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< h1 class = "title" > Amplified Piezoelectric Stack Actuator< / h1 >
< div id = "table-of-contents" >
< h2 > Table of Contents< / h2 >
< div id = "text-table-of-contents" >
< ul >
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< li > < a href = "#orga1734d6" > 1. Simplified Model< / a >
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< ul >
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< li > < a href = "#org9543c57" > 1.1. Parameters< / a > < / li >
< li > < a href = "#org20bf7c7" > 1.2. Identification< / a > < / li >
< li > < a href = "#org1f0fec3" > 1.3. Root Locus< / a > < / li >
< li > < a href = "#orgc967e9c" > 1.4. Analytical Model< / a > < / li >
< li > < a href = "#orgb2d3e3a" > 1.5. Analytical Analysis< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org51b7142" > 2. Rotating X-Y platform< / a >
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< ul >
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< li > < a href = "#orgf847a9d" > 2.1. Parameters< / a > < / li >
< li > < a href = "#orga0c7065" > 2.2. Identification< / a > < / li >
< li > < a href = "#org0815f8b" > 2.3. Root Locus< / a > < / li >
< li > < a href = "#org6964694" > 2.4. Analysis< / a > < / li >
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< / ul >
< / li >
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< li > < a href = "#org630b8fc" > 3. Stewart Platform with Amplified Actuators< / a >
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< ul >
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< li > < a href = "#org0b7c221" > 3.1. Initialization< / a > < / li >
< li > < a href = "#orgff8fe24" > 3.2. Identification< / a > < / li >
< li > < a href = "#org58ae516" > 3.3. Controller Design< / a > < / li >
< li > < a href = "#org1c125d1" > 3.4. Effect of the Low Authority Control on the Primary Plant< / a > < / li >
< li > < a href = "#org3afc321" > 3.5. Effect of the Low Authority Control on the Sensibility to Disturbances< / a > < / li >
< li > < a href = "#orgface252" > 3.6. Optimal Stiffnesses< / a > < / li >
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< / li >
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< / div >
< p >
The presented model is based on < a class = 'org-ref-reference' href = "#souleille18_concep_activ_mount_space_applic" > souleille18_concep_activ_mount_space_applic< / a > .
< / p >
< p >
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The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure < a href = "#org9bfac50" > 1< / a > ).
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The parameters are shown in the table below.
< / p >
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< div id = "org9bfac50" class = "figure" >
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< p > < img src = "./figs/souleille18_model_piezo.png" alt = "souleille18_model_piezo.png" / >
< / p >
< p > < span class = "figure-number" > Figure 1: < / span > Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator< / p >
< / div >
< table border = "2" cellspacing = "0" cellpadding = "6" rules = "groups" frame = "hsides" >
< caption class = "t-above" > < span class = "table-number" > Table 1:< / span > Parameters used for the model of the APA 100M< / caption >
< colgroup >
< col class = "org-left" / >
< col class = "org-left" / >
< col class = "org-left" / >
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< thead >
< tr >
< th scope = "col" class = "org-left" >   < / th >
< th scope = "col" class = "org-left" > Value< / th >
< th scope = "col" class = "org-left" > Meaning< / th >
< / tr >
< / thead >
< tbody >
< tr >
< td class = "org-left" > \(m\)< / td >
< td class = "org-left" > \(1\,[kg]\)< / td >
< td class = "org-left" > Payload mass< / td >
< / tr >
< tr >
< td class = "org-left" > \(k_e\)< / td >
< td class = "org-left" > \(4.8\,[N/\mu m]\)< / td >
< td class = "org-left" > Stiffness used to adjust the pole of the isolator< / td >
< / tr >
< tr >
< td class = "org-left" > \(k_1\)< / td >
< td class = "org-left" > \(0.96\,[N/\mu m]\)< / td >
< td class = "org-left" > Stiffness of the metallic suspension when the stack is removed< / td >
< / tr >
< tr >
< td class = "org-left" > \(k_a\)< / td >
< td class = "org-left" > \(65\,[N/\mu m]\)< / td >
< td class = "org-left" > Stiffness of the actuator< / td >
< / tr >
< tr >
< td class = "org-left" > \(c_1\)< / td >
< td class = "org-left" > \(10\,[N/(m/s)]\)< / td >
< td class = "org-left" > Added viscous damping< / td >
