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content/article/alkhatib03_activ_struc_vibrat_contr.md
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@@ -9,7 +9,7 @@ Tags
Reference
: ([Alkhatib and Golnaraghi 2003](#org701171b))
: ([Alkhatib and Golnaraghi 2003](#org9ef6ccc))
Author(s)
: Alkhatib, R., & Golnaraghi, M. F.
@@ -123,12 +123,12 @@ Uncertainty can be divided into four types:
- neglected nonlinearities
The \\(\mathcal{H}\_\infty\\) controller is developed to address uncertainty by systematic means.
A general block diagram of the control system is shown figure [1](#orgb5f10b2).
A general block diagram of the control system is shown figure [1](#org3926a79).
A **frequency shaped filter** \\(W(s)\\) coupled to selected inputs and outputs of the plant is included.
The outputs of this frequency shaped filter define the error ouputs used to evaluate the system performance and generate the **cost** that will be used in the design process.
<a id="orgb5f10b2"></a>
<a id="org3926a79"></a>
{{< figure src="/ox-hugo/alkhatib03_hinf_control.png" caption="Figure 1: Block diagram for robust control" >}}
@@ -200,11 +200,11 @@ Two different methods
## Active Control Effects on the System {#active-control-effects-on-the-system}
<a id="orgb195fbc"></a>
<a id="org07db29f"></a>
{{< figure src="/ox-hugo/alkhatib03_1dof_control.png" caption="Figure 2: 1 DoF control of a spring-mass-damping system" >}}
Consider the control system figure [2](#orgb195fbc), the equation of motion of the system is:
Consider the control system figure [2](#org07db29f), the equation of motion of the system is:
\\[ m\ddot{x} + c\dot{x} + kx = f\_a + f \\]
The controller force can be expressed as: \\(f\_a = -g\_a \ddot{x} + g\_v \dot{x} + g\_d x\\). The equation of motion becomes:
@@ -227,4 +227,4 @@ If the actuator is placed at the wrong location, the system will require a great
## Bibliography {#bibliography}
<a id="org701171b"></a>Alkhatib, Rabih, and M. F. Golnaraghi. 2003. “Active Structural Vibration Control: A Review.” _The Shock and Vibration Digest_ 35 (5):367–83. <https://doi.org/10.1177/05831024030355002>.
<a id="org9ef6ccc"></a>Alkhatib, Rabih, and M. F. Golnaraghi. 2003. “Active Structural Vibration Control: A Review.” _The Shock and Vibration Digest_ 35 (5):367–83. <https://doi.org/10.1177/05831024030355002>.

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content/article/bibel92_guidel_h.md
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@@ -8,7 +8,7 @@ Tags
: [H Infinity Control]({{< relref "h_infinity_control" >}})
Reference
: ([Bibel and Malyevac 1992](#org47391fd))
: ([Bibel and Malyevac 1992](#org50b9640))
Author(s)
: Bibel, J. E., & Malyevac, D. S.
@@ -19,11 +19,11 @@ Year
## Properties of feedback control {#properties-of-feedback-control}
<a id="org55b0783"></a>
<a id="org1554fcc"></a>
{{< figure src="/ox-hugo/bibel92_control_diag.png" caption="Figure 1: Control System Diagram" >}}
From the figure [1](#org55b0783), we have:
From the figure [1](#org1554fcc), we have:
\begin{align\*}
y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\\\
@@ -77,11 +77,11 @@ Usually, reference signals and disturbances occur at low frequencies, while nois
</div>
<a id="orgbbca2ea"></a>
<a id="orgce70b5f"></a>
{{< figure src="/ox-hugo/bibel92_general_plant.png" caption="Figure 2: \\(\mathcal{H}\_\infty\\) control framework" >}}
New design framework (figure [2](#orgbbca2ea)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
New design framework (figure [2](#orgce70b5f)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
- \\(w\\): exogenous inputs
- \\(z\\): regulated performance output
@@ -108,9 +108,9 @@ The \\(H\_\infty\\) control problem is to find a controller that minimizes \\(\\
## Weights for inputs/outputs signals {#weights-for-inputs-outputs-signals}
Since \\(S\\) and \\(T\\) cannot be minimized together at all frequency, **weights are introduced to shape the solutions**. Not only can \\(S\\) and \\(T\\) be weighted, but other regulated performance variables and inputs (figure [3](#org75a0ac3)).
Since \\(S\\) and \\(T\\) cannot be minimized together at all frequency, **weights are introduced to shape the solutions**. Not only can \\(S\\) and \\(T\\) be weighted, but other regulated performance variables and inputs (figure [3](#orgca469d2)).
<a id="org75a0ac3"></a>
<a id="orgca469d2"></a>
{{< figure src="/ox-hugo/bibel92_hinf_weights.png" caption="Figure 3: Input and Output weights in \\(\mathcal{H}\_\infty\\) framework" >}}
@@ -154,15 +154,15 @@ When using both \\(W\_S\\) and \\(W\_T\\), it is important to make sure that the
## Unmodeled dynamics weighting function {#unmodeled-dynamics-weighting-function}
Another method of limiting the controller bandwidth and providing high frequency gain attenuation is to use a high pass weight on an **unmodeled dynamics uncertainty block** that may be added from the plant input to the plant output (figure [4](#orgd3e0294)).
Another method of limiting the controller bandwidth and providing high frequency gain attenuation is to use a high pass weight on an **unmodeled dynamics uncertainty block** that may be added from the plant input to the plant output (figure [4](#org96cc166)).
<a id="orgd3e0294"></a>
<a id="org96cc166"></a>
{{< figure src="/ox-hugo/bibel92_unmodeled_dynamics.png" caption="Figure 4: Unmodeled dynamics model" >}}
The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown figure [5](#org6d5884c).
The weight is chosen to cover the expected worst case magnitude of the unmodeled dynamics. A typical unmodeled dynamics weighting function is shown figure [5](#orgab966f3).
<a id="org6d5884c"></a>
<a id="orgab966f3"></a>
{{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="Figure 5: Example of unmodeled dynamics weight" >}}
@@ -184,4 +184,4 @@ Typically actuator input weights are constant over frequency and set at the inve
## Bibliography {#bibliography}
<a id="org47391fd"></a>Bibel, John E, and D Stephen Malyevac. 1992. “Guidelines for the Selection of Weighting Functions for H-Infinity Control.” NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA.
<a id="org50b9640"></a>Bibel, John E, and D Stephen Malyevac. 1992. “Guidelines for the Selection of Weighting Functions for H-Infinity Control.” NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA.

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content/article/bryson93_contr_spacec_aircr.md
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@@ -8,7 +8,7 @@ Tags
: [HAC-HAC]({{< relref "hac_hac" >}})
Reference
: ([Bryson 1993](#orgcd1f1a3))
: ([Bryson 1993](#org93399a5))
Author(s)
: Bryson, A. E.
@@ -41,7 +41,7 @@ Year
> LAC uses a co-located rate sensor to add damping to all the vibratory modes (but not the rigid-body mode).
> HAC uses a separated displacement sensor to stabilize the rigid body mode, which slightly decreases the damping of the vibratory modes but not enough to produce instability (called "spillover")
<a id="org3548362"></a>
<a id="orgee09f4a"></a>
{{< figure src="/ox-hugo/bryson93_hac_lac.png" caption="Figure 1: HAC-LAC control concept" >}}
@@ -50,4 +50,4 @@ Year
## Bibliography {#bibliography}
<a id="orgcd1f1a3"></a>Bryson, Arthur Earl. 1993. _Control of Spacecraft and Aircraft_. Princeton university press Princeton, New Jersey.
<a id="org93399a5"></a>Bryson, Arthur Earl. 1993. _Control of Spacecraft and Aircraft_. Princeton university press Princeton, New Jersey.

