From 9e41241e4504ee44cb2b4804085ffc8c4028515b Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 25 Oct 2019 12:33:08 +0200 Subject: [PATCH] Update Analysis --- figs/uniaxial-model-act_damp.png | Bin 0 -> 51256 bytes ...ial-model-nass-flexible-active-damping.png | Bin 0 -> 73576 bytes figs/uniaxial-model-nass-flexible-dvf.png | Bin 0 -> 30838 bytes figs/uniaxial-model-nass-flexible-iff.png | Bin 0 -> 30714 bytes figs/uniaxial-model-nass-flexible-rmc.png | Bin 0 -> 32402 bytes figs/uniaxial-model-nass-flexible.png | Bin 0 -> 29912 bytes figs/uniaxial-plant.png | Bin 0 -> 68597 bytes figs/uniaxial-sensitivity-disturbances.png | Bin 0 -> 113939 bytes figs/uniaxial-sensitivity-force-dist.png | Bin 0 -> 77955 bytes figs/uniaxial_dvf_loop_gain.png | Bin 0 -> 100655 bytes figs/uniaxial_dvf_plant.png | Bin 0 -> 96767 bytes figs/uniaxial_iff_open_loop.png | Bin 0 -> 96010 bytes figs/uniaxial_iff_plant.png | Bin 0 -> 106989 bytes figs/uniaxial_plant_damped_comp.png | Bin 0 -> 103345 bytes 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    @@ -258,51 +280,48 @@ for the JavaScript code in this tag.

    Table of Contents

    @@ -317,52 +336,102 @@ The idea is to use the same model as the full Simscape Model but to restrict the This is done in order to more easily study the system and evaluate control techniques.

    -
    -

    1 Undamped System

    +
    +

    1 Simscape Model

    +

    +A schematic of the uniaxial model used for simulations is represented in figure 1. +

    + +

    +The perturbations \(w\) are: +

    +
      +
    • \(F_s\): direct forces applied to the sample such as inertia forces and cable forces
    • +
    • \(F_{rz}\): parasitic forces due to the rotation of the spindle
    • +
    • \(F_{ty}\): parasitic forces due to scans with the translation stage
    • +
    • \(D_w\): ground motion
    • +
    + +

    +The quantity to \(z\) to control is: +

    +
      +
    • \(D\): the position of the sample with respect to the granite
    • +
    + +

    +The measured quantities \(v\) are: +

    +
      +
    • \(D\): the position of the sample with respect to the granite
    • +
    + +

    +We study the use of an additional sensor: +

    +
      +
    • \(F_n\): a force sensor located in the nano-hexapod
    • +
    • \(v_n\): an absolute velocity sensor located on the top platform of the nano-hexapod
    • +
    • \(d_r\): a relative motion sensor located in the nano-hexapod
    • +
    + +

    +The control signal \(u\) is: +

    +
      +
    • \(F\) the force applied by the nano-hexapod actuator
    • +
    + + +
    +

    uniaxial-model-nass-flexible.png +

    +

    Figure 1: Schematic of the uniaxial model used

    -
    -

    1.1 Init

    -
    + + +

    +Few active damping techniques will be compared in order to decide which sensor is to be included in the system. +Schematics of the active damping techniques are displayed in figure 2. +

    + + +
    +

    uniaxial-model-nass-flexible-active-damping.png +

    +

    Figure 2: Comparison of used active damping techniques

    +
    +
    +
    + +
    +

    2 Undamped System

    +
    +

    +Let's start by study the undamped system. +

    +
    + +
    +

    2.1 Init

    +

    We initialize all the stages with the default parameters. The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg.

    -
    -
    initializeGround();
    -initializeGranite();
    -initializeTy();
    -initializeRy();
    -initializeRz();
    -initializeMicroHexapod();
    -initializeAxisc();
    -initializeMirror();
    -initializeNanoHexapod(struct('actuator', 'piezo'));
    -initializeSample(struct('mass', 50));
    -
    -
    -

    -All the controllers are set to 0. +All the controllers are set to 0 (Open Loop).

    -
    -
    K = tf(0);
    -save('./mat/controllers.mat', 'K', '-append');
    -K_iff = tf(0);
    -save('./mat/controllers.mat', 'K_iff', '-append');
    -K_rmc = tf(0);
    -save('./mat/controllers.mat', 'K_rmc', '-append');
    -K_dvf = tf(0);
    -save('./mat/controllers.mat', 'K_dvf', '-append');
    -
    -
    - -
    -

    1.2 Identification

    -
    +
    +

    2.2 Identification

    +
    +

    +We identify the dynamics of the system. +

    %% Options for Linearized
     options = linearizeOptions;
    @@ -373,13 +442,16 @@ mdl = 'sim_nano_station_uniaxial';
     
    +

    +The inputs and outputs are defined below and corresponds to the name of simulink blocks. +

    %% Input/Output definition
    -io(1)  = linio([mdl, '/Dw'],    1, 'input');  % Ground Motion
    -io(2)  = linio([mdl, '/Fs'],    1, 'input');  % Force applied on the sample
    -io(3)  = linio([mdl, '/Fnl'],   1, 'input');  % Force applied by the NASS
    -io(4)  = linio([mdl, '/Fdty'],  1, 'input');  % Parasitic force Ty
    -io(5)  = linio([mdl, '/Fdrz'],  1, 'input');  % Parasitic force Rz
    +io(1)  = linio([mdl, '/Dw'],   1, 'input'); % Ground Motion
    +io(2)  = linio([mdl, '/Fs'],   1, 'input'); % Force applied on the sample
    +io(3)  = linio([mdl, '/Fnl'],  1, 'input'); % Force applied by the NASS
    +io(4)  = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty
    +io(5)  = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz
     
     io(6)  = linio([mdl, '/Dsm'],  1, 'output'); % Displacement of the sample
     io(7)  = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs
    @@ -389,6 +461,9 @@ io(
     
    %% Run the linearization
     G = linearize(mdl, io, options);
    @@ -404,62 +479,79 @@ G.OutputName = {'Vlm'}; ... % Absolute Velocity of NASS [m/s]
     
    -
    -
    -
    -

    1.3 Sensitivity to Disturbances

    -
    - -
    -

    uniaxial-sensitivity-disturbances.png +

    +Finally, we save the identified system dynamics for further analysis.

