diff --git a/simscape-nano-hexapod.org b/simscape-nano-hexapod.org index 134f1a0..2e967eb 100644 --- a/simscape-nano-hexapod.org +++ b/simscape-nano-hexapod.org @@ -260,6 +260,9 @@ It should be the exact model reference that will be included in the NASS model ( - [ ] Make sure they are all defined in correct order - [ ] Make sure all vectors and matrices are bold +** TODO [#A] Should I include the effect of rotation somewhere? +Similar to what was done with the 3DoF model? + ** DONE [#C] Better understand principle of virtual work CLOSED: [2025-02-10 Mon 15:51] @@ -798,15 +801,15 @@ While a reasonable geometric configuration will be used to validate the NASS dur <> ** 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 -- Model definition (Section ref:ssec:nhexa_model_def) -- Validation of the model by comparing with analytical equations (Section ref:ssec:nhexa_model_validation) -- Dynamics of the nano-hexapod used for conceptual analysis: (Section ref:ssec:nhexa_model_dynamics) +The dynamic modeling of Stewart platforms has traditionally relied on analytical approaches. +However, these analytical models become increasingly complex when the full dynamic behavior of struts and joints must be captured. +To overcome these limitations, a flexible multi-body approach has been developed that can be readily integrated into the broader NASS system model. +Through this multi-body modeling approach, each component model (including joints, actuators, and sensors) can be progressively refined. + +The analysis is structured in three parts. +First, the multi-body model is developed, wherein detailed geometric parameters, inertial properties, and actuator characteristics are established (Section ref:ssec:nhexa_model_def). +The model is then validated through comparison with analytical equations in a simplified configuration (Section ref:ssec:nhexa_model_validation). +Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section ref:ssec:nhexa_model_dynamics). ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) @@ -837,11 +840,13 @@ While a reasonable geometric configuration will be used to validate the NASS dur <> **** Geometry -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. +The Stewart platform's geometry is defined by two principal coordinate frames (Figure ref:fig:nhexa_stewart_model_def): a fixed base frame $\{F\}$ and a moving platform frame $\{M\}$. +The joints connecting the actuators to these frames are located at positions ${}^Fa_i$ and ${}^Mb_i$ respectively. +The point of interest, denoted by frame $\{A\}$, is situated $150\,mm$ above the moving platform frame $\{M\}$. -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. +The geometric parameters of the nano-hexapod are summarized in Table ref:tab:nhexa_stewart_model_geometry. +These parameters define the positions of all connection points in their respective coordinate frames. +From these parameters, key kinematic properties can be derived: the strut orientations $\hat{s}_i$, strut lengths $l_i$, and the system's Jacobian matrix $\bm{J}$. #+attr_latex: :options [b]{0.6\linewidth} #+begin_minipage @@ -879,30 +884,30 @@ From this, the orientation $\hat{s}_i$ and length $l_i$ of the struts can be com **** 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 $5\,kg$. +The fixed base and moving platform are modeled as solid cylindrical bodies. +The base platform is characterized by a radius of $120\,mm$ and thickness of $15\,mm$, matching the dimensions of the micro-hexapod's top platform. +The moving platform is similarly modeled with a radius of $110\,mm$ and thickness of $15\,mm$. +Both platforms are assigned 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. +The platform's joints play a crucial role in its dynamic behavior. +At both the upper and lower connection points, various degrees of freedom can be modeled, including universal joints, spherical joints, and configurations with additional axial and lateral stiffness components. +For each degree of freedom, stiffness characteristics can be incorporated into the model. -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. +In the conceptual design phase, a simplified joint configuration is employed: the bottom joints are modeled as two-degree-of-freedom universal joints, while the top joints are represented as three-degree-of-freedom spherical joints. +These joints are considered massless and exhibit no stiffness along their degrees of freedom. **** Actuators -In its simplest form, the actuators are modelled with one prismatic joint having some internal stiffness $k_a$ and damping $c_a$, and a force source $f$. +The actuator model comprises several key elements (Figure