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b/docs/figs/nano_hexapod_iff_plant_bode_plot.png new file mode 100644 index 0000000..07d444c Binary files /dev/null and b/docs/figs/nano_hexapod_iff_plant_bode_plot.png differ diff --git a/docs/figs/nano_hexapod_iff_root_locus.pdf b/docs/figs/nano_hexapod_iff_root_locus.pdf new file mode 100644 index 0000000..a778977 Binary files /dev/null and b/docs/figs/nano_hexapod_iff_root_locus.pdf differ diff --git a/docs/figs/nano_hexapod_iff_root_locus.png b/docs/figs/nano_hexapod_iff_root_locus.png new file mode 100644 index 0000000..18d6bfc Binary files /dev/null and b/docs/figs/nano_hexapod_iff_root_locus.png differ diff --git a/docs/index.html b/docs/index.html index 796ca60..088e896 100644 --- a/docs/index.html +++ b/docs/index.html @@ -3,11 +3,11 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Simscape Model of the Nano-Active-Stabilization-System - + @@ -39,7 +39,8 @@

  • 15. Dynamic Noise Budgeting (link)
  • 16. Active Damping Techniques on the full Simscape Model (link)
  • 17. Control of the Nano-Active-Stabilization-System (link)
  • -
  • 18. Useful Matlab Functions (link)
  • +
  • 18. Nano-Hexapod Simscape Model (link)
  • +
  • 19. Useful Matlab Functions (link)
  • @@ -239,9 +240,12 @@ In this file are gathered all studies about the control the Nano-Active-Stabiliz +
    +

    18 Nano-Hexapod Simscape Model (link)

    +
    -

    18 Useful Matlab Functions (link)

    -
    +

    19 Useful Matlab Functions (link)

    +

    Many matlab functions are shared among all the files of the projects.

    @@ -254,7 +258,7 @@ These functions are all defined here.

    Author: Dehaeze Thomas

    -

    Created: 2021-02-20 sam. 23:08

    +

    Created: 2021-04-23 ven. 13:22

    diff --git a/docs/nano_hexapod.html b/docs/nano_hexapod.html index f74a75c..9bbc06c 100644 --- a/docs/nano_hexapod.html +++ b/docs/nano_hexapod.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + Nano-Hexapod @@ -41,30 +41,57 @@
    +

    +In this document, a Simscape model of the nano-hexapod is developed. +

    + +
      +
    • Section 1:
    • +
    • Section 2:
    • +
    • Section 3:
    • +
    • Section 4:
    • +
    +

    1 Nano-Hexapod

    +

    + +

    -
    -

    1.1 Nano Hexapod object

    +
    +

    1.1 Nano Hexapod - Configuration

    -The nano-hexapod can be initialized and configured using the initializeNanoHexapodFinal function. + +

    + +

    +The nano-hexapod can be initialized and configured using the initializeNanoHexapodFinal function (link).

    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
                                            'flex_top_type', '3dof', ...
                                            'motion_sensor_type', 'struts', ...
    -                                       'actuator_type', '2dof');
    +                                       'actuator_type', '2dof', ...
    +                                       'MO_B', 150e-3);
    +
    +
    + +

    +We initialize the identification parameters. +

    +
    +
    %% Options for Linearized
    +options = linearizeOptions;
    +options.SampleTime = 0;
    +
    +%% Name of the Simulink File
    +mdl = 'nano_hexapod';
    +
    +%% Input/Output definition
    +clear io; io_i = 1;
    +io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
    +io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
     
    -
    -

    1.2 Jacobian from Solidworks Model

    +
    +

    1.2 Effect of encoders on the decentralized plant

    -Center of joints a_i with respect to {F}: + +

    + +

    +We here wish to compare the plant from actuators to the encoders when the encoders are either fixed on the struts or on the plates. +

    + +

    +Identify the plant when the encoders are on the struts:

    -
    Fa = [[-86.05,  -74.78, 22.49],
    -      [86.05,   -74.78, 22.49],
    -      [107.79,  -37.13, 22.49],
    -      [21.74,   111.91, 22.49],
    -      [-21.74,  111.91, 22.49],
    -      [-107.79, -37.13, 22.49]]'*1e-3;
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof');
    +
    +Gs = linearize(mdl, io, 0.0, options);
    +Gs.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gs.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
     

    -Center of joints a_i with respect to {M}: +And identify the plant when the encoders are fixed on the plates:

    -
    Mb = [[-28.47, -106.25, -22.50],
    -      [28.47,  -106.25, -22.50],
    -      [106.25,  28.47,  -22.50],
    -      [77.78,   77.78,  -22.50],
    -      [-77.78,  77.78,  -22.50],
    -      [-106.25, 28.47,  -22.50]]'*1e-3;
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'plates', ...
    +                                       'actuator_type', '2dof');
    +
    +Gp = linearize(mdl, io, 0.0, options);
    +Gp.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gp.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
     
    +

    +The obtained plants are compared in Figure 1. +

    + +
    +

    nano_hexapod_effect_encoder.png +

    +

    Figure 1: Comparison of the plants from actuator to associated encoder when the encoders are either fixed to the struts or to the plates

    +
    + +
    +

    +Why do we have zeros at 400Hz and 800Hz when the encoders are fixed on the struts? +

    + +
    +
    +
    + +
    +

    1.3 Effect of APA flexibility

    +
    +

    + +

    + +

    +First identify the plant for APA represented by 2DoF system: +

    -
    Fb = Mb + [0; 0; 95e-3];
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof', ...
    +                                       'actuator_Ga', 2);
     
    -si = Fb - Fa;
    -si = si./vecnorm(si);
    +Gs = linearize(mdl, io, 0.0, options);
    +Gs.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gs.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
     
    +

    +First identify the plant for APA represented by a flexible element: +

    -
    ki = ones(1,6);
    -bi = Fb;
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', 'flexible');
    +
    +Gf = linearize(mdl, io, 0.0, options);
    +Gf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gf.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
     
    + +
    +

    nano_hexapod_effect_flexible_apa.png +

    +

    Figure 2: Comparison of the plants from actuator to associated strut encoder when the APA are modelled with a 2DoF system of with a flexible one

    +
    + +
    +

    +The first resonance is strange when using the flexible APA model (Figure 2). +Moreover the system is unstable. +Otherwise, the 2DoF model matches quite well the flexible model considering its simplicity. +

    + +
    +
    +
    + +
    +

    1.4 Nano Hexapod - Number of DoF

    +
    +

    + +

    + +

    +In this section, we wish to see how the configuration of each element changes the number of the states of the obtained system. +

    + +

    +The most minimalist model is the following: +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '3dof', ...
    +                                       'flex_top_type', '2dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof');
    +G = linearize(mdl, io, 0.0, options);
    +
    +
    + +
    +There are 24 states.
    +
    + + +

    +These states are summarized on table 1. +

    + + + + +++ ++ + + + + + + + + + + + + + + + + + + + + + + + +
    Table 1: Number of states for the minimalist model
    ElementStates
    Struts2*6
    Top Plate12
    Total:24
    + +

    +If we add axial stiffness on the top joints, we should add 2 states for each struts. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '2dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof');
    +G = linearize(mdl, io, 0.0, options);
    +
    +
    + +
    +There are 36 states.
    +
    + + +

    +If we add torsional stiffness on the bottom joints, we should again add 2 states for each struts. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof');
    +G = linearize(mdl, io, 0.0, options);
    +
    +
    + +
    +There are 48 states.
    +
    + + +

    +Finally, if we add axial stiffness on the bottom joint, we should add 2 states for each struts. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '4dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof');
    +G = linearize(mdl, io, 0.0, options);
    +
    +
    + +
    +There are 60 states.
    +
    + + +
    +

    +Obtained number of states is very comprehensible. +Depending on the physical effects we want to model, we therefore know how many states are added when configuring the model. +

    + +
    +
    +
    + +
    +

    1.5 Direct Velocity Feedback Plant

    +
    +

    + +

    + +

    +The transfer function from actuator forces \(\tau_i\) to the encoder measurements \(\mathcal{L}_i\) is now identified both when the encoders are fixed to the struts. +

    +
    +
    %% Options for Linearized
    +options = linearizeOptions;
    +options.SampleTime = 0;
    +
    +%% Name of the Simulink File
    +mdl = 'nano_hexapod';
    +
    +%% Input/Output definition
    +clear io; io_i = 1;
    +io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1; % Actuator Inputs [N]
    +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative displacements [m]
    +
    +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof');
    +
    +Gdvf = linearize(mdl, io, 0.0, options);
    +
    +
    + +

    +The DC gain from actuator to relative motion sensor should be equal to (for the 2dof APA): +\[ \frac{1}{k + k_a + kk_a/k_e} \] +

    + +

    +Which is equal to: +

    +
    +DCgain = 1.87e-08 [m/N]
    +
    + + +

    +Let’s verify that by looking at the DC gain of the \(6 \times 6\) DVF plant in Table 2. +

    + + + +++ ++ ++ ++ ++ ++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    Table 2: DC gain of the DVF plant
    1.8617e-08-1.0408e-101.3034e-103.2559e-11-1.1188e-109.0385e-11
    -5.1839e-111.8593e-08-4.4868e-118.016e-114.3558e-11-1.1164e-10
    5.1963e-12-6.9001e-121.8564e-083.0844e-114.0097e-11-3.4387e-11
    1.9359e-111.7432e-10-5.0928e-111.855e-081.6406e-104.5757e-12
    -2.1185e-112.1724e-101.5333e-12-8.802e-111.8803e-08-4.6946e-11
    -1.1728e-11-5.7682e-111.6213e-102.1934e-12-1.6237e-101.8715e-08
    + +

    +And the bode plot of the DVF plant is shown in Figure 3. +

    + +
    +

    nano_hexapod_struts_2dof_dvf_plant.png +

    +

    Figure 3: Bode plot of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors attached to the struts \(\mathcal{L}_i\). Diagonal terms are shown in blue, and off-diagonal terms in black.

    +
    +
    +
    + +
    +

    1.6 Integral Force Feedback Plant

    +
    +

    + +

    + +

    +The transfer function from actuators to force sensors is identified. +

    +
    +
    %% Options for Linearized
    +options = linearizeOptions;
    +options.SampleTime = 0;
    +
    +%% Name of the Simulink File
    +mdl = 'nano_hexapod';
    +
    +%% Input/Output definition
    +clear io; io_i = 1;
    +io(io_i) = linio([mdl, '/F'],  1, 'openinput');   io_i = io_i + 1; % Actuator Inputs
    +io(io_i) = linio([mdl, '/Fm'],  1, 'openoutput'); io_i = io_i + 1; % Force Sensors
    +
    +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof');
    +
    +Giff = linearize(mdl, io, 0.0, options);
    +
    +
    + +

    +This is corresponding to the dynamics for the Integral Force Feedback (IFF) control strategy. +

    + +

    +The bode plot is shown in Figure 4. +

    + +
    +

    nano_hexapod_struts_2dof_iff_plant.png +

    +

    Figure 4: Bode plot of the transfer functions from actuator forces \(\tau_i\) to force sensors \(F_{m,i}\). Diagonal terms are shown in blue, and off-diagonal terms in black.

    +
    +
    +
    + +
    +

    1.7 Decentralized Plant - Cartesian coordinates

    +
    +

    + +

    +

    +Consider the plant shown in Figure 5 with: +

    +
      +
    • \(\tau\) the 6 input forces (APA)
    • +
    • \(d\mathcal{L}\) the relative motion sensor outputs (encoders)
    • +
    • \(\mathcal{X}\) the motion of the top platform measured with “perfect” 6-dof sensor
    • +
    • \(J_a\) and \(J_s\) the Jacobians for the actuators and sensors
    • +
    + + +
    +

    nano_hexapod_decentralized_schematic.png +

    +

    Figure 5: Plant in the cartesian Frame

    +
    +
    +
    +

    1.7.1 Verification of the Sensor Jacobian

    +
    +

    +The “perfect” sensor output \(\mathcal{X}\) is used to verify that the sensor Jacobian is working correctly both when the encoders are fixed to the struts and to the plates. +

    + +

    +Let’s then identify the plant for both configuration, and compare the transfer functions from \(\mathcal{F}\) to \(d\mathcal{X}\) and to \(\mathcal{X}\). +

    +
    +
    %% Options for Linearized
    +options = linearizeOptions;
    +options.SampleTime = 0;
    +
    +%% Name of the Simulink File
    +mdl = 'nano_hexapod';
    +
    +%% Input/Output definition
    +clear io; io_i = 1;
    +io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
    +io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
    +io(io_i) = linio([mdl, '/X'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
    +
    +
    + +

    +Start when the encoders are fixed on the struts. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof', ...
    +                                       'MO_B', 150e-3);
    +Gs = linearize(mdl, io, 0.0, options);
    +Gs.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gs.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6', ...
    +                 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
    +
    +% Cartesian plant using the Jacobians
    +Gsc = inv(n_hexapod.geometry.Js)*Gs({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
    +% Cartesian plant using the perfect sensor
    +Gsp = -Gs({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
    +
    +
    + +

    +The diagonal elements of the plant are shown in Figure 6. +

    + +
    +

    nano_hexapod_comp_cartesian_plants_struts.png +

    +

    Figure 6: Bode plot of the diagonal elements of the decentralized (cartesian) plant when using the sensor Jacobian (solid) and when using “perfect” 6dof sensor (dashed). The encoders are fixed on the struts.

