From 25278519ef4359af3e19c4595a9e7cd194ff6099 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 12 Feb 2025 10:24:57 +0100 Subject: [PATCH] Check all notations --- figs/nhexa_actuator_model.pdf | Bin 21426 -> 21075 bytes figs/nhexa_actuator_model.png | Bin 16651 -> 16362 bytes figs/nhexa_actuator_model.svg | 164 +++-- figs/nhexa_comp_multi_body_analytical.pdf | 689 +++++++++--------- figs/nhexa_comp_multi_body_analytical.png | Bin 154116 -> 152210 bytes figs/nhexa_control_cartesian.pdf | Bin 72367 -> 76591 bytes figs/nhexa_control_cartesian.png | Bin 10127 -> 10220 bytes figs/nhexa_control_cartesian.svg | 10 +- figs/nhexa_control_strut.pdf | Bin 85827 -> 90050 bytes figs/nhexa_control_strut.png | Bin 9478 -> 9577 bytes figs/nhexa_control_strut.svg | 10 +- .../nhexa_decentralized_hac_iff_loop_gain.pdf | Bin 46017 -> 46032 bytes .../nhexa_decentralized_hac_iff_loop_gain.png | Bin 65985 -> 65825 bytes ...exa_decentralized_hac_iff_plant_damped.pdf | Bin 379462 -> 380403 bytes 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eO0>-y2fi5(JFDb2QUUgbK*-4`Nf$~QzWG0|W<-eq diff --git a/figs/nhexa_stewart_model_input_outputs.svg b/figs/nhexa_stewart_model_input_outputs.svg index 6adb2f9..166d4eb 100644 --- a/figs/nhexa_stewart_model_input_outputs.svg +++ b/figs/nhexa_stewart_model_input_outputs.svg @@ -24,16 +24,16 @@ borderopacity="1.0" inkscape:pageopacity="0.0" inkscape:pageshadow="2" - inkscape:zoom="22.627417" - inkscape:cx="155.69607" - inkscape:cy="141.35507" + inkscape:zoom="4" + inkscape:cx="187.75" + inkscape:cy="149.875" inkscape:document-units="mm" - inkscape:current-layer="layer8" + inkscape:current-layer="layer9" showgrid="false" inkscape:window-width="2534" - inkscape:window-height="1367" + inkscape:window-height="1387" inkscape:window-x="11" - inkscape:window-y="60" + inkscape:window-y="38" inkscape:window-maximized="1" inkscape:document-rotation="0" inkscape:showpageshadow="2" @@ -1982,6 +1982,153 @@ style="fill:context-stroke;fill-rule:evenodd;stroke:none" id="path21-6" /> + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + @@ -1998,7 +2145,7 @@ inkscape:groupmode="layer" id="layer8" inkscape:label="Plant" - transform="translate(17.27609,-30.156658)"> + transform="translate(17.276082,-30.156619)"> + transform="translate(17.2761, -30.1567)"> - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/simscape-nano-hexapod.org b/simscape-nano-hexapod.org index f1f9802..277d859 100644 --- a/simscape-nano-hexapod.org +++ b/simscape-nano-hexapod.org @@ -265,11 +265,27 @@ It should be the exact model reference that will be included in the NASS model ( ** TODO [#A] For simplicity, maybe not talk at all about parallel stiffness with the force sensor This could be the topic of the NASS section. -** TODO [#B] Check all notations +** DONE [#B] Check all notations +CLOSED: [2025-02-12 Wed 10:21] - [ ] Make sure they are all defined in correct order - [ ] Make sure all vectors and matrices are bold +| | uniaxial model | rotating model | HERE | Experimental | +|----------------------+----------------+----------------+-----------------------------------------------------+-----------------------------------------------------| +| Strut force | $f$ | $F_u$, $F_v$ | $\bm{f}$, $f_i$ | plant input $\bm{u}$, stack voltages $\bm{V}_a$ | +| Strut displacement | $d\mathcal{L}$ | $d_u$, $d_v$ | $\bm{\mathcal{L}}$, $l_i$ | | +| Error in strut frame | | | $\bm{\epsilon\mathcal{L}}$, $\epsilon\mathcal{L}_i$ | $\bm{\epsilon\mathcal{L}}$, $\epsilon\mathcal{L}_i$ | +| Encoder | - | | | $\bm{d}_e$ | +| Force Sensor | $f_n$ | $f_u$, $f_v$ | $\bm{f}_n$ | voltages: $\bm{V}_s$ | +| Sample motion | $d$ | | $\bm{\mathcal{X}}$ | $D_x$, ..., $R_y$ | +| Damped plant input | $u^\prime$ | | $\bm{f}^\prime$ | $\mathcal{u}^\prime$ | + +Conclusion: +- [X] Actuator forces: $\bm{\tau}$ => $\bm{f}$ and $f_i$ +- [X] Strut displacement: $d_L$ => $\bm{\mathcal{L}}$ and $l_i$ +- [X] Force