ref:fig:nhexa_actuator_model). +At its core, each actuator is modeled as a prismatic joint with internal stiffness $k_a$ and damping $c_a$, driven by a force source $f$. +Similarly to what was found using the rotating 3-DoF model, a parallel stiffness $k_p$ is added in parallel with the force sensor to ensure stability when considering spindle rotation effects. -As was shown using the 3DoF rotating model, having a parallel stiffness $k_p$ with the force sensor permits to regain the guaranteed stability of decentralized IFF when the spindle is rotating. +Each actuator is equipped with two sensors: a force sensor providing measurements $f_m$ and a relative motion sensor measuring displacement $d_L$. +The actuator parameters used in the conceptual phase are presented in Table ref:tab:nhexa_actuator_parameters. -A force sensor with output $f_m$ is added as well as a relative motion sensor with output $d_L$. -The model of the nano-hexapod actuators used during the conceptual phase are shown in Figure ref:fig:nhexa_actuator_model with the parameters summarized in Table ref:tab:nhexa_actuator_parameters. - -Thanks to the flexibility of the multi-body model, the model of the actuators can later be refined. +This modular approach to actuator modeling allows for future refinements as the design evolves, enabling the incorporation of additional dynamic effects or sensor characteristics as needed. #+attr_latex: :options [b]{0.6\linewidth} #+begin_minipage @@ -930,8 +935,9 @@ Thanks to the flexibility of the multi-body model, the model of the actuators ca ** Validation of the multi-body model <> -The obtained multi-body model can schematically be represented as in Figure ref:fig:nhexa_stewart_model_input_outputs with actuator inputs $\bm{f}$, force sensor outputs $\bm{f}_m$ and relative displacement outputs $\bm{d}_L$. -The 3D representation of the Stewart platform using the multi-body model is shown in Figure ref:fig:nhexa_simscape_screenshot. +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 $\bm{f}_m$, and relative displacement measurements $\bm{d}_L$. +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 @@ -949,31 +955,28 @@ The 3D representation of the Stewart platform using the multi-body model is show [[file:figs/nhexa_simscape_screenshot.jpg]] #+end_minipage -To validate the multi-body model of the Stewart platform, the simplest Stewart platform configuration is used to compare the multi-body dynamics with the analytical transfer functions obtained in Section ref:ssec:nhexa_stewart_platform_dynamics. +The validation of the multi-body model is 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. -The bottom joints are universal joints while the top joints are spherical joints. All joints mass-less and have zero stiffness in the free DoF. -The struts are modelled with a stiffness equal to $k_a = 1\,N/\mu m$, damping $c_a = 10\,N/(m/s)$, and are mass-less. -The geometry used is shown 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 $h = 300\,mm$. -The top platform is considered mass-less, but a payload with mass $m = 10\,kg$ is added on top of the Stewart platform. -The payload is cylindrical with a radius of $r = 110\,mm$ and a height $h = 300\,mm$ such that its center of mass coincide with $\{A\}$. - -Stiffness, damping and mass matrices for the analytical equations are summarized in eqref:eq:nhexa_analytical_matrices. -The transfer functions from actuator forces to displacement of each strut can then be computed using eqref:eq:nhexa_transfer_function_struts. +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{K} &= \text{diag}(k_a,\ k_a,\ k_a,\ k_a,\ k_a,\ k_a) \\ -\bm{C} &= \text{diag}(c_a,\ c_a,\ c_a,\ c_a,\ c_a,\ c_a) \\ +\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 same transfer functions are extracted from the multi-body model. -The obtained state-space model has 12 states which corresponds to the 6-DoF of the top platform. +The transfer functions from actuator forces to 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 multi-body model yields a state-space representation with 12 states, corresponding to the six degrees of freedom of the moving platform. -The transfer functions from the first actuator to the displacement of the 6 struts are compared in Figure ref:fig:nhexa_comp_multi_body_analytical. -A good match can be observed between the analytical formulas and the multi-body model therefore validating the multi-body model. +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. #+begin_src matlab %% Plant using Analytical Equations @@ -1077,22 +1080,23 @@ exportFig('figs/nhexa_comp_multi_body_analytical.pdf', 'width', 'wide', 'height' ** Nano Hexapod Dynamics <> -Now that the multi-body