From a581bc9dbd9773c0354435eaaa3e73bc1a740c14 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 3 Apr 2020 18:04:47 +0200 Subject: [PATCH] Update html pages --- docs/optimal_stiffness.html | 541 ++++++++++++++++++++- docs/simscape_subsystems.html | 869 ++++++++++++++++++---------------- 2 files changed, 983 insertions(+), 427 deletions(-) diff --git a/docs/optimal_stiffness.html b/docs/optimal_stiffness.html index 9602925..59f7174 100644 --- a/docs/optimal_stiffness.html +++ b/docs/optimal_stiffness.html @@ -1,11 +1,10 @@ - - + Determination of the optimal nano-hexapod’s stiffness @@ -226,6 +225,16 @@ /*]]>*///--> // @license-end + +
@@ -238,9 +247,35 @@

Table of Contents

@@ -267,14 +302,147 @@ We wish here to see if we can determine an optimal stiffness of the nano-hexapod
  • Section 3: the change of payload impedance has acceptable effect on the plant dynamics
    • +

    +The overall goal is to design a nano-hexapod that will allow the highest possible control bandwidth. +

    +

    1 Spindle Rotation Speed

    +

    +In this section, we look at the effect of the spindle rotation speed on the plant dynamics. +

    + +

    +The rotation speed will have an effect due to the Coriolis effect. +

    +
    + +
    +

    1.1 Initialization

    +
    +

    +We initialize all the stages with the default parameters. +

    +
    +
    initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod();
+initializeAxisc();
+initializeMirror();
+
    +
    + +

    +We use a sample mass of 10kg. +

    +
    +
    initializeSample('mass', 10);
+
    +
    + +

    +We don’t include disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics. +We however include gravity. +

    +
    +
    initializeSimscapeConfiguration('gravity', true);
+initializeDisturbances('enable', false);
+initializeLoggingConfiguration('log', 'none');
+initializeController();
+
    +
    + +
    +

    1.2 Identification when rotating at maximum speed

    +
    +

    +We identify the dynamics for the following spindle rotation speeds Rz_rpm: +

    +
    +
    Rz_rpm = linspace(0, 60, 6);
+
    +
    + +

    +And for the following nano-hexapod actuator stiffness Ks: +

    +
    +
    Ks = logspace(3,9,7); % [N/m]
+
    +
    +
    +
    + +
    +

    1.3 Change of dynamics

    +
    +

    +We plot the change of dynamics due to the change of the spindle rotation speed (from 0rpm to 60rpm): +

    +
      +
    • Figure 1: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)
      • +
    • Figure 2: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)
      • +
    • Figure 3: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)
      • +
    • Figure 4: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_y\) (coupling of the centralized positioning plant)
      • +
    + + +
    +

    opt_stiffness_wz_iff.png +

    +

    Figure 1: Change of dynamics from actuator \(\tau\) to actuator force sensor \(\tau_m\) for a spindle rotation speed from 0rpm to 60rpm (png, pdf)

    +
    + + +
    +

    opt_stiffness_wz_dvf.png +

    +

    Figure 2: Change of dynamics from actuator force \(\tau\) to actuator displacement \(d\mathcal{L}\) for a spindle rotation speed from 0rpm to 60rpm (png, pdf)

    +
    + + +
    +

    opt_stiffness_wz_fx_dx.png +

    +

    Figure 3: Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm (png, pdf)

    +
    + + +
    +

    opt_stiffness_wz_coupling.png +

    +

    Figure 4: Change of Coupling from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_y\) for a spindle rotation speed from 0rpm to 60rpm (png, pdf)

    +
    +
    +
    + +
    +
    +

    +The leg stiffness should be at higher than \(k_i = 10^4\ [N/m]\) such that the main resonance frequency does not shift too much when rotating. +For the coupling, it is more difficult to conclude about the minimum required leg stiffness. +

    + +
    + +
    +

    +Note that we can use very soft nano-hexapod if we limit the spindle rotating speed. +

