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content/article/butler11_posit_contr_lithog_equip.md
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@ -8,7 +8,7 @@ Tags
: [Multivariable Control]({{< relref "multivariable_control" >}}), [Positioning Stations]({{< relref "positioning_stations" >}}) : [Multivariable Control]({{< relref "multivariable_control" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
Reference Reference
: ([Butler 2011](#org79c48f7)) : ([Butler 2011](#org10c4eb8))
Author(s) Author(s)
: Butler, H. : Butler, H.
@ -17,6 +17,7 @@ Year
: 2011 : 2011
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org79c48f7"></a>Butler, Hans. 2011. “Position Control in Lithographic Equipment.” _IEEE Control Systems_ 31 (5):28–47. <https://doi.org/10.1109/mcs.2011.941882>. <a id="org10c4eb8"></a>Butler, Hans. 2011. “Position Control in Lithographic Equipment.” _IEEE Control Systems_ 31 (5):28–47. <https://doi.org/10.1109/mcs.2011.941882>.

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content/article/chen00_ident_decoup_contr_flexur_joint_hexap.md
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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference Reference
: ([Chen and McInroy 2000](#orgc009af3)) : ([Chen and McInroy 2000](#orgfcb31bd))
Author(s) Author(s)
: Chen, Y., & McInroy, J. : Chen, Y., & McInroy, J.
@ -31,7 +31,7 @@ Year
## Introduction {#introduction} ## Introduction {#introduction}
Typical decoupling algorithm impose two constraints: Typical decoupling algorithm ([Decoupled Control]({{< relref "decoupled_control" >}})) impose two constraints:
- the payload mass/inertia matrix is diagonal - the payload mass/inertia matrix is diagonal
- the geometry of the platform and attachment of the payload must be carefully chosen - the geometry of the platform and attachment of the payload must be carefully chosen
@ -43,9 +43,9 @@ The algorithm derived herein removes these constraints, thus greatly expanding t
## Dynamic Model of Flexure Jointed Hexapods {#dynamic-model-of-flexure-jointed-hexapods} ## Dynamic Model of Flexure Jointed Hexapods {#dynamic-model-of-flexure-jointed-hexapods}
The derivation of the dynamic model is done in ([McInroy 1999](#org0e9e807)) ([Notes]({{< relref "mcinroy99_dynam" >}})). The derivation of the dynamic model is done in ([McInroy 1999](#orgaddeeaf)) ([Notes]({{< relref "mcinroy99_dynam" >}})).
<a id="orge595e9c"></a> <a id="org3326cce"></a>
{{< figure src="/ox-hugo/chen00_flexure_hexapod.png" caption="Figure 1: A flexured joint Hexapod. {P} is a cartesian coordiante frame located at (and rigidly connected to) the payload's center of mass. {B} is a frame attached to the (possibly moving) base, and {U} is a universal inertial frame of reference" >}} {{< figure src="/ox-hugo/chen00_flexure_hexapod.png" caption="Figure 1: A flexured joint Hexapod. {P} is a cartesian coordiante frame located at (and rigidly connected to) the payload's center of mass. {B} is a frame attached to the (possibly moving) base, and {U} is a universal inertial frame of reference" >}}
@ -100,8 +100,9 @@ where
## Experimental Results {#experimental-results} ## Experimental Results {#experimental-results}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgc009af3"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>. <a id="orgfcb31bd"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
<a id="org0e9e807"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>. <a id="orgaddeeaf"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.

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content/article/dasgupta00_stewar_platf_manip.md
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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference Reference
: ([Dasgupta and Mruthyunjaya 2000](#orgab59d3a)) : ([Dasgupta and Mruthyunjaya 2000](#orgb54a8e7))
Author(s) Author(s)
: Dasgupta, B., & Mruthyunjaya, T. : Dasgupta, B., & Mruthyunjaya, T.
@ -24,16 +24,17 @@ Year
| | **Advantages** | **Disadvantages** | | | **Advantages** | **Disadvantages** |
|--------------|---------------------------|-----------------------| |--------------|---------------------------|-----------------------|
| **Serial** | Manoeuverability | Poor precision | | **Serial** | Maneuverability | Poor precision |
| | Large workspace | Bends under high load | | | Large workspace | Bends under high load |
| | | Vibrate at high speed | | | | Vibrate at high speed |
| **Parallel** | High stiffness | Small workspace | | **Parallel** | High stiffness | Small workspace |
| | Good dynamic performances | | | | Good dynamic performances | |
| | Precise positioning | | | | Precise positioning | |
The generalized Stewart platforms consists of two rigid bodies (referred to as the base and the platoform) connected through six extensible legs, each with sherical joints at both ends. The generalized Stewart platforms consists of two rigid bodies (referred to as the base and the platform) connected through six extensible legs, each with spherical joints at both ends.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgab59d3a"></a>Dasgupta, Bhaskar, and T.S. Mruthyunjaya. 2000. “The Stewart Platform Manipulator: A Review.” _Mechanism and Machine Theory_ 35 (1):15–40. <https://doi.org/10.1016/s0094-114x(99)>00006-3. <a id="orgb54a8e7"></a>Dasgupta, Bhaskar, and T.S. Mruthyunjaya. 2000. “The Stewart Platform Manipulator: A Review.” _Mechanism and Machine Theory_ 35 (1):15–40. <https://doi.org/10.1016/s0094-114x(99)>00006-3.

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content/article/fleming13_review_nanom_resol_posit_sensor.md
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@ -4,15 +4,11 @@ author = ["Thomas Dehaeze"]
draft = false draft = false
+++ +++
Backlinks:
- [Position Sensors]({{< relref "position_sensors" >}})
Tags Tags
: [Position Sensors]({{< relref "position_sensors" >}}) : [Position Sensors]({{< relref "position_sensors" >}})
Reference Reference
: ([Fleming 2013](#org35f9cea)) : ([Fleming 2013](#org336a947))
Author(s) Author(s)
: Fleming, A. J. : Fleming, A. J.
@ -37,7 +33,7 @@ Usually quoted as a percentage of the fill-scale range (FSR):
With \\(e\_m(v)\\) is the mapping error. With \\(e\_m(v)\\) is the mapping error.
<a id="orge06f384"></a> <a id="org3c27d5a"></a>
{{< figure src="/ox-hugo/fleming13_mapping_error.png" caption="Figure 1: The actual position versus the output voltage of a position sensor. The calibration function \\(f\_{cal}(v)\\) is an approximation of the sensor mapping function \\(f\_a(v)\\) where \\(v\\) is the voltage resulting from a displacement \\(x\\). \\(e\_m(v)\\) is the residual error." >}} {{< figure src="/ox-hugo/fleming13_mapping_error.png" caption="Figure 1: The actual position versus the output voltage of a position sensor. The calibration function \\(f\_{cal}(v)\\) is an approximation of the sensor mapping function \\(f\_a(v)\\) where \\(v\\) is the voltage resulting from a displacement \\(x\\). \\(e\_m(v)\\) is the residual error." >}}
@ -46,7 +42,7 @@ With \\(e\_m(v)\\) is the mapping error.
If the shape of the mapping function actually varies with time, the maximum error due to drift must be evaluated by finding the worst-case mapping error. If the shape of the mapping function actually varies with time, the maximum error due to drift must be evaluated by finding the worst-case mapping error.
<a id="orgc484965"></a> <a id="org69dcb2d"></a>
{{< figure src="/ox-hugo/fleming13_drift_stability.png" caption="Figure 2: The worst case range of a linear mapping function \\(f\_a(v)\\) for a given error in sensitivity and offset." >}} {{< figure src="/ox-hugo/fleming13_drift_stability.png" caption="Figure 2: The worst case range of a linear mapping function \\(f\_a(v)\\) for a given error in sensitivity and offset." >}}
@ -151,9 +147,9 @@ The empirical rule states that there is a \\(99.7\%\\) probability that a sample
This if we define the resolution as \\(\delta = 6 \sigma\\), we will referred to as the \\(6\sigma\text{-resolution}\\). This if we define the resolution as \\(\delta = 6 \sigma\\), we will referred to as the \\(6\sigma\text{-resolution}\\).
Another important parameter that must be specified when quoting resolution is the sensor bandwidth. Another important parameter that must be specified when quoting resolution is the sensor bandwidth.
There is usually a trade-off between bandwidth and resolution (figure [3](#org2ee752d)). There is usually a trade-off between bandwidth and resolution (figure [3](#orge95682f)).
<a id="org2ee752d"></a> <a id="orge95682f"></a>
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}} {{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}}
@ -186,6 +182,7 @@ A convenient method for reporting this ratio is in parts-per-million (ppm):
| Encoder | Meters | | 6 nm | >100kHz | 5 ppm FSR | | Encoder | Meters | | 6 nm | >100kHz | 5 ppm FSR |
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org35f9cea"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” _Sensors and Actuators a: Physical_ 190 (nil):106–26. <https://doi.org/10.1016/j.sna.2012.10.016>. <a id="org336a947"></a>Fleming, Andrew J. 2013. “A Review of Nanometer Resolution Position Sensors: Operation and Performance.” _Sensors and Actuators a: Physical_ 190 (nil):106–26. <https://doi.org/10.1016/j.sna.2012.10.016>.

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content/article/furqan17_studies_stewar_platf_manip.md
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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference Reference
: ([Furqan, Suhaib, and Ahmad 2017](#org7520991)) : ([Furqan, Suhaib, and Ahmad 2017](#orgd310f71))
Author(s) Author(s)
: Furqan, M., Suhaib, M., & Ahmad, N. : Furqan, M., Suhaib, M., & Ahmad, N.
@ -19,6 +19,7 @@ Year
Lots of references. Lots of references.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org7520991"></a>Furqan, Mohd, Mohd Suhaib, and Nazeer Ahmad. 2017. “Studies on Stewart Platform Manipulator: A Review.” _Journal of Mechanical Science and Technology_ 31 (9):4459–70. <https://doi.org/10.1007/s12206-017-0846-1>. <a id="orgd310f71"></a>Furqan, Mohd, Mohd Suhaib, and Nazeer Ahmad. 2017. “Studies on Stewart Platform Manipulator: A Review.” _Journal of Mechanical Science and Technology_ 31 (9):4459–70. <https://doi.org/10.1007/s12206-017-0846-1>.

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content/article/furutani04_nanom_cuttin_machin_using_stewar.md
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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference Reference
: ([Furutani, Suzuki, and Kudoh 2004](#orgaf9f119)) : ([Furutani, Suzuki, and Kudoh 2004](#org37254be))
Author(s) Author(s)
: Furutani, K., Suzuki, M., & Kudoh, R. : Furutani, K., Suzuki, M., & Kudoh, R.
@ -35,6 +35,7 @@ To minimize the errors, a calibration is done between the required leg length an
Then, it is fitted with 4th order polynomial and included in the control architecture. Then, it is fitted with 4th order polynomial and included in the control architecture.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgaf9f119"></a>Furutani, Katsushi, Michio Suzuki, and Ryusei Kudoh. 2004. “Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism.” _Measurement Science and Technology_ 15 (2):467–74. <https://doi.org/10.1088/0957-0233/15/2/022>. <a id="org37254be"></a>Furutani, Katsushi, Michio Suzuki, and Ryusei Kudoh. 2004. “Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism.” _Measurement Science and Technology_ 15 (2):467–74. <https://doi.org/10.1088/0957-0233/15/2/022>.

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content/article/gao15_measur_techn_precis_posit.md
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@ -8,7 +8,7 @@ Tags
: [Position Sensors]({{< relref "position_sensors" >}}) : [Position Sensors]({{< relref "position_sensors" >}})
Reference Reference
: ([Gao et al. 2015](#org7f49efc)) : ([Gao et al. 2015](#orgb71276a))
Author(s) Author(s)
: Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., … : Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …
@ -17,6 +17,7 @@ Year
: 2015 : 2015
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org7f49efc"></a>Gao, W., S.W. Kim, H. Bosse, H. Haitjema, Y.L. Chen, X.D. Lu, W. Knapp, A. Weckenmann, W.T. Estler, and H. Kunzmann. 2015. “Measurement Technologies for Precision Positioning.” _CIRP Annals_ 64 (2):773–96. <https://doi.org/10.1016/j.cirp.2015.05.009>. <a id="orgb71276a"></a>Gao, W., S.W. Kim, H. Bosse, H. Haitjema, Y.L. Chen, X.D. Lu, W. Knapp, A. Weckenmann, W.T. Estler, and H. Kunzmann. 2015. “Measurement Technologies for Precision Positioning.” _CIRP Annals_ 64 (2):773–96. <https://doi.org/10.1016/j.cirp.2015.05.009>.

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content/article/garg07_implem_chall_multiv_contr.md
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@ -8,7 +8,7 @@ Tags
: [Multivariable Control]({{< relref "multivariable_control" >}}) : [Multivariable Control]({{< relref "multivariable_control" >}})
Reference Reference
: ([Garg 2007](#orga5ede8d)) : ([Garg 2007](#orgae24e63))
Author(s) Author(s)
: Garg, S. : Garg, S.
@ -35,6 +35,7 @@ The control rate should be weighted appropriately in order to not saturate the s
- importance of scaling the plant prior to synthesis and also replacing pure integrators with slow poles - importance of scaling the plant prior to synthesis and also replacing pure integrators with slow poles
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orga5ede8d"></a>Garg, Sanjay. 2007. “Implementation Challenges for Multivariable Control: What You Did Not Learn in School!” In _AIAA Guidance, Navigation and Control Conference and Exhibit_, nil. <https://doi.org/10.2514/6.2007-6334>. <a id="orgae24e63"></a>Garg, Sanjay. 2007. “Implementation Challenges for Multivariable Control: What You Did Not Learn in School!” In _AIAA Guidance, Navigation and Control Conference and Exhibit_, nil. <https://doi.org/10.2514/6.2007-6334>.

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content/article/hanieh03_activ_stewar.md
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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Active Damping]({{< relref "active_damping" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Active Damping]({{< relref "active_damping" >}})
Reference Reference
: ([Hanieh 2003](#org5da54db)) : ([Hanieh 2003](#orgd0b61f4))
Author(s) Author(s)
: Hanieh, A. A. : Hanieh, A. A.
@ -17,6 +17,7 @@ Year
: 2003 : 2003
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org5da54db"></a>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium. <a id="orgd0b61f4"></a>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.

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content/article/holler12_instr_x_ray_nano_imagin.md
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@ -8,7 +8,7 @@ Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}}) : [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
Reference Reference
: ([Holler et al. 2012](#org289d119)) : ([Holler et al. 2012](#org76dfce6))
Author(s) Author(s)
: Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., & Bunk, O. : Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., & Bunk, O.
@ -19,7 +19,7 @@ Year
Instrument similar to the NASS. Instrument similar to the NASS.
Obtain position stability of 10nm (standard deviation). Obtain position stability of 10nm (standard deviation).
<a id="orgfed3898"></a> <a id="orgd10cf36"></a>
{{< figure src="/ox-hugo/holler12_station.png" caption="Figure 1: Schematic of the tomography setup" >}} {{< figure src="/ox-hugo/holler12_station.png" caption="Figure 1: Schematic of the tomography setup" >}}
@ -39,6 +39,7 @@ Obtain position stability of 10nm (standard deviation).
- **Feedback Loop**: Using the signals from the 2 interferometers, the loop is closed to compensate low frequency vibrations and thermal drifts. - **Feedback Loop**: Using the signals from the 2 interferometers, the loop is closed to compensate low frequency vibrations and thermal drifts.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org289d119"></a>Holler, M., J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk. 2012. “An Instrument for 3d X-Ray Nano-Imaging.” _Review of Scientific Instruments_ 83 (7):073703. <https://doi.org/10.1063/1.4737624>. <a id="org76dfce6"></a>Holler, M., J. Raabe, A. Diaz, M. Guizar-Sicairos, C. Quitmann, A. Menzel, and O. Bunk. 2012. “An Instrument for 3d X-Ray Nano-Imaging.” _Review of Scientific Instruments_ 83 (7):073703. <https://doi.org/10.1063/1.4737624>.

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content/article/holterman05_activ_dampin_based_decoup_colloc_contr.md
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@ -8,7 +8,7 @@ Tags
: [Active Damping]({{< relref "active_damping" >}}) : [Active Damping]({{< relref "active_damping" >}})
Reference Reference
: ([Holterman and deVries 2005](#orgfb04a8c)) : ([Holterman and deVries 2005](#org69e08df))
Author(s) Author(s)
: Holterman, J., & deVries, T. : Holterman, J., & deVries, T.
@ -17,6 +17,7 @@ Year
: 2005 : 2005
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgfb04a8c"></a>Holterman, J., and T.J.A. deVries. 2005. “Active Damping Based on Decoupled Collocated Control.” _IEEE/ASME Transactions on Mechatronics_ 10 (2):135–45. <https://doi.org/10.1109/tmech.2005.844702>. <a id="org69e08df"></a>Holterman, J., and T.J.A. deVries. 2005. “Active Damping Based on Decoupled Collocated Control.” _IEEE/ASME Transactions on Mechatronics_ 10 (2):135–45. <https://doi.org/10.1109/tmech.2005.844702>.

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content/article/jiao18_dynam_model_exper_analy_stewar.md
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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference Reference
: ([Jiao et al. 2018](#orga067c93)) : ([Jiao et al. 2018](#org72b03e6))
Author(s) Author(s)
: Jiao, J., Wu, Y., Yu, K., & Zhao, R. : Jiao, J., Wu, Y., Yu, K., & Zhao, R.
@ -17,6 +17,7 @@ Year
: 2018 : 2018
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orga067c93"></a>Jiao, Jian, Ying Wu, Kaiping Yu, and Rui Zhao. 2018. “Dynamic Modeling and Experimental Analyses of Stewart Platform with Flexible Hinges.” _Journal of Vibration and Control_ 25 (1):151–71. <https://doi.org/10.1177/1077546318772474>. <a id="org72b03e6"></a>Jiao, Jian, Ying Wu, Kaiping Yu, and Rui Zhao. 2018. “Dynamic Modeling and Experimental Analyses of Stewart Platform with Flexible Hinges.” _Journal of Vibration and Control_ 25 (1):151–71. <https://doi.org/10.1177/1077546318772474>.

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content/article/legnani12_new_isotr_decoup_paral_manip.md
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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference Reference
: ([Legnani et al. 2012](#org4d21607)) : ([Legnani et al. 2012](#orga1dea1c))
Author(s) Author(s)
: Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D. : Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D.
@ -22,15 +22,16 @@ Year
Example of generated isotropic manipulator (not decoupled). Example of generated isotropic manipulator (not decoupled).
<a id="org584231c"></a> <a id="org7df2b7f"></a>
{{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="Figure 1: Location of the leg axes using an isotropy generator" >}} {{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="Figure 1: Location of the leg axes using an isotropy generator" >}}
<a id="orgfad58c6"></a> <a id="org4803974"></a>
{{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="Figure 2: Isotropic configuration" >}} {{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="Figure 2: Isotropic configuration" >}}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org4d21607"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” _Mechanism and Machine Theory_ 58 (nil):64–81. <https://doi.org/10.1016/j.mechmachtheory.2012.07.008>. <a id="orga1dea1c"></a>Legnani, G., I. Fassi, H. Giberti, S. Cinquemani, and D. Tosi. 2012. “A New Isotropic and Decoupled 6-Dof Parallel Manipulator.” _Mechanism and Machine Theory_ 58 (nil):64–81. <https://doi.org/10.1016/j.mechmachtheory.2012.07.008>.

