From 02943f0f2881807dbaf2f5d12e712553a5fbccf2 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 28 Feb 2020 17:35:44 +0100 Subject: [PATCH] Work on HAC-LAC, Control architectures --- docs/active-damping.html | 95 +- docs/control-study.html | 924 ++++++++++++- docs/control-tracking.html | 370 +++++ docs/cubic-configuration.html | 38 +- docs/dynamics-study.html | 18 +- docs/figs/control_arch_hac_dvf.pdf | Bin 0 -> 84131 bytes docs/figs/control_arch_hac_dvf.png | Bin 0 -> 10602 bytes docs/figs/control_arch_hac_iff.pdf | Bin 0 -> 78198 bytes docs/figs/control_arch_hac_iff.png | Bin 0 -> 9994 bytes docs/figs/control_arch_static_decoupling.pdf | Bin 0 -> 26157 bytes docs/figs/control_arch_static_decoupling.png | Bin 0 -> 8963 bytes .../figs/control_arch_static_decoupling_K.pdf | Bin 0 -> 26484 bytes .../figs/control_arch_static_decoupling_K.png | Bin 0 -> 8831 bytes docs/figs/general_control_names.pdf | Bin 0 -> 34475 bytes docs/figs/general_control_names.png | Bin 0 -> 11957 bytes 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-> 91333 bytes docs/figs/hac_lac_plant_dvf.png | Bin 0 -> 96891 bytes docs/figs/hac_lac_plant_iff.pdf | Bin 0 -> 92024 bytes docs/figs/hac_lac_plant_iff.png | Bin 0 -> 97390 bytes .../static_decoupling_C_T_frobenius_norm.pdf | Bin 0 -> 99083 bytes .../static_decoupling_C_T_frobenius_norm.png | Bin 0 -> 201052 bytes .../figs/static_decoupling_diagonal_plant.pdf | Bin 0 -> 79679 bytes .../figs/static_decoupling_diagonal_plant.png | Bin 0 -> 104272 bytes docs/identification.html | 75 +- matlab/stewart_platform_model.slx | Bin 52630 -> 61486 bytes org/active-damping.org | 46 +- org/control-study.org | 1186 +++++++++++++++-- org/control-tracking.org | 257 ++++ org/cubic-configuration.org | 4 + org/dynamics-study.org | 2 + org/identification.org | 11 +- simscape_subsystems/Stewart_Platform.slx | Bin 32221 -> 32248 bytes simscape_subsystems/stewart_strut.slx | Bin 37179 -> 37201 bytes src/computeCompliance.m | 2 +- src/computeTransmissibility.m | 6 +- src/initializeController.m | 6 +- 52 files changed, 2706 insertions(+), 334 deletions(-) create mode 100644 docs/control-tracking.html create mode 100644 docs/figs/control_arch_hac_dvf.pdf create mode 100644 docs/figs/control_arch_hac_dvf.png create mode 100644 docs/figs/control_arch_hac_iff.pdf create mode 100644 docs/figs/control_arch_hac_iff.png create mode 100644 docs/figs/control_arch_static_decoupling.pdf create mode 100644 docs/figs/control_arch_static_decoupling.png create mode 100644 docs/figs/control_arch_static_decoupling_K.pdf create mode 100644 docs/figs/control_arch_static_decoupling_K.png create mode 100644 docs/figs/general_control_names.pdf create mode 100644 docs/figs/general_control_names.png create mode 100644 docs/figs/hac_lac_C_T_comparison.pdf create mode 100644 docs/figs/hac_lac_C_T_comparison.png create mode 100644 docs/figs/hac_lac_C_T_dvf.pdf create mode 100644 docs/figs/hac_lac_C_T_dvf.png create mode 100644 docs/figs/hac_lac_C_T_iff.pdf create mode 100644 docs/figs/hac_lac_C_T_iff.png create mode 100644 docs/figs/hac_lac_C_full_comparison.pdf create mode 100644 docs/figs/hac_lac_C_full_comparison.png create mode 100644 docs/figs/hac_lac_T_full_comparison.pdf create mode 100644 docs/figs/hac_lac_T_full_comparison.png create mode 100644 docs/figs/hac_lac_coupling_jacobian.pdf create mode 100644 docs/figs/hac_lac_coupling_jacobian.png create mode 100644 docs/figs/hac_lac_loop_gain_dvf.pdf create mode 100644 docs/figs/hac_lac_loop_gain_dvf.png create mode 100644 docs/figs/hac_lac_loop_gain_iff.pdf create mode 100644 docs/figs/hac_lac_loop_gain_iff.png create mode 100644 docs/figs/hac_lac_plant_dvf.pdf create mode 100644 docs/figs/hac_lac_plant_dvf.png create mode 100644 docs/figs/hac_lac_plant_iff.pdf create mode 100644 docs/figs/hac_lac_plant_iff.png create mode 100644 docs/figs/static_decoupling_C_T_frobenius_norm.pdf create mode 100644 docs/figs/static_decoupling_C_T_frobenius_norm.png create mode 100644 docs/figs/static_decoupling_diagonal_plant.pdf create mode 100644 docs/figs/static_decoupling_diagonal_plant.png create mode 100644 org/control-tracking.org diff --git a/docs/active-damping.html b/docs/active-damping.html index 3daefd6..c597a72 100644 --- a/docs/active-damping.html +++ b/docs/active-damping.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Stewart Platform - Decentralized Active Damping @@ -249,25 +249,25 @@
  • 1. Inertial Control
  • 2. Integral Force Feedback
  • 3. Direct Velocity Feedback
  • 4. Compliance and Transmissibility Comparison @@ -330,6 +330,7 @@ stewart = initializeInertialSensor(stewart, 'type'
    ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
     payload = initializePayload('type', 'none');
    +controller = initializeController('type', 'open-loop');
     
    @@ -365,8 +366,8 @@ The transfer function from actuator forces to force sensors is shown in Figure < -
    -

    1.2 Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics

    +
    +

    1.2 Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics

    We add some stiffness and damping in the flexible joints and we re-identify the dynamics. @@ -402,8 +403,8 @@ The new dynamics from force actuator to force sensor is shown in Figure

    -
    -

    1.3 Obtained Damping

    +
    -
    -

    1.4 Conclusion

    +
    +

    1.4 Conclusion

    @@ -460,8 +461,8 @@ To run the script, open the Simulink Project, and type run active_damping_

    -
    -

    2.1 Identification of the Dynamics with perfect Joints

    +
    +

    2.1 Identification of the Dynamics with perfect Joints

    We first initialize the Stewart platform without joint stiffness. @@ -484,11 +485,7 @@ stewart = initializeInertialSensor(stewart, 'type'

    ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
     payload = initializePayload('type', 'none');
    -
    -
    - -
    -
    controller = initializeController('type', 'open-loop');
    +controller = initializeController('type', 'open-loop');
     
    @@ -496,11 +493,7 @@ payload = initializePayload('type',
    -
    %% Options for Linearized
    -options = linearizeOptions;
    -options.SampleTime = 0;
    -
    -%% Name of the Simulink File
    +
    %% Name of the Simulink File
     mdl = 'stewart_platform_model';
     
     %% Input/Output definition
    @@ -509,7 +502,7 @@ io(io_i) = linio([mdl, '/Controller'],        1,
     io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'Taum'); io_i = io_i + 1; % Force Sensor Outputs [N]
     
     %% Run the linearization
    -G = linearize(mdl, io, options);
    +G = linearize(mdl, io);
     G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
     G.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
     
    @@ -527,15 +520,15 @@ The transfer function from actuator forces to force sensors is shown in Figure <
    -
    -

    2.2 Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics

    +
    +

    2.2 Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics

    We add some stiffness and damping in the flexible joints and we re-identify the dynamics.

    stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical');
    -Gf = linearize(mdl, io, options);
    +Gf = linearize(mdl, io);
     Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
     Gf.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
     
    @@ -546,7 +539,7 @@ We now use the amplified actuators and re-identify the dynamics

    stewart = initializeAmplifiedStrutDynamics(stewart);
    -Ga = linearize(mdl, io, options);
    +Ga = linearize(mdl, io);
     Ga.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
     Ga.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
     
    @@ -564,8 +557,8 @@ The new dynamics from force actuator to force sensor is shown in Figure
    -
    -

    2.3 Obtained Damping

    +
    -
    -

    2.4 Conclusion

    +
    +

    2.4 Conclusion

    @@ -630,8 +623,8 @@ To run the script, open the Simulink Project, and type run active_damping_

    -
    -

    3.1 Identification of the Dynamics with perfect Joints

    +
    +

    3.1 Identification of the Dynamics with perfect Joints

    We first initialize the Stewart platform without joint stiffness. @@ -654,6 +647,7 @@ stewart = initializeInertialSensor(stewart, 'type'

    ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
     payload = initializePayload('type', 'none');
    +controller = initializeController('type', 'open-loop');
     
    @@ -693,8 +687,8 @@ The transfer function from actuator forces to relative motion sensors is shown i
    -
    -

    3.2 Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics

    +
    +

    3.2 Effect of the Flexible Joint stiffness and Actuator amplification on the Dynamics

    We add some stiffness and damping in the flexible joints and we re-identify the dynamics. @@ -730,8 +724,8 @@ The new dynamics from force actuator to relative motion sensor is shown in Figur

    -
    -

    3.3 Obtained Damping

    +
    +

    3.3 Obtained Damping

    The control is a performed in a decentralized manner. @@ -756,8 +750,8 @@ The root locus is shown in figure 10.

    -
    -

    3.4 Conclusion

    +
    +

    3.4 Conclusion

    @@ -799,6 +793,7 @@ The rotation point of the ground is located at the origin of frame \(\{A\}\).

    ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
     payload = initializePayload('type', 'none');
    +controller = initializeController('type', 'open-loop');
     
    @@ -822,7 +817,7 @@ Now, let’s identify the transmissibility and compliance for the Integral F

    controller = initializeController('type', 'iff');
    -G_iff = (2e4/s)*eye(6);
    +K_iff = (1e4/s)*eye(6);
     
     [T_iff, T_norm_iff, ~] = computeTransmissibility();
     [C_iff, C_norm_iff, ~] = computeCompliance();
    @@ -834,7 +829,7 @@ And for the Direct Velocity Feedback.
     

    controller = initializeController('type', 'dvf');
    -G_dvf = 1e4*s/(1+s/2/pi/5000)*eye(6);
    +K_dvf = 1e4*s/(1+s/2/pi/5000)*eye(6);
     
     [T_dvf, T_norm_dvf, ~] = computeTransmissibility();
     [C_dvf, C_norm_dvf, ~] = computeCompliance();
    @@ -872,7 +867,7 @@ G_dvf = 1e4*s/(1
     

    Author: Dehaeze Thomas

    -

    Created: 2020-02-27 jeu. 14:16

    +

    Created: 2020-02-28 ven. 17:33

    diff --git a/docs/control-study.html b/docs/control-study.html index a5c830b..fc2face 100644 --- a/docs/control-study.html +++ b/docs/control-study.html @@ -4,10 +4,10 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + -Stewart Platform - Control Study +Stewart Platform - Vibration Isolation + + + + + + + + + + + +
    + UP + | + HOME +
    +

    Stewart Platform - Tracking Control

    + + +
    +

    1 First Control Architecture

    +
    +
    +
    +

    1.1 Control Schematic

    +
    + +
    +

    control_measure_rotating_2dof.png +

    +
    +
    +
    + +
    +

    1.2 Initialize the Stewart platform

    +
    +
    +
    stewart = initializeStewartPlatform();
    +stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3);
    +stewart = generateGeneralConfiguration(stewart);
    +stewart = computeJointsPose(stewart);
    +stewart = initializeStrutDynamics(stewart);
    +stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p');
    +stewart = initializeCylindricalPlatforms(stewart);
    +stewart = initializeCylindricalStruts(stewart);
    +stewart = computeJacobian(stewart);
    +stewart = initializeStewartPose(stewart);
    +stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3);
    +
    +
    + +
    +
    ground = initializeGround('type', 'none');
    +payload = initializePayload('type', 'none');
    +
    +
    +
    +
    + +
    +

    1.3 Identification of the plant

    +
    +

    +Let’s identify the transfer function from \(\bm{\tau}\) to \(\bm{L}\). +

    +
    +
    %% Name of the Simulink File
    +mdl = 'stewart_platform_model';
    +
    +%% Input/Output definition
    +clear io; io_i = 1;
    +io(io_i) = linio([mdl, '/Controller'],        1, 'openinput');  io_i = io_i + 1; % Actuator Force Inputs [N]
    +io(io_i) = linio([mdl, '/Stewart Platform'],  1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m]
    +
    +%% Run the linearization
    +G = linearize(mdl, io);
    +G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'};
    +
    +
    +
    +
    + +
    +

    1.4 Plant Analysis

    +
    +

    +Diagonal terms +Compare to off-diagonal terms +

    +
    +
    +
    +

    1.5 Controller Design

    +
    +

    +One integrator should be present in the controller. +

    + +

    +A lead is added around the crossover frequency which is set to be around 500Hz. +

    + +
    +
    % wint = 2*pi*100; % Integrate until [rad]
    +% wlead = 2*pi*500; % Location of the lead [rad]
    +% hlead = 2; % Lead strengh
    +
    +% Kl = 1e6 * ... % Gain
    +%      (s + wint)/(s) * ... % Integrator until 100Hz
    +%      (1 + s/(wlead/hlead)/(1 + s/(wlead*hlead))); % Lead
    +
    +wc = 2*pi*100;
    +Kl = 1/abs(freqresp(G(1,1), wc)) * wc/s * 1/(1 + s/(3*wc));
    +Kl = Kl * eye(6);
    +
    +
    +
    +
    +
    +
    +
    +

    Author: Dehaeze Thomas

    +

    Created: 2020-02-28 ven. 17:37

    +
    + + diff --git a/docs/cubic-configuration.html b/docs/cubic-configuration.html index 7c2a76b..ac02ba6 100644 --- a/docs/cubic-configuration.html +++ b/docs/cubic-configuration.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Cubic configuration for the Stewart Platform @@ -252,33 +252,33 @@
  • 1.2. Cubic Stewart platform centered with the cube center - Jacobian not estimated at the cube center
  • 1.3. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the cube center
  • 1.4. Cubic Stewart platform not centered with the cube center - Jacobian estimated at the Stewart platform center
  • -
  • 1.5. Conclusion
  • +
  • 1.5. Conclusion
  • 2. Configuration with the Cube’s center above the mobile platform
  • 3. Cubic size analysis
  • 4. Dynamic Coupling in the Cartesian Frame
  • 5. Dynamic Coupling between actuators and sensors of each strut
  • 6. Functions @@ -826,8 +826,8 @@ stewart = initializeCylindricalPlatforms(stewart, 'Fpr' -
    -

    1.5 Conclusion

    +
    +

    1.5 Conclusion

    @@ -1164,8 +1164,8 @@ FOc = H + MO_B; % Cente

    -
    -

    2.2 Conclusion

    +
    +

    2.2 Conclusion

    @@ -1251,8 +1251,8 @@ We also find that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) are varyi

    -
    -

    3.2 Conclusion

    +
    +

    3.2 Conclusion

    We observe that \(k_{\theta_x} = k_{\theta_y}\) and \(k_{\theta_z}\) increase linearly with the cube size. @@ -1391,6 +1391,7 @@ No flexibility below the Stewart platform and no payload.

    ground = initializeGround('type', 'none');
     payload = initializePayload('type', 'none');
    +controller = initializeController('type', 'open-loop');
     
    @@ -1535,6 +1536,7 @@ No flexibility below the Stewart platform and no payload.
    ground = initializeGround('type', 'none');
     payload = initializePayload('type', 'none');
    +controller = initializeController('type', 'open-loop');
     
    @@ -1607,8 +1609,8 @@ This was expected as the mass matrix is not diagonal (the Center of Mass of the
    -
    -

    4.3 Conclusion

    +
    +

    4.3 Conclusion

    @@ -1693,6 +1695,7 @@ No flexibility below the Stewart platform and no payload.

    ground = initializeGround('type', 'none');
     payload = initializePayload('type', 'none');
    +controller = initializeController('type', 'open-loop');
     
    @@ -1760,6 +1763,7 @@ No flexibility below the Stewart platform and no payload.
    ground = initializeGround('type', 'none');
     payload = initializePayload('type', 'none');
    +controller = initializeController('type', 'open-loop');
     
    @@ -1790,8 +1794,8 @@ And we identify the dynamics from the actuator forces \(\tau_{i}\) to the relati
    -
    -

    5.3 Conclusion

    +
    +

    5.3 Conclusion

    @@ -1962,7 +1966,7 @@ stewart.platform_M.Mb = Mb;

    Author: Dehaeze Thomas

    -

    Created: 2020-02-27 jeu. 14:16

    +

    Created: 2020-02-28 ven. 17:34

    diff --git a/docs/dynamics-study.html b/docs/dynamics-study.html index 8e273f8..6efc38f 100644 --- a/docs/dynamics-study.html +++ b/docs/dynamics-study.html @@ -4,7 +4,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Stewart Platform - Dynamics Study @@ -250,13 +250,13 @@
  • 2. Comparison of the static transfer function and the Compliance matrix
  • @@ -299,6 +299,7 @@ We also don’t put any payload on top of the Stewart platform.
    ground = initializeGround('type', 'none');
     payload = initializePayload('type', 'none');
    +controller = initializeController('type', 'open-loop');
     
    @@ -441,8 +442,8 @@ And thus \(\mathcal{F}_{x}\) and \(\mathcal{F}_{x,\text{ext}}\) have clearly -
    -

    1.3 Conclusion

    +
    +

    1.3 Conclusion

    @@ -489,6 +490,7 @@ No flexibility below the Stewart platform and no payload.

    ground = initializeGround('type', 'none');
     payload = initializePayload('type', 'none');
    +controller = initializeController('type', 'open-loop');
     
    @@ -675,8 +677,8 @@ And now at the Compliance matrix.
    -
    -

    2.2 Conclusion

    +
    +

    2.2 Conclusion

    @@ -690,7 +692,7 @@ The low frequency transfer function matrix from \(\mathcal{\bm{F}}\) to \(\mathc

    Author: Dehaeze Thomas

    -

    Created: 2020-02-27 jeu. 14:16

    +

    Created: 2020-02-28 ven. 17:34

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    @@ -608,8 +609,8 @@ Save the movie of the mode shape.

    -
    -

    2.1 Initialize the Stewart platform

    +
    +

    2.1 Initialize the Stewart platform

    stewart = initializeStewartPlatform();
    @@ -632,6 +633,7 @@ We set the rotation point of the ground to be at the same point at frames \(\{A\
     
    ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A);
     payload = initializePayload('type', 'rigid');
    +controller = initializeController('type', 'open-loop');
     
    @@ -729,8 +731,8 @@ plot(freqs, Gamma)

    -
    -

    3.1 Initialize the Stewart platform

    +
    +

    3.1 Initialize the Stewart platform

    stewart = initializeStewartPlatform();
    @@ -753,6 +755,7 @@ We set the rotation point of the ground to be at the same point at frames \(\{A\
     
    ground = initializeGround('type', 'none');
     payload = initializePayload('type', 'rigid');
    +controller = initializeController('type', 'open-loop');
     
    @@ -841,9 +844,9 @@ plot(freqs, C_norm)

    -
    -

    Function description

    -
    +
    +

    Function description

    +
    function [T, T_norm, freqs] = computeTransmissibility(args)
     % computeTransmissibility -
    @@ -864,9 +867,9 @@ plot(freqs, C_norm)
     
    -
    -

    Optional Parameters

    -
    +
    +

    Optional Parameters

    +
    arguments
       args.plots logical {mustBeNumericOrLogical} = false
    @@ -896,7 +899,7 @@ mdl = 'stewart_platform_model';
     %% Input/Output definition
     clear io; io_i = 1;
     io(io_i) = linio([mdl, '/Disturbances/D_w'],        1, 'openinput');  io_i = io_i + 1; % Base Motion [m, rad]
    -io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
    +io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]
     
     %% Run the linearization
     T = linearize(mdl, io, options);
    @@ -935,17 +938,17 @@ If wanted, the 6x6 transmissibility matrix is plotted.
       han = axes(fig, 'visible', 'off');
       han.XLabel.Visible = 'on';
       han.YLabel.Visible = 'on';
    -  ylabel(han, 'Frequency [Hz]');
    -  xlabel(han, 'Transmissibility [m/m]');
    +  xlabel(han, 'Frequency [Hz]');
    +  ylabel(han, 'Transmissibility [m/m]');
     end
     
    -
    -

    Computation of the Frobenius norm

    -
    +
    +

    Computation of the Frobenius norm

    +
    T_norm = zeros(length(freqs), 1);
     
    @@ -982,9 +985,9 @@ If wanted, the 6x6 transmissibility matrix is plotted.
     

