From 23125d2abc4c566a757b5b51b5610a171d2fa66f Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 30 Nov 2021 11:45:05 +0100 Subject: [PATCH] Add Relative Active Damping study --- dcm-simscape.html | 307 ++-- dcm-simscape.org | 180 ++- dcm-simscape.pdf | Bin 863908 -> 1104973 bytes ...comp_damp_undamped_plant_rad_bode_plot.pdf | Bin 0 -> 138861 bytes ...comp_damp_undamped_plant_rad_bode_plot.png | Bin 0 -> 136031 bytes figs/relative_damping_root_locus.pdf | Bin 0 -> 30376 bytes figs/relative_damping_root_locus.png | Bin 0 -> 40735 bytes figs/strain_gauge_plant_bode_plot.pdf | 1386 +++++++++++++++++ figs/strain_gauge_plant_bode_plot.png | Bin 0 -> 100919 bytes matlab/simscape_dcm.slx | Bin 54625 -> 56339 bytes 10 files changed, 1757 insertions(+), 116 deletions(-) create mode 100644 figs/comp_damp_undamped_plant_rad_bode_plot.pdf create mode 100644 figs/comp_damp_undamped_plant_rad_bode_plot.png create mode 100644 figs/relative_damping_root_locus.pdf create mode 100644 figs/relative_damping_root_locus.png create mode 100644 figs/strain_gauge_plant_bode_plot.pdf create mode 100644 figs/strain_gauge_plant_bode_plot.png diff --git a/dcm-simscape.html b/dcm-simscape.html index 834116d..b3c5612 100644 --- a/dcm-simscape.html +++ b/dcm-simscape.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + DCM - Dynamical Multi-Body Model @@ -39,39 +39,41 @@

Table of Contents

@@ -87,22 +89,22 @@ In this document, a Simscape (.e.g. multi-body) model of the ESRF Double Crystal It is structured as follow:

-
-

1. System Kinematics

+
+

1. System Kinematics

- +

-
-

1.1. Bragg Angle

+
+

1.1. Bragg Angle

%% Tested bragg angles
@@ -118,7 +120,7 @@ dz = d_off./(2*
 
 
-
+

jack_motion_bragg_angle.png

Figure 1: Jack motion as a function of Bragg angle

@@ -136,34 +138,34 @@ dz = d_off./(2*
-
-

1.2. Kinematics (111 Crystal)

+
+

1.2. Kinematics (111 Crystal)

The reference frame is taken at the center of the 111 second crystal.

-
-

1.2.1. Interferometers - 111 Crystal

+
+

1.2.1. Interferometers - 111 Crystal

Three interferometers are pointed to the bottom surface of the 111 crystal.

-The position of the measurement points are shown in Figure 2 as well as the origin where the motion of the crystal is computed. +The position of the measurement points are shown in Figure 2 as well as the origin where the motion of the crystal is computed.

-
+

sensor_111_crystal_points.png

Figure 2: Bottom view of the second crystal 111. Position of the measurement points.

-The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure 3): +The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure 3):

\begin{equation} \begin{bmatrix} @@ -177,7 +179,7 @@ d_z \\ r_y \\ r_x \end{equation} -
+

schematic_sensor_jacobian_inverse_kinematics.png

Figure 3: Inverse Kinematics - Interferometers

@@ -185,7 +187,7 @@ d_z \\ r_y \\ r_x

-From the Figure 2, the inverse kinematics can be solved as follow (for small motion): +From the Figure 2, the inverse kinematics can be solved as follow (for small motion):

\begin{equation} \bm{J}_{s,111} @@ -205,7 +207,7 @@ J_s_111 = [1, 0.07, -0.015
- +
@@ -237,7 +239,7 @@ J_s_111 = [1, 0.07, -0.015
Table 1: Sensor Jacobian \(\bm{J}_{s,111}\)

-The forward kinematics is solved by inverting the Jacobian matrix (see Figure 4). +The forward kinematics is solved by inverting the Jacobian matrix (see Figure 4).

\begin{equation} \begin{bmatrix} @@ -251,13 +253,13 @@ x_1 \\ x_2 \\ x_3 \end{equation} -
+

schematic_sensor_jacobian_forward_kinematics.png

Figure 4: Forward Kinematics - Interferometers

- +
@@ -290,15 +292,15 @@ x_1 \\ x_2 \\ x_3 -
-

1.2.2. Piezo - 111 Crystal

+
+

1.2.2. Piezo - 111 Crystal

-The location of the actuators with respect with the center of the 111 second crystal are shown in Figure 5. +The location of the actuators with respect with the center of the 111 second crystal are shown in Figure 5.

