1 Gravimeter - Simscape Model
+1 Gravimeter - Simscape Model
1.1 Introduction
+1.1 Introduction
Figure 1: Model of the gravimeter
@@ -95,8 +95,8 @@1.2 Simscape Model - Parameters
+1.2 Simscape Model - Parameters
open('gravimeter.slx') @@ -127,8 +127,8 @@ g = 0; % Gravity [m/s2]
1.3 System Identification - Without Gravity
+1.3 System Identification - Without Gravity
%% Name of the Simulink File @@ -175,7 +175,7 @@ State-space model with 4 outputs, 3 inputs, and 6 states. -+-
Figure 2: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers
@@ -183,8 +183,8 @@ State-space model with 4 outputs, 3 inputs, and 6 states.-1.4 System Identification - With Gravity
++1.4 System Identification - With Gravity
g = 9.80665; % Gravity [m/s2] @@ -213,7 +213,7 @@ ans =-+-
Figure 3: Open Loop Transfer Function from 3 Actuators to 4 Accelerometers with an without gravity
@@ -221,12 +221,12 @@ ans =-1.5 Analytical Model
++1.5 Analytical Model
---1.5.1 Parameters
++1.5.1 Parameters
@@ -259,30 +258,62 @@ Frequency vector.Bode options. @@ -245,7 +245,6 @@ P.TickLabel.FontSize = 12; P.Xlim = [1e-1,1e2]; P.MagLowerLimMode = 'manual'; P.MagLowerLim= 1e-3; -%P.PhaseVisible = 'off';
-1.5.2 generation of the state space model
++1.5.2 Generation of the State Space Model
+-+Mass matrix +
-%Jacobian of the bottom sensor -Js1 = [1 0 h/2 +M = [m 0 0 0 m 0 0 0 I]; +++Jacobian of the bottom sensor +
++-%Jacobian of the actuators -Ja = [1 0 ha % Left horizontal actuator +Js1 = [1 0 h/2 0 1 -l/2]; -%Jacobian of the top sensor -Js2 = [1 0 -h/2 - 0 1 0]; +++Jacobian of the top sensor +
+++ +Js2 = [1 0 -h/2 + 0 1 0]; +++Jacobian of the actuators +
++-E = [1 0 0 +Ja = [1 0 ha % Left horizontal actuator 0 1 -la % Left vertical actuator 0 1 la]; % Right vertical actuator Jta = Ja'; -K = k*Jta*Ja; -C = c*Jta*Ja; +++Stiffness and Damping matrices +
+++ +K = k*Jta*Ja; +C = c*Jta*Ja; +++State Space Matrices +
+++E = [1 0 0 0 1 0 0 0 1]; %projecting ground motion in the directions of the legs @@ -292,12 +323,6 @@ AA = [zeros(3) eye(3) BB = [zeros(3,6) M\Jta M\(k*Jta*E)]; -% BB = [zeros(3,3) -% M\Jta ]; -% -% CC = [Ja zeros(3)]; -% DD = zeros(3,3); - CC = [[Js1;Js2] zeros(4,3); zeros(2,6) (Js1+Js2)./2 zeros(2,3) @@ -307,16 +332,23 @@ CC = [[Js1;Js2] zeros(4,3); DD = [zeros(4,6) zeros(2,3) eye(2,3) zeros(6,6)]; - -system_dec = ss(AA,BB,CC,DD);+State Space model: +
+
- Input = three actuators and three ground motions
- Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation
++ +system_dec = ss(AA,BB,CC,DD); ++size(system_dec)@@ -328,11 +360,11 @@ State-space model with 12 outputs, 6 inputs, and 6 states.-1.5.3 Comparison with the Simscape Model
++1.5.3 Comparison with the Simscape Model
-+-
Figure 4: Comparison of the analytical and the Simscape models
@@ -340,8 +372,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.-1.5.4 Analysis
++1.5.4 Analysis
-% figure @@ -409,8 +441,8 @@ State-space model with 12 outputs, 6 inputs, and 6 states.-1.5.5 Control Section
++1.5.5 Control Section
-system_dec_10Hz = freqresp(system_dec,2*pi*10); @@ -550,8 +582,8 @@ legend('Control OFF','D-1.5.6 Greshgorin radius
++1.5.6 Greshgorin radius
-system_dec_freq = freqresp(system_dec,w); @@ -597,8 +629,8 @@ ylabel('Greshgorin radius [-]');-1.5.7 Injecting ground motion in the system to have the output
++1.5.7 Injecting ground motion in the system to have the output
-Fr = logspace(-2,3,1e3); @@ -654,15 +686,15 @@ rot = PHI(:,11,11);-2 Gravimeter - Functions
++2 Gravimeter - Functions
--2.1
+align+2.1
align-@@ -691,11 +723,11 @@ This Matlab function is accessible here.
-2.2
+pzmap_testCL+-2.2
pzmap_testCL@@ -744,12 +776,12 @@ This Matlab function is accessible here.