    +
    + +

    +The same if performed when the encoders are fixed to the plates. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'plates', ...
    +                                       'actuator_type', '2dof', ...
    +                                       'MO_B', 150e-3);
    +Gp = linearize(mdl, io, 0.0, options);
    +Gp.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gp.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6', ...
    +                 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
    +
    +% Cartesian plant using the Jacobians
    +Gpc = inv(n_hexapod.geometry.Js)*Gp({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
    +% Cartesian plant using the perfect sensor
    +Gpp = -Gp({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)';
    +
    +
    + +

    +The obtained bode plots are shown in Figure 7. +

    + +
    +

    nano_hexapod_comp_cartesian_plants_plates.png +

    +

    Figure 7: Bode plot of the diagonal elements of the decentralized (cartesian) plant when using the sensor Jacobian (solid) and when using “perfect” 6dof sensor (dashed). The encoders are fixed on the plates.

    +
    + +
    +

    +The Jacobian for the encoders is working properly both when the encoders are fixed to the plates or to the struts. +

    + +

    +However, then the encoders are fixed to the struts, there is a mismatch between the estimated motion and the measured motion above 100Hz due to a complex conjugate zero. +

    + +
    +
    +
    + +
    +

    1.7.2 Comparison of the decentralized plants

    +
    +

    +The decentralized plants are now compared whether the encoders are fixed on the struts or on the plates in Figure 8. +

    + +
    +

    nano_hexapod_cartesian_plant_encoder_comp.png +

    +

    Figure 8: Bode plot of the “cartesian” plant (transfer function from \(\mathcal{F}\) to \(d\mathcal{X}\)) when the encoders are fixed on the struts (solid) and on the plates (dashed)

    +
    +
    +
    +
    + +
    +

    1.8 Decentralized Plant - Decoupling at the Center of Stiffness

    +
    +

    + +

    +
    + +
    +

    1.8.1 Center of Stiffness

    +
    +

    + +

    + +

    +Let’s define some parameters that will be used for the computation of the stiffness matrix: +

    +
    +
    si = n_hexapod.geometry.si; % Orientation of struts
    +bi = n_hexapod.geometry.Fb; % Location of bi w.r.t. {F}
    +ki = ones(1,6); % Normalized strut stiffness
    +
    +
    + +

    +In order to find is the Center of Stiffness (CoK) exists, we have to verify is the following is diagonal: +

    ki.*si*si'
     
    @@ -163,7 +779,7 @@ bi = Fb; 1.8977 2.4659e-17 -1.3383e-18 +5.1838e-19 @@ -173,13 +789,16 @@ bi = Fb; -1.3383e-18 +5.1838e-19 -2.3143e-05 2.2046 +

    +And we can find the location of the CoK with respect to {F}: +

    OkX = (ki.*cross(bi, si)*si')/(ki.*si*si');
     Ok = [OkX(3,2);OkX(1,3);OkX(2,1)]
    @@ -208,27 +827,31 @@ Ok = [OkX(3,2);OkX(1,3);OkX(2,1)]
     
     
     

    -The center of the cube is therefore 52.7mm above the bottom platform {F} frame. +The “center of stiffness” is therefore 52.7mm above the bottom platform {F} frame.

    -Let’s compute the Jacobian at this location: +Let’s initialize the hexapod with frame {A} and {B} at the CoK:

    -
    Jk = [si', cross(bi - Ok, si)'];
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof', ...
    +                                       'MO_B', Ok(3)-95e-3);
     

    -And the stiffness matrix: +And the (normalized) stiffness matrix is computed as follows:

    -
    Jk'*diag(ki)*Jk
    +
    n_hexapod.geometry.J'*diag(ki)*n_hexapod.geometry.J
     
    - - +
    +@@ -249,16 +872,16 @@ And the stiffness matrix: - - + + - - + + @@ -266,62 +889,52 @@ And the stiffness matrix: - - + + - - + + - + - + + + - - - - - - + + + +
    Table 3: Normalized Stiffness Matrix - Center of Stiffness
    0 0 0-3.4694e-182.5509e-06-2.0817e-17-1.5311e-06
    0 1.8977 -2.3143e-054.1751e-061.3878e-174.175e-060 0
    0 -2.3143e-05 2.2046-5.0916e-11-3.4694e-184.7422e-060 0
    04.1751e-06-5.0916e-114.175e-064.7422e-06 0.012594 2.1684e-198.6736e-19-8.6736e-19
    -3.2704e-18-1.8521e-1700 0-4.206e-183.9451e-19 0.012594 -9.3183e-08
    2.5509e-06008.6736e-19-1.5311e-06-6.9389e-182.7756e-17-8.6736e-19 -9.3183e-08 0.043362
    -
    -
    - -
    -

    1.3 Jacobian for encoders on the plates

    -
    -

    -Goal: Compute the 6-DoF motion of the top platform (relative to the bottom platform) from the 6 encoder measurements. -

    -The main differences with the case where the encoders are parallel with the strut are: +And we indeed obtain a diagonal stiffness matrix.

    -
      -
    • the encoder and ruler may not be parallel anymore as the top platform is moving
    • -
    • the derivation of the Jacobian (if possible) will not be the same
    • -
    -
    -

    1.4 Compare encoders on the struts and encoders on the platforms

    -
    +
    +

    1.8.2 Obtained plant

    +
    +

    +Let’s identify the transfer function from \(\tau\) to \(d\mathcal{L}\) and from \(\tau\) to \(\mathcal{X}\). +

    %% Options for Linearized
     options = linearizeOptions;
    @@ -334,133 +947,30 @@ mdl = 'nano_hexapod';
     clear io; io_i = 1;
     io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
     io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
    -
    -
    +io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % 6DoF perfect measurement -
    -
    n_hexapod.motion_sensor = struct();
    -n_hexapod.motion_sensor.type = 1; % 1: Leg / 2: Plate
    -
    -
    - -
    -
    Gs = linearize(mdl, io, 0.0, options);
    -Gs.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    -Gs.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    -
    -
    - -
    -
    n_hexapod.motion_sensor = struct();
    -n_hexapod.motion_sensor.type = 2; % 1: Leg / 2: Plate
    -
    -
    - -
    -
    Gp = linearize(mdl, io, 0.0, options);
    -Gp.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    -Gp.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    -
    -
    -
    -
    - -
    -

    1.5 Nano Hexapod

    -
    -
    -
    %% Options for Linearized
    -options = linearizeOptions;
    -options.SampleTime = 0;
    -
    -%% Name of the Simulink File
    -mdl = 'nano_hexapod';
    -
    -%% Input/Output definition
    -clear io; io_i = 1;
    -io(io_i) = linio([mdl, '/F'],  1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
    -io(io_i) = linio([mdl, '/Fe'], 1, 'openinput');  io_i = io_i + 1; % External Force
    -io(io_i) = linio([mdl, '/D'],  1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
    -io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors
    -io(io_i) = linio([mdl, '/X'],  1, 'openoutput'); io_i = io_i + 1; % Absolute Motion of top platform
    -
    -%% Run the linearization
     G = linearize(mdl, io, 0.0, options);
    -G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6', ...
    -                'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
    +G.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
     G.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6', ...
    -                'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6' ...
                     'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
     
    -
    -size(G)
    -State-space model with 18 outputs, 12 inputs, and 60 states.
    -
    - -

    -Verify the number of DOF: +Then use the Jacobian matrices to obtain the “cartesian” centralized plant.

    - - - --- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    ElementDoF
    Bot Plate0
    Struts6*4
    Top Plate + Sample12
    Total:36
    +
    +
    Gc = inv(n_hexapod.geometry.J)*...
    +     G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*...
    +     inv(n_hexapod.geometry.J');
    +
    -
    -

    1.5.1 DVF Plant

    -

    -The DC gain from actuator to relative motion sensor should be equal to (for the 2dof APA): -\[ \frac{1}{k + k_a + kk_a/k_e} \] +The DC gain of the obtained plant is shown in Table 4.

    - -
    -1.8732e-08
    -
    - - -

    -Which is almost the case: -

    - - - +
    +@@ -477,344 +987,82 @@ Which is almost the case: - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + +
    Table 4: DC gain of the centralized plant at the center of stiffness
    -1.8676e-085.2396e-11-8.8935e-112.6392e-119.7629e-11-1.5609e-109.8602e-097.8692e-111.4426e-11-2.663e-102.8e-10-4.7559e-11
    -1.4754e-11-1.8725e-081.4348e-11-5.7189e-11-3.3327e-117.0799e-118.457e-119.8788e-09-2.4002e-11-2.9502e-11-1.3262e-10-8.7346e-11
    -3.4924e-11-7.7445e-11-1.8607e-08-4.6578e-11-1.9592e-114.0299e-11-4.3244e-112.4075e-138.4775e-091.1442e-11-2.5809e-102.8796e-11
    6.1699e-11-7.1962e-111.3374e-11-1.8623e-08-1.7898e-105.4541e-11-1.8326e-09-9.318e-106.8188e-101.4697e-065.5936e-098.7632e-10
    9.735e-11-1.0698e-10-1.6424e-112.3718e-11-1.8826e-088.387e-114.6906e-101.5911e-091.6989e-10-5.223e-091.4729e-06-2.6059e-10
    -3.94e-112.6436e-11-1.1338e-105.0793e-111.2358e-10-1.8792e-08-6.5754e-11-3.0408e-125.394e-11-1.0917e-106.9479e-104.2979e-07

    -Other (off-diagonal) elements should be close to zero (probably limited to joint’s stiffness). -However, when setting joint’s axial stiffness to infinity, I obtain better diagonal. +As the rotations and translations have very different gains, we normalize each motion to one.

    - - -
    -

    nano_hexapod_struts_2dof_dvf_plant.png -

    -

    Figure 1: IFF Plant

    -
    -
    -
    - -
    -

    1.5.2 IFF Plant

    -
    - -
    -

    nano_hexapod_struts_2dof_iff_plant.png -

    -

    Figure 2: DVF Plant

    -
    -
    -
    -
    - -
    -

    1.6 Decoupling at the CoK

    -
    -
    Gx = inv(Jk)*G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(Jk');
    -Gx.inputname  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
    -Gx.outputname = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
    -
    -
    - - - - --- -- -- -- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    -9.8399e-09-8.0785e-11-1.5723e-112.6356e-10-4.9857e-105.2177e-11
    -8.7554e-11-9.8585e-092.6292e-112.0485e-102.0385e-108.2942e-11
    4.3837e-11-5.5862e-14-8.4859e-09-9.0036e-122.9325e-10-3.104e-11
    1.901e-095.3078e-10-7.5477e-10-1.4759e-06-7.7918e-09-7.491e-10
    -7.1795e-11-1.6333e-09-1.8359e-105.2707e-09-1.4794e-063.2304e-10
    7.0322e-111.7713e-12-5.8019e-117.4838e-11-8.9677e-10-4.2938e-07
    -
    -
    - -
    -

    1.7 Jacobian

    -
    -
    -
    -

    1.7.1 Compute Jacobian

    -
    -
    -
    Ai = [out.b1.Data(1,:)
    -      out.b2.Data(1,:)
    -      out.b3.Data(1,:)
    -      out.b4.Data(1,:)
    -      out.b5.Data(1,:)
    -      out.b6.Data(1,:)]';
    -
    -
    -Bi = [out.a1.Data(1,:)
    -      out.a2.Data(1,:)
    -      out.a3.Data(1,:)
    -      out.a4.Data(1,:)
    -      out.a5.Data(1,:)
    -      out.a6.Data(1,:)]';
    -
    -
    - -
    -
    stewart = initializeStewartPlatform();
    -stewart = initializeFramesPositions(stewart, 'H', 95e-3, 'MO_B', 150e-3);
    -stewart = generateGeneralConfiguration(stewart);
    -stewart.platform_F.Fa = Ai;
    -stewart.platform_M.Mb = Bi - [0;0;1]*stewart.geometry.H;
    -stewart = computeJointsPose(stewart);
    -stewart = initializeStewartPose(stewart, 'AP', [0;0;0], 'ARB', eye(3));
    -
    -
    - -
    -
    stewart = initializeCylindricalPlatforms(stewart);
    -stewart = initializeCylindricalStruts(stewart);
    -
    -
    - -
    -
    stewart = initializeStrutDynamics(stewart, 'K', 1e6*ones(6,1), 'C', 1e2*ones(6,1));
    -stewart = initializeAmplifiedStrutDynamics(stewart);
    -stewart = initializeJointDynamics(stewart);
    -
    -
    - -
    -
    displayArchitecture(stewart, 'views', 'all');
    -
    -
    - -
    -describeStewartPlatform(stewart)
    -GEOMETRY:
    -- The height between the fixed based and the top platform is 95 [mm].
    -- Frame {A} is located 150 [mm] above the top platform.
    -- The initial length of the struts are:
    -	 82.8, 82.8, 82.8, 82.8, 82.8, 82.8 [mm]
    -
    -ACTUATORS:
    -- The actuators are mechanicaly amplified.
    -- The vertical stiffness and damping contribution of the piezoelectric stack is:
    -	 ka = 2e+07 [N/m] 	 ca = 1e+01 [N/(m/s)]
    -- Vertical stiffness when the piezoelectric stack is removed is:
    -	 kr = 5e+06 [N/m] 	 cr = 1e+01 [N/(m/s)]
    -
    -JOINTS:
    -- The joints on the fixed based are universal joints
    -- The joints on the mobile based are spherical joints
    -- The position of the joints on the fixed based with respect to {F} are (in [mm]):
    -	 -86.2 	 -74.7 	  22.3
    -	  86.3 	 -74.6 	  22.3
    -	  108 	 -37.3 	  22.3
    -	  21.5 	  112 	  22.3
    -	 -21.5 	  112 	  22.3
    -	 -108 	 -37.5 	  22.2
    -- The position of the joints on the mobile based with respect to {M} are (in [mm]):
    -	 -28.4 	 -106 	 -22.4
    -	  28.5 	 -106 	 -22.5
    -	  106 	  28.5 	 -22.5
    -	  77.8 	  77.8 	 -22.5
    -	 -77.8 	  77.8 	 -22.5
    -	 -106 	  28.4 	 -22.5
    -
    - -
    -
    stewart = computeJacobian(stewart);
    -
    -
    -
    -
    - -
    -

    1.7.2 Verify Jacobian

    -
    -
    -
    load('./mat/nano_hexapod_object.mat', 'stewart');
    -J = stewart.kinematics.J;
    +
    Gc = diag(1./diag(dcgain(Gc)))*Gc;
     

    -Transformation of motion: +The diagonal and off-diagonal elements are shown in Figure 9, and we can see good decoupling at low frequency.

