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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](#org9ef6ccc))
: ([Alkhatib and Golnaraghi 2003](#orgdec9959))
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](#org3926a79).
A general block diagram of the control system is shown figure [1](#orgd2fc896).
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="org3926a79"></a>
<a id="orgd2fc896"></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="org07db29f"></a>
<a id="org4678494"></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](#org07db29f), the equation of motion of the system is:
Consider the control system figure [2](#org4678494), 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:
@ -225,6 +225,7 @@ The problem of optimizing the locations of the actuators can be more significant
If the actuator is placed at the wrong location, the system will require a greater force control. In that case, the system is said to have a **low degree of controllability**.
## Bibliography {#bibliography}
<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>.
<a id="orgdec9959"></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](#org50b9640))
: ([Bibel and Malyevac 1992](#org395ccd3))
Author(s)
: Bibel, J. E., & Malyevac, D. S.
@ -19,11 +19,11 @@ Year
## Properties of feedback control {#properties-of-feedback-control}
<a id="org1554fcc"></a>
<a id="orgd464a3c"></a>
{{< figure src="/ox-hugo/bibel92_control_diag.png" caption="Figure 1: Control System Diagram" >}}
From the figure [1](#org1554fcc), we have:
From the figure [1](#orgd464a3c), 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="orgce70b5f"></a>
<a id="orgf088f75"></a>
{{< figure src="/ox-hugo/bibel92_general_plant.png" caption="Figure 2: \\(\mathcal{H}\_\infty\\) control framework" >}}
New design framework (figure [2](#orgce70b5f)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
New design framework (figure [2](#orgf088f75)): \\(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](#orgca469d2)).
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](#orgff0b295)).
<a id="orgca469d2"></a>
<a id="orgff0b295"></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](#org96cc166)).
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](#orgc150230)).
<a id="org96cc166"></a>
<a id="orgc150230"></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](#orgab966f3).
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](#org42e3b7d).
<a id="orgab966f3"></a>
<a id="org42e3b7d"></a>
{{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="Figure 5: Example of unmodeled dynamics weight" >}}
@ -182,6 +182,7 @@ Typically actuator input weights are constant over frequency and set at the inve
**The order of the weights should be kept reasonably low** to reduce the order of th resulting optimal compensator and avoid potential convergence problems in the DK interactions.
## Bibliography {#bibliography}
<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.
<a id="org395ccd3"></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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@ -5,10 +5,11 @@ draft = false
+++
Tags
: [HAC-HAC]({{< relref "hac_hac" >}})
:
Reference
: ([Bryson 1993](#org93399a5))
: ([Bryson 1993](#org14ecce3))
Author(s)
: Bryson, A. E.
@ -19,6 +20,8 @@ Year
## 9.2.3 Roll-Off Filters {#9-dot-2-dot-3-roll-off-filters}
[Spillover Effect]({{< relref "spillover_effect" >}})
> Synthesizing control logic using only one vibration mode means we are consciously **neglecting the higher-order vibration modes**.
> When doing this, it is a good idea to insert "roll-off" into the control logic, so that the loop-transfer gain decreases rapidly with frequency beyond the control bandwidth.
> This reduces the possibility of destabilizing the unmodelled higher frequency dynamics ("**spillover**").
@ -35,19 +38,20 @@ Year
> If a rate sensor is not co-located with an actuator on a flexible body, ans its signal is fed back to the actuator, some vibration modes are stabilized and others are destabilized, depending on the location of the sensor relative to the actuator.
## 9.5.2 Low-Authority Control/High-Authority Control {#9-dot-5-dot-2-low-authority-control-high-authority-control}
## 9.5.2 Low-Authority Control/High-Authority Control [HAC-HAC]({{< relref "hac_hac" >}}) {#9-dot-5-dot-2-low-authority-control-high-authority-control-hac-hac--hac-hac-dot-md}
> Figure [fig:bryson93_hac_lac](#fig:bryson93_hac_lac) shows the concept of Low-Authority Control/High-Authority Control (LAC/HAC) is the s-plane.
> 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="orgee09f4a"></a>
<a id="orgc9c1915"></a>
{{< figure src="/ox-hugo/bryson93_hac_lac.png" caption="Figure 1: HAC-LAC control concept" >}}
> LAC/HAC is usually insensitive to small deviation of the plant dynamics away from the design values, that is, it is **robust** to plant parameter changes.
## Bibliography {#bibliography}
<a id="org93399a5"></a>Bryson, Arthur Earl. 1993. _Control of Spacecraft and Aircraft_. Princeton university press Princeton, New Jersey.
<a id="org14ecce3"></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](#org10c4eb8))
: ([Butler 2011](#org338ffef))
Author(s)
: Butler, H.
@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="org338ffef"></a>Butler, Hans. 2011. “Position Control in Lithographic Equipment.” _IEEE Control Systems_ 31 (5):28–47. <https://doi.org/10.1109/mcs.2011.941882>.

10
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](#orgfcb31bd))
: ([Chen and McInroy 2000](#org1c74a9c))
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](#orgaddeeaf)) ([Notes]({{< relref "mcinroy99_dynam" >}})).
The derivation of the dynamic model is done in ([McInroy 1999](#orgebf33dd)) ([Notes]({{< relref "mcinroy99_dynam" >}})).
<a id="org3326cce"></a>
<a id="orga594879"></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" >}}
@ -103,6 +103,6 @@ where
## Bibliography {#bibliography}
<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="org1c74a9c"></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="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>.
<a id="orgebf33dd"></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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@ -5,11 +5,10 @@ draft = false
+++
Tags
:
: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
Reference
: ([Claeyssen et al. 2007](#org8c33a63))
: ([Claeyssen et al. 2007](#org66395f6))
Author(s)
: Claeyssen, F., Letty, R. L., Barillot, F., & Sosnicki, O.
@ -35,6 +34,7 @@ The maximum dynamic force achievable by the actuator is determined by the prestr
The prestress design allows a peak force equal to half the blocked force.
## Bibliography {#bibliography}
<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>.
<a id="org66395f6"></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](#org7b1a927))
: ([Collette, Janssens, and Artoos 2011](#orgc3712d7))
Author(s)
: Collette, C., Janssens, S., & Artoos, K.
@ -22,12 +22,11 @@ Year
### Passive Isolation Tradeoffs {#passive-isolation-tradeoffs}
<div class="cbox">
<div></div>
1DoF Equations:
\\[ X(s) = \underbrace{\frac{cs + k}{ms^2 + cs + k}}\_{T\_{wx}(s)} W(s) + \underbrace{\frac{1}{ms^2 + cs + k}}\_{T\_{Fx}(s)} F(s) \\]
</div>
\begin{equation}
\boxed{X(s) = \underbrace{\frac{cs + k}{ms^2 + cs + k}}\_{T\_{wx}(s)} W(s) + \underbrace{\frac{1}{ms^2 + cs + k}}\_{T\_{Fx}(s)} F(s)}
\end{equation}
- \\(T\_{wx}(s)\\) is called the **transmissibility** of the isolator. It characterize the way seismic vibrations \\(w\\) are transmitted to the equipment.
- \\(T\_{Fx}(s)\\) is called the **compliance**. It characterize the capacity of disturbing forces \\(F\\) to create motion \\(x\\) of the equipment.
@ -71,11 +70,12 @@ The general expression of the force delivered by the actuator is \\(f = g\_a \dd
## Conclusions {#conclusions}
<a id="org0035120"></a>
<a id="orgdceedb5"></a>
{{< figure src="/ox-hugo/collette11_comp_isolation_strategies.png" caption="Figure 1: Comparison of Active Vibration Isolation Strategies" >}}
## Bibliography {#bibliography}
<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>.
<a id="orgc3712d7"></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](#orged1e033))
: ([Collette and Matichard 2014](#org6b92a7c))
Author(s)
: Collette, C., & Matichard, F.
@ -19,7 +19,7 @@ Year
## Introduction {#introduction}
Sensor fusion is used to combine the benefits of different types of sensors:
[Sensor Fusion]({{< relref "sensor_fusion" >}}) is used to combine the benefits of different types of sensors:
- Relative sensor for DC positioning capability at low frequency
- Inertial sensors for isolation at high frequency
@ -44,7 +44,7 @@ In this paper, three types of sensors are used. Their advantages and disadvantag
| Sensors | Advantages | Disadvantages |
|------------------|----------------------------------|---------------------------------------|
| Relative motion | Servo-position | No isolation from gorund motion |
| Relative motion | Servo-position | No isolation from ground motion |
| Force sensors | Improve isolation | Increase compliance |
| Inertial sensors | Improve isolation and compliance | AC couple and noisy at high frequency |
@ -100,6 +100,7 @@ Three types of sensors have been considered for the high frequency part of the f
- The fusion with a **force sensor** can be used to increase the loop gain with little effect on the compliance and passive isolation, provided that the blend is possible and that no active damping of flexible modes is required.
## Bibliography {#bibliography}
<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)_.
<a id="org6b92a7c"></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](#org5c9aaf9))
: ([Collette and Matichard 2015](#orgdf378e9))
Author(s)
: Collette, C., & Matichard, F.
@ -16,7 +16,7 @@ Author(s)
Year
: 2015
In order to have good stability margins, it is common practice to collocate sensors and actuators. This ensures alternating poles and zeros along the imaginary axis. Then, each phase lag introduced by the poles is compensed by phase leag introduced by the zeroes. This guarantees stability and such system is referred to as **hyperstable**.
In order to have good stability margins, it is common practice to collocate sensors and actuators. This ensures alternating poles and zeros along the imaginary axis. Then, each phase lag introduced by the poles is compensated by phase lead introduced by the zeroes. This guarantees stability and such system is referred to as **hyperstable**.
In this paper, we study and compare different sensor fusion methods combining inertial sensors at low frequency with sensors adding stability at high frequency.
The stability margins of the controller can be significantly increased with no or little effect on the low-frequency active isolation, provided that the two following conditions are fulfilled:
@ -25,6 +25,7 @@ The stability margins of the controller can be significantly increased with no o
- there exists a bandwidth where we can superimpose the open loop transfer functions obtained with the two sensors.
## Bibliography {#bibliography}
<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>.
<a id="orgdf378e9"></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](#orgb54a8e7))
: ([Dasgupta and Mruthyunjaya 2000](#org9c198f3))
Author(s)
: Dasgupta, B., & Mruthyunjaya, T.
@ -37,4 +37,4 @@ The generalized Stewart platforms consists of two rigid bodies (referred to as t
## Bibliography {#bibliography}
<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.
<a id="org9c198f3"></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](#org4f17944))
: ([Devasia, Eleftheriou, and Moheimani 2007](#orgfa66307))
Author(s)
: Devasia, S., Eleftheriou, E., & Moheimani, S. R.