    + +
    +
    +
    +

    2 Micro-Station Compliance Effect

    @@ -286,7 +454,135 @@ We wish here to see if we can determine an optimal stiffness of the nano-hexapod
  • simple model + uncertainty
    • + +
    +

    2.1 Identification of the micro-station compliance

    +
    +

    +We initialize all the stages with the default parameters. +

    +
    +
    initializeGround();
+initializeGranite();
+initializeTy();
+initializeRy();
+initializeRz();
+initializeMicroHexapod('type', 'compliance');
+
    + +

    +We put nothing on top of the micro-hexapod. +

    +
    +
    initializeAxisc('type', 'none');
+initializeMirror('type', 'none');
+initializeNanoHexapod('type', 'none');
+initializeSample('type', 'none');
+
    +
    + +

    +And we identify the dynamics from forces/torques applied on the micro-hexapod top platform to the motion of the micro-hexapod top platform at the same point. +The diagonal element of the identified Micro-Station compliance matrix are shown in Figure 5. +

    + + +
    +

    opt_stiff_micro_station_compliance.png +

    +

    Figure 5: Identified Compliance of the Micro-Station (png, pdf)

    +
    +
    +
    + +
    +

    2.2 Identification of the dynamics with a rigid micro-station

    +
    +

    +We now identify the dynamics when the micro-station is rigid. +This is equivalent of identifying the dynamics of the nano-hexapod when fixed to a rigid ground. +We also choose the sample to be rigid and to have a mass of 10kg. +

    +
    +
    initializeSample('type', 'rigid', 'mass', 10);
+
    +
    + +

    +As before, we identify the dynamics for the following actuator stiffnesses: +

    +
    +
    Ks = logspace(3,9,7); % [N/m]
+
    +
    +
    +
    + +
    +

    2.3 Identification of the dynamics with a flexible micro-station

    +
    +

    +We now initialize all the micro-station stages to be flexible. +And we identify the dynamics of the nano-hexapod. +

    +
    +
    +
    +

    2.4 Obtained Dynamics

    +
    +

    +We plot the change of dynamics due to the compliance of the Micro-Station. +The solid curves are corresponding to the nano-hexapod without the micro-station, and the dashed curves with the micro-station: +

    +
      +
    • Figure 6: from actuator force \(\tau\) to force sensor \(\tau_m\) (IFF plant)
      • +
    • Figure 7: from actuator force \(\tau\) to actuator relative displacement \(d\mathcal{L}\) (Decentralized positioning plant)
      • +
    • Figure 8: from force in the task space \(\mathcal{F}_x\) to sample displacement \(\mathcal{X}_x\) (Centralized positioning plant)
      • +
    • Figure 9: from force in the task space \(\mathcal{F}_z\) to sample displacement \(\mathcal{X}_z\) (Centralized positioning plant)
      • +
    + + +
    +

    opt_stiffness_micro_station_iff.png +

    +

    Figure 6: Change of dynamics from actuator \(\tau\) to actuator force sensor \(\tau_m\) for a spindle rotation speed from 0rpm to 60rpm (png, pdf)

    +
    + + +
    +

    opt_stiffness_micro_station_dvf.png +

    +

    Figure 7: Change of dynamics from actuator force \(\tau\) to actuator displacement \(d\mathcal{L}\) for a spindle rotation speed from 0rpm to 60rpm (png, pdf)

    +
    + + +
    +

    opt_stiffness_micro_station_fx_dx.png +

    +

    Figure 8: Change of dynamics from force \(\mathcal{F}_x\) to displacement \(\mathcal{X}_x\) for a spindle rotation speed from 0rpm to 60rpm (png, pdf)

    +
    + + +
    +

    opt_stiffness_micro_station_fz_dz.png +

    +

    Figure 9: Change of dynamics from force \(\mathcal{F}_z\) to displacement \(\mathcal{X}_z\) for a spindle rotation speed from 0rpm to 60rpm (png, pdf)

