From 6b225a90ea5cf178fa33ebcc12b698eb90459f8e Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Fri, 23 Apr 2021 17:37:19 +0200 Subject: [PATCH] Update Analysis --- docs/nano_hexapod.html | 343 ++++++++++++++------------- docs/nano_hexapod.pdf | Bin 5136237 -> 5136129 bytes matlab/nano_hexapod/nano_hexapod.slx | Bin 46175 -> 46146 bytes org/index.org | 2 + org/nano_hexapod.org | 47 +++- 5 files changed, 218 insertions(+), 174 deletions(-) diff --git a/docs/nano_hexapod.html b/docs/nano_hexapod.html index 47e3671..fa16bc6 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 @@ -76,26 +76,26 @@
  • 2. Active Damping using Integral Force Feedback
  • 3. Active Damping using Direct Velocity Feedback - Encoders on the struts
  • 4. Active Damping using Direct Velocity Feedback - Encoders on the plates
  • 5. Function - Initialize Nano Hexapod @@ -129,31 +129,31 @@ In this document, a Simscape model of the nano-hexapod is developed and studied. It is structured as follows:

    1 Nano-Hexapod

    - +

    1.1 Nano Hexapod - Configuration

    - +

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

    -The following code would produce the model shown in Figure 1. +The following code would produce the model shown in Figure 1. 

    n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
@@ -165,7 +165,7 @@ The following code would produce the model shown in Figure 
 
 
-
    
+
    
 
    nano_hexapod_simscape_encoder_struts.png
 
    
 
    Figure 1: 3D view of the Sismcape model for the Nano-Hexapod
    
@@ -175,21 +175,21 @@ The following code would produce the model shown in Figure     
 
 
    
 
    
 
    1.1.1 Flexible Joints
    
 
    
 
    
-
+
 
    
 
 
    
-The model of the flexible joint is composed of 3 solid bodies as shown in Figure 2 which are connected by joints representing the flexibility of the joint.
+The model of the flexible joint is composed of 3 solid bodies as shown in Figure 2 which are connected by joints representing the flexibility of the joint.
 
    
 
 
    
@@ -202,10 +202,10 @@ We can represent:
 
 
 
    
-The configurations and the represented flexibilities are summarized in Table 1.
+The configurations and the represented flexibilities are summarized in Table 1.
 
    
 
-
+
    @@ -254,7 +254,7 @@ Of course, adding more DoF for the flexible joint will induce an addition of man
 
    
 
 
-
    
+
    
 
    simscape_model_flexible_joint.png
 
    
 
    Figure 2: 3D view of the Sismcape model for the Flexible joint (4DoF configuration)
    
@@ -266,7 +266,7 @@ Of course, adding more DoF for the flexible joint will induce an addition of man
 
    1.1.2 Amplified Piezoelectric Actuators
    
 
    
 
    
-
+
 
    
 
 
    
@@ -278,11 +278,11 @@ The APA can be modeled in different ways which can be configured with the 
 
    
 
 
    
-The simplest model is a 2-DoF system shown in Figure 3.
+The simplest model is a 2-DoF system shown in Figure 3.
 
    
 
 
-
    
+
    
 
    2dof_apa_model.png
 
    
 
    Figure 3: Schematic of the 2DoF model for the Amplified Piezoelectric Actuator
    
@@ -298,11 +298,11 @@ Then, a more complex model based on a Finite Element Model can be used.
 
    1.1.3 Encoders
    
 
    
 
    
-
+
 
    
 
 
    
-The encoders can be either fixed directly on the struts (Figure 4) or on the two plates (Figure 5).
+The encoders can be either fixed directly on the struts (Figure 4) or on the two plates (Figure 5).
 
    
 
 
    
@@ -310,32 +310,32 @@ This can be configured with the motion_sensor_type parameters which
 
    
 
 
-
    
+
    
 
    encoder_struts.png
 
    
 
    Figure 4: 3D view of the Encoders fixed on the struts
    
 
    
 
 
-
    
+
    
 
    encoders_plates_with_apa.png
 
    
 
    Figure 5: 3D view of the Encoders fixed on the plates
    
 
    
 
 
    
-A complete view of the nano-hexapod with encoders fixed to the struts is shown in Figure 1 while it is shown in Figure 6 when the encoders are fixed to the plates.
+A complete view of the nano-hexapod with encoders fixed to the struts is shown in Figure 1 while it is shown in Figure 6 when the encoders are fixed to the plates.
 