sensor: $f_m$ => $\bm{f}_n$ and $f_{ni}$ + ** TODO [#B] Remove all un-used control architecture - [ ] Make sure HAC-IFF works as explained in the document @@ -371,7 +387,7 @@ Instead, a multi-body model of the Stewart platform is developed (Section ref:se 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 +* TODO Active Vibration Platforms <> ** Introduction :ignore: @@ -498,7 +514,7 @@ On the other hand, frames $\{A\}$ and $\{B\}$ are used to describe the relative 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 center of rotation for the joint fixed to the base is noted $\bm{a}_i$, while $\bm{b}_i$ 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. @@ -565,14 +581,14 @@ As was shown in Section ref:ssec:nhexa_stewart_platform_kinematics, the strut le 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} + {}^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{\bm{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. +By multiplying both sides by ${}^A\hat{\bm{s}}_i$, 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} @@ -620,7 +636,7 @@ As the inverse kinematic can be easily solved exactly this is not much useful, h **** 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. +For a series of platform positions along the $x\text{-axis}$, 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. @@ -682,25 +698,25 @@ exportFig('figs/nhexa_forward_kinematics_approximate_errors.pdf', 'width', 'wide 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 $\bm{O}_B$. +Let $\bm{f} = [f_1, f_2, \cdots, f_6]^T$ 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 actuator forces through virtual strut displacements $\delta \bm{\mathcal{L}}$: $\bm{f}^T \delta \bm{\mathcal{L}}$ - 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 +\delta W = \bm{f}^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 +\left( \bm{f}^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}}} +\bm{f}^T \bm{J} - \bm{\mathcal{F}}^T = 0 \quad \Rightarrow \quad \boxed{\bm{\mathcal{F}} = \bm{J}^T \bm{f}} \quad \text{and} \quad \boxed{\bm{f} = \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. @@ -711,31 +727,31 @@ These equations establish that the transpose of the Jacobian matrix maps actuato 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 $k_i$: +Starting from the individual actuators, the relationship between applied force $f_i$ 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 +f_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 $\mathcal{K}$: +These individual relationships can be combined into a matrix form using the diagonal stiffness matrix $\bm{\mathcal{K}}$: \begin{equation} -\bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}}, \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right] +\bm{f} = \bm{\mathcal{K}} \cdot \delta \bm{\mathcal{L}}, \quad \bm{\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}} +\bm{\mathcal{F}} = \underbrace{\bm{J}^T \bm{\mathcal{K}} \bm{J}}_{\bm{K}} \cdot \delta \bm{\mathcal{X}} \end{equation} -where $\bm{K} = \bm{J}^T \mathcal{K} \bm{J}$ is identified as the platform stiffness matrix. +where $\bm{K} = \bm{J}^T \bm{\mathcal{K}} \bm{J}$ 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}} +\delta \bm{\mathcal{X}} = \underbrace{(\bm{J}^T \bm{\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 individual actuator stiffnesses represented by $\bm{\mathcal{K}}$ - 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. @@ -756,33 +772,33 @@ For initial analysis, we consider a simplified model with the following assumpti Under these assumptions, the system dynamics can be expressed in the Cartesian space as: \begin{equation} -M s^2 \mathcal{X} = \Sigma \mathcal{F} +\bm{M} s^2 \bm{\mathcal{X}} = \Sigma \bm{\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. +where $\bm{M}$ represents the platform mass matrix, $\bm{\mathcal{X}}$ the platform pose, and $\Sigma \bm{\mathcal{F}}$ 