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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" >}})
Reference
: ([Butler 2011](#orgc5e178d))
: ([Butler 2011](#org79c48f7))
Author(s)
: Butler, H.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
<a id="orgc5e178d"></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="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>.

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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" >}})
Reference
: ([Chen and McInroy 2000](#org9682dcf))
: ([Chen and McInroy 2000](#orgc009af3))
Author(s)
: Chen, Y., & McInroy, J.
@@ -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}
The derivation of the dynamic model is done in ([McInroy 1999](#org629610e)) ([Notes]({{< relref "mcinroy99_dynam" >}})).
The derivation of the dynamic model is done in ([McInroy 1999](#org0e9e807)) ([Notes]({{< relref "mcinroy99_dynam" >}})).
<a id="org69c9396"></a>
<a id="orge595e9c"></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" >}}
@@ -102,6 +102,6 @@ where
## Bibliography {#bibliography}
<a id="org9682dcf"></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="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="org629610e"></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="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>.

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content/article/claeyssen07_amplif_piezoel_actuat.md
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@@ -9,7 +9,7 @@ Tags
Reference
: ([Claeyssen et al. 2007](#org28b5d9a))
: ([Claeyssen et al. 2007](#org8c33a63))
Author(s)
: Claeyssen, F., Letty, R. L., Barillot, F., & Sosnicki, O.
@@ -37,4 +37,4 @@ The prestress design allows a peak force equal to half the blocked force.
## Bibliography {#bibliography}
<a id="org28b5d9a"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static & Dynamic Applications.” _Ferroelectrics_ 351 (1):3–14. <https://doi.org/10.1080/00150190701351865>.
<a id="org8c33a63"></a>Claeyssen, Frank, R. Le Letty, F. Barillot, and O. Sosnicki. 2007. “Amplified Piezoelectric Actuators: Static & Dynamic Applications.” _Ferroelectrics_ 351 (1):3–14. <https://doi.org/10.1080/00150190701351865>.

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content/article/collette11_review_activ_vibrat_isolat_strat.md
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@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Collette, Janssens, and Artoos 2011](#org94d4ab9))
: ([Collette, Janssens, and Artoos 2011](#org7b1a927))
Author(s)
: Collette, C., Janssens, S., & Artoos, K.
@@ -71,11 +71,11 @@ The general expression of the force delivered by the actuator is \\(f = g\_a \dd
## Conclusions {#conclusions}
<a id="org9eb39a5"></a>
<a id="org0035120"></a>
{{< figure src="/ox-hugo/collette11_comp_isolation_strategies.png" caption="Figure 1: Comparison of Active Vibration Isolation Strategies" >}}
## Bibliography {#bibliography}
<a id="org94d4ab9"></a>Collette, Christophe, Stef Janssens, and Kurt Artoos. 2011. “Review of Active Vibration Isolation Strategies.” _Recent Patents on Mechanical Engineeringe_ 4 (3):212–19. <https://doi.org/10.2174/2212797611104030212>.
<a id="org7b1a927"></a>Collette, Christophe, Stef Janssens, and Kurt Artoos. 2011. “Review of Active Vibration Isolation Strategies.” _Recent Patents on Mechanical Engineeringe_ 4 (3):212–19. <https://doi.org/10.2174/2212797611104030212>.

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content/article/collette14_vibrat.md
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@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Sensor Fusion]({{< relref "sensor_fusion" >}})
Reference
: ([Collette and Matichard 2014](#org6ed3ef2))
: ([Collette and Matichard 2014](#orged1e033))
Author(s)
: Collette, C., & Matichard, F.
@@ -102,4 +102,4 @@ Three types of sensors have been considered for the high frequency part of the f
## Bibliography {#bibliography}
<a id="org6ed3ef2"></a>Collette, C., and F Matichard. 2014. “Vibration Control of Flexible Structures Using Fusion of Inertial Sensors and Hyper-Stable Actuator-Sensor Pairs.” In _International Conference on Noise and Vibration Engineering (ISMA2014)_.
<a id="orged1e033"></a>Collette, C., and F Matichard. 2014. “Vibration Control of Flexible Structures Using Fusion of Inertial Sensors and Hyper-Stable Actuator-Sensor Pairs.” In _International Conference on Noise and Vibration Engineering (ISMA2014)_.

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content/article/collette15_sensor_fusion_method_high_perfor.md
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@@ -8,7 +8,7 @@ Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Collette and Matichard 2015](#orgec71140))
: ([Collette and Matichard 2015](#org5c9aaf9))
Author(s)
: Collette, C., & Matichard, F.
@@ -27,4 +27,4 @@ The stability margins of the controller can be significantly increased with no o
## Bibliography {#bibliography}
<a id="orgec71140"></a>Collette, C., and F. Matichard. 2015. “Sensor Fusion Methods for High Performance Active Vibration Isolation Systems.” _Journal of Sound and Vibration_ 342 (nil):1–21. <https://doi.org/10.1016/j.jsv.2015.01.006>.
<a id="org5c9aaf9"></a>Collette, C., and F. Matichard. 2015. “Sensor Fusion Methods for High Performance Active Vibration Isolation Systems.” _Journal of Sound and Vibration_ 342 (nil):1–21. <https://doi.org/10.1016/j.jsv.2015.01.006>.

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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" >}})
Reference
: ([Dasgupta and Mruthyunjaya 2000](#org4f597ec))
: ([Dasgupta and Mruthyunjaya 2000](#orgab59d3a))
Author(s)
: Dasgupta, B., & Mruthyunjaya, T.
@@ -36,4 +36,4 @@ The generalized Stewart platforms consists of two rigid bodies (referred to as t
## Bibliography {#bibliography}
<a id="org4f597ec"></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="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.

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content/article/devasia07_survey_contr_issues_nanop.md
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@@ -9,7 +9,7 @@ Tags
Reference
: ([Devasia, Eleftheriou, and Moheimani 2007](#org967030a))
: ([Devasia, Eleftheriou, and Moheimani 2007](#org4f17944))
Author(s)
: Devasia, S., Eleftheriou, E., & Moheimani, S. R.
@@ -22,11 +22,11 @@ Year
- Interesting analysis about Bandwidth-Precision-Range tradeoffs
- Control approaches for piezoelectric actuators: feedforward, Feedback, Iterative, Sensorless controls
<a id="org178295f"></a>
<a id="org47d33b5"></a>
{{< figure src="/ox-hugo/devasia07_piezoelectric_tradeoff.png" caption="Figure 1: Tradeoffs between bandwidth, precision and range" >}}
## Bibliography {#bibliography}
<a id="org967030a"></a>Devasia, Santosh, Evangelos Eleftheriou, and SO Reza Moheimani. 2007. “A Survey of Control Issues in Nanopositioning.” _IEEE Transactions on Control Systems Technology_ 15 (5). IEEE:802–23.
<a id="org4f17944"></a>Devasia, Santosh, Evangelos Eleftheriou, and SO Reza Moheimani. 2007. “A Survey of Control Issues in Nanopositioning.” _IEEE Transactions on Control Systems Technology_ 15 (5). IEEE:802–23.

