2 Simulation Configuration - Configuration reference
--As multiple simulink files will be used for simulation and tests, it is very useful to determine good simulation configuration that will be shared among all the simulink files. +In this document is explained how the Simscape model of the Stewart Platform is implemented.
-This is done using something called “Configuration Reference” (documentation). +It is divided in the following sections: +
+-
+
- section 1: is explained how the parameters of the Stewart platform are set for the Simscape model +
- section 2: the Simulink configuration (solver, simulation time, …) is shared among all the Simulink files. It is explain how this is done. +
- section 3: All the elements (platforms, struts, sensors, …) are saved in separate files and imported in Simulink files using “subsystem referenced”. +
- section 4: The simscape model for the fixed base and mobile platform are described in this section. +
- section 5: The simscape model for the Stewart platform struts is described in this section. +
1 Parameters used for the Simscape Model
+
+
+The Simscape Model of the Stewart Platform is working with the stewart structure generated using the functions described here.
+
+All the geometry and inertia of the mechanical elements are defined in the stewart structure.
+
+By updating the stewart structure in the workspace, the Simscape model will be automatically updated.
+
+Thus, nothing should be changed by hand inside the Simscape model. +
+ ++The main advantage to have all the parameters defined in one structure (and not hard-coded in some simulink blocs) it that we can easily change the Stewart architecture/parameters in a Matlab script to perform some parametric study for instance. +
+2 Simulation Configuration - Configuration reference
++ +As multiple simulink files will be used for simulation and tests, it is very useful to determine good simulation configuration that will be shared among all the simulink files. +
+ ++This is done using something called “Configuration Reference” (documentation).
@@ -298,7 +372,7 @@ It is automatically loaded when the Simulink project is open. It can be loaded m
-It is however possible to modify specific parameters just for one Simulink file using the set_param command:
+It is however possible to modify specific parameters just for one simulation using the set_param command:
set_param(conf_simscape, 'StopTime', 1); @@ -307,10 +381,11 @@ It is however possible to modify specific parameters just for one Simulink file
3 Subsystem Reference
+3 Subsystem Reference
+ Several Stewart platform models are used, for instance one is use to study the dynamics while the other is used to apply active damping techniques.
@@ -328,29 +403,163 @@ These shared subsystems are:
-These subsystems are referenced from another subsystem called Stewart_Platform.slx, that basically connect them correctly.
-This subsystem is then referenced in other simulink models for various purposes.
+These subsystems are referenced from another subsystem called Stewart_Platform.slx shown in figure 1, that basically connect them correctly.
+This subsystem is then referenced in other simulink models for various purposes (control, analysis, simulation, …).
+
+
Figure 1: Simscape Subsystem of the Stewart platform. Encapsulate the Subsystems corresponding to the fixed base, mobile platform and all the struts.
+4 Subsystem - Fixed base and Mobile Platform
++ +Both the fixed base and the mobile platform simscape models share many similarities. +
+ ++Their are both composed of: +
+-
+
- a solid body representing the platform +
- 6 rigid transform blocks to go from the frame \(\{F\}\) (resp. \(\{M\}\)) to the location of the joints. +These rigid transform are using \({}^F\bm{a}_i\) (resp. \({}^M\bm{b}_i\)) for the position of the joint and \({}^F\bm{R}_{a_i}\) (resp. \({}^M\bm{R}_{b_i}\)) for the orientation of the joint. +
+As always, the parameters that define the geometry are taken from the stewart structure.
+
+
Figure 2: Simscape Model of the Fixed base
+
+
Figure 3: Simscape Model of the Mobile platform
+5 Subsystem - Struts
+ +5.1 Strut Configuration
+
+For the Stewart platform, the 6 struts are identical.
+Thus, all the struts used in the Stewart platform are referring to the same subsystem called stewart_strut.slx and shown in Figure 4.
+
+This strut as the following structure: +
+-
+
- Universal Joint* connected on the Fixed base +
- Prismatic Joint* for the actuator +
- Spherical Joint* connected on the Mobile platform +
+This configuration is called UPS. +
+ ++The other common configuration SPS has the disadvantage of having additional passive degrees-of-freedom corresponding to the rotation of the strut around its main axis. +This is why the UPS configuration is used, but other configuration can be easily implemented. +
+ + +
+
Figure 4: Simscape model of the Stewart platform’s strut
++Several sensors are included in the strut that may or may not be used for control: +
+-
+
- Relative Displacement sensor: gives the relative displacement of the strut. +
- Force sensor: measure the total force applied by the force actuator, the stiffness and damping forces in the direction of the strut. +
- Inertial sensor: measure the absolute motion (velocity) of the top part of the strut in the direction of the strut. +
+There is two main types of inertial sensor that can be used to measure the absolute motion of the top part of the strut in the direction of the strut: +
+-
+
- a geophone that measures the absolute velocity above some frequency +
- an accelerometer that measures the absolute acceleration below some frequency +
+Both inertial sensors are described bellow.
4 Basic configuration for the Fixed base, Mobile Platform and Struts
+5.2 Z-Axis Geophone
+5 Sensors included in the Struts and on the Mobile Platform
-6 Inertial Sensors
-6.1 Z-Axis Geophone
-5.2.1 Working Principle
+- +From the schematic of the Z-axis geophone shown in Figure 5, we can write the transfer function from the support velocity \(\dot{w}\) to the relative velocity of the inertial mass \(\dot{d}\): +\[ \frac{\dot{d}}{\dot{w}} = \frac{-\frac{s^2}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \] +with: +
+-
+
- \(\omega_0 = \sqrt{\frac{k}{m}}\) +
- \(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\) +
+
Figure 5: Schematic of a Z-Axis geophone
++We see that at frequencies above \(\omega_0\): +\[ \frac{\dot{d}}{\dot{w}} \approx -1 \] +
+ ++And thus, the measurement of the relative velocity of the mass with respect to its support gives the absolute velocity of the support. +
+ ++We generally want to have the smallest resonant frequency \(\omega_0\) to measure low frequency absolute velocity, however there is a trade-off between \(\omega_0\) and the mass of the inertial mass. +
+5.2.2 Initialization function
+@@ -380,12 +589,56 @@ This Matlab function is accessible he
6.2 Z-Axis Accelerometer
-5.3 Z-Axis Accelerometer
+5.3.1 Working Principle
+- +From the schematic of the Z-axis accelerometer shown in Figure 6, we can write the transfer function from the support acceleration \(\ddot{w}\) to the relative position of the inertial mass \(d\): +\[ \frac{d}{\ddot{w}} = \frac{-\frac{1}{{\omega_0}^2}}{\frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1} \] +with: +
+-
+
- \(\omega_0 = \sqrt{\frac{k}{m}}\) +
- \(\xi = \frac{1}{2} \sqrt{\frac{m}{k}}\) +
+
Figure 6: Schematic of a Z-Axis geophone
++We see that at frequencies below \(\omega_0\): +\[ \frac{d}{\ddot{w}} \approx -\frac{1}{{\omega_0}^2} \] +
+ ++And thus, the measurement of the relative displacement of the mass with respect to its support gives the absolute acceleration of the support. +
+ ++Note that there is trade-off between: +
+-
+
- the highest measurable acceleration \(\omega_0\) +
- the sensitivity of the accelerometer which is equal to \(-\frac{1}{{\omega_0}^2}\) +
5.3.2 Initialization function
+
@@ -420,9 +673,10 @@ This Matlab function is accessible
- Created: 2020-01-28 mar. 17:37 Created: 2020-01-29 mer. 12:02