape of the Mobile platform.
#+begin_src matlab
if args.platforms && stewart.platform_M.type == 1
    theta = [0:0.01:2*pi+0.01]; % Angles [rad]
    v = null((FTm(1:3,1:3)*[0;0;1])'); % Two vectors that are perpendicular to the circle normal
    center = Fm(1:3); % Center of the circle
    radius = stewart.platform_M.R; % Radius of the circle [m]

    points = center*ones(1, length(theta)) + radius*(v(:,1)*cos(theta) + v(:,2)*sin(theta));

    plot3(points(1,:), ...
          points(2,:), ...
          points(3,:), '-', 'Color', args.M_color);
end
#+end_src

Plot the position and labels of the rotation joints fixed to the mobile platform.
#+begin_src matlab
if args.joints
    Fb = FTm*[Mb;ones(1,6)];

    scatter3(Fb(1,:), ...
             Fb(2,:), ...
             Fb(3,:), 'MarkerEdgeColor', args.M_color);

    if args.labels
        for i = 1:size(Fb,2)
            text(Fb(1,i) + d_label, ...
                 Fb(2,i), ...
                 Fb(3,i), sprintf('$b_{%i}$', i), 'Color', args.M_color);
        end
    end
end
#+end_src

**** Legs
:PROPERTIES:
:UNNUMBERED: t
:END:
Plot the legs connecting the joints of the fixed base to the joints of the mobile platform.
#+begin_src matlab
if args.legs
    for i = 1:6
        plot3([Fa(1,i), Fb(1,i)], ...
              [Fa(2,i), Fb(2,i)], ...
              [Fa(3,i), Fb(3,i)], '-', 'Color', args.L_color);

        if args.labels
            text((Fa(1,i)+Fb(1,i))/2 + d_label, ...
                 (Fa(2,i)+Fb(2,i))/2, ...
                 (Fa(3,i)+Fb(3,i))/2, sprintf('$%i$', i), 'Color', args.L_color);
        end
    end
end
#+end_src

**** Figure parameters
#+begin_src matlab
switch args.views
  case 'default'
    view([1 -0.6 0.4]);
  case 'xy'
    view([0 0 1]);
  case 'xz'
    view([0 -1 0]);
  case 'yz'
    view([1 0 0]);
end
axis equal;
axis off;
#+end_src

**** Subplots
#+begin_src matlab
if strcmp(args.views, 'all')
    hAx = findobj('type', 'axes');

    figure;
    s1 = subplot(2,2,1);
    copyobj(get(hAx(1), 'Children'), s1);
    view([0 0 1]);
    axis equal;
    axis off;
    title('Top')

    s2 = subplot(2,2,2);
    copyobj(get(hAx(1), 'Children'), s2);
    view([1 -0.6 0.4]);
    axis equal;
    axis off;

    s3 = subplot(2,2,3);
    copyobj(get(hAx(1), 'Children'), s3);
    view([1 0 0]);
    axis equal;
    axis off;
    title('Front')

    s4 = subplot(2,2,4);
    copyobj(get(hAx(1), 'Children'), s4);
    view([0 -1 0]);
    axis equal;
    axis off;
    title('Side')

    close(f);
end
#+end_src


*** =describeStewartPlatform=: Display some text describing the current defined Stewart Platform
:PROPERTIES:
:header-args:matlab+: :tangle matlab/src/describeStewartPlatform.m
:header-args:matlab+: :comments none :mkdirp yes :eval no
:END:
<<sec:describeStewartPlatform>>

This Matlab function is accessible [[file:../src/describeStewartPlatform.m][here]].