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content/article/fleming10_nanop_system_with_force_feedb.md
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@@ -6,14 +6,14 @@ draft = false
Backlinks:
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
- [Force Sensors]({{< relref "force_sensors" >}})
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Force Sensors]({{< relref "force_sensors" >}})
Reference
: ([Fleming 2010](#org5c1a566))
: ([Fleming 2010](#org28aeac7))
Author(s)
: Fleming, A.
@@ -36,7 +36,7 @@ Year
## Model of a multi-layer monolithic piezoelectric stack actuator {#model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator}
<a id="org8318193"></a>
<a id="org7145764"></a>
{{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="Figure 1: Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
@@ -121,11 +121,11 @@ The capacitance of a piezoelectric stack is typically between \\(1 \mu F\\) and
## Tested feedback control strategies {#tested-feedback-control-strategies}
<a id="orgab9a9d3"></a>
<a id="org9ecbbcf"></a>
{{< figure src="/ox-hugo/fleming10_fb_control_strats.png" caption="Figure 3: Comparison of: (a) basic integral control. (b) direct tracking control. (c) dual-sensor feedback. (d) low frequency bypass" >}}
## Bibliography {#bibliography}
<a id="org5c1a566"></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="org28aeac7"></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>.

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@@ -9,7 +9,7 @@ Tags
Reference
: ([Fleming 2012](#org8c231e4))
: ([Fleming 2012](#orga5f3c24))
Author(s)
: Fleming, A. J.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<a id="org8c231e4"></a>Fleming, Andrew J. 2012. “Estimating the Resolution of Nanopositioning Systems from Frequency Domain Data.” In _2012 IEEE International Conference on Robotics and Automation_, nil. <https://doi.org/10.1109/icra.2012.6224850>.
<a id="orga5f3c24"></a>Fleming, Andrew J. 2012. “Estimating the Resolution of Nanopositioning Systems from Frequency Domain Data.” In _2012 IEEE International Conference on Robotics and Automation_, nil. <https://doi.org/10.1109/icra.2012.6224850>.

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@@ -4,7 +4,7 @@ author = ["Thomas Dehaeze"]
draft = false
+++
### Backlinks {#backlinks}
Backlinks:
- [Position Sensors]({{< relref "position_sensors" >}})
@@ -12,7 +12,7 @@ Tags
: [Position Sensors]({{< relref "position_sensors" >}})
Reference
: ([Fleming 2013](#orgd570a34))
: ([Fleming 2013](#org35f9cea))
Author(s)
: Fleming, A. J.
@@ -37,7 +37,7 @@ Usually quoted as a percentage of the fill-scale range (FSR):
With \\(e\_m(v)\\) is the mapping error.
<a id="org4080fd5"></a>
<a id="orge06f384"></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." >}}
@@ -46,7 +46,7 @@ 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.
<a id="org119a172"></a>
<a id="orgc484965"></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." >}}
@@ -151,9 +151,9 @@ 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}\\).
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](#org690dbcf)).
There is usually a trade-off between bandwidth and resolution (figure [3](#org2ee752d)).
<a id="org690dbcf"></a>
<a id="org2ee752d"></a>
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}}
@@ -188,4 +188,4 @@ A convenient method for reporting this ratio is in parts-per-million (ppm):
## Bibliography {#bibliography}
<a id="orgd570a34"></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="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>.

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Reference
: ([Fleming, Teo, and Leang 2015](#orgdcfeb10))
: ([Fleming, Teo, and Leang 2015](#orgd9b657d))
Author(s)
: Fleming, A. J., Teo, Y. R., & Leang, K. K.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<a id="orgdcfeb10"></a>Fleming, Andrew J., Yik Ren Teo, and Kam K. Leang. 2015. “Low-Order Damping and Tracking Control for Scanning Probe Systems.” _Frontiers in Mechanical Engineering_ 1 (nil):nil. <https://doi.org/10.3389/fmech.2015.00014>.
<a id="orgd9b657d"></a>Fleming, Andrew J., Yik Ren Teo, and Kam K. Leang. 2015. “Low-Order Damping and Tracking Control for Scanning Probe Systems.” _Frontiers in Mechanical Engineering_ 1 (nil):nil. <https://doi.org/10.3389/fmech.2015.00014>.

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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" >}})
Reference
: ([Furqan, Suhaib, and Ahmad 2017](#org12ef37a))
: ([Furqan, Suhaib, and Ahmad 2017](#org7520991))
Author(s)
: Furqan, M., Suhaib, M., & Ahmad, N.
@@ -21,4 +21,4 @@ Lots of references.
## Bibliography {#bibliography}
<a id="org12ef37a"></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="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>.

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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: ([Furutani, Suzuki, and Kudoh 2004](#org9e7f4c5))
: ([Furutani, Suzuki, and Kudoh 2004](#orgaf9f119))
Author(s)
: Furutani, K., Suzuki, M., & Kudoh, R.
@@ -37,4 +37,4 @@ Then, it is fitted with 4th order polynomial and included in the control archite
## Bibliography {#bibliography}
<a id="org9e7f4c5"></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="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>.

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@@ -8,7 +8,7 @@ Tags
: [Position Sensors]({{< relref "position_sensors" >}})
Reference
: ([Gao et al. 2015](#org0a9ecf7))
: ([Gao et al. 2015](#org7f49efc))
Author(s)
: Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
<a id="org0a9ecf7"></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="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>.

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@@ -8,7 +8,7 @@ Tags
: [Multivariable Control]({{< relref "multivariable_control" >}})
Reference
: ([Garg 2007](#org1da9ec5))
: ([Garg 2007](#orga5ede8d))
Author(s)
: Garg, S.
@@ -37,4 +37,4 @@ The control rate should be weighted appropriately in order to not saturate the s
## Bibliography {#bibliography}
<a id="org1da9ec5"></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="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>.

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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Geng et al. 1995](#org8ae89dc))
: ([Geng et al. 1995](#orgec21e7f))
Author(s)
: Geng, Z. J., Pan, G. G., Haynes, L. S., Wada, B. K., & Garba, J. A.
@@ -16,11 +16,11 @@ Author(s)
Year
: 1995
<a id="orgc1d6599"></a>
<a id="org757d5ce"></a>
{{< figure src="/ox-hugo/geng95_control_structure.png" caption="Figure 1: Local force feedback and adaptive acceleration feedback for active isolation" >}}
## Bibliography {#bibliography}
<a id="org8ae89dc"></a>Geng, Z. Jason, George G. Pan, Leonard S. Haynes, Ben K. Wada, and John A. Garba. 1995. “An Intelligent Control System for Multiple Degree-of-Freedom Vibration Isolation.” _Journal of Intelligent Material Systems and Structures_ 6 (6):787–800. <https://doi.org/10.1177/1045389x9500600607>.
<a id="orgec21e7f"></a>Geng, Z. Jason, George G. Pan, Leonard S. Haynes, Ben K. Wada, and John A. Garba. 1995. “An Intelligent Control System for Multiple Degree-of-Freedom Vibration Isolation.” _Journal of Intelligent Material Systems and Structures_ 6 (6):787–800. <https://doi.org/10.1177/1045389x9500600607>.

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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Active Damping]({{< relref "active_damping" >}})
Reference
: ([Hanieh 2003](#orga4be8f4))
: ([Hanieh 2003](#orgcfd0d1f))
Author(s)
: Hanieh, A. A.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
<a id="orga4be8f4"></a>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.
<a id="orgcfd0d1f"></a>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.