@ -18,15 +18,16 @@ Year
: 2007
- Talks about Scanning Tunneling Microscope (STM) and Scanning Probe Microscope (SPM)
- Piezoelectric actuators: Creep, Hysteresis, Vibrations, Modeling errors
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}): Creep, Hysteresis, Vibrations, Modeling errors
- Interesting analysis about Bandwidth-Precision-Range tradeoffs
- Control approaches for piezoelectric actuators: feedforward, Feedback, Iterative, Sensorless controls
<a id="org47d33b5"></a>
<a id="orgd34b44a"></a>
{{< figure src="/ox-hugo/devasia07_piezoelectric_tradeoff.png" caption="Figure 1: Tradeoffs between bandwidth, precision and range" >}}
## Bibliography {#bibliography}
<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.
<a id="orgfa66307"></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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@ -4,16 +4,11 @@ author = ["Thomas Dehaeze"]
draft = false
+++
Backlinks:
- [Force Sensors]({{< relref "force_sensors" >}})
- [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}})
Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Force Sensors]({{< relref "force_sensors" >}})
Reference
: ([Fleming 2010](#org28aeac7))
: ([Fleming 2010](#org21788cf))
Author(s)
: Fleming, A.
@ -36,7 +31,7 @@ Year
## Model of a multi-layer monolithic piezoelectric stack actuator {#model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator}
<a id="org7145764"></a>
<a id="org699947b"></a>
{{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="Figure 1: Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
@ -121,11 +116,12 @@ The capacitance of a piezoelectric stack is typically between \\(1 \mu F\\) and
## Tested feedback control strategies {#tested-feedback-control-strategies}
<a id="org9ecbbcf"></a>
<a id="orgc6b14a0"></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="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>.
<a id="org21788cf"></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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content/article/fleming12_estim.md
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@ -9,7 +9,7 @@ Tags
Reference
: ([Fleming 2012](#orga5f3c24))
: ([Fleming 2012](#orgc7d7404))
Author(s)
: Fleming, A. J.
@ -18,6 +18,7 @@ Year
: 2012
## Bibliography {#bibliography}
<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>.
<a id="orgc7d7404"></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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@ -8,7 +8,7 @@ Tags
: [Position Sensors]({{< relref "position_sensors" >}})
Reference
: ([Fleming 2013](#org336a947))
: ([Fleming 2013](#org687716f))
Author(s)
: Fleming, A. J.
@ -33,7 +33,7 @@ Usually quoted as a percentage of the fill-scale range (FSR):
With \\(e\_m(v)\\) is the mapping error.
<a id="org3c27d5a"></a>
<a id="org0a1d321"></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." >}}
@ -42,7 +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.
<a id="org69dcb2d"></a>
<a id="orgc781e90"></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." >}}
@ -147,9 +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}\\).
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](#orge95682f)).
There is usually a trade-off between bandwidth and resolution (figure [3](#org86a5909)).
<a id="orge95682f"></a>
<a id="org86a5909"></a>
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}}
@ -185,4 +185,4 @@ A convenient method for reporting this ratio is in parts-per-million (ppm):
## Bibliography {#bibliography}
<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>.
<a id="org687716f"></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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@ -9,7 +9,7 @@ Tags
Reference
: ([Fleming, Teo, and Leang 2015](#orgd9b657d))
: ([Fleming, Teo, and Leang 2015](#org0b5cc88))
Author(s)
: Fleming, A. J., Teo, Y. R., & Leang, K. K.
@ -18,6 +18,7 @@ Year
: 2015
## Bibliography {#bibliography}
<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>.
<a id="org0b5cc88"></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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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference
: ([Furqan, Suhaib, and Ahmad 2017](#orgd310f71))
: ([Furqan, Suhaib, and Ahmad 2017](#org1144495))
Author(s)
: Furqan, M., Suhaib, M., & Ahmad, N.
@ -22,4 +22,4 @@ Lots of references.
## Bibliography {#bibliography}
<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>.
<a id="org1144495"></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](#org37254be))
: ([Furutani, Suzuki, and Kudoh 2004](#org9d14335))
Author(s)
: Furutani, K., Suzuki, M., & Kudoh, R.
@ -38,4 +38,4 @@ Then, it is fitted with 4th order polynomial and included in the control archite
## Bibliography {#bibliography}
<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>.
<a id="org9d14335"></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](#orgb71276a))
: ([Gao et al. 2015](#org07ae1a8))
Author(s)
: Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …
@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="org07ae1a8"></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](#orgae24e63))
: ([Garg 2007](#org18482cb))
Author(s)
: Garg, S.
@ -38,4 +38,4 @@ The control rate should be weighted appropriately in order to not saturate the s
## Bibliography {#bibliography}
<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>.
<a id="org18482cb"></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](#orgec21e7f))
: ([Geng et al. 1995](#orgb245b96))
Author(s)
: Geng, Z. J., Pan, G. G., Haynes, L. S., Wada, B. K., & Garba, J. A.
@ -16,11 +16,12 @@ Author(s)
Year
: 1995
<a id="org757d5ce"></a>
<a id="orgec71c1f"></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="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>.
<a id="orgb245b96"></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](#orgd0b61f4))
: ([Hanieh 2003](#org2d21f87))
Author(s)
: Hanieh, A. A.
@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<a id="orgd0b61f4"></a>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.
<a id="org2d21f87"></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](#orge4a516e))
: ([Hauge and Campbell 2004](#org186272b))
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="org59ac043"></a>
<a id="org37bf22a"></a>
{{< figure src="/ox-hugo/hauge04_stewart_platform.png" caption="Figure 1: Hexapod for active vibration isolation" >}}
**Stewart platform** (Figure [1](#org59ac043)):
**Stewart platform** (Figure [1](#org37bf22a)):
- Low corner frequency
- Large actuator stroke (\\(\pm5mm\\))
- Sensors in each strut (Figure [2](#org59c0ee0)):
- Sensors in each strut (Figure [2](#org8b97871)):
- three-axis load cell
- base and payload geophone in parallel with the struts
- LVDT
<a id="org59c0ee0"></a>
<a id="org8b97871"></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="org5b75feb"></a>
<a id="org1bec2a6"></a>
{{< figure src="/ox-hugo/hauge04_strut_model.png" caption="Figure 3: Strut model" >}}
@ -136,11 +136,12 @@ 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="org4a7177a"></a>
<a id="org0a496f7"></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="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.
<a id="org186272b"></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](#org76dfce6))
: ([Holler et al. 2012](#orgacde90c))
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="orgd10cf36"></a>
<a id="org03c494c"></a>
{{< figure src="/ox-hugo/holler12_station.png" caption="Figure 1: Schematic of the tomography setup" >}}
@ -42,4 +42,4 @@ Obtain position stability of 10nm (standard deviation).
## Bibliography {#bibliography}
<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>.
<a id="orgacde90c"></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](#org69e08df))
: ([Holterman and deVries 2005](#orge1d0ea7))
Author(s)
: Holterman, J., & deVries, T.
@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="orge1d0ea7"></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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@ -4,15 +4,11 @@ author = ["Thomas Dehaeze"]
draft = false
+++
Backlinks:
- [Actuators]({{< relref "actuators" >}})
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Actuators]({{< relref "actuators" >}})
Reference
: ([Ito and Schitter 2016](#orgbaa452e))
: ([Ito and Schitter 2016](#org3484be8))
Author(s)
: Ito, S., & Schitter, G.
@ -45,7 +41,7 @@ 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="orgcd249fb"></a>
<a id="org7e94abb"></a>
{{< figure src="/ox-hugo/ito16_low_high_stiffness_actuators.png" caption="Figure 1: Definition of low-stiffness and high-stiffness actuator" >}}
@ -58,7 +54,7 @@ In this paper, the piezoelectric actuator/electronics adds a time delay which is
## Controller Design {#controller-design}
<a id="orgde6bd83"></a>
<a id="org02696ae"></a>
{{< figure src="/ox-hugo/ito16_transmissibility.png" caption="Figure 2: Obtained transmissibility" >}}
@ -71,6 +67,7 @@ In practice, this is difficult to achieve with piezoelectric actuators as their
In contrast, the frequency band between the first and the other resonances of Lorentz actuators can be broad by design making them more suitable to construct a low-stiffness actuators.
## Bibliography {#bibliography}
<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>.
<a id="org3484be8"></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](#org72b03e6))
: ([Jiao et al. 2018](#org9f472e3))
Author(s)
: Jiao, J., Wu, Y., Yu, K., & Zhao, R.
@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="org9f472e3"></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](#orga1dea1c))
: ([Legnani et al. 2012](#orga1e3bf2))
Author(s)
: Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D.
@ -22,11 +22,11 @@ Year
Example of generated isotropic manipulator (not decoupled).
<a id="org7df2b7f"></a>
<a id="org0cc8ba8"></a>
{{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="Figure 1: Location of the leg axes using an isotropy generator" >}}
<a id="org4803974"></a>
<a id="org0474665"></a>
{{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="Figure 2: Isotropic configuration" >}}
@ -34,4 +34,4 @@ Example of generated isotropic manipulator (not decoupled).
## Bibliography {#bibliography}
<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>.
<a id="orga1e3bf2"></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](#org7799941))
: ([Li 2001](#orgb27ac3b))
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="org6847e54"></a>
<a id="org4770c80"></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](#org7d1b81f) or in Figure [3](#orgc8972f9), we obtain a decoupled plant provided that:
If we refine the (force) inputs and (displacement) outputs as shown in Figure [2](#org63bb176) or in Figure [3](#orgd88dae0), 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="org7d1b81f"></a>
<a id="org63bb176"></a>
{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 2: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
<a id="orgc8972f9"></a>
<a id="orgd88dae0"></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](#orgabd1d61).
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [4](#org51e3a97).
<a id="orgabd1d61"></a>
<a id="org51e3a97"></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](#orgabd1d61)
One of the subsystem plant transfer function is shown in Figure [4](#org51e3a97)
<a id="org5a1c162"></a>
<a id="org9c55360"></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](#orga9c8d31).
A block diagram of the pointing control system is shown in Figure [6](#org5fb43ce).
<a id="orga9c8d31"></a>
<a id="org5fb43ce"></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](#org78d2b0e).
A feedforward control is added as shown in Figure [7](#orgea70803).
<a id="org78d2b0e"></a>
<a id="orgea70803"></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](#org94ed578) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
Figure [8](#orgc0af645) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
<a id="org94ed578"></a>
<a id="orgc0af645"></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](#orgae8c117).
The effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [9](#orgdfaf4bf).
<a id="orgae8c117"></a>
<a id="orgdfaf4bf"></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](#org00362f7) and [11](#org77c322d))
- affect the closed loop transmission of the loop first closed (see Figures [10](#orge4bfb77) and [11](#org2bba93e))
As shown in Figure [10](#org00362f7), the peak resonance of the pointing loop increase after the isolation loop is closed.