**** Function description
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
function [] = describeStewartPlatform(stewart)
% describeStewartPlatform - Display some text describing the current defined Stewart Platform
%
% Syntax: [] = describeStewartPlatform(args)
%
% Inputs:
%    - stewart
%
% Outputs:
#+end_src

**** Optional Parameters
:PROPERTIES:
:UNNUMBERED: t
:END:
#+begin_src matlab
arguments
    stewart
end
#+end_src

**** Geometry
#+begin_src matlab
fprintf('GEOMETRY:\n')
fprintf('- The height between the fixed based and the top platform is %.3g [mm].\n', 1e3*stewart.geometry.H)

if stewart.platform_M.MO_B(3) > 0
    fprintf('- Frame {A} is located %.3g [mm] above the top platform.\n',  1e3*stewart.platform_M.MO_B(3))
else
    fprintf('- Frame {A} is located %.3g [mm] below the top platform.\n', - 1e3*stewart.platform_M.MO_B(3))
end

fprintf('- The initial length of the struts are:\n')
fprintf('\t %.3g, %.3g, %.3g, %.3g, %.3g, %.3g [mm]\n', 1e3*stewart.geometry.l)
fprintf('\n')
#+end_src

**** Actuators
#+begin_src matlab
fprintf('ACTUATORS:\n')
if stewart.actuators.type == 1
    fprintf('- The actuators are modelled as 1DoF.\n')
    fprintf('- The Stiffness and Damping of each actuators is:\n')
    fprintf('\t k = %.0e [N/m] \t c = %.0e [N/(m/s)]\n', stewart.actuators.k(1), stewart.actuators.c(1))
    if stewart.actuators.kp > 0
    fprintf('\t Added parallel stiffness: kp = %.0e [N/m] \t c = %.0e [N/(m/s)]\n', stewart.actuators.kp(1))
    end
elseif stewart.actuators.type == 2
    fprintf('- The actuators are modelled as 2DoF (APA).\n')
    fprintf('- The vertical stiffness and damping contribution of the piezoelectric stack is:\n')
    fprintf('\t ka = %.0e [N/m] \t ca = %.0e [N/(m/s)]\n', stewart.actuators.ka(1), stewart.actuators.ca(1))
    fprintf('- Vertical stiffness when the piezoelectric stack is removed is:\n')
    fprintf('\t kr = %.0e [N/m] \t cr = %.0e [N/(m/s)]\n', stewart.actuators.kr(1), stewart.actuators.cr(1))
elseif stewart.actuators.type == 3
    fprintf('- The actuators are modelled with a flexible element (FEM).\n')
end
fprintf('\n')
#+end_src

**** Joints
#+begin_src matlab
fprintf('JOINTS:\n')
#+end_src

Type of the joints on the fixed base.
#+begin_src matlab
switch stewart.joints_F.type
  case 1
    fprintf('- The joints on the fixed based are universal joints (2DoF)\n')
  case 2
    fprintf('- The joints on the fixed based are spherical joints (3DoF)\n')
end
#+end_src

Type of the joints on the mobile platform.
#+begin_src matlab
switch stewart.joints_M.type
  case 1
    fprintf('- The joints on the mobile based are universal joints (2DoF)\n')
  case 2
    fprintf('- The joints on the mobile based are spherical joints (3DoF)\n')
end
#+end_src

Position of the fixed joints
#+begin_src matlab
fprintf('- The position of the joints on the fixed based with respect to {F} are (in [mm]):\n')
fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_F.Fa)
#+end_src

Position of the mobile joints
#+begin_src matlab
fprintf('- The position of the joints on the mobile based with respect to {M} are (in [mm]):\n')
fprintf('\t % .3g \t % .3g \t % .3g\n', 1e3*stewart.platform_M.Mb)
fprintf('\n')
#+end_src

**** Kinematics
#+begin_src matlab
fprintf('KINEMATICS:\n')

if isfield(stewart.kinematics, 'K')
    fprintf('- The Stiffness matrix K is (in [N/m]):\n')
    fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.K)
end

if isfield(stewart.kinematics, 'C')
    fprintf('- The Damping matrix C is (in [m/N]):\n')
    fprintf('\t % .0e \t % .0e \t % .0e \t % .0e \t % .0e \t % .0e\n', stewart.kinematics.C)
end
#+end_src

* Footnotes

[fn:1]Different architecture exists, typically referred as "6-SPS" (Spherical, Prismatic, Spherical) or "6-UPS" (Universal, Prismatic, Spherical)