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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
Reference
: ([Hauge and Campbell 2004](#org03befb5))
: ([Hauge and Campbell 2004](#orge4a516e))
Author(s)
: Hauge, G., & Campbell, M.
@@ -24,20 +24,20 @@ Year
- Vibration isolation using a Stewart platform
- Experimental comparison of Force sensor and Inertial Sensor and associated control architecture for vibration isolation
<a id="orge348607"></a>
<a id="org59ac043"></a>
{{< figure src="/ox-hugo/hauge04_stewart_platform.png" caption="Figure 1: Hexapod for active vibration isolation" >}}
**Stewart platform** (Figure [1](#orge348607)):
**Stewart platform** (Figure [1](#org59ac043)):
- Low corner frequency
- Large actuator stroke (\\(\pm5mm\\))
- Sensors in each strut (Figure [2](#orge668964)):
- Sensors in each strut (Figure [2](#org59c0ee0)):
- three-axis load cell
- base and payload geophone in parallel with the struts
- LVDT
<a id="orge668964"></a>
<a id="org59c0ee0"></a>
{{< figure src="/ox-hugo/hauge05_struts.png" caption="Figure 2: Strut" >}}
@@ -64,7 +64,7 @@ With \\(|T(\omega)|\\) is the Frobenius norm of the transmissibility matrix and
- single strut axis as the cubic Stewart platform can be decomposed into 6 single-axis systems
<a id="orgb65b964"></a>
<a id="org5b75feb"></a>
{{< figure src="/ox-hugo/hauge04_strut_model.png" caption="Figure 3: Strut model" >}}
@@ -136,11 +136,11 @@ And we find that for \\(u\\) and \\(y\\) to be an acceptable pair for high gain
- The performance requirements are met
- Good robustness
<a id="org19e7d49"></a>
<a id="org4a7177a"></a>
{{< figure src="/ox-hugo/hauge04_obtained_transmissibility.png" caption="Figure 4: Experimental open loop (solid) and closed loop six-axis transmissibility using the geophone only controller (dotted), and combined geophone/load cell controller (dashed)" >}}
## Bibliography {#bibliography}
<a id="org03befb5"></a>Hauge, G.S., and M.E. Campbell. 2004. “Sensors and Control of a Space-Based Six-Axis Vibration Isolation System.” _Journal of Sound and Vibration_ 269 (3-5):913–31. <https://doi.org/10.1016/s0022-460x(03)>00206-2.
<a id="orge4a516e"></a>Hauge, G.S., and M.E. Campbell. 2004. “Sensors and Control of a Space-Based Six-Axis Vibration Isolation System.” _Journal of Sound and Vibration_ 269 (3-5):913–31. <https://doi.org/10.1016/s0022-460x(03)>00206-2.

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@@ -8,7 +8,7 @@ Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
Reference
: ([Holler et al. 2012](#org28ec5cd))
: ([Holler et al. 2012](#org289d119))
Author(s)
: 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.
Obtain position stability of 10nm (standard deviation).
<a id="orgd35b9ca"></a>
<a id="orgfed3898"></a>
{{< figure src="/ox-hugo/holler12_station.png" caption="Figure 1: Schematic of the tomography setup" >}}
@@ -41,4 +41,4 @@ Obtain position stability of 10nm (standard deviation).
## Bibliography {#bibliography}
<a id="org28ec5cd"></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="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>.

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@@ -8,7 +8,7 @@ Tags
: [Active Damping]({{< relref "active_damping" >}})
Reference
: ([Holterman and deVries 2005](#org13eedae))
: ([Holterman and deVries 2005](#orgfb04a8c))
Author(s)
: Holterman, J., & deVries, T.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
<a id="org13eedae"></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="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>.

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@@ -4,7 +4,7 @@ author = ["Thomas Dehaeze"]
draft = false
+++
### Backlinks {#backlinks}
Backlinks:
- [Actuators]({{< relref "actuators" >}})
@@ -12,7 +12,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Actuators]({{< relref "actuators" >}})
Reference
: ([Ito and Schitter 2016](#orga31805f))
: ([Ito and Schitter 2016](#orgbaa452e))
Author(s)
: Ito, S., & Schitter, G.
@@ -45,7 +45,7 @@ In this paper, the piezoelectric actuator/electronics adds a time delay which is
- **Low Stiffness** actuator is defined as the ones where the transmissibility stays below 0dB at all frequency
- **High Stiffness** actuator is defined as the ones where the transmissibility goes above 0dB at some frequency
<a id="org0fdae5f"></a>
<a id="orgcd249fb"></a>
{{< figure src="/ox-hugo/ito16_low_high_stiffness_actuators.png" caption="Figure 1: Definition of low-stiffness and high-stiffness actuator" >}}
@@ -58,7 +58,7 @@ In this paper, the piezoelectric actuator/electronics adds a time delay which is
## Controller Design {#controller-design}
<a id="org6565e90"></a>
<a id="orgde6bd83"></a>
{{< figure src="/ox-hugo/ito16_transmissibility.png" caption="Figure 2: Obtained transmissibility" >}}
@@ -73,4 +73,4 @@ In contrast, the frequency band between the first and the other resonances of Lo
## Bibliography {#bibliography}
<a id="orga31805f"></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="orgbaa452e"></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>.

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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: ([Jiao et al. 2018](#org142ce8a))
: ([Jiao et al. 2018](#orga067c93))
Author(s)
: Jiao, J., Wu, Y., Yu, K., & Zhao, R.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
<a id="org142ce8a"></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="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>.

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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference
: ([Legnani et al. 2012](#orgb569080))
: ([Legnani et al. 2012](#org4d21607))
Author(s)
: Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D.
@@ -22,15 +22,15 @@ Year
Example of generated isotropic manipulator (not decoupled).
<a id="orgb13b47f"></a>
<a id="org584231c"></a>
{{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="Figure 1: Location of the leg axes using an isotropy generator" >}}
<a id="org0b17ad4"></a>
<a id="orgfad58c6"></a>
{{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="Figure 2: Isotropic configuration" >}}
## Bibliography {#bibliography}
<a id="orgb569080"></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="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>.

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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" >}})
Reference
: ([Li 2001](#orgb070307))
: ([Li 2001](#org9c3830b))
Author(s)
: Li, X.
@@ -24,7 +24,7 @@ Year
- Cubic (mutually orthogonal)
- Flexure Joints => eliminate friction and backlash but add complexity to the dynamics
<a id="org7d51783"></a>
<a id="org0b311b0"></a>
{{< 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.
**Decoupling**:
If we refine the (force) inputs and (displacement) outputs as shown in Figure [2](#org88939ed) or in Figure [3](#org0b68d74), we obtain a decoupled plant provided that:
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:
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
> 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="org88939ed"></a>
<a id="orgdda87ef"></a>
{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 2: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
<a id="org0b68d74"></a>
<a id="orga45a9ca"></a>
{{< 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}
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [4](#org8a3c3a5).
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [4](#org2bd4cf2).
<a id="org8a3c3a5"></a>
<a id="org2bd4cf2"></a>
{{< 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](#org8a3c3a5)
One of the subsystem plant transfer function is shown in Figure [4](#org2bd4cf2)
<a id="org9632b34"></a>
<a id="org16f5e77"></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" >}}
@@ -97,9 +97,9 @@ The unity control bandwidth of the isolation loop is designed to be from **5Hz t
### Pointing Control {#pointing-control}
A block diagram of the pointing control system is shown in Figure [6](#orgd000413).
A block diagram of the pointing control system is shown in Figure [6](#orge508ebf).
<a id="orgd000413"></a>
<a id="orge508ebf"></a>
{{< 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**.
A feedforward control is added as shown in Figure [7](#orgb2f844f).
A feedforward control is added as shown in Figure [7](#org68e9af0).
<a id="orgb2f844f"></a>
<a id="org68e9af0"></a>
{{< 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
2. the reverse design order
Figure [8](#org03ff7c9) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
Figure [8](#orgdb53c99) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
<a id="org03ff7c9"></a>
<a id="orgdb53c99"></a>
{{< 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 effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [9](#orgbec95ff).
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).
<a id="orgbec95ff"></a>
<a id="org3ec32bc"></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)" >}}
@@ -143,19 +143,19 @@ The effect of pointing control on the isolation plant has not much effect.
The dynamic interaction effect:
- 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](#org77dbe42) and [11](#org70ff0f3))
- affect the closed loop transmission of the loop first closed (see Figures [10](#org4a1beff) and [11](#org945793e))
As shown in Figure [10](#org77dbe42), the peak resonance of the pointing loop increase after the isolation loop is closed.
As shown in Figure [10](#org4a1beff), 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.
<a id="org77dbe42"></a>
<a id="org4a1beff"></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" >}}
The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [11](#org70ff0f3)).
The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [11](#org945793e)).
The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
<a id="org70ff0f3"></a>
<a id="org945793e"></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" >}}
@@ -165,18 +165,18 @@ The first peak resonance of the vibration isolation loop at 15Hz is increased wh
### Experimental results {#experimental-results}
Two hexapods are stacked (Figure [12](#org3e27211)):
Two hexapods are stacked (Figure [12](#orgbdc7900)):
- the bottom hexapod is used to generate disturbances matching candidate applications
- the top hexapod provide simultaneous vibration isolation and pointing control
<a id="org3e27211"></a>
<a id="orgbdc7900"></a>
{{< 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](#org217639e).
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [13](#org7a6d899).
<a id="org217639e"></a>
<a id="org7a6d899"></a>
{{< 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 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](#org2cdf6ad) where the bandwidth of the isolation control is expanded to very low frequency.
The results of simultaneous control is shown in Figure [14](#org15d93e5) where the bandwidth of the isolation control is expanded to very low frequency.
<a id="org2cdf6ad"></a>
<a id="org15d93e5"></a>
{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 14: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
@@ -218,4 +218,4 @@ Proposed future research areas include:
## Bibliography {#bibliography}
<a id="orgb070307"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.
<a id="org9c3830b"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.