    -
    -
    G1 = -inv(J)*G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'});
    -G2 = G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'});
    -
    -
    -

    -Transformation of Forces and Torques +

    +

    nano_hexapod_diagonal_plant_cok.png

    -
    -
    G1 = G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'});
    -G2 = G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(J');
    -
    +

    Figure 9: Diagonal and off-diagonal elements of the (normalized) decentralized plant with the Jacobians estimated at the “center of stiffness”

    +

    -Not working because the forces applied on the APA are not really forces applied on the top platform (reduced by a factor ~ka/ke) -

    -
    -
    - -
    -

    1.7.3 Plant from APA forces to sample’s displacement

    -
    -
    -
    Gx = G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(J');
    -Gx.inputname  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
    -
    -
    -
    -
    - -
    -

    1.7.4 Plant in the Cartesian frame

    -
    -

    -Transfer function from forces/torque to displacement/rotation: -

    -
    -
    GJ = inv(J)*G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(J');
    -GJ.inputname  = {'Fx', 'Fy', 'Fz', 'Mx', 'My', 'Mz'};
    -GJ.outputname = {'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'};
    -
    -
    -
    -
    - -
    -

    1.7.5 Plant in the frame of the legs => Verification of Jacobian for displacements

    -
    -

    -Transfer Function in the frame of the legs -

    -
    -
    G1 = -J*G({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
    -G1.outputname = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    -
    -G2 = G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'});
    -
    -
    - -
    -

    -It is working fine (until internal resonance of strut). +The Jacobian matrices can be used to decoupled the plant at low frequency.

    @@ -823,8 +1071,20 @@ It is working fine (until internal resonance of strut).
    -

    1.8 Stiffness matrix

    -
    +

    1.9 Stiffness matrix

    +
    +

    + +

    +

    +The stiffness matrix of the nano-hexapod describes its induced static displacement/rotation when a force/torque is applied on its top platform. +The location of the applied force/torque and the expressed displacement/rotation can be defined as wanted. +Such location (or frame) is then used for the computation of the Jacobian which in turns is used to compute the stiffness matrix. +

    +
    +
    +

    1.9.1 Compute the theoretical stiffness of the nano-hexapod

    +

    Neglecting stiffness of the joints, we have: \[ K = J^t \mathcal{K} J \] @@ -839,22 +1099,86 @@ Let’s note the axial stiffness of the APA:

    Them axial stiffness of the struts \(k_s\): \[ k_s = \frac{k_z k_{\text{APA}}}{k_z + 2 k_{\text{APA}}} \] +with \(k_z\) the axial stiffness of the flexible joints.

    +

    +Let’s initialize the nano-hexapod. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '4dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof');
    +
    +
    + +

    +The axial stiffness of the joints and stiffnesses of the 2-DoF actuators are defined below. +

    +
    +
    k  = n_hexapod.actuator.k(1);
    +ke = n_hexapod.actuator.ke(1);
    +ka = n_hexapod.actuator.ka(1);
    +kz = n_hexapod.flex_top.kz(1); % Joint's axial stiffness [m/N]
    +
    +
    + +

    +The total axial stiffness of the APA is: +

    kAPA = k + ke*ka/(ke + ka);
    -ks = kz*kAPA/(kz + 2*kAPA);
     
    +
    +kAPA = 1.799e+06 [N/m]
    +
    + + +

    +And the total axial stiffness of the struts is: +

    -
    Ks = J'*(ks*eye(6))*J
    +
    ks = kz*kAPA/(kz + 2*kAPA);
     
    - +
    +ks = 1.737e+06 [N/m]
    +
    +
    +

    +We can see that the axial stiffness of the flexible joint as little impact on the total axial stiffness of the struts. +

    + +
    + +

    +Let’s now compute the stiffness matrix corresponding to an hexapod with perfect joints and the above computed axial strut stiffness: +

    +
    +
    Ks = n_hexapod.geometry.J'*(ks*eye(6))*n_hexapod.geometry.J;
    +
    +
    + +

    +And the compliance matrix can be computed as the inverse of the stiffness matrix. +

    +
    +
    C = inv(Ks);
    +
    +
    + +

    +The obtained compliance matrix is shown in Table 5. +

    +
    + + @@ -870,140 +1194,97 @@ ks = kz*kAPA/(kz - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + + - - - - - + + + + +
    Table 5: Compliance Matrix - Perfect Joints
    1.9937e-067.7027e-12-1.5885e-11-1.6811e-108.7902e-063.0476e-111.9938e-06-2.3138e-223.3403e-231.0202e-218.7906e-062.9603e-11
    7.7027e-121.9937e-062.7221e-11-8.7901e-066.5081e-122.8619e-11-3.1875e-231.9938e-062.2094e-11-8.7909e-06-1.6576e-22-3.5622e-28
    -1.5885e-112.7221e-112.6114e-07-1.3209e-10-6.1725e-114.0402e-116.6811e-232.2094e-112.6115e-07-9.8337e-113.4744e-227.4663e-28
    -1.6811e-10-8.7901e-06-1.3209e-104.5712e-05-5.7781e-10-4.5817e-101.4054e-22-8.7909e-06-9.8337e-114.5715e-057.3086e-221.5706e-27
    8.7902e-066.5081e-12-6.1725e-11-5.7781e-104.5712e-055.1628e-108.7906e-06-1.0202e-211.7371e-224.498e-214.5714e-059.8237e-11
    3.0476e-112.8619e-114.0402e-11-4.5817e-105.1628e-102.9603e-11-1.9261e-22-1.7611e-278.4925e-229.8237e-11 1.3277e-05
    +
    +
    - +
    +

    1.9.2 Comparison with Simscape Model

    +
    +

    +Let’s now identify the compliance matrix using Simscape. +

    +
    +
    %% Options for Linearized
    +options = linearizeOptions;
    +options.SampleTime = 0;
     
    +%% Name of the Simulink File
    +mdl = 'nano_hexapod';
     
    -
    -+%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Fe'], 1, 'openinput'); io_i = io_i + 1; % External forces [N, Nm/rad] +io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % Induced motion [m, rad] -- -- -- -- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1.9869e-062.043e-081.6444e-09-4.094e-088.7579e-06-3.5002e-09
    3.746e-091.9841e-06-5.5274e-09-8.7257e-06-5.5082e-08-7.6978e-09
    -3.2056e-09-6.4295e-112.6079e-073.4103e-10-9.3873e-099.6146e-10
    -1.1677e-08-8.736e-062.4322e-084.5343e-052.52e-072.5932e-08
    8.7439e-068.441e-085.9144e-09-1.6879e-074.546e-05-9.8986e-09
    3.3453e-094.508e-101.8718e-09-2.7294e-092.8776e-081.3193e-05
    +G = linearize(mdl, io, 0.0, options); + +

    -We can see that we have almost the same stiffness: +The DC gain of the identified plant is therefore the compliance matrix of the nano-hexapod. +It takes into account the bending and torsional stiffness of the flexible joints.

    - +

    +The obtained compliance matrix is shown in Table 6. +

    +
    +@@ -1020,67 +1301,715 @@ We can see that we have almost the same stiffness: - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + + - - - - - - + + + + + +
    Table 6: Compliance Matrix - Estimated from Simscape
    1.00340.00037703-0.00965970.00410631.0037-0.00870711.9863e-062.4683e-082.2859e-09-6.4768e-088.7307e-063.2754e-10
    0.00205621.0048-0.00492471.0074-0.00011815-0.0037178-3.693e-091.9663e-06-7.4503e-09-8.615e-06-6.7926e-08-2.0193e-08
    0.0049552-0.423381.0013-0.387340.00657540.042021-3.5525e-09-8.8671e-102.6042e-074.7504e-09-9.9677e-091.7242e-10
    0.0143971.0062-0.0054311.0081-0.0022929-0.0176682.1133e-08-8.6554e-063.2841e-084.4849e-053.0873e-078.1525e-08
    1.00537.7101e-05-0.0104360.00342331.0055-0.0521578.7375e-069.9165e-088.135e-09-2.5208e-074.5331e-053.0602e-09
    0.00911020.0634850.0215840.167860.0179411.00636.1611e-096.1733e-092.6778e-09-3.3188e-083.3887e-081.3212e-05
    + +
    +

    +The bending and torsional stiffness of the flexible joints induces a lot of coupling between forces/torques applied to the to platform to its displacement/rotation. +It can be seen by comparison the compliance matrices in Tables 5 and 6. +

    + +
    +
    +
    +
    +
    + +
    +

    2 Active Damping using Integral Force Feedback

    +
    +

    + +

    +

    +In this section Integral Force Feedback (IFF) strategy is used to damp the nano-hexapod resonances. +

    + +

    +It is structured as follows: +

    +
      +
    • Section 2.1: the IFF plant is identified
    • +
    • Section 2.2: the optimal control gain is identified using the Root Locus plot
    • +
    • Section 2.3: the IFF is applied, and the effect on the damped plant is identified and compared with the un-damped one
    • +
    • Section 2.4: the IFF is applied, and the effect on the compliance is identified
    • +
    +
    +
    +

    2.1 Plant Identification

    +
    +

    + +

    + +

    +The nano-hexapod is initialized as usual. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '4dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof');
    +
    +
    + +

    +The transfer function from actuator inputs to force sensors outputs is identified. +

    +
    +
    %% Options for Linearized
    +options = linearizeOptions;
    +options.SampleTime = 0;
    +
    +%% Name of the Simulink File
    +mdl = 'nano_hexapod';
    +
    +%% Input/Output definition
    +clear io; io_i = 1;
    +io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1;  % Actuator Inputs
    +io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors
    +
    +Giff = linearize(mdl, io, 0.0, options);
    +Giff.InputName  = {'F1',  'F2',  'F3',  'F4',  'F5',  'F6'};
    +Giff.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'};
    +
    +
    + +

    +Its bode plot is shown in Figure +

    + +
    +

    nano_hexapod_iff_plant_bode_plot.png +

    +

    Figure 10: Integral Force Feedback plant

    +
    +
    +
    + +
    +

    2.2 Root Locus

    +
    +

    + +

    + +

    +The controller is a diagonal (i.e. decentralized) controller with simple low pass filters (i.e. pseudo integrators) on the diagonal: +

    +\begin{equation} +K_{\text{IFF}} = \frac{g}{s + \omega_c} \bm{I}_{6 \times 6} +\end{equation} + +

    +The value of \(\omega_c\) has quite a large impact both on the attainable damping and on the compliance degradation at low frequency. +

    + +

    +It is here chosen to have quite a large \(\omega_c\) in order to not modify the plant at low frequency. +

    +
    +
    wc = 2*pi*20;
    +
    +
    + +

    +The obtained Root Locus is shown in Figure 11. +The control gain chosen for future plots is shown by the red crosses. +

    + +
    +

    nano_hexapod_iff_root_locus.png +

    +

    Figure 11: Root locus for the decentralized IFF control strategy

    +
    + + +

    +The obtained controller is then: +

    +
    +
    Kiff = 1.5e4/(s + 2*pi*20)*eye(6); % IFF Controller
    +
    +
    + +

    +The corresponding loop gain of the diagonal terms are shown in Figure 12. +It is shown that the loop gain is quite large around resonances (which allows to add lots of damping) and less than one at low frequency thanks to the large value of \(\omega_c\). +

    + +
    +

    nano_hexapod_iff_loop_gain.png +

    +

    Figure 12: Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{IFF}}(i,i)\)

    +
    +
    +
    + +
    +

    2.3 Effect of IFF on the plant

    +
    +

    + +

    + +

    +Let’s now see how the IFF control strategy effectively damps the plant and how it affects the transfer functions from the actuator forces to the relative motion sensors (encoders). +First identify the plant in open-loop. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof', ...
    +                                       'controller_type', 'none');
    +
    +Gol = linearize(mdl, io, 0.0, options);
    +Gol.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gol.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    +
    +
    + +

    +And then with the IFF controller. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '3dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof', ...
    +                                       'controller_type', 'iff');
    +
    +Giff = linearize(mdl, io, 0.0, options);
    +Giff.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Giff.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    +
    +
    + +

    +The obtained plants are compared in Figure 13. +

    + +
    +

    nano_hexapod_effect_iff_plant.png +

    +

    Figure 13: Bode plots of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors \(\mathcal{L}_i\) with and without the IFF controller.