    
 
 
-
    
+
    
 
    nano_hexapod_simscape_encoder_plates.png
 
    
 
    Figure 6: Nano-Hexapod with encoders fixed to the plates
    
 
    
 
 
    
-The encoder model is schematically represented in Figure 7:
+The encoder model is schematically represented in Figure 7:
 
    
 
      
 
    • a frame {B}, fixed to the ruler is positioned on its top surface
      • 
@@ -347,18 +347,18 @@ The output measurement is then the x displacement of the origin of the frame {F}
 
      
 
 
-
      
+
      
 
      simscape_encoder_model.png
 
      
 
      Figure 7: Schematic of the encoder model
      
 
      
 
 
      
-If the encoder is experiencing some tilt, it is then “converted” into a measured displacement as shown in Figure 8.
+If the encoder is experiencing some tilt, it is then “converted” into a measured displacement as shown in Figure 8.
 
      
 
 
-
      
+
      
 
      simscape_encoder_model_disp.png
 
      
 
      Figure 8: Schematic of the encoder model
      
@@ -370,7 +370,7 @@ If the encoder is experiencing some tilt, it is then “converted” int
 
      1.1.4 Jacobians
      
 
      
 
      
-
+
 
      
 
 
      
@@ -390,7 +390,7 @@ Same thing can be done for the measured encoder displacements.
 
      1.2 Effect of encoders on the decentralized plant
      
 
      
 
      
-
+
 
      
 
 
      
@@ -446,18 +446,19 @@ Gp.OutputName = {'D1', 
 
      
 
 
      
-The obtained plants are compared in Figure 9.
+The obtained plants are compared in Figure 9.
 
      
 
-
      
+
      
 
      nano_hexapod_effect_encoder.png
 
      
 
      Figure 9: 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?
+The zeros at  400Hz and 800Hz should corresponds to resonances of the system when one of the APA is blocked.
+It is linked to the axial stiffness of the flexible joints: increasing the axial stiffness of the joints will increase the frequency of the zeros.
 
      
 
 
      
@@ -468,7 +469,7 @@ Why do we have zeros at 400Hz and 800Hz when the encoders are fixed on the strut
 
      1.3 Effect of APA flexibility
      
 
      
 
      
-
+
 
      
 
 
      
@@ -503,15 +504,15 @@ Gf.OutputName = {'D1', 
 
      
 
 
-
      
+
      
 
      nano_hexapod_effect_flexible_apa.png
 
      
 
      Figure 10: 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 10).
+The first resonance is strange when using the flexible APA model (Figure 10).
 Moreover the system is unstable.
 Otherwise, the 2DoF model matches quite well the flexible model considering its simplicity.
 
      
@@ -524,7 +525,7 @@ Otherwise, the 2DoF model matches quite well the flexible model considering its
 
      1.4 Nano Hexapod - Number of DoF
      
 
      
 
      
-
+
 
      
 
 
      
@@ -549,10 +550,10 @@ There are 24 states.
 
 
 
      
-These states are summarized on table 2.
+These states are summarized on table 2.
 
      
 
-
    
 Table 1: Flexible joint’s configuration and associated represented flexibility
 
 

    
+
    @@ -636,7 +637,7 @@ 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.
@@ -650,7 +651,7 @@ Depending on the physical effects we want to model, we therefore know how many s
 
    1.5 Direct Velocity Feedback Plant
    
 
    
 
    
-
+
 
    
 
 
    
@@ -692,9 +693,9 @@ 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 3.
+Let’s verify that by looking at the DC gain of the \(6 \times 6\) DVF plant in Table 3.
 
    
-
    
 Table 2: Number of states for the minimalist model
 
 

    
+
    @@ -768,10 +769,10 @@ Let’s verify that by looking at the DC gain of the \(6 \times 6\) DVF plan
 
    
 Table 3: DC gain of the DVF plant
 
 

    
 
 
    
-And the bode plot of the DVF plant is shown in Figure 11.
+And the bode plot of the DVF plant is shown in Figure 11.
 
    
 
-
    
+
    
 
    nano_hexapod_struts_2dof_dvf_plant.png
 
    
 
    Figure 11: 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.
    
@@ -783,7 +784,7 @@ And the bode plot of the DVF plant is shown in Figure 111.6 Integral Force Feedback Plant
 
    
 
    
-
+
 
    
 
 
    
@@ -816,10 +817,10 @@ This is corresponding to the dynamics for the Integral Force Feedback (IFF) cont
 
    
 
 
    
-The bode plot is shown in Figure 12.
+The bode plot is shown in Figure 12.
 
    
 
-
    
+
    
 
    nano_hexapod_struts_2dof_iff_plant.png
 
    
 
    Figure 12: 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.
    
@@ -831,10 +832,10 @@ The bode plot is shown in Figure 12.
 
