diff --git a/docs/control-tracking.html b/docs/control-tracking.html index 6f71fe6..2b89ab4 100644 --- a/docs/control-tracking.html +++ b/docs/control-tracking.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Stewart Platform - Tracking Control @@ -248,42 +248,42 @@
  • 3. Hybrid Control Architecture - HAC-LAC with relative DVF
  • 4. Comparison of all the methods
    • @@ -351,8 +351,8 @@ The control configuration are compare in section 4.

    -
    -

    1.1 Control Schematic

    +
    +

    1.1 Control Schematic

    The control architecture is shown in Figure 1. @@ -375,8 +375,8 @@ Then, a diagonal (decentralized) controller \(\bm{K}_\mathcal{L}\) is used such

    -
    -

    1.2 Initialize the Stewart platform

    +
    +

    1.2 Initialize the Stewart platform

    % Stewart Platform
@@ -407,8 +407,8 @@ references = initializeReferences(stewart);
    -
    -

    1.3 Identification of the plant

    +
    +

    1.3 Identification of the plant

    Let’s identify the transfer function from \(\bm{\tau}\) to \(\bm{\mathcal{L}}\). @@ -431,8 +431,8 @@ G.OutputName = {'L1', '

    -
    -

    1.4 Plant Analysis

    +
    +

    1.4 Plant Analysis

    The diagonal terms of the plant is shown in Figure 2. @@ -466,8 +466,8 @@ We see that the plant is decoupled at low frequency which indicate that decentra

    -
    -

    1.5 Controller Design

    +
    +

    1.5 Controller Design

    The controller consists of: @@ -496,8 +496,8 @@ Kl = diag(1./diag(abs(freqresp(G, wc))))

    -
    -

    1.6 Simulation

    +
    +

    1.6 Simulation

    Let’s define some reference path to follow. @@ -596,8 +596,8 @@ zlim([0, inf]);

    -
    -

    1.8 Conclusion

    +
    +

    1.8 Conclusion

    Such control architecture is easy to implement and give good results. @@ -618,8 +618,8 @@ However, as \(\mathcal{X}\) is not directly measured, it is possible that import

    -
    -

    2.1 Control Schematic

    +
    +

    2.1 Control Schematic

    The centralized controller takes the position error \(\bm{\epsilon}_\mathcal{X}\) as an inputs and generate actuator forces \(\bm{\tau}\) (see Figure 9). @@ -657,8 +657,8 @@ It is indeed a more complex computation explained in section -

    2.2 Initialize the Stewart platform

    +
    +

    2.2 Initialize the Stewart platform

    % Stewart Platform
@@ -689,8 +689,8 @@ references = initializeReferences(stewart);
    -
    -

    2.3 Identification of the plant

    +
    +

    2.3 Identification of the plant

    Let’s identify the transfer function from \(\bm{\tau}\) to \(\bm{\mathcal{X}}\). @@ -720,8 +720,8 @@ G.OutputName = {'Dx', '

    -
    -

    2.4.1 Control Architecture

    +
    +

    2.4.1 Control Architecture

    The pose error \(\bm{\epsilon}_\mathcal{X}\) is first converted in the frame of the leg by using the Jacobian matrix. @@ -742,8 +742,8 @@ Note here that the transformation from the pose error \(\bm{\epsilon}_\mathcal{X

    -
    -

    2.4.2 Plant Analysis

    +
    +

    2.4.2 Plant Analysis

    We now multiply the plant by the Jacobian matrix as shown in Figure 10 to obtain a more diagonal plant. @@ -793,8 +793,8 @@ Thus \(J \cdot G(\omega = 0) = J \cdot \frac{\delta\bm{\mathcal{X}}}{\delta\bm{\

    -
    -

    2.4.3 Controller Design

    +
    +

    2.4.3 Controller Design

    The controller consists of: @@ -831,8 +831,8 @@ The controller \(\bm{K} = \bm{K}_\mathcal{L} \bm{J}\) is computed.