    +
    + +
    +

    +The Integral Force Feedback Strategy is very effective to damp the 6 suspension modes of the nano-hexapod. +

    + +
    +
    +
    + +
    +

    2.4 Effect of IFF on the compliance

    +
    +

    + +

    + +

    +The IFF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large. +Let’s quantify that for the nano-hexapod. +The obtained compliances are compared in Figure +

    + +
    +

    nano_hexapod_iff_compare_compliance.png +

    +

    Figure 14: Comparison of the compliances in Open Loop and with Integral Force Feedback controller

    +
    + +
    +

    +The use of IFF induces a degradation of the compliance. +This degradation is limited due to the use of a pseudo integrator (instead of a pure integrator). +Also, it should not be a major problem for the NASS, as no direct forces should be applied to the top platform. +

    + +
    +
    +
    +
    + +
    +

    3 Active Damping using Direct Velocity Feedback - Encoders on the struts

    +
    +

    + +

    +

    +In this section, the Direct Velocity Feedback (DVF) strategy is used to damp the nano-hexapod resonances. +

    + +

    +It is structured as follows: +

    +
      +
    • Section 3.1: the DVF plant is identified
    • +
    • Section 3.2: the optimal control gain is identified using the Root Locus plot
    • +
    • Section 3.3: the DVF is applied, and the effect on the damped plant is identified and compared with the un-damped one
    • +
    • Section 3.4: the DVF is applied, and the effect on the compliance is identified
    • +
    +
    +
    +

    3.1 Plant Identification

    +
    +

    + +

    + +

    +The nano-hexapod is initialized as usual. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '4dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof');
    +
    +
    + +

    +The transfer function from actuator inputs to force sensors outputs is identified. +

    +
    +
    %% Options for Linearized
    +options = linearizeOptions;
    +options.SampleTime = 0;
    +
    +%% Name of the Simulink File
    +mdl = 'nano_hexapod';
    +
    +%% Input/Output definition
    +clear io; io_i = 1;
    +io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
    +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
    +
    +Gdvf = linearize(mdl, io, 0.0, options);
    +Gdvf.InputName =  {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gdvf.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    +
    +
    + +

    +Its bode plot is shown in Figure 15. +

    + +
    +

    nano_hexapod_dvf_plant_bode_plot_struts.png +

    +

    Figure 15: Direct Velocity Feedback plant

    +
    +
    +
    + +
    +

    3.2 Root Locus

    +
    +

    + +

    + +

    +The controller is a diagonal (i.e. decentralized) controller with simple high pass filters (i.e. pseudo derivators) on the diagonal: +

    +\begin{equation} +K_{\text{DVF}} = g \frac{s}{s + \omega_d} \bm{I}_{6 \times 6} +\end{equation} + +

    +The value of \(\omega_d\) sets the frequency above high the derivative action is stopped. +

    + +
    +
    wd = 2*pi*150;
    +
    +
    + +

    +The obtained Root Locus is shown in Figure 16. +The control gain chosen for future plots is shown by the red crosses. +

    + +
    +

    nano_hexapod_dvf_root_locus_struts.png +

    +

    Figure 16: Root locus for the decentralized DVF control strategy

    +
    + + +

    +The obtained controller is then: +

    +
    +
    Kdvf = 5e8*s/(s + wd)*eye(6); % DVF Controller
    +
    +
    + +

    +The corresponding loop gain of the diagonal terms are shown in Figure 17. +It is shown that the loop gain is quite large around resonances (which allows to add lots of damping) and less than one at low frequency thanks to the large value of \(\omega_c\). +

    + +
    +

    nano_hexapod_dvf_loop_gain_struts.png +

    +

    Figure 17: Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{DVF}}(i,i)\)

    +
    +
    +
    + +
    +

    3.3 Effect of DVF on the plant

    +
    +

    + +

    + +

    +Let’s now see how the DVF control strategy effectively damps the plant and how it affects the transfer functions from the actuator forces to the relative motion sensors (encoders). +First identify the plant in open-loop. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '4dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof', ...
    +                                       'controller_type', 'none');
    +
    +Gol = linearize(mdl, io, 0.0, options);
    +Gol.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gol.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    +
    +
    + +

    +And then with the DVF controller. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '4dof', ...
    +                                       'motion_sensor_type', 'struts', ...
    +                                       'actuator_type', '2dof', ...
    +                                       'controller_type', 'dvf');
    +
    +Gdvf = linearize(mdl, io, 0.0, options);
    +Gdvf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gdvf.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    +
    +
    + +

    +The obtained plants are compared in Figure 18. +

    + +
    +

    nano_hexapod_effect_dvf_plant_struts.png +

    +

    Figure 18: Bode plots of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors \(\mathcal{L}_i\) with and without the DVF controller.

    +
    + +
    +

    +The Direct Velocity Feedback Strategy is very effective to damp the 6 suspension modes of the nano-hexapod. +

    + +
    +
    +
    + +
    +

    3.4 Effect of DVF on the compliance

    +
    +

    + +

    + +

    +The DVF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large. +Let’s quantify that for the nano-hexapod. +The obtained compliances are compared in Figure 19. +

    + +
    +

    nano_hexapod_dvf_compare_compliance_struts.png +

    +

    Figure 19: Comparison of the compliances in Open Loop and with Direct Velocity Feedback controller

    +
    +
    +
    +
    + +
    +

    4 Active Damping using Direct Velocity Feedback - Encoders on the plates

    +
    +

    + +

    +

    +In this section, the Direct Velocity Feedback (DVF) strategy is used to damp the nano-hexapod resonances. +

    + +

    +It is structured as follows: +

    +
      +
    • Section 4.1: the DVF plant is identified
    • +
    • Section 4.2: the optimal control gain is identified using the Root Locus plot
    • +
    • Section 4.3: the DVF is applied, and the effect on the damped plant is identified and compared with the un-damped one
    • +
    • Section 4.4: the DVF is applied, and the effect on the compliance is identified
    • +
    +
    +
    +

    4.1 Plant Identification

    +
    +

    + +

    + +

    +The nano-hexapod is initialized as usual. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '4dof', ...
    +                                       'motion_sensor_type', 'plates', ...
    +                                       'actuator_type', '2dof');
    +
    +
    + +

    +The transfer function from actuator inputs to force sensors outputs is identified. +

    +
    +
    %% Options for Linearized
    +options = linearizeOptions;
    +options.SampleTime = 0;
    +
    +%% Name of the Simulink File
    +mdl = 'nano_hexapod';
    +
    +%% Input/Output definition
    +clear io; io_i = 1;
    +io(io_i) = linio([mdl, '/F'], 1, 'openinput');  io_i = io_i + 1; % Actuator Inputs
    +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
    +
    +Gdvf = linearize(mdl, io, 0.0, options);
    +Gdvf.InputName =  {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gdvf.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    +
    +
    + +

    +Its bode plot is shown in Figure 20. +

    + +
    +

    nano_hexapod_dvf_plant_bode_plot_plates.png +

    +

    Figure 20: Direct Velocity Feedback plant

    +
    +
    +
    + +
    +

    4.2 Root Locus

    +
    +

    + +

    + +

    +The controller is a diagonal (i.e. decentralized) controller with simple high pass filters (i.e. pseudo derivators) on the diagonal: +

    +\begin{equation} +K_{\text{DVF}} = g \frac{s}{s + \omega_d} \bm{I}_{6 \times 6} +\end{equation} + +

    +The value of \(\omega_d\) sets the frequency above high the derivative action is stopped. +

    + +
    +
    wd = 2*pi*150;
    +
    +
    + +

    +The obtained Root Locus is shown in Figure 21. +The control gain chosen for future plots is shown by the red crosses. +

    + +
    +

    nano_hexapod_dvf_root_locus_plates.png +

    +

    Figure 21: Root locus for the decentralized DVF control strategy

    +
    + + +

    +The obtained controller is then: +

    +
    +
    Kdvf = 2e8*s/(s + wd)*eye(6); % DVF Controller
    +
    +
    + +

    +The corresponding loop gain of the diagonal terms are shown in Figure 22. +It is shown that the loop gain is quite large around resonances (which allows to add lots of damping). +

    + +
    +

    nano_hexapod_dvf_loop_gain_plates.png +

    +

    Figure 22: Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{DVF}}(i,i)\)

    +
    +
    +
    + +
    +

    4.3 Effect of DVF on the plant

    +
    +

    + +

    + +

    +Let’s now see how the DVF control strategy effectively damps the plant and how it affects the transfer functions from the actuator forces to the relative motion sensors (encoders). +First identify the plant in open-loop. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '4dof', ...
    +                                       'motion_sensor_type', 'plates', ...
    +                                       'actuator_type', '2dof', ...
    +                                       'controller_type', 'none');
    +
    +Gol = linearize(mdl, io, 0.0, options);
    +Gol.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gol.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    +
    +
    + +

    +And then with the DVF controller. +

    +
    +
    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
    +                                       'flex_top_type', '4dof', ...
    +                                       'motion_sensor_type', 'plates', ...
    +                                       'actuator_type', '2dof', ...
    +                                       'controller_type', 'dvf');
    +
    +Gdvf = linearize(mdl, io, 0.0, options);
    +Gdvf.InputName  = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'};
    +Gdvf.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'};
    +
    +
    + +

    +The obtained plants are compared in Figure 23. +

    + +
    +

    nano_hexapod_effect_dvf_plant_plates.png +

    +

    Figure 23: Bode plots of the transfer functions from actuator forces \(\tau_i\) to relative motion sensors \(\mathcal{L}_i\) with and without the DVF controller.

    +
    + +
    +

    +The Direct Velocity Feedback Strategy is very effective in damping the 6 suspension modes of the nano-hexapod. +

    + +
    +
    +
    + +
    +

    4.4 Effect of DVF on the compliance

    +
    +

    + +

    + +

    +The DVF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large. +Let’s quantify that for the nano-hexapod. +The obtained compliances are compared in Figure 24. +

    + +
    +

    nano_hexapod_dvf_compare_compliance_plates.png +

    +

    Figure 24: Comparison of the compliances in Open Loop and with Direct Velocity Feedback controller

    +
    -

    2 Function - Initialize Nano Hexapod

    -
    +

    5 Function - Initialize Nano Hexapod

    +

    @@ -1090,7 +2019,7 @@ We can see that we have almost the same stiffness:

    Function description

    -
      function [nano_hexapod] = initializeNanoHexapodFinal(args)
    +
    function [nano_hexapod] = initializeNanoHexapodFinal(args)
     
    @@ -1100,47 +2029,51 @@ We can see that we have almost the same stiffness:

    Optional Parameters

    -
      arguments
    -      %% Bottom Flexible Joints
    -      args.flex_bot_type      char {mustBeMember(args.flex_bot_type,{'2dof', '3dof', '4dof', 'flexible'})} = '4dof'
    -      args.flex_bot_kRx (6,1) double {mustBeNumeric} = ones(6,1)*5 % X bending stiffness [Nm/rad]
    -      args.flex_bot_kRy (6,1) double {mustBeNumeric} = ones(6,1)*5 % Y bending stiffness [Nm/rad]
    -      args.flex_bot_kRz (6,1) double {mustBeNumeric} = ones(6,1)*260 % Torsionnal stiffness [Nm/rad]
    -      args.flex_bot_kz  (6,1) double {mustBeNumeric} = ones(6,1)*1e8 % Axial Stiffness [N/m]
    -      args.flex_bot_cRx (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % X bending Damping [Nm/(rad/s)]
    -      args.flex_bot_cRy (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Y bending Damping [Nm/(rad/s)]
    -      args.flex_bot_cRz (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Torsionnal Damping [Nm/(rad/s)]
    -      args.flex_bot_cz  (6,1) double {mustBeNumeric} = ones(6,1)*1e2 % Axial Damping [N/(m/s)]
    -      %% Top Flexible Joints
    -      args.flex_top_type      char {mustBeMember(args.flex_top_type,{'2dof', '3dof', '4dof', 'flexible'})} = '4dof'
    -      args.flex_top_kRx (6,1) double {mustBeNumeric} = ones(6,1)*5 % X bending stiffness [Nm/rad]
    -      args.flex_top_kRy (6,1) double {mustBeNumeric} = ones(6,1)*5 % Y bending stiffness [Nm/rad]
    -      args.flex_top_kRz (6,1) double {mustBeNumeric} = ones(6,1)*260 % Torsionnal stiffness [Nm/rad]
    -      args.flex_top_kz  (6,1) double {mustBeNumeric} = ones(6,1)*1e8 % Axial Stiffness [N/m]
    -      args.flex_top_cRx (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % X bending Damping [Nm/(rad/s)]
    -      args.flex_top_cRy (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Y bending Damping [Nm/(rad/s)]
    -      args.flex_top_cRz (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Torsionnal Damping [Nm/(rad/s)]
    -      args.flex_top_cz  (6,1) double {mustBeNumeric} = ones(6,1)*1e2 % Axial Damping [N/(m/s)]
    -      %% Relative Motion Sensor
    -      args.motion_sensor_type char {mustBeMember(args.motion_sensor_type,{'struts', 'plates'})} = 'struts'
    -      %% Actuators
    -      args.actuator_type char {mustBeMember(args.actuator_type,{'2dof', 'flexible frame', 'flexible'})} = 'flexible'
    -      args.actuator_Ga  (6,1) double {mustBeNumeric} = ones(6,1)*1 % Actuator gain [N/V]
    -      args.actuator_Gs  (6,1) double {mustBeNumeric} = ones(6,1)*1 % Sensor gain [V/m]
    -      % For 2DoF
    -      args.actuator_k   (6,1) double {mustBeNumeric} = ones(6,1)*0.35e6 % [N/m]
    -      args.actuator_ke  (6,1) double {mustBeNumeric} = ones(6,1)*1.5e6 % [N/m]
    -      args.actuator_ka  (6,1) double {mustBeNumeric} = ones(6,1)*43e6 % [N/m]
    -      args.actuator_c   (6,1) double {mustBeNumeric} = ones(6,1)*3e1 % [N/(m/s)]
    -      args.actuator_ce  (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % [N/(m/s)]
    -      args.actuator_ca  (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % [N/(m/s)]
    -      args.actuator_Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056 % [m]
    -      % For Flexible Frame
    -      args.actuator_ks (6,1) double {mustBeNumeric} = ones(6,1)*235e6 % Stiffness of one stack [N/m]
    -      args.actuator_cs (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % Stiffness of one stack [N/m]
    -      % For Flexible
    -      args.actuator_xi  (1,1) double {mustBeNumeric} = 0.01 % Sensor gain [V/m]
    -  end
    +
    arguments
    +    %% Bottom Flexible Joints
    +    args.flex_bot_type      char {mustBeMember(args.flex_bot_type,{'2dof', '3dof', '4dof', 'flexible'})} = '4dof'
    +    args.flex_bot_kRx (6,1) double {mustBeNumeric} = ones(6,1)*5 % X bending stiffness [Nm/rad]
    +    args.flex_bot_kRy (6,1) double {mustBeNumeric} = ones(6,1)*5 % Y bending stiffness [Nm/rad]
    +    args.flex_bot_kRz (6,1) double {mustBeNumeric} = ones(6,1)*260 % Torsionnal stiffness [Nm/rad]
    +    args.flex_bot_kz  (6,1) double {mustBeNumeric} = ones(6,1)*1e8 % Axial Stiffness [N/m]
    +    args.flex_bot_cRx (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % X bending Damping [Nm/(rad/s)]
    +    args.flex_bot_cRy (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Y bending Damping [Nm/(rad/s)]
    +    args.flex_bot_cRz (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Torsionnal Damping [Nm/(rad/s)]
    +    args.flex_bot_cz  (6,1) double {mustBeNumeric} = ones(6,1)*1e2 % Axial Damping [N/(m/s)]
    +    %% Top Flexible Joints
    +    args.flex_top_type      char {mustBeMember(args.flex_top_type,{'2dof', '3dof', '4dof', 'flexible'})} = '4dof'
    +    args.flex_top_kRx (6,1) double {mustBeNumeric} = ones(6,1)*5 % X bending stiffness [Nm/rad]
    +    args.flex_top_kRy (6,1) double {mustBeNumeric} = ones(6,1)*5 % Y bending stiffness [Nm/rad]
    +    args.flex_top_kRz (6,1) double {mustBeNumeric} = ones(6,1)*260 % Torsionnal stiffness [Nm/rad]
    +    args.flex_top_kz  (6,1) double {mustBeNumeric} = ones(6,1)*1e8 % Axial Stiffness [N/m]
    +    args.flex_top_cRx (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % X bending Damping [Nm/(rad/s)]
    +    args.flex_top_cRy (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Y bending Damping [Nm/(rad/s)]
    +    args.flex_top_cRz (6,1) double {mustBeNumeric} = ones(6,1)*0.1 % Torsionnal Damping [Nm/(rad/s)]
    +    args.flex_top_cz  (6,1) double {mustBeNumeric} = ones(6,1)*1e2 % Axial Damping [N/(m/s)]
    +    %% Jacobian - Location of frame {A} and {B}
    +    args.MO_B (1,1) double {mustBeNumeric} = 150e-3 % Height of {B} w.r.t. {M} [m]
    +    %% Relative Motion Sensor
    +    args.motion_sensor_type char {mustBeMember(args.motion_sensor_type,{'struts', 'plates'})} = 'struts'
    +    %% Actuators
    +    args.actuator_type char {mustBeMember(args.actuator_type,{'2dof', 'flexible frame', 'flexible'})} = 'flexible'
    +    args.actuator_Ga  (6,1) double {mustBeNumeric} = ones(6,1)*1 % Actuator gain [N/V]
    +    args.actuator_Gs  (6,1) double {mustBeNumeric} = ones(6,1)*1 % Sensor gain [V/m]
    +                                                                 % For 2DoF
    +    args.actuator_k   (6,1) double {mustBeNumeric} = ones(6,1)*0.35e6 % [N/m]
    +    args.actuator_ke  (6,1) double {mustBeNumeric} = ones(6,1)*1.5e6 % [N/m]
    +    args.actuator_ka  (6,1) double {mustBeNumeric} = ones(6,1)*43e6 % [N/m]
    +    args.actuator_c   (6,1) double {mustBeNumeric} = ones(6,1)*3e1 % [N/(m/s)]
    +    args.actuator_ce  (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % [N/(m/s)]
    +    args.actuator_ca  (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % [N/(m/s)]
    +    args.actuator_Leq (6,1) double {mustBeNumeric} = ones(6,1)*0.056 % [m]
    +                                                                     % For Flexible Frame
    +    args.actuator_ks (6,1) double {mustBeNumeric} = ones(6,1)*235e6 % Stiffness of one stack [N/m]
    +    args.actuator_cs (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % Stiffness of one stack [N/m]
    +                                                                  % For Flexible
    +    args.actuator_xi  (1,1) double {mustBeNumeric} = 0.01 % Damping Ratio
    +    %% Controller
    +    args.controller_type char {mustBeMember(args.controller_type,{'none', 'iff', 'dvf'})} = 'none'
    +end
     
    @@ -1284,9 +2217,9 @@ Flexible frame and fully flexible nano_hexapod.actuator.M = readmatrix('APA300ML_b_mat_M.CSV'); % Mass Matrix nano_hexapod.actuator.P = extractNodes('APA300ML_b_out_nodes_3D.txt'); % Node coordinates [m] case 'flexible' - nano_hexapod.actuator.K = readmatrix('APA300ML_full_mat_K.CSV'); % Stiffness Matrix - nano_hexapod.actuator.M = readmatrix('APA300ML_full_mat_M.CSV'); % Mass Matrix - nano_hexapod.actuator.P = extractNodes('APA300ML_full_out_nodes_3D.txt'); % Node coordiantes [m] + nano_hexapod.actuator.K = readmatrix('full_APA300ML_K.CSV'); % Stiffness Matrix + nano_hexapod.actuator.M = readmatrix('full_APA300ML_M.CSV'); % Mass Matrix + nano_hexapod.actuator.P = extractNodes('full_APA300ML_out_nodes_3D.txt'); % Node coordiantes [m] end nano_hexapod.actuator.xi = args.actuator_xi; % Damping ratio @@ -1298,6 +2231,129 @@ nano_hexapod.actuator.cs = args.actuator_cs; % Damping
    +
    +

    Geometry

    +
    +
    +
    nano_hexapod.geometry = struct();
    +
    +
    + +

    +Center of joints \(a_i\) with respect to {F}: +

    +
    +
    Fa = [[-86.05,  -74.78, 22.49],
    +      [ 86.05,  -74.78, 22.49],
    +      [ 107.79, -37.13, 22.49],
    +      [ 21.74,  111.91, 22.49],
    +      [-21.74,  111.91, 22.49],
    +      [-107.79, -37.13, 22.49]]'*1e-3; % Ai w.r.t. {F} [m]
    +
    +
    + +

    +Center of joints \(b_i\) with respect to {M}: +

    +
    +
    Mb = [[-28.47, -106.25, -22.50],
    +      [ 28.47, -106.25, -22.50],
    +      [ 106.25, 28.47,  -22.50],
    +      [ 77.78,  77.78,  -22.50],
    +      [-77.78,  77.78,  -22.50],
    +      [-106.25, 28.47,  -22.50]]'*1e-3; % Bi w.r.t. {M} [m]
    +
    +
    + +

    +Now compute the positions \(b_i\) with respect to {F}: +

    +
    +
    Fb = Mb + [0; 0; 95e-3]; % Bi w.r.t. {F} [m]
    +
    +
    + +

    +The unit vector representing the orientation of the struts can then be computed: +

    +
    +
    si = Fb - Fa;
    +si = si./vecnorm(si); % Normalize
    +
    +
    + +

    +Location of encoder measurement points when fixed on the plates: +

    +
    +
    Fc = [[-29.362, -105.765, 52.605]
    +      [ 29.362, -105.765, 52.605]
    +      [ 106.276, 27.454,  52.605]
    +      [ 76.914,  78.31,   52.605]
    +      [-76.914,  78.31,   52.605]
    +      [-106.276, 27.454,  52.605]]'*1e-3; % Meas pos w.r.t. {F}
    +Mc = Fc - [0; 0; 95e-3]; % Meas pos w.r.t. {M}
    +
    +
    + +
    +
    nano_hexapod.geometry.Fa = Fa;
    +nano_hexapod.geometry.Fb = Fb;
    +nano_hexapod.geometry.Fc = Fc;
    +nano_hexapod.geometry.Mb = Mb;
    +nano_hexapod.geometry.Mc = Mc;
    +nano_hexapod.geometry.si = si;
    +nano_hexapod.geometry.MO_B = args.MO_B;
    +
    +
    +
    +
    + +
    +

    Jacobian for Actuators

    +
    +
    +
    Bb = Mb - [0; 0; args.MO_B];
    +
    +nano_hexapod.geometry.J = [nano_hexapod.geometry.si', cross(Bb, nano_hexapod.geometry.si)'];
    +
    +
    +
    +
    + +
    +

    Jacobian for Sensors

    +
    +
    +
    switch args.motion_sensor_type
    +  case 'struts'
    +    nano_hexapod.geometry.Js = nano_hexapod.geometry.J;
    +  case 'plates'
    +    Bc = Mc - [0; 0; args.MO_B];
    +    nano_hexapod.geometry.Js = [nano_hexapod.geometry.si', cross(Bc, nano_hexapod.geometry.si)'];
    +end
    +
    +
    +
    +
    + +
    +

    Controller

    +
    +
    +
    switch args.controller_type
    +  case 'none'
    +    nano_hexapod.controller.type = 1;
    +  case 'iff'
    +    nano_hexapod.controller.type = 2;
    +  case 'dvf'
    +    nano_hexapod.controller.type = 3;
    +end
    +
    +
    +
    +
    +