< / tr >
< / tbody >
< / table >
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< div id = "outline-container-orga1734d6" class = "outline-2" >
< h2 id = "orga1734d6" > < span class = "section-number-2" > 1< / span > Simplified Model< / h2 >
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< div class = "outline-text-2" id = "text-1" >
< / div >
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< div id = "outline-container-org9543c57" class = "outline-3" >
< h3 id = "org9543c57" > < span class = "section-number-3" > 1.1< / span > Parameters< / h3 >
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< div class = "outline-text-3" id = "text-1-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > m = 1; % [kg]
ke = 4.8e6; % [N/m]
ce = 5; % [N/(m/s)]
me = 0.001; % [kg]
k1 = 0.96e6; % [N/m]
c1 = 10; % [N/(m/s)]
ka = 65e6; % [N/m]
ca = 5; % [N/(m/s)]
ma = 0.001; % [kg]
h = 0.2; % [m]
< / pre >
< / div >
< p >
IFF Controller:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kiff = -8000/s;
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-org20bf7c7" class = "outline-3" >
< h3 id = "org20bf7c7" > < span class = "section-number-3" > 1.2< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-1-2" >
< p >
Identification in open-loop.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'amplified_piezo_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion
io(io_i) = linio([mdl, '/f'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % External Force
io(io_i) = linio([mdl, '/Fs'], 3, 'openoutput'); io_i = io_i + 1; % Force Sensors
io(io_i) = linio([mdl, '/x1'], 1, 'openoutput'); io_i = io_i + 1; % Mass displacement
G = linearize(mdl, io, 0);
G.InputName = {'w', 'f', 'F'};
G.OutputName = {'Fs', 'x1'};
< / pre >
< / div >
< p >
Identification in closed-loop.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'amplified_piezo_model';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/w'], 1, 'input'); io_i = io_i + 1; % Base Motion
io(io_i) = linio([mdl, '/f'], 1, 'input'); io_i = io_i + 1; % Actuator Inputs
io(io_i) = linio([mdl, '/F'], 1, 'input'); io_i = io_i + 1; % External Force
io(io_i) = linio([mdl, '/Fs'], 3, 'output'); io_i = io_i + 1; % Force Sensors
io(io_i) = linio([mdl, '/x1'], 1, 'output'); io_i = io_i + 1; % Mass displacement
Giff = linearize(mdl, io, 0);
Giff.InputName = {'w', 'f', 'F'};
Giff.OutputName = {'Fs', 'x1'};
< / pre >
< / div >
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< div id = "org3c557c6" class = "figure" >
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< p > < img src = "figs/amplified_piezo_tf_ol_and_cl.png" alt = "amplified_piezo_tf_ol_and_cl.png" / >
< / p >
< p > < span class = "figure-number" > Figure 2: < / span > Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org1f0fec3" class = "outline-3" >
< h3 id = "org1f0fec3" > < span class = "section-number-3" > 1.3< / span > Root Locus< / h3 >
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< div class = "outline-text-3" id = "text-1-3" >
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< div id = "org6370599" class = "figure" >
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< p > < img src = "figs/amplified_piezo_root_locus.png" alt = "amplified_piezo_root_locus.png" / >
< / p >
< p > < span class = "figure-number" > Figure 3: < / span > Root Locus< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgc967e9c" class = "outline-3" >
< h3 id = "orgc967e9c" > < span class = "section-number-3" > 1.4< / span > Analytical Model< / h3 >
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< div class = "outline-text-3" id = "text-1-4" >
< p >
If we apply the Newton’ s second law of motion on the top mass, we obtain:
\[ ms^2 x_1 = F + k_1 (w - x_1) + k_e (x_e - x_1) \]
< / p >
< p >
Then, we can write that the measured force \(F_s\) is equal to:
\[ F_s = k_a(w - x_e) + f = -k_e (x_1 - x_e) \]
which gives:
\[ x_e = \frac{k_a}{k_e + k_a} w + \frac{1}{k_e + k_a} f + \frac{k_e}{k_e + k_a} x_1 \]
< / p >
< p >
Re-injecting that into the previous equations gives:
\[ x_1 = F \frac{1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + f \frac{\frac{k_e}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \]
\[ F_s = - F \frac{\frac{k_e k_a}{k_e + k_a}}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} + w \frac{k_e k_a}{k_e + k_a} \Big( \frac{ms^2}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) - f \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Ga = 1/(m*s^2 + k1 + ke*ka/(ke + ka)) * ...