    -
    -

    Function description

    -
    +
    +

    Function description

    +
    function [C, C_norm, freqs] = computeCompliance(args)
     % computeCompliance -
    @@ -1005,9 +1008,9 @@ If wanted, the 6x6 transmissibility matrix is plotted.
     
    -
    -

    Optional Parameters

    -
    +
    +

    Optional Parameters

    +
    arguments
       args.plots logical {mustBeNumericOrLogical} = false
    @@ -1037,7 +1040,7 @@ mdl = 'stewart_platform_model';
     %% Input/Output definition
     clear io; io_i = 1;
     io(io_i) = linio([mdl, '/Disturbances/F_ext'],      1, 'openinput');  io_i = io_i + 1; % External forces [N, N*m]
    -io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad]
    +io(io_i) = linio([mdl, '/Absolute Motion Sensor'],  1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad]
     
     %% Run the linearization
     C = linearize(mdl, io, options);
    @@ -1083,9 +1086,9 @@ If wanted, the 6x6 transmissibility matrix is plotted.
     
    -
    -

    Computation of the Frobenius norm

    -
    +
    +

    Computation of the Frobenius norm

    +
    freqs = args.freqs;
     
    @@ -1114,7 +1117,7 @@ C_norm = zeros(length(freqs), 1);
     