model is validated, it can be used to study the dynamics of the nano-hexapod. -The model is initialized as described in Section ref:ssec:nhexa_model_def with a $10\,kg$ payload, and the transfer functions from $\bm{f}$ to $\bm{f}_m$ and $\bm{d}_e$ are computed from the multi-body model. +Following the validation of the multi-body model, a detailed analysis of the nano-hexapod dynamics has been performed. +The model parameters are set according to the specifications outlined in Section ref:ssec:nhexa_model_def, with a payload mass of $10\,kg$. +Transfer functions from actuator forces $\bm{f}$ to both strut displacements $\bm{d}_L$ and force measurements $\bm{f}_m$ are 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 $d_{Li}$ of the same strut) exhibit identical behavior. +While the system possesses six degrees of freedom, only four distinct resonance frequencies are 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: $G(j\omega) \xrightarrow[\omega \to 0]{} \mathcal{K}^{-1}$. +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: $G(j\omega) \xrightarrow[\omega \to \infty]{} J M^{-T} J^T \frac{-1}{\omega^2}$ -- [X] What payload to use? - 10kg payload -- [ ] Transfer function from f to de: - - [ ] All diagonal terms equal (thanks to symmetry and having the same struts) - - [ ] 4 observed modes (due to symmetry, in reality 6 modes) - - [ ] Decoupled at low frequency $G(j\omega) \xrightarrow[\omega \to 0]{} \mathcal{K}^{-1}$ (low frequency gain is K) - - [ ] High frequency gain is $G(j\omega) \xrightarrow[\omega \to \infty]{} J M^{-T} J^T \frac{-1}{\omega^2}$, which is in general not diagonal -- [ ] Transfer function from f to fm: - - [ ] Alternating poles and zeros - - [ ] Effect of parallel stiffness on IFF plant? -- [ ] Validation of compliance matrix? +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, a feature previously observed in the three-degree-of-freedom rotating model. #+begin_src matlab %% Multi-Body model of the Nano-Hexapod @@ -1230,13 +1234,13 @@ exportFig('figs/nhexa_multi_body_plant_fm.pdf', 'width', 'half', 'height', 600); #+caption: Bode plot of the transfer functions computed from the nano-hexapod multi-body model #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_dL}$f_i$ to $d_{Li}$} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_dL}$\bm{f}$ to $\bm{d}_{L}$} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth [[file:figs/nhexa_multi_body_plant_dL.png]] #+end_subfigure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_fm}$f_i$ to $f_{mi}$} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_fm}$\bm{f}$ to $\bm{f}_{m}$} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth @@ -1249,13 +1253,13 @@ exportFig('figs/nhexa_multi_body_plant_fm.pdf', 'width', 'half', 'height', 600); :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 +The multi-body modeling approach presented in this section provides a comprehensive framework for analyzing the dynamics of the nano-hexapod system. +Through comparison with analytical solutions in a simplified configuration, the model's accuracy has been validated, demonstrating its ability to capture the essential dynamic behavior of the Stewart platform. + +A key advantage of this modeling approach lies in its flexibility for future refinements. +While the current implementation employs idealized joints for the conceptual design phase, the framework readily accommodates the incorporation of joint stiffness and other non-ideal effects. +The joint stiffness, known to impact the performance of decentralized IFF control strategy [[cite:&preumont07_six_axis_singl_stage_activ]], can be studied as the design evolved and will be optimized during the detail design phase. +The validated multi-body model will serve as a valuable tool for predicting system behavior and evaluating control performance throughout the design process. * Control of Stewart Platforms :PROPERTIES: diff --git a/simscape-nano-hexapod.pdf b/simscape-nano-hexapod.pdf index 99a6c4a..2b01afc 100644 Binary files a/simscape-nano-hexapod.pdf and b/simscape-nano-hexapod.pdf differ diff --git a/simscape-nano-hexapod.tex b/simscape-nano-hexapod.tex index 151157d..a46dab5 100644 --- a/simscape-nano-hexapod.tex +++ b/simscape-nano-hexapod.tex @@ -1,4 +1,4 @@ -% Created 2025-02-10 Mon 19:03 +% Created 2025-02-11 Tue 09:54 % Intended LaTeX compiler: pdflatex \documentclass[a4paper, 10pt, DIV=12, parskip=full, bibliography=totoc]{scrreprt} @@ -87,6 +87,7 @@ Maybe not the topic here. \begin{table}[htbp] +\caption{\label{tab:nhexa_serial_vs_parallel}Advantages and Disadvantages of both serial and parallel robots} \centering \begin{tabularx}{\linewidth}{lXX} \toprule @@ -97,8 +98,6 @@ Disadvantages & Low Stiffness & Small Workspace\\ Kinematic Struture & Open & Closed-loop\\ \bottomrule \end{tabularx} -\caption{\label{tab:nhexa_serial_vs_parallel}Advantages and Disadvantages of both serial and parallel robots} - \end{table} \chapter{The Stewart platform} @@ -425,26 +424,26 @@ While a reasonable geometric configuration will be used to validate the NASS dur \chapter{Multi-Body Model} \label{sec:nhexa_model} -\textbf{Goal}: -\begin{itemize} -\item Study the dynamics of Stewart platform -\item 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 -\item Model definition (Section \ref{ssec:nhexa_model_def}) -\item Validation of the model by comparing with analytical equations (Section \ref{ssec:nhexa_model_validation}) -\item Dynamics of the nano-hexapod used for conceptual analysis: (Section \ref{ssec:nhexa_model_dynamics}) -\end{itemize} +The dynamic modeling of Stewart platforms has traditionally relied on analytical approaches. +However, these analytical models become increasingly complex when the full dynamic behavior of struts and joints must be captured. +To overcome these limitations, a flexible multi-body approach has been developed that can be readily integrated into the broader NASS system model. +Through this multi-body modeling approach, each component model (including joints, actuators, and sensors) can be progressively refined. + +The analysis is structured in three parts. +First, the multi-body model is developed, wherein detailed geometric parameters, inertial properties, and actuator characteristics are established (Section \ref{ssec:nhexa_model_def}). +The model is then validated through comparison with analytical equations in a simplified configuration (Section \ref{ssec:nhexa_model_validation}). +Finally, the validated model is employed to analyze the nano-hexapod dynamics, from which insights for the control system design are derived (Section \ref{ssec:nhexa_model_dynamics}). \section{Model Definition} \label{ssec:nhexa_model_def} \paragraph{Geometry} -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}. +The Stewart platform's geometry is defined by two principal coordinate frames (Figure \ref{fig:nhexa_stewart_model_def}): a fixed base frame \(\{F\}\) and a moving platform frame \(\{M\}\). +The joints connecting the actuators to these frames are located at positions \({}^Fa_i\) and \({}^Mb_i\) respectively. +The point of interest, denoted by frame \(\{A\}\), is situated \(150\,mm\) above the moving platform frame \(\{M\}\). -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. +The geometric parameters of the nano-hexapod are summarized in Table \ref{tab:nhexa_stewart_model_geometry}. +These parameters define the positions of all connection points in their respective coordinate frames. +From these parameters, key kinematic properties can be derived: the strut orientations \(\hat{s}_i\), strut lengths \(l_i\), and the system's Jacobian matrix \(\bm{J}\). \begin{minipage}[b]{0.6\linewidth} \begin{center} @@ -482,30 +481,30 @@ From this, the orientation \(\hat{s}_i\) and length \(l_i\) of the struts can be \paragraph{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 \(5\,kg\). +The fixed base and moving platform are modeled as solid cylindrical bodies. +The base platform is characterized by a radius of \(120\,mm\) and thickness of \(15\,mm\), matching the dimensions of the micro-hexapod's top platform. +The moving platform is similarly modeled with a radius of \(110\,mm\) and thickness of \(15\,mm\). +Both platforms are assigned a mass of \(5\,kg\). \paragraph{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. +The platform's joints play a crucial role in its dynamic behavior. +At both the upper and lower connection points, various degrees of freedom can be modeled, including universal joints, spherical joints, and configurations with additional axial and lateral stiffness components. +For each degree of freedom, stiffness characteristics can be incorporated into the model. -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. +In the conceptual design phase, a simplified joint configuration is employed: the bottom joints are modeled as two-degree-of-freedom universal joints, while the top joints are represented as three-degree-of-freedom spherical joints. +These joints are considered massless and exhibit no stiffness along their degrees of freedom. \paragraph{Actuators} -In its simplest form, the actuators are modelled with one prismatic joint having some internal stiffness \(k_a\) and damping \(c_a\), and a force source \(f\). +The actuator model comprises several key elements (Figure \ref{fig:nhexa_actuator_model}). +At its core, each actuator is modeled as