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content/article/li01_simul_fault_vibrat_isolat_point.md
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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Multivariable Control]({{< relref "multivariable_control" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Multivariable Control]({{< relref "multivariable_control" >}})
Reference Reference
: ([Li 2001](#org9c3830b)) : ([Li 2001](#org7799941))
Author(s) Author(s)
: Li, X. : Li, X.
@ -24,7 +24,7 @@ Year
- Cubic (mutually orthogonal) - Cubic (mutually orthogonal)
- Flexure Joints => eliminate friction and backlash but add complexity to the dynamics - Flexure Joints => eliminate friction and backlash but add complexity to the dynamics
<a id="org0b311b0"></a> <a id="org6847e54"></a>
{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 1: Flexure jointed Stewart platform used for analysis and control" >}} {{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 1: Flexure jointed Stewart platform used for analysis and control" >}}
@ -38,18 +38,18 @@ Year
The origin of \\(\\{P\\}\\) is taken as the center of mass of the payload. The origin of \\(\\{P\\}\\) is taken as the center of mass of the payload.
**Decoupling**: **Decoupling**:
If we refine the (force) inputs and (displacement) outputs as shown in Figure [2](#orgdda87ef) or in Figure [3](#orga45a9ca), we obtain a decoupled plant provided that: If we refine the (force) inputs and (displacement) outputs as shown in Figure [2](#org7d1b81f) or in Figure [3](#orgc8972f9), we obtain a decoupled plant provided that:
1. the payload mass/inertia matrix must be diagonal (the CoM is coincident with the origin of frame \\(\\{P\\}\\)) 1. the payload mass/inertia matrix must be diagonal (the CoM is coincident with the origin of frame \\(\\{P\\}\\))
2. the geometry of the hexapod and the attachment of the payload to the hexapod must be carefully chosen 2. the geometry of the hexapod and the attachment of the payload to the hexapod must be carefully chosen
> For instance, if the hexapod has a mutually orthogonal geometry (cubic configuration), the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts. > For instance, if the hexapod has a mutually orthogonal geometry (cubic configuration), the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts.
<a id="orgdda87ef"></a> <a id="org7d1b81f"></a>
{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 2: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}} {{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 2: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
<a id="orga45a9ca"></a> <a id="orgc8972f9"></a>
{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 3: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}} {{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 3: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
@ -75,15 +75,15 @@ The control bandwidth is divided as follows:
### Vibration Isolation {#vibration-isolation} ### Vibration Isolation {#vibration-isolation}
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [4](#org2bd4cf2). The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [4](#orgabd1d61).
<a id="org2bd4cf2"></a> <a id="orgabd1d61"></a>
{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 4: Figure caption" >}} {{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 4: Figure caption" >}}
One of the subsystem plant transfer function is shown in Figure [4](#org2bd4cf2) One of the subsystem plant transfer function is shown in Figure [4](#orgabd1d61)
<a id="org16f5e77"></a> <a id="org5a1c162"></a>
{{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="Figure 5: Plant transfer function of one of the SISO subsystem for Vibration Control" >}} {{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="Figure 5: Plant transfer function of one of the SISO subsystem for Vibration Control" >}}
@ -97,9 +97,9 @@ The unity control bandwidth of the isolation loop is designed to be from **5Hz t
### Pointing Control {#pointing-control} ### Pointing Control {#pointing-control}
A block diagram of the pointing control system is shown in Figure [6](#orge508ebf). A block diagram of the pointing control system is shown in Figure [6](#orga9c8d31).
<a id="orge508ebf"></a> <a id="orga9c8d31"></a>
{{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 6: Figure caption" >}} {{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 6: Figure caption" >}}
@ -108,9 +108,9 @@ The compensators are design with inverse-dynamics methods.
The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**. The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**.
A feedforward control is added as shown in Figure [7](#org68e9af0). A feedforward control is added as shown in Figure [7](#org78d2b0e).
<a id="org68e9af0"></a> <a id="org78d2b0e"></a>
{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 7: Feedforward control" >}} {{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 7: Feedforward control" >}}
@ -122,17 +122,17 @@ The simultaneous vibration isolation and pointing control is approached in two w
1. design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators 1. design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators
2. the reverse design order 2. the reverse design order
Figure [8](#orgdb53c99) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length. Figure [8](#org94ed578) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
<a id="orgdb53c99"></a> <a id="org94ed578"></a>
{{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 8: A parallel scheme" >}} {{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 8: A parallel scheme" >}}
The transfer function matrix for the pointing loop after the vibration isolation is closed is still decoupled. The same happens when closing the pointing loop first and looking at the transfer function matrix of the vibration isolation. The transfer function matrix for the pointing loop after the vibration isolation is closed is still decoupled. The same happens when closing the pointing loop first and looking at the transfer function matrix of the vibration isolation.
The effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [9](#org3ec32bc). The effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [9](#orgae8c117).
<a id="org3ec32bc"></a> <a id="orgae8c117"></a>
{{< figure src="/ox-hugo/li01_effect_isolation_loop_closed.png" caption="Figure 9: \\(\theta\_x/\theta\_{x\_d}\\) transfer function with the isolation loop closed (simulation)" >}} {{< figure src="/ox-hugo/li01_effect_isolation_loop_closed.png" caption="Figure 9: \\(\theta\_x/\theta\_{x\_d}\\) transfer function with the isolation loop closed (simulation)" >}}
@ -143,19 +143,19 @@ The effect of pointing control on the isolation plant has not much effect.
The dynamic interaction effect: The dynamic interaction effect:
- only happens in the unity bandwidth of the loop transmission of the first closed loop. - only happens in the unity bandwidth of the loop transmission of the first closed loop.
- affect the closed loop transmission of the loop first closed (see Figures [10](#org4a1beff) and [11](#org945793e)) - affect the closed loop transmission of the loop first closed (see Figures [10](#org00362f7) and [11](#org77c322d))
As shown in Figure [10](#org4a1beff), the peak resonance of the pointing loop increase after the isolation loop is closed. As shown in Figure [10](#org00362f7), the peak resonance of the pointing loop increase after the isolation loop is closed.
The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness. The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness.
<a id="org4a1beff"></a> <a id="org00362f7"></a>
{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="Figure 10: Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}} {{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="Figure 10: Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}}
The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [11](#org945793e)). The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [11](#org77c322d)).
The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop. The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
<a id="org945793e"></a> <a id="org77c322d"></a>
{{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="Figure 11: Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}} {{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="Figure 11: Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}}
@ -165,18 +165,18 @@ The first peak resonance of the vibration isolation loop at 15Hz is increased wh
### Experimental results {#experimental-results} ### Experimental results {#experimental-results}
Two hexapods are stacked (Figure [12](#orgbdc7900)): Two hexapods are stacked (Figure [12](#org5977cb3)):
- the bottom hexapod is used to generate disturbances matching candidate applications - the bottom hexapod is used to generate disturbances matching candidate applications
- the top hexapod provide simultaneous vibration isolation and pointing control - the top hexapod provide simultaneous vibration isolation and pointing control
<a id="orgbdc7900"></a> <a id="org5977cb3"></a>
{{< figure src="/ox-hugo/li01_test_bench.png" caption="Figure 12: Stacked Hexapods" >}} {{< figure src="/ox-hugo/li01_test_bench.png" caption="Figure 12: Stacked Hexapods" >}}
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [13](#org7a6d899). Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [13](#org4707691).
<a id="org7a6d899"></a> <a id="org4707691"></a>
{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 13: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}} {{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 13: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}}
@ -185,9 +185,9 @@ The simultaneous control is of dual use:
- it provide simultaneous pointing and isolation control - it provide simultaneous pointing and isolation control
- it can also be used to expand the bandwidth of the isolation control to low frequencies because the pointing loops suppress pointing errors due to both base vibrations and tracking - it can also be used to expand the bandwidth of the isolation control to low frequencies because the pointing loops suppress pointing errors due to both base vibrations and tracking
The results of simultaneous control is shown in Figure [14](#org15d93e5) where the bandwidth of the isolation control is expanded to very low frequency. The results of simultaneous control is shown in Figure [14](#orge4b1f73) where the bandwidth of the isolation control is expanded to very low frequency.
<a id="org15d93e5"></a> <a id="orge4b1f73"></a>
{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 14: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}} {{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 14: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
@ -216,6 +216,7 @@ Proposed future research areas include:
- **Geophones** to provide payload and base velocity information - **Geophones** to provide payload and base velocity information
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org9c3830b"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming. <a id="org7799941"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.

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@ -9,7 +9,7 @@ Tags
Reference Reference
: ([McInroy and Hamann 2000](#orgb4fc604)) : ([McInroy and Hamann 2000](#org6ce0f37))
Author(s) Author(s)
: McInroy, J., & Hamann, J. : McInroy, J., & Hamann, J.
@ -18,6 +18,7 @@ Year
: 2000 : 2000
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgb4fc604"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>. <a id="org6ce0f37"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>.

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@ -4,15 +4,11 @@ author = ["Thomas Dehaeze"]
draft = false draft = false
+++ +++
Backlinks:
- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})
Tags Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference Reference
: ([McInroy 1999](#orgfc7fa52)) : ([McInroy 1999](#org788f3dd))
Author(s) Author(s)
: McInroy, J. : McInroy, J.
@ -20,7 +16,7 @@ Author(s)
Year Year
: 1999 : 1999
This conference paper has been further published in a journal as a short note ([McInroy 2002](#org7752c60)). This conference paper has been further published in a journal as a short note ([McInroy 2002](#org6bd1808)).
## Abstract {#abstract} ## Abstract {#abstract}
@ -42,22 +38,22 @@ The actuators for FJHs can be divided into two categories:
1. soft (voice coil), which employs a spring flexure mount 1. soft (voice coil), which employs a spring flexure mount
2. hard (piezoceramic or magnetostrictive), which employs a compressive load spring. 2. hard (piezoceramic or magnetostrictive), which employs a compressive load spring.
<a id="orgd835559"></a> <a id="orge71c3a4"></a>
{{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="Figure 1: A general Stewart Platform" >}} {{< figure src="/ox-hugo/mcinroy99_general_hexapod.png" caption="Figure 1: A general Stewart Platform" >}}
Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#org26f1840). Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#orgc6987ef).
In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs. In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
<a id="org26f1840"></a> <a id="orgc6987ef"></a>
{{< figure src="/ox-hugo/mcinroy99_strut_model.png" caption="Figure 2: The dynamics of the i'th strut. A parallel spring, damper and actuator drives the moving mass of the strut and a payload" >}} {{< figure src="/ox-hugo/mcinroy99_strut_model.png" caption="Figure 2: The dynamics of the i'th strut. A parallel spring, damper and actuator drives the moving mass of the strut and a payload" >}}
<a id="table--tab:mcinroy99-strut-model"></a> <a id="table--tab:mcinroy99-strut-model"></a>
<div class="table-caption"> <div class="table-caption">
<span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a></span>: <span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a></span>:
Definition of quantities on Figure <a href="#org26f1840">2</a> Definition of quantities on Figure <a href="#orgc6987ef">2</a>
</div> </div>
| **Symbol** | **Meaning** | | **Symbol** | **Meaning** |
@ -74,11 +70,11 @@ In order to provide low frequency passive vibration isolation, the hard actuator
| \\(v\_i = p\_i - q\_i\\) | vector pointing from the bottom to the top | | \\(v\_i = p\_i - q\_i\\) | vector pointing from the bottom to the top |
| \\(\hat{u}\_i = v\_i/l\_i\\) | unit direction of the strut | | \\(\hat{u}\_i = v\_i/l\_i\\) | unit direction of the strut |
It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in ([McInroy 2002](#org7752c60))). It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in ([McInroy 2002](#org6bd1808))).
This is a good approximation unless the spherical joints and extremely stiff or massive, of high inertia struts are used. This is a good approximation unless the spherical joints and extremely stiff or massive, of high inertia struts are used.
This allows to reduce considerably the complexity of the model. This allows to reduce considerably the complexity of the model.
From Figure [2](#org26f1840) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term): From Figure [2](#orgc6987ef) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
\begin{equation} \begin{equation}
m\_i \hat{u}\_i^T \ddot{p}\_i = f\_{m\_i} - f\_{p\_i} - m\_i \hat{u}\_i^Tg - k\_i(l\_i - l\_{r\_i}) - b\_i \dot{l}\_i m\_i \hat{u}\_i^T \ddot{p}\_i = f\_{m\_i} - f\_{p\_i} - m\_i \hat{u}\_i^Tg - k\_i(l\_i - l\_{r\_i}) - b\_i \dot{l}\_i
@ -166,8 +162,9 @@ In the next section, a connection between the two will be found to complete the
## Control Example {#control-example} ## Control Example {#control-example}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgfc7fa52"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>. <a id="org788f3dd"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
<a id="org7752c60"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>. <a id="org6bd1808"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.

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@ -8,7 +8,7 @@ Tags
: [Motion Control]({{< relref "motion_control" >}}) : [Motion Control]({{< relref "motion_control" >}})
Reference Reference
: ([Oomen 2018](#org18923fa)) : ([Oomen 2018](#org93151b9))
Author(s) Author(s)
: Oomen, T. : Oomen, T.
@ -16,11 +16,12 @@ Author(s)
Year Year
: 2018 : 2018
<a id="orgf64e727"></a> <a id="orgcc3437a"></a>
{{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="Figure 1: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth." >}} {{< figure src="/ox-hugo/oomen18_next_gen_loop_gain.png" caption="Figure 1: Envisaged developments in motion systems. In traditional motion systems, the control bandwidth takes place in the rigid-body region. In the next generation systemes, flexible dynamics are foreseen to occur within the control bandwidth." >}}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org18923fa"></a>Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” _IEEJ Journal of Industry Applications_ 7 (2):127–40. <https://doi.org/10.1541/ieejjia.7.127>. <a id="org93151b9"></a>Oomen, Tom. 2018. “Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems.” _IEEJ Journal of Industry Applications_ 7 (2):127–40. <https://doi.org/10.1541/ieejjia.7.127>.

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@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) : [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference Reference
: ([Preumont et al. 2007](#org3b370bf)) : ([Preumont et al. 2007](#org66383b9))
Author(s) Author(s)
: Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., … : Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …
@ -18,34 +18,35 @@ Year
Summary: Summary:
- **Cubic** Stewart platform (Figure [3](#org0274a43)) - **Cubic** Stewart platform (Figure [3](#orga6bbb17))
- Provides uniform control capability - Provides uniform control capability
- Uniform stiffness in all directions - Uniform stiffness in all directions
- minimizes the cross-coupling among actuators and sensors of different legs - minimizes the cross-coupling among actuators and sensors of different legs
- Flexible joints (Figure [2](#orge9ceb91)) - Flexible joints (Figure [2](#orga37017c))
- Piezoelectric force sensors - Piezoelectric force sensors
- Voice coil actuators - Voice coil actuators
- Decentralized feedback control approach for vibration isolation - Decentralized feedback control approach for vibration isolation
- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#orgb3c0578)) - Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#orgec64e08))
- The Stewart platform has 6 suspension modes at different frequencies. - The Stewart platform has 6 suspension modes at different frequencies.
Thus the gain of the IFF controller cannot be optimal for all the modes. Thus the gain of the IFF controller cannot be optimal for all the modes.
It is better if all the modes of the platform are near to each other. It is better if all the modes of the platform are near to each other.
- Discusses the design of the legs in order to maximize the natural frequency of the local modes. - Discusses the design of the legs in order to maximize the natural frequency of the local modes.
- To estimate the isolation performance of the Stewart platform, a scalar indicator is defined as the Frobenius norm of the transmissibility matrix - To estimate the isolation performance of the Stewart platform, a scalar indicator is defined as the Frobenius norm of the transmissibility matrix
<a id="orgb3c0578"></a> <a id="orgec64e08"></a>
{{< figure src="/ox-hugo/preumont07_iff_effect_stiffness.png" caption="Figure 1: Root locus with IFF with no parasitic stiffness and with parasitic stiffness" >}} {{< figure src="/ox-hugo/preumont07_iff_effect_stiffness.png" caption="Figure 1: Root locus with IFF with no parasitic stiffness and with parasitic stiffness" >}}
<a id="orge9ceb91"></a> <a id="orga37017c"></a>
{{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="Figure 2: Flexible joints used for the Stewart platform" >}} {{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="Figure 2: Flexible joints used for the Stewart platform" >}}
<a id="org0274a43"></a> <a id="orga6bbb17"></a>
{{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="Figure 3: Stewart platform" >}} {{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="Figure 3: Stewart platform" >}}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org3b370bf"></a>Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” _Journal of Sound and Vibration_ 300 (3-5):644–61. <https://doi.org/10.1016/j.jsv.2006.07.050>. <a id="org66383b9"></a>Preumont, A., M. Horodinca, I. Romanescu, B. de Marneffe, M. Avraam, A. Deraemaeker, F. Bossens, and A. Abu Hanieh. 2007. “A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform.” _Journal of Sound and Vibration_ 300 (3-5):644–61. <https://doi.org/10.1016/j.jsv.2006.07.050>.

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@ -8,7 +8,7 @@ Tags
: [Complementary Filters]({{< relref "complementary_filters" >}}), [Virtual Sensor Fusion]({{< relref "virtual_sensor_fusion" >}}) : [Complementary Filters]({{< relref "complementary_filters" >}}), [Virtual Sensor Fusion]({{< relref "virtual_sensor_fusion" >}})
Reference Reference
: ([Saxena and Hote 2012](#org0284b71)) : ([Saxena and Hote 2012](#org6d87fa3))
Author(s) Author(s)
: Saxena, S., & Hote, Y. : Saxena, S., & Hote, Y.
@ -85,6 +85,7 @@ Issues:
The interesting feature regarding IMC is that the design scheme is identical to the open-loop control design procedure and the implementation of IMC results in a feedback system, thereby copying the disturbances and parameter uncertainties, while open-loop control is not. The interesting feature regarding IMC is that the design scheme is identical to the open-loop control design procedure and the implementation of IMC results in a feedback system, thereby copying the disturbances and parameter uncertainties, while open-loop control is not.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org0284b71"></a>Saxena, Sahaj, and YogeshV Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” _IETE Technical Review_ 29 (6):461. <https://doi.org/10.4103/0256-4602.105001>. <a id="org6d87fa3"></a>Saxena, Sahaj, and YogeshV Hote. 2012. “Advances in Internal Model Control Technique: A Review and Future Prospects.” _IETE Technical Review_ 29 (6):461. <https://doi.org/10.4103/0256-4602.105001>.

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@ -9,7 +9,7 @@ Tags
Reference Reference
: ([Sayed and Kailath 2001](#orgaf03ac4)) : ([Sayed and Kailath 2001](#org9b5be86))
Author(s) Author(s)
: Sayed, A. H., & Kailath, T. : Sayed, A. H., & Kailath, T.
@ -18,6 +18,7 @@ Year
: 2001 : 2001
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgaf03ac4"></a>Sayed, A. H., and T. Kailath. 2001. “A Survey of Spectral Factorization Methods.” _Numerical Linear Algebra with Applications_ 8 (6-7):467–96. <https://doi.org/10.1002/nla.250>. <a id="org9b5be86"></a>Sayed, A. H., and T. Kailath. 2001. “A Survey of Spectral Factorization Methods.” _Numerical Linear Algebra with Applications_ 8 (6-7):467–96. <https://doi.org/10.1002/nla.250>.