    Author: Dehaeze Thomas

    -

    Created: 2020-02-27 jeu. 14:16

    +

    Created: 2020-02-28 ven. 17:34

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z(`$rcOX65*>*8JQ4YiQy$z!%?8ol#~zwH2oaG>pCMgAPPlsSJ0P#p|NxerpPtju{6 zbkejrTX4hVdoH?{bMpidy-R3U^vmBIII#Lc>u2KTEmv74;vE{$DSj-bd_4jY;P8uw zEI(RF?Covt1)U!w%nnuD9~o`#TO|x3-96>k@67vl+;1vu<>=C4j3&ap*AkbV=vt<^ zix<(08Tt&wD4uxMzEF(NbqEZs9BkgX{)Y6+a|_zEsN#OUm)R`h@Ow`5L)__b-xHXv zf-v-LC8`Rf?#$2KCyBL?P)S-i)&9zewm42D_DhrDFU zci(-h3jto;(G3zDQ{dOkXHua62CO2-h_}W>iARYya>k;xK*2aBDFz3)0rgUGy<-^+ z4^iAu(>%P6f9w67Lj6XqKWMfnVhU?74|@4KuMNi^uw5B50!uusKdULiz=A+{FC_$I z(BHj8TCsCK4`DzcHc0p`@85z!$1Yfe>;+`%X6@w8!T!(n|JVAbmSE#iU1R*B)cOJg zzu5Yxz1+UEPB_Ru92taj5B;w#i9Il^76K#~5e#A7!~QFZdJ$hf z`isg!!eaSvZ`A}JU^b~ipnDn+h~Tf6|93HjsO{5H{u9w}twt-qmuPqqLI3dL{%!w> z$ejg}ywC90b=N)^mX;l|w9oWcr!v+}I_3o=g#-f8{!52|3nIr!4^ce$PZkCTVAxG= z2rD-OB>MoJ;vaul-pvp#WDqEn>gAaAf9Wnl1`g=|V*ZP(_y3E@3W9ma_%|e>Lon<` zAcXY`8S8(i{Xe_BGXLAfZ488~m>AN0i2gSZQ=oUv_r-eNODFkPEQe(;Ks?CpA>Lo$ z|JL?FAc}thv&tbWP%4P>5jyfe+5IiDnI93rYS%z~kC^`||8ME(zh}$479uiC4~aYZ zPdHiRWe@r<)`MO;)4#l+*FnaHSs;?f|Cv_RV=!!a9mHjX8M1fuA6@tuM|8Cp9m|(a z`7d3CdI;ty_5T3lUFyE7z5vW#`oB||tA}8pu>GA}$rCWFRRhHMMC@+{|E_@j$NJ0T Ze`#2aFZD2LOyZYU$M)rAAcp=U{6F> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes +<> +#+end_src + +#+begin_src matlab + simulinkproject('../'); +#+end_src + +#+begin_src matlab + open('stewart_platform_model.slx') +#+end_src + ** Initialization We first initialize the Stewart platform without joint stiffness. #+begin_src matlab @@ -798,6 +811,7 @@ The rotation point of the ground is located at the origin of frame $\{A\}$. #+begin_src matlab ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A); payload = initializePayload('type', 'none'); + controller = initializeController('type', 'open-loop'); #+end_src ** Identification @@ -811,7 +825,7 @@ Let's first identify the transmissibility and compliance in the open-loop case. Now, let's identify the transmissibility and compliance for the Integral Force Feedback architecture. #+begin_src matlab controller = initializeController('type', 'iff'); - G_iff = (2e4/s)*eye(6); + K_iff = (1e4/s)*eye(6); [T_iff, T_norm_iff, ~] = computeTransmissibility(); [C_iff, C_norm_iff, ~] = computeCompliance(); @@ -820,7 +834,7 @@ Now, let's identify the transmissibility and compliance for the Integral Force F And for the Direct Velocity Feedback. #+begin_src matlab controller = initializeController('type', 'dvf'); - G_dvf = 1e4*s/(1+s/2/pi/5000)*eye(6); + K_dvf = 1e4*s/(1+s/2/pi/5000)*eye(6); [T_dvf, T_norm_dvf, ~] = computeTransmissibility(); [C_dvf, C_norm_dvf, ~] = computeCompliance(); @@ -857,8 +871,8 @@ And for the Direct Velocity Feedback. han = axes(fig, 'visible', 'off'); han.XLabel.Visible = 'on'; han.YLabel.Visible = 'on'; - ylabel(han, 'Frequency [Hz]'); - xlabel(han, 'Transmissibility'); + xlabel(han, 'Frequency [Hz]'); + ylabel(han, 'Transmissibility'); #+end_src #+header: :tangle no :exports results :results none :noweb yes @@ -900,8 +914,8 @@ And for the Direct Velocity Feedback. han = axes(fig, 'visible', 'off'); han.XLabel.Visible = 'on'; han.YLabel.Visible = 'on'; - ylabel(han, 'Frequency [Hz]'); - xlabel(han, 'Compliance'); + xlabel(han, 'Frequency [Hz]'); + ylabel(han, 'Compliance'); #+end_src #+header: :tangle no :exports results :results none :noweb yes diff --git a/org/control-study.org b/org/control-study.org index 7638365..913f7ac 100644 --- a/org/control-study.org +++ b/org/control-study.org @@ -1,4 +1,4 @@ -#+TITLE: Stewart Platform - Control Study +#+TITLE: Stewart Platform - Vibration Isolation :DRAWER: #+STARTUP: overview @@ -38,7 +38,81 @@ #+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") :END: -* First Control Architecture +* HAC-LAC (Cascade) Control - Integral Control +** Introduction +In this section, we wish to study the use of the High Authority Control - Low Authority Control (HAC-LAC) architecture on the Stewart platform. + +The control architectures are shown in Figures [[fig:control_arch_hac_iff]] and [[fig:control_arch_hac_dvf]]. + +First, the LAC loop is closed (the LAC control is described [[file:active-damping.org][here]]), and then the HAC controller is designed and the outer loop is closed. + +#+begin_src latex :file control_arch_hac_iff.pdf + \begin{tikzpicture} + % Blocs + \node[block={2.0cm}{2.0cm}] (P) {}; + \node[above] at (P.north) {Plant}; + \node[block, below=0.7 of P] (Kiff) {$\bm{K}_\text{IFF}$}; + \node[block, below=0.7 of Kiff] (Khac) {$\bm{K}_\text{HAC}$}; + + % Add + \node[addb, left=1 of P] (add) {}; + + \node[block, left=1 of add] (J) {$\bm{J}^{-T}$}; + + % Input and outputs coordinates + \coordinate[] (outputhac) at ($(P.south east)!0.75!(P.north east)$); + \coordinate[] (outputiff) at ($(P.south east)!0.25!(P.north east)$); + + \draw[->] (outputiff) node[above right]{$\bm{\tau}_m$} -- ++(0.8, 0) |- (Kiff.east); + \draw[->] (outputhac) node[above right]{$\bm{\mathcal{X}}$} -- ++(1.6, 0) |- (Khac.east); + + \draw[->] (Kiff.west) -| (add.south); + + \draw[->] (J.east) -- (add.west); + \draw[<-] (J.west) node[above left]{$\bm{\mathcal{F}}$} -- ++(-0.8, 0) |- (Khac.west); + \draw[->] (add.east) -- (P.west) node[above left]{$\bm{\tau}$}; + \end{tikzpicture} +#+end_src + +#+name: fig:control_arch_hac_iff +#+caption: HAC-LAC architecture with IFF +#+RESULTS: +[[file:figs/control_arch_hac_iff.png]] + + +#+begin_src latex :file control_arch_hac_dvf.pdf + \begin{tikzpicture} + % Blocs + \node[block={2.0cm}{2.0cm}] (P) {}; + \node[above] at (P.north) {Plant}; + \node[block, below=0.7 of P] (Kdvf) {$\bm{K}_\text{DVF}$}; + \node[block, below=0.7 of Kdvf] (Khac) {$\bm{K}_\text{HAC}$}; + + % Add + \node[addb, left=1 of P] (add) {}; + + \node[block, left=1 of add] (J) {$\bm{J}^{-T}$}; + + % Input and outputs coordinates + \coordinate[] (outputhac) at ($(P.south east)!0.75!(P.north east)$); + \coordinate[] (outputdvf) at ($(P.south east)!0.25!(P.north east)$); + + \draw[->] (outputdvf) node[above right]{$\delta \bm{\mathcal{L}}_m$} -- ++(0.8, 0) |- (Kdvf.east); + \draw[->] (outputhac) node[above right]{$\bm{\mathcal{X}}$} -- ++(1.6, 0) |- (Khac.east); + + \draw[->] (Kdvf.west) -| (add.south); + + \draw[->] (J.east) -- (add.west); + \draw[<-] (J.west) node[above left]{$\bm{\mathcal{F}}$} -- ++(-0.8, 0) |- (Khac.west); + \draw[->] (add.east) -- (P.west) node[above left]{$\bm{\tau}$}; + \end{tikzpicture} +#+end_src + +#+name: fig:control_arch_hac_dvf +#+caption: HAC-LAC architecture with DVF +#+RESULTS: +[[file:figs/control_arch_hac_dvf.png]] + ** Matlab Init :noexport: #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> @@ -52,169 +126,189 @@ simulinkproject('../'); #+end_src -** Control Schematic -#+begin_src latex :file control_measure_rotating_2dof.pdf - \begin{tikzpicture} - % Blocs - \node[block] (J) at (0, 0) {$J$}; - \node[addb={+}{}{}{}{-}, right=1 of J] (subr) {}; - \node[block, right=0.8 of subr] (K) {$K_{L}$}; - \node[block, right=1 of K] (G) {$G_{L}$}; - - % Connections and labels - \draw[<-] (J.west)node[above left]{$\bm{r}_{n}$} -- ++(-1, 0); - \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{L}$}; - \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{L}$}; - \draw[->] (K.east) -- (G.west) node[above left]{$\bm{\tau}$}; - \draw[->] (G.east) node[above right]{$\bm{L}$} -| ($(G.east)+(1, -1)$) -| (subr.south); - \end{tikzpicture} +#+begin_src matlab + open('stewart_platform_model.slx') #+end_src -#+RESULTS: -[[file:figs/control_measure_rotating_2dof.png]] - -** Initialize the Stewart platform +** Initialization +We first initialize the Stewart platform. #+begin_src matlab stewart = initializeStewartPlatform(); stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3); stewart = generateGeneralConfiguration(stewart); stewart = computeJointsPose(stewart); stewart = initializeStrutDynamics(stewart); - stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p'); + stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical'); stewart = initializeCylindricalPlatforms(stewart); stewart = initializeCylindricalStruts(stewart); stewart = computeJacobian(stewart); stewart = initializeStewartPose(stewart); - stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3); + stewart = initializeInertialSensor(stewart, 'type', 'none'); #+end_src +The rotation point of the ground is located at the origin of frame $\{A\}$. #+begin_src matlab - ground = initializeGround('type', 'none'); + ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A); payload = initializePayload('type', 'none'); #+end_src -** Identification of the plant -Let's identify the transfer function from $\bm{\tau}}$ to $\bm{L}$. -#+begin_src matlab - %% Options for Linearized - options = linearizeOptions; - options.SampleTime = 0; +** Identification +*** Introduction :ignore: +We identify the transfer function from the actuator forces $\bm{\tau}$ to the absolute displacement of the mobile platform $\bm{\mathcal{X}}$ in three different cases: +- Open Loop plant +- Already damped plant using Integral Force Feedback +- Already damped plant using Direct velocity feedback +*** HAC - Without LAC +#+begin_src matlab + controller = initializeController('type', 'open-loop'); +#+end_src + +#+begin_src matlab %% Name of the Simulink File mdl = 'stewart_platform_model'; %% Input/Output definition clear io; io_i = 1; - io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N] - io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m] + io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N] + io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad] %% Run the linearization - G = linearize(mdl, io, options); - G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; - G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'}; + G_ol = linearize(mdl, io); + G_ol.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; + G_ol.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; #+end_src -** Plant Analysis -Diagonal terms -#+begin_src matlab :exports none - freqs = logspace(1, 4, 1000); - - figure; - - ax1 = subplot(2, 1, 1); - hold on; - for i = 1:6 - plot(freqs, abs(squeeze(freqresp(G(i, i), freqs, 'Hz')))); - end - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); - - ax2 = subplot(2, 1, 2); - hold on; - for i = 1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, i), freqs, 'Hz')))); - end - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); - ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); - ylim([-180, 180]); - yticks([-180, -90, 0, 90, 180]); - - linkaxes([ax1,ax2],'x'); +*** HAC - IFF +#+begin_src matlab + controller = initializeController('type', 'iff'); + K_iff = -(1e4/s)*eye(6); #+end_src -Compare to off-diagonal terms -#+begin_src matlab :exports none - freqs = logspace(1, 4, 1000); - - figure; - - ax1 = subplot(2, 1, 1); - hold on; - for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end - end - set(gca,'ColorOrderIndex',1); - plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); - - ax2 = subplot(2, 1, 2); - hold on; - for i = 1:5 - for j = i+1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end - end - set(gca,'ColorOrderIndex',1); - plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz')))); - hold off; - set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); - ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); - ylim([-180, 180]); - yticks([-180, -90, 0, 90, 180]); - - linkaxes([ax1,ax2],'x'); -#+end_src - -** Controller Design -One integrator should be present in the controller. - -A lead is added around the crossover frequency which is set to be around 500Hz. - #+begin_src matlab - % wint = 2*pi*100; % Integrate until [rad] - % wlead = 2*pi*500; % Location of the lead [rad] - % hlead = 2; % Lead strengh + %% Name of the Simulink File + mdl = 'stewart_platform_model'; - % Kl = 1e6 * ... % Gain - % (s + wint)/(s) * ... % Integrator until 100Hz - % (1 + s/(wlead/hlead)/(1 + s/(wlead*hlead))); % Lead + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N] + io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad] - wc = 2*pi*100; - Kl = 1/abs(freqresp(G(1,1), wc)) * wc/s * 1/(1 + s/(3*wc)); - Kl = Kl * eye(6); + %% Run the linearization + G_iff = linearize(mdl, io); + G_iff.