[ 1 , k1 + ke*ka/(ke + ka) , ke/(ke + ka) ;
-ke*ka/(ke + ka), ke*ka/(ke + ka)*m*s^2 , -ke/(ke+ka)*(m*s^2 + k1)];
Ga.InputName = {'F', 'w', 'f'};
Ga.OutputName = {'x1', 'Fs'};
< / pre >
< / div >
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< div id = "orgf78178e" class = "figure" >
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< p > < img src = "figs/comp_simscape_analytical.png" alt = "comp_simscape_analytical.png" / >
< / p >
< p > < span class = "figure-number" > Figure 4: < / span > Comparison of the Identified Simscape Dynamics (solid) and the Analytical Model (dashed)< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-orgb2d3e3a" class = "outline-3" >
< h3 id = "orgb2d3e3a" > < span class = "section-number-3" > 1.5< / span > Analytical Analysis< / h3 >
< div class = "outline-text-3" id = "text-1-5" >
< p >
For Integral Force Feedback Control, the plant is:
\[ \frac{F_s}{f} = \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \]
< / p >
< p >
As high frequency, this converge to:
\[ \frac{F_s}{f} \underset{\omega\to\infty}{\longrightarrow} \frac{k_e}{k_e + k_a} \]
And at low frequency:
\[ \frac{F_s}{f} \underset{\omega\to 0}{\longrightarrow} \frac{k_e}{k_e + k_a} \frac{k_1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \]
< / p >
< p >
It has two complex conjugate zeros at:
\[ z = \pm j \sqrt{\frac{k_1}{m}} \]
And two complex conjugate poles at:
\[ p = \pm j \sqrt{\frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{m}} \]
< / p >
< p >
If maximal damping is to be attained with IFF, the distance between the zero and the pole is to be maximized.
Thus, we wish to maximize \(p/z\), which is equivalent as to minimize \(k_1\) and have \(k_e \approx k_a\) (supposing \(k_e + k_a \approx \text{cst}\)).
< / p >
< / div >
< / div >
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< / div >
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< div id = "outline-container-org51b7142" class = "outline-2" >
< h2 id = "org51b7142" > < span class = "section-number-2" > 2< / span > Rotating X-Y platform< / h2 >
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< div class = "outline-text-2" id = "text-2" >
< / div >
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< div id = "outline-container-orgf847a9d" class = "outline-3" >
< h3 id = "orgf847a9d" > < span class = "section-number-3" > 2.1< / span > Parameters< / h3 >
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< div class = "outline-text-3" id = "text-2-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > m = 1; % [kg]
ke = 4.8e6; % [N/m]
ce = 5; % [N/(m/s)]
me = 0.001; % [kg]
k1 = 0.96e6; % [N/m]
c1 = 10; % [N/(m/s)]
ka = 65e6; % [N/m]
ca = 5; % [N/(m/s)]
ma = 0.001; % [kg]
h = 0.2; % [m]
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kiff = tf(0);
< / pre >
< / div >
< / div >
< / div >
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< div id = "outline-container-orga0c7065" class = "outline-3" >
< h3 id = "orga0c7065" > < span class = "section-number-3" > 2.2< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-2-2" >
< p >
Rotating speed in rad/s:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Ws = 2*pi*[0, 1, 10, 100];
< / pre >
< / div >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Gs = {zeros(length(Ws), 1)};
< / pre >
< / div >
< p >
Identification in open-loop.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'amplified_piezo_xy_rotating_stage';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
for i = 1:length(Ws)
ws = Ws(i);
G = linearize(mdl, io, 0);
G.InputName = {'fx', 'fy'};
G.OutputName = {'Fsx', 'Fsy'};
Gs(i) = {G};
end
< / pre >
< / div >
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< div id = "org49fe0d0" class = "figure" >
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< p > < img src = "figs/amplitifed_piezo_xy_rotation_plant_iff.png" alt = "amplitifed_piezo_xy_rotation_plant_iff.png" / >
< / p >
< p > < span class = "figure-number" > Figure 5: < / span > Transfer function matrix from forces to force sensors for multiple rotation speed< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org0815f8b" class = "outline-3" >
< h3 id = "org0815f8b" > < span class = "section-number-3" > 2.3< / span > Root Locus< / h3 >
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< div class = "outline-text-3" id = "text-2-3" >
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< div id = "orgbf543c5" class = "figure" >
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< p > < img src = "figs/amplified_piezo_xy_rotation_root_locus.png" alt = "amplified_piezo_xy_rotation_root_locus.png" / >
< / p >
< p > < span class = "figure-number" > Figure 6: < / span > Root locus for 3 rotating speed< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org6964694" class = "outline-3" >
< h3 id = "org6964694" > < span class = "section-number-3" > 2.4< / span > Analysis< / h3 >
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< div class = "outline-text-3" id = "text-2-4" >
< p >
The negative stiffness induced by the rotation is equal to \(m \omega_0^2\).