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@ -9,7 +9,7 @@ Tags
Reference Reference
: ([Schroeck, Messner, and McNab 2001](#orgf8182bc)) : ([Schroeck, Messner, and McNab 2001](#org5e2e067))
Author(s) Author(s)
: Schroeck, S., Messner, W., & McNab, R. : Schroeck, S., Messner, W., & McNab, R.
@ -18,6 +18,7 @@ Year
: 2001 : 2001
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgf8182bc"></a>Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” _IEEE/ASME Transactions on Mechatronics_ 6 (1):50–57. <https://doi.org/10.1109/3516.914391>. <a id="org5e2e067"></a>Schroeck, S.J., W.C. Messner, and R.J. McNab. 2001. “On Compensator Design for Linear Time-Invariant Dual-Input Single-Output Systems.” _IEEE/ASME Transactions on Mechatronics_ 6 (1):50–57. <https://doi.org/10.1109/3516.914391>.

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@ -8,7 +8,7 @@ Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}) : [Sensor Fusion]({{< relref "sensor_fusion" >}})
Reference Reference
: ([Sebastian and Pantazi 2012](#org03bb39f)) : ([Sebastian and Pantazi 2012](#orgacf93b4))
Author(s) Author(s)
: Sebastian, A., & Pantazi, A. : Sebastian, A., & Pantazi, A.
@ -17,6 +17,7 @@ Year
: 2012 : 2012
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org03bb39f"></a>Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” _IEEE Transactions on Control Systems Technology_ 20 (2):382–94. <https://doi.org/10.1109/tcst.2011.2177982>. <a id="orgacf93b4"></a>Sebastian, Abu, and Angeliki Pantazi. 2012. “Nanopositioning with Multiple Sensors: A Case Study in Data Storage.” _IEEE Transactions on Control Systems Technology_ 20 (2):382–94. <https://doi.org/10.1109/tcst.2011.2177982>.

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@ -8,7 +8,7 @@ Tags
: [Active Damping]({{< relref "active_damping" >}}) : [Active Damping]({{< relref "active_damping" >}})
Reference Reference
: ([Souleille et al. 2018](#org5546d0c)) : ([Souleille et al. 2018](#org5548942))
Author(s) Author(s)
: Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gonccalo, & Collette, C. : Souleille, A., Lampert, T., Lafarga, V., Hellegouarch, S., Rondineau, A., Rodrigues, Gonccalo, & Collette, C.
@ -23,10 +23,10 @@ This article discusses the use of Integral Force Feedback with amplified piezoel
## Single degree-of-freedom isolator {#single-degree-of-freedom-isolator} ## Single degree-of-freedom isolator {#single-degree-of-freedom-isolator}
Figure [1](#org8634178) shows a picture of the amplified piezoelectric stack. Figure [1](#org0952ca2) shows a picture of the amplified piezoelectric stack.
The piezoelectric actuator is divided into two parts: one is used as an actuator, and the other one is used as a force sensor. The piezoelectric actuator is divided into two parts: one is used as an actuator, and the other one is used as a force sensor.
<a id="org8634178"></a> <a id="org0952ca2"></a>
{{< figure src="/ox-hugo/souleille18_model_piezo.png" caption="Figure 1: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator" >}} {{< figure src="/ox-hugo/souleille18_model_piezo.png" caption="Figure 1: Picture of an APA100M from Cedrat Technologies. Simplified model of a one DoF payload mounted on such isolator" >}}
@ -61,38 +61,39 @@ and the control force is given by:
f = F\_s G(s) = F\_s \frac{g}{s} f = F\_s G(s) = F\_s \frac{g}{s}
\end{equation} \end{equation}
The effect of the controller are shown in Figure [2](#orgcb733df): The effect of the controller are shown in Figure [2](#orgb36ba37):
- the resonance peak is almost critically damped - the resonance peak is almost critically damped
- the passive isolation \\(\frac{x\_1}{w}\\) is not degraded at high frequencies - the passive isolation \\(\frac{x\_1}{w}\\) is not degraded at high frequencies
- the degradation of the compliance \\(\frac{x\_1}{F}\\) induced by feedback is limited at \\(\frac{1}{k\_1}\\) - the degradation of the compliance \\(\frac{x\_1}{F}\\) induced by feedback is limited at \\(\frac{1}{k\_1}\\)
- the fraction of the force transmitted to the payload that is measured by the force sensor is reduced at low frequencies - the fraction of the force transmitted to the payload that is measured by the force sensor is reduced at low frequencies
<a id="orgcb733df"></a> <a id="orgb36ba37"></a>
{{< figure src="/ox-hugo/souleille18_tf_iff_result.png" caption="Figure 2: Matrix of transfer functions from input (w, f, F) to output (Fs, x1) in open loop (blue curves) and closed loop (dashed red curves)" >}} {{< figure src="/ox-hugo/souleille18_tf_iff_result.png" caption="Figure 2: Matrix of transfer functions from input (w, f, F) to output (Fs, x1) in open loop (blue curves) and closed loop (dashed red curves)" >}}
<a id="orga434456"></a> <a id="org5119227"></a>
{{< figure src="/ox-hugo/souleille18_root_locus.png" caption="Figure 3: Single DoF system. Comparison between the theoretical (solid curve) and the experimental (crosses) root-locus" >}} {{< figure src="/ox-hugo/souleille18_root_locus.png" caption="Figure 3: Single DoF system. Comparison between the theoretical (solid curve) and the experimental (crosses) root-locus" >}}
## Flexible payload mounted on three isolators {#flexible-payload-mounted-on-three-isolators} ## Flexible payload mounted on three isolators {#flexible-payload-mounted-on-three-isolators}
A heavy payload is mounted on a set of three isolators (Figure [4](#org09ac00a)). A heavy payload is mounted on a set of three isolators (Figure [4](#orgbe1030b)).
The payload consists of two masses, connected through flexible blades such that the flexible resonance of the payload in the vertical direction is around 65Hz. The payload consists of two masses, connected through flexible blades such that the flexible resonance of the payload in the vertical direction is around 65Hz.
<a id="org09ac00a"></a> <a id="orgbe1030b"></a>
{{< figure src="/ox-hugo/souleille18_setup_flexible_payload.png" caption="Figure 4: Right: picture of the experimental setup. It consists of a flexible payload mounted on a set of three isolators. Left: simplified sketch of the setup, showing only the vertical direction" >}} {{< figure src="/ox-hugo/souleille18_setup_flexible_payload.png" caption="Figure 4: Right: picture of the experimental setup. It consists of a flexible payload mounted on a set of three isolators. Left: simplified sketch of the setup, showing only the vertical direction" >}}
As shown in Figure [5](#org2dcbc51), both the suspension modes and the flexible modes of the payload can be critically damped. As shown in Figure [5](#orgbb573b8), both the suspension modes and the flexible modes of the payload can be critically damped.
<a id="org2dcbc51"></a> <a id="orgbb573b8"></a>
{{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="Figure 5: Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}} {{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="Figure 5: Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org5546d0c"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” _CEAS Space Journal_ 10 (2). Springer:157–65. <a id="org5548942"></a>Souleille, Adrien, Thibault Lampert, V Lafarga, Sylvain Hellegouarch, Alan Rondineau, Gonçalo Rodrigues, and Christophe Collette. 2018. “A Concept of Active Mount for Space Applications.” _CEAS Space Journal_ 10 (2). Springer:157–65.

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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference Reference
: ([Tang, Cao, and Yu 2018](#org45ebb6f)) : ([Tang, Cao, and Yu 2018](#org44aa05e))
Author(s) Author(s)
: Tang, J., Cao, D., & Yu, T. : Tang, J., Cao, D., & Yu, T.
@ -17,6 +17,7 @@ Year
: 2018 : 2018
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org45ebb6f"></a>Tang, Jie, Dengqing Cao, and Tianhu Yu. 2018. “Decentralized Vibration Control of a Voice Coil Motor-Based Stewart Parallel Mechanism: Simulation and Experiments.” _Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science_ 233 (1):132–45. <https://doi.org/10.1177/0954406218756941>. <a id="org44aa05e"></a>Tang, Jie, Dengqing Cao, and Tianhu Yu. 2018. “Decentralized Vibration Control of a Voice Coil Motor-Based Stewart Parallel Mechanism: Simulation and Experiments.” _Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science_ 233 (1):132–45. <https://doi.org/10.1177/0954406218756941>.

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@ -8,7 +8,7 @@ Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}) : [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}})
Reference Reference
: ([Wang et al. 2012](#org187cf70)) : ([Wang et al. 2012](#org172c2d6))
Author(s) Author(s)
: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M. : Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M.
@ -26,6 +26,7 @@ There is a need for markerless nano-tomography
It uses calibrated metrology disc and capacitive sensors It uses calibrated metrology disc and capacitive sensors
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org187cf70"></a>Wang, Jun, Yu-chen Karen Chen, Qingxi Yuan, Andrei Tkachuk, Can Erdonmez, Benjamin Hornberger, and Michael Feser. 2012. “Automated Markerless Full Field Hard X-Ray Microscopic Tomography at Sub-50 Nm 3-Dimension Spatial Resolution.” _Applied Physics Letters_ 100 (14):143107. <https://doi.org/10.1063/1.3701579>. <a id="org172c2d6"></a>Wang, Jun, Yu-chen Karen Chen, Qingxi Yuan, Andrei Tkachuk, Can Erdonmez, Benjamin Hornberger, and Michael Feser. 2012. “Automated Markerless Full Field Hard X-Ray Microscopic Tomography at Sub-50 Nm 3-Dimension Spatial Resolution.” _Applied Physics Letters_ 100 (14):143107. <https://doi.org/10.1063/1.3701579>.

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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference Reference
: ([Wang et al. 2016](#orgda82aa7)) : ([Wang et al. 2016](#org6a7c8f9))
Author(s) Author(s)
: Wang, C., Xie, X., Chen, Y., & Zhang, Z. : Wang, C., Xie, X., Chen, Y., & Zhang, Z.
@ -25,7 +25,7 @@ Year
The model is compared with a Finite Element model and is shown to give the same results. The model is compared with a Finite Element model and is shown to give the same results.
The proposed model is thus effective. The proposed model is thus effective.
<a id="orgc0b4cf5"></a> <a id="org5acb23c"></a>
{{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="Figure 1: Stewart Platform" >}} {{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
@ -35,11 +35,11 @@ Combines:
- the FxLMS-based adaptive inverse control => suppress transmission of periodic vibrations - the FxLMS-based adaptive inverse control => suppress transmission of periodic vibrations
- direct feedback of integrated forces => dampen vibration of inherent modes and thus reduce random vibrations - direct feedback of integrated forces => dampen vibration of inherent modes and thus reduce random vibrations
Force Feedback (Figure [2](#orgaf56f94)). Force Feedback (Figure [2](#org25780ff)).
- the force sensor is mounted **between the base and the strut** - the force sensor is mounted **between the base and the strut**
<a id="orgaf56f94"></a> <a id="org25780ff"></a>
{{< figure src="/ox-hugo/wang16_force_feedback.png" caption="Figure 2: Feedback of integrated forces in the platform" >}} {{< figure src="/ox-hugo/wang16_force_feedback.png" caption="Figure 2: Feedback of integrated forces in the platform" >}}
@ -54,6 +54,7 @@ Sorts of HAC-LAC control:
- Effectiveness of control methods are shown - Effectiveness of control methods are shown
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgda82aa7"></a>Wang, Chaoxin, Xiling Xie, Yanhao Chen, and Zhiyi Zhang. 2016. “Investigation on Active Vibration Isolation of a Stewart Platform with Piezoelectric Actuators.” _Journal of Sound and Vibration_ 383 (November). Elsevier BV:1–19. <https://doi.org/10.1016/j.jsv.2016.07.021>. <a id="org6a7c8f9"></a>Wang, Chaoxin, Xiling Xie, Yanhao Chen, and Zhiyi Zhang. 2016. “Investigation on Active Vibration Isolation of a Stewart Platform with Piezoelectric Actuators.” _Journal of Sound and Vibration_ 383 (November). Elsevier BV:1–19. <https://doi.org/10.1016/j.jsv.2016.07.021>.

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@ -9,7 +9,7 @@ Tags
Reference Reference
: ([Yun et al. 2020](#orgd3a0930)) : ([Yun et al. 2020](#org2a2f02a))
Author(s) Author(s)
: Yun, H., Liu, L., Li, Q., & Yang, H. : Yun, H., Liu, L., Li, Q., & Yang, H.
@ -18,6 +18,7 @@ Year
: 2020 : 2020
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgd3a0930"></a>Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” _Aerospace Science and Technology_ 96 (nil):105543. <https://doi.org/10.1016/j.ast.2019.105543>. <a id="org2a2f02a"></a>Yun, Hai, Lei Liu, Qing Li, and Hongjie Yang. 2020. “Investigation on Two-Stage Vibration Suppression and Precision Pointing for Space Optical Payloads.” _Aerospace Science and Technology_ 96 (nil):105543. <https://doi.org/10.1016/j.ast.2019.105543>.

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@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}) : [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference Reference
: ([Zuo 2004](#orgb4186fb)) : ([Zuo 2004](#orgd9c4f73))
Author(s) Author(s)
: Zuo, L. : Zuo, L.
@ -40,23 +40,24 @@ Year
> They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback. > They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback.
> Note that reaction force actuators can also work with soft mounts or hard mounts. > Note that reaction force actuators can also work with soft mounts or hard mounts.
<a id="orgd66c057"></a> <a id="orgc88908d"></a>
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}} {{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}}
<a id="org1008b43"></a> <a id="orgb14b6c0"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}} {{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}}
<a id="orgab03e30"></a> <a id="org96eaaed"></a>
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}} {{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}}
<a id="orgc03f8c8"></a> <a id="orgaf13ec6"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}} {{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgb4186fb"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology. <a id="orgd9c4f73"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.

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@ -8,7 +8,7 @@ Tags
: [Multivariable Control]({{< relref "multivariable_control" >}}) : [Multivariable Control]({{< relref "multivariable_control" >}})
Reference Reference
: ([Albertos and Antonio 2004](#org911b8ba)) : ([Albertos and Antonio 2004](#orgb06343d))
Author(s) Author(s)
: Albertos, P., & Antonio, S. : Albertos, P., & Antonio, S.
@ -17,6 +17,116 @@ Year
: 2004 : 2004
## Introduction to Multivariable Control {#introduction-to-multivariable-control}
## Linear System Representation: Models and Equivalence {#linear-system-representation-models-and-equivalence}
## Linear Systems Analysis {#linear-systems-analysis}
## Solutions to the Control Problem {#solutions-to-the-control-problem}
## Decentralised and Decoupled Control {#decentralised-and-decoupled-control}
### Decoupling {#decoupling}
In cases when multi-loop control is not effective in reaching the desired specifications, a possible strategy for tackling the MIMO control could be to transform the transfer function matrix into a diagonal dominant one.
This strategy is called **decoupling**.
[Decoupled Control]({{< relref "decoupled_control" >}}) can be achieved in two ways:
- feedforward cancellation of the cross-coupling terms
- based on state measurements, via a feedback law
#### Feedforward Decoupling {#feedforward-decoupling}
A pre-compensator can be added to transform the open-loop characteristics into a new one as chosen by the designer.
This decoupler can be taken as the inverse of the plant provided it does not include RHP-zeros.
**Approximate decoupling**:
To design low-bandwidth loops, insertion of the inverse DC-gain before the loop ensures decoupling at least at steady-state.
If further bandwidth extension is desired, an approximation of \\(G^{-1}\\) valid in low frequencies can be used.
#### Feedback Decoupling {#feedback-decoupling}
Although at first glance, decoupling seems an appealing idea, there are some drawbacks:
- as decoupling is achieved via the coordination of sensors and actuators to achieve an "apparent" diagonal behavior, the failure of one the actuators may heavily affects all loops.
- a decoupling design (inverse-based controller) may not be desirable for all disturbance-rejection tasks.
- many MIMO non-minimum phase systems, when feedforward decoupled, increase the RHP-zero multiplicity so performance limitations due to its presence are exacerbated.
- decoupling may be very sensitive to modeling errors, specially for ill-conditionned plants
- feedback decoupling needs full state measurements
#### SVD Decoupling {#svd-decoupling}
A matrix \\(M\\) can be expressed, using the [Singular Value Decomposition]({{< relref "singular_value_decomposition" >}}) as:
\begin{equation}
M = U \Sigma V^T
\end{equation}
where \\(U\\) and \\(V\\) are orthogonal matrices and \\(\Sigma\\) is diagonal.
The SVD can be used to obtain decoupled equations between linear combinations of sensors and linear combinations of actuators.
In this way, although losing part of its intuitive sense, a decoupled design can be carried out even for non-square plants.
If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [1](#org3d5b40c), then the loop, in the transformed variables, is decoupled, so a diagonal controller \\(K\_D\\) can be used.
Usually, the sensor and actuator transformations are obtained using the DC gain, or a real approximation of \\(G(j\omega)\\), where \\(\omega\\) is around the desired closed-loop bandwidth.
<a id="org3d5b40c"></a>
{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
The transformed sensor-actuator pair corresponding to the maximum singular value is the direction with biggest "gain" on the plant, that is, the combination of variables being "easiest to control".
In ill-conditioned plants, the ratio between the biggest and lower singular value is large (for reference, greater than 20).
They are very sensitive to input uncertainty as some "input directions" have much bigger gain than other ones.
SVD decoupling produces the most suitable combinations for independent "multi-loop" control in the transformed variables, so its performance may be better than RGA-based design (at the expense of losing physical interpretability).
If some of the vectors in \\(V\\) (input directions) have a significant component on a particular input, and the corresponding output direction is also significantly pointing to a particular output, that combination is a good candidate for an independent multi-loop control.
## Fundamentals of Centralised Closed-loop Control {#fundamentals-of-centralised-closed-loop-control}
## Optimisation-based Control {#optimisation-based-control}
## Designing for Robustness {#designing-for-robustness}
## Implementation and Other Issues {#implementation-and-other-issues}
## Appendices {#appendices}
### Summary of SISO System Analysis {#summary-of-siso-system-analysis}
### Matrices {#matrices}
### Signal and System Norms {#signal-and-system-norms}
### Optimisation {#optimisation}
### Multivariable Statistics {#multivariable-statistics}
### Robust Control Analysis and Synthesis {#robust-control-analysis-and-synthesis}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org911b8ba"></a>Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>. <a id="orgb06343d"></a>Albertos, P., and S. Antonio. 2004. _Multivariable Control Systems: An Engineering Approach_. Advanced Textbooks in Control and Signal Processing. Springer-Verlag. <https://doi.org/10.1007/b97506>.