    Save the Structure

    @@ -1313,7 +2369,7 @@ nano_hexapod.actuator.cs = args.actuator_cs; % Damping

    Author: Dehaeze Thomas

    -

    Created: 2021-04-19 lun. 16:22

    +

    Created: 2021-04-23 ven. 13:22

    diff --git a/mat/stages.mat b/mat/stages.mat index 278ac2f..f878984 100644 Binary files a/mat/stages.mat and b/mat/stages.mat differ diff --git a/matlab/nano_hexapod/nano_hexapod.slx b/matlab/nano_hexapod/nano_hexapod.slx index 705b3ff..db13e21 100644 Binary files a/matlab/nano_hexapod/nano_hexapod.slx and b/matlab/nano_hexapod/nano_hexapod.slx differ diff --git a/org/index.org b/org/index.org index 980bbdc..2d4a65b 100644 --- a/org/index.org +++ b/org/index.org @@ -81,6 +81,7 @@ Active damping techniques are applied to the full Simscape model. * Control of the Nano-Active-Stabilization-System ([[file:control.org][link]]) In this file are gathered all studies about the control the Nano-Active-Stabilization-System. +* Nano-Hexapod Simscape Model ([[file:nano_hexapod.org][link]]) * Useful Matlab Functions ([[./functions.org][link]]) Many matlab functions are shared among all the files of the projects. diff --git a/org/nano_hexapod.org b/org/nano_hexapod.org index fb8e357..447a518 100644 --- a/org/nano_hexapod.org +++ b/org/nano_hexapod.org @@ -3,7 +3,15 @@ * Introduction :ignore: +In this document, a Simscape model of the nano-hexapod is developed. + +- Section [[sec:nano_hexapod]]: +- Section [[sec:integral_force_feedback]]: +- Section [[sec:direct_velocity_feedback_struts]]: +- Section [[sec:direct_velocity_feedback_plates]]: + * Nano-Hexapod +<> ** Introduction :ignore: ** Matlab Init :noexport:ignore: @@ -233,10 +241,11 @@ exportFig('figs/nano_hexapod_effect_flexible_apa.pdf', 'width', 'full', 'height' #+RESULTS: [[file:figs/nano_hexapod_effect_flexible_apa.png]] -#+begin_important +#+begin_question The first resonance is strange when using the flexible APA model (Figure [[fig:nano_hexapod_effect_flexible_apa]]). +Moreover the system is unstable. Otherwise, the 2DoF model matches quite well the flexible model considering its simplicity. -#+end_important +#+end_question ** Nano Hexapod - Number of DoF <> @@ -263,7 +272,7 @@ These states are summarized on table [[tab:num_states_nano_hexapod]]. #+name: tab:num_states_nano_hexapod #+caption: Number of states for the minimalist model -#+attr_latex: :environment tabularx :width 0.3\linewidth :align cc +#+attr_latex: :environment tabularx :width 0.25\linewidth :align Xc #+attr_latex: :center t :booktabs t :float t | Element | States | |-----------+--------| @@ -340,8 +349,8 @@ mdl = 'nano_hexapod'; %% Input/Output definition clear io; io_i = 1; -io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs -io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs [N] +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative displacements [m] n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '3dof', ... @@ -369,7 +378,7 @@ dcgain(Gdvf) #+name: tab:dvf_dc_gain #+caption: DC gain of the DVF plant -#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc +#+attr_latex: :environment tabularx :width 0.8\linewidth :align cccccc #+attr_latex: :center t :booktabs t :float t #+RESULTS: | 1.8617e-08 | -1.0408e-10 | 1.3034e-10 | 3.2559e-11 | -1.1188e-10 | 9.0385e-11 | @@ -397,7 +406,7 @@ for i = 1:6 end % Off diagonal terms plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... - 'DisplayName', '$F_{m,i}/\tau_j \quad i \neq j$'); + 'DisplayName', '$\mathcal{L}_{i}/\tau_j \quad i \neq j$'); for i = 1:5 for j = i+1:6 plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... @@ -784,7 +793,7 @@ exportFig('figs/nano_hexapod_cartesian_plant_encoder_comp.pdf', 'width', 'full', *** Center of Stiffness <> -Let's define some parameters: +Let's define some parameters that will be used for the computation of the stiffness matrix: #+begin_src matlab si = n_hexapod.geometry.si; % Orientation of struts bi = n_hexapod.geometry.Fb; % Location of bi w.r.t. {F} @@ -816,7 +825,7 @@ ans = Ok | 2.1511e-06 | | 0.052707 | -The center of the cube is therefore 52.7mm above the bottom platform {F} frame. +The "center of stiffness" is therefore 52.7mm above the bottom platform {F} frame. Let's initialize the hexapod with frame {A} and {B} at the CoK: #+begin_src matlab @@ -834,7 +843,7 @@ n_hexapod.geometry.J'*diag(ki)*n_hexapod.geometry.J #+name: tab:stiffness_matrix_cok #+caption: Normalized Stiffness Matrix - Center of Stiffness -#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc +#+attr_latex: :environment tabularx :width 0.85\linewidth :align cccccc #+attr_latex: :center t :booktabs t :float t #+RESULTS: | 1.8977 | 0 | 0 | 0 | -2.0817e-17 | -1.5311e-06 | @@ -847,7 +856,7 @@ n_hexapod.geometry.J'*diag(ki)*n_hexapod.geometry.J And we indeed obtain a diagonal stiffness matrix. *** Obtained plant - +Let's identify the transfer function from $\tau$ to $d\mathcal{L}$ and from $\tau$ to $\mathcal{X}$. #+begin_src matlab %% Options for Linearized options = linearizeOptions; @@ -860,29 +869,30 @@ mdl = 'nano_hexapod'; clear io; io_i = 1; io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs -io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs -#+end_src +io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % 6DoF perfect measurement -#+begin_src matlab G = linearize(mdl, io, 0.0, options); G.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; G.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6', ... 'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}; - -% % Cartesian plant using the Jacobians -% Gsc = inv(n_hexapod.geometry.Js)*Gs({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)'; -% % Cartesian plant using the perfect sensor -% Gsp = -Gs({'Dx', 'Dy', 'Dz', 'Rx', 'Ry', 'Rz'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J)'; #+end_src +Then use the Jacobian matrices to obtain the "cartesian" centralized plant. #+begin_src matlab -Gc = inv(n_hexapod.geometry.J)*G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*inv(n_hexapod.geometry.J'); +Gc = inv(n_hexapod.geometry.J)*... + G({'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}, {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'})*... + inv(n_hexapod.geometry.J'); #+end_src +The DC gain of the obtained plant is shown in Table [[tab:dc_gain_cartesian_cok_plan]]. #+begin_src matlab :results value replace :exports results :tangle no dcgain(Gc) #+end_src +#+name: tab:dc_gain_cartesian_cok_plan +#+caption: DC gain of the centralized plant at the center of stiffness +#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc +#+attr_latex: :center t :booktabs t :float t #+RESULTS: | 9.8602e-09 | 7.8692e-11 | 1.4426e-11 | -2.663e-10 | 2.8e-10 | -4.7559e-11 | | 8.457e-11 | 9.8788e-09 | -2.4002e-11 | -2.9502e-11 | -1.3262e-10 | -8.7346e-11 | @@ -938,14 +948,13 @@ The Jacobian matrices can be used to decoupled the plant at low frequency. #+end_important ** Stiffness matrix +<> *** Introduction :ignore: - The stiffness matrix of the nano-hexapod describes its induced static displacement/rotation when a force/torque is applied on its top platform. The location of the applied force/torque and the expressed displacement/rotation can be defined as wanted. Such location (or frame) is then used for the computation of the Jacobian which in turns is used to compute the stiffness matrix. *** Compute the theoretical stiffness of the nano-hexapod - Neglecting stiffness of the joints, we have: \[ K = J^t \mathcal{K} J \] where $\mathcal{K}$ is a diagonal 6x6 matrix with axial stiffness of the struts on the diagonal. @@ -957,6 +966,7 @@ Them axial stiffness of the struts $k_s$: \[ k_s = \frac{k_z k_{\text{APA}}}{k_z + 2 k_{\text{APA}}} \] with $k_z$ the axial stiffness of the flexible joints. +Let's initialize the nano-hexapod. #+begin_src matlab n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'flex_top_type', '4dof', ... @@ -964,13 +974,15 @@ n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... 'actuator_type', '2dof'); #+end_src +The axial stiffness of the joints and stiffnesses of the 2-DoF actuators are defined below. #+begin_src matlab k = n_hexapod.actuator.k(1); ke = n_hexapod.actuator.ke(1); ka = n_hexapod.actuator.ka(1); -kz = n_hexapod.flex_top.kz(1); +kz = n_hexapod.flex_top.kz(1); % Joint's axial stiffness [m/N] #+end_src +The total axial stiffness of the APA is: #+begin_src matlab kAPA = k + ke*ka/(ke + ka); #+end_src @@ -982,6 +994,7 @@ sprintf('kAPA = %.3e [N/m]', kAPA) #+RESULTS: : kAPA = 1.799e+06 [N/m] +And the total axial stiffness of the struts is: #+begin_src matlab ks = kz*kAPA/(kz + 2*kAPA); #+end_src @@ -993,25 +1006,28 @@ sprintf('ks = %.3e [N/m]', ks) #+RESULTS: : ks = 1.737e+06 [N/m] +#+begin_important We can see that the axial stiffness of the flexible joint as little impact on the total axial stiffness of the struts. +#+end_important -Let's now compute the stiffness matrix corresponding to an hexapod with perfect joints and the above computed axial stiffness: +Let's now compute the stiffness matrix corresponding to an hexapod with perfect joints and the above computed axial strut stiffness: #+begin_src matlab Ks = n_hexapod.geometry.J'*(ks*eye(6))*n_hexapod.geometry.J; #+end_src -And the compliance matrix can be computed as the inverse of the stiffness matrix: +And the compliance matrix can be computed as the inverse of the stiffness matrix. #+begin_src matlab C = inv(Ks); #+end_src +The obtained compliance matrix is shown in Table [[tab:compliance_matrix_perfect_joints]]. #+begin_src matlab :results value replace :exports results :tangle no ans = C #+end_src #+name: tab:compliance_matrix_perfect_joints #+caption: Compliance Matrix - Perfect Joints -#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc +#+attr_latex: :environment tabularx :width 0.85\linewidth :align cccccc #+attr_latex: :center t :booktabs t :float t #+RESULTS: | 1.9938e-06 | -2.3138e-22 | 3.3403e-23 | 1.0202e-21 | 8.7906e-06 | 2.9603e-11 | @@ -1022,6 +1038,7 @@ ans = C | 2.9603e-11 | -1.9261e-22 | -1.7611e-27 | 8.4925e-22 | 9.8237e-11 | 1.3277e-05 | *** Comparison with Simscape Model +Let's now identify the compliance matrix using Simscape. #+begin_src matlab %% Options for Linearized options = linearizeOptions; @@ -1032,19 +1049,24 @@ mdl = 'nano_hexapod'; %% Input/Output definition clear io; io_i = 1; -io(io_i) = linio([mdl, '/Fe'], 1, 'openinput'); io_i = io_i + 1; -io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; +io(io_i) = linio([mdl, '/Fe'], 1, 'openinput'); io_i = io_i + 1; % External forces [N, Nm/rad] +io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % Induced motion [m, rad] G = linearize(mdl, io, 0.0, options); #+end_src +The DC gain of the identified plant is therefore the compliance matrix of the nano-hexapod. +It takes into account the bending and torsional stiffness of the flexible joints. + +The obtained compliance matrix is shown in Table [[tab:compliance_matrix_simscape_joints]]. + #+begin_src matlab :results value replace :exports results :tangle no dcgain(G) #+end_src #+name: tab:compliance_matrix_simscape_joints #+caption: Compliance Matrix - Estimated from Simscape -#+attr_latex: :environment tabularx :width 0.9\linewidth :align cccccc +#+attr_latex: :environment tabularx :width 0.85\linewidth :align cccccc #+attr_latex: :center t :booktabs t :float t #+RESULTS: | 1.9863e-06 | 2.4683e-08 | 2.2859e-09 | -6.4768e-08 | 8.7307e-06 | 3.2754e-10 | @@ -1054,6 +1076,1263 @@ dcgain(G) | 8.7375e-06 | 9.9165e-08 | 8.135e-09 | -2.5208e-07 | 4.5331e-05 | 3.0602e-09 | | 6.1611e-09 | 6.1733e-09 | 2.6778e-09 | -3.3188e-08 | 3.3887e-08 | 1.3212e-05 | +#+begin_important +The bending and torsional stiffness of the flexible joints induces a lot of coupling between forces/torques applied to the to platform to its displacement/rotation. +It can be seen by comparison the compliance matrices in Tables [[tab:compliance_matrix_perfect_joints]] and [[tab:compliance_matrix_simscape_joints]]. +#+end_important + +* Active Damping using Integral Force Feedback +<> +** Introduction :ignore: +In this section *Integral Force Feedback* (IFF) strategy is used to damp the nano-hexapod resonances. + +It is structured as follows: +- Section [[sec:iff_identification]]: the IFF plant is identified +- Section [[sec:iff_controller]]: the optimal control gain is identified using the Root Locus plot +- Section [[sec:iff_effect_plant]]: the IFF is applied, and the effect on the damped plant is identified and compared with the un-damped one +- Section [[sec:iff_effect_compliance]]: the IFF is applied, and the effect on the compliance is identified + +** Matlab Init :noexport:ignore: +#+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 :tangle no +simulinkproject('../'); +#+end_src + +#+begin_src matlab :eval no +addpath('nano_hexapod'); +open('nano_hexapod/nano_hexapod.slx') +#+end_src + +#+begin_src matlab :tangle no +addpath('matlab/nano_hexapod'); +open('matlab/nano_hexapod/nano_hexapod.slx') +#+end_src + +** Plant Identification +<> + +The nano-hexapod is initialized as usual. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +#+end_src + +The transfer function from actuator inputs to force sensors outputs is identified. +#+begin_src matlab +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/Fm'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors + +Giff = linearize(mdl, io, 0.0, options); +Giff.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Giff.OutputName = {'Fm1', 'Fm2', 'Fm3', 'Fm4', 'Fm5', 'Fm6'}; +#+end_src + +Its bode plot is shown in Figure +#+begin_src matlab :exports none +freqs = 5*logspace(0, 2, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$F_{m,i}/\tau_i$') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +% Off diagonal terms +plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$F_{m,i}/\tau_j \quad i \neq j$'); +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [N/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(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'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_iff_plant_bode_plot.