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; + G_iff.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; #+end_src +*** HAC - DVF +#+begin_src matlab + controller = initializeController('type', 'dvf'); + K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6); +#+end_src + +#+begin_src matlab + %% Name of the Simulink File + mdl = 'stewart_platform_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N] + io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad] + + %% Run the linearization + G_dvf = linearize(mdl, io); + G_dvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; + G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +#+end_src + +** Control Architecture +We use the Jacobian to express the actuator forces in the cartesian frame, and thus we obtain the transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$. + +#+begin_src matlab + Gc_ol = minreal(G_ol)/stewart.kinematics.J'; + Gc_ol.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + + Gc_iff = minreal(G_iff)/stewart.kinematics.J'; + Gc_iff.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + + Gc_dvf = minreal(G_dvf)/stewart.kinematics.J'; + Gc_dvf.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; +#+end_src + +We then design a controller based on the transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$, finally, we will pre-multiply the controller by $\bm{J}^{-T}$. + +** 6x6 Plant Comparison #+begin_src matlab :exports none - freqs = logspace(1, 3, 1000); + p_handle = zeros(6*6,1); + + fig = figure; + for ix = 1:6 + for iy = 1:6 + p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy); + hold on; + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gc_ol(ix, iy), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Gc_iff(ix, iy), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',3); + plot(freqs, abs(squeeze(freqresp(Gc_dvf(ix, iy), freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + if ix < 6 + xticklabels({}); + end + if iy > 1 + yticklabels({}); + end + end + end + + linkaxes(p_handle, 'xy') + xlim([freqs(1), freqs(end)]); + ylim([1e-9 1e-3]); + + han = axes(fig, 'visible', 'off'); + han.XLabel.Visible = 'on'; + han.YLabel.Visible = 'on'; + xlabel(han, 'Frequency [Hz]'); + ylabel(han, 'Plant'); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hac_lac_coupling_jacobian.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:hac_lac_coupling_jacobian +#+caption: Norm of the transfer functions from $\bm{\mathcal{F}}$ to $\bm{\mathcal{X}}$ ([[./figs/hac_lac_coupling_jacobian.png][png]], [[./figs/hac_lac_coupling_jacobian.pdf][pdf]]) +[[file:figs/hac_lac_coupling_jacobian.png]] + +** HAC - DVF +*** Plant +#+begin_src matlab :exports none + freqs = logspace(1, 4, 1000); figure; ax1 = subplot(2, 1, 1); hold on; - plot(freqs, abs(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_dvf('Dx', 'Fx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_dvf('Dy', 'Fy'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_dvf('Dz', 'Fz'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_dvf('Rx', 'Mx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_dvf('Ry', 'My'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_dvf('Rz', 'Mz'), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); - ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); ax2 = subplot(2, 1, 2); hold on; - plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz')))); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_dvf('Dx', 'Fx'), freqs, 'Hz'))), 'DisplayName', 'Dx/Fx'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_dvf('Dy', 'Fy'), freqs, 'Hz'))), 'DisplayName', 'Dy/Fy'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_dvf('Dz', 'Fz'), freqs, 'Hz'))), 'DisplayName', 'Dz/Fz'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_dvf('Rx', 'Mx'), freqs, 'Hz'))), 'DisplayName', 'Rx/Mx'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_dvf('Ry', 'My'), freqs, 'Hz'))), 'DisplayName', 'Ry/My'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_dvf('Rz', 'Mz'), freqs, 'Hz'))), 'DisplayName', 'Rz/Mz'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); @@ -222,6 +316,29 @@ A lead is added around the crossover frequency which is set to be around 500Hz. yticks([-180, -90, 0, 90, 180]); linkaxes([ax1,ax2],'x'); + legend('location', 'northeast'); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hac_lac_plant_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:hac_lac_plant_dvf +#+caption: Diagonal elements of the plant for HAC control when DVF is previously applied ([[./figs/hac_lac_plant_dvf.png][png]], [[./figs/hac_lac_plant_dvf.pdf][pdf]]) +[[file:figs/hac_lac_plant_dvf.png]] + +*** Controller Design +We design a diagonal controller with equal bandwidth for the 6 terms. +The controller is a pure integrator with a small lead near the crossover. + +#+begin_src matlab + wc = 2*pi*300; % Wanted Bandwidth [rad/s] + + h = 1.2; + H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h)); + + Kd_dvf = diag(1./abs(diag(freqresp(1/s*Gc_dvf, wc)))) .* H_lead .* 1/s; #+end_src #+begin_src matlab :exports none @@ -231,26 +348,730 @@ A lead is added around the crossover frequency which is set to be around 500Hz. ax1 = subplot(2, 1, 1); hold on; - for i = 1:5 - for j = i+1:6 - plot(freqs, abs(squeeze(freqresp(Kl(i,i)*G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); + plot(freqs, abs(squeeze(freqresp(Kd_dvf(1,1)*Gc_dvf('Dx', 'Fx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Kd_dvf(2,2)*Gc_dvf('Dy', 'Fy'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Kd_dvf(3,3)*Gc_dvf('Dz', 'Fz'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Kd_dvf(4,4)*Gc_dvf('Rx', 'Mx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Kd_dvf(5,5)*Gc_dvf('Ry', 'My'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Kd_dvf(6,6)*Gc_dvf('Rz', 'Mz'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_dvf(1,1)*Gc_dvf('Dx', 'Fx'), freqs, 'Hz'))), 'DisplayName', 'Dx/Fx'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_dvf(2,2)*Gc_dvf('Dy', 'Fy'), freqs, 'Hz'))), 'DisplayName', 'Dy/Fy'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_dvf(3,3)*Gc_dvf('Dz', 'Fz'), freqs, 'Hz'))), 'DisplayName', 'Dz/Fz'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_dvf(4,4)*Gc_dvf('Rx', 'Mx'), freqs, 'Hz'))), 'DisplayName', 'Rx/Mx'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_dvf(5,5)*Gc_dvf('Ry', 'My'), freqs, 'Hz'))), 'DisplayName', 'Ry/My'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_dvf(6,6)*Gc_dvf('Rz', 'Mz'), freqs, 'Hz'))), 'DisplayName', 'Rz/Mz'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); + legend('location', 'northeast'); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hac_lac_loop_gain_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:hac_lac_loop_gain_dvf +#+caption: Diagonal elements of the Loop Gain for the HAC control ([[./figs/hac_lac_loop_gain_dvf.png][png]], [[./figs/hac_lac_loop_gain_dvf.pdf][pdf]]) +[[file:figs/hac_lac_loop_gain_dvf.png]] + + +Finally, we pre-multiply the diagonal controller by $\bm{J}^{-T}$ prior implementation. +#+begin_src matlab + K_hac_dvf = inv(stewart.kinematics.J')*Kd_dvf; +#+end_src + +*** Obtained Performance +We identify the transmissibility and compliance of the system. + +#+begin_src matlab + controller = initializeController('type', 'open-loop'); + [T_ol, T_norm_ol, freqs] = computeTransmissibility(); + [C_ol, C_norm_ol, ~] = computeCompliance(); +#+end_src + +#+begin_src matlab + controller = initializeController('type', 'dvf'); + [T_dvf, T_norm_dvf, ~] = computeTransmissibility(); + [C_dvf, C_norm_dvf, ~] = computeCompliance(); +#+end_src + +#+begin_src matlab + controller = initializeController('type', 'hac-dvf'); + [T_hac_dvf, T_norm_hac_dvf, ~] = computeTransmissibility(); + [C_hac_dvf, C_norm_hac_dvf, ~] = computeCompliance(); +#+end_src + +#+begin_src matlab :exports none + figure; + + subplot(1,2,1); + hold on; + plot(freqs, T_norm_ol) + plot(freqs, T_norm_dvf) + plot(freqs, T_norm_hac_dvf) + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Transmissibility - Frobenius Norm'); + + subplot(1,2,2); + hold on; + plot(freqs, C_norm_ol, 'DisplayName', 'OL') + plot(freqs, C_norm_dvf, 'DisplayName', 'DVF') + plot(freqs, C_norm_hac_dvf, 'DisplayName', 'HAC-DVF') + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Compliance - Frobenius Norm'); + legend(); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hac_lac_C_T_dvf.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:hac_lac_C_T_dvf +#+caption: Obtained Compliance and Transmissibility ([[./figs/hac_lac_C_T_dvf.png][png]], [[./figs/hac_lac_C_T_dvf.pdf][pdf]]) +[[file:figs/hac_lac_C_T_dvf.png]] + +** HAC - IFF +*** Plant +#+begin_src matlab :exports none + freqs = logspace(1, 4, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(Gc_iff('Dx', 'Fx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_iff('Dy', 'Fy'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_iff('Dz', 'Fz'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_iff('Rx', 'Mx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_iff('Ry', 'My'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Gc_iff('Rz', 'Mz'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_iff('Dx', 'Fx'), freqs, 'Hz'))), 'DisplayName', 'Dx/Fx'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_iff('Dy', 'Fy'), freqs, 'Hz'))), 'DisplayName', 'Dy/Fy'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_iff('Dz', 'Fz'), freqs, 'Hz'))), 'DisplayName', 'Dz/Fz'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_iff('Rx', 'Mx'), freqs, 'Hz'))), 'DisplayName', 'Rx/Mx'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_iff('Ry', 'My'), freqs, 'Hz'))), 'DisplayName', 'Ry/My'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gc_iff('Rz', 'Mz'), freqs, 'Hz'))), 'DisplayName', 'Rz/Mz'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); + legend('location', 'northeast'); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hac_lac_plant_iff.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:hac_lac_plant_iff +#+caption: Diagonal elements of the plant for HAC control when IFF is previously applied ([[./figs/hac_lac_plant_iff.png][png]], [[./figs/hac_lac_plant_iff.pdf][pdf]]) +[[file:figs/hac_lac_plant_iff.png]] + +*** Controller Design +We design a diagonal controller with equal bandwidth for the 6 terms. +The controller is a pure integrator with a small lead near the crossover. + +#+begin_src matlab + wc = 2*pi*300; % Wanted Bandwidth [rad/s] + + h = 1.2; + H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h)); + + Kd_iff = diag(1./abs(diag(freqresp(1/s*Gc_iff, wc)))) .* H_lead .