Thus, the maximum rotation speed where IFF can be applied is:
\[ \omega_\text{max} = \sqrt{\frac{k_1}{m}} \approx 156\,[Hz] \]
< / p >
< p >
Let’ s verify that.
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Ws = 2*pi*[140, 160];
< / pre >
< / div >
< p >
Identification
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > %% Name of the Simulink File
mdl = 'amplified_piezo_xy_rotating_stage';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/fx'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/fy'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fs'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Fs'], 2, 'openoutput'); io_i = io_i + 1;
for i = 1:length(Ws)
ws = Ws(i);
G = linearize(mdl, io, 0);
G.InputName = {'fx', 'fy'};
G.OutputName = {'Fsx', 'Fsy'};
Gs(i) = {G};
end
< / pre >
< / div >
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< div id = "org1b0c04f" class = "figure" >
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< p > < img src = "figs/amplified_piezo_xy_rotating_unstable_root_locus.png" alt = "amplified_piezo_xy_rotating_unstable_root_locus.png" / >
< / p >
< p > < span class = "figure-number" > Figure 7: < / span > Root Locus for the two considered rotation speed. For the red curve, the system is unstable.< / p >
< / div >
< / div >
< / div >
< / div >
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< div id = "outline-container-org630b8fc" class = "outline-2" >
< h2 id = "org630b8fc" > < span class = "section-number-2" > 3< / span > Stewart Platform with Amplified Actuators< / h2 >
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< div class = "outline-text-2" id = "text-3" >
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< div id = "outline-container-org0b7c221" class = "outline-3" >
< h3 id = "org0b7c221" > < span class = "section-number-3" > 3.1< / span > Initialization< / h3 >
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< div class = "outline-text-3" id = "text-3-1" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeGround();
initializeGranite();
initializeTy();
initializeRy();
initializeRz();
initializeMicroHexapod();
initializeAxisc();
initializeMirror();
initializeSimscapeConfiguration();
initializeDisturbances('enable', false);
initializeLoggingConfiguration('log', 'none');
initializeController('type', 'hac-iff');
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< p >
We set the stiffness of the payload fixation:
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Kp = 1e8; % [N/m]
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< div id = "outline-container-orgff8fe24" class = "outline-3" >
< h3 id = "orgff8fe24" > < span class = "section-number-3" > 3.2< / span > Identification< / h3 >
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< div class = "outline-text-3" id = "text-3-2" >
< div class = "org-src-container" >
< pre class = "src src-matlab" > K = tf(zeros(6));
Kiff = tf(zeros(6));
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< p >
We identify the system for the following payload masses:
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< div class = "org-src-container" >
< pre class = "src src-matlab" > Ms = [1, 10, 50];
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< p >
The nano-hexapod has the following leg’ s stiffness and damping.