    +
    +
    +
    + +
    +
    +

    +The dynamics of the nano-hexapod is not affected by the micro-station dynamics (compliance) when the stiffness of the legs is less than \(10^6\ [N/m]\). +When the nano-hexapod is stiff (\(k>10^7\ [N/m]\)), the compliance of the micro-station appears in the primary plant. +

    + +
    +
    +
    +

    3 Payload “Impedance” Effect

    @@ -294,11 +590,244 @@ We wish here to see if we can determine an optimal stiffness of the nano-hexapod

    + +
    +

    3.1 Initialization

    +
    +

    +We initialize all the stages with the default parameters. +We don’t include disturbances in this model as it adds complexity to the simulations and does not alter the obtained dynamics. :exports none +

    +
    +
    initializeDisturbances('enable', false);
+
    +
    + +

    +We set the controller type to Open-Loop, and we do not need to log any signal. +

    +
    +
    initializeSimscapeConfiguration('gravity', true);
+initializeController();
+initializeLoggingConfiguration('log', 'none');
+initializeReferences();
+
    +
    +
    +
    + +
    +

    3.2 Identification of the dynamics while change the payload dynamics

    +
    +

    +We make the following change of payload dynamics: +

    +
      +
    • Change of mass: from 1kg to 50kg
      • +
    • Change of resonance frequency: from 50Hz to 500Hz
      • +
    • The damping ratio of the payload is fixed to \(\xi = 0.2\)
      • +
    + +

    +We identify the dynamics for the following payload masses Ms and nano-hexapod leg’s stiffnesses Ks: +

    +
    +
    Ms = [1, 20, 50]; % [Kg]
+Ks = logspace(3,9,7); % [N/m]
+
    +
    + +

    +We then identify the dynamics for the following payload resonance frequencies Fs: +

    +
    +
    Fs = [50, 200, 500]; % [Hz]
+
    +
    +
    +
    + +
    +

    3.3 Change of dynamics for the primary controller

    +
    +
    +
    +

    3.3.1 Frequency variation

    +
    +

    +We here compare the dynamics for the same payload mass, but different stiffness resulting in different resonance frequency of the payload: +

    +
      +
    • Figure 10: dynamics from a force \(\mathcal{F}_z\) applied in the task space in the vertical direction to the vertical displacement of the sample \(\mathcal{X}_z\) for both a very soft and a very stiff nano-hexapod.
      • +
    • Figure 11: same, but for all tested nano-hexapod stiffnesses
      • +
    + +

    +We can see two mass lines for the soft nano-hexapod (Figure 10): +

    +
      +
    • The first mass line corresponds to \(\frac{1}{(m_n + m_p)s^2}\) where \(m_p = 10\ [kg]\) is the mass of the payload and \(m_n = 15\ [Kg]\) is the mass of the nano-hexapod top platform and attached mirror
      • +
    • The second mass line corresponds to \(\frac{1}{m_n s^2}\)
      • +
    • The zero corresponds to the resonance of the payload alone (fixed nano-hexapod’s top platform)
      • +
    + + +
    +

    opt_stiffness_payload_freq_fz_dz.png +

    +

    Figure 10: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency, both for a soft nano-hexapod and a stiff nano-hexapod (png, pdf)

    +
    + + +
    +

    opt_stiffness_payload_freq_all.png +

    +

    Figure 11: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload resonance frequency (png, pdf)

    +
    +
    +
    + +
    +

    3.3.2 Mass variation

    +
    +

    +We here compare the dynamics for different payload mass with the same resonance frequency (100Hz): +

    +
      +
    • Figure 12: dynamics from a force \(\mathcal{F}_z\) applied in the task space in the vertical direction to the vertical displacement of the sample \(\mathcal{X}_z\) for both a very soft and a very stiff nano-hexapod.
      • +
    • Figure 13: same, but for all tested nano-hexapod stiffnesses
      • +
    + +