-
+

actuator_jacobian_111_points.png

Figure 5: Location of actuators with respect to the center of the 111 second crystal (bottom view)

@@ -319,14 +321,14 @@ d_z \\ r_y \\ r_x \end{equation} -
+

schematic_actuator_jacobian_inverse_kinematics.png

Figure 6: Inverse Kinematics - Actuators

-Based on the geometry in Figure 5, we obtain: +Based on the geometry in Figure 5, we obtain:

\begin{equation} \bm{J}_{a,111} @@ -346,7 +348,7 @@ J_a_111 = [1, 0.14, -0.1525
-
Table 2: Inverse of the sensor Jacobian \(\bm{J}_{s,111}^{-1}\)
+
@@ -392,13 +394,13 @@ d_{u_r} \\ d_{u_h} \\ d_{d} \end{equation} -
+

schematic_actuator_jacobian_forward_kinematics.png

Figure 7: Forward Kinematics - Actuators for 111 crystal

-
Table 3: Actuator Jacobian \(\bm{J}_{a,111}\)
+
@@ -432,8 +434,8 @@ d_{u_r} \\ d_{u_h} \\ d_{d} -
-

1.3. Save Kinematics

+
+

1.3. Save Kinematics

save('mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
@@ -443,15 +445,15 @@ d_{u_r} \\ d_{u_h} \\ d_{d}
 
-
-

2. Open Loop System Identification

+
+

2. Open Loop System Identification

- +

-
-

2.1. Identification

+
+

2.1. Identification

Let’s considered the system \(\bm{G}(s)\) with: @@ -462,11 +464,11 @@ Let’s considered the system \(\bm{G}(s)\) with:

-It is schematically shown in Figure 8. +It is schematically shown in Figure 8.

-
+

schematic_system_inputs_outputs.png

Figure 8: Dynamical system with inputs and outputs

@@ -508,8 +510,8 @@ State-space model with 3 outputs, 3 inputs, and 24 states.
-
-

2.2. Plant in the frame of the fastjacks

+
+

2.2. Plant in the frame of the fastjacks

load('mat/dcm_kinematics.mat');
@@ -517,11 +519,11 @@ State-space model with 3 outputs, 3 inputs, and 24 states.
 

-Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure 9). +Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure 9).

-
+

schematic_jacobian_frame_fastjack.png

Figure 9: Use of Jacobian matrices to obtain the system in the frame of the fastjacks

@@ -542,7 +544,7 @@ The DC gain of the new system shows that the system is well decoupled at low fre
-
Table 4: Inverse of the actuator Jacobian \(\bm{J}_{a,111}^{-1}\)
+
@@ -574,17 +576,17 @@ The DC gain of the new system shows that the system is well decoupled at low fre
Table 5: DC gain of the plant in the frame of the fast jacks \(\bm{G}_{\text{fj}}\)

-The bode plot of \(\bm{G}_{\text{fj}}(s)\) is shown in Figure 10. +The bode plot of \(\bm{G}_{\text{fj}}(s)\) is shown in Figure 10.

-
+

bode_plot_plant_fj.png

Figure 10: Bode plot of the diagonal and off-diagonal elements of the plant in the frame of the fast jacks

-
+

Computing the system in the frame of the fastjack gives good decoupling at low frequency (until the first resonance of the system).

@@ -593,11 +595,11 @@ Computing the system in the frame of the fastjack gives good decoupling at low f
-
-

2.3. Plant in the frame of the crystal

+
+

2.3. Plant in the frame of the crystal

-
+

schematic_jacobian_frame_crystal.png

Figure 11: Use of Jacobian matrices to obtain the system in the frame of the crystal

@@ -652,18 +654,18 @@ The main reason is that, as we map forces to the center of the 111 crystal and n
-
-

3. Active Damping Plant (Strain gauges)

+
+

3. Active Damping Plant (Strain gauges)

- +

In this section, we wish to see whether if strain gauges fixed to the piezoelectric actuator can be used for active damping.