    1.7 Decentralized Plant - Cartesian coordinates
    
 
    
 
    
-
+
 
    
 
    
-Consider the plant shown in Figure 13 with:
+Consider the plant shown in Figure 13 with:
 
    
 
      
 
    • \(\tau\) the 6 input forces (APA)
      • 
@@ -844,7 +845,7 @@ Consider the plant shown in Figure 13 with:
 
    
 
 
-
    
+
    
 
    nano_hexapod_decentralized_schematic.png
 
    
 
    Figure 13: Plant in the cartesian Frame
    
@@ -898,10 +899,10 @@ Gsp = -Gs({'Dx', <
 
    
 
 
    
-The diagonal elements of the plant are shown in Figure 14.
+The diagonal elements of the plant are shown in Figure 14.
 
    
 
-
    
+
    
 
    nano_hexapod_comp_cartesian_plants_struts.png
 
    
 
    Figure 14: 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.
    
@@ -929,16 +930,16 @@ Gpp = -Gp({'Dx', <
 
    
 
 
    
-The obtained bode plots are shown in Figure 15.
+The obtained bode plots are shown in Figure 15.
 
    
 
-
    
+
    
 
    nano_hexapod_comp_cartesian_plants_plates.png
 
    
 
    Figure 15: 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.
 
    
@@ -955,10 +956,10 @@ However, then the encoders are fixed to the struts, there is a mismatch between
 
    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 16.
+The decentralized plants are now compared whether the encoders are fixed on the struts or on the plates in Figure 16.
 
    
 
-
    
+
    
 
    nano_hexapod_cartesian_plant_encoder_comp.png
 
    
 
    Figure 16: 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)
    
@@ -971,7 +972,7 @@ The decentralized plants are now compared whether the encoders are fixed on the
 
    1.8 Decentralized Plant - Decoupling at the Center of Stiffness
    
 
    
 
    
-
+
 
    
 
    
 
@@ -979,7 +980,7 @@ The decentralized plants are now compared whether the encoders are fixed on the
 
    1.8.1 Center of Stiffness
    
 
    
 
    
-
+
 
    
 
 
    
@@ -1085,7 +1086,7 @@ And the (normalized) stiffness matrix is computed as follows:
    - +
    @@ -1202,9 +1203,9 @@ Then use the Jacobian matrices to obtain the “cartesian” centralized

    -The DC gain of the obtained plant is shown in Table 5. +The DC gain of the obtained plant is shown in Table 5.

    -
    Table 4: Normalized Stiffness Matrix - Center of Stiffness
    +
    @@ -1286,16 +1287,16 @@ As the rotations and translations have very different gains, we normalize each m

    -The diagonal and off-diagonal elements are shown in Figure 17, and we can see good decoupling at low frequency. +The diagonal and off-diagonal elements are shown in Figure 17, and we can see good decoupling at low frequency.

    -
    +

    nano_hexapod_diagonal_plant_cok.png

    Figure 17: Diagonal and off-diagonal elements of the (normalized) decentralized plant with the Jacobians estimated at the “center of stiffness”

    -
    +

    The Jacobian matrices can be used to decoupled the plant at low frequency.

    @@ -1309,7 +1310,7 @@ The Jacobian matrices can be used to decoupled the plant at low frequency.

    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. @@ -1385,7 +1386,7 @@ 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.

    @@ -1409,9 +1410,9 @@ And the compliance matrix can be computed as the inverse of the stiffness matrix

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

    -
    Table 5: DC gain of the centralized plant at the center of stiffness
    +
    @@ -1515,10 +1516,10 @@ It takes into account the bending and torsional stiffness of the flexible joints

    -The obtained compliance matrix is shown in Table 7. +The obtained compliance matrix is shown in Table 7.

    -
    Table 6: Compliance Matrix - Perfect Joints
    +
    @@ -1591,10 +1592,10 @@ The obtained compliance matrix is shown in Table 7.
    Table 7: Compliance Matrix - Estimated from Simscape
    -
    +

    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 6 and 7. +It can be seen by comparison the compliance matrices in Tables 6 and 7.