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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}) : [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference Reference
: ([Du and Xie 2010](#orge0a6379)) : ([Du and Xie 2010](#org5093366))
Author(s) Author(s)
: Du, C., & Xie, L. : Du, C., & Xie, L.
@ -16,9 +16,524 @@ Author(s)
Year Year
: 2010 : 2010
Read Chapter 1 and 3.
## 1. Mechanical Systems and Vibration {#1-dot-mechanical-systems-and-vibration}
### 1.1 Magnetic recording system {#1-dot-1-magnetic-recording-system}
### 1.2 Stewart platform {#1-dot-2-stewart-platform}
### 1.3 Vibration sources and descriptions {#1-dot-3-vibration-sources-and-descriptions}
### 1.4 Types of vibration {#1-dot-4-types-of-vibration}
#### 1.4.1 Free and forced vibration {#1-dot-4-dot-1-free-and-forced-vibration}
#### 1.4.2 Damped and undamped vibration {#1-dot-4-dot-2-damped-and-undamped-vibration}
#### 1.4.3 Linear and nonlinear vibration {#1-dot-4-dot-3-linear-and-nonlinear-vibration}
#### 1.4.4 Deterministic and random vibration {#1-dot-4-dot-4-deterministic-and-random-vibration}
#### 1.4.5 Periodic and nonperiodic vibration {#1-dot-4-dot-5-periodic-and-nonperiodic-vibration}
#### 1.4.6 Broad-band and narrow-band vibration {#1-dot-4-dot-6-broad-band-and-narrow-band-vibration}
### 1.5 Random vibration {#1-dot-5-random-vibration}
#### 1.5.1 Random process {#1-dot-5-dot-1-random-process}
#### 1.5.2 Stationary random process {#1-dot-5-dot-2-stationary-random-process}
#### 1.5.3 Gaussian random process {#1-dot-5-dot-3-gaussian-random-process}
### 1.6 Vibration analysis {#1-dot-6-vibration-analysis}
#### 1.6.1 Fourier transform and spectrum analysis {#1-dot-6-dot-1-fourier-transform-and-spectrum-analysis}
#### 1.6.2 Relationship between the Fourier and Laplace transforms {#1-dot-6-dot-2-relationship-between-the-fourier-and-laplace-transforms}
#### 1.6.3 Spectral analysis {#1-dot-6-dot-3-spectral-analysis}
## 2. Modeling of Disk Drive System and Its Vibration {#2-dot-modeling-of-disk-drive-system-and-its-vibration}
### 2.1 Introduction {#2-dot-1-introduction}
### 2.2 System description {#2-dot-2-system-description}
### 2.3 System modeling {#2-dot-3-system-modeling}
#### 2.3.1 Modeling of a VCM actuator {#2-dot-3-dot-1-modeling-of-a-vcm-actuator}
#### 2.3.2 Modeling of friction {#2-dot-3-dot-2-modeling-of-friction}
#### 2.3.3 Modeling of a PZT microactuator {#2-dot-3-dot-3-modeling-of-a-pzt-microactuator}
#### 2.3.4 An example {#2-dot-3-dot-4-an-example}
### 2.4 Vibration modeling {#2-dot-4-vibration-modeling}
#### 2.4.1 Spectrum-based vibration modeling {#2-dot-4-dot-1-spectrum-based-vibration-modeling}
#### 2.4.2 Adaptive modeling of disturbance {#2-dot-4-dot-2-adaptive-modeling-of-disturbance}
### 2.5 Conclusion {#2-dot-5-conclusion}
## 3. Modeling of [Stewart Platforms]({{< relref "stewart_platforms" >}}) {#3-dot-modeling-of-stewart-platforms--stewart-platforms-dot-md}
### 3.1 Introduction {#3-dot-1-introduction}
### 3.2 System description and governing equations {#3-dot-2-system-description-and-governing-equations}
### 3.3 Modeling using adaptive filtering approach {#3-dot-3-modeling-using-adaptive-filtering-approach}
#### 3.3.1 Adaptive filtering theory {#3-dot-3-dot-1-adaptive-filtering-theory}
#### 3.3.2 Modeling of a Stewart platform {#3-dot-3-dot-2-modeling-of-a-stewart-platform}
### 3.4 Conclusion {#3-dot-4-conclusion}
## 4. Classical Vibration Control {#4-dot-classical-vibration-control}
### 4.1 Introduction {#4-dot-1-introduction}
### 4.2 Passive control {#4-dot-2-passive-control}
#### 4.2.1 Isolators {#4-dot-2-dot-1-isolators}
#### 4.2.2 Absorbers {#4-dot-2-dot-2-absorbers}
#### 4.2.3 Resonators {#4-dot-2-dot-3-resonators}
#### 4.2.4 Suspension {#4-dot-2-dot-4-suspension}
#### 4.2.5 An application example &#8211; Disk vibration reduction via stacked disks {#4-dot-2-dot-5-an-application-example-and-8211-disk-vibration-reduction-via-stacked-disks}
### 4.3 Self-adapting systems {#4-dot-3-self-adapting-systems}
### 4.4 Active vibration control {#4-dot-4-active-vibration-control}
#### 4.4.1 Actuators {#4-dot-4-dot-1-actuators}
#### 4.4.2 Active systems {#4-dot-4-dot-2-active-systems}
#### 4.4.3 Control strategy {#4-dot-4-dot-3-control-strategy}
### 4.5 Conclusion {#4-dot-5-conclusion}
## 5. Introduction to Optimal and Robust Control {#5-dot-introduction-to-optimal-and-robust-control}
### 5.1 Introduction {#5-dot-1-introduction}
### 5.2 H2 and H&#8734; norms {#5-dot-2-h2-and-h-and-8734-norms}
#### 5.2.1 H2 norm {#5-dot-2-dot-1-h2-norm}
#### 5.2.2 H&#8734; norm {#5-dot-2-dot-2-h-and-8734-norm}
### 5.3 H2 optimal control {#5-dot-3-h2-optimal-control}
#### 5.3.1 Continuous-time case {#5-dot-3-dot-1-continuous-time-case}
#### 5.3.2 Discrete-time case {#5-dot-3-dot-2-discrete-time-case}
### 5.4 H&#8734; control {#5-dot-4-h-and-8734-control}
#### 5.4.1 Continuous-time case {#5-dot-4-dot-1-continuous-time-case}
#### 5.4.2 Discrete-time case {#5-dot-4-dot-2-discrete-time-case}
### 5.5 Robust control {#5-dot-5-robust-control}
### 5.6 Controller parametrization {#5-dot-6-controller-parametrization}
### 5.7 Performance limitation {#5-dot-7-performance-limitation}
#### 5.7.1 Bode integral constraint {#5-dot-7-dot-1-bode-integral-constraint}
#### 5.7.2 Relationship between system gain and phase {#5-dot-7-dot-2-relationship-between-system-gain-and-phase}
#### 5.7.3 Sampling {#5-dot-7-dot-3-sampling}
### 5.8 Conclusion {#5-dot-8-conclusion}
## 6. Mixed H2/H&#8734; Control Design for Vibration Rejection {#6-dot-mixed-h2-h-and-8734-control-design-for-vibration-rejection}
### 6.1 Introduction {#6-dot-1-introduction}
### 6.2 Mixed H2/H&#8734; control problem {#6-dot-2-mixed-h2-h-and-8734-control-problem}
### 6.3 Method 1: slack variable approach {#6-dot-3-method-1-slack-variable-approach}
### 6.4 Method 2: an improved slack variable approach {#6-dot-4-method-2-an-improved-slack-variable-approach}
### 6.5 Application in servo loop design for hard disk drives {#6-dot-5-application-in-servo-loop-design-for-hard-disk-drives}
#### 6.5.1 Problem formulation {#6-dot-5-dot-1-problem-formulation}
#### 6.5.2 Design results {#6-dot-5-dot-2-design-results}
### 6.6 Conclusion {#6-dot-6-conclusion}
## 7. Low-Hump Sensitivity Control Design for Hard Disk Drive Systems {#7-dot-low-hump-sensitivity-control-design-for-hard-disk-drive-systems}
### 7.1 Introduction {#7-dot-1-introduction}
### 7.2 Problem statement {#7-dot-2-problem-statement}
### 7.3 Design in continuous-time domain {#7-dot-3-design-in-continuous-time-domain}
#### 7.3.1 H&#8734; loop shaping for low-hump sensitivity functions {#7-dot-3-dot-1-h-and-8734-loop-shaping-for-low-hump-sensitivity-functions}
#### 7.3.2 Application examples {#7-dot-3-dot-2-application-examples}
#### 7.3.3 Implementation on a hard disk drive {#7-dot-3-dot-3-implementation-on-a-hard-disk-drive}
### 7.4 Design in discrete-time domain {#7-dot-4-design-in-discrete-time-domain}
#### 7.4.1 Synthesis method for low-hump sensitivity function {#7-dot-4-dot-1-synthesis-method-for-low-hump-sensitivity-function}
#### 7.4.2 An application example {#7-dot-4-dot-2-an-application-example}
#### 7.4.3 Implementation on a hard disk drive {#7-dot-4-dot-3-implementation-on-a-hard-disk-drive}
### 7.5 Conclusion {#7-dot-5-conclusion}
## 8. Generalized KYP Lemma-Based Loop Shaping Control Design {#8-dot-generalized-kyp-lemma-based-loop-shaping-control-design}
### 8.1 Introduction {#8-dot-1-introduction}
### 8.2 Problem description {#8-dot-2-problem-description}
### 8.3 Generalized KYP lemma-based control design method {#8-dot-3-generalized-kyp-lemma-based-control-design-method}
### 8.4 Peak filter {#8-dot-4-peak-filter}
#### 8.4.1 Conventional peak filter {#8-dot-4-dot-1-conventional-peak-filter}
#### 8.4.2 Phase lead peak filter {#8-dot-4-dot-2-phase-lead-peak-filter}
#### 8.4.3 Group peak filter {#8-dot-4-dot-3-group-peak-filter}
### 8.5 Application in high frequency vibration rejection {#8-dot-5-application-in-high-frequency-vibration-rejection}
### 8.6 Application in mid-frequency vibration rejection {#8-dot-6-application-in-mid-frequency-vibration-rejection}
### 8.7 Conclusion {#8-dot-7-conclusion}
## 9. Combined H2 and KYP Lemma-Based Control Design {#9-dot-combined-h2-and-kyp-lemma-based-control-design}
### 9.1 Introduction {#9-dot-1-introduction}
### 9.2 Problem formulation {#9-dot-2-problem-formulation}
### 9.3 Controller design for specific disturbance rejection and overall error minimization {#9-dot-3-controller-design-for-specific-disturbance-rejection-and-overall-error-minimization}
#### 9.3.1 Q parametrization to meet specific specifications {#9-dot-3-dot-1-q-parametrization-to-meet-specific-specifications}
#### 9.3.2 Q parametrization to minimize H2 performance {#9-dot-3-dot-2-q-parametrization-to-minimize-h2-performance}
#### 9.3.3 Design steps {#9-dot-3-dot-3-design-steps}
### 9.4 Simulation and implementation results {#9-dot-4-simulation-and-implementation-results}
#### 9.4.1 System models {#9-dot-4-dot-1-system-models}
#### 9.4.2 Rejection of specific disturbance and H2 performance minimization {#9-dot-4-dot-2-rejection-of-specific-disturbance-and-h2-performance-minimization}
#### 9.4.3 Rejection of two disturbances with H[sub(2)] performance minimization {#9-dot-4-dot-3-rejection-of-two-disturbances-with-h-sub--2--performance-minimization}
### 9.5 Conclusion {#9-dot-5-conclusion}
## 10. Blending Control for Multi-Frequency Disturbance Rejection {#10-dot-blending-control-for-multi-frequency-disturbance-rejection}
### 10.1 Introduction {#10-dot-1-introduction}
### 10.2 Control blending {#10-dot-2-control-blending}
#### 10.2.1 State feedback control blending {#10-dot-2-dot-1-state-feedback-control-blending}
#### 10.2.2 Output feedback control blending {#10-dot-2-dot-2-output-feedback-control-blending}
### 10.3 Control blending application in multi-frequency disturbance rejection {#10-dot-3-control-blending-application-in-multi-frequency-disturbance-rejection}
#### 10.3.1 Problem formulation {#10-dot-3-dot-1-problem-formulation}
#### 10.3.2 Controller design via the control blending technique {#10-dot-3-dot-2-controller-design-via-the-control-blending-technique}
### 10.4 Simulation and experimental results {#10-dot-4-simulation-and-experimental-results}
#### 10.4.1 Rejecting high-frequency disturbances {#10-dot-4-dot-1-rejecting-high-frequency-disturbances}
#### 10.4.2 Rejecting a combined mid and high frequency disturbance {#10-dot-4-dot-2-rejecting-a-combined-mid-and-high-frequency-disturbance}
### 10.5 Conclusion {#10-dot-5-conclusion}
## 11. H&#8734;-Based Design for Disturbance Observer {#11-dot-h-and-8734-based-design-for-disturbance-observer}
### 11.1 Introduction {#11-dot-1-introduction}
### 11.2 Conventional disturbance observer {#11-dot-2-conventional-disturbance-observer}
### 11.3 A general form of disturbance observer {#11-dot-3-a-general-form-of-disturbance-observer}
### 11.4 Application results {#11-dot-4-application-results}
### 11.5 Conclusion {#11-dot-5-conclusion}
## 12. Two-Dimensional H2 Control for Error Minimization {#12-dot-two-dimensional-h2-control-for-error-minimization}
### 12.1 Introduction {#12-dot-1-introduction}
### 12.2 2-D stabilization control {#12-dot-2-2-d-stabilization-control}
### 12.3 2-D H2 control {#12-dot-3-2-d-h2-control}
### 12.4 SSTW process and modeling {#12-dot-4-sstw-process-and-modeling}
#### 12.4.1 SSTW servo loop {#12-dot-4-dot-1-sstw-servo-loop}
#### 12.4.2 Two-dimensional model {#12-dot-4-dot-2-two-dimensional-model}
### 12.5 Feedforward compensation method {#12-dot-5-feedforward-compensation-method}
### 12.6 2-D control formulation for SSTW {#12-dot-6-2-d-control-formulation-for-sstw}
### 12.7 2-D stabilization control for error propagation containment {#12-dot-7-2-d-stabilization-control-for-error-propagation-containment}
#### 12.7.1 Simulation results {#12-dot-7-dot-1-simulation-results}
### 12.8 2-D H2 control for error minimization {#12-dot-8-2-d-h2-control-for-error-minimization}
#### 12.8.1 Simulation results {#12-dot-8-dot-1-simulation-results}
#### 12.8.2 Experimental results {#12-dot-8-dot-2-experimental-results}
### 12.9 Conclusion {#12-dot-9-conclusion}
## 13. Nonlinearity Compensation and Nonlinear Control {#13-dot-nonlinearity-compensation-and-nonlinear-control}
### 13.1 Introduction {#13-dot-1-introduction}
### 13.2 Nonlinearity compensation {#13-dot-2-nonlinearity-compensation}
### 13.3 Nonlinear control {#13-dot-3-nonlinear-control}
#### 13.3.1 Design of a composite control law {#13-dot-3-dot-1-design-of-a-composite-control-law}
#### 13.3.2 Experimental results in hard disk drives {#13-dot-3-dot-2-experimental-results-in-hard-disk-drives}
### 13.4 Conclusion {#13-dot-4-conclusion}
## 14. Quantization Effect on Vibration Rejection and Its Compensation {#14-dot-quantization-effect-on-vibration-rejection-and-its-compensation}
### 14.1 Introduction {#14-dot-1-introduction}
### 14.2 Description of control system with quantizer {#14-dot-2-description-of-control-system-with-quantizer}
### 14.3 Quantization effect on error rejection {#14-dot-3-quantization-effect-on-error-rejection}
#### 14.3.1 Quantizer frequency response measurement {#14-dot-3-dot-1-quantizer-frequency-response-measurement}
#### 14.3.2 Quantization effect on error rejection {#14-dot-3-dot-2-quantization-effect-on-error-rejection}
### 14.4 Compensation of quantization effect on error rejection {#14-dot-4-compensation-of-quantization-effect-on-error-rejection}
### 14.5 Conclusion {#14-dot-5-conclusion}
## 15. Adaptive Filtering Algorithms for Active Vibration Control {#15-dot-adaptive-filtering-algorithms-for-active-vibration-control}
### 15.1 Introduction {#15-dot-1-introduction}
### 15.2 Adaptive feedforward algorithm {#15-dot-2-adaptive-feedforward-algorithm}
### 15.3 Adaptive feedback algorithm {#15-dot-3-adaptive-feedback-algorithm}
### 15.4 Comparison between feedforward and feedback controls {#15-dot-4-comparison-between-feedforward-and-feedback-controls}
### 15.5 Application in Stewart platform {#15-dot-5-application-in-stewart-platform}
#### 15.5.1 Multi-channel adaptive feedback AVC system {#15-dot-5-dot-1-multi-channel-adaptive-feedback-avc-system}
#### 15.5.2 Multi-channel adaptive feedback algorithm for hexapod platform {#15-dot-5-dot-2-multi-channel-adaptive-feedback-algorithm-for-hexapod-platform}
#### 15.5.3 Simulation and implementation {#15-dot-5-dot-3-simulation-and-implementation}
### 15.6 Conclusion {#15-dot-6-conclusion}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orge0a6379"></a>Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. <https://doi.org/10.1201/9781439817995>. <a id="org5093366"></a>Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. <https://doi.org/10.1201/9781439817995>.