    -
    -

    2.4.4 Simulation

    +
    +

    2.4.4 Simulation

    We specify the reference path to follow. @@ -876,8 +876,8 @@ save('./mat/control_tracking.mat',

    -
    -

    2.5.1 Control Architecture

    +
    +

    2.5.1 Control Architecture

    A diagonal controller \(\bm{K}_\mathcal{X}\) take the pose error \(\bm{\epsilon}_\mathcal{X}\) and generate cartesian forces \(\bm{\mathcal{F}}\) that are then converted to actuators forces using the Jacobian as shown in Figure e 14. @@ -896,8 +896,8 @@ The final implemented controller is \(\bm{K} = \bm{J}^{-T} \cdot \bm{K}_\mathcal

    -
    -

    2.5.2 Plant Analysis

    +
    +

    2.5.2 Plant Analysis

    We now multiply the plant by the Jacobian matrix as shown in Figure 14 to obtain a more diagonal plant. @@ -1019,8 +1019,8 @@ This control architecture can also give a dynamically decoupled plant if the Cen

    -
    -

    2.5.3 Controller Design

    +
    +

    2.5.3 Controller Design

    The controller consists of: @@ -1057,8 +1057,8 @@ The controller \(\bm{K} = \bm{J}^{-T} \bm{K}_\mathcal{X}\) is computed.

    -
    -

    2.5.4 Simulation

    +
    +

    2.5.4 Simulation

    We specify the reference path to follow. @@ -1102,8 +1102,8 @@ save('./mat/control_tracking.mat',

    -
    -

    2.6.1 Control Architecture

    +
    +

    2.6.1 Control Architecture

    The plant \(\bm{G}\) is pre-multiply by \(\bm{G}^{-1}(\omega = 0)\) such that the “shaped plant” \(\bm{G}_0 = \bm{G} \bm{G}^{-1}(\omega = 0)\) is diagonal at low frequency. @@ -1126,8 +1126,8 @@ The control architecture is shown in Figure 18.

    -
    -

    2.6.2 Plant Analysis

    +
    +

    2.6.2 Plant Analysis

    The plant is pre-multiplied by \(\bm{G}^{-1}(\omega = 0)\). @@ -1155,8 +1155,8 @@ The diagonal elements of the shaped plant are shown in Figure -

    2.6.3 Controller Design

    +
    +

    2.6.3 Controller Design

    We have that: @@ -1222,8 +1222,8 @@ This error is much lower when using the diagonal control in the frame of the leg

    -
    -

    2.8 Conclusion

    +
    +

    2.8 Conclusion

    Both control architecture gives similar results even tough the control in the Leg’s frame gives slightly better results. @@ -1306,8 +1306,8 @@ Thus, this method should be quite robust against parameter variation (e.g. the p

    -
    -

    3.1 Control Schematic

    +
    +

    3.1 Control Schematic

    Let’s consider the control schematic shown in Figure 23. @@ -1348,8 +1348,8 @@ This second loop is responsible for the reference tracking.

    -
    -

    3.2 Initialize the Stewart platform

    +
    +

    3.2 Initialize the Stewart platform

    % Stewart Platform
@@ -1384,8 +1384,8 @@ references = initializeReferences(stewart);
 
    3.3 First Control Loop - \(\bm{K}_\mathcal{L}\)
    
 
    
 
    
-
    
-
    3.3.1 Identification
    
+
    
+
    3.3.1 Identification
    
 
    
 
    
 Let’s identify the transfer function from \(\bm{\tau}\) to \(\bm{L}\).
@@ -1408,8 +1408,8 @@ Gl.OutputName = {'L1', 
 
    
 
    
 
-
    
-
    3.3.2 Obtained Plant
    
+
    
+
    3.3.2 Obtained Plant
    
 
    
 
    
 The diagonal elements of the plant are shown in Figure 24 while the off diagonal terms are shown in Figure 25.
@@ -1431,8 +1431,8 @@ The diagonal elements of the plant are shown in Figure 24<
 
    
 
    
 
-
    
-
    3.3.3 Controller Design
    
+
    
+
    3.3.3 Controller Design
    
 
    
 
    
 We apply a decentralized (diagonal) direct velocity feedback.
@@ -1464,8 +1464,8 @@ The obtain loop gain is shown in Figure 26.
 