-
-

3.1. Identification

+
+

3.1. Identification

%% Input/Output definition
@@ -671,13 +673,11 @@ clear io; io_i = 1;
 
 %% Inputs
 % Control Input {3x1} [N]
-io(io_i) = linio([mdl, '/u'], 1, 'openinput');  io_i = io_i + 1;
-% % Stepper Displacement {3x1} [m]
-% io(io_i) = linio([mdl, '/d'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/control_system'], 1, 'openinput');  io_i = io_i + 1;
 
 %% Outputs
 % Strain Gauges {3x1} [m]
-io(io_i) = linio([mdl, '/sg'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/DCM'], 2, 'openoutput'); io_i = io_i + 1;
 
@@ -704,33 +704,128 @@ G_sg = linearize(mdl, io); --1.4113e-13 +4.4443e-09 1.0339e-13 3.774e-14 1.0339e-13 --1.4113e-13 +4.4443e-09 3.774e-14 3.7792e-14 3.7792e-14 --7.5585e-14 +4.4444e-09 + + +
+

strain_gauge_plant_bode_plot.png +

+

Figure 12: Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements

+
+ +
+

+As the distance between the poles and zeros in Figure 15 is very small, little damping can be actively added using the strain gauges. +This will be confirmed using a Root Locus plot. +

+
-
-

4. Active Damping Plant (Force Sensors)

+
+

3.2. Relative Active Damping

+
+
+
Krad_g1 = eye(3)*s/(s^2/(2*pi*500)^2 + 2*s/(2*pi*500) + 1);
+
+
+ +

+As can be seen in Figure 13, very little damping can be added using relative damping strategy using strain gauges. +

+ + +
+

relative_damping_root_locus.png +

+

Figure 13: Root Locus for the relative damping control

+
+ +
+
Krad = -g*Krad_g1;
+
+
+
+
+ +
+

3.3. Damped Plant

+
+

+The controller is implemented on Simscape, and the damped plant is identified. +

+ +
+
%% Input/Output definition
+clear io; io_i = 1;
+
+%% Inputs
+% Control Input {3x1} [N]
+io(io_i) = linio([mdl, '/control_system'], 1, 'input');  io_i = io_i + 1;
+
+%% Outputs
+% Force Sensor {3x1} [m]
+io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
+
+
+ +
+
%% DCM Kinematics
+load('mat/dcm_kinematics.mat');
+
+
+ +
+
%% Identification of the Open Loop plant
+controller.type = 0; % Open Loop
+G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io);
+G_ol.InputName  = {'u_ur',  'u_uh',  'u_d'};
+G_ol.OutputName = {'d_ur',  'd_uh',  'd_d'};
+
+
+ +
+
%% Identification of the damped plant with Relative Active Damping
+controller.type = 2; % RAD
+G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io);
+G_dp.InputName  = {'u_ur',  'u_uh',  'u_d'};
+G_dp.OutputName = {'d_ur',  'd_uh',  'd_d'};
+
+
+ + +
+

comp_damp_undamped_plant_rad_bode_plot.png +

+

Figure 14: Bode plot of both the open-loop plant and the damped plant using relative active damping

+
+
+
+
+ +
+

4. Active Damping Plant (Force Sensors)

- +

Force sensors are added above the piezoelectric actuators. @@ -738,8 +833,8 @@ They can consists of a simple piezoelectric ceramic stack. See for instance fleming10_integ_strain_force_feedb_high.

-
-

4.1. Identification

+
+

4.1. Identification

%% Input/Output definition
@@ -798,16 +893,16 @@ G_fs = linearize(mdl, io);
 
 
 
-
+

iff_plant_bode_plot.png

-

Figure 12: Bode plot of IFF Plant

+

Figure 15: Bode plot of IFF Plant

-
-

4.2. Controller - Root Locus

+
+

4.2. Controller - Root Locus

Kiff_g1 = eye(3)*1/(1 + s/2/pi/20);
@@ -815,10 +910,10 @@ G_fs = linearize(mdl, io);
 
-
+

iff_root_locus.png

-

Figure 13: Root Locus plot for the IFF Control strategy

+

Figure 16: Root Locus plot for the IFF Control strategy

@@ -829,8 +924,8 @@ Kiff = g*Kiff_g1;
-
-

4.3. Damped Plant

+
+

4.3. Damped Plant

%% Input/Output definition
@@ -871,13 +966,13 @@ G_dp.OutputName = {'d_ur',  
+

comp_damped_undamped_plant_iff_bode_plot.png

-

Figure 14: Bode plot of both the open-loop plant and the damped plant using IFF

+

Figure 17: Bode plot of both the open-loop plant and the damped plant using IFF

-
+

The Integral Force Feedback control strategy is very effective in damping the suspension modes of the DCM.