    -

    Figure 1: Sensitivity to disturbances (png, pdf)

    -
    - - - -
    -

    uniaxial-sensitivity-force-dist.png -

    -

    Figure 2: Sensitivity to disturbances (png, pdf)

    -
    -
    -
    - -
    -

    1.4 Plant

    -
    - -
    -

    uniaxial-plant.png -

    -

    Figure 3: Bode plot of the Plant (png, pdf)

    -
    -
    -
    - -
    -

    1.5 Save

    -
    save('./uniaxial/mat/plants.mat', 'G');
     
    + +
    +

    2.3 Sensitivity to Disturbances

    +
    +

    +We show several plots representing the sensitivity to disturbances: +

    +
      +
    • in figure 3 the transfer functions from ground motion \(D_w\) to the sample position \(D\) and the transfer function from direct force on the sample \(F_s\) to the sample position \(D\) are shown
    • +
    • in figure 4, it is the effect of parasitic forces of the positioning stages (\(F_{ty}\) and \(F_{rz}\)) on the position \(D\) of the sample that are shown
    • +
    + + +
    +

    uniaxial-sensitivity-disturbances.png +

    +

    Figure 3: Sensitivity to disturbances (png, pdf)

    -
    -

    2 Integral Force Feedback

    -
    -

    - + + +

    +

    uniaxial-sensitivity-force-dist.png

    +

    Figure 4: Sensitivity to disturbances (png, pdf)

    -
    -

    2.1 Control Design

    -
    +
    +
    + +
    +

    2.4 Plant

    +
    +

    +The transfer function from the force \(F\) applied by the nano-hexapod to the position of the sample \(D\) is shown in figure 5. +It corresponds to the plant to control. +

    + + +
    +

    uniaxial-plant.png +

    +

    Figure 5: Bode plot of the Plant (png, pdf)

    +
    +
    +
    +
    + +
    +

    3 Integral Force Feedback

    +
    +

    + +

    + +
    +

    uniaxial-model-nass-flexible-iff.png +

    +

    Figure 6: Uniaxial IFF Control Schematic

    +
    +
    +
    +

    3.1 Control Design

    +
    load('./uniaxial/mat/plants.mat', 'G');
     
    @@ -470,10 +562,10 @@ Let's look at the transfer function from actuator forces in the nano-hexapod to

    -
    +

    uniaxial_iff_plant.png

    -

    Figure 4: Transfer function from forces applied in the legs to force sensor (png, pdf)

    +

    Figure 7: Transfer function from forces applied in the legs to force sensor (png, pdf)

    @@ -485,17 +577,17 @@ The controller for each pair of actuator/sensor is:

    -
    +

    uniaxial_iff_open_loop.png

    -

    Figure 5: Loop Gain for the Integral Force Feedback (png, pdf)

    +

    Figure 8: Loop Gain for the Integral Force Feedback (png, pdf)

    -
    -

    2.2 Identification

    -
    +
    +

    3.2 Identification

    +

    Let's initialize the system prior to identification.

    @@ -569,43 +661,7 @@ G_iff.OutputName = {'Vlm'}; ... % Absolute Velocity of NASS [m/s]
    -
    -
    -
    -

    2.3 Sensitivity to Disturbance

    -
    - -
    -

    uniaxial_sensitivity_dist_iff.png -

    -

    Figure 6: Sensitivity to disturbance once the IFF controller is applied to the system (png, pdf)

    -
    - - -
    -

    uniaxial_sensitivity_dist_stages_iff.png -

    -

    Figure 7: Sensitivity to force disturbances in various stages when IFF is applied (png, pdf)

    -
    -
    -
    - -
    -

    2.4 Damped Plant

    -
    - -
    -

    uniaxial_plant_iff_damped.png -

    -

    Figure 8: Damped Plant after IFF is applied (png, pdf)

    -
    -
    -
    - -
    -

    2.5 Save

    -
    save('./uniaxial/mat/plants.mat', 'G_iff', '-append');
     
    @@ -613,9 +669,40 @@ G_iff.OutputName = {
    -
    -

    2.6 Conclusion

    -
    +
    +

    3.3 Sensitivity to Disturbance

    +
    + +
    +

    uniaxial_sensitivity_dist_iff.png +

    +

    Figure 9: Sensitivity to disturbance once the IFF controller is applied to the system (png, pdf)

    +
    + + +
    +

    uniaxial_sensitivity_dist_stages_iff.png +

    +

    Figure 10: Sensitivity to force disturbances in various stages when IFF is applied (png, pdf)

    +
    +
    +
    + +
    +

    3.4 Damped Plant

    +
    + +
    +

    uniaxial_plant_iff_damped.png +

    +

    Figure 11: Damped Plant after IFF is applied (png, pdf)

    +
    +
    +
    + +
    +

    3.5 Conclusion

    +

    Integral Force Feedback: @@ -626,19 +713,26 @@ Integral Force Feedback:

    -
    -

    3 Relative Motion Control

    -
    +
    +

    4 Relative Motion Control

    +

    - +

    In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input.

    + + +
    +

    uniaxial-model-nass-flexible-rmc.png +

    +

    Figure 12: Uniaxial RMC Control Schematic

    -
    -

    3.1 Control Design

    -
    +
    +
    +

    4.1 Control Design

    +
    load('./uniaxial/mat/plants.mat', 'G');
     
    @@ -649,10 +743,10 @@ Let's look at the transfer function from actuator forces in the nano-hexapod to

    -
    +

    uniaxial_rmc_plant.png

    -

    Figure 9: Transfer function from forces applied in the legs to leg displacement sensor (png, pdf)

    +

    Figure 13: Transfer function from forces applied in the legs to leg displacement sensor (png, pdf)

    @@ -665,17 +759,17 @@ A Low pass Filter is added to make the controller transfer function proper.

    -
    +

    uniaxial_rmc_open_loop.png

    -

    Figure 10: Loop Gain for the Integral Force Feedback (png, pdf)

    +

    Figure 14: Loop Gain for the Integral Force Feedback (png, pdf)

    -
    -

    3.2 Identification

    -
    +
    +

    4.2 Identification

    +

    Let's initialize the system prior to identification.

    @@ -749,44 +843,7 @@ G_rmc.OutputName = {'Vlm'}; ... % Absolute Velocity of NASS [m/s]
    -
    -
    - -
    -

    3.3 Sensitivity to Disturbance

    -
    - -
    -

    uniaxial_sensitivity_dist_rmc.png -

    -

    Figure 11: Sensitivity to disturbance once the RMC controller is applied to the system (png, pdf)

    -
    - - -
    -

    uniaxial_sensitivity_dist_stages_rmc.png -

    -

    Figure 12: Sensitivity to force disturbances in various stages when RMC is applied (png, pdf)

    -
    -
    -
    - -
    -

    3.4 Damped Plant

    -
    - -
    -

    uniaxial_plant_rmc_damped.png -

    -

    Figure 13: Damped Plant after RMC is applied (png, pdf)

    -
    -
    -
    - -
    -

    3.5 Save

    -
    save('./uniaxial/mat/plants.mat', 'G_rmc', '-append');
     
    @@ -794,9 +851,41 @@ G_rmc.OutputName = {
    -
    -

    3.6 Conclusion

    -
    + +
    +

    4.3 Sensitivity to Disturbance

    +
    + +
    +

    uniaxial_sensitivity_dist_rmc.png +

    +

    Figure 15: Sensitivity to disturbance once the RMC controller is applied to the system (png, pdf)

    +
    + + +
    +

    uniaxial_sensitivity_dist_stages_rmc.png +

    +

    Figure 16: Sensitivity to force disturbances in various stages when RMC is applied (png, pdf)

    +
    +
    +
    + +
    +

    4.4 Damped Plant

    +
    + +
    +

    uniaxial_plant_rmc_damped.png +

    +

    Figure 17: Damped Plant after RMC is applied (png, pdf)

    +
    +
    +
    + +
    +

    4.5 Conclusion

    +

    Relative Motion Control: @@ -807,29 +896,36 @@ Relative Motion Control:

    -
    -

    4 Direct Velocity Feedback

    -
    +
    +

    5 Direct Velocity Feedback

    +

    - +

    In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input.