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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Li, Hamann, and McInroy 2001](#orgd4f4d69))
: ([Li, Hamann, and McInroy 2001](#org98b01e6))
Author(s)
: Li, X., Hamann, J. C., & McInroy, J. E.
@@ -21,4 +21,4 @@ Year
## Bibliography {#bibliography}
<a id="orgd4f4d69"></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="org98b01e6"></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>.

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@@ -9,7 +9,7 @@ Tags
Reference
: ([Lin and McInroy 2006](#org7a24d0e))
: ([Lin and McInroy 2006](#orge05f299))
Author(s)
: Lin, H., & McInroy, J. E.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<a id="org7a24d0e"></a>Lin, Haomin, and John E. McInroy. 2006. “Disturbance Attenuation in Precise Hexapod Pointing Using Positive Force Feedback.” _Control Engineering Practice_ 14 (11):1377–86. <https://doi.org/10.1016/j.conengprac.2005.10.002>.
<a id="orge05f299"></a>Lin, Haomin, and John E. McInroy. 2006. “Disturbance Attenuation in Precise Hexapod Pointing Using Positive Force Feedback.” _Control Engineering Practice_ 14 (11):1377–86. <https://doi.org/10.1016/j.conengprac.2005.10.002>.

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@@ -9,7 +9,7 @@ Tags
Reference
: ([McInroy and Hamann 2000](#orgd22225e))
: ([McInroy and Hamann 2000](#orgb4fc604))
Author(s)
: McInroy, J., & Hamann, J.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<a id="orgd22225e"></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="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>.

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@@ -9,7 +9,7 @@ Tags
Reference
: ([McInroy 2002](#org8c59af1))
: ([McInroy 2002](#orgf7c9a88))
Author(s)
: McInroy, J.
@@ -17,7 +17,7 @@ Author(s)
Year
: 2002
This short paper is very similar to ([McInroy 1999](#org69da46e)).
This short paper is very similar to ([McInroy 1999](#org4526c4b)).
> This paper develops guidelines for designing the flexure joints to facilitate closed-loop control.
@@ -36,15 +36,15 @@ This short paper is very similar to ([McInroy 1999](#org69da46e)).
## Flexure Jointed Hexapod Dynamics {#flexure-jointed-hexapod-dynamics}
<a id="orgccb4fed"></a>
<a id="orgd884ef4"></a>
{{< figure src="/ox-hugo/mcinroy02_leg_model.png" caption="Figure 1: The dynamics of the ith strut. A parallel spring, damper, and actautor drives the moving mass of the strut and a payload" >}}
The strut can be modeled as consisting of a parallel arrangement of an actuator force, a spring and some damping driving a mass (Figure [1](#orgccb4fed)).
The strut can be modeled as consisting of a parallel arrangement of an actuator force, a spring and some damping driving a mass (Figure [1](#orgd884ef4)).
Thus, **the strut does not output force directly, but rather outputs a mechanically filtered force**.
The model of the strut are shown in Figure [1](#orgccb4fed) with:
The model of the strut are shown in Figure [1](#orgd884ef4) with:
- \\(m\_{s\_i}\\) moving strut mass
- \\(k\_i\\) spring constant
@@ -132,16 +132,16 @@ Many prior hexapod dynamic formulations assume that the strut exerts force only
The flexure joints Hexapods transmit forces (or torques) proportional to the deflection of the joints.
This section establishes design guidelines for the spherical flexure joint to guarantee that the dynamics remain tractable for control.
<a id="orgaffe5ce"></a>
<a id="org54c99c4"></a>
{{< figure src="/ox-hugo/mcinroy02_model_strut_joint.png" caption="Figure 2: A simplified dynamic model of a strut and its joint" >}}
Figure [2](#orgaffe5ce) depicts a strut, along with the corresponding force diagram.
Figure [2](#org54c99c4) depicts a strut, along with the corresponding force diagram.
The force diagram is obtained using standard finite element assumptions (\\(\sin \theta \approx \theta\\)).
Damping terms are neglected.
\\(k\_r\\) denotes the rotational stiffness of the spherical joint.
From Figure [2](#orgaffe5ce) (b), Newton's second law yields:
From Figure [2](#org54c99c4) (b), Newton's second law yields:
\begin{equation}
f\_p = \begin{bmatrix}
@@ -188,7 +188,7 @@ The first part depends on the mechanical terms and the frequency of the movement
x\_{\text{gain}\_\omega} = \frac{|-m\_s \omega^2 + k|}{|-m\_s \omega^2 + \frac{k\_r}{l^2}|}
\end{equation}
<div class="important">
<div class="bred">
<div></div>
In order to get dominance at low frequencies, the hexapod must be designed so that:
@@ -206,7 +206,7 @@ By satisfying \eqref{eq:cond_stiff}, \\(f\_p\\) can be aligned with the strut fo
At frequencies much above the strut's resonance mode, \\(f\_p\\) is not dominated by its \\(x\\) component:
\\[ \omega \gg \sqrt{\frac{k}{m\_s}} \rightarrow x\_{\text{gain}\_\omega} \approx 1 \\]
<div class="important">
<div class="bred">
<div></div>
To ensure that the control system acts only in the band of frequencies where dominance is retained, the control bandwidth can be selected so that:
@@ -225,7 +225,7 @@ In this case, it is reasonable to use:
\text{control bandwidth} \ll \sqrt{\frac{k}{m\_s}}
\end{equation}
<div class="important">
<div class="bred">
<div></div>
By designing the flexure jointed hexapod and its controller so that both \eqref{eq:cond_stiff} and \eqref{eq:cond_bandwidth} are met, the dynamics of the hexapod can be greatly reduced in complexity.
@@ -271,6 +271,6 @@ By using the vector triple identity \\(a \cdot (b \times c) = b \cdot (c \times
## Bibliography {#bibliography}
<a id="org69da46e"></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="org4526c4b"></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="org8c59af1"></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="orgf7c9a88"></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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@@ -4,7 +4,7 @@ author = ["Thomas Dehaeze"]
draft = false
+++
### Backlinks {#backlinks}
Backlinks:
- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})
@@ -12,7 +12,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: ([McInroy 1999](#org5efb28a))
: ([McInroy 1999](#orgfc7fa52))
Author(s)
: McInroy, J.
@@ -20,7 +20,7 @@ Author(s)
Year
: 1999
This conference paper has been further published in a journal as a short note ([McInroy 2002](#org4990a96)).
This conference paper has been further published in a journal as a short note ([McInroy 2002](#org7752c60)).
## Abstract {#abstract}
@@ -42,22 +42,22 @@ The actuators for FJHs can be divided into two categories:
1. soft (voice coil), which employs a spring flexure mount
2. hard (piezoceramic or magnetostrictive), which employs a compressive load spring.
<a id="org5279430"></a>
<a id="orgd835559"></a>
{{< 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](#org6b356c7).
Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#org26f1840).
In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
<a id="org6b356c7"></a>
<a id="org26f1840"></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" >}}
<a id="table--tab:mcinroy99-strut-model"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:mcinroy99-strut-model">Table 1</a></span>:
Definition of quantities on Figure <a href="#org6b356c7">2</a>
Definition of quantities on Figure <a href="#org26f1840">2</a>
</div>
| **Symbol** | **Meaning** |
@@ -74,11 +74,11 @@ 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 |
| \\(\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](#org4990a96))).
It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in ([McInroy 2002](#org7752c60))).
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.
From Figure [2](#org6b356c7) (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](#org26f1840) (b), forces along the strut direction are summed to yield (projected along the strut direction, hence the \\(\hat{u}\_i^T\\) term):
\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
@@ -168,6 +168,6 @@ In the next section, a connection between the two will be found to complete the
## Bibliography {#bibliography}
<a id="org5efb28a"></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="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="org4990a96"></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="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>.