@@ -886,8 +981,8 @@ The Integral Force Feedback control strategy is very effective in damping the su
-
-

4.4. Save

+
+

4.4. Save

save('mat/Kiff.mat', 'Kiff');
@@ -897,18 +992,18 @@ The Integral Force Feedback control strategy is very effective in damping the su
 
-
-

5. HAC-LAC (IFF) architecture

+
+

5. HAC-LAC (IFF) architecture

- +

Author: Dehaeze Thomas

-

Created: 2021-11-30 mar. 11:26

+

Created: 2021-11-30 mar. 11:44

diff --git a/dcm-simscape.org b/dcm-simscape.org index 88720e2..57b6f9c 100644 --- a/dcm-simscape.org +++ b/dcm-simscape.org @@ -738,13 +738,11 @@ clear io; io_i = 1; %% Inputs % Control Input {3x1} [N] -io(io_i) = linio([mdl, '/u'], 1, 'openinput'); io_i = io_i + 1; -% % Stepper Displacement {3x1} [m] -% io(io_i) = linio([mdl, '/d'], 1, 'openinput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/control_system'], 1, 'openinput'); io_i = io_i + 1; %% Outputs % Strain Gauges {3x1} [m] -io(io_i) = linio([mdl, '/sg'], 1, 'openoutput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/DCM'], 2, 'openoutput'); io_i = io_i + 1; #+end_src #+begin_src matlab @@ -762,12 +760,12 @@ dcgain(G_sg) #+end_src #+RESULTS: -| -1.4113e-13 | 1.0339e-13 | 3.774e-14 | -| 1.0339e-13 | -1.4113e-13 | 3.774e-14 | -| 3.7792e-14 | 3.7792e-14 | -7.5585e-14 | +| 4.4443e-09 | 1.0339e-13 | 3.774e-14 | +| 1.0339e-13 | 4.4443e-09 | 3.774e-14 | +| 3.7792e-14 | 3.7792e-14 | 4.4444e-09 | #+begin_src matlab :exports none -%% Bode plot for the plant +%% Bode plot for the plant (strain gauge output) figure; tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); @@ -788,7 +786,8 @@ end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); -legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-14, 1e-7]); ax2 = nexttile; hold on; @@ -800,12 +799,173 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); hold off; yticks(-360:90:360); -ylim([-180, 180]); +ylim([-180, 0]); linkaxes([ax1,ax2],'x'); xlim([freqs(1), freqs(end)]); #+end_src +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/strain_gauge_plant_bode_plot.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:strain_gauge_plant_bode_plot +#+caption: Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements +#+RESULTS: +[[file:figs/strain_gauge_plant_bode_plot.png]] + +#+begin_important +As the distance between the poles and zeros in Figure [[fig:iff_plant_bode_plot]] is very small, little damping can be actively added using the strain gauges. +This will be confirmed using a Root Locus plot. +#+end_important + +** Relative Active Damping +#+begin_src matlab +Krad_g1 = eye(3)*s/(s^2/(2*pi*500)^2 + 2*s/(2*pi*500) + 1); +#+end_src + +As can be seen in Figure [[fig:relative_damping_root_locus]], very little damping can be added using relative damping strategy using strain gauges. + +#+begin_src matlab :exports none +%% Root Locus for IFF +gains = logspace(3, 8, 200); + +figure; + +hold on; +plot(real(pole(G_sg)), imag(pole(G_sg)), 'x', 'color', colors(1,:), ... + 'DisplayName', '$g = 0$'); +plot(real(tzero(G_sg)), imag(tzero(G_sg)), 'o', 'color', colors(1,:), ... + 'HandleVisibility', 'off'); + +for g = gains + clpoles = pole(feedback(G_sg, g*Krad_g1, -1)); + plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ... + 'HandleVisibility', 'off'); +end + +% Optimal gain +g = 2e5; +clpoles = pole(feedback(G_sg, g*Krad_g1, -1)); +plot(real(clpoles), imag(clpoles), 'x', 'color', colors(2,:), ... + 'DisplayName', sprintf('$g=%.0e$', g)); +hold off; +xlim([-6, 0]); ylim([0, 2700]); +xlabel('Real Part'); ylabel('Imaginary Part'); +legend('location', 'northwest'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/relative_damping_root_locus.