diff --git a/org/amplified_piezoelectric_stack.html b/org/amplified_piezoelectric_stack.html index 0ee6ffb..a12c877 100644 --- a/org/amplified_piezoelectric_stack.html +++ b/org/amplified_piezoelectric_stack.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Amplified Piezoelectric Stack Actuator @@ -34,30 +34,31 @@

Table of Contents

@@ -69,12 +70,12 @@ The presented model is based on 1). +The model represents the amplified piezo APA100M from Cedrat-Technologies (Figure 1). The parameters are shown in the table below.

-
+

souleille18_model_piezo.png

Figure 1: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator

@@ -130,12 +131,12 @@ The parameters are shown in the table below. -
-

1 Simplified Model

+
+

1 Simplified Model

-
-

1.1 Parameters

+
+

1.1 Parameters

m = 1; % [kg]
@@ -165,8 +166,8 @@ IFF Controller:
-
-

1.2 Identification

+
+

1.2 Identification

Identification in open-loop. @@ -213,7 +214,7 @@ Giff.OutputName = {'Fs', 'x1'};

-
+

amplified_piezo_tf_ol_and_cl.png

Figure 2: Matrix of transfer functions from input to output in open loop (blue) and closed loop (red)

@@ -221,11 +222,11 @@ Giff.OutputName = {'Fs', 'x1'};
-
-

1.3 Root Locus

+
+

1.3 Root Locus

-
+

amplified_piezo_root_locus.png

Figure 3: Root Locus

@@ -233,8 +234,8 @@ Giff.OutputName = {'Fs', 'x1'};
-
-

1.4 Analytical Model

+
+

1.4 Analytical Model

If we apply the Newton’s second law of motion on the top mass, we obtain: @@ -264,21 +265,50 @@ Ga.OutputName = {'x1', 'Fs'};

-
+

comp_simscape_analytical.png

Figure 4: Comparison of the Identified Simscape Dynamics (solid) and the Analytical Model (dashed)

+ +
+

1.5 Analytical Analysis

+
+

+For Integral Force Feedback Control, the plant is: +\[ \frac{F_s}{f} = \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \] +

+ +

+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: +\[ \frac{F_s}{f} \underset{\omega\to 0}{\longrightarrow} \frac{k_e}{k_e + k_a} \frac{k_1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \] +

+ +

+It has two complex conjugate zeros at: +\[ z = \pm j \sqrt{\frac{k_1}{m}} \] +And two complex conjugate poles at: +\[ p = \pm j \sqrt{\frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{m}} \] +

+ +

+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 \(k_e + k_a \approx \text{cst}\)). +

+
+
-
-

2 Rotating X-Y platform

+
+

2 Rotating X-Y platform

-
-

2.1 Parameters

+
+

2.1 Parameters

m = 1; % [kg]
@@ -305,8 +335,8 @@ h = 0.2; % [m]
-
-

2.2 Identification

+
+

2.2 Identification

Rotating speed in rad/s: @@ -347,7 +377,7 @@ end

-
+

amplitifed_piezo_xy_rotation_plant_iff.png

Figure 5: Transfer function matrix from forces to force sensors for multiple rotation speed

@@ -355,11 +385,11 @@ end
-
-

2.3 Root Locus

+
+

2.3 Root Locus

-
+

amplified_piezo_xy_rotation_root_locus.png

Figure 6: Root locus for 3 rotating speed

@@ -367,8 +397,8 @@ end
-
-

2.4 Analysis

+
+

2.4 Analysis

The negative stiffness induced by the rotation is equal to \(m \omega_0^2\). @@ -410,7 +440,7 @@ end

-
+

amplified_piezo_xy_rotating_unstable_root_locus.png

Figure 7: Root Locus for the two considered rotation speed. For the red curve, the system is unstable.