As shown in Figure [10](#orge4bfb77), 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="org00362f7"></a>
<a id="orge4bfb77"></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](#org77c322d)).
The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [11](#org2bba93e)).
The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
<a id="org77c322d"></a>
<a id="org2bba93e"></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](#org5977cb3)):
Two hexapods are stacked (Figure [12](#orgd81de76)):
- the bottom hexapod is used to generate disturbances matching candidate applications
- the top hexapod provide simultaneous vibration isolation and pointing control
<a id="org5977cb3"></a>
<a id="orgd81de76"></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](#org4707691).
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [13](#orgd54b851).
<a id="org4707691"></a>
<a id="orgd54b851"></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](#orge4b1f73) where the bandwidth of the isolation control is expanded to very low frequency.
The results of simultaneous control is shown in Figure [14](#orgb3f240b) where the bandwidth of the isolation control is expanded to very low frequency.
<a id="orge4b1f73"></a>
<a id="orgb3f240b"></a>
{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 14: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
@ -219,4 +219,4 @@ Proposed future research areas include:
## Bibliography {#bibliography}
<a id="org7799941"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.
<a id="orgb27ac3b"></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](#org98b01e6))
: ([Li, Hamann, and McInroy 2001](#orgd01725e))
Author(s)
: Li, X., Hamann, J. C., & McInroy, J. E.
@ -19,6 +19,7 @@ Year
- if the hexapod is designed such that the payload mass/inertia matrix (\\(M\_x\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\\) to \\(y\\) are decoupled.
## Bibliography {#bibliography}
<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>.
<a id="orgd01725e"></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](#orge05f299))
: ([Lin and McInroy 2006](#org0bfd86d))
Author(s)
: Lin, H., & McInroy, J. E.
@ -18,6 +18,7 @@ Year
: 2006
## Bibliography {#bibliography}
<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>.
<a id="org0bfd86d"></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](#org6ce0f37))
: ([McInroy and Hamann 2000](#org0b673fd))
Author(s)
: McInroy, J., & Hamann, J.
@ -21,4 +21,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="org0b673fd"></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](#org48d21a1))
: ([McInroy 2002](#org2871bf9))
Author(s)
: McInroy, J.
@ -17,7 +17,7 @@ Author(s)
Year
: 2002
This short paper is very similar to ([McInroy 1999](#org287d886)).
This short paper is very similar to ([McInroy 1999](#org1d169f9)).
> 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](#org287d886)).
## Flexure Jointed Hexapod Dynamics {#flexure-jointed-hexapod-dynamics}
<a id="org66e9285"></a>
<a id="org4ea1e8b"></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](#org66e9285)).
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](#org4ea1e8b)).
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](#org66e9285) with:
The model of the strut are shown in Figure [1](#org4ea1e8b) 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="org343afcb"></a>
<a id="org5bc5fa8"></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](#org343afcb) depicts a strut, along with the corresponding force diagram.
Figure [2](#org5bc5fa8) 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](#org343afcb) (b), Newton's second law yields:
From Figure [2](#org5bc5fa8) (b), Newton's second law yields:
\begin{equation}
f\_p = \begin{bmatrix}
@ -269,8 +269,9 @@ By using the vector triple identity \\(a \cdot (b \times c) = b \cdot (c \times
\end{equation}
## Bibliography {#bibliography}
<a id="org287d886"></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="org1d169f9"></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="org48d21a1"></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="org2871bf9"></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
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: ([McInroy 1999](#org788f3dd))
: ([McInroy 1999](#orgc5d256d))
Author(s)
: McInroy, J.
@ -16,7 +16,7 @@ Author(s)
Year
: 1999
This conference paper has been further published in a journal as a short note ([McInroy 2002](#org6bd1808)).
This conference paper has been further published in a journal as a short note ([McInroy 2002](#orge25929e)).
## Abstract {#abstract}
@ -38,22 +38,22 @@ 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="orge71c3a4"></a>
<a id="org89aa8b3"></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](#orgc6987ef).
Since both actuator types employ force production in parallel with a spring, they can both be modeled as shown in Figure [2](#org0b2b1e5).
In order to provide low frequency passive vibration isolation, the hard actuators are sometimes placed in series with additional passive springs.
<a id="orgc6987ef"></a>
<a id="org0b2b1e5"></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="#orgc6987ef">2</a>
Definition of quantities on Figure <a href="#org0b2b1e5">2</a>
</div>
| **Symbol** | **Meaning** |
@ -70,11 +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 |
| \\(\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](#org6bd1808))).
It is here supposed that \\(f\_{p\_i}\\) is predominantly in the strut direction (explained in ([McInroy 2002](#orge25929e))).
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](#orgc6987ef) (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](#org0b2b1e5) (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
@ -165,6 +165,6 @@ In the next section, a connection between the two will be found to complete the
## Bibliography {#bibliography}
<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="orgc5d256d"></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="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>.
<a id="orge25929e"></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](#org93151b9))
: ([Oomen 2018](#org733608c))
Author(s)
: Oomen, T.
@ -16,7 +16,7 @@ Author(s)
Year
: 2018
<a id="orgcc3437a"></a>
<a id="orgee4800c"></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." >}}
@ -24,4 +24,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="org733608c"></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](#org913a900))
: ([Poel 2010](#orge212f89))
Author(s)
: van der Poel, G. W.
@ -17,6 +17,7 @@ Year
: 2010
## Bibliography {#bibliography}
<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>.
<a id="orge212f89"></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](#org35c3663))
: ([Preumont et al. 2002](#orgbec44eb))
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](#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
- For light payload (Figure [1](#orga040a9a)), the acceleration feedback gives larger damping on the higher mode.
- For heavy payload (Figure [2](#org1916ab2)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
<a id="org93ee8cd"></a>
<a id="orga040a9a"></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="orgec90b36"></a>
<a id="org1916ab2"></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" >}}
@ -46,6 +46,7 @@ The same is true for the transfer function from the force actuator to the relati
> According to physical interpretation of the zeros, they represent the resonances of the subsystem constrained by the sensor and the actuator.
## Bibliography {#bibliography}
<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>.
<a id="orgbec44eb"></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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@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: ([Preumont et al. 2007](#org66383b9))
: ([Preumont et al. 2007](#org003735a))
Author(s)
: Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …
@ -18,30 +18,30 @@ Year
Summary:
- **Cubic** Stewart platform (Figure [3](#orga6bbb17))
- **Cubic** Stewart platform (Figure [3](#org144c76e))
- Provides uniform control capability
- Uniform stiffness in all directions
- minimizes the cross-coupling among actuators and sensors of different legs
- Flexible joints (Figure [2](#orga37017c))
- Flexible joints (Figure [2](#org04bd941))
- 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](#orgec64e08))
- Effect of parasitic stiffness of the flexible joints on the IFF performance (Figure [1](#org06a63d6))
- 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="orgec64e08"></a>
<a id="org06a63d6"></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="orga37017c"></a>
<a id="org04bd941"></a>
{{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="Figure 2: Flexible joints used for the Stewart platform" >}}
<a id="orga6bbb17"></a>
<a id="org144c76e"></a>
{{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="Figure 3: Stewart platform" >}}
@ -49,4 +49,4 @@ Summary:
## Bibliography {#bibliography}
<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>.
<a id="org003735a"></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](#org6d87fa3))
: ([Saxena and Hote 2012](#org13b6614))
Author(s)
: Saxena, S., & Hote, Y.
@ -88,4 +88,4 @@ The interesting feature regarding IMC is that the design scheme is identical to
## Bibliography {#bibliography}
<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>.
<a id="org13b6614"></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
: ([Sayed and Kailath 2001](#org9b5be86))
: ([Sayed and Kailath 2001](#org337ccf9))
Author(s)
: Sayed, A. H., & Kailath, T.
@ -21,4 +21,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="org337ccf9"></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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content/article/schellekens98_desig_precis.md
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@ -8,7 +8,7 @@ Tags
: [Precision Engineering]({{< relref "precision_engineering" >}})
Reference
: ([Schellekens et al. 1998](#orge27f1a2))
: ([Schellekens et al. 1998](#org035ecc6))
Author(s)
: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W.
@ -17,6 +17,7 @@ Year
: 1998
## Bibliography {#bibliography}
<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.
<a id="org035ecc6"></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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@ -9,7 +9,7 @@ Tags
Reference
: ([Schroeck, Messner, and McNab 2001](#org5e2e067))
: ([Schroeck, Messner, and McNab 2001](#orga580bdc))
Author(s)
: Schroeck, S., Messner, W., & McNab, R.
@ -21,4 +21,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="orga580bdc"></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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content/article/sebastian12_nanop_with_multip_sensor.md
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@ -8,7 +8,7 @@ Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}})
Reference
: ([Sebastian and Pantazi 2012](#orgacf93b4))
: ([Sebastian and Pantazi 2012](#orge399d74))
Author(s)
: Sebastian, A., & Pantazi, A.
@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="orge399d74"></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](#org5548942))
: ([Souleille et al. 2018](#org34cc88f))
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](#org0952ca2) shows a picture of the amplified piezoelectric stack.
Figure [1](#org5ad28c2) 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="org0952ca2"></a>
<a id="org5ad28c2"></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" >}}
@ -61,34 +61,34 @@ 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](#orgb36ba37):
The effect of the controller are shown in Figure [2](#org985b671):
- 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="orgb36ba37"></a>
<a id="org985b671"></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="org5119227"></a>
<a id="orgd326cea"></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](#orgbe1030b)).
A heavy payload is mounted on a set of three isolators (Figure [4](#org62f47a3)).
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="orgbe1030b"></a>
<a id="org62f47a3"></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](#orgbb573b8), both the suspension modes and the flexible modes of the payload can be critically damped.
As shown in Figure [5](#org5b0f55b), both the suspension modes and the flexible modes of the payload can be critically damped.
<a id="orgbb573b8"></a>
<a id="org5b0f55b"></a>
{{< figure src="/ox-hugo/souleille18_result_damping_transmissibility.png" caption="Figure 5: Transmissibility between the table top \\(w\\) and \\(m\_1\\)" >}}
@ -96,4 +96,4 @@ As shown in Figure [5](#orgbb573b8), both the suspension modes and the flexible
## Bibliography {#bibliography}
<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.
<a id="org34cc88f"></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](#orgfa35a11))
: ([Spanos, Rahman, and Blackwood 1995](#org2800cc5))
Author(s)
: Spanos, J., Rahman, Z., & Blackwood, G.
@ -16,14 +16,14 @@ Author(s)
Year
: 1995
**Stewart Platform** (Figure [1](#org04a652d)):
**Stewart Platform** (Figure [1](#orgcac471d)):
- Voice Coil
- Flexible joints (cross-blades)
- Force Sensors
- Cubic Configuration
<a id="org04a652d"></a>
<a id="orgcac471d"></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="org0b2816d"></a>
<a id="org5cb89c4"></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 +52,14 @@ 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](#org3083c42).
