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matlab/nhexa_2_model.m

@@ -21,53 +21,6 @@ colors = colororder;
 %% Frequency Vector [Hz]
 freqs = logspace(0, 3, 1000);
 
-% Validation of the multi-body model
-% <<ssec:nhexa_model_validation>>
-
-% The developed multi-body model of the Stewart platform is represented schematically in Figure ref:fig:nhexa_stewart_model_input_outputs, highlighting the key inputs and outputs: actuator forces $\bm{f}$, force sensor measurements $\bm{f}_n$, and relative displacement measurements $\bm{\mathcal{L}}$.
-% The frames $\{F\}$ and $\{M\}$ serve as interfaces for integration with other elements in the multi-body system.
-% A three-dimensional visualization of the model is presented in Figure ref:fig:nhexa_simscape_screenshot.
-
-% #+attr_latex: :options [b]{0.6\linewidth}
-% #+begin_minipage
-% #+name: fig:nhexa_stewart_model_input_outputs
-% #+caption: Nano-Hexapod plant with inputs and outputs. Frames $\{F\}$ and $\{M\}$ can be connected to other elements in the multi-body models.
-% #+attr_latex: :scale 1 :float nil
-% [[file:figs/nhexa_stewart_model_input_outputs.png]]
-% #+end_minipage
-% \hfill
-% #+attr_latex: :options [b]{0.35\linewidth}
-% #+begin_minipage
-% #+name: fig:nhexa_simscape_screenshot
-% #+caption: 3D representation of the multi-body model
-% #+attr_latex: :width 0.90\linewidth :float nil
-% [[file:figs/nhexa_simscape_screenshot.jpg]]
-% #+end_minipage
-
-% The validation of the multi-body model was performed using the simplest Stewart platform configuration, enabling direct comparison with the analytical transfer functions derived in Section ref:ssec:nhexa_stewart_platform_dynamics.
-% This configuration consists of massless universal joints at the base, massless spherical joints at the top platform, and massless struts with stiffness $k_a = 1\,\text{N}/\mu\text{m}$ and damping $c_a = 10\,\text{N}/({\text{m}/\text{s}})$.
-% The geometric parameters remain as specified in Table ref:tab:nhexa_actuator_parameters.
-
-% While the moving platform itself is considered massless, a $10\,\text{kg}$ cylindrical payload is mounted on top with a radius of $r = 110\,mm$ and a height $h = 300\,mm$.
-
-% For the analytical model, the stiffness, damping, and mass matrices are defined in eqref:eq:nhexa_analytical_matrices.
-
-% \begin{subequations}\label{eq:nhexa_analytical_matrices}
-% \begin{align}
-% \bm{\mathcal{K}} &= \text{diag}(k_a,\ k_a,\ k_a,\ k_a,\ k_a,\ k_a) \\
-% \bm{\mathcal{C}} &= \text{diag}(c_a,\ c_a,\ c_a,\ c_a,\ c_a,\ c_a) \\
-% \bm{M} &= \text{diag}\left(m,\ m,\ m,\ \frac{1}{12}m(3r^2 + h^2),\ \frac{1}{12}m(3r^2 + h^2),\ \frac{1}{2}mr^2\right)
-% \end{align}
-% \end{subequations}
-
-% The transfer functions from the actuator forces to the strut displacements are computed using these matrices according to equation eqref:eq:nhexa_transfer_function_struts.
-% These analytical transfer functions are then compared with those extracted from the multi-body model.
-% The developed multi-body model yields a state-space representation with 12 states, corresponding to the six degrees of freedom of the moving platform.
-
-% Figure ref:fig:nhexa_comp_multi_body_analytical presents a comparison between the analytical and multi-body transfer functions, specifically showing the response from the first actuator force to all six strut displacements.
-% The close agreement between both approaches across the frequency spectrum validates the multi-body model's accuracy in capturing the system's dynamic behavior.
-
-
 %% Plant using Analytical Equations
 % Stewart platform definition
 k = 1e6; % Actuator stiffness [N/m]
@@ -154,28 +107,6 @@ yticks([-180, -90, 0, 90, 180]);
 linkaxes([ax1,ax2],'x');
 xlim([freqs(1), freqs(end)]);
 
-% Nano Hexapod Dynamics
-% <<ssec:nhexa_model_dynamics>>
-
-% Following the validation of the multi-body model, a detailed analysis of the nano-hexapod dynamics was performed.
-% The model parameters were set according to the specifications outlined in Section ref:ssec:nhexa_model_def, with a payload mass of $10\,kg$.
-% The transfer functions from actuator forces $\bm{f}$ to both strut displacements $\bm{\mathcal{L}}$ and force measurements $\bm{f}_n$ were derived from the multi-body model.
-
-% The transfer functions relating actuator forces to strut displacements are presented in Figure ref:fig:nhexa_multi_body_plant_dL.
-% Due to the system's symmetrical design and identical strut configurations, all diagonal terms (transfer functions from force $f_i$ to displacement $l_i$ of the same strut) exhibit identical behavior.
-% While the system has six degrees of freedom, only four distinct resonance frequencies were observed in the frequency response.
-% This reduction from six to four observable modes is attributed to the system's symmetry, where two pairs of resonances occur at identical frequencies.
-
-% The system's behavior can be characterized in three frequency regions.
-% At low frequencies, well below the first resonance, the plant demonstrates good decoupling between actuators, with the response dominated by the strut stiffness: $\bm{G}(j\omega) \xrightarrow[\omega \to 0]{} \bm{\mathcal{K}}^{-1}$.
-% In the mid-frequency range, the system exhibits coupled dynamics through its resonant modes, reflecting the complex interactions between the platform's degrees of freedom.
-% At high frequencies, above the highest resonance, the response is governed by the payload's inertia mapped to the strut coordinates: $\bm{G}(j\omega) \xrightarrow[\omega \to \infty]{} \bm{J} \bm{M}^{-T} \bm{J}^T \frac{-1}{\omega^2}$
-
-% The force sensor transfer functions, shown in Figure ref:fig:nhexa_multi_body_plant_fm, display characteristics typical of collocated actuator-sensor pairs.
-% Each actuator's transfer function to its associated force sensor exhibits alternating complex conjugate poles and zeros.
-% The inclusion of parallel stiffness introduces an additional complex conjugate zero at low frequency, which was previously observed in the three-degree-of-freedom rotating model.
-
-
 %% Multi-Body model of the Nano-Hexapod
 % Initialize 1DoF
 initializeSimplifiedNanoHexapod('flex_type_F', '2dof', 'flex_type_M', '3dof', 'actuator_type', '1dof');