Describe simscape model of nano-hexapod elements
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* Introduction :ignore:
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In this document, a Simscape model of the nano-hexapod is developed.
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In this document, a Simscape model of the nano-hexapod is developed and studied (shown in Figure [[fig:nano_hexapod_simscape]]).
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- Section [[sec:nano_hexapod]]:
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- Section [[sec:integral_force_feedback]]:
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- Section [[sec:direct_velocity_feedback_struts]]:
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- Section [[sec:direct_velocity_feedback_plates]]:
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It is structured as follows:
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- Section [[sec:nano_hexapod]]: the simscape model of the nano-hexapod is presented. Few of its elements can be configured as wanted. The effect of the configuration on the obtained dynamics is studied.
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- Section [[sec:integral_force_feedback]]: Direct Velocity Feedback is applied and the obtained damping is derived.
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- Section [[sec:direct_velocity_feedback_struts]]: the encoders are fixed to the struts, and Integral Force Feedback is applied. The obtained damping is computed.
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- Section [[sec:direct_velocity_feedback_plates]]: the same is done when the encoders are fixed on the plates
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#+name: fig:nano_hexapod_simscape
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#+caption: 3D view of the Sismcape model for the Nano-Hexapod
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#+attr_latex: :width \linewidth
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[[file:figs/nano_hexapod_simscape_encoder_struts.png]]
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* Nano-Hexapod
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<<sec:nano_hexapod>>
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@@ -39,9 +45,10 @@ open('matlab/nano_hexapod/nano_hexapod.slx')
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** Nano Hexapod - Configuration
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<<sec:nano_hexapod_conf>>
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*** Introduction :ignore:
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The nano-hexapod can be initialized and configured using the =initializeNanoHexapodFinal= function ([[sec:initializeNanoHexapodFinal][link]]).
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The following code would produce the model shown in Figure [[fig:nano_hexapod_simscape_encoder_struts]].
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#+begin_src matlab
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n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
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'flex_top_type', '3dof', ...
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@@ -50,6 +57,115 @@ n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
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'MO_B', 150e-3);
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#+end_src
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#+name: fig:nano_hexapod_simscape_encoder_struts
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#+caption: 3D view of the Sismcape model for the Nano-Hexapod
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#+attr_latex: :width \linewidth
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[[file:figs/nano_hexapod_simscape_encoder_struts.png]]
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Several elements on the nano-hexapod can be configured:
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- The flexible joints (Section [[sec:conf_flexible_joint]])
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- The amplified piezoelectric actuators (Section [[sec:conf_apa]])
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- The encoders (Section [[sec:conf_encoders]])
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- The Jacobian matrices (Section [[sec:conf_jacobian]])
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*** Flexible Joints
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<<sec:conf_flexible_joint>>
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The model of the flexible joint is composed of 3 solid bodies as shown in Figure [[fig:simscape_model_flexible_joint]] which are connected by joints representing the flexibility of the joint.
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We can represent:
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- the bending flexibility $k_{R_x}$, $k_{R_y}$
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- the torsional flexibility $k_{R_z}$
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- the axial flexibility $k_z$
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The configurations and the represented flexibilities are summarized in Table [[tab:flex_type_conf]].
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#+name: tab:flex_type_conf
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#+caption: Flexible joint's configuration and associated represented flexibility
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#+attr_latex: :environment tabularx :width 0.6\linewidth :align lXXX
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#+attr_latex: :center t :booktabs t :float t
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| =flex_type= | Bending | Torsional | Axial |
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|-------------+---------+-----------+-------|
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| =2dof= | x | | |
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| =3dof= | x | x | |
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| =4dof= | x | x | x |
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Of course, adding more DoF for the flexible joint will induce an addition of many states for the nano-hexapod simscape model.
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#+name: fig:simscape_model_flexible_joint
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#+caption: 3D view of the Sismcape model for the Flexible joint (4DoF configuration)
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#+attr_latex: :width 0.8\linewidth
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[[file:figs/simscape_model_flexible_joint.png]]
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*** Amplified Piezoelectric Actuators
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<<sec:conf_apa>>
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The nano-hexapod's struts are containing one amplified piezoelectric actuator (APA300ML from Cedrat Technologies).