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_iff_plant_bode_plot +#+caption: Integral Force Feedback plant +#+RESULTS: +[[file:figs/nano_hexapod_iff_plant_bode_plot.png]] + +** Root Locus +<> + +The controller is a diagonal (i.e. decentralized) controller with simple low pass filters (i.e. pseudo integrators) on the diagonal: +\begin{equation} +K_{\text{IFF}} = \frac{g}{s + \omega_c} \bm{I}_{6 \times 6} +\end{equation} + +The value of $\omega_c$ has quite a large impact both on the attainable damping and on the compliance degradation at low frequency. + +It is here chosen to have quite a large $\omega_c$ in order to not modify the plant at low frequency. +#+begin_src matlab +wc = 2*pi*20; +#+end_src + +The obtained Root Locus is shown in Figure [[fig:nano_hexapod_iff_root_locus]]. +The control gain chosen for future plots is shown by the red crosses. +#+begin_src matlab :exports none +gains = logspace(2, 6, 200); + +figure; + +hold on; +% Pure Integrator +set(gca,'ColorOrderIndex',1); +plot(real(pole(Giff)), imag(pole(Giff)), 'x', 'DisplayName', '$g = 0$'); +set(gca,'ColorOrderIndex',1); +plot(real(tzero(Giff)), imag(tzero(Giff)), 'o', 'HandleVisibility', 'off'); +for g = gains + clpoles = pole(feedback(Giff, (g/(s + wc))*eye(6))); + set(gca,'ColorOrderIndex',1); + plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off'); +end + +g = 1.5e4 +clpoles = pole(feedback(Giff, (g/(s + wc))*eye(6))); +set(gca,'ColorOrderIndex',1); +plot(real(clpoles), imag(clpoles), 'rx', 'DisplayName', '$g = 1.5 \cdot 10^4$'); +axis square; +xlim([-225, 5]); ylim([-10, 450]); +xlabel('Real Part'); ylabel('Imaginary Part'); +legend('location', 'northwest'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_iff_root_locus.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_iff_root_locus +#+caption: Root locus for the decentralized IFF control strategy +#+RESULTS: +[[file:figs/nano_hexapod_iff_root_locus.png]] + + +The obtained controller is then: +#+begin_src matlab +Kiff = 1.5e4/(s + 2*pi*20)*eye(6); % IFF Controller +#+end_src + +The corresponding loop gain of the diagonal terms are shown in Figure [[fig:nano_hexapod_iff_loop_gain]]. +It is shown that the loop gain is quite large around resonances (which allows to add lots of damping) and less than one at low frequency thanks to the large value of $\omega_c$. +#+begin_src matlab :exports none +Piff = Giff*Kiff; % Loop Gain + +freqs = 5*logspace(0, 2, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Piff(i,i), freqs, 'Hz'))), '-'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Piff(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'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_iff_loop_gain.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_iff_loop_gain +#+caption: Loop gain of the diagonal terms $G(i,i) \cdot K_{\text{IFF}}(i,i)$ +#+RESULTS: +[[file:figs/nano_hexapod_iff_loop_gain.png]] + +** Effect of IFF on the plant +<> + +Let's now see how the IFF control strategy effectively damps the plant and how it affects the transfer functions from the actuator forces to the relative motion sensors (encoders). +#+begin_src matlab :exports none +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motions +#+end_src + +First identify the plant in open-loop. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'none'); + +Gol = linearize(mdl, io, 0.0, options); +Gol.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gol.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; +#+end_src + +And then with the IFF controller. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'iff'); + +Giff = linearize(mdl, io, 0.0, options); +Giff.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Giff.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; +#+end_src + +The obtained plants are compared in Figure [[fig:nano_hexapod_effect_iff_plant]]. +#+begin_src matlab :exports none +freqs = logspace(0, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(Gol(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{L}_{i}/\tau_i$ - OL') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gol(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{L}_{i}/\tau_i$ - IFF') +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +% Off diagonal terms +plot(freqs, abs(squeeze(freqresp(Giff(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$\mathcal{L}_{i}/\tau_j \quad i \neq j$'); +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(Giff(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gol(i,i), freqs, 'Hz'))), '-'); +end +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, 180/pi*angle(squeeze(freqresp(Giff(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'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_effect_iff_plant.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_effect_iff_plant +#+caption: Bode plots of the transfer functions from actuator forces $\tau_i$ to relative motion sensors $\mathcal{L}_i$ with and without the IFF controller. +#+RESULTS: +[[file:figs/nano_hexapod_effect_iff_plant.png]] + +#+begin_important +The Integral Force Feedback Strategy is very effective to damp the 6 suspension modes of the nano-hexapod. +#+end_important + +** Effect of IFF on the compliance +<> + +The IFF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large. +Let's quantify that for the nano-hexapod. +#+begin_src matlab :exports none +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Fe'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motions +#+end_src + +#+begin_src matlab :exports none +%% Identification of the compliance in Open Loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'none'); + +Gol = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +%% Identification of the compliance with the IFF controller +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '3dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'iff'); + +Giff = linearize(mdl, io, 0.0, options); +#+end_src + +The obtained compliances are compared in Figure +#+begin_src matlab :exports none +freqs = logspace(0, 3, 1000); + +figure; +hold on; +plot(freqs, abs(squeeze(freqresp(Gol(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{X}/F_e$ - OL') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gol(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Giff(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{X}/F_e$ - IFF') +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Giff(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N, rad/Nm]'); +legend('location', 'southeast'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_iff_compare_compliance.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_iff_compare_compliance +#+caption: Comparison of the compliances in Open Loop and with Integral Force Feedback controller +#+RESULTS: +[[file:figs/nano_hexapod_iff_compare_compliance.png]] + +#+begin_important +The use of IFF induces a degradation of the compliance. +This degradation is limited due to the use of a pseudo integrator (instead of a pure integrator). +Also, it should not be a major problem for the NASS, as no direct forces should be applied to the top platform. +#+end_important + +* Active Damping using Direct Velocity Feedback - Encoders on the struts +<> +** Introduction :ignore: +In this section, the *Direct Velocity Feedback* (DVF) strategy is used to damp the nano-hexapod resonances. + +It is structured as follows: +- Section [[sec:dvf_identification_struts]]: the DVF plant is identified +- Section [[sec:dvf_controller_struts]]: the optimal control gain is identified using the Root Locus plot +- Section [[sec:dvf_effect_plant_struts]]: the DVF is applied, and the effect on the damped plant is identified and compared with the un-damped one +- Section [[sec:dvf_effect_compliance_struts]]: the DVF is applied, and the effect on the compliance is identified + +** Matlab Init :noexport:ignore: +#+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 :tangle no +simulinkproject('../'); +#+end_src + +#+begin_src matlab :eval no +addpath('nano_hexapod'); +open('nano_hexapod/nano_hexapod.slx') +#+end_src + +#+begin_src matlab :tangle no +addpath('matlab/nano_hexapod'); +open('matlab/nano_hexapod/nano_hexapod.slx') +#+end_src + +** Plant Identification +<> + +The nano-hexapod is initialized as usual. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof'); +#+end_src + +The transfer function from actuator inputs to force sensors outputs is identified. +#+begin_src matlab +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs + +Gdvf = linearize(mdl, io, 0.0, options); +Gdvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gdvf.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; +#+end_src + +Its bode plot is shown in Figure [[fig:nano_hexapod_dvf_plant_bode_plot_struts]]. +#+begin_src matlab :exports none +freqs = 5*logspace(0, 2, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{L}_i/\tau_i$') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +% Off diagonal terms +plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$\mathcal{L}_i/\tau_j \quad i \neq j$'); +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(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'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_dvf_plant_bode_plot_struts.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_dvf_plant_bode_plot_struts +#+caption: Direct Velocity Feedback plant +#+RESULTS: +[[file:figs/nano_hexapod_dvf_plant_bode_plot_struts.png]] + +** Root Locus +<> + +The controller is a diagonal (i.e. decentralized) controller with simple high pass filters (i.e. pseudo derivators) on the diagonal: +\begin{equation} +K_{\text{DVF}} = g \frac{s}{s + \omega_d} \bm{I}_{6 \times 6} +\end{equation} + +The value of $\omega_d$ sets the frequency above high the derivative action is stopped. + +#+begin_src matlab +wd = 2*pi*150; +#+end_src + +The obtained Root Locus is shown in Figure [[fig:nano_hexapod_dvf_root_locus_struts]]. +The control gain chosen for future plots is shown by the red crosses. +#+begin_src matlab :exports none +gains = logspace(6, 11, 200); + +figure; + +hold on; +% Pure Integrator +set(gca,'ColorOrderIndex',1); +plot(real(pole(Gdvf)), imag(pole(Gdvf)), 'x', 'DisplayName', '$g = 0$'); +set(gca,'ColorOrderIndex',1); +plot(real(tzero(Gdvf)), imag(tzero(Gdvf)), 'o', 'HandleVisibility', 'off'); +for g = gains + clpoles = pole(feedback(Gdvf, (g*s/(s + wd))*eye(6))); + set(gca,'ColorOrderIndex',1); + plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off'); +end + +g = 5e8; +clpoles = pole(feedback(Gdvf, (g*s/(s + wd))*eye(6))); +set(gca,'ColorOrderIndex',1); +plot(real(clpoles), imag(clpoles), 'rx', 'DisplayName', '$g = 5 \cdot 10^8$'); +axis square; +xlim([-350, 5]); ylim([-10, 2500]); +xlabel('Real Part'); ylabel('Imaginary Part'); +legend('location', 'northwest'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_dvf_root_locus_struts.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_dvf_root_locus_struts +#+caption: Root locus for the decentralized DVF control strategy +#+RESULTS: +[[file:figs/nano_hexapod_dvf_root_locus_struts.png]] + + +The obtained controller is then: +#+begin_src matlab +Kdvf = 5e8*s/(s + wd)*eye(6); % DVF Controller +#+end_src + +The corresponding loop gain of the diagonal terms are shown in Figure [[fig:nano_hexapod_dvf_loop_gain_struts]]. +It is shown that the loop gain is quite large around resonances (which allows to add lots of damping) and less than one at low frequency thanks to the large value of $\omega_c$. +#+begin_src matlab :exports none +Pdvf = Gdvf*Kdvf; % Loop Gain + +freqs = 5*logspace(0, 2, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Pdvf(i,i), freqs, 'Hz'))), '-'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Pdvf(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'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_dvf_loop_gain_struts.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_dvf_loop_gain_struts +#+caption: Loop gain of the diagonal terms $G(i,i) \cdot K_{\text{DVF}}(i,i)$ +#+RESULTS: +[[file:figs/nano_hexapod_dvf_loop_gain_struts.png]] + +** Effect of DVF on the plant +<> + +Let's now see how the DVF control strategy effectively damps the plant and how it affects the transfer functions from the actuator forces to the relative motion sensors (encoders). +#+begin_src matlab :exports none +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/D'], 1, 'output'); io_i = io_i + 1; % Relative Motions +#+end_src + +First identify the plant in open-loop. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'none'); + +Gol = linearize(mdl, io, 0.0, options); +Gol.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gol.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; +#+end_src + +And then with the DVF controller. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'dvf'); + +Gdvf = linearize(mdl, io, 0.0, options); +Gdvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gdvf.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; +#+end_src + +The obtained plants are compared in Figure [[fig:nano_hexapod_effect_dvf_plant_struts]]. +#+begin_src matlab :exports none +freqs = logspace(0, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(Gol(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{L}_{i}/\tau_i$ - OL') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gol(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{L}_{i}/\tau_i$ - DVF') +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +% Off diagonal terms +plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$\mathcal{L}_{i}/\tau_j \quad i \neq j$'); +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gol(i,i), freqs, 'Hz'))), '-'); +end +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(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'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_effect_dvf_plant_struts.