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 $\mathcal{C} \dot{\mathcal{L}}$. +The primary forces acting on the system are actuator forces $\bm{f}$, elastic forces due to strut stiffness $-\bm{\mathcal{K}} \bm{\mathcal{L}}$ and damping forces in the struts $\bm{\mathcal{C}} \dot{\bm{\mathcal{L}}}$. \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) +\Sigma \bm{\mathcal{F}} = \bm{J}^T (\bm{f} - \bm{\mathcal{K}} \bm{\mathcal{L}} - s \bm{\mathcal{C}} \bm{\mathcal{L}}), \quad \bm{\mathcal{K}} = \text{diag}(k_1\,\dots\,k_6),\ \bm{\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} -\bm{M} s^2 \bm{\mathcal{X}} = \bm{\mathcal{F}} - \bm{J}^T \bm{\mathcal{K}} \bm{J} \bm{\mathcal{X}} - \bm{J}^T \bm{\mathcal{C}} \bm{J} s \bm{\mathcal{X}} + \bm{M} s^2 \bm{\mathcal{X}} = \bm{\mathcal{F}} - \bm{J}^T \bm{\mathcal{K}} \bm{J} \bm{\mathcal{X}} - \bm{J}^T \bm{\mathcal{C}} \bm{J} s \bm{\mathcal{X}} \end{equation} -The transfer function in the Cartesian frame becomes eqref:eq:nhexa_transfer_function_cart. +The transfer function matrix 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} + \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{T} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{T} \bm{\mathcal{K}} \bm{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} + \frac{\bm{\mathcal{L}}}{\bm{f}}(s) = ( \bm{J}^{-T} \bm{M} \bm{J}^{-1} s^2 + \bm{\mathcal{C}} + \bm{\mathcal{K}} )^{-1} \end{equation} While this simplified model provides useful insights, real Stewart platforms exhibit more complex behaviors. @@ -858,12 +874,12 @@ Finally, the validated model is employed to analyze the nano-hexapod dynamics, f **** 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 joints connecting the actuators to these frames are located at positions ${}^F\bm{a}_i$ and ${}^M\bm{b}_i$ respectively. The point of interest, denoted by frame $\{A\}$, is situated $150\,mm$ above the moving platform frame $\{M\}$. 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 $\bm{J}$. +From these parameters, key kinematic properties can be derived: the strut orientations $\hat{\bm{s}}_i$, strut lengths $l_i$, and the system's Jacobian matrix $\bm{J}$. #+attr_latex: :options [b]{0.6\linewidth} #+begin_minipage @@ -881,20 +897,20 @@ From these parameters, key kinematic properties can be derived: the strut orient #+attr_latex: :booktabs t :float nil :center nil | | $\bm{x}$ | $\bm{y}$ | $\bm{z}$ | |-----------+----------+----------+----------| -| ${}^MO_B$ | $0$ | $0$ | $150$ | -| ${}^FO_M$ | $0$ | $0$ | $95$ | -| ${}^Fa_1$ | $-92$ | $-77$ | $20$ | -| ${}^Fa_2$ | $92$ | $-77$ | $20$ | -| ${}^Fa_3$ | $113$ | $-41$ | $20$ | -| ${}^Fa_4$ | $21$ | $118$ | $20$ | -| ${}^Fa_5$ | $-21$ | $118$ | $20$ | -| ${}^Fa_6$ | $-113$ | $-41$ | $20$ | -| ${}^Mb_1$ | $-28$ | $-106$ | $-20$ | -| ${}^Mb_2$ | $28$ | $-106$ | $-20$ | -| ${}^Mb_3$ | $106$ | $28$ | $-20$ | -| ${}^Mb_4$ | $78$ | $78$ | $-20$ | -| ${}^Mb_5$ | $-78$ | $78$ | $-20$ | -| ${}^Mb_6$ | $-106$ | $28$ | $-20$ | +| ${}^M\bm{O}_B$ | $0$ | $0$ | $150$ | +| ${}^F\bm{O}_M$ | $0$ | $0$ | $95$ | +| ${}^F\bm{a}_1$ | $-92$ | $-77$ | $20$ | +| ${}^F\bm{a}_2$ | $92$ | $-77$ | $20$ | +| ${}^F\bm{a}_3$ | $113$ | $-41$ | $20$ | +| ${}^F\bm{a}_4$ | $21$ | $118$ | $20$ | +| ${}^F\bm{a}_5$ | $-21$ | $118$ | $20$ | +| ${}^F\bm{a}_6$ | $-113$ | $-41$ | $20$ | +| ${}^M\bm{b}_1$ | $-28$ | $-106$ | $-20$ | +| ${}^M\bm{b}_2$ | $28$ | $-106$ | $-20$ | +| ${}^M\bm{b}_3$ | $106$ | $28$ | $-20$ | +| ${}^M\bm{b}_4$ | $78$ | $78$ | $-20$ | +| ${}^M\bm{b}_5$ | $-78$ | $78$ | $-20$ | +| ${}^M\bm{b}_6$ | $-106$ | $28$ | $-20$ | #+latex: \captionof{table}{\label{tab:nhexa_stewart_model_geometry}Parameter values in [mm]} #+end_scriptsize #+end_minipage @@ -921,7 +937,7 @@ The actuator model comprises several key elements (Figure ref:fig:nhexa_actuator 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. -Each actuator is equipped with two sensors: a force sensor providing measurements $f_m$ and a relative motion sensor measuring displacement $d_L$. +Each actuator is equipped with two sensors: a force sensor providing measurements $f_n$ and a relative motion sensor measuring strut length $l_i$. The actuator parameters used in the conceptual phase are presented in Table ref:tab:nhexa_actuator_parameters. 