* 1/s; +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(1, 4, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(Kd_iff(1,1)*Gc_iff('Dx', 'Fx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Kd_iff(2,2)*Gc_iff('Dy', 'Fy'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Kd_iff(3,3)*Gc_iff('Dz', 'Fz'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Kd_iff(4,4)*Gc_iff('Rx', 'Mx'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Kd_iff(5,5)*Gc_iff('Ry', 'My'), freqs, 'Hz')))); + plot(freqs, abs(squeeze(freqresp(Kd_iff(6,6)*Gc_iff('Rz', 'Mz'), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_iff(1,1)*Gc_iff('Dx', 'Fx'), freqs, 'Hz'))), 'DisplayName', 'Dx/Fx'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_iff(2,2)*Gc_iff('Dy', 'Fy'), freqs, 'Hz'))), 'DisplayName', 'Dy/Fy'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_iff(3,3)*Gc_iff('Dz', 'Fz'), freqs, 'Hz'))), 'DisplayName', 'Dz/Fz'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_iff(4,4)*Gc_iff('Rx', 'Mx'), freqs, 'Hz'))), 'DisplayName', 'Rx/Mx'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_iff(5,5)*Gc_iff('Ry', 'My'), freqs, 'Hz'))), 'DisplayName', 'Ry/My'); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kd_iff(6,6)*Gc_iff('Rz', 'Mz'), freqs, 'Hz'))), 'DisplayName', 'Rz/Mz'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); + legend('location', 'northeast'); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hac_lac_loop_gain_iff.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:hac_lac_loop_gain_iff +#+caption: Diagonal elements of the Loop Gain for the HAC control ([[./figs/hac_lac_loop_gain_iff.png][png]], [[./figs/hac_lac_loop_gain_iff.pdf][pdf]]) +[[file:figs/hac_lac_loop_gain_iff.png]] + + +Finally, we pre-multiply the diagonal controller by $\bm{J}^{-T}$ prior implementation. +#+begin_src matlab + K_hac_iff = inv(stewart.kinematics.J')*Kd_iff; +#+end_src + +*** Obtained Performance +We identify the transmissibility and compliance of the system. + +#+begin_src matlab + controller = initializeController('type', 'open-loop'); + [T_ol, T_norm_ol, freqs] = computeTransmissibility(); + [C_ol, C_norm_ol, ~] = computeCompliance(); +#+end_src + +#+begin_src matlab + controller = initializeController('type', 'iff'); + [T_iff, T_norm_iff, ~] = computeTransmissibility(); + [C_iff, C_norm_iff, ~] = computeCompliance(); +#+end_src + +#+begin_src matlab + controller = initializeController('type', 'hac-iff'); + [T_hac_iff, T_norm_hac_iff, ~] = computeTransmissibility(); + [C_hac_iff, C_norm_hac_iff, ~] = computeCompliance(); +#+end_src + +#+begin_src matlab :exports none + figure; + + subplot(1,2,1); + hold on; + plot(freqs, T_norm_ol) + plot(freqs, T_norm_iff) + plot(freqs, T_norm_hac_iff) + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Transmissibility - Frobenius Norm'); + + subplot(1,2,2); + hold on; + plot(freqs, C_norm_ol, 'DisplayName', 'OL') + plot(freqs, C_norm_iff, 'DisplayName', 'IFF') + plot(freqs, C_norm_hac_iff, 'DisplayName', 'HAC-IFF') + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Compliance - Frobenius Norm'); + legend(); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hac_lac_C_T_iff.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:hac_lac_C_T_iff +#+caption: Obtained Compliance and Transmissibility ([[./figs/hac_lac_C_T_iff.png][png]], [[./figs/hac_lac_C_T_iff.pdf][pdf]]) +[[file:figs/hac_lac_C_T_iff.png]] + +** Comparison +#+begin_src matlab :exports none + p_handle = zeros(6*6,1); + + fig = figure; + for ix = 1:6 + for iy = 1:6 + p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy); + hold on; + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(C_ol(ix, iy), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(C_hac_dvf(ix, iy), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',3); + plot(freqs, abs(squeeze(freqresp(C_hac_iff(ix, iy), freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + if ix < 6 + xticklabels({}); + end + if iy > 1 + yticklabels({}); + end end end - set(gca,'ColorOrderIndex',1); - plot(freqs, abs(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz')))); + + linkaxes(p_handle, 'xy') + ylim([1e-10, 1e-3]); + xlim([freqs(1), freqs(end)]); + + han = axes(fig, 'visible', 'off'); + han.XLabel.Visible = 'on'; + han.YLabel.Visible = 'on'; + xlabel(han, 'Frequency [Hz]'); + ylabel(han, 'Compliance'); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hac_lac_C_full_comparison.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:hac_lac_C_full_comparison +#+caption: Comparison of the norm of the Compliance matrices for the HAC-LAC architecture ([[./figs/hac_lac_C_full_comparison.png][png]], [[./figs/hac_lac_C_full_comparison.pdf][pdf]]) +[[file:figs/hac_lac_C_full_comparison.png]] + +#+begin_src matlab :exports none + p_handle = zeros(6*6,1); + + fig = figure; + for ix = 1:6 + for iy = 1:6 + p_handle((ix-1)*6 + iy) = subplot(6, 6, (ix-1)*6 + iy); + hold on; + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(T_ol(ix, iy), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(T_hac_dvf(ix, iy), freqs, 'Hz')))); + set(gca,'ColorOrderIndex',3); + plot(freqs, abs(squeeze(freqresp(T_hac_iff(ix, iy), freqs, 'Hz')))); + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + if ix < 6 + xticklabels({}); + end + if iy > 1 + yticklabels({}); + end + end + end + + linkaxes(p_handle, 'xy') + ylim([1e-5, 10]); + xlim([freqs(1), freqs(end)]); + + han = axes(fig, 'visible', 'off'); + han.XLabel.Visible = 'on'; + han.YLabel.Visible = 'on'; + xlabel(han, 'Frequency [Hz]'); + ylabel(han, 'Transmissibility'); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hac_lac_T_full_comparison.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:hac_lac_T_full_comparison +#+caption: Comparison of the norm of the Transmissibility matrices for the HAC-LAC architecture ([[./figs/hac_lac_T_full_comparison.png][png]], [[./figs/hac_lac_T_full_comparison.pdf][pdf]]) +[[file:figs/hac_lac_T_full_comparison.png]] + +#+begin_src matlab :exports none + figure; + + subplot(1,2,1); + hold on; + plot(freqs, T_norm_ol) + plot(freqs, T_norm_hac_dvf) + plot(freqs, T_norm_hac_iff) + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Transmissibility - Frobenius Norm'); + + subplot(1,2,2); + hold on; + plot(freqs, C_norm_ol, 'DisplayName', 'OL') + plot(freqs, C_norm_hac_dvf, 'DisplayName', 'HAC-DVF') + plot(freqs, C_norm_hac_iff, 'DisplayName', 'HAC-IFF') + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Compliance - Frobenius Norm'); + legend(); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/hac_lac_C_T_comparison.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:hac_lac_C_T_comparison +#+caption: Comparison of the Frobenius norm of the Compliance and Transmissibility for the HAC-LAC architecture with both IFF and DVF ([[./figs/hac_lac_C_T_comparison.png][png]], [[./figs/hac_lac_C_T_comparison.pdf][pdf]]) +[[file:figs/hac_lac_C_T_comparison.png]] + +* MIMO Analysis +** Introduction :ignore: +Let's define the system as shown in figure [[fig:general_control_names]]. + +#+begin_src latex :file general_control_names.pdf + \begin{tikzpicture} + + % Blocs + \node[block={2.0cm}{2.0cm}] (P) {$P$}; + \node[block={1.5cm}{1.5cm}, below=0.7 of P] (K) {$K$}; + + % Input and outputs coordinates + \coordinate[] (inputw) at ($(P.south west)!0.75!(P.north west)$); + \coordinate[] (inputu) at ($(P.south west)!0.25!(P.north west)$); + \coordinate[] (outputz) at ($(P.south east)!0.75!(P.north east)$); + \coordinate[] (outputv) at ($(P.south east)!0.25!(P.north east)$); + + % Connections and labels + \draw[<-] (inputw) node[above left, align=right]{(weighted)\\exogenous inputs\\$w$} -- ++(-1.5, 0); + \draw[<-] (inputu) -- ++(-0.8, 0) |- node[left, near start, align=right]{control signals\\$u$} (K.west); + + \draw[->] (outputz) node[above right, align=left]{(weighted)\\exogenous outputs\\$z$} -- ++(1.5, 0); + \draw[->] (outputv) -- ++(0.8, 0) |- node[right, near start, align=left]{sensed output\\$v$} (K.east); + \end{tikzpicture} +#+end_src + +#+name: fig:general_control_names +#+caption: General Control Architecture +#+RESULTS: +[[file:figs/general_control_names.png]] + +#+name: tab:general_plant_signals +#+caption: Signals definition for the generalized plant +| *Exogenous Inputs* | $\bm{\mathcal{X}}_w$ | Ground motion | +| | $\bm{\mathcal{F}}_d$ | External Forces applied to the Payload | +| | $\bm{r}$ | Reference signal for tracking | +|---------------------+-----------------------------+----------------------------------------| +| *Exogenous Outputs* | $\bm{\mathcal{X}}$ | Absolute Motion of the Payload | +| | $\bm{\tau}$ | Actuator Rate | +|---------------------+-----------------------------+----------------------------------------| +| *Sensed Outputs* | $\bm{\tau}_m$ | Force Sensors in each leg | +| | $\delta \bm{\mathcal{L}}_m$ | Measured displacement of each leg | +| | $\bm{\mathcal{X}}$ | Absolute Motion of the Payload | +|---------------------+-----------------------------+----------------------------------------| +| *Control Signals* | $\bm{\tau}$ | Actuator Inputs | + +** Matlab Init :noexport: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + simulinkproject('../'); +#+end_src + +#+begin_src matlab + open('stewart_platform_model.slx') +#+end_src + +** Initialization +We first initialize the Stewart platform. +#+begin_src matlab + stewart = initializeStewartPlatform(); + stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3); + stewart = generateGeneralConfiguration(stewart); + stewart = computeJointsPose(stewart); + stewart = initializeStrutDynamics(stewart); + stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical'); + stewart = initializeCylindricalPlatforms(stewart); + stewart = initializeCylindricalStruts(stewart); + stewart = computeJacobian(stewart); + stewart = initializeStewartPose(stewart); + stewart = initializeInertialSensor(stewart, 'type', 'none'); +#+end_src + +The rotation point of the ground is located at the origin of frame $\{A\}$. +#+begin_src matlab + ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A); + payload = initializePayload('type', 'none'); +#+end_src + +** Identification +*** HAC - Without LAC +#+begin_src matlab + controller = initializeController('type', 'open-loop'); +#+end_src + +#+begin_src matlab + %% Name of the Simulink File + mdl = 'stewart_platform_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N] + io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad] + + %% Run the linearization + G_ol = linearize(mdl, io); + G_ol.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; + G_ol.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +#+end_src + +*** HAC - DVF +#+begin_src matlab + controller = initializeController('type', 'dvf'); + K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6); +#+end_src + +#+begin_src matlab + %% Name of the Simulink File + mdl = 'stewart_platform_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N] + io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad] + + %% Run the linearization + G_dvf = linearize(mdl, io); + G_dvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; + G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +#+end_src + +*** Cartesian Frame +#+begin_src matlab + Gc_ol = minreal(G_ol)/stewart.kinematics.J'; + Gc_ol.