    +We can see here that for the soft nano-hexapod: +

    +
      +
    • the first resonance \(\omega_n\) is changing with the mass of the payload as \(\omega_n = \sqrt{\frac{k_n}{m_p + m_n}}\) with \(k_p\) the stiffness of the nano-hexapod, \(m_p\) the payload’s mass and \(m_n\) the mass of the nano-hexapod top platform
      • +
    • the first mass line corresponding to \(\frac{1}{(m_p + m_n)s^2}\) is changing with the payload mass
      • +
    • the zero at 100Hz is not changing as it corresponds to the resonance of the payload itself
      • +
    • the second mass line does not change
      • +
    + + +
    +

    opt_stiffness_payload_mass_fz_dz.png +

    +

    Figure 12: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass, both for a soft nano-hexapod and a stiff nano-hexapod (png, pdf)

    +
    + + +
    +

    opt_stiffness_payload_mass_all.png +

    +

    Figure 13: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload mass (png, pdf)

    +
    +
    +
    + +
    +

    3.3.3 Total variation

    +
    +

    +We now plot the total change of dynamics due to change of the payload (Figure 14): +

    +
      +
    • mass from 1kg to 50kg
      • +
    • main resonance from 50Hz to 500Hz
      • +
    + + +
    +

    opt_stiffness_payload_impedance_fz_dz.png +

    +

    Figure 14: Dynamics from \(\mathcal{F}_z\) to \(\mathcal{X}_z\) for varying payload dynamics, both for a soft nano-hexapod and a stiff nano-hexapod (png, pdf)

    +
    +
    +
    +
    + +
    +
    +

    + +

    + +
    +
    +
    + +
    +

    4 Total Change of dynamics

    +
    +

    +We now consider the total change of nano-hexapod dynamics due to: +

    +
      +
    • Gk_wz_err - Change of spindle rotation speed
      • +
    • Gf_err and Gm_err - Change of payload resonance
      • +
    • Gmf_err and Gmr_err - Micro-Station compliance
      • +
    + +

    +The obtained dynamics are shown: +

    +
      +
    • Figure 15 for a stiffness \(k = 10^3\ [N/m]\)
      • +
    • Figure 16 for a stiffness \(k = 10^5\ [N/m]\)
      • +
    • Figure 17 for a stiffness \(k = 10^7\ [N/m]\)
      • +
    • Figure 18 for a stiffness \(k = 10^9\ [N/m]\)
      • +
    + +

    +And finally, in Figures 19 and 20 are shown an animation of the change of dynamics with the nano-hexapod’s stiffness. +

    + + +
    +

    opt_stiffness_plant_dynamics_fx_dx_k_1e3.png +

    +

    Figure 15: Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^3\ [N/m]\) (png, pdf)

    +
    + + +
    +

    opt_stiffness_plant_dynamics_fx_dx_k_1e5.png +

    +

    Figure 16: Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^5\ [N/m]\) (png, pdf)

    +
    + + +
    +

    opt_stiffness_plant_dynamics_fx_dx_k_1e7.png +

    +

    Figure 17: Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^7\ [N/m]\) (png, pdf)

    +
    + + +
    +

    opt_stiffness_plant_dynamics_fx_dx_k_1e9.png +

    +

    Figure 18: Total variation of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\). Nano-hexapod leg’s stiffness is equal to \(k = 10^9\ [N/m]\) (png, pdf)

    +
    + + +
    +

    opt_stiffness_plant_dynamics_task_space.gif +

    +

    Figure 19: Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness

    +
    + + +
    +

    opt_stiffness_plant_dynamics_task_space_colors.gif +

    +

    Figure 20: Variability of the dynamics from \(\mathcal{F}_x\) to \(\mathcal{X}_x\) with varying nano-hexapod stiffness

    +
    +

    Author: Dehaeze Thomas

    -

    Created: 2020-04-01 mer. 17:19

    +

    Created: 2020-04-03 ven. 17:55

    diff --git a/docs/simscape_subsystems.html b/docs/simscape_subsystems.html index b34ee11..4e147c3 100644 --- a/docs/simscape_subsystems.html +++ b/docs/simscape_subsystems.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Subsystems used for the Simscape Models @@ -239,170 +239,170 @@