    @@ -1607,7 +1608,7 @@ It can be seen by comparison the compliance matrices in Tables 2 Active Damping using Integral Force Feedback

    - +

    In this section Integral Force Feedback (IFF) strategy is used to damp the nano-hexapod resonances. @@ -1617,17 +1618,17 @@ In this section Integral Force Feedback (IFF) strategy is used to damp th 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
      • +
    • 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

    +
    +

    2.1 Plant Identification

    - +

    @@ -1667,7 +1668,7 @@ Giff.OutputName = {'Fm1', +

    nano_hexapod_iff_plant_bode_plot.png

    Figure 18: Integral Force Feedback plant

    @@ -1675,11 +1676,11 @@ Its bode plot is shown in Figure
    -
    -

    2.2 Root Locus

    +
    +

    2.2 Root Locus

    - +

    @@ -1702,11 +1703,11 @@ It is here chosen to have quite a large \(\omega_c\) in order to not modify the

    -The obtained Root Locus is shown in Figure 19. +The obtained Root Locus is shown in Figure 19. The control gain chosen for future plots is shown by the red crosses.

    -
    +

    nano_hexapod_iff_root_locus.png

    Figure 19: Root locus for the decentralized IFF control strategy

    @@ -1722,11 +1723,11 @@ The obtained controller is then:

    -The corresponding loop gain of the diagonal terms are shown in Figure 20. +The corresponding loop gain of the diagonal terms are shown in Figure 20. 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 20: Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{IFF}}(i,i)\)

    @@ -1738,7 +1739,7 @@ It is shown that the loop gain is quite large around resonances (which allows to

    2.3 Effect of IFF on the plant

    - +

    @@ -1775,16 +1776,16 @@ Giff.OutputName = {'D1', 21. +The obtained plants are compared in Figure 21.

    -
    +

    nano_hexapod_effect_iff_plant.png

    Figure 21: 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.

    @@ -1797,7 +1798,7 @@ The Integral Force Feedback Strategy is very effective to damp the 6 suspension

    2.4 Effect of IFF on the compliance

    - +

    @@ -1806,13 +1807,13 @@ Let’s quantify that for the nano-hexapod. The obtained compliances are compared in Figure

    -
    +

    nano_hexapod_iff_compare_compliance.png

    Figure 22: 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). @@ -1828,7 +1829,7 @@ Also, it should not be a major problem for the NASS, as no direct forces should

    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. @@ -1838,17 +1839,17 @@ In this section, the Direct Velocity Feedback (DVF) strategy is used to d 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
      • +
    • 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

    +
    +

    3.1 Plant Identification

    - +

    @@ -1885,10 +1886,10 @@ Gdvf.OutputName = {'D1', 23. +Its bode plot is shown in Figure 23.

    -
    +

    nano_hexapod_dvf_plant_bode_plot_struts.png

    Figure 23: Direct Velocity Feedback plant

    @@ -1896,11 +1897,11 @@ Its bode plot is shown in Figure 23.
    -
    -

    3.2 Root Locus

    +
    +

    3.2 Root Locus

    - +

    @@ -1920,11 +1921,11 @@ The value of \(\omega_d\) sets the frequency above high the derivative action is

    -The obtained Root Locus is shown in Figure 24. +The obtained Root Locus is shown in Figure 24. The control gain chosen for future plots is shown by the red crosses.

    -
    +

    nano_hexapod_dvf_root_locus_struts.png

    Figure 24: Root locus for the decentralized DVF control strategy

    @@ -1940,11 +1941,11 @@ The obtained controller is then:

    -The corresponding loop gain of the diagonal terms are shown in Figure 25. +The corresponding loop gain of the diagonal terms are shown in Figure 25. 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 25: Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{DVF}}(i,i)\)

    @@ -1952,11 +1953,11 @@ It is shown that the loop gain is quite large around resonances (which allows to
    -
    -

    3.3 Effect of DVF on the plant

    +
    +

    3.3 Effect of DVF on the plant

    - +

    @@ -1993,16 +1994,16 @@ Gdvf.OutputName = {'D1', 26. +The obtained plants are compared in Figure 26.

    -
    +

    nano_hexapod_effect_dvf_plant_struts.png

    Figure 26: 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.

    @@ -2011,20 +2012,20 @@ The Direct Velocity Feedback Strategy is very effective to damp the 6 suspension
    -
    -

    3.4 Effect of DVF on the compliance

    +
    +

    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 27. +The obtained compliances are compared in Figure 27.

    -
    +

    nano_hexapod_dvf_compare_compliance_struts.png

    Figure 27: Comparison of the compliances in Open Loop and with Direct Velocity Feedback controller

    @@ -2037,7 +2038,7 @@ The obtained compliances are compared in Figure 27.

    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. @@ -2047,17 +2048,17 @@ In this section, the Direct Velocity Feedback (DVF) strategy is used to d 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
      • +
    • 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

    +
    +

    4.1 Plant Identification

    - +

    @@ -2094,10 +2095,10 @@ Gdvf.OutputName = {'D1', 28. +Its bode plot is shown in Figure 28.