The results in terms of transmissibility are shown in Figure [3](#orgd8726b9).
<a id="org3083c42"></a>
<a id="orgd8726b9"></a>
{{< figure src="/ox-hugo/spanos95_results.png" caption="Figure 3: Experimentally measured Frobenius norm of the 6-axis transmissibility" >}}
## Bibliography {#bibliography}
<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>.
<a id="org2800cc5"></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](#org5a36a6f))
: ([Stankevic et al. 2017](#org125690d))
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="orgbfe970a"></a>
<a id="org5481c46"></a>
{{< figure src="/ox-hugo/stankevic17_station.png" caption="Figure 1: Positioning Station" >}}
@ -30,6 +30,7 @@ Year
- Result: 40nm runout error
## Bibliography {#bibliography}
<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>.
<a id="org125690d"></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](#org44aa05e))
: ([Tang, Cao, and Yu 2018](#orgb3d3aa7))
Author(s)
: Tang, J., Cao, D., & Yu, T.
@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="orgb3d3aa7"></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>.

5
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](#org349e155))
: ([Tjepkema, Dijk, and Soemers 2012](#org06c1cb7))
Author(s)
: Tjepkema, D., Dijk, J. v., & Soemers, H.
@ -47,6 +47,7 @@ Heavier sensor => lower noise but it is harder to maintain collocation with the
There is a compromise between sensor noise and the influence of the sensor size on the system's design and on the control bandwidth.
## Bibliography {#bibliography}
<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>.
<a id="org06c1cb7"></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>.

4
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](#org172c2d6))
: ([Wang et al. 2012](#orgf2371c9))
Author(s)
: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M.
@ -29,4 +29,4 @@ It uses calibrated metrology disc and capacitive sensors
## Bibliography {#bibliography}
<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>.
<a id="orgf2371c9"></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](#org6a7c8f9))
: ([Wang et al. 2016](#org89f2008))
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="org5acb23c"></a>
<a id="org0d482b7"></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](#org25780ff)).
Force Feedback (Figure [2](#org1b645a1)).
- the force sensor is mounted **between the base and the strut**
<a id="org25780ff"></a>
<a id="org1b645a1"></a>
{{< figure src="/ox-hugo/wang16_force_feedback.png" caption="Figure 2: Feedback of integrated forces in the platform" >}}
@ -57,4 +57,4 @@ Sorts of HAC-LAC control:
## Bibliography {#bibliography}
<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>.
<a id="org89f2008"></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](#org8fbcee2))
: ([Yang et al. 2019](#orgb15122e))
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](#org006c2df)):
**Stewart platform** (Figure [1](#org479da8d)):
- piezoelectric actuators
- flexible joints (Figure [2](#org8725dbf))
- flexible joints (Figure [2](#org83afe99))
- force sensors (used for vibration isolation)
- displacement sensors (used to decouple the dynamics)
- cubic (even though not said explicitly)
<a id="org006c2df"></a>
<a id="org479da8d"></a>
{{< figure src="/ox-hugo/yang19_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
<a id="org8725dbf"></a>
<a id="org83afe99"></a>
{{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="Figure 2: 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).
The stiffness of the flexible joints (Figure [2](#org83afe99)) 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](#org820f661).
The block diagram of the control strategy is represented in Figure [3](#orgd526d94).
<a id="org820f661"></a>
<a id="orgd526d94"></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](#org990744b).
The results are shown in Figure [4](#orge73e046).
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="org990744b"></a>
<a id="orge73e046"></a>
{{< figure src="/ox-hugo/yang19_results.png" caption="Figure 4: Frequency response of the acceleration ratio between the paylaod and excitation (Transmissibility)" >}}
@ -134,6 +134,7 @@ In theory, the vibration performance can be improved, however in practice, incre
> The proportional and integral gains in the sub-controller are used to separately regulate the vibration isolation bandwidth and active damping simultaneously for the six vibration modes.
## Bibliography {#bibliography}
<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>.
<a id="orgb15122e"></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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content/article/yun20_inves_two_stage_vibrat_suppr.md
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@ -9,7 +9,7 @@ Tags
Reference
: ([Yun et al. 2020](#org2a2f02a))
: ([Yun et al. 2020](#org70fb5c6))
Author(s)
: Yun, H., Liu, L., Li, Q., & Yang, H.
@ -21,4 +21,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="org70fb5c6"></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](#orgcea0d62))
: ([Zhang et al. 2011](#org293b885))
Author(s)
: Zhang, Z., Liu, J., Mao, J., Guo, Y., & Ma, Y.
@ -25,11 +25,12 @@ Year
- **Accelerometers** for active isolation
- Adaptive FIR filters for active isolation control
<a id="orgd24035d"></a>
<a id="orgf49a13c"></a>
{{< figure src="/ox-hugo/zhang11_platform.png" caption="Figure 1: Prototype of the non-cubic stewart platform" >}}
## Bibliography {#bibliography}
<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>.
<a id="org293b885"></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](#orgd9c4f73))
: ([Zuo 2004](#orgf5a4502))
Author(s)
: Zuo, L.
@ -40,19 +40,19 @@ 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="orgc88908d"></a>
<a id="org9c33e29"></a>
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}}
<a id="orgb14b6c0"></a>
<a id="org141cdb3"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}}
<a id="org96eaaed"></a>
<a id="org2ed63ef"></a>
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}}
<a id="orgaf13ec6"></a>
<a id="orgc14af87"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}}
@ -60,4 +60,4 @@ Year
## Bibliography {#bibliography}
<a id="orgd9c4f73"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.
<a id="orgf5a4502"></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" >}})
Reference
: ([Albertos and Antonio 2004](#orgb06343d))
: ([Albertos and Antonio 2004](#orgbcb0991))
Author(s)
: Albertos, P., & Antonio, S.
@ -77,10 +77,10 @@ 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.
If sensors are multiplied by \\(U^T\\) and control actions multiplied by \\(V\\), as in Figure [1](#org335191d), 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>
<a id="org335191d"></a>
{{< figure src="/ox-hugo/albertos04_svd_decoupling.png" caption="Figure 1: SVD decoupling: \\(K\_D\\) is a diagonal controller designed for \\(\Sigma\\)" >}}
@ -129,4 +129,4 @@ If some of the vectors in \\(V\\) (input directions) have a significant componen
## Bibliography {#bibliography}
<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>.
<a id="orgbcb0991"></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" >}})
Reference
: ([Du and Xie 2010](#org5093366))
: ([Du and Xie 2010](#orge4a6b7f))
Author(s)
: Du, C., & Xie, L.
@ -536,4 +536,4 @@ Year
## Bibliography {#bibliography}
<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>.
<a id="orge4a6b7f"></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/du19_multi_actuat_system_contr.md
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@ -9,7 +9,7 @@ Tags
Reference
: ([Du and Pang 2019](#orgfa82dec))
: ([Du and Pang 2019](#orge6fd258))
Author(s)
: Du, C., & Pang, C. K.
@ -41,11 +41,11 @@ When high bandwidth, high position accuracy and long stroke are required simulta
Popular choices for coarse actuator are:
- DC motor
- Voice coil motor (VCM)
- [Voice Coil Motors]({{< relref "voice_coil_actuators" >}}) (VCM)
- Permanent magnet stepper motor
- Permanent magnet linear synchronous motor
As fine actuators, most of the time piezoelectric actuator are used.
As fine actuators, most of the time [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}) are used.
In order to overcome fine actuator stringent stroke limitation and increase control bandwidth, three-stage actuation systems are necessary in practical applications.
@ -81,9 +81,9 @@ and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following fo
#### Secondary Actuators {#secondary-actuators}
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org2501c9c)) and the microactuator.
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org5b375a0)) and the microactuator.
<a id="org2501c9c"></a>
<a id="org5b375a0"></a>
{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}}
@ -105,9 +105,9 @@ There characteristics are shown on table [1](#table--tab:microactuator).
### Single-Stage Actuation Systems {#single-stage-actuation-systems}
A typical closed-loop control system is shown on figure [2](#org3f9b6d4), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
A typical closed-loop control system is shown on figure [2](#orgeb3a161), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
<a id="org3f9b6d4"></a>
<a id="orgeb3a161"></a>
{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}}
@ -117,7 +117,7 @@ A typical closed-loop control system is shown on figure [2](#org3f9b6d4), where
Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.
<a id="org59cb446"></a>
<a id="org1f9ca75"></a>
{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}}
@ -143,7 +143,7 @@ In view of this, the controller design for dual-stage actuation systems adopts a
### Control Schemes {#control-schemes}
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#org371fae9).
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#orgd6f5782).
- \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
- \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
@ -151,7 +151,7 @@ A popular control scheme for dual-stage actuation system is the **decoupled stru
- \\(n\\) is the measurement noise
- \\(d\_u\\) stands for external vibration
<a id="org371fae9"></a>
<a id="orgd6f5782"></a>
{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}}
@ -173,14 +173,14 @@ The sensitivity functions of the VCM loop and the secondary actuator loop are
And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
Thus, the dual-stage system control design can be decoupled into two independent controller designs.
Another type of control scheme is the **parallel structure** as shown in figure [5](#org3d4cd09).
Another type of control scheme is the **parallel structure** as shown in figure [5](#org3fd07ea).
The open-loop transfer function from \\(pes\\) to \\(y\\) is
\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
\\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]
<a id="org3d4cd09"></a>
<a id="org3fd07ea"></a>
{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}}
@ -190,7 +190,7 @@ Because of the limited displacement range of the secondary actuator, the control
### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
\\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org6dcd465) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org7277927) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
@ -200,11 +200,11 @@ For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such tha
is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
<a id="org6dcd465"></a>
<a id="org7277927"></a>
{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="Figure 6: Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
Equation [1](#orgd1210d7) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
Equation [1](#org2fdb8dc) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
One form of \\(W(s)\\) is taken as
\begin{equation}
@ -217,16 +217,16 @@ The controller can then be synthesis using the linear matrix inequality (LMI) ap
The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
As seen in figure [7](#orgbe7f7d1), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
As seen in figure [7](#org038c9b4), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
<a id="orgbe7f7d1"></a>
<a id="org038c9b4"></a>
{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="Figure 7: Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
The sensitivity functions are shown in figure [8](#orgfed486e), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
The sensitivity functions are shown in figure [8](#orged3fc33), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
<a id="orgfed486e"></a>
<a id="orged3fc33"></a>
{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
@ -259,13 +259,13 @@ A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a P
### Control Strategy and Controller Design {#control-strategy-and-controller-design}
Figure [9](#orga4074a3) shows the control structure for the three-stage actuation system.
Figure [9](#orgd7a95ee) shows the control structure for the three-stage actuation system.
The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.
<a id="orga4074a3"></a>
<a id="orgd7a95ee"></a>
{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}}
@ -294,15 +294,15 @@ The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandw
The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
The open-loop frequency responses of the three stages are shown on figure [10](#org11a6581).