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@@ -8,7 +8,7 @@ Tags
: [Motion Control]({{< relref "motion_control" >}})
Reference
: ([Oomen 2018](#orga862567))
: ([Oomen 2018](#org18923fa))
Author(s)
: Oomen, T.
@@ -16,11 +16,11 @@ Author(s)
Year
: 2018
<a id="org883ebf7"></a>
<a id="orgf64e727"></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." >}}
## Bibliography {#bibliography}
<a id="orga862567"></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="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>.

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@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Poel 2010](#org0c02ed0))
: ([Poel 2010](#org913a900))
Author(s)
: van der Poel, G. W.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
<a id="org0c02ed0"></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="org913a900"></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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@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Preumont et al. 2002](#org1f14c5d))
: ([Preumont et al. 2002](#org35c3663))
Author(s)
: Preumont, A., A. Francois, Bossens, F., & Abu-Hanieh, A.
@@ -26,14 +26,14 @@ The force applied to a **rigid body** is proportional to its acceleration, thus
Thus force feedback and acceleration feedback are equivalent for solid bodies.
When there is a flexible payload, the two sensing options are not longer equivalent.
- For light payload (Figure [1](#orgbbc4740)), the acceleration feedback gives larger damping on the higher mode.
- For heavy payload (Figure [2](#orgdf8d8cc)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
- For light payload (Figure [1](#org93ee8cd)), the acceleration feedback gives larger damping on the higher mode.
- For heavy payload (Figure [2](#orgec90b36)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
<a id="orgbbc4740"></a>
<a id="org93ee8cd"></a>
{{< figure src="/ox-hugo/preumont02_force_acc_fb_light.png" caption="Figure 1: Root locus for **light** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
<a id="orgdf8d8cc"></a>
<a id="orgec90b36"></a>
{{< figure src="/ox-hugo/preumont02_force_acc_fb_heavy.png" caption="Figure 2: Root locus for **heavy** flexible payload, (a) Force feedback, (b) acceleration feedback" >}}
@@ -48,4 +48,4 @@ The same is true for the transfer function from the force actuator to the relati
## Bibliography {#bibliography}
<a id="org1f14c5d"></a>Preumont, A., A. François, F. Bossens, and A. Abu-Hanieh. 2002. “Force Feedback Versus Acceleration Feedback in Active Vibration Isolation.” _Journal of Sound and Vibration_ 257 (4):605–13. <https://doi.org/10.1006/jsvi.2002.5047>.
<a id="org35c3663"></a>Preumont, A., A. François, F. Bossens, and A. Abu-Hanieh. 2002. “Force Feedback Versus Acceleration Feedback in Active Vibration Isolation.” _Journal of Sound and Vibration_ 257 (4):605–13. <https://doi.org/10.1006/jsvi.2002.5047>.

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: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: ([Preumont et al. 2007](#orgce12c3f))
: ([Preumont et al. 2007](#org3b370bf))
Author(s)
: Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …
@@ -18,34 +18,34 @@ Year
Summary:
- **Cubic** Stewart platform (Figure [3](#orgb093d6d))
- **Cubic** Stewart platform (Figure [3](#org0274a43))
- Provides uniform control capability
- Uniform stiffness in all directions
- minimizes the cross-coupling among actuators and sensors of different legs
- Flexible joints (Figure [2](#org2531c05))
- Flexible joints (Figure [2](#orge9ceb91))
- Piezoelectric force sensors
- Voice coil actuators
- Decentralized feedback control approach for vibration isolation
- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#orgffaf2cb))
- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#orgb3c0578))
- The Stewart platform has 6 suspension modes at different frequencies.
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.
- 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
<a id="orgffaf2cb"></a>
<a id="orgb3c0578"></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" >}}
<a id="org2531c05"></a>
<a id="orge9ceb91"></a>
{{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="Figure 2: Flexible joints used for the Stewart platform" >}}
<a id="orgb093d6d"></a>
<a id="org0274a43"></a>
{{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="Figure 3: Stewart platform" >}}
## Bibliography {#bibliography}
<a id="orgce12c3f"></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="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>.

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@@ -8,7 +8,7 @@ Tags
: [Complementary Filters]({{< relref "complementary_filters" >}}), [Virtual Sensor Fusion]({{< relref "virtual_sensor_fusion" >}})
Reference
: ([Saxena and Hote 2012](#org7fbcfc6))
: ([Saxena and Hote 2012](#org0284b71))
Author(s)
: Saxena, S., & Hote, Y.
@@ -87,4 +87,4 @@ The interesting feature regarding IMC is that the design scheme is identical to
## Bibliography {#bibliography}
<a id="org7fbcfc6"></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="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>.

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Reference
: ([Sayed and Kailath 2001](#org48c8405))
: ([Sayed and Kailath 2001](#orgaf03ac4))
Author(s)
: Sayed, A. H., & Kailath, T.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<a id="org48c8405"></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="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>.

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@@ -8,7 +8,7 @@ Tags
: [Precision Engineering]({{< relref "precision_engineering" >}})
Reference
: ([Schellekens et al. 1998](#org5a7db02))
: ([Schellekens et al. 1998](#orge27f1a2))
Author(s)
: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
<a id="org5a7db02"></a>Schellekens, P., N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril. 1998. “Design for Precision: Current Status and Trends.” _Cirp Annals_, no. 2:557–86. <https://doi.org/10.1016/s0007-8506(07)>63243-0.
<a id="orge27f1a2"></a>Schellekens, P., N. Rosielle, H. Vermeulen, M. Vermeulen, S. Wetzels, and W. Pril. 1998. “Design for Precision: Current Status and Trends.” _Cirp Annals_, no. 2:557–86. <https://doi.org/10.1016/s0007-8506(07)>63243-0.