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< div class = "org-src-container" >
< pre class = "src src-matlab" > initializeNanoHexapod('actuator', 'amplified');
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< div id = "outline-container-org58ae516" class = "outline-3" >
< h3 id = "org58ae516" > < span class = "section-number-3" > 3.3< / span > Controller Design< / h3 >
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< div class = "outline-text-3" id = "text-3-3" >
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< div id = "orgd36ec15" class = "figure" >
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< p > < img src = "figs/amplified_piezo_iff_loop_gain.png" alt = "amplified_piezo_iff_loop_gain.png" / >
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< p > < span class = "figure-number" > Figure 8: < / span > Dynamics for the Integral Force Feedback for three payload masses< / p >
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< div id = "org7816aa3" class = "figure" >
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< p > < img src = "figs/amplified_piezo_iff_root_locus.png" alt = "amplified_piezo_iff_root_locus.png" / >
< / p >
< p > < span class = "figure-number" > Figure 9: < / span > Root Locus for the IFF control for three payload masses< / p >
< / div >
< p >
Damping as function of the gain
< / p >
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< div id = "orgfe47a62" class = "figure" >
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< p > < img src = "figs/amplified_piezo_iff_damping_gain.png" alt = "amplified_piezo_iff_damping_gain.png" / >
< / p >
< p > < span class = "figure-number" > Figure 10: < / span > Damping ratio of the poles as a function of the IFF gain< / p >
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< p >
Finally, we use the following controller for the Decentralized Direct Velocity Feedback:
< / p >
< div class = "org-src-container" >
< pre class = "src src-matlab" > Kiff = -1e4/s*eye(6);
< / pre >
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< / div >
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< div id = "outline-container-org1c125d1" class = "outline-3" >
< h3 id = "org1c125d1" > < span class = "section-number-3" > 3.4< / span > Effect of the Low Authority Control on the Primary Plant< / h3 >
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< div class = "outline-text-3" id = "text-3-4" >
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< div id = "orgd199337" class = "figure" >
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< p > < img src = "figs/amplified_piezo_iff_plant_damped_X.png" alt = "amplified_piezo_iff_plant_damped_X.png" / >
< / p >
< p > < span class = "figure-number" > Figure 11: < / span > Primary plant in the task space with (dashed) and without (solid) IFF< / p >
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< div id = "org7b6b3d9" class = "figure" >
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< p > < img src = "figs/amplified_piezo_iff_damped_plant_L.png" alt = "amplified_piezo_iff_damped_plant_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 12: < / span > Primary plant in the space of the legs with (dashed) and without (solid) IFF< / p >
< / div >
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< div id = "org0bd0e56" class = "figure" >
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< p > < img src = "figs/amplified_piezo_iff_damped_coupling_X.png" alt = "amplified_piezo_iff_damped_coupling_X.png" / >
< / p >
< p > < span class = "figure-number" > Figure 13: < / span > Coupling in the primary plant in the task with (dashed) and without (solid) IFF< / p >
< / div >
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< div id = "org1358a80" class = "figure" >
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< p > < img src = "figs/amplified_piezo_iff_damped_coupling_L.png" alt = "amplified_piezo_iff_damped_coupling_L.png" / >
< / p >
< p > < span class = "figure-number" > Figure 14: < / span > Coupling in the primary plant in the space of the legs with (dashed) and without (solid) IFF< / p >
< / div >
< / div >
< / div >
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< div id = "outline-container-org3afc321" class = "outline-3" >
< h3 id = "org3afc321" > < span class = "section-number-3" > 3.5< / span > Effect of the Low Authority Control on the Sensibility to Disturbances< / h3 >
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< div class = "outline-text-3" id = "text-3-5" >
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< div id = "org71868a4" class = "figure" >
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< p > < img src = "figs/amplified_piezo_iff_disturbances.png" alt = "amplified_piezo_iff_disturbances.png" / >
< / p >
< p > < span class = "figure-number" > Figure 15: < / span > Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Integral Force Feedback applied< / p >
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< div class = "important" >
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< div id = "outline-container-orgface252" class = "outline-3" >
< h3 id = "orgface252" > < span class = "section-number-3" > 3.6< / span > Optimal Stiffnesses< / h3 >
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< div id = "postamble" class = "status" >
< p class = "author" > Author: Dehaeze Thomas< / p >
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< p class = "date" > Created: 2020-05-25 lun. 11:05< / p >
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