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The APA can be modeled in different ways which can be configured with the =actuator_type= argument.
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The simplest model is a 2-DoF system shown in Figure [[fig:2dof_apa_model]].
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#+name: fig:2dof_apa_model
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#+caption: Schematic of the 2DoF model for the Amplified Piezoelectric Actuator
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[[file:figs/2dof_apa_model.png]]
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Then, a more complex model based on a Finite Element Model can be used.
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*** Encoders
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<<sec:conf_encoders>>
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The encoders can be either fixed directly on the struts (Figure [[fig:encoder_struts]]) or on the two plates (Figure [[fig:encoders_plates_with_apa]]).
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This can be configured with the =motion_sensor_type= parameters which can be equal to ='struts'= or ='plates'=.
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#+name: fig:encoder_struts
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#+caption: 3D view of the Encoders fixed on the struts
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#+attr_latex: :width 0.8\linewidth
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[[file:figs/encoder_struts.png]]
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#+name: fig:encoders_plates_with_apa
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#+caption: 3D view of the Encoders fixed on the plates
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#+attr_latex: :width 0.6\linewidth
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[[file:figs/encoders_plates_with_apa.png]]
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A complete view of the nano-hexapod with encoders fixed to the struts is shown in Figure [[fig:nano_hexapod_simscape_encoder_struts]] while it is shown in Figure [[fig:nano_hexapod_simscape_encoder_plates]] when the encoders are fixed to the plates.
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#+name: fig:nano_hexapod_simscape_encoder_plates
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#+caption: Nano-Hexapod with encoders fixed to the plates
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#+attr_latex: :width \linewidth
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[[file:figs/nano_hexapod_simscape_encoder_plates.png]]
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The encoder model is schematically represented in Figure [[fig:simscape_encoder_model]]:
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- a frame {B}, fixed to the ruler is positioned on its top surface
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- a frame {F}, rigidly fixed to the encoder is initially positioned such that its origin is aligned with the x axis of frame {B}
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The output measurement is then the x displacement of the origin of the frame {F} expressed in frame {B}.
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#+name: fig:simscape_encoder_model
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#+caption: Schematic of the encoder model
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[[file:figs/simscape_encoder_model.png]]
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If the encoder is experiencing some tilt, it is then "converted" into a measured displacement as shown in Figure [[fig:simscape_encoder_model_disp]].
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#+name: fig:simscape_encoder_model_disp
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#+caption: Schematic of the encoder model
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[[file:figs/simscape_encoder_model_disp.png]]
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*** Jacobians
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<<sec:conf_jacobian>>
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While the Jacobian configuration will not change the physical system, it is still quite an important part of the model.
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This configuration consists on defining the location of the frame {B} in which the Jacobian will be computed.
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This Jacobian is then used to transform the actuator forces to forces/torques applied on the payload and expressed in frame {B}.
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Same thing can be done for the measured encoder displacements.
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** Effect of encoders on the decentralized plant
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<<sec:effect_encoder_location>>
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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.
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We initialize the identification parameters.
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#+begin_src matlab
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%% Options for Linearized
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@@ -65,11 +181,6 @@ io(io_i) = linio([mdl, '/F'], 1, 'openinput'); io_i = io_i + 1; % Actuator Inp
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io(io_i) = linio([mdl, '/D'], 1, 'openoutput'); io_i = io_i + 1; % Relative Motion Outputs
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#+end_src
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** Effect of encoders on the decentralized plant
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<<sec:effect_encoder_location>>
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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.
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Identify the plant when the encoders are on the struts:
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#+begin_src matlab
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n_hexapod = initializeNanoHexapodFinal('flex_bot_type', '4dof', ...
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@@ -3183,7 +3294,7 @@ arguments
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% For Flexible Frame
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args.actuator_ks (6,1) double {mustBeNumeric} = ones(6,1)*235e6 % Stiffness of one stack [N/m]
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args.actuator_cs (6,1) double {mustBeNumeric} = ones(6,1)*1e1 % Stiffness of one stack [N/m]
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% For Flexible
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args.actuator_xi (1,1) double {mustBeNumeric} = 0.01 % Damping Ratio
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%% Controller
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args.controller_type char {mustBeMember(args.controller_type,{'none', 'iff', 'dvf'})} = 'none'
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