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. @@ -952,7 +968,7 @@ This modular approach to actuator modeling allows for future refinements as the ** Validation of the multi-body model <> -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 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. @@ -1054,11 +1070,11 @@ ax1 = nexttile([2,1]); hold on; for i = 1:6 plot(freqs, abs(squeeze(freqresp(G_simscape(i,1), freqs, 'Hz'))), 'color', [colors(i,:), 0.5], ... - 'DisplayName', sprintf('$d_{L%i}/f_1$ - Multi-Body', i)) + 'DisplayName', sprintf('$l_%i/f_1$ - Multi-Body', i)) end for i = 1:6 plot(freqs, abs(squeeze(freqresp(G_analytical(i,1), freqs, 'Hz'))), '--', 'color', [colors(i,:)], ... - 'DisplayName', sprintf('$d_{L%i}/f_1$ - Analytical', i)) + 'DisplayName', sprintf('$l_%i/f_1$ - Analytical', i)) end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); @@ -1099,17 +1115,17 @@ exportFig('figs/nhexa_comp_multi_body_analytical.pdf', 'width', 'wide', 'height' 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. +Transfer functions from actuator forces $\bm{f}$ to both strut displacements $\bm{\mathcal{L}}$ and force measurements $\bm{f}_n$ 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. +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 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: $G(j\omega) \xrightarrow[\omega \to 0]{} \mathcal{K}^{-1}$. +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: $G(j\omega) \xrightarrow[\omega \to \infty]{} J M^{-T} J^T \frac{-1}{\omega^2}$ +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. @@ -1209,9 +1225,9 @@ for i = 1:5 end end plot(freqs, abs(squeeze(freqresp(G(7,1), freqs, 'Hz'))), 'color', colors(1,:), ... - 'DisplayName', '$f_{mi}/f_i$') + 'DisplayName', '$f_{ni}/f_i$') plot(freqs, abs(squeeze(freqresp(G_no_kp(7,1), freqs, 'Hz'))), 'color', colors(2,:), ... - 'DisplayName', '$f_{mi}/f_i$ (no $k_p$)') + 'DisplayName', '$f_{ni}/f_i$ (no $k_p$)') for i = 2:6 plot(freqs, abs(squeeze(freqresp(G(6+i,i), freqs, 'Hz'))), 'color', colors(1,:), ... 'HandleVisibility', 'off'); @@ -1219,7 +1235,7 @@ for i = 2:6 'HandleVisibility', 'off'); end plot(freqs, abs(squeeze(freqresp(G(7,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... - 'DisplayName', '$f_{mi}/f_j$') + 'DisplayName', '$f_{ni}/f_j$') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); @@ -1251,13 +1267,13 @@ 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}$\bm{f}$ to $\bm{d}_{L}$} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_dL}$\bm{f}$ to $\bm{\mathcal{L}}$} #+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}$\bm{f}$ to $\bm{f}_{m}$} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_multi_body_plant_fm}$\bm{f}$ to $\bm{f}_{n}$} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :width \linewidth @@ -1376,7 +1392,7 @@ Furthermore, at low frequencies, the plant exhibits good decoupling between stru \draw[->] ($(outputX) + (0.3, 0)$)node[branch]{} node[above]{$\bm{\mathcal{X}}$} -- ++(0, -1.2) -| (subr.south); \draw[->] (subr.east) -- node[midway, above]{$\bm{\epsilon}_{\mathcal{X}}$} (J.west); \draw[->] (J.east) -- node[midway, above]{$\bm{\epsilon}_{\mathcal{L}}$} (K.west); - \draw[->] (K.east) -- node[midway, above]{$\bm{\tau}$} (inputF); + \draw[->] (K.east) -- node[midway, above]{$\bm{f}$} (inputF); \draw[<-] (subr.