The open-loop frequency responses of the three stages are shown on figure [10](#org50938c4).
<a id="org11a6581"></a>
<a id="org50938c4"></a>
{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="Figure 10: Frequency response of the open-loop transfer function" >}}
The obtained sensitivity function is shown on figure [11](#org58e9561).
The obtained sensitivity function is shown on figure [11](#org47b7e75).
<a id="org58e9561"></a>
<a id="org47b7e75"></a>
{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="Figure 11: Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
@ -317,7 +317,7 @@ Otherwise, saturation will occur in the control loop and the control system perf
Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
For the three-stage control architecture as shown on figure [9](#orga4074a3), the position error is
For the three-stage control architecture as shown on figure [9](#orgd7a95ee), the position error is
\\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
The control signals and positions of the actuators are given by
@ -333,11 +333,11 @@ Higher bandwidth/higher level of disturbance generally means high stroke needed.
### Different Configurations of the Control System {#different-configurations-of-the-control-system}
A decoupled control structure can be used for the three-stage actuation system (see figure [12](#org066c259)).
A decoupled control structure can be used for the three-stage actuation system (see figure [12](#orgc297a81)).
The overall sensitivity function is
\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org7c7e2b1) and
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org77f79f5) and
\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
Denote the dual-stage open-loop transfer function as \\(G\_d\\)
@ -346,7 +346,7 @@ Denote the dual-stage open-loop transfer function as \\(G\_d\\)
The open-loop transfer function of the overall system is
\\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]
<a id="org066c259"></a>
<a id="orgc297a81"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}}
@ -358,9 +358,9 @@ The control signals and the positions of the three actuators are
u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
\end{align\*}
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org8c9fc90)).
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org10ab429)).
<a id="org8c9fc90"></a>
<a id="org10ab429"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="Figure 13: Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
@ -669,6 +669,7 @@ Using PZT elements as a sensor to deal with high-frequency vibration beyond the
As a more advanced concept, PZT elements being used as actuator and sensor simultaneously has also been addressed in this book with detailed scheme and controller design methodology for effective utilization.
## Bibliography {#bibliography}
<a id="orgfa82dec"></a>Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.
<a id="orge6fd258"></a>Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.

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@ -8,7 +8,7 @@ Tags
: [System Identification]({{< relref "system_identification" >}}), [Reference Books]({{< relref "reference_books" >}}), [Modal Analysis]({{< relref "modal_analysis" >}})
Reference
: ([Ewins 2000](#org300069f))
: ([Ewins 2000](#org15876a9))
Author(s)
: Ewins, D.
@ -159,9 +159,9 @@ Indeed, we shall see later how these predictions can be quite detailed, to the p
The main measurement technique studied are those which will permit to make **direct measurements of the various FRF** properties of the test structure.
The type of test best suited to FRF measurement is shown in figure [1](#orge1f0d37).
The type of test best suited to FRF measurement is shown in figure [1](#org8f0a8f0).
<a id="orge1f0d37"></a>
<a id="org8f0a8f0"></a>
{{< figure src="/ox-hugo/ewins00_modal_analysis_schematic.png" caption="Figure 1: Basic components of FRF measurement system" >}}
@ -231,11 +231,11 @@ Thus there is **no single modal analysis method**, but rater a selection, each b
One of the most widespread and useful approaches is known as the **single-degree-of-freedom curve-fit**, or often as the **circle fit** procedure.
This method uses the fact that **at frequencies close to a natural frequency**, the FRF can often be **approximated to that of a single degree-of-freedom system** plus a constant offset term (which approximately accounts for the existence of other modes).
This assumption allows us to use the circular nature of a modulus/phase polar plot of the frequency response function of a SDOF system (see figure [2](#orgecf1743)).
This assumption allows us to use the circular nature of a modulus/phase polar plot of the frequency response function of a SDOF system (see figure [2](#org703f940)).
This process can be **repeated** for each resonance individually until the whole curve has been analyzed.
At this stage, a theoretical regeneration of the FRF is possible using the set of coefficients extracted.
<a id="orgecf1743"></a>
<a id="org703f940"></a>
{{< figure src="/ox-hugo/ewins00_sdof_modulus_phase.png" caption="Figure 2: Curve fit to resonant FRF data" >}}
@ -270,10 +270,10 @@ Even though the same overall procedure is always followed, there will be a **dif
Theoretical foundations of modal testing are of paramount importance to its successful implementation.
The three phases through a typical theoretical vibration analysis progresses are shown on figure [3](#orge69e740).
The three phases through a typical theoretical vibration analysis progresses are shown on figure [3](#orga0bcee3).
Generally, we start with a description of the structure's physical characteristics (mass, stiffness and damping properties), this is referred to as the **Spatial model**.
<a id="orge69e740"></a>
<a id="orga0bcee3"></a>
{{< figure src="/ox-hugo/ewins00_vibration_analysis_procedure.png" caption="Figure 3: Theoretical route to vibration analysis" >}}
@ -295,7 +295,7 @@ Thus our response model will consist of a set of **frequency response functions
<div class="important">
<div></div>
As indicated in figure [3](#orge69e740), it is also possible to do an analysis in the reverse directly: from a description of the response properties (FRFs), we can deduce modal properties and the spatial properties: this is the **experimental route** to vibration analysis.
As indicated in figure [3](#orga0bcee3), it is also possible to do an analysis in the reverse directly: from a description of the response properties (FRFs), we can deduce modal properties and the spatial properties: this is the **experimental route** to vibration analysis.
</div>
@ -315,10 +315,10 @@ Three classes of system model will be described:
</div>
The basic model for the SDOF system is shown in figure [4](#org115fa16) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The basic model for the SDOF system is shown in figure [4](#org863f8fd) where \\(f(t)\\) and \\(x(t)\\) are general time-varying force and displacement response quantities.
The spatial model consists of a **mass** \\(m\\), a **spring** \\(k\\) and (when damped) either a **viscous dashpot** \\(c\\) or **hysteretic damper** \\(d\\).
<a id="org115fa16"></a>
<a id="org863f8fd"></a>
{{< figure src="/ox-hugo/ewins00_sdof_model.png" caption="Figure 4: Single degree-of-freedom system" >}}
@ -394,9 +394,9 @@ which is a single mode of vibration with a complex natural frequency having two
- **An imaginary or oscillatory part**
- **A real or decay part**
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [5](#orgb1d4a98)
The physical significance of these two parts is illustrated in the typical free response plot shown in figure [5](#org777e04b)
<a id="orgb1d4a98"></a>
<a id="org777e04b"></a>
{{< figure src="/ox-hugo/ewins00_sdof_response.png" caption="Figure 5: Oscillatory and decay part" >}}
@ -427,7 +427,7 @@ which is now complex, containing both magnitude and phase information:
All structures exhibit a degree of damping due to the **hysteresis properties** of the material(s) from which they are made.
A typical example of this effect is shown in the force displacement plot in figure [1](#org640f8a0) in which the **area contained by the loop represents the energy lost in one cycle of vibration** between the extremities shown.
A typical example of this effect is shown in the force displacement plot in figure [1](#orgf0a4ea9) in which the **area contained by the loop represents the energy lost in one cycle of vibration** between the extremities shown.
The maximum energy stored corresponds to the elastic energy of the structure at the point of maximum deflection.
The damping effect of such a component can conveniently be defined by the ratio of these two:
\\[ \tcmbox{\text{damping capacity} = \frac{\text{energy lost per cycle}}{\text{maximum energy stored}}} \\]
@ -440,13 +440,13 @@ The damping effect of such a component can conveniently be defined by the ratio
| ![](/ox-hugo/ewins00_material_histeresis.png) | ![](/ox-hugo/ewins00_dry_friction.png) | ![](/ox-hugo/ewins00_viscous_damper.png) |
|-----------------------------------------------|----------------------------------------|------------------------------------------|
| <a id="org640f8a0"></a> Material hysteresis | <a id="org915cf01"></a> Dry friction | <a id="orgc37b9e9"></a> Viscous damper |
| <a id="orgf0a4ea9"></a> Material hysteresis | <a id="org2134b3d"></a> Dry friction | <a id="org82f9a69"></a> Viscous damper |
| height=2cm | height=2cm | height=2cm |
Another common source of energy dissipation in practical structures, is the **friction** which exist in joints between components of the structure.
It may be described very roughly by the simple **dry friction model** shown in figure [1](#org915cf01).
It may be described very roughly by the simple **dry friction model** shown in figure [1](#org2134b3d).
The mathematical model of the **viscous damper** which we have used can be compared with these more physical effects by plotting the corresponding force-displacement diagram for it, and this is shown in figure [1](#orgc37b9e9).
The mathematical model of the **viscous damper** which we have used can be compared with these more physical effects by plotting the corresponding force-displacement diagram for it, and this is shown in figure [1](#org82f9a69).
Because the relationship is linear between force and velocity, it is necessary to suppose harmonic motion, at frequency \\(\omega\\), in order to construct a force-displacement diagram.
The resulting diagram shows the nature of the approximation provided by the viscous damper model and the concept of the **effective or equivalent viscous damping coefficient** for any of the actual phenomena as being which provides the **same energy loss per cycle** as the real thing.
@ -567,7 +567,7 @@ Bode plot are usually displayed using logarithmic scales as shown on figure [3](
| ![](/ox-hugo/ewins00_bode_receptance.png) | ![](/ox-hugo/ewins00_bode_mobility.png) | ![](/ox-hugo/ewins00_bode_accelerance.png) |
|-------------------------------------------|-----------------------------------------|--------------------------------------------|
| <a id="org98a8b4c"></a> Receptance FRF | <a id="orga72dec1"></a> Mobility FRF | <a id="org92b33d3"></a> Accelerance FRF |
| <a id="orgf6df26a"></a> Receptance FRF | <a id="org58db881"></a> Mobility FRF | <a id="org1c64176"></a> Accelerance FRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
Each plot can be divided into three regimes:
@ -590,7 +590,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
| ![](/ox-hugo/ewins00_plot_receptance_real.png) | ![](/ox-hugo/ewins00_plot_receptance_imag.png) |
|------------------------------------------------|------------------------------------------------|
| <a id="org731c2d0"></a> Real part | <a id="orgfb5a86a"></a> Imaginary part |
| <a id="orgb3efe5a"></a> Real part | <a id="org6c0e23c"></a> Imaginary part |
| width=\linewidth | width=\linewidth |
@ -598,7 +598,7 @@ This type of display is not widely used as we cannot use logarithmic axes (as we
It can be seen from the expression of the inverse receptance \eqref{eq:dynamic_stiffness} that the Real part depends entirely on the mass and stiffness properties while the Imaginary part is a only function of the damping.