    + + +
    +

    uniaxial-model-nass-flexible-dvf.png +

    +

    Figure 18: Uniaxial DVF Control Schematic

    -
    -

    4.1 Control Design

    -
    +
    +
    +

    5.1 Control Design

    +
    load('./uniaxial/mat/plants.mat', 'G');
     
    -
    +

    uniaxial_dvf_plant.png

    -

    Figure 14: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

    +

    Figure 19: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

    @@ -838,17 +934,17 @@ In the Relative Motion Control (RMC), a feedback is applied between the measured
    -
    +

    uniaxial_dvf_loop_gain.png

    -

    Figure 15: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

    +

    Figure 20: Transfer function from forces applied in the legs to leg velocity sensor (png, pdf)

    -
    -

    4.2 Identification

    -
    +
    +

    5.2 Identification

    +

    Let's initialize the system prior to identification.

    @@ -922,43 +1018,7 @@ G_dvf.OutputName = {'Vlm'}; ... % Absolute Velocity of NASS [m/s]
    -
    -
    -
    -

    4.3 Sensitivity to Disturbance

    -
    - -
    -

    uniaxial_sensitivity_dist_dvf.png -

    -

    Figure 16: Sensitivity to disturbance once the DVF controller is applied to the system (png, pdf)

    -
    - - -
    -

    uniaxial_sensitivity_dist_stages_dvf.png -

    -

    Figure 17: Sensitivity to force disturbances in various stages when DVF is applied (png, pdf)

    -
    -
    -
    - -
    -

    4.4 Damped Plant

    -
    - -
    -

    uniaxial_plant_dvf_damped.png -

    -

    Figure 18: Damped Plant after DVF is applied (png, pdf)

    -
    -
    -
    - -
    -

    4.5 Save

    -
    save('./uniaxial/mat/plants.mat', 'G_dvf', '-append');
     
    @@ -966,9 +1026,40 @@ G_dvf.OutputName = {
    -
    -

    4.6 Conclusion

    -
    +
    +

    5.3 Sensitivity to Disturbance

    +
    + +
    +

    uniaxial_sensitivity_dist_dvf.png +

    +

    Figure 21: Sensitivity to disturbance once the DVF controller is applied to the system (png, pdf)

    +
    + + +
    +

    uniaxial_sensitivity_dist_stages_dvf.png +

    +

    Figure 22: Sensitivity to force disturbances in various stages when DVF is applied (png, pdf)

    +
    +
    +
    + +
    +

    5.4 Damped Plant

    +
    + +
    +

    uniaxial_plant_dvf_damped.png +

    +

    Figure 23: Damped Plant after DVF is applied (png, pdf)

    +
    +
    +
    + +
    +

    5.5 Conclusion

    +

    Direct Velocity Feedback: @@ -978,16 +1069,16 @@ Direct Velocity Feedback:

    -
    -

    5 Comparison of Active Damping Techniques

    -
    +
    +

    6 Comparison of Active Damping Techniques

    +

    - +

    -
    -

    5.1 Load the plants

    -
    +
    +

    6.1 Load the plants

    +
    load('./uniaxial/mat/plants.mat', 'G', 'G_iff', 'G_rmc', 'G_dvf');
     
    @@ -995,44 +1086,126 @@ Direct Velocity Feedback:
    -
    -

    5.2 Sensitivity to Disturbance

    -
    +
    +

    6.2 Sensitivity to Disturbance

    +
    -
    -

    uniaxial_sensitivity_dist_comp.png +

    +

    uniaxial_sensitivity_ground_motion.png

    -

    Figure 19: Sensitivity to disturbance - Comparison (png, pdf)

    +

    Figure 24: Sensitivity to Ground Motion - Comparison (png, pdf)

    -
    -

    uniaxial_sensitivity_dist_stages_comp.png + +

    +

    uniaxial_sensitivity_direct_force.png

    -

    Figure 20: Sensitivity to force disturbances - Comparison (png, pdf)

    +

    Figure 25: Sensitivity to disturbance - Comparison (png, pdf)

    +
    + + +
    +

    uniaxial_sensitivity_fty.png +

    +

    Figure 26: Sensitivity to force disturbances - Comparison (png, pdf)

    +
    + + +
    +

    uniaxial_sensitivity_frz.png +

    +

    Figure 27: Sensitivity to force disturbances - Comparison (png, pdf)

    -
    -

    5.3 Damped Plant

    -
    +
    +

    6.3 Damped Plant

    +
    -
    +

    uniaxial_plant_damped_comp.png

    -

    Figure 21: Damped Plant - Comparison (png, pdf)

    +

    Figure 28: Damped Plant - Comparison (png, pdf)

    -
    -

    5.4 Conclusion

    + +
    +

    6.4 Conclusion

    +
    + + + +++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    Table 1: Comparison of proposed active damping techniques
     IFFRMCDVF
    Sensor TypeForce sensorRelative MotionInertial
    Guaranteed Stability++-
    Sensitivity (\(D_w\))-+-
    Sensitivity (\(F_s\))- (at low freq)++
    Sensitivity (\(F_{ty,rz}\))+-+
    + +
    +

    +The next step is to take into account the power spectral density of each disturbance. +