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Reference
: ([Schroeck, Messner, and McNab 2001](#orgd39dd43))
: ([Schroeck, Messner, and McNab 2001](#orgf8182bc))
Author(s)
: Schroeck, S., Messner, W., & McNab, R.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<a id="orgd39dd43"></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="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>.

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: [Sensor Fusion]({{< relref "sensor_fusion" >}})
Reference
: ([Sebastian and Pantazi 2012](#org3eeb20a))
: ([Sebastian and Pantazi 2012](#org03bb39f))
Author(s)
: Sebastian, A., & Pantazi, A.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
<a id="org3eeb20a"></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="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>.

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content/article/souleille18_concep_activ_mount_space_applic.md
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@@ -8,7 +8,7 @@ Tags
: [Active Damping]({{< relref "active_damping" >}})
Reference
: ([Souleille et al. 2018](#orgaa465de))
: ([Souleille et al. 2018](#org5546d0c))
Author(s)
: 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}
Figure [1](#org024c118) shows a picture of the amplified piezoelectric stack.
Figure [1](#org8634178) 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.
<a id="org024c118"></a>
<a id="org8634178"></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" >}}
@@ -61,38 +61,38 @@ and the control force is given by:
f = F\_s G(s) = F\_s \frac{g}{s}
\end{equation}
The effect of the controller are shown in Figure [2](#orge334eeb):
The effect of the controller are shown in Figure [2](#orgcb733df):
- the resonance peak is almost critically damped
- 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 fraction of the force transmitted to the payload that is measured by the force sensor is reduced at low frequencies
<a id="orge334eeb"></a>
<a id="orgcb733df"></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)" >}}
<a id="orgcccf310"></a>
<a id="orga434456"></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" >}}
## 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](#orga53fff7)).
A heavy payload is mounted on a set of three isolators (Figure [4](#org09ac00a)).
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="orga53fff7"></a>
<a id="org09ac00a"></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" >}}
As shown in Figure [5](#orge070a1d), both the suspension modes and the flexible modes of the payload can be critically damped.
As shown in Figure [5](#org2dcbc51), both the suspension modes and the flexible modes of the payload can be critically damped.
<a id="orge070a1d"></a>
<a id="org2dcbc51"></a>
{{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="Figure 5: Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}}
## Bibliography {#bibliography}
<a id="orgaa465de"></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="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.

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content/article/spanos95_soft_activ_vibrat_isolat.md
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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Spanos, Rahman, and Blackwood 1995](#orgad76892))
: ([Spanos, Rahman, and Blackwood 1995](#orgfa35a11))
Author(s)
: Spanos, J., Rahman, Z., & Blackwood, G.
@@ -16,14 +16,14 @@ Author(s)
Year
: 1995
**Stewart Platform** (Figure [1](#org6d10ec2)):
**Stewart Platform** (Figure [1](#org04a652d)):
- Voice Coil
- Flexible joints (cross-blades)
- Force Sensors
- Cubic Configuration
<a id="org6d10ec2"></a>
<a id="org04a652d"></a>
{{< figure src="/ox-hugo/spanos95_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
@@ -41,7 +41,7 @@ After redesign of the struts:
- low frequency zero at 2.6Hz but non-minimum phase (not explained).
Small viscous damping material in the cross blade flexures made the zero minimum phase again.
<a id="orgfef5d56"></a>
<a id="org0b2816d"></a>
{{< figure src="/ox-hugo/spanos95_iff_plant.png" caption="Figure 2: Experimentally measured transfer function from voice coil drive voltage to collocated load cell output voltage" >}}
@@ -52,13 +52,13 @@ The controller used consisted of:
- first order lag filter to provide adequate phase margin at the low frequency crossover
- a first order high pass filter to attenuate the excess gain resulting from the low frequency zero
The results in terms of transmissibility are shown in Figure [3](#orgec62915).
The results in terms of transmissibility are shown in Figure [3](#org3083c42).
<a id="orgec62915"></a>
<a id="org3083c42"></a>
{{< figure src="/ox-hugo/spanos95_results.png" caption="Figure 3: Experimentally measured Frobenius norm of the 6-axis transmissibility" >}}
## Bibliography {#bibliography}
<a id="orgad76892"></a>Spanos, J., Z. Rahman, and G. Blackwood. 1995. “A Soft 6-Axis Active Vibration Isolator.” In _Proceedings of 1995 American Control Conference - ACC’95_, nil. <https://doi.org/10.1109/acc.1995.529280>.
<a id="orgfa35a11"></a>Spanos, J., Z. Rahman, and G. Blackwood. 1995. “A Soft 6-Axis Active Vibration Isolator.” In _Proceedings of 1995 American Control Conference - ACC’95_, nil. <https://doi.org/10.1109/acc.1995.529280>.

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content/article/stankevic17_inter_charac_rotat_stages_x_ray_nanot.md
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@@ -8,7 +8,7 @@ Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
Reference
: ([Stankevic et al. 2017](#org7b0b97a))
: ([Stankevic et al. 2017](#org5a36a6f))
Author(s)
: Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …
@@ -19,7 +19,7 @@ Year
- Similar Station than the NASS
- Similar Metrology with fiber based interferometers and cylindrical reference mirror
<a id="org89f4881"></a>
<a id="orgbfe970a"></a>
{{< figure src="/ox-hugo/stankevic17_station.png" caption="Figure 1: Positioning Station" >}}
@@ -32,4 +32,4 @@ Year
## Bibliography {#bibliography}
<a id="org7b0b97a"></a>Stankevic, Tomas, Christer Engblom, Florent Langlois, Filipe Alves, Alain Lestrade, Nicolas Jobert, Gilles Cauchon, Ulrich Vogt, and Stefan Kubsky. 2017. “Interferometric Characterization of Rotation Stages for X-Ray Nanotomography.” _Review of Scientific Instruments_ 88 (5):053703. <https://doi.org/10.1063/1.4983405>.
<a id="org5a36a6f"></a>Stankevic, Tomas, Christer Engblom, Florent Langlois, Filipe Alves, Alain Lestrade, Nicolas Jobert, Gilles Cauchon, Ulrich Vogt, and Stefan Kubsky. 2017. “Interferometric Characterization of Rotation Stages for X-Ray Nanotomography.” _Review of Scientific Instruments_ 88 (5):053703. <https://doi.org/10.1063/1.4983405>.

4
content/article/tang18_decen_vibrat_contr_voice_coil.md
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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference
: ([Tang, Cao, and Yu 2018](#orge54a32c))
: ([Tang, Cao, and Yu 2018](#org45ebb6f))
Author(s)
: Tang, J., Cao, D., & Yu, T.
@@ -19,4 +19,4 @@ Year
## Bibliography {#bibliography}
<a id="orge54a32c"></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="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>.

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content/article/tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip.md
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@@ -8,7 +8,7 @@ Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Tjepkema, Dijk, and Soemers 2012](#org04bf993))
: ([Tjepkema, Dijk, and Soemers 2012](#org349e155))
Author(s)
: Tjepkema, D., Dijk, J. v., & Soemers, H.
@@ -49,4 +49,4 @@ There is a compromise between sensor noise and the influence of the sensor size
## Bibliography {#bibliography}
<a id="org04bf993"></a>Tjepkema, D., J. van Dijk, and H.M.J.R. Soemers. 2012. “Sensor Fusion for Active Vibration Isolation in Precision Equipment.” _Journal of Sound and Vibration_ 331 (4):735–49. <https://doi.org/10.1016/j.jsv.2011.09.022>.
<a id="org349e155"></a>Tjepkema, D., J. van Dijk, and H.M.J.R. Soemers. 2012. “Sensor Fusion for Active Vibration Isolation in Precision Equipment.” _Journal of Sound and Vibration_ 331 (4):735–49. <https://doi.org/10.1016/j.jsv.2011.09.022>.