Figure [5](#org50f821d) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
Figure [5](#org0339be1) shows an example of a plot of a system with a combination of both viscous and structural damping. The imaginary part is a straight line whose slope is given by the viscous damping rate \\(c\\) and whose intercept at \\(\omega = 0\\) is provided by the structural damping coefficient \\(d\\).
<a id="table--fig:inverse-frf"></a>
<div class="table-caption">
@ -608,7 +608,7 @@ Figure [5](#org50f821d) shows an example of a plot of a system with a combinatio
| ![](/ox-hugo/ewins00_inverse_frf_mixed.png) | ![](/ox-hugo/ewins00_inverse_frf_viscous.png) |
|---------------------------------------------|-----------------------------------------------|
| <a id="org50f821d"></a> Mixed | <a id="orgd67281f"></a> Viscous |
| <a id="org0339be1"></a> Mixed | <a id="org03893ea"></a> Viscous |
| width=\linewidth | width=\linewidth |
@ -625,7 +625,7 @@ The missing information (in this case, the frequency) must be added by identifyi
| ![](/ox-hugo/ewins00_nyquist_receptance_viscous.png) | ![](/ox-hugo/ewins00_nyquist_receptance_structural.png) |
|------------------------------------------------------|---------------------------------------------------------|
| <a id="org09f1bd3"></a> Viscous damping | <a id="org82484ab"></a> Structural damping |
| <a id="org6baff82"></a> Viscous damping | <a id="orgfadfd34"></a> Structural damping |
| width=\linewidth | width=\linewidth |
The Nyquist plot has the particularity of distorting the plot so as to focus on the resonance area.
@ -1130,9 +1130,9 @@ Equally, in a real mode, all parts of the structure pass through their **zero de
</div>
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [6](#org9e29afc)).
While the real mode has the appearance of a **standing wave**, the complex mode is better described as exhibiting **traveling waves** (illustrated on figure [6](#org64f75be)).
<a id="org9e29afc"></a>
<a id="org64f75be"></a>
{{< figure src="/ox-hugo/ewins00_real_complex_modes.png" caption="Figure 6: Real and complex mode shapes displays" >}}
@ -1147,7 +1147,7 @@ Note that the almost-real mode shape does not necessarily have vector elements w
| ![](/ox-hugo/ewins00_argand_diagram_a.png) | ![](/ox-hugo/ewins00_argand_diagram_b.png) | ![](/ox-hugo/ewins00_argand_diagram_c.png) |
|--------------------------------------------|--------------------------------------------|-----------------------------------------------|
| <a id="orgc5f0718"></a> Almost-real mode | <a id="org8caca26"></a> Complex Mode | <a id="org4118fbf"></a> Measure of complexity |
| <a id="orgf1cdf2d"></a> Almost-real mode | <a id="orgf730e4e"></a> Complex Mode | <a id="orgdb80ebd"></a> Measure of complexity |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -1157,7 +1157,7 @@ There exist few indicators of the modal complexity.
The first one, a simple and crude one, called **MCF1** consists of summing all the phase differences between every combination of two eigenvector elements:
\\[ \text{MCF1} = \sum\_{j=1}^N \sum\_{k=1 \neq j}^N (\theta\_{rj} - \theta\_{rk}) \\]
The second measure is shown on figure [7](#org4118fbf) where a polygon is drawn around the extremities of the individual vectors.
The second measure is shown on figure [7](#orgdb80ebd) where a polygon is drawn around the extremities of the individual vectors.
The obtained area of this polygon is then compared with the area of the circle which is based on the length of the largest vector element. The resulting ratio is used as an indication of the complexity of the mode, and is defined as **MCF2**.
@ -1253,7 +1253,7 @@ We write \\(\alpha\_{11}\\) the point FRF and \\(\alpha\_{21}\\) the transfer FR
It can be seen that the only difference between the point and transfer receptance is in the sign of the modal constant of the second mode.
Consider the first point mobility (figure [9](#org1acadea)), between the two resonances, the two components have opposite signs so that they are substractive rather than additive, and indeed, at the point where they cross, their sum is zero.
Consider the first point mobility (figure [9](#orgc4a7fb9)), between the two resonances, the two components have opposite signs so that they are substractive rather than additive, and indeed, at the point where they cross, their sum is zero.
On a logarithmic plot, this produces the antiresonance characteristic which reflects that of the resonance.
<a id="table--fig:mobility-frf-mdof"></a>
@ -1264,10 +1264,10 @@ On a logarithmic plot, this produces the antiresonance characteristic which refl
| ![](/ox-hugo/ewins00_mobility_frf_mdof_point.png) | ![](/ox-hugo/ewins00_mobility_frf_mdof_transfer.png) |
|---------------------------------------------------|------------------------------------------------------|
| <a id="org1acadea"></a> Point FRF | <a id="orgff3476c"></a> Transfer FRF |
| <a id="orgc4a7fb9"></a> Point FRF | <a id="org55cabb4"></a> Transfer FRF |
| width=\linewidth | width=\linewidth |
For the plot in figure [9](#orgff3476c), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
For the plot in figure [9](#org55cabb4), between the two resonances, the two components have the same sign and they add up, no antiresonance is present.
##### FRF modulus plots for MDOF systems {#frf-modulus-plots-for-mdof-systems}
@ -1283,13 +1283,13 @@ If they have apposite signs, there will not be an antiresonance.
##### Bode plots {#bode-plots}
The resonances and antiresonances are blunted by the inclusion of damping, and the phase angles are no longer exactly \\(\SI{0}{\degree}\\) or \\(\SI{180}{\degree}\\), but the general appearance of the plot is a natural extension of that for the system without damping.
Figure [7](#orga72841a) shows a plot for the same mobility as appears in figure [9](#org1acadea) but here for a system with added damping.
Figure [7](#org6fd9292) shows a plot for the same mobility as appears in figure [9](#orgc4a7fb9) but here for a system with added damping.
Most mobility plots have this general form as long as the modes are relatively well-separated.
This condition is satisfied unless the separation between adjacent natural frequencies is of the same order as, or less than, the modal damping factors, in which case it becomes difficult to distinguish the individual modes.
<a id="orga72841a"></a>
<a id="org6fd9292"></a>
{{< figure src="/ox-hugo/ewins00_frf_damped_system.png" caption="Figure 7: Mobility plot of a damped system" >}}
@ -1298,9 +1298,9 @@ This condition is satisfied unless the separation between adjacent natural frequ
Each of the frequency response of a MDOF system in the Nyquist plot is composed of a number of SDOF components.
Figure [10](#org498fea6) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
Figure [10](#org4caa691) shows the result of plotting the point receptance \\(\alpha\_{11}\\) for the 2DOF system described above.
The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [10](#org6052b7b) where it may be seen that the opposing signs of the modal constants of the two modes have caused one of the modal circle to be in the upper half of the complex plane.
The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [10](#org5060284) where it may be seen that the opposing signs of the modal constants of the two modes have caused one of the modal circle to be in the upper half of the complex plane.
<a id="table--fig:nyquist-frf-plots"></a>
<div class="table-caption">
@ -1310,10 +1310,10 @@ The plot for the transfer receptance \\(\alpha\_{21}\\) is presented in figure [
| ![](/ox-hugo/ewins00_nyquist_point.png) | ![](/ox-hugo/ewins00_nyquist_transfer.png) |
|------------------------------------------|---------------------------------------------|
| <a id="org498fea6"></a> Point receptance | <a id="org6052b7b"></a> Transfer receptance |
| <a id="org4caa691"></a> Point receptance | <a id="org5060284"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
In the two figures [11](#org6a1d778) and [11](#org58ff327), we show corresponding data for **non-proportional** damping.
In the two figures [11](#org7d25d6c) and [11](#org9e70037), we show corresponding data for **non-proportional** damping.
In this case, a relative phase has been introduced between the first and second elements of the eigenvectors: of \\(\SI{30}{\degree}\\) in mode 1 and of \\(\SI{150}{\degree}\\) in mode 2.
Now we find that the individual modal circles are no longer "upright" but are **rotated by an amount dictated by the complexity of the modal constants**.
@ -1325,7 +1325,7 @@ Now we find that the individual modal circles are no longer "upright" but are **
| ![](/ox-hugo/ewins00_nyquist_nonpropdamp_point.png) | ![](/ox-hugo/ewins00_nyquist_nonpropdamp_transfer.png) |
|-----------------------------------------------------|--------------------------------------------------------|
| <a id="org6a1d778"></a> Point receptance | <a id="org58ff327"></a> Transfer receptance |
| <a id="org7d25d6c"></a> Point receptance | <a id="org9e70037"></a> Transfer receptance |
| width=\linewidth | width=\linewidth |
@ -1481,7 +1481,7 @@ Examples of random signals, autocorrelation function and power spectral density
| ![](/ox-hugo/ewins00_random_time.png) | ![](/ox-hugo/ewins00_random_autocorrelation.png) | ![](/ox-hugo/ewins00_random_psd.png) |
|---------------------------------------|--------------------------------------------------|------------------------------------------------|
| <a id="orgc7de431"></a> Time history | <a id="orge5f7f25"></a> Autocorrelation Function | <a id="org023b27f"></a> Power Spectral Density |
| <a id="org5355634"></a> Time history | <a id="org618edae"></a> Autocorrelation Function | <a id="org363a29a"></a> Power Spectral Density |
| width=\linewidth | width=\linewidth | width=\linewidth |
A similar concept can be applied to a pair of functions such as \\(f(t)\\) and \\(x(t)\\) to produce **cross correlation** and **cross spectral density** functions.
@ -1566,8 +1566,8 @@ The existence of two equations presents an opportunity to **check the quality**
There are difficulties to implement some of the above formulae in practice because of noise and other limitations concerned with the data acquisition and processing.
One technique involves **three quantities**, rather than two, in the definition of the output/input ratio.
The system considered can best be described with reference to figure [13](#table--fig:frf-determination) which shows first in [13](#orgc413fea) the traditional single-input single-output model upon which the previous formulae are based.
Then in [13](#org1dd02c9) is given a more detailed and representative model of the system which is used in a modal test.
The system considered can best be described with reference to figure [13](#table--fig:frf-determination) which shows first in [13](#org400650f) the traditional single-input single-output model upon which the previous formulae are based.
Then in [13](#org7285276) is given a more detailed and representative model of the system which is used in a modal test.
<a id="table--fig:frf-determination"></a>
<div class="table-caption">
@ -1577,7 +1577,7 @@ Then in [13](#org1dd02c9) is given a more detailed and representative model of t
| ![](/ox-hugo/ewins00_frf_siso_model.png) | ![](/ox-hugo/ewins00_frf_feedback_model.png) |
|------------------------------------------|--------------------------------------------------|
| <a id="orgc413fea"></a> Basic SISO model | <a id="org1dd02c9"></a> SISO model with feedback |
| <a id="org400650f"></a> Basic SISO model | <a id="org7285276"></a> SISO model with feedback |
| width=\linewidth | width=\linewidth |
In this configuration, it can be seen that there are two feedback mechanisms which apply.