a prismatic joint with internal stiffness \(k_a\) and damping \(c_a\), driven by a force source \(f\). +Similarly to what was found using the rotating 3-DoF model, a parallel stiffness \(k_p\) is added in parallel with the force sensor to ensure stability when considering spindle rotation effects. -As was shown using the 3DoF rotating model, having a parallel stiffness \(k_p\) with the force sensor permits to regain the guaranteed stability of decentralized IFF when the spindle is rotating. +Each actuator is equipped with two sensors: a force sensor providing measurements \(f_m\) and a relative motion sensor measuring displacement \(d_L\). +The actuator parameters used in the conceptual phase are presented in Table \ref{tab:nhexa_actuator_parameters}. -A force sensor with output \(f_m\) is added as well as a relative motion sensor with output \(d_L\). -The model of the nano-hexapod actuators used during the conceptual phase are shown in Figure \ref{fig:nhexa_actuator_model} with the parameters summarized in Table \ref{tab:nhexa_actuator_parameters}. - -Thanks to the flexibility of the multi-body model, the model of the actuators can later be refined. +This modular approach to actuator modeling allows for future refinements as the design evolves, enabling the incorporation of additional dynamic effects or sensor characteristics as needed. \begin{minipage}[b]{0.6\linewidth} \begin{center} @@ -533,8 +532,9 @@ Thanks to the flexibility of the multi-body model, the model of the actuators ca \section{Validation of the multi-body model} \label{ssec:nhexa_model_validation} -The obtained multi-body model can schematically be represented as in Figure \ref{fig:nhexa_stewart_model_input_outputs} with actuator inputs \(\bm{f}\), force sensor outputs \(\bm{f}_m\) and relative displacement outputs \(\bm{d}_L\). -The 3D representation of the Stewart platform using the multi-body model is shown in Figure \ref{fig:nhexa_simscape_screenshot}. +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 \(\bm{f}_m\), and relative displacement measurements \(\bm{d}_L\). +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}. \begin{minipage}[b]{0.6\linewidth} \begin{center} @@ -550,31 +550,28 @@ The 3D representation of the Stewart platform using the multi-body model is show \end{center} \end{minipage} -To validate the multi-body model of the Stewart platform, the simplest Stewart platform configuration is used to compare the multi-body dynamics with the analytical transfer functions obtained in Section \ref{ssec:nhexa_stewart_platform_dynamics}. +The validation of the multi-body model is 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}. -The bottom joints are universal joints while the top joints are spherical joints. All joints mass-less and have zero stiffness in the free DoF. -The struts are modelled with a stiffness equal to \(k_a = 1\,N/\mu m\), damping \(c_a = 10\,N/(m/s)\), and are mass-less. -The geometry used is shown 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 \(h = 300\,mm\). -The top platform is considered mass-less, but a payload with mass \(m = 10\,kg\) is added on top of the Stewart platform. -The payload is cylindrical with a radius of \(r = 110\,mm\) and a height \(h = 300\,mm\) such that its center of mass coincide with \(\{A\}\). - -Stiffness, damping and mass matrices for the analytical equations are summarized in \eqref{eq:nhexa_analytical_matrices}. -The transfer functions from actuator forces to displacement of each strut can then be computed using \eqref{eq:nhexa_transfer_function_struts}. +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{K} &= \text{diag}(k_a,\ k_a,\ k_a,\ k_a,\ k_a,\ k_a) \\ -\bm{C} &= \text{diag}(c_a,\ c_a,\ c_a,\ c_a,\ c_a,\ c_a) \\ +\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 same transfer functions are extracted from the multi-body model. -The obtained state-space model has 12 states which corresponds to the 6-DoF of the top platform. +The transfer functions from actuator forces to 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 multi-body model yields a state-space representation with 12 states, corresponding to the six degrees of freedom of the moving platform. -The transfer functions from the first actuator to the displacement of the 6 struts are compared in Figure \ref{fig:nhexa_comp_multi_body_analytical}. -A good match can be observed between the analytical formulas and the multi-body model therefore validating the multi-body model. +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. \begin{figure}[htbp] \centering @@ -585,57 +582,48 @@ A good match can be observed between the analytical formulas and the multi-body \section{Nano Hexapod