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-0.8, 0); \end{tikzpicture} @@ -1399,7 +1415,7 @@ Furthermore, at low frequencies, the plant exhibits good decoupling between stru \draw[->] (subr.east) -- node[midway, above]{$\bm{\epsilon}_{\mathcal{X}}$} (K.west); \draw[->] (K.east) -- node[midway, above]{$\bm{\mathcal{F}}$} (J.west); - \draw[->] (J.east) -- node[midway, above]{$\bm{\tau}$} (inputF.west); + \draw[->] (J.east) -- node[midway, above]{$\bm{f}$} (inputF.west); \draw[<-] (subr.west)node[above left]{$\bm{r}_{\mathcal{X}}$} -- ++(-0.8, 0); \end{tikzpicture} #+end_src @@ -1479,13 +1495,13 @@ for i = 1:5 end end plot(freqs, abs(squeeze(freqresp(G_struts(1,1), freqs, 'Hz'))), 'color', colors(1,:), ... - 'DisplayName', '$-\epsilon_{\mathcal{L}i}/\tau_i$') + 'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_i$') for i = 2:6 plot(freqs, abs(squeeze(freqresp(G_struts(i,i), freqs, 'Hz'))), 'color', colors(1,:), ... 'HandleVisibility', 'off'); end plot(freqs, abs(squeeze(freqresp(G_struts(1,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... - 'DisplayName', '$-\epsilon_{\mathcal{L}i}/\tau_j$') + 'DisplayName', '$-\epsilon_{\mathcal{L}i}/f_j$') hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); @@ -1604,11 +1620,11 @@ The corresponding block diagram of the control loop is shown in Figure ref:fig:n % Connections and labels \draw[->] (outputL) -- ++(0.7, 0) coordinate(eastlac) |- (Klac.east); - \node[above right] at (outputL){$\bm{f}_m$}; + \node[above right] at (outputL){$\bm{f}_n$}; \draw[->] (Klac.west) -| (addF.north); \draw[->] (addF.east) -- (input) node[above left]{$\bm{f}$}; - \draw[->] (outputH) -- ++(1.7, 0) node[above left]{$\bm{d_L}$}; + \draw[->] (outputH) -- ++(1.7, 0) node[above left]{$\bm{\mathcal{L}}$}; \draw[<-] (addF.west) -- ++(-0.8, 0) node[above right]{$\bm{f}^{\prime}$}; \begin{scope}[on background layer] @@ -1640,7 +1656,7 @@ The Root Locus analysis, shown in Figure ref:fig:nhexa_decentralized_iff_root_lo A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability [[cite:&preumont08_trans_zeros_struc_contr_with]]. This property is particularly valuable as the coupling is very large around resonance frequencies, enabling control of modes that would be difficult to include within the bandwidth using position feedback alone. -The bode plot of an individual loop gain (i.e. the loop gain of $K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)$), presented in Figure ref:fig:nhexa_decentralized_iff_loop_gain, exhibits the typical characteristics of integral force feedback of having a phase bounded between $-90^o$ and $+90^o$. +The bode plot of an individual loop gain (i.e. the loop gain of $K_{\text{IFF}}(s) \cdot \frac{f_{ni}}{f_i}(s)$), presented in Figure ref:fig:nhexa_decentralized_iff_loop_gain, exhibits the typical characteristics of integral force feedback of having a phase bounded between $-90^o$ and $+90^o$. The loop-gain is high around the resonance frequencies, indicating that the decentralized IFF provides significant control authority over these modes. This high gain, combined with the bounded phase, enables effective damping of the resonant modes while maintaining stability. @@ -1741,7 +1757,7 @@ exportFig('figs/nhexa_decentralized_iff_loop_gain.pdf', 'width', 'half', 'height #+caption: Decentralized IFF #+attr_latex: :options [htbp] #+begin_figure -#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain: bode plot of $K_{\text{IFF}}(s) \frac{f_{m1}}{f_1}(s)$} +#+attr_latex: :caption \subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain: bode plot of $K_{\text{IFF}}(s) \frac{f_{n1}}{f_1}(s)$} #+attr_latex: :options {0.48\textwidth} #+begin_subfigure #+attr_latex: :scale 0.85 @@ -1783,7 +1799,7 @@ A diagonal High Authority Controller $\bm{K}_{\text{HAC}}$ then processes these % Connections and labels \draw[->] (outputL) -- ++(0.7, 0) coordinate(eastlac) |- (Klac.east); - \node[above right] at (outputL){$\bm{f}_m$}; + \node[above right] at (outputL){$\bm{f}_n$}; \draw[->] (Klac.west) -| 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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} @@ -147,7 +148,7 @@ On the