    + +
    +

    Author: Dehaeze Thomas

    -

    Created: 2019-10-24 jeu. 17:44

    +

    Created: 2019-10-25 ven. 12:32

    Validate

    diff --git a/uniaxial/index.org b/uniaxial/index.org index 874e239..ca57bb0 100644 --- a/uniaxial/index.org +++ b/uniaxial/index.org @@ -46,7 +46,586 @@ The idea is to use the same model as the full Simscape Model but to restrict the This is done in order to more easily study the system and evaluate control techniques. +* Simscape Model +A schematic of the uniaxial model used for simulations is represented in figure [[fig:uniaxial-model-nass-flexible]]. + +The perturbations $w$ are: +- $F_s$: direct forces applied to the sample such as inertia forces and cable forces +- $F_{rz}$: parasitic forces due to the rotation of the spindle +- $F_{ty}$: parasitic forces due to scans with the translation stage +- $D_w$: ground motion + +The quantity to $z$ to control is: +- $D$: the position of the sample with respect to the granite + +The measured quantities $v$ are: +- $D$: the position of the sample with respect to the granite + +We study the use of an additional sensor: +- $F_n$: a force sensor located in the nano-hexapod +- $v_n$: an absolute velocity sensor located on the top platform of the nano-hexapod +- $d_r$: a relative motion sensor located in the nano-hexapod + +The control signal $u$ is: +- $F$ the force applied by the nano-hexapod actuator + +#+begin_src latex :file uniaxial-model-nass-flexible.pdf :post pdf2svg(file=*this*, ext="png") :exports results + \begin{tikzpicture} + % ==================== + % Parameters + % ==================== + \def\massw{2.2} % Width of the masses + \def\massh{0.8} % Height of the masses + \def\spaceh{1.2} % Height of the springs/dampers + \def\dispw{0.4} % Width of the dashed line for the displacement + \def\disph{0.3} % Height of the arrow for the displacements + \def\bracs{0.05} % Brace spacing vertically + \def\brach{-12pt} % Brace shift horizontaly + \def\fsensh{0.2} % Height of the force sensor + \def\velsize{0.2} % Size of the velocity sensor + % ==================== + + + % ==================== + % Ground + % ==================== + \draw (-0.5*\massw, 0) -- (0.5*\massw, 0); + \draw[dashed] (0.5*\massw, 0) -- ++(1, 0); + \draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$}; + % ==================== + + + % ==================== + % Marble + \begin{scope}[shift={(0, 0)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Marble}; + + % Displacements + \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xm) -- ++(2.2*\dispw, 0) coordinate(dbot); + \end{scope} + % ==================== + + + % ==================== + % Ty + \begin{scope}[shift={(0, \spaceh+\massh)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$}; + \draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Ty}; + \end{scope} + % ==================== + + + % ==================== + % Rz + \begin{scope}[shift={(0, 2*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Rz}; + \end{scope} + % ==================== + + + % ==================== + % Hexapod + \begin{scope}[shift={(0, 3*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$}; + + % Displacements + % \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Hexapod}; + \end{scope} + % ==================== + + + % ==================== + % NASS + \begin{scope}[shift={(0, 4*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$}; + + % % Force Sensor + % \node[forcesensor={\massw}{\fsensh}] (fsensn) at (0, \spaceh-\fsensh){}; + % \node[right] at (fsensn.east) {$F_n$}; + + % % Velocity Sensor + % \node[inertialsensor={\velsize}] (veln) at (0.5*\massw, \spaceh+\massh) {}; + % \node[right] at (veln.north east) {$v_n$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{n}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{n}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F$}; + + % % Displacements + % \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{NASS}; + \end{scope} + % ==================== + + % ==================== + % sample + \begin{scope}[shift={(0, 5*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$}; + + % External Force + \draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$}; + + % Displacements + \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xs) -- ++(2.2*\dispw, 0) coordinate(dtop); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Sample}; + \end{scope} + % ==================== + + + \draw[<->] ($(dbot)+(-0.3,0)$) --node[midway, right]{$D$} ($(dtop)+(-0.3,0)$); + \end{tikzpicture} +#+end_src + +#+name: fig:uniaxial-model-nass-flexible +#+caption: Schematic of the uniaxial model used +#+RESULTS: +[[file:figs/uniaxial-model-nass-flexible.png]] + + +Few active damping techniques will be compared in order to decide which sensor is to be included in the system. +Schematics of the active damping techniques are displayed in figure [[fig:uniaxial-model-nass-flexible-active-damping]]. + +#+begin_src latex :file uniaxial-model-nass-flexible-active-damping.pdf :post pdf2svg(file=*this*, ext="png") :exports results + \begin{tikzpicture} + % ==================== + % Parameters + % ==================== + \def\massw{2.2} % Width of the masses + \def\massh{0.8} % Height of the masses + \def\spaceh{1.2} % Height of the springs/dampers + \def\dispw{0.4} % Width of the dashed line for the displacement + \def\disph{0.3} % Height of the arrow for the displacements + \def\bracs{0.05} % Brace spacing vertically + \def\brach{-12pt} % Brace shift horizontaly + \def\fsensh{0.2} % Height of the force sensor + \def\velsize{0.2} % Size of the velocity sensor + % ==================== + + + \begin{scope} + % ==================== + % Ground + % ==================== + \draw (-0.5*\massw, 0) -- (0.5*\massw, 0); + \draw[dashed] (0.5*\massw, 0) -- ++(1, 0); + \draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$}; + % ==================== + + + % ==================== + % Marble + \begin{scope}[shift={(0, 0)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Marble}; + \end{scope} + % ==================== + + + % ==================== + % Ty + \begin{scope}[shift={(0, \spaceh+\massh)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$}; + \draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Ty}; + \end{scope} + % ==================== + + + % ==================== + % Rz + \begin{scope}[shift={(0, 2*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Rz}; + \end{scope} + % ==================== + + + % ==================== + % Hexapod + \begin{scope}[shift={(0, 3*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$}; + + % Displacements + % \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Hexapod}; + \end{scope} + % ==================== + + + % ==================== + % NASS + \begin{scope}[shift={(0, 4*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$}; + + % Force Sensor + \node[forcesensor={\massw}{\fsensh}] (fsensn) at (0, \spaceh-\fsensh){}; + \node[right] (fn) at (fsensn.east) {$F_n$}; + + % % Velocity Sensor + % \node[inertialsensor={\velsize}] (veln) at (0.5*\massw, \spaceh+\massh) {}; + % \node[right] at (veln.north east) {$v_n$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh-\fsensh) node[midway, left=0.1]{$k_{n}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh-\fsensh) node[midway, left=0.2]{$c_{n}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh-\fsensh) node[midway, right=0.1](F){$F$}; + + % % Displacements + % \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{NASS}; + \end{scope} + % ==================== + + % ==================== + % sample + \begin{scope}[shift={(0, 5*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$}; + + % External Force + \draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Sample}; + \end{scope} + % ==================== + + % ==================== + % IFF Control + \node[block={2em}{2em}, right=0.3 of fn] (iff) {$K_{\text{IFF}}$}; + \draw[->] (fn.east) -- (iff.west); + \draw[->] (iff.south) |- (F.east); + % ==================== + \end{scope} + + + + \begin{scope}[shift={(5,0)}] + % ==================== + % Ground + % ==================== + \draw (-0.5*\massw, 0) -- (0.5*\massw, 0); + \draw[dashed] (0.5*\massw, 0) -- ++(1, 0); + \draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$}; + % ==================== + + + % ==================== + % Marble + \begin{scope}[shift={(0, 0)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$}; + \end{scope} + % ==================== + + + % ==================== + % Ty + \begin{scope}[shift={(0, \spaceh+\massh)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$}; + \draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$}; + \end{scope} + % ==================== + + + % ==================== + % Rz + \begin{scope}[shift={(0, 2*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$}; + \end{scope} + % ==================== + + + % ==================== + % Hexapod + \begin{scope}[shift={(0, 3*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$}; + + % Displacements + % \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot); + \end{scope} + % ==================== + + + % ==================== + % NASS + \begin{scope}[shift={(0, 4*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$}; + + % % Force Sensor + % \node[forcesensor={\massw}{\fsensh}] (fsensn) at (0, \spaceh-\fsensh){}; + % \node[right] (fn) at (fsensn.east) {$F_n$}; + + % Velocity Sensor + \node[inertialsensor={\velsize}] (veln) at (0.5*\massw, \spaceh+\massh) {}; + \node[above right] at (veln.east) {$v_n$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{n}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{n}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F$}; + + % % Displacements + % \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop); + \end{scope} + % ==================== + + % ==================== + % sample + \begin{scope}[shift={(0, 5*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$}; + + % External Force + \draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$}; + \end{scope} + % ==================== + + % ==================== + % DVF Control + \node[block={2em}{2em}, below right=0.4 and 0.4 of veln] (ppf) {$K_{\text{DVF}}$}; + \draw[->] (veln.east) -| (ppf.north); + \draw[->] (ppf.south) |- (F.east); + % ==================== + \end{scope} + + + + \begin{scope}[shift={(10,0)}] + % ==================== + % Ground + % ==================== + \draw (-0.5*\massw, 0) -- (0.5*\massw, 0); + \draw[dashed] (0.5*\massw, 0) -- ++(1, 0); + \draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$}; + % ==================== + + + % ==================== + % Marble + \begin{scope}[shift={(0, 0)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$}; + \end{scope} + % ==================== + + + % ==================== + % Ty + \begin{scope}[shift={(0, \spaceh+\massh)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$}; + \draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$}; + \end{scope} + % ==================== + + + % ==================== + % Rz + \begin{scope}[shift={(0, 2*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$}; + \end{scope} + % ==================== + + + % ==================== + % Hexapod + \begin{scope}[shift={(0, 3*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$}; + + % Displacements + \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot); + \end{scope} + % ==================== + + + % ==================== + % NASS + \begin{scope}[shift={(0, 4*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{n}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{n}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F$}; + + % Displacements + \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop); + \end{scope} + % ==================== + + % ==================== + % sample + \begin{scope}[shift={(0, 5*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$}; + + % External Force + \draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$}; + \end{scope} + % ==================== + + % ==================== + % RMC Control + \draw[<->, dashed] ($(drbot)+(-0.1, 0)$) -- ($(drtop)+(-0.1, 0)$) node[right](dr){$d_r$}; + \node[block={2em}{2em}, below right=0.2 and 0.2 of dr] (rmc) {$K_{\text{RMC}}$}; + \draw[->] (dr.east) -| (rmc.north); + \draw[->] (rmc.south) |- (F.east); + % ==================== + \end{scope} + \end{tikzpicture} +#+end_src + +#+name: fig:uniaxial-model-nass-flexible-active-damping +#+caption: Comparison of used active damping techniques +#+RESULTS: +[[file:figs/uniaxial-model-nass-flexible-active-damping.png]] + * Undamped System +** Introduction :ignore: +Let's start by study the undamped system. + ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -68,7 +647,7 @@ This is done in order to more easily study the system and evaluate control techn We initialize all the stages with the default parameters. The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg. -#+begin_src matlab +#+begin_src matlab :exports none initializeGround(); initializeGranite(); initializeTy(); @@ -81,8 +660,8 @@ The nano-hexapod is a piezoelectric hexapod and the sample has a mass of 50kg. initializeSample(struct('mass', 50)); #+end_src -All the controllers are set to 0. -#+begin_src matlab +All the controllers are set to 0 (Open Loop). +#+begin_src matlab :exports none K = tf(0); save('./mat/controllers.mat', 'K', '-append'); K_iff = tf(0); @@ -94,6 +673,7 @@ All the controllers are set to 0. #+end_src ** Identification +We identify the dynamics of the system. #+begin_src matlab %% Options for Linearized options = linearizeOptions; @@ -103,13 +683,14 @@ All the controllers are set to 0. mdl = 'sim_nano_station_uniaxial'; #+end_src +The inputs and outputs are defined below and corresponds to the name of simulink blocks. #+begin_src matlab %% Input/Output definition - io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion - io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample - io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS - io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty - io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz + io(1) = linio([mdl, '/Dw'], 1, 'input'); % Ground Motion + io(2) = linio([mdl, '/Fs'], 1, 'input'); % Force applied on the sample + io(3) = linio([mdl, '/Fnl'], 1, 'input'); % Force applied by the NASS + io(4) = linio([mdl, '/Fdty'], 1, 'input'); % Parasitic force Ty + io(5) = linio([mdl, '/Fdrz'], 1, 'input'); % Parasitic force Rz io(6) = linio([mdl, '/Dsm'], 1, 'output'); % Displacement of the sample io(7) = linio([mdl, '/Fnlm'], 1, 'output'); % Force sensor in NASS's legs @@ -118,6 +699,7 @@ All the controllers are set to 0. io(10) = linio([mdl, '/Vlm'], 1, 'output'); % Measured absolute velocity of the top NASS platform #+end_src +Finally, we use the =linearize= Matlab function to extract a state space model from the simscape model. #+begin_src matlab %% Run the linearization G = linearize(mdl, io, options); @@ -133,7 +715,16 @@ All the controllers are set to 0. 