Dynamics} \label{ssec:nhexa_model_dynamics} -Now that the multi-body model is validated, it can be used to study the dynamics of the nano-hexapod. -The model is initialized as described in Section \ref{ssec:nhexa_model_def} with a \(10\,kg\) payload, and the transfer functions from \(\bm{f}\) to \(\bm{f}_m\) and \(\bm{d}_e\) are computed from the multi-body model. +Following the validation of the multi-body model, a detailed analysis of the nano-hexapod dynamics has been performed. +The model parameters are set according to the specifications outlined in Section \ref{ssec:nhexa_model_def}, with a payload mass of \(10\,kg\). +Transfer functions from actuator forces \(\bm{f}\) to both strut displacements \(\bm{d}_L\) and force measurements \(\bm{f}_m\) are 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 \(d_{Li}\) of the same strut) exhibit identical behavior. +While the system possesses six degrees of freedom, only four distinct resonance frequencies are 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: \(G(j\omega) \xrightarrow[\omega \to 0]{} \mathcal{K}^{-1}\). +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: \(G(j\omega) \xrightarrow[\omega \to \infty]{} J M^{-T} J^T \frac{-1}{\omega^2}\) -\begin{itemize} -\item[{$\boxtimes$}] What payload to use? -10kg payload -\item[{$\square$}] Transfer function from f to de: -\begin{itemize} -\item[{$\square$}] All diagonal terms equal (thanks to symmetry and having the same struts) -\item[{$\square$}] 4 observed modes (due to symmetry, in reality 6 modes) -\item[{$\square$}] Decoupled at low frequency \(G(j\omega) \xrightarrow[\omega \to 0]{} \mathcal{K}^{-1}\) (low frequency gain is K) -\item[{$\square$}] High frequency gain is \(G(j\omega) \xrightarrow[\omega \to \infty]{} J M^{-T} J^T \frac{-1}{\omega^2}\), which is in general not diagonal -\end{itemize} -\item[{$\square$}] Transfer function from f to fm: -\begin{itemize} -\item[{$\square$}] Alternating poles and zeros -\item[{$\square$}] Effect of parallel stiffness on IFF plant? -\end{itemize} -\item[{$\square$}] Validation of compliance matrix? -\end{itemize} +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, a feature previously observed in the three-degree-of-freedom rotating model. \begin{figure}[htbp] \begin{subfigure}{0.48\textwidth} \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/nhexa_multi_body_plant_dL.png} \end{center} -\subcaption{\label{fig:nhexa_multi_body_plant_dL}$f_i$ to $d_{Li}$} +\subcaption{\label{fig:nhexa_multi_body_plant_dL}$\bm{f}$ to $\bm{d}_{L}$} \end{subfigure} \begin{subfigure}{0.48\textwidth} \begin{center} \includegraphics[scale=1,width=\linewidth]{figs/nhexa_multi_body_plant_fm.png} \end{center} -\subcaption{\label{fig:nhexa_multi_body_plant_fm}$f_i$ to $f_{mi}$} +\subcaption{\label{fig:nhexa_multi_body_plant_fm}$\bm{f}$ to $\bm{f}_{m}$} \end{subfigure} \caption{\label{fig:nhexa_multi_body_plant}Bode plot of the transfer functions computed from the nano-hexapod multi-body model} \end{figure} \section*{Conclusion} -\begin{itemize} -\item Validation of multi-body model in a simple case -\item Possible to increase the model complexity when required -\begin{itemize} -\item If considered 6dof joint stiffness, model order increases -\item Can have an effect on IFF performances: \cite{preumont07_six_axis_singl_stage_activ} -\item Conclusion: during the conceptual design, we consider a perfect, but will be taken into account later -\item Optimization of the Flexible joint will be performed in Chapter 2.2 -\end{itemize} -\item MIMO system: how to control? => next section -\end{itemize} +The multi-body modeling approach presented in this section provides a comprehensive framework for analyzing the dynamics of the nano-hexapod system. +Through comparison with analytical solutions in a simplified configuration, the model's accuracy has been validated, demonstrating its ability to capture the essential dynamic behavior of the Stewart platform. + +A key advantage of this modeling approach lies in its flexibility for future refinements. +While the current implementation employs idealized joints for the conceptual design phase, the framework readily accommodates the incorporation of joint stiffness and other non-ideal effects. +The joint stiffness, known to impact the performance of decentralized IFF control strategy \cite{preumont07_six_axis_singl_stage_activ}, can be studied as the design evolved and will be optimized during the detail design phase. +The validated multi-body model will serve as a valuable tool for predicting system behavior and evaluating control performance throughout the design process. \chapter{Control of Stewart Platforms} \label{sec:nhexa_control}