-3 Stewart Platform - Simscape Model
++3 Stewart Platform - Simscape Model
--3.1 Jacobian
++-3.1 Jacobian
First, the position of the “joints” (points of force application) are estimated and the Jacobian computed. @@ -791,8 +823,8 @@ save('./jacobian.mat',
-3.2 Simscape Model
++3.2 Simscape Model
-open('stewart_platform/drone_platform.slx'); @@ -823,8 +855,8 @@ We load the Jacobian.-3.3 Identification of the plant
++3.3 Identification of the plant
-The dynamics is identified from forces applied by each legs to the measured acceleration of the top platform. @@ -881,32 +913,32 @@ Gl.OutputName = {'A1',
-3.4 Obtained Dynamics
++3.4 Obtained Dynamics
-+-
Figure 5: Stewart Platform Plant from forces applied by the legs to the acceleration of the platform
+-
Figure 6: Stewart Platform Plant from torques applied by the legs to the angular acceleration of the platform
+-
Figure 7: Stewart Platform Plant from forces applied by the legs to displacement of the legs
+-
Figure 8: Transmissibility
@@ -914,8 +946,8 @@ Gl.OutputName = {'A1',-3.5 Real Approximation of \(G\) at the decoupling frequency
++-3.5 Real Approximation of \(G\) at the decoupling frequency
Let’s compute a real approximation of the complex matrix \(H_1\) which corresponds to the the transfer function \(G_c(j\omega_c)\) from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency \(\omega_c\). @@ -941,8 +973,8 @@ H1 = inv(D*real(H1'*
-3.6 Verification of the decoupling using the “Gershgorin Radii”
++3.6 Verification of the decoupling using the “Gershgorin Radii”
-First, the Singular Value Decomposition of \(H_1\) is performed: @@ -1010,7 +1042,7 @@ H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));
+-
Figure 9: Gershgorin Radii of the Coupled and Decoupled plants
@@ -1018,8 +1050,8 @@ H = abs(squeeze(freqresp(Gj, freqs, 'Hz')));-3.7 Decoupled Plant
++3.7 Decoupled Plant
Let’s see the bode plot of the decoupled plant \(G_d(s)\). @@ -1027,14 +1059,14 @@ Let’s see the bode plot of the decoupled plant \(G_d(s)\).
-+-
Figure 10: Decoupled Plant using SVD
+-
Figure 11: Decoupled Plant using the Jacobian
@@ -1042,8 +1074,8 @@ Let’s see the bode plot of the decoupled plant \(G_d(s)\).-3.8 Diagonal Controller
++3.8 Diagonal Controller
-The controller \(K\) is a diagonal controller consisting a low pass filters with a crossover frequency \(\omega_c\) and a DC gain \(C_g\). @@ -1059,8 +1091,8 @@ K = eye(6)*C_g/(s
3.9 Centralized Control
++-3.9 Centralized Control
The control diagram for the centralized control is shown below. @@ -1084,8 +1116,8 @@ The Jacobian is used to convert forces in the cartesian frame to forces applied
-3.10 SVD Control
++-3.10 SVD Control
The SVD control architecture is shown below. @@ -1108,8 +1140,8 @@ SVD Control
-3.11 Results
++3.11 Results
Let’s first verify the stability of the closed-loop systems: @@ -1139,11 +1171,11 @@ ans =
-The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 14. +The obtained transmissibility in Open-loop, for the centralized control as well as for the SVD control are shown in Figure 14.
-+
Figure 14: Obtained Transmissibility
@@ -1154,7 +1186,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well-diff --git a/index.org b/index.org index 6b80cfa..65efaa0 100644 --- a/index.org +++ b/index.org @@ -226,7 +226,6 @@ Bode options. P.Xlim = [1e-1,1e2]; P.MagLowerLimMode = 'manual'; P.MagLowerLim= 1e-3; - %P.PhaseVisible = 'off'; #+end_src Frequency vector. @@ -234,27 +233,42 @@ Frequency vector. w = 2*pi*logspace(-1,2,1000); % [rad/s] #+end_src -*** generation of the state space model +*** Generation of the State Space Model +Mass matrix #+begin_src matlab M = [m 0 0 0 m 0 0 0 I]; +#+end_src - %Jacobian of the bottom sensor +Jacobian of the bottom sensor +#+begin_src matlab Js1 = [1 0 h/2 0 1 -l/2]; - %Jacobian of the top sensor +#+end_src + +Jacobian of the top sensor +#+begin_src matlab Js2 = [1 0 -h/2 0 1 0]; +#+end_src - %Jacobian of the actuators +Jacobian of the actuators +#+begin_src matlab Ja = [1 0 ha % Left horizontal actuator 0 1 -la % Left vertical actuator 0 1 la]; % Right vertical actuator Jta = Ja'; +#+end_src + +Stiffness and Damping matrices +#+begin_src matlab K = k*Jta*Ja; C = c*Jta*Ja; +#+end_src +State Space Matrices +#+begin_src matlab E = [1 0 0 0 1 0 0 0 1]; %projecting ground motion in the directions of the legs @@ -265,12 +279,6 @@ Frequency vector. BB = [zeros(3,6) M\Jta M\(k*Jta*E)]; - % BB = [zeros(3,3) - % M\Jta ]; - % - % CC = [Ja zeros(3)]; - % DD = zeros(3,3); - CC = [[Js1;Js2] zeros(4,3); zeros(2,6) (Js1+Js2)./2 zeros(2,3) @@ -280,12 +288,16 @@ Frequency vector. DD = [zeros(4,6) zeros(2,3) eye(2,3) zeros(6,6)]; +#+end_src +State Space model: +- Input = three actuators and three ground motions +- Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation + +#+begin_src matlab system_dec = ss(AA,BB,CC,DD); #+end_src -- Input = three actuators and three ground motions -- Output = the bottom sensor; the top sensor; the ground motion; the half sum; the half difference; the rotation #+begin_src matlab :results output replace size(system_dec)Created: 2020-10-05 lun. 18:06
+Created: 2020-10-05 lun. 18:28