    3.4 Second Control Loop - \(\bm{K}_\mathcal{X}\)
    
 
    
 
    
-
    
-
    3.4.1 Identification
    
+
    
+
    3.4.1 Identification
    
 
    
 
    
 
    Kx = tf(zeros(6));
@@ -1492,8 +1492,8 @@ G.OutputName = {'Dx', '
 
    
 
    
 
-
    
-
    3.4.2 Obtained Plant
    
+
    
+
    3.4.2 Obtained Plant
    
 
    
 
    
 We use the Jacobian matrix to apply forces in the cartesian frame.
@@ -1516,15 +1516,17 @@ The obtained plant is shown in Figure 27.
 
    
 
    
 
-
    
-
    3.4.3 Controller Design
    
+
    
+
    3.4.3 Controller Design
    
 
    
 
    
 The controller consists of:
 
    
 
      
 
    • A pure integrator
      • 
-
    • A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin
      • 
+
    • A Second integrator up to half the wanted bandwidth
      • 
+
    • A Lead around the cross-over frequency
      • 
+
    • A low pass filter with a cut-off equal to three times the wanted bandwidth
      • 
 
    
 
 
    
@@ -1532,7 +1534,7 @@ The controller consists of:
 
 h = 3; % Lead parameter
 
-Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc/2 + 1)/(s/wc/2));
+Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc*2 + 1)/(s/wc*2)) * (1/(1 + s/wc/3));
 
 % Normalization of the gain of have a loop gain of 1 at frequency wc
 Kx = Kx.*diag(1./diag(abs(freqresp(Gx*Kx, wc))));
@@ -1600,8 +1602,8 @@ The obtained position error is shown in Figure 29.
 
    
 
    
 
-
    
-
    3.6 Conclusion
    
+
    
+
    3.6 Conclusion
    
 
    
 
    
 
@@ -1765,7 +1767,7 @@ Erz = atan2(-T(1, 2)/
 
    
 
    Author: Dehaeze Thomas
    
-
    Created: 2020-03-13 ven. 13:12
    
+
    Created: 2020-03-13 ven. 13:23
    
 
    
 
 
diff --git a/docs/figs/hybrid_control_Kx_loop_gain.pdf b/docs/figs/hybrid_control_Kx_loop_gain.pdf
index bae1e32..3753188 100644
Binary files a/docs/figs/hybrid_control_Kx_loop_gain.pdf and b/docs/figs/hybrid_control_Kx_loop_gain.pdf differ
diff --git a/docs/figs/hybrid_control_Kx_loop_gain.png b/docs/figs/hybrid_control_Kx_loop_gain.png
index 4812dcb..0cbb07b 100644
Binary files a/docs/figs/hybrid_control_Kx_loop_gain.png and b/docs/figs/hybrid_control_Kx_loop_gain.png differ
diff --git a/org/control-tracking.org b/org/control-tracking.org
index 0a1cb88..9c5daee 100644
--- a/org/control-tracking.org
+++ b/org/control-tracking.org
@@ -1529,14 +1529,16 @@ The obtained plant is shown in Figure [[fig:hybrid_control_Kx_plant]].
 *** Controller Design
 The controller consists of:
 - A pure integrator
-- A low pass filter with a cut-off frequency 3 times the crossover to increase the gain margin
+- A Second integrator up to half the wanted bandwidth
+- A Lead around the cross-over frequency
+- A low pass filter with a cut-off equal to three times the wanted bandwidth
 
 #+begin_src matlab
   wc = 2*pi*200; % Bandwidth Bandwidth [rad/s]
 
   h = 3; % Lead parameter
 
-  Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc/2 + 1)/(s/wc/2));
+  Kx = (1/h) * (1 + s/wc*h)/(1 + s/wc/h) * wc/s * ((s/wc*2 + 1)/(s/wc*2)) * (1/(1 + s/wc/3));
 
   % Normalization of the gain of have a loop gain of 1 at frequency wc
   Kx = Kx.*diag(1./diag(abs(freqresp(Gx*Kx, wc))));