@ -1597,7 +1597,7 @@ where \\(v\\) is a third signal in the system.
##### Derivation of FRF from MIMO data {#derivation-of-frf-from-mimo-data}
A diagram for the general n-input case is shown in figure [8](#orgad01713).
A diagram for the general n-input case is shown in figure [8](#org8f4df84).
We obtain two alternative formulas:
@ -1608,7 +1608,7 @@ We obtain two alternative formulas:
In practical application of both of these formulae, care must be taken to ensure the non-singularity of the spectral density matrix which is to be inverted, and it is in this respect that the former version may be found to be more reliable.
<a id="orgad01713"></a>
<a id="org8f4df84"></a>
{{< figure src="/ox-hugo/ewins00_frf_mimo.png" caption="Figure 8: System for FRF determination via MIMO model" >}}
@ -1878,9 +1878,9 @@ The experimental setup used for mobility measurement contains three major items:
2. **A transduction system**. For the most part, piezoelectric transducer are used, although lasers and strain gauges are convenient because of their minimal interference with the test object. Conditioning amplifiers are used depending of the transducer used
3. **An analyzer**
A typical layout for the measurement system is shown on figure [9](#org5658974).
A typical layout for the measurement system is shown on figure [9](#org7f3a496).
<a id="org5658974"></a>
<a id="org7f3a496"></a>
{{< figure src="/ox-hugo/ewins00_general_frf_measurement_setup.png" caption="Figure 9: General layout of FRF measurement system" >}}
@ -1934,21 +1934,21 @@ However, we need a direct measurement of the force applied to the structure (we
The shakers are usually stiff in the orthogonal directions to the excitation.
This can modify the response of the system in those directions.
In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on figure [10](#org227d83a).
In order to avoid that, a drive rod which is stiff in one direction and flexible in the other five directions is attached between the shaker and the structure as shown on figure [10](#orge1056cd).
Typical size for the rod are \\(5\\) to \\(\SI{10}{mm}\\) long and \\(\SI{1}{mm}\\) in diameter, if the rod is longer, it may introduce the effect of its own resonances.
<a id="org227d83a"></a>
<a id="orge1056cd"></a>
{{< figure src="/ox-hugo/ewins00_shaker_rod.png" caption="Figure 10: Exciter attachment and drive rod assembly" >}}
The support of shaker is also of primary importance.
The setup shown on figure [14](#org0d724dc) presents the most satisfactory arrangement in which the shaker is fixed to ground while the test structure is supported by a soft spring.
The setup shown on figure [14](#org2ce9b2d) presents the most satisfactory arrangement in which the shaker is fixed to ground while the test structure is supported by a soft spring.
Figure [14](#orgf8ba17d) shows an alternative configuration in which the shaker itself is supported.
Figure [14](#orgaf570a9) shows an alternative configuration in which the shaker itself is supported.
It may be necessary to add an additional inertia mass to the shaker in order to generate sufficient excitation forces at low frequencies.
Figure [14](#orge6e0404) shows an unsatisfactory setup. Indeed, the response measured at \\(A\\) would not be due solely to force applied at \\(B\\), but would also be caused by the forces applied at \\(C\\).
Figure [14](#orgc943938) shows an unsatisfactory setup. Indeed, the response measured at \\(A\\) would not be due solely to force applied at \\(B\\), but would also be caused by the forces applied at \\(C\\).
<a id="table--fig:shaker-mount"></a>
<div class="table-caption">
@ -1958,7 +1958,7 @@ Figure [14](#orge6e0404) shows an unsatisfactory setup. Indeed, the response mea
| ![](/ox-hugo/ewins00_shaker_mount_1.png) | ![](/ox-hugo/ewins00_shaker_mount_2.png) | ![](/ox-hugo/ewins00_shaker_mount_3.png) |
|---------------------------------------------|-------------------------------------------------|------------------------------------------|
| <a id="org0d724dc"></a> Ideal Configuration | <a id="orgf8ba17d"></a> Suspended Configuration | <a id="orge6e0404"></a> Unsatisfactory |
| <a id="org2ce9b2d"></a> Ideal Configuration | <a id="orgaf570a9"></a> Suspended Configuration | <a id="orgc943938"></a> Unsatisfactory |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -1973,10 +1973,10 @@ The magnitude of the impact is determined by the mass of the hammer head and its
The frequency range which is effectively excited is controlled by the stiffness of the contacting surface and the mass of the impactor head: there is a resonance at a frequency given by \\(\sqrt{\frac{\text{contact stiffness}}{\text{impactor mass}}}\\) above which it is difficult to deliver energy into the test structure.
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [11](#orgf285619).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [11](#orgf285619).
When the hammer tip impacts the test structure, this will experience a force pulse as shown on figure [11](#orgb47b9bd).
A pulse of this type (half-sine shape) has a frequency content of the form illustrated on figure [11](#orgb47b9bd).
<a id="orgf285619"></a>
<a id="orgb47b9bd"></a>
{{< figure src="/ox-hugo/ewins00_hammer_impulse.png" caption="Figure 11: Typical impact force pulse and spectrum" >}}
@ -2005,9 +2005,9 @@ By suitable design, such a material may be incorporated into a device which **in
#### Force Transducers {#force-transducers}
The force transducer is the simplest type of piezoelectric transducer.
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [12](#org7500151)).
The transmitter force \\(F\\) is applied directly across the crystal, which thus generates a corresponding charge \\(q\\), proportional to \\(F\\) (figure [12](#org930ef4e)).
<a id="org7500151"></a>
<a id="org930ef4e"></a>
{{< figure src="/ox-hugo/ewins00_piezo_force_transducer.png" caption="Figure 12: Force transducer" >}}
@ -2016,11 +2016,11 @@ There exists an undesirable possibility of a cross sensitivity, i.e. an electric
#### Accelerometers {#accelerometers}
In an accelerometer, transduction is indirect and is achieved using a seismic mass (figure [13](#org8d7da26)).
In an accelerometer, transduction is indirect and is achieved using a seismic mass (figure [13](#orga075bcf)).
In this configuration, the force exerted on the crystals is the inertia force of the seismic mass (\\(m\ddot{z}\\)).
Thus, so long as the body and the seismic mass move together, the output of the transducer will be proportional to the acceleration of its body \\(x\\).
<a id="org8d7da26"></a>
<a id="orga075bcf"></a>
{{< figure src="/ox-hugo/ewins00_piezo_accelerometer.png" caption="Figure 13: Compression-type of piezoelectric accelerometer" >}}
@ -2056,9 +2056,9 @@ However, they cannot be used at such low frequencies as the charge amplifiers an
The correct installation of transducers, especially accelerometers is important.
There are various means of fixing the transducers to the surface of the test structure, some more convenient than others.
Some of these methods are illustrated in figure [15](#orgf956916).
Some of these methods are illustrated in figure [15](#orge053903).
Shown on figure [15](#orga10bae6) are typical high frequency limits for each type of attachment.
Shown on figure [15](#org1b85602) are typical high frequency limits for each type of attachment.
<a id="table--fig:transducer-mounting"></a>
<div class="table-caption">
@ -2068,7 +2068,7 @@ Shown on figure [15](#orga10bae6) are typical high frequency limits for each typ
| ![](/ox-hugo/ewins00_transducer_mounting_types.png) | ![](/ox-hugo/ewins00_transducer_mounting_response.png) |
|-----------------------------------------------------|------------------------------------------------------------|
| <a id="orgf956916"></a> Attachment methods | <a id="orga10bae6"></a> Frequency response characteristics |
| <a id="orge053903"></a> Attachment methods | <a id="org1b85602"></a> Frequency response characteristics |
| width=\linewidth | width=\linewidth |
@ -2153,9 +2153,9 @@ That however requires \\(N\\) to be an integral power of \\(2\\).
Aliasing originates from the discretisation of the originally continuous time history.
With this discretisation process, the **existence of very high frequencies in the original signal may well be misinterpreted if the sampling rate is too slow**.
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [14](#org88605d5).
These high frequencies will be **indistinguishable** from genuine low frequency components as shown on figure [14](#org91dbe3e).
<a id="org88605d5"></a>
<a id="org91dbe3e"></a>
{{< figure src="/ox-hugo/ewins00_aliasing.png" caption="Figure 14: The phenomenon of aliasing. On top: Low-frequency signal, On the bottom: High frequency signal" >}}
@ -2172,7 +2172,7 @@ This is illustrated on figure [16](#table--fig:effect-aliasing).
| ![](/ox-hugo/ewins00_aliasing_no_distortion.png) | ![](/ox-hugo/ewins00_aliasing_distortion.png) |
|--------------------------------------------------|-----------------------------------------------------|
| <a id="org86930d6"></a> True spectrum of signal | <a id="orgc744a70"></a> Indicated spectrum from DFT |
| <a id="orgb4560b8"></a> True spectrum of signal | <a id="orgd413cee"></a> Indicated spectrum from DFT |
| width=\linewidth | width=\linewidth |
The solution of the problem is to use an **anti-aliasing filter** which subjects the original time signal to a low-pass, sharp cut-off filter.
@ -2193,12 +2193,12 @@ Leakage is a problem which is a direct **consequence of the need to take only a
| ![](/ox-hugo/ewins00_leakage_ok.png) | ![](/ox-hugo/ewins00_leakage_nok.png) |
|--------------------------------------|----------------------------------------|
| <a id="org5c4b348"></a> Ideal signal | <a id="org9371337"></a> Awkward signal |
| <a id="orgd54be6b"></a> Ideal signal | <a id="org95b6cdc"></a> Awkward signal |
| width=\linewidth | width=\linewidth |
The problem is illustrated on figure [17](#table--fig:leakage).
In the first case (figure [17](#org5c4b348)), the signal is perfectly periodic and the resulting spectrum is just a single line at the frequency of the sine wave.
In the second case (figure [17](#org9371337)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
In the first case (figure [17](#orgd54be6b)), the signal is perfectly periodic and the resulting spectrum is just a single line at the frequency of the sine wave.
In the second case (figure [17](#org95b6cdc)), the periodicity assumption is not strictly valid as there is a discontinuity at each end of the sample.
As a result, the spectrum produced for this case does not indicate the single frequency which the original time signal possessed.
Energy has "leaked" into a number of the spectral lines close to the true frequency and the spectrum is spread over several lines.
@ -2216,14 +2216,14 @@ Leakage is a serious problem in many applications, **ways of avoiding its effect
Windowing involves the imposition of a prescribed profile on the time signal prior to performing the Fourier transform.
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [15](#org8903f56).
The profiles, or "windows" are generally depicted as a time function \\(w(t)\\) as shown in figure [15](#org105c7d0).