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:relative_damping_root_locus +#+caption: Root Locus for the relative damping control +#+RESULTS: +[[file:figs/relative_damping_root_locus.png]] + +#+begin_src matlab +Krad = -g*Krad_g1; +#+end_src + +** Damped Plant +The controller is implemented on Simscape, and the damped plant is identified. + +#+begin_src matlab +%% Input/Output definition +clear io; io_i = 1; + +%% Inputs +% Control Input {3x1} [N] +io(io_i) = linio([mdl, '/control_system'], 1, 'input'); io_i = io_i + 1; + +%% Outputs +% Force Sensor {3x1} [m] +io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1; +#+end_src + +#+begin_src matlab +%% DCM Kinematics +load('mat/dcm_kinematics.mat'); +#+end_src + +#+begin_src matlab +%% Identification of the Open Loop plant +controller.type = 0; % Open Loop +G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io); +G_ol.InputName = {'u_ur', 'u_uh', 'u_d'}; +G_ol.OutputName = {'d_ur', 'd_uh', 'd_d'}; +#+end_src + +#+begin_src matlab +%% Identification of the damped plant with Relative Active Damping +controller.type = 2; % RAD +G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io); +G_dp.InputName = {'u_ur', 'u_uh', 'u_d'}; +G_dp.OutputName = {'d_ur', 'd_uh', 'd_d'}; +#+end_src + +#+begin_src matlab :exports none +%% Comparison of the damped and undamped plant +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))), ... + 'DisplayName', 'd - OL'); +plot(freqs, abs(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))), ... + 'DisplayName', 'uh - OL'); +plot(freqs, abs(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))), ... + 'DisplayName', 'ur - OL'); +set(gca,'ColorOrderIndex',1) +plot(freqs, abs(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '--', ... + 'DisplayName', 'd - IFF'); +plot(freqs, abs(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '--', ... + 'DisplayName', 'uh - IFF'); +plot(freqs, abs(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '--', ... + 'DisplayName', 'ur - IFF'); +for i = 1:2 + for j = i+1:3 + plot(freqs, abs(squeeze(freqresp(G_dp(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-12, 1e-6]); + +ax2 = nexttile; +hold on; +plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(1,1), freqs, 'Hz')))); +plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(2,2), freqs, 'Hz')))); +plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(3,3), freqs, 'Hz')))); +set(gca,'ColorOrderIndex',1) +plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '--'); +plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '--'); +plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '--'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); +xlabel('Frequency [Hz]'); ylabel('Phase [deg]'); +hold off; +yticks(-360:90:360); +ylim([-180, 0]); + +linkaxes([ax1,ax2],'x'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/comp_damp_undamped_plant_rad_bode_plot.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:comp_damp_undamped_plant_rad_bode_plot +#+caption: Bode plot of both the open-loop plant and the damped plant using relative active damping +#+RESULTS: +[[file:figs/comp_damp_undamped_plant_rad_bode_plot.png]] + * Active Damping Plant (Force Sensors) :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_active_damping_iff.m diff --git a/dcm-simscape.pdf b/dcm-simscape.pdf index 6732f54f3752908949852dd54893d420ae8aeb8a..595ee5c07e23c2ab99283090fe465afd37fb16dd 100644 GIT binary patch delta 273270 zcmZsi1z1$i_y1i$5RmSW5Tykcc9%sEBm_|e1Zj~L3F)p&cPWhs(kY;%(k)6#3Q{87 zAsxTF@PYk4|L5=XtZ-)Toq5mPJBK;vHJdR8>#US4*1?Mtx4J<7mWL_w3Rahb zqOr)k6t{RAylZ&AO!OU8_dk0wU-f+pX`=41Y=9^I)yOTeSvigeetp5_pW_hsq%r!o zOufmM@!?^8t)$xAPB~UiJw3%v;kCU`;A)bRqvY~_FbndqhUee+tz6=Dv&03JXIa(& zxhIsIJ+~v{D~9l^f|q8=`u}K1AZg6&P(t!K`)^*jv@8;n8YH)aa1)iB5^q1Qp}?)& z;n3q-cMI=9iU``36aG-N@2$nleNy?_B_pfijWII{MG9dDPYF52eNDe2ogSz^g2x+4d6~6XgE^X5iLVTj8vgyCV4sgK{t0io1$qj$m zb-l%mJaYUg*Syoe7Jz%jO81JOYryv*8EJa&&^nb{cNjI7hPglA-h3rinqJZ-{?TE| 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