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content/book/hatch00_vibrat_matlab_ansys.md
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@ -8,7 +8,7 @@ Tags
: [Finite Element Model]({{< relref "finite_element_model" >}}) : [Finite Element Model]({{< relref "finite_element_model" >}})
Reference Reference
: ([Hatch 2000](#orgebf8ccb)) : ([Hatch 2000](#orgf661cf4))
Author(s) Author(s)
: Hatch, M. R. : Hatch, M. R.
@ -21,14 +21,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction} ## Introduction {#introduction}
<a id="org4115a7e"></a> <a id="org1a34f33"></a>
The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models. The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models.
### Modal Analysis {#modal-analysis} ### Modal Analysis {#modal-analysis}
The diagram in Figure [1](#org242701e) shows the methodology for analyzing a lightly damped structure using normal modes. The diagram in Figure [1](#org3a8e4cc) shows the methodology for analyzing a lightly damped structure using normal modes.
<div class="important"> <div class="important">
<div></div> <div></div>
@ -46,7 +46,7 @@ The steps are:
</div> </div>
<a id="org242701e"></a> <a id="org3a8e4cc"></a>
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}} {{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}}
@ -58,7 +58,7 @@ Because finite element models usually have a very large number of states, an imp
<div class="important"> <div class="important">
<div></div> <div></div>
Figure [2](#org398c443) shows such process, the steps are: Figure [2](#org0a7e8db) shows such process, the steps are:
- start with the finite element model - start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model) - compute the eigenvalues and eigenvectors (as many as dof in the model)
@ -71,14 +71,14 @@ Figure [2](#org398c443) shows such process, the steps are:
</div> </div>
<a id="org398c443"></a> <a id="org0a7e8db"></a>
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}} {{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}
### Notations {#notations} ### Notations {#notations}
Tables [3](#org02d84e8), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document. Tables [3](#orgc82b5d8), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
<a id="table--tab:notations-modes-nodes"></a> <a id="table--tab:notations-modes-nodes"></a>
<div class="table-caption"> <div class="table-caption">
@ -127,22 +127,22 @@ Tables [3](#org02d84e8), [2](#table--tab:notations-eigen-vectors-values) and [3]
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems} ## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
<a id="orga20292c"></a> <a id="orgea88319"></a>
The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency. The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency.
We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**. We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**.
Figure [3](#org02d84e8) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs. Figure [3](#orgc82b5d8) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\). The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
<a id="org02d84e8"></a> <a id="orgc82b5d8"></a>
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}} {{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
<div class="important"> <div class="important">
<div></div> <div></div>
([Miu 1993](#org39eead7)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function. ([Miu 1993](#org849cfe4)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
The resonances of the "overhanging appendages" of the constrained system create the zeros. The resonances of the "overhanging appendages" of the constrained system create the zeros.
@ -151,12 +151,12 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis} ## State Space Analysis {#state-space-analysis}
<a id="org24eb004"></a> <a id="org79a4830"></a>
## Modal Analysis {#modal-analysis} ## Modal Analysis {#modal-analysis}
<a id="orgc6df38d"></a> <a id="orgb718e30"></a>
Lightly damped structures are typically analyzed with the "normal mode" method described in this section. Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
@ -196,9 +196,9 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and
#### Equation of Motion {#equation-of-motion} #### Equation of Motion {#equation-of-motion}
Let's consider the model shown in Figure [4](#org0c2921d) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\). Let's consider the model shown in Figure [4](#orgebf4457) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
<a id="org0c2921d"></a> <a id="orgebf4457"></a>
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}} {{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}}
@ -297,17 +297,17 @@ One then find:
\end{bmatrix} \end{bmatrix}
\end{equation} \end{equation}
Virtual interpretation of the eigenvectors are shown in Figures [5](#orgc90fe3a), [6](#orgfd8222c) and [7](#orgaf9cc36). Virtual interpretation of the eigenvectors are shown in Figures [5](#org520a99d), [6](#org722a9ff) and [7](#org9e25b28).
<a id="orgc90fe3a"></a> <a id="org520a99d"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}} {{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
<a id="orgfd8222c"></a> <a id="org722a9ff"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}} {{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}
<a id="orgaf9cc36"></a> <a id="org9e25b28"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}} {{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}}
@ -346,9 +346,9 @@ There are many options for change of basis, but we will show that **when eigenve
The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems. The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems.
The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate. The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate.
This procedure is schematically shown in Figure [8](#orgf9a2963). This procedure is schematically shown in Figure [8](#orgfbabf08).
<a id="orgf9a2963"></a> <a id="orgfbabf08"></a>
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}} {{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}
@ -696,7 +696,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
## Frequency Response: Modal Form {#frequency-response-modal-form} ## Frequency Response: Modal Form {#frequency-response-modal-form}
<a id="orgfd97109"></a> <a id="org3f5ad6c"></a>
The procedure to obtain the frequency response from a modal form is as follow: The procedure to obtain the frequency response from a modal form is as follow:
@ -704,9 +704,9 @@ The procedure to obtain the frequency response from a modal form is as follow:
- use Laplace transform to obtain the transfer functions in principal coordinates - use Laplace transform to obtain the transfer functions in principal coordinates
- back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident - back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident
This will be applied to the model shown in Figure [9](#org48b68a4). This will be applied to the model shown in Figure [9](#orge102983).
<a id="org48b68a4"></a> <a id="orge102983"></a>
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}} {{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}
@ -888,9 +888,9 @@ Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows tha
</div> </div>
Figure [10](#org87763b9) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\). Figure [10](#org3024448) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
<a id="org87763b9"></a> <a id="org3024448"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}} {{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@ -899,16 +899,16 @@ The zeros for SISO transfer functions are the roots of the numerator, however, f
## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model} ## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
<a id="org031e9ac"></a> <a id="org6842a3c"></a>
### Introduction {#introduction} ### Introduction {#introduction}
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orga66d597). In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org2292476).
A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output. A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics. The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
<a id="orga66d597"></a> <a id="org2292476"></a>
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}} {{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}}
@ -987,7 +987,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model} ## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
<a id="org9a76f4b"></a> <a id="orgc13e165"></a>
### Model Description {#model-description} ### Model Description {#model-description}
@ -1001,25 +1001,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model} ## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
<a id="orga8b2a2f"></a> <a id="org03aa6d1"></a>
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org94e126d)). In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org143e4e8)).
### Actuator Description {#actuator-description} ### Actuator Description {#actuator-description}
<a id="org94e126d"></a> <a id="org143e4e8"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}} {{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}}
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#org4a20950)). The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#orgc294fc5)).
<a id="org4a20950"></a> <a id="orgc294fc5"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}} {{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required. For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
Figure [13](#org4a20950) shows the nodes used for the reduced Matlab model. Figure [13](#orgc294fc5) shows the nodes used for the reduced Matlab model.
The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force. The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
The arrows at the nodes indicate the direction of forces. The arrows at the nodes indicate the direction of forces.
@ -1087,7 +1087,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction} ## Balanced Reduction {#balanced-reduction}
<a id="org56fcc2f"></a> <a id="org1f06bfa"></a>
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods. In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
@ -1202,14 +1202,14 @@ The **states to be kept are the states with the largest diagonal terms**.
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model} ## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
<a id="orga7cf69e"></a> <a id="orgfc560f8"></a>
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org1453e17)). In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org7003388)).
### Actuator Description {#actuator-description} ### Actuator Description {#actuator-description}
<a id="org1453e17"></a> <a id="org7003388"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}} {{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
@ -1231,9 +1231,9 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description} ### Ansys Model Description {#ansys-model-description}
In Figure [15](#orge94bde1) are shown the principal nodes used for the model. In Figure [15](#org472d510) are shown the principal nodes used for the model.
<a id="orge94bde1"></a> <a id="org472d510"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}} {{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
@ -1352,11 +1352,11 @@ And we note:
G = zn * Gp; G = zn * Gp;
``` ```
<a id="org22a3db4"></a> <a id="orgfd8bb64"></a>
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}} {{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
<a id="org33c49a2"></a> <a id="org8f08d6b"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}} {{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@ -1454,13 +1454,13 @@ State Space Model
### Simple mode truncation {#simple-mode-truncation} ### Simple mode truncation {#simple-mode-truncation}
Let's plot the frequency of the modes (Figure [18](#orga04e866)). Let's plot the frequency of the modes (Figure [18](#org6e52a4a)).
<a id="orga04e866"></a> <a id="org6e52a4a"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
<a id="org0c4b8bc"></a> <a id="org52d380e"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
@ -1529,7 +1529,7 @@ Let's sort the modes by their DC gains and plot their sorted DC gains.
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend'); [dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
``` ```
<a id="orga62ba4f"></a> <a id="org873b074"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
@ -1873,7 +1873,7 @@ Then, we compute the controllability and observability gramians.
And we plot the diagonal terms And we plot the diagonal terms
<a id="orgab7d0ba"></a> <a id="org295f621"></a>
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}} {{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
@ -1891,7 +1891,7 @@ We use `balreal` to rank oscillatory states.
[G_b, G, T, Ti] = balreal(G_m); [G_b, G, T, Ti] = balreal(G_m);
``` ```
<a id="orgb25e36e"></a> <a id="org2867cfa"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}} {{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
@ -2134,8 +2134,9 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
``` ```
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgebf8ccb"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press. <a id="orgf661cf4"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="org39eead7"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York. <a id="org849cfe4"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.

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@ -8,7 +8,7 @@ Tags
: [Metrology]({{< relref "metrology" >}}) : [Metrology]({{< relref "metrology" >}})
Reference Reference
: ([Leach 2014](#orgc132434)) : ([Leach 2014](#org023e404))
Author(s) Author(s)
: Leach, R. : Leach, R.
@ -87,6 +87,7 @@ The measurement of angles is then relative.
This type of angular interferometer is used to measure small angles (less than \\(10deg\\)). This type of angular interferometer is used to measure small angles (less than \\(10deg\\)).
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgc132434"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>. <a id="org023e404"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.

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@ -8,7 +8,7 @@ Tags
: [Precision Engineering]({{< relref "precision_engineering" >}}) : [Precision Engineering]({{< relref "precision_engineering" >}})
Reference Reference
: ([Leach and Smith 2018](#org50ae2e1)) : ([Leach and Smith 2018](#orgdc805b5))
Author(s) Author(s)
: Leach, R., & Smith, S. T. : Leach, R., & Smith, S. T.
@ -17,6 +17,7 @@ Year
: 2018 : 2018
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org50ae2e1"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press. <a id="orgdc805b5"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.

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content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
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@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}}) : [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
Reference Reference
: ([Preumont 2018](#orgd83c544)) : ([Preumont 2018](#org29acb4a))
Author(s) Author(s)
: Preumont, A. : Preumont, A.
@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback
#### Feedback {#feedback} #### Feedback {#feedback}
<a id="orgda21dda"></a> <a id="orga09f785"></a>
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}} {{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
The principle of feedback is represented on figure [1](#orgda21dda). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance. The principle of feedback is represented on figure [1](#orga09f785). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner. The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner.
In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**. In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
#### Feedforward {#feedforward} #### Feedforward {#feedforward}
<a id="orgf75c047"></a> <a id="org57ee378"></a>
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}} {{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}}
The method relies on the availability of a **reference signal correlated to the primary disturbance**. The method relies on the availability of a **reference signal correlated to the primary disturbance**.
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#orgf75c047). The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#org57ee378).
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized. The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
@ -123,11 +123,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
### The Various Steps of the Design {#the-various-steps-of-the-design} ### The Various Steps of the Design {#the-various-steps-of-the-design}
<a id="org1939c0d"></a> <a id="org8ea735d"></a>
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}} {{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
The various steps of the design of a controlled structure are shown in figure [3](#org1939c0d). The various steps of the design of a controlled structure are shown in figure [3](#org8ea735d).
The **starting point** is: The **starting point** is:
@ -154,14 +154,14 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget} ### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
From the block diagram of the control system (figure [4](#orgaf01f6c)): From the block diagram of the control system (figure [4](#orga135390)):
\begin{align\*} \begin{align\*}
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\ y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
\end{align\*} \end{align\*}
<a id="orgaf01f6c"></a> <a id="orga135390"></a>
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}} {{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
@ -186,12 +186,12 @@ Even more interesting for the design is the **Cumulative Mean Square** response
It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\). It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
\\(\sigma\_z(0)\\) is then the global RMS response. \\(\sigma\_z(0)\\) is then the global RMS response.
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org7ddcf2a). A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orge835b98).
It is useful to **identify the critical modes** in a design, at which the effort should be targeted. It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration. The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
<a id="org7ddcf2a"></a> <a id="orge835b98"></a>
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}} {{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
@ -398,11 +398,11 @@ With:
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i} D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
\end{equation} \end{equation}
<a id="orga618336"></a> <a id="orga21e5bb"></a>
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}} {{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#orga618336)). If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#orga21e5bb)).
And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**: And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
\\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\] \\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
@ -441,9 +441,9 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real: If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
\\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\] \\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgecdb253)). \\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#org4ad84e0)).
<a id="orgecdb253"></a> <a id="org4ad84e0"></a>
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}} {{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
@ -457,9 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
</div> </div>
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org2e6ee6b). If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org0d5b542).
<a id="org2e6ee6b"></a> <a id="org0d5b542"></a>
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}} {{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
@ -474,9 +474,9 @@ The open-loop poles are independant of the actuator and sensor configuration whi
</div> </div>
By looking at figure [7](#orgecdb253), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right. By looking at figure [7](#org4ad84e0), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
<a id="org8e5acfb"></a> <a id="org6f76f34"></a>
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}} {{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
@ -486,7 +486,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)} G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
\end{equation} \end{equation}
The corresponding Bode plot is represented in figure [9](#org8e5acfb). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\). The corresponding Bode plot is represented in figure [9](#org6f76f34). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property. In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
@ -508,12 +508,12 @@ Two broad categories of actuators can be distinguish:
A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa. A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
The system consists of (see figure [10](#org5b9842b)): The system consists of (see figure [10](#orgc872907)):
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap - A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
- A coil which is free to move axially - A coil which is free to move axially
<a id="org5b9842b"></a> <a id="orgc872907"></a>
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}} {{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
@ -551,9 +551,9 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
#### Proof-Mass Actuator {#proof-mass-actuator} #### Proof-Mass Actuator {#proof-mass-actuator}
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#org608f53f)). A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#org4783db3)).
<a id="org608f53f"></a> <a id="org4783db3"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}} {{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
@ -583,9 +583,9 @@ with:
</div> </div>
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org21ce10b)). Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org1a21332)).
<a id="org21ce10b"></a> <a id="org1a21332"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}} {{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
@ -610,7 +610,7 @@ By using the two equations, we obtain:
Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\). Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
<a id="orgb548c88"></a> <a id="orgb1e0d40"></a>
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}} {{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
@ -619,9 +619,9 @@ Designing geophones with very low corner frequency is in general difficult. Acti
### General Electromechanical Transducer {#general-electromechanical-transducer} ### General Electromechanical Transducer {#general-electromechanical-transducer}
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org98492c9). The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#orgf2af0aa).
<a id="org98492c9"></a> <a id="orgf2af0aa"></a>
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}} {{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
@ -646,7 +646,7 @@ With:
Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals. Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained. Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
To do so, the bridge circuit as shown on figure [15](#org3b85763) can be used. To do so, the bridge circuit as shown on figure [15](#orgeb5cb84) can be used.
We can show that We can show that
@ -656,7 +656,7 @@ We can show that
which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals. which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
<a id="org3b85763"></a> <a id="orgeb5cb84"></a>
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}} {{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
@ -664,9 +664,9 @@ which is indeed a linear function of the velocity \\(v\\) at the mechanical term
### Smart Materials {#smart-materials} ### Smart Materials {#smart-materials}
Smart materials have the ability to respond significantly to stimuli of different physical nature. Smart materials have the ability to respond significantly to stimuli of different physical nature.
Figure [16](#org6279c77) lists various effects that are observed in materials in response to various inputs. Figure [16](#org1e5bcfc) lists various effects that are observed in materials in response to various inputs.
<a id="org6279c77"></a> <a id="org1e5bcfc"></a>
{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}} {{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
@ -761,7 +761,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
</div> </div>
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org8006b4a)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer: If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#orgffdc1af)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer:
\begin{equation} \begin{equation}
\begin{bmatrix}Q\\\Delta\end{bmatrix} \begin{bmatrix}Q\\\Delta\end{bmatrix}
@ -782,7 +782,7 @@ where
- \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\)) - \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
- \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\)) - \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
<a id="org8006b4a"></a> <a id="orgffdc1af"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}} {{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
@ -802,7 +802,7 @@ Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
#### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer} #### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org7c30411). Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org890e9f3).
The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is
@ -810,7 +810,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
\end{equation} \end{equation}
<a id="org7c30411"></a> <a id="org890e9f3"></a>
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}} {{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
@ -844,10 +844,10 @@ The ratio between the remaining stored energy and the initial stored energy is
#### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer} #### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
Consider the system of figure [19](#org5060008), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\). Consider the system of figure [19](#org87aa6cd), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\). The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\).
<a id="org5060008"></a> <a id="org87aa6cd"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}} {{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
@ -866,9 +866,9 @@ And one can see that
\frac{z^2 - p^2}{z^2} = k^2 \frac{z^2 - p^2}{z^2} = k^2
\end{equation} \end{equation}
Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#org7f3b3bf)). Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#org12f8fb9)).
<a id="org7f3b3bf"></a> <a id="org12f8fb9"></a>
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}} {{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
@ -1566,7 +1566,7 @@ Their design requires a model of the structure, and there is usually a trade-off
When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC. When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC.
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org62e1395). The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgca40454).
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure. The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
This approach has the following advantages: This approach has the following advantages:
@ -1574,7 +1574,7 @@ This approach has the following advantages:
- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin) - The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
- The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin) - The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
<a id="org62e1395"></a> <a id="orgca40454"></a>
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}} {{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
@ -1816,6 +1816,7 @@ This approach has the following advantages:
### Problems {#problems} ### Problems {#problems}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgd83c544"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>. <a id="org29acb4a"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.

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@ -8,7 +8,7 @@ Tags
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}) : [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
Reference Reference
: ([Monkhorst 2004](#orgf3d75f1)) : ([Monkhorst 2004](#org5371372))
Author(s) Author(s)
: Monkhorst, W. : Monkhorst, W.
@ -95,9 +95,9 @@ Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_
In order to synthesize an \\(\mathcal{H}\_2\\) controller that will minimize the output error, the total system including disturbances needs to be modeled as a system with zero mean white noise inputs. In order to synthesize an \\(\mathcal{H}\_2\\) controller that will minimize the output error, the total system including disturbances needs to be modeled as a system with zero mean white noise inputs.
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#org43cbab3)). This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#orgbc6c1df)).
<a id="org43cbab3"></a> <a id="orgbc6c1df"></a>
{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="Figure 1: The use of a weighting filter \\(V\_w(f)\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}} {{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="Figure 1: The use of a weighting filter \\(V\_w(f)\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
@ -108,23 +108,23 @@ The PSD \\(S\_w(f)\\) of the weighted signal is:
Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_. Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
However, this can be avoided if the modelling of the disturbances is directly done in terms of weighting filters. However, this can be avoided if the modelling of the disturbances is directly done in terms of weighting filters.
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#orge8654a7)). Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#org5bbc30c)).
<a id="orge8654a7"></a> <a id="org5bbc30c"></a>
{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="Figure 2: The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}} {{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="Figure 2: The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}}
#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve} #### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
In this research, the outputs of the closed loop system (Figure [3](#org55e6862)) are: In this research, the outputs of the closed loop system (Figure [3](#orgff3035c)) are:
- the performance (error) signal \\(e\\) - the performance (error) signal \\(e\\)
- the controller output \\(u\\) - the controller output \\(u\\)
In this way, the designer can analyze how much control effort is used to achieve the performance level at the performance output. In this way, the designer can analyze how much control effort is used to achieve the performance level at the performance output.
<a id="org55e6862"></a> <a id="orgff3035c"></a>
{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="Figure 3: The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}} {{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="Figure 3: The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}}
@ -148,6 +148,7 @@ When an \\(\mathcal{H}\_2\\) controller is synthesized for a particular system,
Drawbacks however are, that no robustness guarantees can be given and that the order of the \\(\mathcal{H}\_2\\) controller will generally be too high for implementation. Drawbacks however are, that no robustness guarantees can be given and that the order of the \\(\mathcal{H}\_2\\) controller will generally be too high for implementation.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgf3d75f1"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University. <a id="org5371372"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.

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+++
author = ["Thomas Dehaeze"]
draft = false
+++
## Homepage for Papers {#main}
Here is the list of papers I took note about.
## Homepage for Books {#main}
Here is the list of books I took note about.
## Homepage for Zettels {#main}
Here is the list of subjects I took note about.