'Vlm'}; ... % Absolute Velocity of NASS [m/s] #+end_src +Finally, we save the identified system dynamics for further analysis. +#+begin_src matlab + save('./uniaxial/mat/plants.mat', 'G'); +#+end_src + ** Sensitivity to Disturbances +We show several plots representing the sensitivity to disturbances: +- in figure [[fig:uniaxial-sensitivity-disturbances]] the transfer functions from ground motion $D_w$ to the sample position $D$ and the transfer function from direct force on the sample $F_s$ to the sample position $D$ are shown +- in figure [[fig:uniaxial-sensitivity-force-dist]], it is the effect of parasitic forces of the positioning stages ($F_{ty}$ and $F_{rz}$) on the position $D$ of the sample that are shown + #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); @@ -195,6 +786,9 @@ All the controllers are set to 0. [[file:figs/uniaxial-sensitivity-force-dist.png]] ** Plant +The transfer function from the force $F$ applied by the nano-hexapod to the position of the sample $D$ is shown in figure [[fig:uniaxial-plant]]. +It corresponds to the plant to control. + #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); @@ -224,14 +818,169 @@ All the controllers are set to 0. #+CAPTION: Bode plot of the Plant ([[./figs/uniaxial-plant.png][png]], [[./figs/uniaxial-plant.pdf][pdf]]) [[file:figs/uniaxial-plant.png]] -** Save -#+begin_src matlab - save('./uniaxial/mat/plants.mat', 'G'); -#+end_src - * Integral Force Feedback <> ** Introduction :ignore: +#+begin_src latex :file uniaxial-model-nass-flexible-iff.pdf :post pdf2svg(file=*this*, ext="png") :exports results + \begin{tikzpicture} + % ==================== + % Parameters + % ==================== + \def\massw{2.2} % Width of the masses + \def\massh{0.8} % Height of the masses + \def\spaceh{1.2} % Height of the springs/dampers + \def\dispw{0.4} % Width of the dashed line for the displacement + \def\disph{0.3} % Height of the arrow for the displacements + \def\bracs{0.05} % Brace spacing vertically + \def\brach{-12pt} % Brace shift horizontaly + \def\fsensh{0.2} % Height of the force sensor + \def\velsize{0.2} % Size of the velocity sensor + % ==================== + + + % ==================== + % Ground + % ==================== + \draw (-0.5*\massw, 0) -- (0.5*\massw, 0); + \draw[dashed] (0.5*\massw, 0) -- ++(1, 0); + \draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$}; + % ==================== + + + % ==================== + % Marble + \begin{scope}[shift={(0, 0)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Marble}; + \end{scope} + % ==================== + + + % ==================== + % Ty + \begin{scope}[shift={(0, \spaceh+\massh)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$}; + \draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Ty}; + \end{scope} + % ==================== + + + % ==================== + % Rz + \begin{scope}[shift={(0, 2*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Rz}; + \end{scope} + % ==================== + + + % ==================== + % Hexapod + \begin{scope}[shift={(0, 3*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$}; + + % Displacements + % \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Hexapod}; + \end{scope} + % ==================== + + + % ==================== + % NASS + \begin{scope}[shift={(0, 4*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$}; + + % Force Sensor + \node[forcesensor={\massw}{\fsensh}] (fsensn) at (0, \spaceh-\fsensh){}; + \node[right] (fn) at (fsensn.east) {$F_n$}; + + % % Velocity Sensor + % \node[inertialsensor={\velsize}] (veln) at (0.5*\massw, \spaceh+\massh) {}; + % \node[right] at (veln.north east) {$v_n$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh-\fsensh) node[midway, left=0.1]{$k_{n}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh-\fsensh) node[midway, left=0.2]{$c_{n}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh-\fsensh) node[midway, right=0.1](F){$F$}; + + % % Displacements + % \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{NASS}; + \end{scope} + % ==================== + + % ==================== + % sample + \begin{scope}[shift={(0, 5*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$}; + + % External Force + \draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Sample}; + \end{scope} + % ==================== + + % ==================== + % IFF Control + \node[block={2em}{2em}, right=0.3 of fn] (iff) {$K_{\text{IFF}}$}; + \draw[->] (fn.east) -- (iff.west); + \draw[->] (iff.south) |- (F.east); + % ==================== + \end{tikzpicture} +#+end_src + +#+name: fig:iff_schematic +#+caption: Uniaxial IFF Control Schematic +#+RESULTS: +[[file:figs/uniaxial-model-nass-flexible-iff.png]] + ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -385,6 +1134,10 @@ All the controllers are set to 0. 'Vlm'}; ... % Absolute Velocity of NASS [m/s] #+end_src +#+begin_src matlab + save('./uniaxial/mat/plants.mat', 'G_iff', '-append'); +#+end_src + ** Sensitivity to Disturbance #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); @@ -446,7 +1199,7 @@ All the controllers are set to 0. #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_iff.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_iff.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") <> #+end_src @@ -491,11 +1244,6 @@ All the controllers are set to 0. #+CAPTION: Damped Plant after IFF is applied ([[./figs/uniaxial_plant_iff_damped.png][png]], [[./figs/uniaxial_plant_iff_damped.pdf][pdf]]) [[file:figs/uniaxial_plant_iff_damped.png]] -** Save -#+begin_src matlab - save('./uniaxial/mat/plants.mat', 'G_iff', '-append'); -#+end_src - ** Conclusion #+begin_important Integral Force Feedback: @@ -506,6 +1254,159 @@ Integral Force Feedback: ** Introduction :ignore: In the Relative Motion Control (RMC), a derivative feedback is applied between the measured actuator displacement to the actuator force input. +#+begin_src latex :file uniaxial-model-nass-flexible-rmc.pdf :post pdf2svg(file=*this*, ext="png") :exports results + \begin{tikzpicture} + % ==================== + % Parameters + % ==================== + \def\massw{2.2} % Width of the masses + \def\massh{0.8} % Height of the masses + \def\spaceh{1.2} % Height of the springs/dampers + \def\dispw{0.4} % Width of the dashed line for the displacement + \def\disph{0.3} % Height of the arrow for the displacements + \def\bracs{0.05} % Brace spacing vertically + \def\brach{-12pt} % Brace shift horizontaly + \def\fsensh{0.2} % Height of the force sensor + \def\velsize{0.2} % Size of the velocity sensor + % ==================== + + + % ==================== + % Ground + % ==================== + \draw (-0.5*\massw, 0) -- (0.5*\massw, 0); + \draw[dashed] (0.5*\massw, 0) -- ++(1, 0); + \draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$}; + % ==================== + + + % ==================== + % Marble + \begin{scope}[shift={(0, 0)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Marble}; + \end{scope} + % ==================== + + + % ==================== + % Ty + \begin{scope}[shift={(0, \spaceh+\massh)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$}; + \draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Ty}; + \end{scope} + % ==================== + + + % ==================== + % Rz + \begin{scope}[shift={(0, 2*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Rz}; + \end{scope} + % ==================== + + + % ==================== + % Hexapod + \begin{scope}[shift={(0, 3*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$}; + + % Displacements + \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Hexapod}; + \end{scope} + % ==================== + + + % ==================== + % NASS + \begin{scope}[shift={(0, 4*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{n}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{n}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F$}; + + % Displacements + \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{NASS}; + \end{scope} + % ==================== + + % ==================== + % sample + \begin{scope}[shift={(0, 5*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$}; + + % External Force + \draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Sample}; + \end{scope} + % ==================== + + % ==================== + % RMC Control + \draw[<->, dashed] ($(drbot)+(-0.1, 0)$) -- ($(drtop)+(-0.1, 0)$) node[right](dr){$d_r$}; + \node[block={2em}{2em}, below right=0.2 and 0.2 of dr] (rmc) {$K_{\text{RMC}}$}; + \draw[->] (dr.east) -| (rmc.north); + \draw[->] (rmc.south) |- (F.east); + % ==================== + \end{tikzpicture} +#+end_src + +#+name: fig:rmc_schematic +#+caption: Uniaxial RMC Control Schematic +#+RESULTS: +[[file:figs/uniaxial-model-nass-flexible-rmc.png]] + ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -660,6 +1561,10 @@ And initialize the controllers. 'Vlm'}; ... % Absolute Velocity of NASS [m/s] #+end_src +#+begin_src matlab + save('./uniaxial/mat/plants.mat', 'G_rmc', '-append'); +#+end_src + ** Sensitivity to Disturbance #+begin_src matlab :exports none @@ -722,7 +1627,7 @@ And initialize the controllers. #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_rmc.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_rmc.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") <> #+end_src @@ -767,11 +1672,6 @@ And initialize the controllers. #+CAPTION: Damped Plant after RMC is applied ([[./figs/uniaxial_plant_rmc_damped.png][png]], [[./figs/uniaxial_plant_rmc_damped.pdf][pdf]]) [[file:figs/uniaxial_plant_rmc_damped.png]] -** Save -#+begin_src matlab - save('./uniaxial/mat/plants.mat', 'G_rmc', '-append'); -#+end_src - ** Conclusion #+begin_important Relative Motion Control: @@ -782,6 +1682,166 @@ Relative Motion Control: ** Introduction :ignore: In the Relative Motion Control (RMC), a feedback is applied between the measured velocity of the platform to the actuator force input. +#+begin_src latex :file uniaxial-model-nass-flexible-dvf.pdf :post pdf2svg(file=*this*, ext="png") :exports results + \begin{tikzpicture} + % ==================== + % Parameters + % ==================== + \def\massw{2.2} % Width of the masses + \def\massh{0.8} % Height of the masses + \def\spaceh{1.2} % Height of the springs/dampers + \def\dispw{0.4} % Width of the dashed line for the displacement + \def\disph{0.3} % Height of the arrow for the displacements + \def\bracs{0.05} % Brace spacing vertically + \def\brach{-12pt} % Brace shift horizontaly + \def\fsensh{0.2} % Height of the force sensor + \def\velsize{0.2} % Size of the velocity sensor + % ==================== + + + % ==================== + % Ground + % ==================== + \draw (-0.5*\massw, 0) -- (0.5*\massw, 0); + \draw[dashed] (0.5*\massw, 0) -- ++(1, 0); + \draw[->, dasehd] (0.5*\massw+0.8, 0) -- ++(0, 0.5) node[right]{$D_{w}$}; + % ==================== + + + % ==================== + % Marble + \begin{scope}[shift={(0, 0)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{m}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{m}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{m}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Marble}; + \end{scope} + % ==================== + + + % ==================== + % Ty + \begin{scope}[shift={(0, \spaceh+\massh)}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{t}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{t}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{t}$}; + \draw[actuator={0.45}{0.2}] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](ft){$F_{ty}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Ty}; + \end{scope} + % ==================== + + + % ==================== + % Rz + \begin{scope}[shift={(0, 2*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{z}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{z}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{z}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F_{rz}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Rz}; + \end{scope} + % ==================== + + + % ==================== + % Hexapod + \begin{scope}[shift={(0, 3*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{h}$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{h}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{h}$}; + + % Displacements + % \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(3*\dispw, 0) coordinate(drbot); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Hexapod}; + \end{scope} + % ==================== + + + % ==================== + % NASS + \begin{scope}[shift={(0, 4*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{n}$}; + + % % Force Sensor + % \node[forcesensor={\massw}{\fsensh}] (fsensn) at (0, \spaceh-\fsensh){}; + % \node[right] (fn) at (fsensn.east) {$F_n$}; + + % Velocity Sensor + \node[inertialsensor={\velsize}] (veln) at (0.5*\massw, \spaceh+\massh) {}; + \node[above right] at (veln.east) {$v_n$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{n}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{n}$}; + \draw[actuator] ( 0.4*\massw, 0) -- ( 0.4*\massw, \spaceh) node[midway, right=0.1](F){$F$}; + + % % Displacements + % \draw[dashed] (0.5*\massw, \spaceh+\massh) -- ++(2*\dispw, 0) coordinate(xn) -- ++(\dispw, 0) coordinate(drtop); + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{NASS}; + \end{scope} + % ==================== + + % ==================== + % sample + \begin{scope}[shift={(0, 5*(\spaceh+\massh))}] + % Mass + \draw[fill=white] (-0.5*\massw, \spaceh) rectangle (0.5*\massw, \spaceh+\massh) node[pos=0.5]{$m_{s}$}; + + % External Force + \draw[->] (0, \spaceh+\massh) node[]{$\bullet$} -- ++(0, 0.5*\massh) node[right]{$F_s$}; + + % Spring, Damper, and Actuator + \draw[spring] (-0.4*\massw, 0) -- (-0.4*\massw, \spaceh) node[midway, left=0.1]{$k_{s}$}; + \draw[damper] (0, 0) -- ( 0, \spaceh) node[midway, left=0.2]{$c_{s}$}; + + % Legend + \draw[decorate, decoration={brace, amplitude=8pt}, xshift=\brach] % + (-0.5*\massw, \bracs) -- (-0.5*\massw, \spaceh+\massh-\bracs) node[midway,rotate=90,anchor=south,yshift=10pt]{Sample}; + \end{scope} + % ==================== + + % ==================== + % DVF Control + \node[block={2em}{2em}, below right=0.4 and 0.4 of veln] (ppf) {$K_{\text{DVF}}$}; + \draw[->] (veln.east) -| (ppf.north); + \draw[->] (ppf.south) |- (F.east); + % ==================== + \end{tikzpicture} +#+end_src + +#+name: fig:dvf_schematic +#+caption: Uniaxial DVF Control Schematic +#+RESULTS: +[[file:figs/uniaxial-model-nass-flexible-dvf.png]] + ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -932,6 +1992,10 @@ And initialize the controllers. 