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_effect_dvf_plant_struts +#+caption: Bode plots of the transfer functions from actuator forces $\tau_i$ to relative motion sensors $\mathcal{L}_i$ with and without the DVF controller. +#+RESULTS: +[[file:figs/nano_hexapod_effect_dvf_plant_struts.png]] + +#+begin_important +The Direct Velocity Feedback Strategy is very effective to damp the 6 suspension modes of the nano-hexapod. +#+end_important + +** Effect of DVF on the compliance +<> + +The DVF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large. +Let's quantify that for the nano-hexapod. +#+begin_src matlab :exports none +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Fe'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motions +#+end_src + +#+begin_src matlab :exports none +%% Identification of the compliance in Open Loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'none'); + +Gol = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +%% Identification of the compliance with the DVF controller +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'struts', ... + 'actuator_type', '2dof', ... + 'controller_type', 'dvf'); + +Gdvf = linearize(mdl, io, 0.0, options); +#+end_src + +The obtained compliances are compared in Figure [[fig:nano_hexapod_dvf_compare_compliance_struts]]. +#+begin_src matlab :exports none +freqs = logspace(0, 3, 1000); + +figure; +hold on; +plot(freqs, abs(squeeze(freqresp(Gol(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{X}/F_e$ - OL') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gol(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{X}/F_e$ - DVF') +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N, rad/Nm]'); +legend('location', 'southeast'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_dvf_compare_compliance_struts.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_dvf_compare_compliance_struts +#+caption: Comparison of the compliances in Open Loop and with Direct Velocity Feedback controller +#+RESULTS: +[[file:figs/nano_hexapod_dvf_compare_compliance_struts.png]] + +* Active Damping using Direct Velocity Feedback - Encoders on the plates +<> +** Introduction :ignore: +In this section, the *Direct Velocity Feedback* (DVF) strategy is used to damp the nano-hexapod resonances. + +It is structured as follows: +- Section [[sec:dvf_identification_plates]]: the DVF plant is identified +- Section [[sec:dvf_controller_plates]]: the optimal control gain is identified using the Root Locus plot +- Section [[sec:dvf_effect_plant_plates]]: the DVF is applied, and the effect on the damped plant is identified and compared with the un-damped one +- Section [[sec:dvf_effect_compliance_plates]]: the DVF is applied, and the effect on the compliance is identified + +** Matlab Init :noexport:ignore: +#+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 :tangle no +simulinkproject('../'); +#+end_src + +#+begin_src matlab :eval no +addpath('nano_hexapod'); +open('nano_hexapod/nano_hexapod.slx') +#+end_src + +#+begin_src matlab :tangle no +addpath('matlab/nano_hexapod'); +open('matlab/nano_hexapod/nano_hexapod.slx') +#+end_src + +** Plant Identification +<> + +The nano-hexapod is initialized as usual. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof'); +#+end_src + +The transfer function from actuator inputs to force sensors outputs is identified. +#+begin_src matlab +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs + +Gdvf = linearize(mdl, io, 0.0, options); +Gdvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gdvf.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; +#+end_src + +Its bode plot is shown in Figure [[fig:nano_hexapod_dvf_plant_bode_plot_plates]]. +#+begin_src matlab :exports none +freqs = 5*logspace(0, 2, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{L}_i/\tau_i$') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +% Off diagonal terms +plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$\mathcal{L}_i/\tau_j \quad i \neq j$'); +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(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'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_dvf_plant_bode_plot_plates.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_dvf_plant_bode_plot_plates +#+caption: Direct Velocity Feedback plant +#+RESULTS: +[[file:figs/nano_hexapod_dvf_plant_bode_plot_plates.png]] + +** Root Locus +<> + +The controller is a diagonal (i.e. decentralized) controller with simple high pass filters (i.e. pseudo derivators) on the diagonal: +\begin{equation} +K_{\text{DVF}} = g \frac{s}{s + \omega_d} \bm{I}_{6 \times 6} +\end{equation} + +The value of $\omega_d$ sets the frequency above high the derivative action is stopped. + +#+begin_src matlab +wd = 2*pi*150; +#+end_src + +The obtained Root Locus is shown in Figure [[fig:nano_hexapod_dvf_root_locus_plates]]. +The control gain chosen for future plots is shown by the red crosses. +#+begin_src matlab :exports none +gains = logspace(6, 11, 200); + +figure; + +hold on; +% Pure Integrator +set(gca,'ColorOrderIndex',1); +plot(real(pole(Gdvf)), imag(pole(Gdvf)), 'x', 'DisplayName', '$g = 0$'); +set(gca,'ColorOrderIndex',1); +plot(real(tzero(Gdvf)), imag(tzero(Gdvf)), 'o', 'HandleVisibility', 'off'); +for g = gains + clpoles = pole(feedback(Gdvf, (g*s/(s + wd))*eye(6))); + set(gca,'ColorOrderIndex',1); + plot(real(clpoles), imag(clpoles), '.', 'HandleVisibility', 'off'); +end + +g = 2e8; +clpoles = pole(feedback(Gdvf, (g*s/(s + wd))*eye(6))); +set(gca,'ColorOrderIndex',1); +plot(real(clpoles), imag(clpoles), 'rx', 'DisplayName', '$g = 5 \cdot 10^8$'); +axis square; +xlim([-600, 5]); ylim([-10, 3000]); +xlabel('Real Part'); ylabel('Imaginary Part'); +legend('location', 'northeast'); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_dvf_root_locus_plates.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_dvf_root_locus_plates +#+caption: Root locus for the decentralized DVF control strategy +#+RESULTS: +[[file:figs/nano_hexapod_dvf_root_locus_plates.png]] + + +The obtained controller is then: +#+begin_src matlab +Kdvf = 2e8*s/(s + wd)*eye(6); % DVF Controller +#+end_src + +The corresponding loop gain of the diagonal terms are shown in Figure [[fig:nano_hexapod_dvf_loop_gain_plates]]. +It is shown that the loop gain is quite large around resonances (which allows to add lots of damping). +#+begin_src matlab :exports none +Pdvf = Gdvf*Kdvf; % Loop Gain + +freqs = 5*logspace(0, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Pdvf(i,i), freqs, 'Hz'))), '-'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Loop Gain'); set(gca, 'XTickLabel',[]); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Pdvf(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'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_dvf_loop_gain_plates.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_dvf_loop_gain_plates +#+caption: Loop gain of the diagonal terms $G(i,i) \cdot K_{\text{DVF}}(i,i)$ +#+RESULTS: +[[file:figs/nano_hexapod_dvf_loop_gain_plates.png]] + +** Effect of DVF on the plant +<> + +Let's now see how the DVF control strategy effectively damps the plant and how it affects the transfer functions from the actuator forces to the relative motion sensors (encoders). +#+begin_src matlab :exports none +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/D'], 1, 'output'); io_i = io_i + 1; % Relative Motions +#+end_src + +First identify the plant in open-loop. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'none'); + +Gol = linearize(mdl, io, 0.0, options); +Gol.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gol.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; +#+end_src + +And then with the DVF controller. +#+begin_src matlab +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'dvf'); + +Gdvf = linearize(mdl, io, 0.0, options); +Gdvf.InputName = {'F1', 'F2', 'F3', 'F4', 'F5', 'F6'}; +Gdvf.OutputName = {'D1', 'D2', 'D3', 'D4', 'D5', 'D6'}; +#+end_src + +The obtained plants are compared in Figure [[fig:nano_hexapod_effect_dvf_plant_plates]]. +#+begin_src matlab :exports none +freqs = logspace(0, 3, 1000); + +figure; +tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(Gol(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{L}_{i}/\tau_i$ - OL') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gol(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{L}_{i}/\tau_i$ - DVF') +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +% Off diagonal terms +plot(freqs, abs(squeeze(freqresp(Gdvf(1, 2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'DisplayName', '$\mathcal{L}_{i}/\tau_j \quad i \neq j$'); +for i = 1:5 + for j = i+1:6 + plot(freqs, abs(squeeze(freqresp(Gdvf(i, j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ... + 'HandleVisibility', 'off'); + end +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +legend('location', 'southeast'); + +ax2 = nexttile; +hold on; +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gol(i,i), freqs, 'Hz'))), '-'); +end +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, 180/pi*angle(squeeze(freqresp(Gdvf(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'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_effect_dvf_plant_plates.pdf', 'width', 'full', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_effect_dvf_plant_plates +#+caption: Bode plots of the transfer functions from actuator forces $\tau_i$ to relative motion sensors $\mathcal{L}_i$ with and without the DVF controller. +#+RESULTS: +[[file:figs/nano_hexapod_effect_dvf_plant_plates.png]] + +#+begin_important +The Direct Velocity Feedback Strategy is very effective in damping the 6 suspension modes of the nano-hexapod. +#+end_important + +** Effect of DVF on the compliance +<> + +The DVF strategy has the well known drawback of degrading the compliance (transfer function from external forces/torques applied to the top platform to the motion of the top platform), especially at low frequency where the control gain is large. +Let's quantify that for the nano-hexapod. +#+begin_src matlab :exports none +%% Options for Linearized +options = linearizeOptions; +options.SampleTime = 0; + +%% Name of the Simulink File +mdl = 'nano_hexapod'; + +%% Input/Output definition +clear io; io_i = 1; +io(io_i) = linio([mdl, '/Fe'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inputs +io(io_i) = linio([mdl, '/X'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motions +#+end_src + +#+begin_src matlab :exports none +%% Identification of the compliance in Open Loop +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'none'); + +Gol = linearize(mdl, io, 0.0, options); +#+end_src + +#+begin_src matlab :exports none +%% Identification of the compliance with the DVF controller +n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ... + 'flex_top_type', '4dof', ... + 'motion_sensor_type', 'plates', ... + 'actuator_type', '2dof', ... + 'controller_type', 'dvf'); + +Gdvf = linearize(mdl, io, 0.0, options); +#+end_src + +The obtained compliances are compared in Figure [[fig:nano_hexapod_dvf_compare_compliance_plates]]. +#+begin_src matlab :exports none +freqs = logspace(0, 3, 1000); + +figure; +hold on; +plot(freqs, abs(squeeze(freqresp(Gol(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{X}/F_e$ - OL') +for i = 1:6 + set(gca,'ColorOrderIndex',1); + plot(freqs, abs(squeeze(freqresp(Gol(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +set(gca,'ColorOrderIndex',2); +plot(freqs, abs(squeeze(freqresp(Gdvf(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', '$\mathcal{X}/F_e$ - DVF') +for i = 1:6 + set(gca,'ColorOrderIndex',2); + plot(freqs, abs(squeeze(freqresp(Gdvf(i,i), freqs, 'Hz'))), '-', ... + 'HandleVisibility', 'off'); +end +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Amplitude [m/N, rad/Nm]'); +legend('location', 'southeast'); +xlim([freqs(1), freqs(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/nano_hexapod_dvf_compare_compliance_plates.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:nano_hexapod_dvf_compare_compliance_plates +#+caption: Comparison of the compliances in Open Loop and with Direct Velocity Feedback controller +#+RESULTS: +[[file:figs/nano_hexapod_dvf_compare_compliance_plates.png]] + * To-order :noexport: ** Verify why unstable strut @@ -1906,6 +3185,8 @@ arguments args.actuator_cs (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % Stiffness of one stack [N/m] % For Flexible args.actuator_xi (1,1) double {mustBeNumeric} = 0.01 % Damping Ratio + %% Controller + args.controller_type char {mustBeMember(args.controller_type,{'none', 'iff', 'dvf'})} = 'none' end #+end_src @@ -2135,6 +3416,22 @@ switch args.motion_sensor_type end #+end_src +** Controller +:PROPERTIES: +:UNNUMBERED: t +:END: + +#+begin_src matlab +switch args.controller_type + case 'none' + nano_hexapod.controller.type = 1; + case 'iff' + nano_hexapod.controller.type = 2; + case 'dvf' + nano_hexapod.controller.type = 3; +end +#+end_src + ** Save the Structure :PROPERTIES: :UNNUMBERED: t diff --git a/org/setup/org-setup-file.org b/org/setup/org-setup-file.org index d4a1f0c..865d80e 100644 --- a/org/setup/org-setup-file.org +++ b/org/setup/org-setup-file.org @@ -10,6 +10,15 @@ #+HTML_HEAD: #+HTML_HEAD: +#+BIND: org-latex-image-default-option "scale=1" +#+BIND: org-latex-image-default-width "" +#+BIND: org-latex-bib-compiler "biber" + +#+LaTeX_CLASS: scrreprt +#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full] +#+LaTeX_HEADER_EXTRA: \input{preamble.tex} +#+LATEX_HEADER_EXTRA: \addbibresource{ref.bib} + #+PROPERTY: header-args:matlab :session *MATLAB* #+PROPERTY: header-args:matlab+ :comments org #+PROPERTY: header-args:matlab+ :results none diff --git a/src/initializeNanoHexapodFinal.m b/src/initializeNanoHexapodFinal.m index 70c8a82..5a1e2ea 100644 --- a/src/initializeNanoHexapodFinal.m +++ b/src/initializeNanoHexapodFinal.m @@ -42,6 +42,8 @@ arguments args.actuator_cs (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % Stiffness of one stack [N/m] % For Flexible args.actuator_xi (1,1) double {mustBeNumeric} = 0.01 % Damping Ratio + %% Controller + args.controller_type char {mustBeMember(args.controller_type,{'none', 'iff', 'dvf'})} = 'none' end nano_hexapod = struct(); @@ -190,6 +192,15 @@ switch args.motion_sensor_type nano_hexapod.geometry.Js = [nano_hexapod.geometry.si', cross(Bc, nano_hexapod.geometry.si)']; end +switch args.controller_type + case 'none' + nano_hexapod.controller.type = 1; + case 'iff' + nano_hexapod.controller.type = 2; + case 'dvf' + nano_hexapod.controller.type = 3; +end + if nargout == 0 save('./mat/stages.mat', 'nano_hexapod', '-append'); end