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@ -179,5 +179,3 @@ Use powerful optimization techniques from machine learning to learn what are goo
### Applications {#applications} ### Applications {#applications}
<./biblio/references.bib>

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@ -15,5 +15,3 @@ Tags
| [Dewesoft](https://dewesoft.com/) | Slovenia | | [Dewesoft](https://dewesoft.com/) | Slovenia |
| [Oros](https://www.oros.com/) | France | | [Oros](https://www.oros.com/) | France |
| [National Instruments](https://www.ni.com/fr-fr/shop/pc-based-measurement-and-control-system.html) | USA | | [National Instruments](https://www.ni.com/fr-fr/shop/pc-based-measurement-and-control-system.html) | USA |
<./biblio/references.bib>

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@ -4,20 +4,17 @@ author = ["Thomas Dehaeze"]
draft = false draft = false
+++ +++
Backlinks:
- [Sensor Fusion]({{< relref "sensor_fusion" >}})
Tags Tags
: [Complementary Filters]({{< relref "complementary_filters" >}}) : [Complementary Filters]({{< relref "complementary_filters" >}})
([Beijen et al. 2019](#orgaff80f9)) ([Beijen et al. 2019](#orgc359149))
([Beijen 2018](#org585205d)) (section 6.3.1)
([Beijen 2018](#org35f402d)) (section 6.3.1)
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org35f402d"></a>Beijen, MA. 2018. “Disturbance Feedforward Control for Vibration Isolation Systems: Analysis, Design, and Implementation.” Technische Universiteit Eindhoven. <a id="org585205d"></a>Beijen, MA. 2018. “Disturbance Feedforward Control for Vibration Isolation Systems: Analysis, Design, and Implementation.” Technische Universiteit Eindhoven.
<a id="orgaff80f9"></a>Beijen, Michiel A., Marcel F. Heertjes, Hans Butler, and Maarten Steinbuch. 2019. “Mixed Feedback and Feedforward Control Design for Multi-Axis Vibration Isolation Systems.” _Mechatronics_ 61:106–16. <https://doi.org/https://doi.org/10.1016/j.mechatronics.2019.06.005>. <a id="orgc359149"></a>Beijen, Michiel A., Marcel F. Heertjes, Hans Butler, and Maarten Steinbuch. 2019. “Mixed Feedback and Feedforward Control Design for Multi-Axis Vibration Isolation Systems.” _Mechatronics_ 61:106–16. <https://doi.org/https://doi.org/10.1016/j.mechatronics.2019.06.005>.

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@ -4,13 +4,6 @@ author = ["Thomas Dehaeze"]
draft = false draft = false
+++ +++
Backlinks:
- [Collocated Control]({{< relref "collocated_control" >}})
- [Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation]({{< relref "ito16_compar_class_high_precis_actuat" >}})
- [Voice Coil Actuators]({{< relref "voice_coil_actuators" >}})
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
Tags Tags
: :
@ -24,18 +17,19 @@ Links to specific actuators:
For vibration isolation: For vibration isolation:
- In ([Ito and Schitter 2016](#org7d0bcd5)), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat" >}})) - In ([Ito and Schitter 2016](#org4bbf168)), the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat" >}}))
## Brush-less DC Motor {#brush-less-dc-motor} ## Brush-less DC Motor {#brush-less-dc-motor}
- ([Yedamale 2003](#org87858d4)) - ([Yedamale 2003](#org1638958))
<https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html> <https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html>
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org7d0bcd5"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” _IEEE/ASME Transactions on Mechatronics_ 21 (2):1169–78. <https://doi.org/10.1109/tmech.2015.2478658>. <a id="org4bbf168"></a>Ito, Shingo, and Georg Schitter. 2016. “Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation.” _IEEE/ASME Transactions on Mechatronics_ 21 (2):1169–78. <https://doi.org/10.1109/tmech.2015.2478658>.
<a id="org87858d4"></a>Yedamale, Padmaraja. 2003. “Brushless Dc (BLDC) Motor Fundamentals.” _Microchip Technology Inc_ 20:3–15. <a id="org1638958"></a>Yedamale, Padmaraja. 2003. “Brushless Dc (BLDC) Motor Fundamentals.” _Microchip Technology Inc_ 20:3–15.

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@ -12,7 +12,7 @@ Tags
<https://dewesoft.com/daq/types-of-adc-converters> <https://dewesoft.com/daq/types-of-adc-converters>
- Delta Sigma ([Baker 2011](#org9db2758)) - Delta Sigma ([Baker 2011](#orgf10fad8))
- Successive Approximation - Successive Approximation
@ -31,9 +31,9 @@ Let's suppose that the ADC is ideal and the only noise comes from the quantizati
Interestingly, the noise amplitude is uniformly distributed. Interestingly, the noise amplitude is uniformly distributed.
The quantization noise can take a value between \\(\pm q/2\\), and the probability density function is constant in this range (i.e., it’s a uniform distribution). The quantization noise can take a value between \\(\pm q/2\\), and the probability density function is constant in this range (i.e., it’s a uniform distribution).
Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 < e < q/2\\) (Fig. [1](#org79dc805)). Since the integral of the probability density function is equal to one, its value will be \\(1/q\\) for \\(-q/2 < e < q/2\\) (Fig. [1](#org0a7db3b)).
<a id="org79dc805"></a> <a id="org0a7db3b"></a>
{{< figure src="/ox-hugo/probability_density_function_adc.png" caption="Figure 1: Probability density function \\(p(e)\\) of the ADC error \\(e\\)" >}} {{< figure src="/ox-hugo/probability_density_function_adc.png" caption="Figure 1: Probability density function \\(p(e)\\) of the ADC error \\(e\\)" >}}
@ -85,6 +85,7 @@ The quantization is:
{{< youtube b9lxtOJj3yU >}} {{< youtube b9lxtOJj3yU >}}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org9db2758"></a>Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” _Analog Applications_ 7. <a id="orgf10fad8"></a>Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” _Analog Applications_ 7.

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| [Capacitec](https://www.capacitec.com/Displacement-Sensing-Systems) | USA | | [Capacitec](https://www.capacitec.com/Displacement-Sensing-Systems) | USA |
| [MTIinstruments](https://www.mtiinstruments.com/products/non-contact-measurement/capacitance-sensors/) | USA | | [MTIinstruments](https://www.mtiinstruments.com/products/non-contact-measurement/capacitance-sensors/) | USA |
| [Althen](https://www.althensensors.com/sensors/linear-position-sensors/capacitive-position-sensors/) | Netherlands | | [Althen](https://www.althensensors.com/sensors/linear-position-sensors/capacitive-position-sensors/) | Netherlands |
<./biblio/references.bib>

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@ -17,19 +17,19 @@ This can be typically used to interface with piezoelectric sensors.
## Basic Circuit {#basic-circuit} ## Basic Circuit {#basic-circuit}
Two basic circuits of charge amplifiers are shown in Figure [1](#org4fccf5a) (taken from ([Fleming 2010](#org17ae69b))) and Figure [2](#orgad97f51) (taken from ([Schmidt, Schitter, and Rankers 2014](#orge90efed))) Two basic circuits of charge amplifiers are shown in Figure [1](#org45de288) (taken from ([Fleming 2010](#org2341229))) and Figure [2](#org8955723) (taken from ([Schmidt, Schitter, and Rankers 2014](#orgf9a1421)))
<a id="org4fccf5a"></a> <a id="org45de288"></a>
{{< figure src="/ox-hugo/charge_amplifier_circuit.png" caption="Figure 1: Electrical model of a piezoelectric force sensor is shown in gray. The op-amp charge amplifier is shown on the right. The output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}} {{< figure src="/ox-hugo/charge_amplifier_circuit.png" caption="Figure 1: Electrical model of a piezoelectric force sensor is shown in gray. The op-amp charge amplifier is shown on the right. The output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}}
<a id="orgad97f51"></a> <a id="org8955723"></a>
{{< figure src="/ox-hugo/charge_amplifier_circuit_bis.png" caption="Figure 2: A piezoelectric accelerometer with a charge amplifier as signal conditioning element" >}} {{< figure src="/ox-hugo/charge_amplifier_circuit_bis.png" caption="Figure 2: A piezoelectric accelerometer with a charge amplifier as signal conditioning element" >}}
The input impedance of the charge amplifier is very small (unlike when using a voltage amplifier). The input impedance of the charge amplifier is very small (unlike when using a voltage amplifier).
The gain of the charge amplified (Figure [1](#org4fccf5a)) is equal to: The gain of the charge amplified (Figure [1](#org45de288)) is equal to:
\\[ \frac{V\_s}{q} = \frac{-1}{C\_s} \\] \\[ \frac{V\_s}{q} = \frac{-1}{C\_s} \\]
@ -47,8 +47,9 @@ The gain of the charge amplified (Figure [1](#org4fccf5a)) is equal to:
| [L-Card](https://en.lcard.ru/products/accesories/le-41) | Rusia | | [L-Card](https://en.lcard.ru/products/accesories/le-41) | Rusia |
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org17ae69b"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>. <a id="org2341229"></a>Fleming, A.J. 2010. “Nanopositioning System with Force Feedback for High-Performance Tracking and Vibration Control.” _IEEE/ASME Transactions on Mechatronics_ 15 (3):433–47. <https://doi.org/10.1109/tmech.2009.2028422>.
<a id="orge90efed"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press. <a id="orgf9a1421"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press.

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## Complementary Filters Synthesis {#complementary-filters-synthesis} ## Complementary Filters Synthesis {#complementary-filters-synthesis}
The shaping of complementary filters can be done using the \\(\mathcal{H}\_\infty\\) synthesis ([Dehaeze, Vermat, and Christophe 2019](#orgc79060a)). The shaping of complementary filters can be done using the \\(\mathcal{H}\_\infty\\) synthesis ([Dehaeze, Vermat, and Christophe 2019](#org0c35169)).
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgc79060a"></a>Dehaeze, Thomas, Mohit Vermat, and Collette Christophe. 2019. “Complementary Filters Shaping Using \\(mathcalH\_Infty\\) Synthesis.” In _7th International Conference on Control, Mechatronics and Automation (ICCMA)_, 459–64. <https://doi.org/10.1109/ICCMA46720.2019.8988642>. <a id="org0c35169"></a>Dehaeze, Thomas, Mohit Vermat, and Collette Christophe. 2019. “Complementary Filters Shaping Using \\(mathcalH\_Infty\\) Synthesis.” In _7th International Conference on Control, Mechatronics and Automation (ICCMA)_, 459–64. <https://doi.org/10.1109/ICCMA46720.2019.8988642>.

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draft = false draft = false
+++ +++
Backlinks:
- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})
- [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})
- [Sensors and control of a space-based six-axis vibration isolation system]({{< relref "hauge04_sensor_contr_space_based_six" >}})
Tags Tags
: :
@ -19,12 +13,13 @@ Tags
## Special Properties {#special-properties} ## Special Properties {#special-properties}
Cubic Stewart Platforms can be decoupled provided that (from ([Chen and McInroy 2000](#org4014064))) Cubic Stewart Platforms can be decoupled provided that (from ([Chen and McInroy 2000](#org0969434)))
> 1. The payload mass-inertia matrix is diagonal > 1. The payload mass-inertia matrix is diagonal
> 2. If a mutually orthogonal geometry has been selected, the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts. > 2. If a mutually orthogonal geometry has been selected, the payload's center of mass must coincide with the center of the cube formed by the orthogonal struts.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org4014064"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>. <a id="org0969434"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.

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Tags Tags
: [Electronics]({{< relref "electronics" >}}) : [Electronics]({{< relref "electronics" >}})
<./biblio/references.bib>

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Tags Tags
: [Active Damping]({{< relref "active_damping" >}}) : [Active Damping]({{< relref "active_damping" >}})
<./biblio/references.bib>

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@ -18,5 +18,3 @@ There are two main types of encoders: optical encoders, and magnetic encoders.
| [MicroE Systems](https://www.celeramotion.com/microe/products/linear-encoders/) | USA | | [MicroE Systems](https://www.celeramotion.com/microe/products/linear-encoders/) | USA |
| [Renishaw](https://www.renishaw.com/en/browse-encoder-range--6440) | UK | | [Renishaw](https://www.renishaw.com/en/browse-encoder-range--6440) | UK |
| [Celera Motion](https://www.celeramotion.com/microe/) | USA | | [Celera Motion](https://www.celeramotion.com/microe/) | USA |
<./biblio/references.bib>

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draft = false draft = false
+++ +++
Backlinks:
- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})
- [Dynamic modeling and experimental analyses of stewart platform with flexible hinges]({{< relref "jiao18_dynam_model_exper_analy_stewar" >}})
- [A six-axis single-stage active vibration isolator based on stewart platform]({{< relref "preumont07_six_axis_singl_stage_activ" >}})
- [Investigation on active vibration isolation of a stewart platform with piezoelectric actuators]({{< relref "wang16_inves_activ_vibrat_isolat_stewar" >}})
- [Nanometre-cutting machine using a stewart-platform parallel mechanism]({{< relref "furutani04_nanom_cuttin_machin_using_stewar" >}})
- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})
- [Dynamic modeling of flexure jointed hexapods for control purposes]({{< relref "mcinroy99_dynam" >}})
- [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})
- [Flexures]({{< relref "flexures" >}})
Tags Tags
: :
@ -24,32 +12,33 @@ Tags
Books: Books:
- ([Lobontiu 2002](#org74b9989)) - ([Lobontiu 2002](#orgf96bd1c))
- ([Henein 2003](#org1491e2e)) - ([Henein 2003](#org77c1a30))
- ([Smith 2005](#orgcdbef5f)) - ([Smith 2005](#orgdf03b02))
- ([Soemers 2011](#org9626592)) - ([Soemers 2011](#orgc441221))
- ([Cosandier 2017](#org9b28dc9)) - ([Cosandier 2017](#orgc637f07))
## Flexure Joints for Stewart Platforms: {#flexure-joints-for-stewart-platforms} ## Flexure Joints for Stewart Platforms: {#flexure-joints-for-stewart-platforms}
From ([Chen and McInroy 2000](#org4bbdddf)): From ([Chen and McInroy 2000](#org26c43a0)):
> To avoid the extremely non-linear micro-dynamics of joint friction and backlash, these hexapods employ flexure joints. > To avoid the extremely non-linear micro-dynamics of joint friction and backlash, these hexapods employ flexure joints.
> A flexure joint bends material to achieve motion, rather than sliding of rolling across two surfaces. > A flexure joint bends material to achieve motion, rather than sliding of rolling across two surfaces.
> This does eliminate friction and backlash, but adds spring dynamics and limits the workspace. > This does eliminate friction and backlash, but adds spring dynamics and limits the workspace.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org4bbdddf"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>. <a id="org26c43a0"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
<a id="org9b28dc9"></a>Cosandier, Florent. 2017. _Flexure Mechanism Design_. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press. <a id="orgc637f07"></a>Cosandier, Florent. 2017. _Flexure Mechanism Design_. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press.
<a id="org1491e2e"></a>Henein, Simon. 2003. _Conception Des Guidages Flexibles_. Lausanne, Suisse: Presses polytechniques et universitaires romandes. <a id="org77c1a30"></a>Henein, Simon. 2003. _Conception Des Guidages Flexibles_. Lausanne, Suisse: Presses polytechniques et universitaires romandes.
<a id="org74b9989"></a>Lobontiu, Nicolae. 2002. _Compliant Mechanisms: Design of Flexure Hinges_. CRC press. <a id="orgf96bd1c"></a>Lobontiu, Nicolae. 2002. _Compliant Mechanisms: Design of Flexure Hinges_. CRC press.
<a id="orgcdbef5f"></a>Smith, Stuart T. 2005. _Foundations of Ultra-Precision Mechanism Design_. Vol. 2. CRC Press. <a id="orgdf03b02"></a>Smith, Stuart T. 2005. _Foundations of Ultra-Precision Mechanism Design_. Vol. 2. CRC Press.
<a id="org9626592"></a>Soemers, Herman. 2011. _Design Principles for Precision Mechanisms_. T-Pointprint. <a id="orgc441221"></a>Soemers, Herman. 2011. _Design Principles for Precision Mechanisms_. T-Pointprint.

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|--------------------------------------------------|---------| |--------------------------------------------------|---------|
| [Microplan](https://www.microplan-group.com/fr/) | France | | [Microplan](https://www.microplan-group.com/fr/) | France |
| [Zali](http://zali-precision.it/en/products/) | Italy | | [Zali](http://zali-precision.it/en/products/) | Italy |
<./biblio/references.bib>

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draft = false draft = false
+++ +++
Backlinks:
- [Guidelines for the selection of weighting functions for h-infinity control]({{< relref "bibel92_guidel_h" >}})
Tags Tags
: :
@ -19,5 +15,3 @@ From _Rosenbrock, H. H. (1974). Computer-Aided Control System Design, Academic P
> Solutions are constrained by so many requirements that it is virtually impossible to list them all. > Solutions are constrained by so many requirements that it is virtually impossible to list them all.
> The designer finds himself threading a maze of such requirements, attempting to reconcile conflicting demands of cost, performance, easy maintenance, and so on. > The designer finds himself threading a maze of such requirements, attempting to reconcile conflicting demands of cost, performance, easy maintenance, and so on.
> A good design usually has strong aesthetic appeal to those who are competent in the subject. > A good design usually has strong aesthetic appeal to those who are competent in the subject.
<./biblio/references.bib>

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## Review of Absolute (inertial) Position Sensors {#review-of-absolute--inertial--position-sensors} ## Review of Absolute (inertial) Position Sensors {#review-of-absolute--inertial--position-sensors}
- Collette, C. et al., Review: inertial sensors for low-frequency seismic vibration measurement ([Collette, Janssens, Fernandez-Carmona, et al. 2012](#orgb31e055)) - Collette, C. et al., Review: inertial sensors for low-frequency seismic vibration measurement ([Collette, Janssens, Fernandez-Carmona, et al. 2012](#orge266e77))
- Collette, C. et al., Comparison of new absolute displacement sensors ([Collette, Janssens, Mokrani, et al. 2012](#orgcd873cb)) - Collette, C. et al., Comparison of new absolute displacement sensors ([Collette, Janssens, Mokrani, et al. 2012](#orga0b31ea))
<a id="org392ac3e"></a> <a id="orgaa8be44"></a>
{{< figure src="/ox-hugo/collette12_absolute_disp_sensors.png" caption="Figure 1: Dynamic range of several types of inertial sensors; Price versus resolution for several types of inertial sensors" >}} {{< figure src="/ox-hugo/collette12_absolute_disp_sensors.png" caption="Figure 1: Dynamic range of several types of inertial sensors; Price versus resolution for several types of inertial sensors" >}}
@ -35,7 +35,7 @@ Wireless Accelerometers
- <https://micromega-dynamics.com/products/recovib/miniature-vibration-recorder/> - <https://micromega-dynamics.com/products/recovib/miniature-vibration-recorder/>
<a id="org4561c6d"></a> <a id="org47441e2"></a>
{{< figure src="/ox-hugo/inertial_sensors_characteristics_accelerometers.png" caption="Figure 2: Characteristics of commercially available accelerometers <sup id=\"642a18d86de4e062c6afb0f5f20501c4\"><a href=\"#collette11_review\" title=\"Collette, Artoos, Guinchard, Janssens, , Carmona Fernandez \&amp; Hauviller, Review of sensors for low frequency seismic vibration measurement, CERN, (2011).\">collette11_review</a></sup>" >}} {{< figure src="/ox-hugo/inertial_sensors_characteristics_accelerometers.png" caption="Figure 2: Characteristics of commercially available accelerometers <sup id=\"642a18d86de4e062c6afb0f5f20501c4\"><a href=\"#collette11_review\" title=\"Collette, Artoos, Guinchard, Janssens, , Carmona Fernandez \&amp; Hauviller, Review of sensors for low frequency seismic vibration measurement, CERN, (2011).\">collette11_review</a></sup>" >}}
@ -52,13 +52,14 @@ Wireless Accelerometers
| [Guralp](https://www.guralp.com/products/surface) | UK | | [Guralp](https://www.guralp.com/products/surface) | UK |
| [Nanometric](https://www.nanometrics.ca/products/seismometers) | Canada | | [Nanometric](https://www.nanometrics.ca/products/seismometers) | Canada |
<a id="orgd74071e"></a> <a id="orga5e26ab"></a>
{{< figure src="/ox-hugo/inertial_sensors_characteristics_geophone.png" caption="Figure 3: Characteristics of commercially available geophones <sup id=\"642a18d86de4e062c6afb0f5f20501c4\"><a href=\"#collette11_review\" title=\"Collette, Artoos, Guinchard, Janssens, , Carmona Fernandez \&amp; Hauviller, Review of sensors for low frequency seismic vibration measurement, CERN, (2011).\">collette11_review</a></sup>" >}} {{< figure src="/ox-hugo/inertial_sensors_characteristics_geophone.png" caption="Figure 3: Characteristics of commercially available geophones <sup id=\"642a18d86de4e062c6afb0f5f20501c4\"><a href=\"#collette11_review\" title=\"Collette, Artoos, Guinchard, Janssens, , Carmona Fernandez \&amp; Hauviller, Review of sensors for low frequency seismic vibration measurement, CERN, (2011).\">collette11_review</a></sup>" >}}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgb31e055"></a>Collette, C., S. Janssens, P. Fernandez-Carmona, K. Artoos, M. Guinchard, C. Hauviller, and A. Preumont. 2012. “Review: Inertial Sensors for Low-Frequency Seismic Vibration Measurement.” _Bulletin of the Seismological Society of America_ 102 (4):1289–1300. <https://doi.org/10.1785/0120110223>. <a id="orge266e77"></a>Collette, C., S. Janssens, P. Fernandez-Carmona, K. Artoos, M. Guinchard, C. Hauviller, and A. Preumont. 2012. “Review: Inertial Sensors for Low-Frequency Seismic Vibration Measurement.” _Bulletin of the Seismological Society of America_ 102 (4):1289–1300. <https://doi.org/10.1785/0120110223>.
<a id="orgcd873cb"></a>Collette, C, S Janssens, B Mokrani, L Fueyo-Roza, K Artoos, M Esposito, P Fernandez-Carmona, M Guinchard, and R Leuxe. 2012. “Comparison of New Absolute Displacement Sensors.” In _International Conference on Noise and Vibration Engineering (ISMA)_. <a id="orga0b31ea"></a>Collette, C, S Janssens, B Mokrani, L Fueyo-Roza, K Artoos, M Esposito, P Fernandez-Carmona, M Guinchard, and R Leuxe. 2012. “Comparison of New Absolute Displacement Sensors.” In _International Conference on Noise and Vibration Engineering (ISMA)_.

