From f2115d6b620c28c116b9e7ad8a58d928f6f2155d Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Wed, 11 Mar 2020 19:01:50 +0100 Subject: [PATCH] Update one latex formula --- docs/control-tracking.html | 144 ++++++++++++++++++------------------- org/control-tracking.org | 2 +- 2 files changed, 73 insertions(+), 73 deletions(-) diff --git a/docs/control-tracking.html b/docs/control-tracking.html index af26d3f..1b1b89c 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. Position Error computation
    • @@ -325,8 +325,8 @@ Depending of the measured quantity, different control architectures can be used:

    -
    -

    1.1 Control Schematic

    +
    +

    1.1 Control Schematic

    The control architecture is shown in Figure 1. @@ -349,8 +349,8 @@ 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 = initializeStewartPlatform();
@@ -382,8 +382,8 @@ 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 \(\bm{L}\). @@ -406,8 +406,8 @@ G.OutputName = {'L1', '

    -
    -

    1.4 Plant Analysis

    +
    +

    1.4 Plant Analysis

    The diagonal terms of the plant is shown in Figure 2. @@ -441,8 +441,8 @@ 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: @@ -472,8 +472,8 @@ Kl = diag(1./diag(abs(freqresp(G, wc))))

    -
    -

    1.6 Simulation

    +
    +

    1.6 Simulation

    t = linspace(0, 10, 1000);
@@ -518,8 +518,8 @@ simout_D = simout;
    -
    -

    1.8 Conclusion

    +
    +

    1.8 Conclusion

    Such control architecture is easy to implement and give good results. @@ -540,8 +540,8 @@ 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 7). @@ -551,7 +551,7 @@ The signals are:

  • reference path \(\bm{r}_\mathcal{X} = \begin{bmatrix} \epsilon_x & \epsilon_y & \epsilon_z & \epsilon_{R_x} & \epsilon_{R_y} & \epsilon_{R_z} \end{bmatrix}\)
  • tracking error \(\bm{\epsilon}_\mathcal{X} = \begin{bmatrix} \epsilon_x & \epsilon_y & \epsilon_z & \epsilon_{R_x} & \epsilon_{R_y} & \epsilon_{R_z} \end{bmatrix}\)
  • actuator forces \(\bm{\tau} = \begin{bmatrix} \tau_1 & \tau_2 & \tau_3 & \tau_4 & \tau_5 & \tau_6 \end{bmatrix}\)
    • -
  • payload pose \(\bm{\mathcal{X}) = \begin{bmatrix} x & y & z & R_x & R_y & R_z \end{bmatrix}\)
    • +
  • payload pose \(\bm{\mathcal{X}} = \begin{bmatrix} x & y & z & R_x & R_y & R_z \end{bmatrix}\)
    • @@ -579,8 +579,8 @@ It is indeed a more complex computation explained in section -

    2.2 Initialize the Stewart platform

    +
    +

    2.2 Initialize the Stewart platform

    stewart = initializeStewartPlatform();
@@ -612,8 +612,8 @@ 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 \(\bm{L}\). @@ -643,8 +643,8 @@ 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. @@ -665,8 +665,8 @@ Note here that the transformation from the pose error \(\bm{\epsilon}_\mathcal{X

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    -

    2.4.2 Plant Analysis

    +
    +

    2.4.2 Plant Analysis

    We now multiply the plant by the Jacobian matrix as shown in Figure 8 to obtain a more diagonal plant. @@ -716,8 +716,8 @@ Thus \(J \cdot G(\omega = 0) = J \cdot \frac{\delta\bm{\mathcal{X}}}{\delta\bm{\

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    -

    2.4.3 Controller Design

    +
    +

    2.4.3 Controller Design

    The controller consists of: @@ -755,8 +755,8 @@ 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. @@ -796,8 +796,8 @@ simout_L = simout;

    -
    -

    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 12. @@ -816,8 +816,8 @@ The final implemented controller is \(\bm{K} = \bm{J}^{-T} \cdot \bm{K}_\mathcal

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    2.5.2 Plant Analysis

    +
    +

    2.5.2 Plant Analysis

    We now multiply the plant by the Jacobian matrix as shown in Figure 12 to obtain a more diagonal plant. @@ -939,8 +939,8 @@ 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: @@ -978,8 +978,8 @@ The controller \(\bm{K} = \bm{J}^{-T} \bm{K}_\mathcal{X}\) is computed.

    -
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    2.5.4 Simulation

    +
    +

    2.5.4 Simulation

    We specify the reference path to follow. @@ -1019,8 +1019,8 @@ simout_X = simout;

    -
    -

    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. @@ -1043,8 +1043,8 @@ The control architecture is shown in Figure 16.

    -
    -

    2.6.2 Plant Analysis

    +
    +

    2.6.2 Plant Analysis

    The plant is pre-multiplied by \(\bm{G}^{-1}(\omega = 0)\). @@ -1072,8 +1072,8 @@ The diagonal elements of the shaped plant are shown in Figure -

    2.6.3 Controller Design

    +
    +

    2.6.3 Controller Design

    We have that: @@ -1147,8 +1147,8 @@ This error is much lower when using the diagonal control in the frame of the leg

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    -

    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. @@ -1231,8 +1231,8 @@ Thus, this method should be quite robust against parameter variation (e.g. the p

    -
    -

    3.1 Control Schematic

    +
    +

    3.1 Control Schematic

    @@ -1359,7 +1359,7 @@ Erz = atan2(-T(1, 2)/

    Author: Dehaeze Thomas

    -

    Created: 2020-03-11 mer. 19:00

    +

    Created: 2020-03-11 mer. 19:01

    diff --git a/org/control-tracking.org b/org/control-tracking.org index 81b4b4b..ff23f44 100644 --- a/org/control-tracking.org +++ b/org/control-tracking.org @@ -434,7 +434,7 @@ The signals are: - reference path $\bm{r}_\mathcal{X} = \begin{bmatrix} \epsilon_x & \epsilon_y & \epsilon_z & \epsilon_{R_x} & \epsilon_{R_y} & \epsilon_{R_z} \end{bmatrix}$ - tracking error $\bm{\epsilon}_\mathcal{X} = \begin{bmatrix} \epsilon_x & \epsilon_y & \epsilon_z & \epsilon_{R_x} & \epsilon_{R_y} & \epsilon_{R_z} \end{bmatrix}$ - actuator forces $\bm{\tau} = \begin{bmatrix} \tau_1 & \tau_2 & \tau_3 & \tau_4 & \tau_5 & \tau_6 \end{bmatrix}$ -- payload pose $\bm{\mathcal{X}) = \begin{bmatrix} x & y & z & R_x & R_y & R_z \end{bmatrix}$ +- payload pose $\bm{\mathcal{X}} = \begin{bmatrix} x & y & z & R_x & R_y & R_z \end{bmatrix}$ #+begin_src latex :file centralized_reference_tracking.pdf \begin{tikzpicture}