other hand, frames \(\{A\}\) and \(\{B\}\) are used to describe the relat 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 center of rotation for the joint fixed to the base is noted \(\bm{a}_i\), while \(\bm{b}_i\) 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}. @@ -211,7 +212,7 @@ As was shown in Section \ref{ssec:nhexa_stewart_platform_kinematics}, the strut By taking the time derivative of the position loop close \eqref{eq:nhexa_loop_closure}, the \emph{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} + {}^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{\bm{a}}_i}_{=0} \end{equation} Moreover, we have: @@ -220,7 +221,7 @@ Moreover, we have: \item \(l_i {}^A\dot{\hat{\bm{s}}}_i = l_i \left( {}^A\bm{\omega}_i \times \hat{\bm{s}}_i \right)\) in which \({}^A\bm{\omega}_i\) is the angular velocity of strut \(i\) express in fixed frame \(\{\bm{A}\}\). \end{itemize} -By multiplying both sides by \({}^A\hat{s}_i\), \eqref{eq:nhexa_loop_closure_velocity_bis} is obtained. +By multiplying both sides by \({}^A\hat{\bm{s}}_i\), \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} @@ -270,7 +271,7 @@ As the inverse kinematic can be easily solved exactly this is not much useful, h \paragraph{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}. +For a series of platform positions along the \(x\text{-axis}\), 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}. @@ -291,27 +292,27 @@ It can be computed once at the rest position and used for both forward and inver 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 \(\bm{O}_B\). +Let \(\bm{f} = [f_1, f_2, \cdots, f_6]^T\) 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: \begin{itemize} -\item The work performed by the actuator forces through virtual strut displacements \(\delta \bm{\mathcal{L}}\): \(\bm{\tau}^T \delta \bm{\mathcal{L}}\) +\item The work performed by the actuator forces through virtual strut displacements \(\delta \bm{\mathcal{L}}\): \(\bm{f}^T \delta \bm{\mathcal{L}}\) \item The work performed by the external wrench through virtual platform displacements \(\delta \bm{\mathcal{X}}\): \(-\bm{\mathcal{F}}^T \delta \bm{\mathcal{X}}\) \end{itemize} 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 +\delta W = \bm{f}^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 +\left( \bm{f}^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}}} +\bm{f}^T \bm{J} - \bm{\mathcal{F}}^T = 0 \quad \Rightarrow \quad \boxed{\bm{\mathcal{F}} = \bm{J}^T \bm{f}} \quad \text{and} \quad \boxed{\bm{f} = \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. @@ -322,32 +323,32 @@ These equations establish that the transpose of the Jacobian matrix maps actuato 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 \(k_i\): +Starting from the individual actuators, the relationship between applied force \(f_i\) 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 +f_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 \(\mathcal{K}\): +These individual relationships can be combined into a matrix form using the diagonal stiffness matrix \(\bm{\mathcal{K}}\): \begin{equation} -\bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}}, \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right] +\bm{f} = \bm{\mathcal{K}} \cdot \delta \bm{\mathcal{L}}, \quad \bm{\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}} +\bm{\mathcal{F}} = \underbrace{\bm{J}^T \bm{\mathcal{K}} \bm{J}}_{\bm{K}} \cdot \delta \bm{\mathcal{X}} \end{equation} -where \(\bm{K} = \bm{J}^T \mathcal{K} \bm{J}\) is identified as the platform stiffness matrix. +where \(\bm{K} = \bm{J}^T \bm{\mathcal{K}} \bm{J}\) 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}} +\delta \bm{\mathcal{X}} = \underbrace{(\bm{J}^T \bm{\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: \begin{itemize} -\item The individual actuator stiffnesses represented by \(\mathcal{K}\) +\item The individual actuator stiffnesses