'Vlm'}; ... % Absolute Velocity of NASS [m/s] #+end_src +#+begin_src matlab + save('./uniaxial/mat/plants.mat', 'G_dvf', '-append'); +#+end_src + ** Sensitivity to Disturbance #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); @@ -993,7 +2057,7 @@ And initialize the controllers. #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_dvf.pdf" :var figsize="full-normal" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") <> #+end_src @@ -1038,11 +2102,6 @@ And initialize the controllers. #+CAPTION: Damped Plant after DVF is applied ([[./figs/uniaxial_plant_dvf_damped.png][png]], [[./figs/uniaxial_plant_dvf_damped.pdf][pdf]]) [[file:figs/uniaxial_plant_dvf_damped.png]] -** Save -#+begin_src matlab - save('./uniaxial/mat/plants.mat', 'G_dvf', '-append'); -#+end_src - ** Conclusion #+begin_important Direct Velocity Feedback: @@ -1077,76 +2136,104 @@ Direct Velocity Feedback: figure; - subplot(2, 1, 1); title('$D_w$ to $D$'); hold on; - plot(freqs, abs(squeeze(freqresp(G('D', 'Dw'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL'); - plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Dw'), freqs, 'Hz'))), 'k:', 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G ('D', 'Dw'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL'); + plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Dw'), freqs, 'Hz'))), 'k:', 'DisplayName', 'IFF'); plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Dw'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Dw'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/m]'); xlabel('Frequency [Hz]'); legend('location', 'northeast'); - - subplot(2, 1, 2); - title('$F_s$ to $D$'); - hold on; - plot(freqs, abs(squeeze(freqresp(G('D', 'Fs'), freqs, 'Hz'))), 'k-'); - plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Fs'), freqs, 'Hz'))), 'k:'); - plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Fs'), freqs, 'Hz'))), 'k--'); - plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Fs'), freqs, 'Hz'))), 'k-.'); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_comp.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_ground_motion.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") <> #+end_src -#+NAME: fig:uniaxial_sensitivity_dist_comp -#+CAPTION: Sensitivity to disturbance - Comparison ([[./figs/uniaxial_sensitivity_dist_comp.png][png]], [[./figs/uniaxial_sensitivity_dist_comp.pdf][pdf]]) -[[file:figs/uniaxial_sensitivity_dist_comp.png]] +#+NAME: fig:uniaxial_sensitivity_ground_motion +#+CAPTION: Sensitivity to Ground Motion - Comparison ([[./figs/uniaxial_sensitivity_ground_motion.png][png]], [[./figs/uniaxial_sensitivity_ground_motion.pdf][pdf]]) +[[file:figs/uniaxial_sensitivity_ground_motion.png]] + + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + title('$F_s$ to $D$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G ('D', 'Fs'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'OL'); + plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Fs'), freqs, 'Hz'))), 'k:' , 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Fs'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); + plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Fs'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_direct_force.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:uniaxial_sensitivity_direct_force +#+CAPTION: Sensitivity to disturbance - Comparison ([[./figs/uniaxial_sensitivity_direct_force.png][png]], [[./figs/uniaxial_sensitivity_direct_force.pdf][pdf]]) +[[file:figs/uniaxial_sensitivity_direct_force.png]] #+begin_src matlab :exports none freqs = logspace(0, 3, 1000); figure; - subplot(2, 1, 1); title('$F_{ty}$ to $D$'); hold on; - plot(freqs, abs(squeeze(freqresp(G('D', 'Fty'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL'); - plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Fty'), freqs, 'Hz'))), 'k:', 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G ('D', 'Fty'), freqs, 'Hz'))), 'k-', 'DisplayName', 'OL'); + plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Fty'), freqs, 'Hz'))), 'k:', 'DisplayName', 'IFF'); plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Fty'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Fty'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); legend('location', 'northeast'); - - subplot(2, 1, 2); - title('$F_{rz}$ to $D$'); - hold on; - plot(freqs, abs(squeeze(freqresp(G('D', 'Frz'), freqs, 'Hz'))), 'k-'); - plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Frz'), freqs, 'Hz'))), 'k'); - plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Frz'), freqs, 'Hz'))), 'k--'); - plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Frz'), freqs, 'Hz'))), 'k-.'); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); #+end_src #+HEADER: :tangle no :exports results :results none :noweb yes -#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_dist_stages_comp.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_fty.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") <> #+end_src -#+NAME: fig:uniaxial_sensitivity_dist_stages_comp -#+CAPTION: Sensitivity to force disturbances - Comparison ([[./figs/uniaxial_sensitivity_dist_stages_comp.png][png]], [[./figs/uniaxial_sensitivity_dist_stages_comp.pdf][pdf]]) -[[file:figs/uniaxial_sensitivity_dist_stages_comp.png]] +#+NAME: fig:uniaxial_sensitivity_fty +#+CAPTION: Sensitivity to force disturbances - Comparison ([[./figs/uniaxial_sensitivity_fty.png][png]], [[./figs/uniaxial_sensitivity_fty.pdf][pdf]]) +[[file:figs/uniaxial_sensitivity_fty.png]] + +#+begin_src matlab :exports none + freqs = logspace(0, 3, 1000); + + figure; + + title('$F_{rz}$ to $D$'); + hold on; + plot(freqs, abs(squeeze(freqresp(G ('D', 'Frz'), freqs, 'Hz'))), 'k-' , 'DisplayName', 'OL'); + plot(freqs, abs(squeeze(freqresp(G_iff('D', 'Frz'), freqs, 'Hz'))), 'k' , 'DisplayName', 'IFF'); + plot(freqs, abs(squeeze(freqresp(G_rmc('D', 'Frz'), freqs, 'Hz'))), 'k--', 'DisplayName', 'RMC'); + plot(freqs, abs(squeeze(freqresp(G_dvf('D', 'Frz'), freqs, 'Hz'))), 'k-.', 'DisplayName', 'DVF'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); xlabel('Frequency [Hz]'); + legend('location', 'northeast'); +#+end_src + +#+HEADER: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/uniaxial_sensitivity_frz.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:uniaxial_sensitivity_frz +#+CAPTION: Sensitivity to force disturbances - Comparison ([[./figs/uniaxial_sensitivity_frz.png][png]], [[./figs/uniaxial_sensitivity_frz.pdf][pdf]]) +[[file:figs/uniaxial_sensitivity_frz.png]] ** Damped Plant #+begin_src matlab :exports none @@ -1188,4 +2275,19 @@ Direct Velocity Feedback: #+NAME: fig:uniaxial_plant_damped_comp #+CAPTION: Damped Plant - Comparison ([[./figs/uniaxial_plant_damped_comp.png][png]], [[./figs/uniaxial_plant_damped_comp.pdf][pdf]]) [[file:figs/uniaxial_plant_damped_comp.png]] + ** Conclusion + +#+name: tab:active_damping_comparison +#+caption: Comparison of proposed active damping techniques +| | IFF | RMC | DVF | +|---------------------------+-----------------+-----------------+----------| +| Sensor Type | Force sensor | Relative Motion | Inertial | +| Guaranteed Stability | + | + | - | +| Sensitivity ($D_w$) | - | + | - | +| Sensitivity ($F_s$) | - (at low freq) | + | + | +| Sensitivity ($F_{ty,rz}$) | + | - | + | + +#+begin_important +The next step is to take into account the power spectral density of each disturbance. +#+end_important diff --git a/uniaxial/mat/plants.mat b/uniaxial/mat/plants.mat index 1e3496555c9c721c5629af606eb46268313a7049..e8d44106ac2d3a137976e43711c3bd58c086408d 100644 GIT binary patch delta 9496 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