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content/article/wang12_autom_marker_full_field_hard.md
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@@ -8,7 +8,7 @@ Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}})
Reference
: ([Wang et al. 2012](#orgc06402b))
: ([Wang et al. 2012](#org187cf70))
Author(s)
: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M.
@@ -28,4 +28,4 @@ It uses calibrated metrology disc and capacitive sensors
## Bibliography {#bibliography}
<a id="orgc06402b"></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="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>.

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content/article/wang16_inves_activ_vibrat_isolat_stewar.md
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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: ([Wang et al. 2016](#orgf512901))
: ([Wang et al. 2016](#orgda82aa7))
Author(s)
: 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 proposed model is thus effective.
<a id="orgce42532"></a>
<a id="orgc0b4cf5"></a>
{{< 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
- direct feedback of integrated forces => dampen vibration of inherent modes and thus reduce random vibrations
Force Feedback (Figure [2](#org6f9719a)).
Force Feedback (Figure [2](#orgaf56f94)).
- the force sensor is mounted **between the base and the strut**
<a id="org6f9719a"></a>
<a id="orgaf56f94"></a>
{{< figure src="/ox-hugo/wang16_force_feedback.png" caption="Figure 2: Feedback of integrated forces in the platform" >}}
@@ -56,4 +56,4 @@ Sorts of HAC-LAC control:
## Bibliography {#bibliography}
<a id="orgf512901"></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="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>.

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content/article/yang19_dynam_model_decoup_contr_flexib.md
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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
Reference
: ([Yang et al. 2019](#org1678fd1))
: ([Yang et al. 2019](#org8fbcee2))
Author(s)
: Yang, X., Wu, H., Chen, B., Kang, S., & Cheng, S.
@@ -25,23 +25,23 @@ Year
The joint stiffness impose a limitation on the control performance using force sensors as it adds a zero at low frequency in the dynamics.
Thus, this stiffness is taken into account in the dynamics and compensated for.
**Stewart platform** (Figure [1](#org082a4f7)):
**Stewart platform** (Figure [1](#org006c2df)):
- piezoelectric actuators
- flexible joints (Figure [2](#org66efbec))
- flexible joints (Figure [2](#org8725dbf))
- force sensors (used for vibration isolation)
- displacement sensors (used to decouple the dynamics)
- cubic (even though not said explicitly)
<a id="org082a4f7"></a>
<a id="org006c2df"></a>
{{< figure src="/ox-hugo/yang19_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
<a id="org66efbec"></a>
<a id="org8725dbf"></a>
{{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="Figure 2: Flexible Joints" >}}
The stiffness of the flexible joints (Figure [2](#org66efbec)) are computed with an FEM model and shown in Table [1](#table--tab:yang19-stiffness-flexible-joints).
The stiffness of the flexible joints (Figure [2](#org8725dbf)) are computed with an FEM model and shown in Table [1](#table--tab:yang19-stiffness-flexible-joints).
<a id="table--tab:yang19-stiffness-flexible-joints"></a>
<div class="table-caption">
@@ -105,9 +105,9 @@ In order to apply this control strategy:
- The jacobian has to be computed
- No information about modal matrix is needed
The block diagram of the control strategy is represented in Figure [3](#orgc6324f9).
The block diagram of the control strategy is represented in Figure [3](#org820f661).
<a id="orgc6324f9"></a>
<a id="org820f661"></a>
{{< figure src="/ox-hugo/yang19_control_arch.png" caption="Figure 3: Control Architecture used" >}}
@@ -121,10 +121,10 @@ Substituting \\(H(s)\\) in the equation of motion gives that:
**Experimental Validation**:
An external Shaker is used to excite the base and accelerometers are located on the base and mobile platforms to measure their motion.
The results are shown in Figure [4](#org45c63bf).
The results are shown in Figure [4](#org990744b).
In theory, the vibration performance can be improved, however in practice, increasing the gain causes saturation of the piezoelectric actuators and then the instability occurs.
<a id="org45c63bf"></a>
<a id="org990744b"></a>
{{< figure src="/ox-hugo/yang19_results.png" caption="Figure 4: Frequency response of the acceleration ratio between the paylaod and excitation (Transmissibility)" >}}
@@ -136,4 +136,4 @@ In theory, the vibration performance can be improved, however in practice, incre
## Bibliography {#bibliography}
<a id="org1678fd1"></a>Yang, XiaoLong, HongTao Wu, Bai Chen, ShengZheng Kang, and ShiLi Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” _Journal of Sound and Vibration_ 439 (January). Elsevier BV:398–412. <https://doi.org/10.1016/j.jsv.2018.10.007>.
<a id="org8fbcee2"></a>Yang, XiaoLong, HongTao Wu, Bai Chen, ShengZheng Kang, and ShiLi Cheng. 2019. “Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation.” _Journal of Sound and Vibration_ 439 (January). Elsevier BV:398–412. <https://doi.org/10.1016/j.jsv.2018.10.007>.

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@@ -9,7 +9,7 @@ Tags
Reference
: ([Yun et al. 2020](#orga992678))
: ([Yun et al. 2020](#orgd3a0930))
Author(s)
: Yun, H., Liu, L., Li, Q., & Yang, H.
@@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<a id="orga992678"></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="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>.

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content/article/zhang11_six_dof.md
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@@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Zhang et al. 2011](#org9aff0f7))
: ([Zhang et al. 2011](#orgcea0d62))
Author(s)
: Zhang, Z., Liu, J., Mao, J., Guo, Y., & Ma, Y.
@@ -25,11 +25,11 @@ Year
- **Accelerometers** for active isolation
- Adaptive FIR filters for active isolation control
<a id="org7c06a5a"></a>
<a id="orgd24035d"></a>
{{< figure src="/ox-hugo/zhang11_platform.png" caption="Figure 1: Prototype of the non-cubic stewart platform" >}}
## Bibliography {#bibliography}
<a id="org9aff0f7"></a>Zhang, Zhen, J Liu, Jq Mao, Yx Guo, and Yh Ma. 2011. “Six DOF Active Vibration Control Using Stewart Platform with Non-Cubic Configuration.” In _2011 6th IEEE Conference on Industrial Electronics and Applications_, nil. <https://doi.org/10.1109/iciea.2011.5975679>.
<a id="orgcea0d62"></a>Zhang, Zhen, J Liu, Jq Mao, Yx Guo, and Yh Ma. 2011. “Six DOF Active Vibration Control Using Stewart Platform with Non-Cubic Configuration.” In _2011 6th IEEE Conference on Industrial Electronics and Applications_, nil. <https://doi.org/10.1109/iciea.2011.5975679>.

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content/article/zuo04_elemen_system_desig_activ_passiv_vibrat_isolat.md
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@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: ([Zuo 2004](#org09e672c))
: ([Zuo 2004](#orgdb2a627))
Author(s)
: Zuo, L.
@@ -26,23 +26,23 @@ Year
> 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.
<a id="org367df78"></a>
<a id="org8018206"></a>
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}}
<a id="orgbb59454"></a>
<a id="org8874676"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}}
<a id="orgb7c9f25"></a>
<a id="orga7046e2"></a>
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}}
<a id="org740bc33"></a>
<a id="org735f298"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}}
## Bibliography {#bibliography}
<a id="org09e672c"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.
<a id="orgdb2a627"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.