    -
    +

    nano_hexapod_dvf_plant_bode_plot_plates.png

    Figure 28: Direct Velocity Feedback plant

    @@ -2105,11 +2106,11 @@ Its bode plot is shown in Figure 28.
    -
    -

    4.2 Root Locus

    +
    +

    4.2 Root Locus

    - +

    @@ -2129,11 +2130,11 @@ The value of \(\omega_d\) sets the frequency above high the derivative action is

    -The obtained Root Locus is shown in Figure 29. +The obtained Root Locus is shown in Figure 29. The control gain chosen for future plots is shown by the red crosses.

    -
    +

    nano_hexapod_dvf_root_locus_plates.png

    Figure 29: Root locus for the decentralized DVF control strategy

    @@ -2149,11 +2150,11 @@ The obtained controller is then:

    -The corresponding loop gain of the diagonal terms are shown in Figure 30. +The corresponding loop gain of the diagonal terms are shown in Figure 30. 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 30: Loop gain of the diagonal terms \(G(i,i) \cdot K_{\text{DVF}}(i,i)\)

    @@ -2161,11 +2162,11 @@ It is shown that the loop gain is quite large around resonances (which allows to
    -
    -

    4.3 Effect of DVF on the plant

    +
    +

    4.3 Effect of DVF on the plant

    - +

    @@ -2202,16 +2203,16 @@ Gdvf.OutputName = {'D1', 31. +The obtained plants are compared in Figure 31.

    -
    +

    nano_hexapod_effect_dvf_plant_plates.png

    Figure 31: 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.

    @@ -2220,20 +2221,20 @@ The Direct Velocity Feedback Strategy is very effective in damping the 6 suspens
    -
    -

    4.4 Effect of DVF on the compliance

    +
    +

    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 32. +The obtained compliances are compared in Figure 32.

    -
    +

    nano_hexapod_dvf_compare_compliance_plates.png

    Figure 32: Comparison of the compliances in Open Loop and with Direct Velocity Feedback controller

    @@ -2246,7 +2247,7 @@ The obtained compliances are compared in Figure 32.

    5 Function - Initialize Nano Hexapod

    - +

    @@ -2604,7 +2605,7 @@ nano_hexapod.geometry.J = [nano_hexapod.geometry.si'

    Author: Dehaeze Thomas

    -

    Created: 2021-04-23 ven. 15:50

    +

    Created: 2021-04-23 ven. 17:35

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In this file are gathered all studies about the control the Nano-Active-Stabilization-System. * Nano-Hexapod Simscape Model ([[file:nano_hexapod.org][link]]) +The nano-hexapod simscape model is described and used for simulations. + * 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 aaac76c..f473324 100644 --- a/org/nano_hexapod.org +++ b/org/nano_hexapod.org @@ -264,9 +264,10 @@ exportFig('figs/nano_hexapod_effect_encoder.pdf', 'width', 'full', 'height', 'ta #+RESULTS: [[file:figs/nano_hexapod_effect_encoder.png]] -#+begin_question -Why do we have zeros at 400Hz and 800Hz when the encoders are fixed on the struts? -#+end_question +#+begin_important +The zeros at 400Hz and 800Hz should corresponds to resonances of the system when one of the APA is blocked. +It is linked to the axial stiffness of the flexible joints: increasing the axial stiffness of the joints will increase the frequency of the zeros. +#+end_important ** Effect of APA flexibility <> @@ -2446,6 +2447,46 @@ exportFig('figs/nano_hexapod_dvf_compare_compliance_plates.pdf', 'width', 'wide' [[file:figs/nano_hexapod_dvf_compare_compliance_plates.png]] * To-order :noexport: +** Why Zero when encoder on the struts +#+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 = 'test_apa300ml'; + +%% 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, '/dLs'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors +io(io_i) = linio([mdl, '/dLp'], 1, 'openoutput'); io_i = io_i + 1; % Force Sensors + +G = linearize(mdl, io, 0.0, options); +G.InputName = {'F'}; +G.OutputName = {'dLs', 'dLp'}; + +bodeFig({G(1), G(2)}, logspace(1,4,1000)) +#+end_src + +The zero seems to be linked to the axial flexibility of the top joint. + +In this mode, the APA does not experience any motion (hence the zero). +The resonance frequency then corresponds to the top mass on top of the axial stiffness of the two joints in series. + +For the nano-hexapod, it corresponds to the resonance of the top mass when all (or *just one*?) of the APA is blocked. + +#+begin_src matlab +sqrt((n_hexapod.flex_bot.kz(1)/2)/(55/3))/2/pi +#+end_src ** Verify why unstable strut #+begin_src matlab :results value replace