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| [PCB](https://www.pcb.com/sensors-for-test-measurement/impact-hammers-electrodynamic-shakers/impact-hammers) | USA | | [PCB](https://www.pcb.com/sensors-for-test-measurement/impact-hammers-electrodynamic-shakers/impact-hammers) | USA |
| [DJB](https://www.djbinstruments.com/products/instrumentation/impact-hammers) | UK | | [DJB](https://www.djbinstruments.com/products/instrumentation/impact-hammers) | UK |
| [Dewesoft](https://dewesoft.com/fr/products/interfaces-and-sensors/accelerometers-and-modal-hammers) | Slovenia | | [Dewesoft](https://dewesoft.com/fr/products/interfaces-and-sensors/accelerometers-and-modal-hammers) | Slovenia |
<./biblio/references.bib>

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## Effect of Refractive Index - Environmental Units {#effect-of-refractive-index-environmental-units} ## Effect of Refractive Index - Environmental Units {#effect-of-refractive-index-environmental-units}
The measured distance is proportional to the refractive index of the air that depends on several quantities as shown in Table [1](#table--tab:index-air) (Taken from ([Thurner et al. 2015](#org68c8bbb))). The measured distance is proportional to the refractive index of the air that depends on several quantities as shown in Table [1](#table--tab:index-air) (Taken from ([Thurner et al. 2015](#org1b86993))).
<a id="table--tab:index-air"></a> <a id="table--tab:index-air"></a>
<div class="table-caption"> <div class="table-caption">
@ -59,16 +59,16 @@ Typical characteristics of commercial environmental units are shown in Table [2]
## Interferometer Precision {#interferometer-precision} ## Interferometer Precision {#interferometer-precision}
Figure [1](#org960bbd9) shows the expected precision as a function of the measured distance due to change of refractive index of the air (taken from ([Jang and Kim 2017](#orgd724d07))). Figure [1](#org3490ef0) shows the expected precision as a function of the measured distance due to change of refractive index of the air (taken from ([Jang and Kim 2017](#org3b0a481))).
<a id="org960bbd9"></a> <a id="org3490ef0"></a>
{{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 1: Expected precision of interferometer as a function of measured distance" >}} {{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 1: Expected precision of interferometer as a function of measured distance" >}}
## Sources of uncertainty {#sources-of-uncertainty} ## Sources of uncertainty {#sources-of-uncertainty}
Sources of error in laser interferometry are well described in ([Ducourtieux 2018](#orgeacbea1)). Sources of error in laser interferometry are well described in ([Ducourtieux 2018](#org588696d)).
It includes: It includes:
@ -78,18 +78,19 @@ It includes:
- Pressure: \\(K\_P \approx 0.27 ppm hPa^{-1}\\) - Pressure: \\(K\_P \approx 0.27 ppm hPa^{-1}\\)
- Humidity: \\(K\_{HR} \approx 0.01 ppm \% RH^{-1}\\) - Humidity: \\(K\_{HR} \approx 0.01 ppm \% RH^{-1}\\)
- These errors can partially be compensated using an environmental unit. - These errors can partially be compensated using an environmental unit.
- Air turbulence (Figure [2](#orgd403994)) - Air turbulence (Figure [2](#orgceb0667))
- Non linearity - Non linearity
<a id="orgd403994"></a> <a id="orgceb0667"></a>
{{< figure src="/ox-hugo/interferometers_air_turbulence.png" caption="Figure 2: Effect of air turbulences on measurement stability" >}} {{< figure src="/ox-hugo/interferometers_air_turbulence.png" caption="Figure 2: Effect of air turbulences on measurement stability" >}}
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgeacbea1"></a>Ducourtieux, Sebastien. 2018. “Toward High Precision Position Control Using Laser Interferometry: Main Sources of Error.” <https://doi.org/10.13140/rg.2.2.21044.35205>. <a id="org588696d"></a>Ducourtieux, Sebastien. 2018. “Toward High Precision Position Control Using Laser Interferometry: Main Sources of Error.” <https://doi.org/10.13140/rg.2.2.21044.35205>.
<a id="orgd724d07"></a>Jang, Yoon-Soo, and Seung-Woo Kim. 2017. “Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: A Review.” _International Journal of Precision Engineering and Manufacturing_ 18 (12):1881–90. <https://doi.org/10.1007/s12541-017-0217-y>. <a id="org3b0a481"></a>Jang, Yoon-Soo, and Seung-Woo Kim. 2017. “Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: A Review.” _International Journal of Precision Engineering and Manufacturing_ 18 (12):1881–90. <https://doi.org/10.1007/s12541-017-0217-y>.
<a id="org68c8bbb"></a>Thurner, Klaus, Francesca Paola Quacquarelli, Pierre-François Braun, Claudio Dal Savio, and Khaled Karrai. 2015. “Fiber-Based Distance Sensing Interferometry.” _Applied Optics_ 54 (10). Optical Society of America:3051–63. <a id="org1b86993"></a>Thurner, Klaus, Francesca Paola Quacquarelli, Pierre-François Braun, Claudio Dal Savio, and Khaled Karrai. 2015. “Fiber-Based Distance Sensing Interferometry.” _Applied Optics_ 54 (10). Optical Society of America:3051–63.

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content/zettels/irr_and_fir_filters.md
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> >
> The primary disadvantage of FIR filters is that they often require a much higher filter order than IIR filters to achieve a given level of performance. Correspondingly, the delay of these filters is often much greater than for an equal performance IIR filter. > The primary disadvantage of FIR filters is that they often require a much higher filter order than IIR filters to achieve a given level of performance. Correspondingly, the delay of these filters is often much greater than for an equal performance IIR filter.
From ([Shaw and Srinivasan 1990](#org9a58282)) From ([Shaw and Srinivasan 1990](#orge62ce0f))
> The FIR are capable of realizing filters with linear phase shift characteristics and furthermore are less susceptible to signal input and filter coefficient quantization effects. > The FIR are capable of realizing filters with linear phase shift characteristics and furthermore are less susceptible to signal input and filter coefficient quantization effects.
> However, their computational demands are excessively large because of the large number of multiplications and additions to be performed at each sampling interval. > However, their computational demands are excessively large because of the large number of multiplications and additions to be performed at each sampling interval.
@ -49,6 +49,7 @@ From <https://dsp.stackexchange.com/a/30999>
> - Feed-forward control. FIR filters are useful for producing filters that approximate arbitrary frequency responses, hence they can be used to shape a reference signal. A typical example is to use an FIR filter with the inverse frequency response of the plant -- trying to counteract the dynamics of the plant in order to get a desired output. Phase/time-delay is not interfering with the stability or performance since the computation can be done offline. FIR filters can often produce higher performance than IIR filters, especially where there are non-minimum phase zeros. > - Feed-forward control. FIR filters are useful for producing filters that approximate arbitrary frequency responses, hence they can be used to shape a reference signal. A typical example is to use an FIR filter with the inverse frequency response of the plant -- trying to counteract the dynamics of the plant in order to get a desired output. Phase/time-delay is not interfering with the stability or performance since the computation can be done offline. FIR filters can often produce higher performance than IIR filters, especially where there are non-minimum phase zeros.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org9a58282"></a>Shaw, F.R., and K. Srinivasan. 1990. “Bandwidth Enhancement of Position Measurements Using Measured Acceleration.” _Mechanical Systems and Signal Processing_ 4 (1):23–38. <https://doi.org/10.1016/0888-3270(90)>90038-m. <a id="orge62ce0f"></a>Shaw, F.R., and K. Srinivasan. 1990. “Bandwidth Enhancement of Position Measurements Using Measured Acceleration.” _Mechanical Systems and Signal Processing_ 4 (1):23–38. <https://doi.org/10.1016/0888-3270(90)>90038-m.

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content/zettels/mass_spring_damper_systems.md
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## Actuated Mass Spring Damper System {#actuated-mass-spring-damper-system} ## Actuated Mass Spring Damper System {#actuated-mass-spring-damper-system}
Let's consider Figure [1](#orgeec8f0f) where: Let's consider Figure [1](#orga358a0b) where:
- \\(m\\) is the mass in [kg] - \\(m\\) is the mass in [kg]
- \\(ḱ\\) is the spring stiffness in [N/m] - \\(ḱ\\) is the spring stiffness in [N/m]
@ -20,7 +20,7 @@ Let's consider Figure [1](#orgeec8f0f) where:
- \\(w\\) is ground motion - \\(w\\) is ground motion
- \\(x\\) is the absolute mass motion - \\(x\\) is the absolute mass motion
<a id="orgeec8f0f"></a> <a id="orga358a0b"></a>
{{< figure src="/ox-hugo/mass_spring_damper_system.png" caption="Figure 1: Mass Spring Damper System" >}} {{< figure src="/ox-hugo/mass_spring_damper_system.png" caption="Figure 1: Mass Spring Damper System" >}}
@ -54,5 +54,3 @@ with:
\begin{equation} \begin{equation}
\frac{x}{F\_d}(s) = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1} \frac{x}{F\_d}(s) = \frac{1/k}{\frac{s^2}{\omega\_0^2} + 2 \xi \frac{s}{\omega\_0} + 1}
\end{equation} \end{equation}
<./biblio/references.bib>

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content/zettels/matlab.md
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draft = false draft = false
+++ +++
Backlinks:
- [Simulink]({{< relref "simulink" >}})
Tags Tags
: [Simulink]({{< relref "simulink" >}}) : [Simulink]({{< relref "simulink" >}})
@ -16,11 +12,11 @@ Tags
Books: Books:
- ([Higham 2017](#org9e841d2)) - ([Higham 2017](#org68f863c))
- ([Attaway 2018](#org1d7c8f3)) - ([Attaway 2018](#org3441bfb))
- ([OverFlow 2018](#orgdf04055)) - ([OverFlow 2018](#org8e0ff2b))
- ([Johnson 2010](#org329fa4a)) - ([Johnson 2010](#org019531d))
- ([Hahn and Valentine 2016](#org7d00587)) - ([Hahn and Valentine 2016](#orgbeacac3))
## Useful Commands {#useful-commands} ## Useful Commands {#useful-commands}
@ -109,14 +105,15 @@ Nice functions:
| `echo` | Display statements during function execution | | `echo` | Display statements during function execution |
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org1d7c8f3"></a>Attaway, Stormy. 2018. _MATLAB : a Practical Introduction to Programming and Problem Solving_. Amsterdam: Butterworth-Heinemann. <a id="org3441bfb"></a>Attaway, Stormy. 2018. _MATLAB : a Practical Introduction to Programming and Problem Solving_. Amsterdam: Butterworth-Heinemann.
<a id="org7d00587"></a>Hahn, Brian, and Daniel T Valentine. 2016. _Essential MATLAB for Engineers and Scientists_. Academic Press. <a id="orgbeacac3"></a>Hahn, Brian, and Daniel T Valentine. 2016. _Essential MATLAB for Engineers and Scientists_. Academic Press.
<a id="org9e841d2"></a>Higham, Desmond. 2017. _MATLAB Guide_. Philadelphia: Society for Industrial and Applied Mathematics. <a id="org68f863c"></a>Higham, Desmond. 2017. _MATLAB Guide_. Philadelphia: Society for Industrial and Applied Mathematics.
<a id="org329fa4a"></a>Johnson, Richard K. 2010. _The Elements of MATLAB Style_. Cambridge University Press. <a id="org019531d"></a>Johnson, Richard K. 2010. _The Elements of MATLAB Style_. Cambridge University Press.
<a id="orgdf04055"></a>OverFlow, Stack. 2018. _MATLAB Notes for Professionals_. GoalKicker.com. <a id="org8e0ff2b"></a>OverFlow, Stack. 2018. _MATLAB Notes for Professionals_. GoalKicker.com.

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content/zettels/metrology.md
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draft = false draft = false
+++ +++
Backlinks:
- [Fundamental principles of engineering nanometrology]({{< relref "leach14_fundam_princ_engin_nanom" >}})
Tags Tags
: :
<./biblio/references.bib>

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content/zettels/modal_analysis.md
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draft = false draft = false
+++ +++
Backlinks:
- [Modal testing: theory, practice and application]({{< relref "ewins00_modal" >}})
- [Force Sensors]({{< relref "force_sensors" >}})
- [Instrumented Hammer]({{< relref "instrumented_hammer" >}})
- [System Identification]({{< relref "system_identification" >}})
Tags Tags
: [Inertial Sensors]({{< relref "inertial_sensors" >}}), [Shaker]({{< relref "shaker" >}}) : [Inertial Sensors]({{< relref "inertial_sensors" >}}), [Shaker]({{< relref "shaker" >}})
<./biblio/references.bib>

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content/zettels/model_predictive_control.md
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Tags Tags
: :
<./biblio/references.bib>

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content/zettels/multivariable_control.md
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@ -7,9 +7,10 @@ draft = false
Tags Tags
: [Norms]({{< relref "norms" >}}) : [Norms]({{< relref "norms" >}})
A very nice book about Multivariable Control is ([Skogestad and Postlethwaite 2007](#org8b835f5)) A very nice book about Multivariable Control is ([Skogestad and Postlethwaite 2007](#org2f5ed44))
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org8b835f5"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley. <a id="org2f5ed44"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.

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content/zettels/norms.md
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Resources: Resources:
- ([Skogestad and Postlethwaite 2007](#orgf64cea0)) - ([Skogestad and Postlethwaite 2007](#org4c8b20e))
- ([Toivonen 2002](#org3ddabae)) - ([Toivonen 2002](#org81db503))
- ([Zhang 2011](#orge43be7a)) - ([Zhang 2011](#orgc4d2be1))
## Definition {#definition} ## Definition {#definition}
@ -176,17 +176,18 @@ In terms of signals, the \\(\mathcal{H}\_\infty\\) norm can be interpreted as fo
The \\(\mathcal{H}\_2\\) is very useful when combined to [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}). The \\(\mathcal{H}\_2\\) is very useful when combined to [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}}).
As explained in ([Monkhorst 2004](#org45aca82)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation: As explained in ([Monkhorst 2004](#orgc401feb)), the \\(\mathcal{H}\_2\\) norm has a stochastic interpretation:
> The squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input. > The squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input.
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org45aca82"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University. <a id="orgc401feb"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
<a id="orgf64cea0"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley. <a id="org4c8b20e"></a>Skogestad, Sigurd, and Ian Postlethwaite. 2007. _Multivariable Feedback Control: Analysis and Design_. John Wiley.
<a id="org3ddabae"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University. <a id="org81db503"></a>Toivonen, Hannu T. 2002. “Robust Control Methods.” Abo Akademi University.
<a id="orge43be7a"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press. <a id="orgc4d2be1"></a>Zhang, Weidong. 2011. _Quantitative Process Control Theory_. CRC Press.

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content/zettels/operational_amplifiers.md
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## Defaults of Operational Amplifiers {#defaults-of-operational-amplifiers} ## Defaults of Operational Amplifiers {#defaults-of-operational-amplifiers}
{{< youtube nF104EvI0HM >}} {{< youtube nF104EvI0HM >}}
<./biblio/references.bib>

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content/zettels/positioning_stations.md
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@ -16,5 +16,3 @@ Tags
| [PI](https://www.physikinstrumente.com/en/) | USA | | [PI](https://www.physikinstrumente.com/en/) | USA |
| [Attocube](https://www.attocube.com/en/products/nanopositioners) | Germany | | [Attocube](https://www.attocube.com/en/products/nanopositioners) | Germany |
| [Newport](https://www.newport.com/c/manual-positioning) | | | [Newport](https://www.newport.com/c/manual-positioning) | |
<./biblio/references.bib>

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content/zettels/power_spectral_density.md
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Tutorial about Power Spectral Density is accessible [here](https://research.tdehaeze.xyz/spectral-analysis/). Tutorial about Power Spectral Density is accessible [here](https://research.tdehaeze.xyz/spectral-analysis/).
A good article about how to use the `pwelch` function with Matlab ([Schmid 2012](#org23bcfe9)). A good article about how to use the `pwelch` function with Matlab ([Schmid 2012](#org6fb2cbe)).
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org23bcfe9"></a>Schmid, Hanspeter. 2012. “How to Use the FFT and Matlab’s Pwelch Function for Signal and Noise Simulations and Measurements.” _Institute of Microelectronics_. <a id="org6fb2cbe"></a>Schmid, Hanspeter. 2012. “How to Use the FFT and Matlab’s Pwelch Function for Signal and Noise Simulations and Measurements.” _Institute of Microelectronics_.