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; + + Gc_dvf = minreal(G_dvf)/stewart.kinematics.J'; + Gc_dvf.InputName = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'}; +#+end_src + +** Singular Value Decomposition +#+begin_src matlab + freqs = logspace(1, 4, 1000); + + U_ol = zeros(6,6,length(freqs)); + S_ol = zeros(6,length(freqs)); + V_ol = zeros(6,6,length(freqs)); + + U_dvf = zeros(6,6,length(freqs)); + S_dvf = zeros(6,length(freqs)); + V_dvf = zeros(6,6,length(freqs)); + + for i = 1:length(freqs) + [U,S,V] = svd(freqresp(Gc_ol, freqs(i), 'Hz')); + U_ol(:,:,i) = U; + S_ol(:,i) = diag(S); + V_ol(:,:,i) = V; + + [U,S,V] = svd(freqresp(Gc_dvf, freqs(i), 'Hz')); + U_dvf(:,:,i) = U; + S_dvf(:,i) = diag(S); + V_dvf(:,:,i) = V; + end +#+end_src + +#+begin_src matlab :exports none + figure; + + ax1 = subplot(1,2,1); + hold on; + plot(freqs, S_ol(1,:), '-'); + plot(freqs, S_ol(2,:), '--'); + plot(freqs, S_ol(3,:), '-.'); + plot(freqs, S_ol(4,:), '--'); + plot(freqs, S_ol(5,:), '-'); + plot(freqs, S_ol(6,:), '-.'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Singular Values'); + title('Undamped Plant'); + + ax2 = subplot(1,2,2); + hold on; + plot(freqs, S_dvf(1,:), '-' , 'DisplayName', '$\sigma_1$'); + plot(freqs, S_dvf(2,:), '--', 'DisplayName', '$\sigma_2$'); + plot(freqs, S_dvf(3,:), '-.', 'DisplayName', '$\sigma_3$'); + plot(freqs, S_dvf(4,:), '-' , 'DisplayName', '$\sigma_4$'); + plot(freqs, S_dvf(5,:), '--', 'DisplayName', '$\sigma_5$'); + plot(freqs, S_dvf(6,:), '-.', 'DisplayName', '$\sigma_6$'); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Singular Values'); + title('Damped Plant - DVF'); + + linkaxes([ax1, ax2], 'xy'); + legend(); +#+end_src + +#+begin_src matlab :exports none + figure; + + ax1 = subplot(1,2,1); + hold on; + for i = 1:6 + plot(freqs, abs(squeeze(V_ol(i,1,:))), '-' , 'DisplayName', Gc_ol.InputName{i}); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [Hz]'); + ylabel('Singular Values'); + legend(); + + ax2 = subplot(1,2,2); + hold on; + for i = 1:6 + plot(freqs, abs(squeeze(U_ol(i,1,:))), '-' , 'DisplayName', Gc_ol.OutputName{i}); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + xlabel('Frequency [Hz]'); + ylabel('Singular Values'); + legend(); + + linkaxes([ax1,ax2], 'x'); +#+end_src + +* Diagonal Control based on the damped plant +** Introduction :ignore: +From cite:skogestad07_multiv_feedb_contr, a simple approach to multivariable control is the following two-step procedure: +1. *Design a pre-compensator* $W_1$, which counteracts the interactions in the plant and results in a new *shaped plant* $G_S(s) = G(s) W_1(s)$ which is *more diagonal and easier to control* than the original plant $G(s)$. +2. *Design a diagonal controller* $K_S(s)$ for the shaped plant using methods similar to those for SISO systems. + +The overall controller is then: +\[ K(s) = W_1(s)K_s(s) \] + +There are mainly three different cases: +1. *Dynamic decoupling*: $G_S(s)$ is diagonal at all frequencies. For that we can choose $W_1(s) = G^{-1}(s)$ and this is an inverse-based controller. +2. *Steady-state decoupling*: $G_S(0)$ is diagonal. This can be obtained by selecting $W_1(s) = G^{-1}(0)$. +3. *Approximate decoupling at frequency $\w_0$*: $G_S(j\w_0)$ is as diagonal as possible. Decoupling the system at $\w_0$ is a good choice because the effect on performance of reducing interaction is normally greatest at this frequency. + +** Initialization +We first initialize the Stewart platform. +#+begin_src matlab + stewart = initializeStewartPlatform(); + stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3); + stewart = generateGeneralConfiguration(stewart); + stewart = computeJointsPose(stewart); + stewart = initializeStrutDynamics(stewart); + stewart = initializeJointDynamics(stewart, 'type_F', 'universal', 'type_M', 'spherical'); + stewart = initializeCylindricalPlatforms(stewart); + stewart = initializeCylindricalStruts(stewart); + stewart = computeJacobian(stewart); + stewart = initializeStewartPose(stewart); + stewart = initializeInertialSensor(stewart, 'type', 'none'); +#+end_src + +The rotation point of the ground is located at the origin of frame $\{A\}$. +#+begin_src matlab + ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A); + payload = initializePayload('type', 'none'); +#+end_src + +** Identification +#+begin_src matlab + controller = initializeController('type', 'dvf'); + K_dvf = -1e4*s/(1+s/2/pi/5000)*eye(6); +#+end_src + +#+begin_src matlab + %% Name of the Simulink File + mdl = 'stewart_platform_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'input'); io_i = io_i + 1; % Actuator Force Inputs [N] + io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Sensor [m, rad] + + %% Run the linearization + G_dvf = linearize(mdl, io); + G_dvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; + G_dvf.OutputName = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; +#+end_src + +** Steady State Decoupling +*** Pre-Compensator Design +We choose $W_1 = G^{-1}(0)$. +#+begin_src matlab + W1 = inv(freqresp(G_dvf, 0)); +#+end_src + +The (static) decoupled plant is $G_s(s) = G(s) W_1$. +#+begin_src matlab + Gs = G_dvf*W1; +#+end_src + +In the case of the Stewart platform, the pre-compensator for static decoupling is equal to $\mathcal{K} \bm{J}$: +\begin{align*} + W_1 &= \left( \frac{\bm{\mathcal{X}}}{\bm{\tau}}(s=0) \right)^{-1}\\ + &= \left( \frac{\bm{\mathcal{X}}}{\bm{\tau}}(s=0) \bm{J}^T \right)^{-1}\\ + &= \left( \bm{C} \bm{J}^T \right)^{-1}\\ + &= \left( \bm{J}^{-1} \mathcal{K}^{-1} \right)^{-1}\\ + &= \mathcal{K} \bm{J} +\end{align*} + +The static decoupled plant is schematic shown in Figure [[fig:control_arch_static_decoupling]] and the bode plots of its diagonal elements are shown in Figure [[fig:static_decoupling_diagonal_plant]]. + +#+begin_src latex :file control_arch_static_decoupling.pdf + \begin{tikzpicture} + % Blocs + \node[block] (G) {$G(s)$}; + \node[block, left=1 of G] (J) {$\mathcal{K}\bm{J}$}; + \node[block, left=1 of J] (Ks) {$\bm{K}_s(s)$}; + + \draw[->] (Ks.east) -- (J.west); + \draw[->] (J.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) node[above right]{$\bm{\mathcal{X}}$} -| ++(0.8, -0.8) -| ($(Ks.west) + (-0.8, 0)$) -- (Ks.west); + + \begin{scope}[on background layer] + \node[fit={(J.north west) (G.south east)}, inner sep=4pt, draw, dashed, fill=black!20!white, label={$G_s(s)$}] {}; + \end{scope} + \end{tikzpicture} +#+end_src + +#+name: fig:control_arch_static_decoupling +#+caption: Static Decoupling of the Stewart platform +#+RESULTS: +[[file:figs/control_arch_static_decoupling.png]] + +#+begin_src matlab :exports none + freqs = logspace(1, 4, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:6 + plot(freqs, abs(squeeze(freqresp(Gs(i, i), freqs, 'Hz')))); + end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); ax2 = subplot(2, 1, 2); hold on; - for i = 1:5 - for j = i+1:6 - plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(i, i)*G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); - end + for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(Gs(i, i), freqs, 'Hz')))); end - set(gca,'ColorOrderIndex',1); - plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz')))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); @@ -260,6 +1081,105 @@ A lead is added around the crossover frequency which is set to be around 500Hz. linkaxes([ax1,ax2],'x'); #+end_src +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/static_decoupling_diagonal_plant.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:static_decoupling_diagonal_plant +#+caption: Bode plot of the diagonal elements of $G_s(s)$ ([[./figs/static_decoupling_diagonal_plant.png][png]], [[./figs/static_decoupling_diagonal_plant.pdf][pdf]]) +[[file:figs/static_decoupling_diagonal_plant.png]] + +*** Diagonal Control Design +We design a diagonal controller $K_s(s)$ that consist of a pure integrator and a lead around the crossover. + +#+begin_src matlab + wc = 2*pi*300; % Wanted Bandwidth [rad/s] + + h = 1.5; + H_lead = 1/h*(1 + s/(wc/h))/(1 + s/(wc*h)); + + Ks_dvf = diag(1./abs(diag(freqresp(1/s*Gs, wc)))) .* H_lead .* 1/s; +#+end_src + +The overall controller is then $K(s) = W_1 K_s(s)$ as shown in Figure [[fig:control_arch_static_decoupling_K]]. + +#+begin_src matlab + K_hac_dvf = W1 * Ks_dvf; +#+end_src + +#+begin_src latex :file control_arch_static_decoupling_K.pdf + \begin{tikzpicture} + % Blocs + \node[block] (G) {$G(s)$}; + \node[block, left=1 of G] (J) {$\mathcal{K}\bm{J}$}; + \node[block, left=1 of J] (Ks) {$\bm{K}_s(s)$}; + + \draw[->] (Ks.east) -- (J.west); + \draw[->] (J.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) node[above right]{$\bm{\mathcal{X}}$} -| ++(0.8, -0.8) -| ($(Ks.west) + (-0.8, 0)$) -- (Ks.west); + + \begin{scope}[on background layer] + \node[fit={(Ks.north west) (J.south east)}, inner sep=4pt, draw, dashed, fill=black!20!white, label={$K(s)$}] {}; + \end{scope} + \end{tikzpicture} +#+end_src + +#+name: fig:control_arch_static_decoupling_K +#+caption: Controller including the static decoupling matrix +#+RESULTS: +[[file:figs/control_arch_static_decoupling_K.png]] + +*** Results +We identify the transmissibility and compliance of the Stewart platform under open-loop and closed-loop control. + +#+begin_src matlab + controller = initializeController('type', 'open-loop'); + [T_ol, T_norm_ol, freqs] = computeTransmissibility(); + [C_ol, C_norm_ol, ~] = computeCompliance(); +#+end_src + +#+begin_src matlab + controller = initializeController('type', 'hac-dvf'); + [T_hac_dvf, T_norm_hac_dvf, ~] = computeTransmissibility(); + [C_hac_dvf, C_norm_hac_dvf, ~] = computeCompliance(); +#+end_src + +The results are shown in figure + +#+begin_src matlab :exports none + figure; + + subplot(1,2,1); + hold on; + plot(freqs, T_norm_ol) + plot(freqs, T_norm_hac_dvf) + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Transmissibility - Frobenius Norm'); + + subplot(1,2,2); + hold on; + plot(freqs, C_norm_ol, 'DisplayName', 'OL') + plot(freqs, C_norm_hac_dvf, 'DisplayName', 'HAC-DVF - Static decoupl.') + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + xlabel('Frequency [Hz]'); + ylabel('Compliance - Frobenius Norm'); + legend(); +#+end_src + +#+header: :tangle no :exports results :results none :noweb yes +#+begin_src matlab :var filepath="figs/static_decoupling_C_T_frobenius_norm.pdf" :var figsize="full-tall" :post pdf2svg(file=*this*, ext="png") +<> +#+end_src + +#+name: fig:static_decoupling_C_T_frobenius_norm +#+caption: Frobenius norm of the Compliance and transmissibility matrices ([[./figs/static_decoupling_C_T_frobenius_norm.png][png]], [[./figs/static_decoupling_C_T_frobenius_norm.pdf][pdf]]) +[[file:figs/static_decoupling_C_T_frobenius_norm.png]] + +** Decoupling at Crossover +- [ ] Find a method for real approximation of a complex matrix + * Functions ** =initializeController=: Initialize the Controller :PROPERTIES: @@ -288,7 +1208,7 @@ A lead is added around the crossover frequency which is set to be around 500Hz. :END: #+begin_src matlab arguments - args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf'})} = 'open-loop' + args.type char {mustBeMember(args.type, {'open-loop', 'iff', 'dvf', 'hac-iff', 'hac-dvf'})} = 'open-loop' end #+end_src @@ -312,5 +1232,9 @@ A lead is added around the crossover frequency which is set to be around 500Hz. controller.type = 1; case 'dvf' controller.type = 2; + case 'hac-iff' + controller.type = 3; + case 'hac-dvf' + controller.type = 4; end #+end_src diff --git a/org/control-tracking.org b/org/control-tracking.org new file mode 100644 index 0000000..d0c177b --- /dev/null +++ b/org/control-tracking.org @@ -0,0 +1,257 @@ +#+TITLE: Stewart Platform - Tracking Control +:DRAWER: +#+STARTUP: overview + +#+LANGUAGE: en +#+EMAIL: dehaeze.thomas@gmail.com +#+AUTHOR: Dehaeze Thomas + +#+HTML_LINK_HOME: ./index.html +#+HTML_LINK_UP: ./index.html + +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: +#+HTML_HEAD: + +#+PROPERTY: header-args:matlab :session *MATLAB* +#+PROPERTY: header-args:matlab+ :comments org +#+PROPERTY: header-args:matlab+ :exports both +#+PROPERTY: header-args:matlab+ :results none +#+PROPERTY: header-args:matlab+ :eval no-export +#+PROPERTY: header-args:matlab+ :noweb yes +#+PROPERTY: header-args:matlab+ :mkdirp yes +#+PROPERTY: header-args:matlab+ :output-dir figs + +#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/thesis/latex/}{config.tex}") +#+PROPERTY: header-args:latex+ :imagemagick t :fit yes +#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 +#+PROPERTY: header-args:latex+ :imoutoptions -quality 100 +#+PROPERTY: header-args:latex+ :results file raw replace +#+PROPERTY: header-args:latex+ :buffer no +#+PROPERTY: header-args:latex+ :eval no-export +#+PROPERTY: header-args:latex+ :exports results +#+PROPERTY: header-args:latex+ :mkdirp yes +#+PROPERTY: header-args:latex+ :output-dir figs +#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png") +:END: + +* First Control Architecture +** Matlab Init :noexport: +#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) + <> +#+end_src + +#+begin_src matlab :exports none :results silent :noweb yes + <> +#+end_src + +#+begin_src matlab + simulinkproject('../'); +#+end_src + +** Control Schematic +#+begin_src latex :file control_measure_rotating_2dof.pdf + \begin{tikzpicture} + % Blocs + \node[block] (J) at (0, 0) {$J$}; + \node[addb={+}{}{}{}{-}, right=1 of J] (subr) {}; + \node[block, right=0.8 of subr] (K) {$K_{L}$}; + \node[block, right=1 of K] (G) {$G_{L}$}; + + % Connections and labels + \draw[<-] (J.west)node[above left]{$\bm{r}_{n}$} -- ++(-1, 0); + \draw[->] (J.east) -- (subr.west) node[above left]{$\bm{r}_{L}$}; + \draw[->] (subr.east) -- (K.west) node[above left]{$\bm{\epsilon}_{L}$}; + \draw[->] (K.east) -- (G.west) node[above left]{$\bm{\tau}$}; + \draw[->] (G.east) node[above right]{$\bm{L}$} -| ($(G.east)+(1, -1)$) -| (subr.south); + \end{tikzpicture} +#+end_src + +#+RESULTS: +[[file:figs/control_measure_rotating_2dof.png]] + +** Initialize the Stewart platform +#+begin_src matlab + stewart = initializeStewartPlatform(); + stewart = initializeFramesPositions(stewart, 'H', 90e-3, 'MO_B', 45e-3); + stewart = generateGeneralConfiguration(stewart); + stewart = computeJointsPose(stewart); + stewart = initializeStrutDynamics(stewart); + stewart = initializeJointDynamics(stewart, 'type_F', 'universal_p', 'type_M', 'spherical_p'); + stewart = initializeCylindricalPlatforms(stewart); + stewart = initializeCylindricalStruts(stewart); + stewart = computeJacobian(stewart); + stewart = initializeStewartPose(stewart); + stewart = initializeInertialSensor(stewart, 'type', 'accelerometer', 'freq', 5e3); +#+end_src + +#+begin_src matlab + ground = initializeGround('type', 'none'); + payload = initializePayload('type', 'none'); +#+end_src + +** Identification of the plant +Let's identify the transfer function from $\bm{\tau}$ to $\bm{L}$. +#+begin_src matlab + %% Name of the Simulink File + mdl = 'stewart_platform_model'; + + %% Input/Output definition + clear io; io_i = 1; + io(io_i) = linio([mdl, '/Controller'], 1, 'openinput'); io_i = io_i + 1; % Actuator Force Inputs [N] + io(io_i) = linio([mdl, '/Stewart Platform'], 1, 'openoutput', [], 'dLm'); io_i = io_i + 1; % Relative Displacement Outputs [m] + + %% Run the linearization + G = linearize(mdl, io); + G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; + G.OutputName = {'L1', 'L2', 'L3', 'L4', 'L5', 'L6'}; +#+end_src + +** Plant Analysis +Diagonal terms +#+begin_src matlab :exports none + freqs = logspace(1, 4, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:6 + plot(freqs, abs(squeeze(freqresp(G(i, i), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, i), freqs, 'Hz')))); + end + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +Compare to off-diagonal terms +#+begin_src matlab :exports none + freqs = logspace(1, 4, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); + end + end + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(G(1, 1), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:5 + for j = i+1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); + end + end + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(G(1, 1), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +** Controller Design +One integrator should be present in the controller. + +A lead is added around the crossover frequency which is set to be around 500Hz. + +#+begin_src matlab + % wint = 2*pi*100; % Integrate until [rad] + % wlead = 2*pi*500; % Location of the lead [rad] + % hlead = 2; % Lead strengh + + % Kl = 1e6 * ... % Gain + % (s + wint)/(s) * ... % Integrator until 100Hz + % (1 + s/(wlead/hlead)/(1 + s/(wlead*hlead))); % Lead + + wc = 2*pi*100; + Kl = 1/abs(freqresp(G(1,1), wc)) * wc/s * 1/(1 + s/(3*wc)); + Kl = Kl * eye(6); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(1, 3, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + plot(freqs, abs(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src + +#+begin_src matlab :exports none + freqs = logspace(1, 4, 1000); + + figure; + + ax1 = subplot(2, 1, 1); + hold on; + for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(Kl(i,i)*G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); + end + end + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); + ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); + + ax2 = subplot(2, 1, 2); + hold on; + for i = 1:5 + for j = i+1:6 + plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(i, i)*G(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2]); + end + end + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Kl(1,1)*G(1, 1), freqs, 'Hz')))); + hold off; + set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin'); + ylabel('Phase [deg]'); xlabel('Frequency [Hz]'); + ylim([-180, 180]); + yticks([-180, -90, 0, 90, 180]); + + linkaxes([ax1,ax2],'x'); +#+end_src diff --git a/org/cubic-configuration.org b/org/cubic-configuration.org index 158fd1a..9f51124 100644 --- a/org/cubic-configuration.org +++ b/org/cubic-configuration.org @@ -695,6 +695,7 @@ No flexibility below the Stewart platform and no payload. #+begin_src matlab ground = initializeGround('type', 'none'); payload = initializePayload('type', 'none'); + controller = initializeController('type', 'open-loop'); #+end_src The obtain geometry is shown in figure [[fig:stewart_cubic_conf_decouple_dynamics]]. @@ -880,6 +881,7 @@ No flexibility below the Stewart platform and no payload. #+begin_src matlab ground = initializeGround('type', 'none'); payload = initializePayload('type', 'none'); + controller = initializeController('type', 'open-loop'); #+end_src The obtain geometry is shown in figure [[fig:stewart_cubic_conf_mass_above]]. @@ -1087,6 +1089,7 @@ No flexibility below the Stewart platform and no payload. #+begin_src matlab ground = initializeGround('type', 'none'); payload = initializePayload('type', 'none'); + controller = initializeController('type', 'open-loop'); #+end_src #+begin_src matlab :exports none @@ -1258,6 +1261,7 @@ No flexibility below the Stewart platform and no payload. #+begin_src matlab ground = initializeGround('type', 'none'); payload = initializePayload('type', 'none'); + controller = initializeController('type', 'open-loop'); #+end_src #+begin_src matlab :exports none diff --git a/org/dynamics-study.org b/org/dynamics-study.org index 3f75e78..bbc6fcd 100644 --- a/org/dynamics-study.org +++ b/org/dynamics-study.org @@ -80,6 +80,7 @@ We also don't put any payload on top of the Stewart platform. #+begin_src matlab ground = initializeGround('type', 'none'); payload = initializePayload('type', 'none'); + controller = initializeController('type', 'open-loop'); #+end_src The transfer function from actuator forces $\bm{\tau}$ to the relative displacement of the mobile platform $\mathcal{\bm{X}}$ is extracted. @@ -327,6 +328,7 @@ No flexibility below the Stewart platform and no payload. #+begin_src matlab ground = initializeGround('type', 'none'); payload = initializePayload('type', 'none'); + controller = initializeController('type', 'open-loop'); #+end_src Estimation of the transfer function from $\mathcal{\bm{F}}$ to $\mathcal{\bm{X}}$: diff --git a/org/identification.org b/org/identification.org index 92d12a9..76df46e 100644 --- a/org/identification.org +++ b/org/identification.org @@ -83,6 +83,7 @@ In this document, we discuss the various methods to identify the behavior of the #+begin_src matlab ground = initializeGround('type', 'none'); payload = initializePayload('type', 'none'); + controller = initializeController('type', 'open-loop'); #+end_src ** Identification @@ -284,6 +285,7 @@ We set the rotation point of the ground to be at the same point at frames $\{A\} #+begin_src matlab ground = initializeGround('type', 'rigid', 'rot_point', stewart.platform_F.FO_A); payload = initializePayload('type', 'rigid'); + controller = initializeController('type', 'open-loop'); #+end_src ** Transmissibility @@ -396,6 +398,7 @@ We set the rotation point of the ground to be at the same point at frames $\{A\} #+begin_src matlab ground = initializeGround('type', 'none'); payload = initializePayload('type', 'rigid'); + controller = initializeController('type', 'open-loop'); #+end_src ** Compliance @@ -517,7 +520,7 @@ We can try to use the Frobenius norm to obtain a scalar value representing the 6 %% Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/Disturbances/D_w'], 1, 'openinput'); io_i = io_i + 1; % Base Motion [m, rad] - io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad] + io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad] %% Run the linearization T = linearize(mdl, io, options); @@ -553,8 +556,8 @@ If wanted, the 6x6 transmissibility matrix is plotted. han = axes(fig, 'visible', 'off'); han.XLabel.Visible = 'on'; han.YLabel.Visible = 'on'; - ylabel(han, 'Frequency [Hz]'); - xlabel(han, 'Transmissibility [m/m]'); + xlabel(han, 'Frequency [Hz]'); + ylabel(han, 'Transmissibility [m/m]'); end #+end_src @@ -642,7 +645,7 @@ If wanted, the 6x6 transmissibility matrix is plotted. %% Input/Output definition clear io; io_i = 1; io(io_i) = linio([mdl, '/Disturbances/F_ext'], 1, 'openinput'); io_i = io_i + 1; % External forces [N, N*m] - io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'openoutput'); io_i = io_i + 1; % Absolute Motion [m, rad] + io(io_i) = linio([mdl, '/Absolute Motion Sensor'], 1, 'output'); io_i = io_i + 1; % Absolute Motion [m, rad] %% Run the linearization C = linearize(mdl, io, options); diff --git a/simscape_subsystems/Stewart_Platform.slx b/simscape_subsystems/Stewart_Platform.slx index fdcb6d0e489a08b0b374a454d4c40a98b07ce3a3..af23f5f1ab49ba9933c15972459cf5aca953f258 100644 GIT binary patch delta 9790 zcmZX4WmH^C(Ct9*AcF^YcL?qp2<{Nv-EGjp2?QV9JwR}G3GNPoV8PuX1bu{ipWOG= zI=$+w>fKdcRo%aOuiQi3-b0}%$-aTX0ssIAfN53TY7|W9H!r%D0y+i?@#RNpF?{^bL52jH#wF>KhdHdKoec=Czlj*N7C3N|*MqPz)JTvoZjWm-6Hla zGPbnRxG}!+LwRDMjdjzVF)7O|YGj>e9}x$=VgsT2z$P0Xb>{f)*IVD~+i71;M)jZc 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