@@ -419,12 +449,12 @@ end
-
-

3 Stewart Platform with Amplified Actuators

+
+

3 Stewart Platform with Amplified Actuators

-
-

3.1 Initialization

+
+

3.1 Initialization

initializeGround();
@@ -454,8 +484,8 @@ We set the stiffness of the payload fixation:
-
-

3.2 Identification

+
+

3.2 Identification

K = tf(zeros(6));
@@ -481,11 +511,11 @@ The nano-hexapod has the following leg’s stiffness and damping.
-
-

3.3 Controller Design

+
+

3.3 Controller Design

-
+

amplified_piezo_iff_loop_gain.png

Figure 8: Dynamics for the Integral Force Feedback for three payload masses

@@ -493,7 +523,7 @@ The nano-hexapod has the following leg’s stiffness and damping. -
+

amplified_piezo_iff_root_locus.png

Figure 9: Root Locus for the IFF control for three payload masses

@@ -503,7 +533,7 @@ The nano-hexapod has the following leg’s stiffness and damping. Damping as function of the gain

-
+

amplified_piezo_iff_damping_gain.png

Figure 10: Damping ratio of the poles as a function of the IFF gain

@@ -519,11 +549,11 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
-
-

3.4 Effect of the Low Authority Control on the Primary Plant

+
+

3.4 Effect of the Low Authority Control on the Primary Plant

-
+

amplified_piezo_iff_plant_damped_X.png

Figure 11: Primary plant in the task space with (dashed) and without (solid) IFF

@@ -531,13 +561,13 @@ Finally, we use the following controller for the Decentralized Direct Velocity F -
+

amplified_piezo_iff_damped_plant_L.png

Figure 12: Primary plant in the space of the legs with (dashed) and without (solid) IFF

-
+

amplified_piezo_iff_damped_coupling_X.png

Figure 13: Coupling in the primary plant in the task with (dashed) and without (solid) IFF

@@ -545,7 +575,7 @@ Finally, we use the following controller for the Decentralized Direct Velocity F -
+

amplified_piezo_iff_damped_coupling_L.png

Figure 14: Coupling in the primary plant in the space of the legs with (dashed) and without (solid) IFF

@@ -553,11 +583,11 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
-
-

3.5 Effect of the Low Authority Control on the Sensibility to Disturbances

+
+

3.5 Effect of the Low Authority Control on the Sensibility to Disturbances

-
+

amplified_piezo_iff_disturbances.png

Figure 15: Norm of the transfer function from vertical disturbances to vertical position error with (dashed) and without (solid) Integral Force Feedback applied

@@ -569,14 +599,14 @@ Finally, we use the following controller for the Decentralized Direct Velocity F
-
-

3.6 Optimal Stiffnesses

+
+

3.6 Optimal Stiffnesses

Author: Dehaeze Thomas

-

Created: 2020-05-25 lun. 10:45

+

Created: 2020-05-25 lun. 11:05

diff --git a/org/amplified_piezoelectric_stack.org b/org/amplified_piezoelectric_stack.org index e5560f1..6663190 100644 --- a/org/amplified_piezoelectric_stack.org +++ b/org/amplified_piezoelectric_stack.org @@ -292,6 +292,23 @@ exportFig('figs/comp_simscape_analytical.pdf', 'width', 'full', 'height', 'full' #+RESULTS: [[file:figs/comp_simscape_analytical.png]] +** Analytical Analysis +For Integral Force Feedback Control, the plant is: +\[ \frac{F_s}{f} = \frac{k_e}{k_e + k_a} \Big( \frac{ms^2 + k_1}{ms^2 + k_1 + \frac{k_e k_a}{k_e + k_a}} \Big) \] + +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: +\[ \frac{F_s}{f} \underset{\omega\to 0}{\longrightarrow} \frac{k_e}{k_e + k_a} \frac{k_1}{k_1 + \frac{k_e k_a}{k_e + k_a}} \] + +It has two complex conjugate zeros at: +\[ z = \pm j \sqrt{\frac{k_1}{m}} \] +And two complex conjugate poles at: +\[ p = \pm j \sqrt{\frac{k_1 + \frac{k_e k_a}{k_e + k_a}}{m}} \] + +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 $k_e + k_a \approx \text{cst}$). + * Rotating X-Y platform ** Matlab Init :noexport:ignore: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)