represented by \(\bm{\mathcal{K}}\) \item The geometric configuration embodied in the Jacobian matrix \(\bm{J}\) \end{itemize} @@ -371,33 +372,33 @@ For initial analysis, we consider a simplified model with the following assumpti Under these assumptions, the system dynamics can be expressed in the Cartesian space as: \begin{equation} -M s^2 \mathcal{X} = \Sigma \mathcal{F} +\bm{M} s^2 \bm{\mathcal{X}} = \Sigma \bm{\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. +where \(\bm{M}\) represents the platform mass matrix, \(\bm{\mathcal{X}}\) the platform pose, and \(\Sigma \bm{\mathcal{F}}\) 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 \(\mathcal{C} \dot{\mathcal{L}}\). +The primary forces acting on the system are actuator forces \(\bm{f}\), elastic forces due to strut stiffness \(-\bm{\mathcal{K}} \bm{\mathcal{L}}\) and damping forces in the struts \(\bm{\mathcal{C}} \dot{\bm{\mathcal{L}}}\). \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) +\Sigma \bm{\mathcal{F}} = \bm{J}^T (\bm{f} - \bm{\mathcal{K}} \bm{\mathcal{L}} - s \bm{\mathcal{C}} \bm{\mathcal{L}}), \quad \bm{\mathcal{K}} = \text{diag}(k_1\,\dots\,k_6),\ \bm{\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} -\bm{M} s^2 \bm{\mathcal{X}} = \bm{\mathcal{F}} - \bm{J}^T \bm{\mathcal{K}} \bm{J} \bm{\mathcal{X}} - \bm{J}^T \bm{\mathcal{C}} \bm{J} s \bm{\mathcal{X}} + \bm{M} s^2 \bm{\mathcal{X}} = \bm{\mathcal{F}} - \bm{J}^T \bm{\mathcal{K}} \bm{J} \bm{\mathcal{X}} - \bm{J}^T \bm{\mathcal{C}} \bm{J} s \bm{\mathcal{X}} \end{equation} -The transfer function in the Cartesian frame becomes \eqref{eq:nhexa_transfer_function_cart}. +The transfer function matrix 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} + \frac{{\mathcal{X}}}{\bm{\mathcal{F}}}(s) = ( \bm{M} s^2 + \bm{J}^{T} \bm{\mathcal{C}} \bm{J} s + \bm{J}^{T} \bm{\mathcal{K}} \bm{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} + \frac{\bm{\mathcal{L}}}{\bm{f}}(s) = ( \bm{J}^{-T} \bm{M} \bm{J}^{-1} s^2 + \bm{\mathcal{C}} + \bm{\mathcal{K}} )^{-1} \end{equation} While this simplified model provides useful insights, real Stewart platforms exhibit more complex behaviors. @@ -438,12 +439,12 @@ Finally, the validated model is employed to analyze the nano-hexapod dynamics, f \paragraph{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 joints connecting the actuators to these frames are located at positions \({}^F\bm{a}_i\) and \({}^M\bm{b}_i\) respectively. The point of interest, denoted by frame \(\{A\}\), is situated \(150\,mm\) above the moving platform frame \(\{M\}\). 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 \(\bm{J}\). +From these parameters, key kinematic properties can be derived: the strut orientations \(\hat{\bm{s}}_i\), strut lengths \(l_i\), and the system's Jacobian matrix \(\bm{J}\). \begin{minipage}[b]{0.6\linewidth} \begin{center} @@ -459,20 +460,20 @@ From these parameters, key kinematic properties can be derived: the strut orient \toprule & \(\bm{x}\) & \(\bm{y}\) & \(\bm{z}\)\\ \midrule -\({}^MO_B\) & \(0\) & \(0\) & \(150\)\\ -\({}^FO_M\) & \(0\) & \(0\) & \(95\)\\ -\({}^Fa_1\) & \(-92\) & \(-77\) & \(20\)\\ -\({}^Fa_2\) & \(92\) & \(-77\) & \(20\)\\ -\({}^Fa_3\) & \(113\) & \(-41\) & \(20\)\\ -\({}^Fa_4\) & \(21\) & \(118\) & \(20\)\\ -\({}^Fa_5\) & \(-21\) & \(118\) & \(20\)\\ -\({}^Fa_6\) & \(-113\) & \(-41\) & \(20\)\\ -\({}^Mb_1\) & \(-28\) & \(-106\) & \(-20\)\\ -\({}^Mb_2\) & \(28\) & \(-106\) & \(-20\)\\ -\({}^Mb_3\) & \(106\) & \(28\) & \(-20\)\\ -\({}^Mb_4\) & \(78\) & \(78\) & \(-20\)\\ -\({}^Mb_5\) & \(-78\) & \(78\) & \(-20\)\\ -\({}^Mb_6\) & \(-106\) & \(28\) & \(-20\)\\ +\({}^M\bm{O}_B\) & \(0\) & \(0\) & \(150\)\\ +\({}^F\bm{O}_M\) & \(0\) & \(0\) & \(95\)\\ +\({}^F\bm{a}_1\) & \(-92\) & \(-77\) & \(20\)\\ +\({}^F\bm{a}_2\) & \(92\) & \(-77\) & \(20\)\\ +\({}^F\bm{a}_3\) & \(113\) & \(-41\) & \(20\)\\ +\({}^F\bm{a}_4\) & \(21\) & \(118\) & \(20\)\\ +\({}^F\bm{a}_5\) & \(-21\) & \(118\) & \(20\)\\ +\({}^F\bm{a}_6\) & \(-113\) & \(-41\) & \(20\)\\ +\({}^M\bm{b}_1\) & \(-28\) & \(-106\) & \(-20\)\\ +\({}^M\bm{b}_2\) & \(28\) & \(-106\) & \(-20\)\\ +\({}^M\bm{b}_3\) & \(106\) & \(28\) & \(-20\)\\ +\({}^M\bm{b}_4\) & \(78\) & \(78\) & \(-20\)\\ +\({}^M\bm{b}_5\) & \(-78\) & \(78\) & \(-20\)\\ +\({}^M\bm{b}_6\) & \(-106\) & \(28\) & \(-20\)\\ \bottomrule \end{tabularx} \captionof{table}{\label{tab:nhexa_stewart_model_geometry}Parameter values in [mm]} @@ -501,7 +502,7 @@ The actuator model comprises several key elements (Figure \ref{fig:nhexa_actuato 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. -Each actuator is equipped with two sensors: a force sensor providing measurements \(f_m\) and a relative motion sensor measuring displacement \(d_L\). +Each actuator is equipped with two sensors: a force sensor providing measurements \(f_n\) and a relative motion sensor measuring strut length \(l_i\). The actuator parameters used in the conceptual phase are presented in Table \ref{tab:nhexa_actuator_parameters}. 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. @@ -532,7 +533,7 @@ This modular approach to actuator modeling allows for future refinements as the \section{Validation of the multi-body model} \label{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 \(\bm{f}_m\), and relative displacement measurements \(\bm{d}_L\). +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}. @@ -584,10 +585,10 @@ The close agreement between both approaches across the frequency spectrum valida 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. +Transfer functions from actuator forces \(\bm{f}\) to both strut displacements \(\bm{\mathcal{L}}\) and force measurements \(\bm{f}_n\) 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. +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 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. @@ -758,7 +759,7 @@ The Root Locus analysis, shown in Figure \ref{fig:nhexa_decentralized_iff_root_l A key characteristic of force feedback control with collocated sensor-actuator pairs is observed: all closed-loop poles are bounded to the left-half plane, indicating guaranteed stability \cite{preumont08_trans_zeros_struc_contr_with}. This property is particularly valuable as the coupling is very large around resonance frequencies, enabling control of modes that would be difficult to include within the bandwidth using position feedback alone. -The bode plot of an individual loop gain (i.e. the loop gain of \(K_{\text{IFF}}(s) \cdot \frac{f_{mi}}{f_i}(s)\)), presented in Figure \ref{fig:nhexa_decentralized_iff_loop_gain}, exhibits the typical characteristics of integral force feedback of having a phase bounded between \(-90^o\) and \(+90^o\). +The bode plot of an individual loop gain (i.e. the loop gain of \(K_{\text{IFF}}(s) \cdot \frac{f_{ni}}{f_i}(s)\)), presented in Figure \ref{fig:nhexa_decentralized_iff_loop_gain}, exhibits the typical characteristics of integral force feedback of having a phase bounded between \(-90^o\) and \(+90^o\). The loop-gain is high around the resonance frequencies, indicating that the decentralized IFF provides significant control authority over these modes. This high gain, combined with the bounded phase, enables effective damping of the resonant modes while maintaining stability. @@ -767,7 +768,7 @@ This high gain, combined with the bounded phase, enables effective damping of th \begin{center} \includegraphics[scale=1,scale=0.85]{figs/nhexa_decentralized_iff_loop_gain.png} \end{center} -\subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain: bode plot of $K_{\text{IFF}}(s) \frac{f_{m1}}{f_1}(s)$} +\subcaption{\label{fig:nhexa_decentralized_iff_loop_gain}Loop Gain: bode plot of $K_{\text{IFF}}(s) \frac{f_{n1}}{f_1}(s)$} \end{subfigure} \begin{subfigure}{0.48\textwidth} \begin{center}

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