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content/zettels/precision_engineering.md
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draft = false draft = false
+++ +++
Backlinks:
- [Basics of precision engineering - 1st edition]({{< relref "leach18_basic_precis_engin_edition" >}})
- [Design for precision: current status and trends]({{< relref "schellekens98_desig_precis" >}})
Tags Tags
: :
<./biblio/references.bib>

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content/zettels/sensor_noise_estimation.md
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@ -12,13 +12,13 @@ Tags
Measuring the noise level of inertial sensors is not easy as the seismic motion is usually much larger than the sensor's noise level. Measuring the noise level of inertial sensors is not easy as the seismic motion is usually much larger than the sensor's noise level.
A technique to estimate the sensor noise in such case is proposed in ([Barzilai, VanZandt, and Kenny 1998](#org7fe766e)) and well explained in ([Poel 2010](#org964c18e)) (Section 6.1.3). A technique to estimate the sensor noise in such case is proposed in ([Barzilai, VanZandt, and Kenny 1998](#org4702c9a)) and well explained in ([Poel 2010](#orgeaef46f)) (Section 6.1.3).
The idea is to mount two inertial sensors closely together such that they should measure the same quantity. The idea is to mount two inertial sensors closely together such that they should measure the same quantity.
This is represented in Figure [1](#org53e9426) where two identical sensors are measuring the same motion \\(x(t)\\). This is represented in Figure [1](#org030f5c0) where two identical sensors are measuring the same motion \\(x(t)\\).
<a id="org53e9426"></a> <a id="org030f5c0"></a>
{{< figure src="/ox-hugo/huddle_test_setup.png" caption="Figure 1: Schematic representation of the setup for measuring the noise of inertial sensors." >}} {{< figure src="/ox-hugo/huddle_test_setup.png" caption="Figure 1: Schematic representation of the setup for measuring the noise of inertial sensors." >}}
@ -76,7 +76,7 @@ Now suppose that:
- sensor noises are modelled as input noises \\(n\_1(t)\\) and \\(n\_2(s)\\) - sensor noises are modelled as input noises \\(n\_1(t)\\) and \\(n\_2(s)\\)
- sensor noises are uncorrelated and each are uncorrelated with \\(x(t)\\) - sensor noises are uncorrelated and each are uncorrelated with \\(x(t)\\)
Then, the system can be represented by the block diagram in Figure [2](#org0e1cf4a), and we can write: Then, the system can be represented by the block diagram in Figure [2](#orgec7c79b), and we can write:
\begin{align} \begin{align}
P\_{y\_1y\_1}(\omega) &= |H\_1(\omega)|^2 ( P\_{x}(\omega) + P\_{n\_1}(\omega) ) \\\\\\ P\_{y\_1y\_1}(\omega) &= |H\_1(\omega)|^2 ( P\_{x}(\omega) + P\_{n\_1}(\omega) ) \\\\\\
@ -90,7 +90,7 @@ And the CSD between \\(y\_1(t)\\) and \\(y\_2(t)\\) is:
\gamma^2\_{y\_1y\_2}(\omega) = \frac{|C\_{y\_1y\_2}(j\omega)|^2}{P\_{y\_1}(\omega) P\_{y\_2}(\omega)} \gamma^2\_{y\_1y\_2}(\omega) = \frac{|C\_{y\_1y\_2}(j\omega)|^2}{P\_{y\_1}(\omega) P\_{y\_2}(\omega)}
\end{equation} \end{equation}
<a id="org0e1cf4a"></a> <a id="orgec7c79b"></a>
{{< figure src="/ox-hugo/huddle_test_block_diagram.png" caption="Figure 2: Huddle test block diagram" >}} {{< figure src="/ox-hugo/huddle_test_block_diagram.png" caption="Figure 2: Huddle test block diagram" >}}
@ -113,8 +113,9 @@ If we assume the two sensor dynamics to be the same \\(H\_1(s) \approx H\_2(s)\\
</div> </div>
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org7fe766e"></a>Barzilai, Aaron, Tom VanZandt, and Tom Kenny. 1998. “Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals.” _Review of Scientific Instruments_ 69 (7):2767–72. <https://doi.org/10.1063/1.1149013>. <a id="org4702c9a"></a>Barzilai, Aaron, Tom VanZandt, and Tom Kenny. 1998. “Technique for Measurement of the Noise of a Sensor in the Presence of Large Background Signals.” _Review of Scientific Instruments_ 69 (7):2767–72. <https://doi.org/10.1063/1.1149013>.
<a id="org964c18e"></a>Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. <https://doi.org/10.3990/1.9789036530163>. <a id="orgeaef46f"></a>Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. <https://doi.org/10.3990/1.9789036530163>.

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@ -12,5 +12,3 @@ Notes about sensors:
- [Force Sensors]({{< relref "force_sensors" >}}) - [Force Sensors]({{< relref "force_sensors" >}})
- [Position Sensors]({{< relref "position_sensors" >}}) - [Position Sensors]({{< relref "position_sensors" >}})
- [Inertial Sensors]({{< relref "inertial_sensors" >}}) - [Inertial Sensors]({{< relref "inertial_sensors" >}})
<./biblio/references.bib>

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content/zettels/signal_conditioner.md
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draft = false draft = false
+++ +++
Backlinks:
- [Position Sensors]({{< relref "position_sensors" >}})
- [Force Sensors]({{< relref "force_sensors" >}})
Tags Tags
: [Sensors]({{< relref "sensors" >}}), [Electronics]({{< relref "electronics" >}}) : [Sensors]({{< relref "sensors" >}}), [Electronics]({{< relref "electronics" >}})
@ -33,5 +28,3 @@ Depending on the electrical quantity that is meaningful for the measurement, dif
- Current to Voltage ([Transimpedance Amplifiers]({{< relref "transimpedance_amplifiers" >}})) - Current to Voltage ([Transimpedance Amplifiers]({{< relref "transimpedance_amplifiers" >}}))
- Charge to Voltage ([Charge Amplifiers]({{< relref "charge_amplifiers" >}})) - Charge to Voltage ([Charge Amplifiers]({{< relref "charge_amplifiers" >}}))
- Voltage to Voltage ([Voltage Amplifier]({{< relref "voltage_amplifier" >}})) - Voltage to Voltage ([Voltage Amplifier]({{< relref "voltage_amplifier" >}}))
<./biblio/references.bib>

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content/zettels/simulink.md
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@ -36,5 +36,3 @@ Tips:
## Linearize portion of Simulink file {#linearize-portion-of-simulink-file} ## Linearize portion of Simulink file {#linearize-portion-of-simulink-file}
<https://in.mathworks.com/help/slcontrol/ug/specify-model-portion-to-linearize.html> <https://in.mathworks.com/help/slcontrol/ug/specify-model-portion-to-linearize.html>
<./biblio/references.bib>

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content/zettels/simulink_real_time_target_machines.md
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@ -34,5 +34,3 @@ Comparison
| Encoder (quadrature) | 2 | 4 | 4 | 2 | | Encoder (quadrature) | 2 | 4 | 4 | 2 |
| Sampling Frequency | ? | ? | 1kHz (USB), 15kHz (Serial) | 2kHz | | Sampling Frequency | ? | ? | 1kHz (USB), 15kHz (Serial) | 2kHz |
| Price | waiting for quote | 1000 | 900 | 1300 | | Price | waiting for quote | 1000 | 900 | 1300 |
<./biblio/references.bib>

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content/zettels/stewart_platforms.md
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Papers by J.E. McInroy: Papers by J.E. McInroy:
- ([O’Brien et al. 1998](#org301ae65)) - ([O’Brien et al. 1998](#org413fc20))
- ([McInroy, O’Brien, and Neat 1999](#org43a0fe2)) - ([McInroy, O’Brien, and Neat 1999](#orgc5005e3))
- ([McInroy 1999](#org41ba097)) - ([McInroy 1999](#orgb4c311e))
- ([McInroy and Hamann 2000](#org73060fc)) - ([McInroy and Hamann 2000](#org8285ab1))
- ([Chen and McInroy 2000](#org2b98584)) - ([Chen and McInroy 2000](#org709b3d5))
- ([McInroy 2002](#org2d6222b)) - ([McInroy 2002](#org349aaf8))
- ([Li, Hamann, and McInroy 2001](#org6598adc)) - ([Li, Hamann, and McInroy 2001](#orgaa83268))
- ([Lin and McInroy 2003](#orgfc1736f)) - ([Lin and McInroy 2003](#org055e9ff))
- ([Jafari and McInroy 2003](#org72de1d8)) - ([Jafari and McInroy 2003](#org26e42d2))
- ([Chen and McInroy 2004](#org6bdfb26)) - ([Chen and McInroy 2004](#orgd92590e))
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="org6bdfb26"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>. <a id="orgd92590e"></a>Chen, Y., and J.E. McInroy. 2004. “Decoupled Control of Flexure-Jointed Hexapods Using Estimated Joint-Space Mass-Inertia Matrix.” _IEEE Transactions on Control Systems Technology_ 12 (3):413–21. <https://doi.org/10.1109/tcst.2004.824339>.
<a id="org2b98584"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>. <a id="org709b3d5"></a>Chen, Yixin, and J.E. McInroy. 2000. “Identification and Decoupling Control of Flexure Jointed Hexapods.” In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065)_, nil. <https://doi.org/10.1109/robot.2000.844878>.
<a id="org72de1d8"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>. <a id="org26e42d2"></a>Jafari, F., and J.E. McInroy. 2003. “Orthogonal Gough-Stewart Platforms for Micromanipulation.” _IEEE Transactions on Robotics and Automation_ 19 (4). Institute of Electrical and Electronics Engineers (IEEE):595–603. <https://doi.org/10.1109/tra.2003.814506>.
<a id="orgfc1736f"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>. <a id="org055e9ff"></a>Lin, Haomin, and J.E. McInroy. 2003. “Adaptive Sinusoidal Disturbance Cancellation for Precise Pointing of Stewart Platforms.” _IEEE Transactions on Control Systems Technology_ 11 (2):267–72. <https://doi.org/10.1109/tcst.2003.809248>.
<a id="org6598adc"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>. <a id="orgaa83268"></a>Li, Xiaochun, Jerry C. Hamann, and John E. McInroy. 2001. “Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” In _Smart Structures and Materials 2001: Smart Structures and Integrated Systems_, nil. <https://doi.org/10.1117/12.436521>.
<a id="org41ba097"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>. <a id="orgb4c311e"></a>McInroy, J.E. 1999. “Dynamic Modeling of Flexure Jointed Hexapods for Control Purposes.” In _Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328)_, nil. <https://doi.org/10.1109/cca.1999.806694>.
<a id="org2d6222b"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>. <a id="org349aaf8"></a>———. 2002. “Modeling and Design of Flexure Jointed Stewart Platforms for Control Purposes.” _IEEE/ASME Transactions on Mechatronics_ 7 (1):95–99. <https://doi.org/10.1109/3516.990892>.
<a id="org73060fc"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>. <a id="org8285ab1"></a>McInroy, J.E., and J.C. Hamann. 2000. “Design and Control of Flexure Jointed Hexapods.” _IEEE Transactions on Robotics and Automation_ 16 (4):372–81. <https://doi.org/10.1109/70.864229>.
<a id="org43a0fe2"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>. <a id="orgc5005e3"></a>McInroy, J.E., J.F. O’Brien, and G.W. Neat. 1999. “Precise, Fault-Tolerant Pointing Using a Stewart Platform.” _IEEE/ASME Transactions on Mechatronics_ 4 (1):91–95. <https://doi.org/10.1109/3516.752089>.
<a id="org301ae65"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>. <a id="org413fc20"></a>O’Brien, J.F., J.E. McInroy, D. Bodtke, M. Bruch, and J.C. Hamann. 1998. “Lessons Learned in Nonlinear Systems and Flexible Robots Through Experiments on a 6 Legged Platform.” In _Proceedings of the 1998 American Control Conference. ACC (IEEE Cat. No.98CH36207)_, nil. <https://doi.org/10.1109/acc.1998.703532>.

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@ -78,13 +78,13 @@ Sed aliquam
Here is a list of links to: Here is a list of links to:
- Figure [3](#org54379fd) - Figure [3](#orgcbf9e46)
- Table [3](#table--tab:table-with-equations) - Table [3](#table--tab:table-with-equations)
- Listing [1](#code-snippet--lst:matlab-figure) - Listing [1](#code-snippet--lst:matlab-figure)
- Specific line of code - Specific line of code
- Equation \eqref{eq:numbered} - Equation \eqref{eq:numbered}
- Section - Section
- Bibliographic Reference ([Stanisic and Legrand 2014](#orgfe85fe6)), and ([Schulte and Davison 2011](#orgdf0380b); [Dominik 2010](#orgb0733db); [Stanisic and Legrand 2014](#orgfe85fe6)) - Bibliographic Reference ([Stanisic and Legrand 2014](#org0ed95e1)), and ([Schulte and Davison 2011](#org7b9fb79); [Dominik 2010](#org4f5b6d0); [Stanisic and Legrand 2014](#org0ed95e1))
### Maths {#maths} ### Maths {#maths}
@ -157,7 +157,7 @@ Some text.
## Headlines {#headlines} ## Headlines {#headlines}
<a id="orgcd5a9a0"></a> <a id="org94d8c54"></a>
### Second level Headline with tags {#second-level-headline-with-tags} ### Second level Headline with tags {#second-level-headline-with-tags}
@ -304,7 +304,7 @@ Cras non mauris ex. Morbi ut eros eu tellus egestas dapibus et et est. Aenean so
xlabel('Time [s]'); ylabel('Voltage [V]'); xlabel('Time [s]'); ylabel('Voltage [V]');
``` ```
<a id="org485b9a9"></a> <a id="org75ab154"></a>
{{< figure src="figs/matlab_fig_example.png" caption="Figure 1: Matlab Figure" >}} {{< figure src="figs/matlab_fig_example.png" caption="Figure 1: Matlab Figure" >}}
@ -375,7 +375,7 @@ Moreover, we can link to specific bode blocks (Listing [1](#code-snippet--lst:ma
Code to produce a nice contour plot Code to produce a nice contour plot
</div> </div>
<a id="org405a280"></a> <a id="orgfcd383d"></a>
{{< figure src="figs/matlab_logo.png" caption="Figure 2: Obtained Contour Plot" >}} {{< figure src="figs/matlab_logo.png" caption="Figure 2: Obtained Contour Plot" >}}
@ -450,7 +450,7 @@ Numbering can be continued by using `+n` option as shown below.
### Normal Image {#normal-image} ### Normal Image {#normal-image}
Figure [3](#org54379fd) shows the results of the Tikz code of listing [4](#code-snippet--lst:tikz-test). Figure [3](#orgcbf9e46) shows the results of the Tikz code of listing [4](#code-snippet--lst:tikz-test).
<a id="code-snippet--lst:tikz-test"></a> <a id="code-snippet--lst:tikz-test"></a>
```latex ```latex
@ -477,10 +477,10 @@ Figure [3](#org54379fd) shows the results of the Tikz code of listing [4](#code-
<div class="src-block-caption"> <div class="src-block-caption">
<span class="src-block-number"><a href="#code-snippet--lst:tikz-test">Code Snippet 4</a></span>: <span class="src-block-number"><a href="#code-snippet--lst:tikz-test">Code Snippet 4</a></span>:
Tikz code that is used to generate Figure <a href="#org54379fd">3</a> Tikz code that is used to generate Figure <a href="#orgcbf9e46">3</a>
</div> </div>
<a id="org54379fd"></a> <a id="orgcbf9e46"></a>
{{< figure src="figs/general_control_names.png" caption="Figure 3: General Control Configuration" >}} {{< figure src="figs/general_control_names.png" caption="Figure 3: General Control Configuration" >}}
@ -493,7 +493,7 @@ Figure [3](#org54379fd) shows the results of the Tikz code of listing [4](#code-
### Wrap Image {#wrap-image} ### Wrap Image {#wrap-image}
<a id="orgc7b3c56"></a> <a id="orge97a9ba"></a>
{{< figure src="figs/general_control_names.png" caption="Figure 4: General Control Configuration" >}} {{< figure src="figs/general_control_names.png" caption="Figure 4: General Control Configuration" >}}
@ -509,7 +509,7 @@ Fusce blandit mauris dui, sed lobortis sapien tincidunt ac. Maecenas vitae moles
[[file:figs/general_control_names.png]] [[file:figs/general_control_names.png]]
``` ```
<a id="org674d2a1"></a> <a id="org669836c"></a>
{{< figure src="figs/general_control_names.png" caption="Figure 5: General Control Configuration" >}} {{< figure src="figs/general_control_names.png" caption="Figure 5: General Control Configuration" >}}
@ -518,7 +518,7 @@ Fusce blandit mauris dui, sed lobortis sapien tincidunt ac. Maecenas vitae moles
### Sub Images {#sub-images} ### Sub Images {#sub-images}
Link to subfigure [2](#orga5ea12b). Link to subfigure [2](#org0dc182a).
```md ```md
#+name: fig:subfigure #+name: fig:subfigure
@ -536,7 +536,7 @@ Link to subfigure [2](#orga5ea12b).
| ![](figs/general_control_names.png) | ![](figs/general_control_names.png) | | ![](figs/general_control_names.png) | ![](figs/general_control_names.png) |
|--------------------------------------------|--------------------------------------------| |--------------------------------------------|--------------------------------------------|
| <a id="orga5ea12b"></a> sub figure caption | <a id="org2c6cb70"></a> sub figure caption | | <a id="org0dc182a"></a> sub figure caption | <a id="org5fce826"></a> sub figure caption |
## Tables {#tables} ## Tables {#tables}
@ -647,11 +647,11 @@ It is approximately **12,742 km**
## Bibliography {#bibliography} ## Bibliography {#bibliography}
<a id="orgb0733db"></a>Dominik, Carsten. 2010. _The Org Mode 7 Reference Manual-Organize Your Life with GNU Emacs_. Network Theory Ltd. <a id="org4f5b6d0"></a>Dominik, Carsten. 2010. _The Org Mode 7 Reference Manual-Organize Your Life with GNU Emacs_. Network Theory Ltd.
<a id="orgdf0380b"></a>Schulte, Eric, and Dan Davison. 2011. “Active Documents with Org-Mode.” _Computing in Science & Engineering_ 13 (3). IEEE Computer Society:66–73. <a id="org7b9fb79"></a>Schulte, Eric, and Dan Davison. 2011. “Active Documents with Org-Mode.” _Computing in Science & Engineering_ 13 (3). IEEE Computer Society:66–73.
<a id="orgfe85fe6"></a>Stanisic, Luka, and Arnaud Legrand. 2014. “Effective Reproducible Research with Org-Mode and Git.” In _European Conference on Parallel Processing_, 475–86. Springer. <a id="org0ed95e1"></a>Stanisic, Luka, and Arnaud Legrand. 2014. “Effective Reproducible Research with Org-Mode and Git.” In _European Conference on Parallel Processing_, 475–86. Springer.
[^fn:1]: A long foot note. Lorem ipsum dolor sit amet, consectetur adipiscing elit. With a reference to Figure [3](#org54379fd). [^fn:1]: A long foot note. Lorem ipsum dolor sit amet, consectetur adipiscing elit. With a reference to Figure [3](#orgcbf9e46).
[^fn:2]: An other footnote. [^fn:2]: An other footnote.

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@ -19,5 +19,3 @@ Mechatronics:
Matlab: Matlab:
- [Mathworks](https://www.mathworks.com/training-schedule/) - [Mathworks](https://www.mathworks.com/training-schedule/)
<./biblio/references.bib>

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@ -13,5 +13,3 @@ Tags
A Transconductance Amplifier converts the control voltage into current with a current source characteristic. A Transconductance Amplifier converts the control voltage into current with a current source characteristic.
Such a converter is called a voltage-to-current converter, also named a voltage-controlled current source or _transconductance_ amplifier. Such a converter is called a voltage-to-current converter, also named a voltage-controlled current source or _transconductance_ amplifier.
<./biblio/references.bib>

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