Rework hexapod specifications + static deflection

index.org

@@ -2027,22 +2027,52 @@ The simulation is considered to be fairly realistic as the model used has been s
 * General Conclusion and Further notes
 <<sec:conclusion_and_further_notes>>
 
+** Introduction                                                      :ignore:
 ** Nano-Hexapod Specifications
-Table summarizing the nano-hexapod wanted characteristics:
+<<sec:nano_hexapod_specifications>>
 
+*** Introduction                                                    :ignore:
+In this section are gathered all the specifications related to the nano-hexapod.
+
+*** Dimensions
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+The wanted dimension of the nano-hexapod are shown in Figure [[fig:nano_hexapod_size]]:
+- Diameter of the bottom platform: 300mm
+- Diameter of the top platform: 200mm
+- Maximum Height: 90mm
+
+#+name: fig:nano_hexapod_size
+#+caption: First implementation of the nano-hexapod / metrology reflector and sample interface
+[[file:figs/nano_hexapod_size.png]]
+
+*** Flexible Joints
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+Flexible joints are located at each end of the six struts.
+These flexible joints should have the following properties:
+- High Axial Stiffness: $K_a > 10^7\,[N/m]$
+- Small Bending Stiffness: $K_b < 50\,[Nm/rad]$
+- Small Torsion Stiffness: $K_t < 50\,[Nm/rad]$
+
+The required angular stroke has not been estimated in this study.
+It is however simple to do so as the angular motion of each joint can easily be measured in the multi-body model used to perform the simulations.
+Typical angular stroke for such flexible joints is expected.
+
+*** Actuators
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+The actuation part is probably the most important part of the Stewart platform.
 
-- Dimensions (Figure [[fig:nano_hexapod_size]]):
-  - Maximum Height: 90mm
-  - Diameter of the bottom platform: 300mm
-  - Diameter of the top platform: 200mm
 - Stiffness:
   - Resonances should stay between 5Hz and 50Hz for payload masses up to 50kg
   - This corresponds to strut stiffnesses of $k \approx 10^5 - 10^6\,[N/m]$
-- Flexible joints:
-  - Axial Stiffness: $K_a > 10^7\,[N/m]$
-  - Bending Stiffness: $K_b < 50\,[Nm/rad]$
-  - Torsion Stiffness: $K_t < 50\,[Nm/rad]$
-  - Required angular stroke: can be estimated with simulations
 - Force:
   - Weight: $60\,kg \rightarrow 600\,N \rightarrow 60\,N$ on each actuator
   - Dynamic: few Newtons
@@ -2054,12 +2084,60 @@ Table summarizing the nano-hexapod wanted characteristics:
       This is probably more difficult to obtain.
       However, by limiting the acceleration of these stages, we may limit the dynamic tracking errors to acceptable levels
   - If the chosen technology allows $\pm 50 \mu m$ that would be safer
-- Sensors to be included:
-  -
 
-#+name: fig:nano_hexapod_size
-#+caption: First implementation of the nano-hexapod / metrology reflector and sample interface
-[[file:figs/nano_hexapod_size.png]]
+*** Sensors
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+A relative displacement sensor must be included in each of the nano-hexapod's legs as explained in Section [[sec:robust_control_architecture]].
+
+The sensors must as the following properties:
+- High bandwidth $> 1\,[kHz]$
+- Fine resolution $< 10\,[nm]$
+- Range of $> 100\,[\mu m]$
+
+Note that the sensor signal will have to pass through the Slip-Ring.
+This adds few constrains:
+- The sensor signal must be immune to some "electrical noise" that could be induced by the slip-ring
+- Limited number of slip-ring channels should be required for the six sensors
+
+Several sensor technology could be used for the nano-hexapod.
+Characteristics of those sensors are shown in Table [[tab:characteristics_relative_sensor]].
+
+#+name: tab:characteristics_relative_sensor
+#+caption: Characteristics of relative measurement sensors cite:collette11_review
+| Technology     | Frequency  | Resolution     | Range        | T Range     |
+|----------------+------------+----------------+--------------+-------------|
+| LVDT           | DC-200 Hz  | 10 nm rms      | 1-10 mm      | -50,100 °C  |
+| Eddy current   | 5 kHz      | 0.1-100 nm rms | 0.5-55 mm    | -50,100 °C  |
+| Capacitive     | DC-100 kHz | 0.05-50 nm rms | 50 nm - 1 cm | -40,100 °C  |
+| Interferometer | 300 kHz    | 0.1 nm rms     | 10 cm        | -250,100 °C |
+| Encoder        | DC-1 MHz   | 1 nm rms       | 7-27 mm      | 0,40 °C     |
+
+*** Architecture
+:PROPERTIES:
+:UNNUMBERED: t
+:END:
+
+** Problem of Static Deflection?
+Let's now consider the problem of static deflection when changing the payload.
+
+The maximum payload's mass is $50\,[kg]$, this corresponds to an added vertical force of $\approx 500\,[N]$.
+
+If this is to be compensated by the nano-hexapod, $\approx 100\,[N]$ should be applied by each of the nano-hexapod's actuators.
+In such case, the nano-hexapod keeps its nominal configuration.
+
+In practice, this could be done using the relative motion sensor included in each leg, and a feedback control keeping the legs displacements at the wanted value.
+
+This might not be a problem is piezoelectric stacks are used, but it is a big issue is voice coils are used.
+
+
+An alternative would be to accept that the nano-hexapod experiences some static deflection.
+
+With a vertical nano-hexapod stiffness $\approx 10^6\,[N/m]$, the maximum static deflection would be:
+\[ k_z \approx 10^6\,[N/m] \longrightarrow \Delta z = \frac{\Delta m g}{k_z} \approx 0.5\,[mm] \]
+This will change a little bit the architecture of the nano-hexapod but this should be too small to change significantly the dynamics.
 
 ** Sensor Noise introduced by the Metrology
 Say that is will introduce noise inside the bandwidth (100Hz)

ref.bib

@@ -63,3 +63,10 @@
   doi = {10.1016/j.jsv.2006.07.050},
   url = {https://doi.org/10.1016/j.jsv.2006.07.050},
 }
+
+@techreport{collette11_review,
+  author = {Collette, C and Artoos, K and Guinchard, M and Janssens, S and Carmona Fernandez, P and Hauviller, C},
+  institution = {cern},
+  title = {Review of sensors for low frequency seismic vibration measurement},
+  year = {2011},
+}