<a id="org8903f56"></a>
<a id="org105c7d0"></a>
{{< figure src="/ox-hugo/ewins00_windowing_examples.png" caption="Figure 15: Different types of window. (a) Boxcar, (b) Hanning, (c) Cosine-taper, (d) Exponential" >}}
The analyzed signal is then \\(x^\prime(t) = x(t) w(t)\\).
The result of using a window is seen in the third column of figure [15](#org8903f56).
The result of using a window is seen in the third column of figure [15](#org105c7d0).
The **Hanning and Cosine Taper windows are typically used for continuous signals**, such as are produced by steady periodic or random vibration, while the **Exponential window is used for transient vibration** applications where much of the important information is concentrated in the initial part of the time record.
@ -2239,7 +2239,7 @@ Common filters are: low-pass, high-pass, band-limited, narrow-band, notch.
#### Improving Resolution {#improving-resolution}
<a id="org2a431e7"></a>
<a id="org4b52cde"></a>
##### Increasing transform size {#increasing-transform-size}
@ -2263,9 +2263,9 @@ The common solution to the need for finer frequency resolution is to zoom on the
There are various ways of achieving this result.
The easiest way is to use a frequency shifting process coupled with a controlled aliasing device.
Suppose the signal to be analyzed \\(x(t)\\) has a spectrum \\(X(\omega)\\) has shown on figure [18](#org41c500b), and that we are interested in a detailed analysis between \\(\omega\_1\\) and \\(\omega\_2\\).
Suppose the signal to be analyzed \\(x(t)\\) has a spectrum \\(X(\omega)\\) has shown on figure [18](#orgfeb63a7), and that we are interested in a detailed analysis between \\(\omega\_1\\) and \\(\omega\_2\\).
If we apply a band-pass filter to the signal, as shown on figure [18](#orgfb95cfe), and perform a DFT between \\(0\\) and \\((\omega\_2 - \omega\_1)\\), then because of the aliasing phenomenon described earlier, the frequency components between \\(\omega\_1\\) and \\(\omega\_2\\) will appear between \\(0\\) and \\((\omega\_2 - \omega\_1)\\) with the advantage of a finer resolution (see figure [16](#org1dc44b8)).
If we apply a band-pass filter to the signal, as shown on figure [18](#org94b4dd9), and perform a DFT between \\(0\\) and \\((\omega\_2 - \omega\_1)\\), then because of the aliasing phenomenon described earlier, the frequency components between \\(\omega\_1\\) and \\(\omega\_2\\) will appear between \\(0\\) and \\((\omega\_2 - \omega\_1)\\) with the advantage of a finer resolution (see figure [16](#org0cfcb53)).
<a id="table--fig:frequency-zoom"></a>
<div class="table-caption">
@ -2275,10 +2275,10 @@ If we apply a band-pass filter to the signal, as shown on figure [18](#orgfb95cf
| ![](/ox-hugo/ewins00_zoom_range.png) | ![](/ox-hugo/ewins00_zoom_bandpass.png) |
|------------------------------------------------|------------------------------------------|
| <a id="org41c500b"></a> Spectrum of the signal | <a id="orgfb95cfe"></a> Band-pass filter |
| <a id="orgfeb63a7"></a> Spectrum of the signal | <a id="org94b4dd9"></a> Band-pass filter |
| width=\linewidth | width=\linewidth |
<a id="org1dc44b8"></a>
<a id="org0cfcb53"></a>
{{< figure src="/ox-hugo/ewins00_zoom_result.png" caption="Figure 16: Effective frequency translation for zoom" >}}
@ -2348,9 +2348,9 @@ For instance, the typical FRF curve has large region of relatively slow changes
This is the traditional method of FRF measurement and involves the use of a sweep oscillator to provide a sinusoidal command signal with a frequency that varies slowly in the range of interest.
It is necessary to check that progress through the frequency range is sufficiently slow to check that steady-state response conditions are attained.
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [17](#orgbb6a3e8).
If excessive sweep rate is used, then distortions of the FRF plot are introduced as shown on figure [17](#orgd1e88bf).
<a id="orgbb6a3e8"></a>
<a id="orgd1e88bf"></a>
{{< figure src="/ox-hugo/ewins00_sweep_distortions.png" caption="Figure 17: FRF measurements by sine sweep test" >}}
@ -2466,9 +2466,9 @@ where \\(v(t)\\) is a third signal in the system, such as the voltage supplied t
It is known that a low coherence can arise in a measurement where the frequency resolution of the analyzer is not fine enough to describe adequately the very rapidly changing functions such as are encountered near resonance and anti-resonance on lightly-damped structures.
This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on figure [18](#org7b09a80).
This is known as a **bias** error and leakage is often the most likely source of low coherence on lightly-damped structures as shown on figure [18](#org2d9ba99).
<a id="org7b09a80"></a>
<a id="org2d9ba99"></a>
{{< figure src="/ox-hugo/ewins00_coherence_resonance.png" caption="Figure 18: Coherence \\(\gamma^2\\) and FRF estimate \\(H\_1(\omega)\\) for a lightly damped structure" >}}
@ -2509,9 +2509,9 @@ For the chirp and impulse excitations, each individual sample is collected and p
##### Burst excitation signals {#burst-excitation-signals}
Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see figure [19](#org98bdc4b)).
Burst excitation signals consist of short sections of an underlying continuous signal (which may be a sine wave, a sine sweep or a random signal), followed by a period of zero output, resulting in a response which shows a transient build-up followed by a decay (see figure [19](#org63d0501)).
<a id="org98bdc4b"></a>
<a id="org63d0501"></a>
{{< figure src="/ox-hugo/ewins00_burst_excitation.png" caption="Figure 19: Example of burst excitation and response signals" >}}
@ -2526,22 +2526,22 @@ In the case of burst random, however, each individual burst will be different to
##### Chirp excitation {#chirp-excitation}
The chirp consist of a short duration signal which has the form shown in figure [20](#orgb584cc5).
The chirp consist of a short duration signal which has the form shown in figure [20](#org3d7182f).
The frequency content of the chirp can be precisely chosen by the starting and finishing frequencies of the sweep.
<a id="orgb584cc5"></a>
<a id="org3d7182f"></a>
{{< figure src="/ox-hugo/ewins00_chirp_excitation.png" caption="Figure 20: Example of chirp excitation and response signals" >}}
##### Impulsive excitation {#impulsive-excitation}
The hammer blow produces an input and response as shown in the figure [21](#org4db89b3).
The hammer blow produces an input and response as shown in the figure [21](#orgee86d4a).
This and the chirp excitation are very similar in the analysis point of view, the main difference is that the chirp offers the possibility of greater control of both amplitude and frequency content of the input and also permits the input of a greater amount of vibration energy.
<a id="org4db89b3"></a>
<a id="orgee86d4a"></a>
{{< figure src="/ox-hugo/ewins00_impulsive_excitation.png" caption="Figure 21: Example of impulsive excitation and response signals" >}}
@ -2549,9 +2549,9 @@ The frequency content of the hammer blow is dictated by the **materials** involv
However, it should be recorded that in the region below the first cut-off frequency induced by the elasticity of the hammer tip structure contact, the spectrum of the force signal tends to be **very flat**.
On some structures, the movement of the structure in response to the hammer blow can be such that it returns and **rebounds** on the hammer tip before the user has had time to move that out of the way.
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [22](#orgb29cf45)).
In such cases, the spectrum of the excitation is seen to have "holes" in it at certain frequencies (figure [22](#org2914aa8)).
<a id="orgb29cf45"></a>
<a id="org2914aa8"></a>
{{< figure src="/ox-hugo/ewins00_double_hits.png" caption="Figure 22: Double hits time domain and frequency content" >}}
@ -2624,9 +2624,9 @@ and so **what is required is the ratio of the two sensitivities**:
The overall sensitivity can be more readily obtained by a calibration process because we can easily make an independent measurement of the quantity now being measured: the ratio of response to force.
Suppose the response parameter is acceleration, then the FRF obtained is inertance which has the units of \\(1/\text{mass}\\), a quantity which can readily be independently measured by other means.
Figure [23](#orgddf08d1) shows a typical calibration setup.
Figure [23](#org1f3d9fc) shows a typical calibration setup.
<a id="orgddf08d1"></a>
<a id="org1f3d9fc"></a>
{{< figure src="/ox-hugo/ewins00_calibration_setup.png" caption="Figure 23: Mass calibration procedure, measurement setup" >}}
@ -2639,9 +2639,9 @@ Thus, frequent checks on the overall calibration factors are strongly recommende
It is very important the ensure that the force is measured directly at the point at which it is applied to the structure, rather than deducing its magnitude from the current flowing in the shaker coil or other similar **indirect** processes.
This is because near resonance, the actual applied force becomes very small and is thus very prone to inaccuracy.
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [24](#orgc770be3).
This same argument applies on a lesser scale as we examine the detail around the attachment to the structure, as shown in figure [24](#org5a54bfb).
<a id="orgc770be3"></a>
<a id="org5a54bfb"></a>
{{< figure src="/ox-hugo/ewins00_mass_cancellation.png" caption="Figure 24: Added mass to be cancelled (crossed area)" >}}
@ -2696,9 +2696,9 @@ There are two problems to be tackled:
1. measurement of rotational responses
2. generation of measurement of rotation excitation
The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on figure [25](#org672cadb).
The first of these is less difficult and techniques usually use a pair a matched conventional accelerometers placed at a short distance apart on the structure to be measured as shown on figure [25](#org7c44a7c).
<a id="org672cadb"></a>
<a id="org7c44a7c"></a>
{{< figure src="/ox-hugo/ewins00_rotational_measurement.png" caption="Figure 25: Measurement of rotational response" >}}
@ -2714,12 +2714,12 @@ The principle of operation is that by measuring both accelerometer signals, the
This approach permits us to measure half of the possible FRFs: all those which are of the \\(X/F\\) and \\(\Theta/F\\) type.
The others can only be measured directly by applying a moment excitation.
Figure [26](#org1e07ef2) shows a device to simulate a moment excitation.
Figure [26](#org69e6665) shows a device to simulate a moment excitation.
First, a single applied excitation force \\(F\_1\\) corresponds to a simultaneous force \\(F\_0 = F\_1\\) and a moment \\(M\_0 = -F\_1 l\_1\\).
Then, the same excitation force is applied at the second position that gives a force \\(F\_0 = F\_2\\) and moment \\(M\_0 = F\_2 l\_2\\).
By adding and subtracting the responses produced by these two separate excitations conditions, we can deduce the translational and rotational responses to the translational force and the rotational moment separately, thus enabling the measurement of all four types of FRF: \\(X/F\\), \\(\Theta/F\\), \\(X/M\\) and \\(\Theta/M\\).
<a id="org1e07ef2"></a>
<a id="org69e6665"></a>
{{< figure src="/ox-hugo/ewins00_rotational_excitation.png" caption="Figure 26: Application of moment excitation" >}}
@ -3043,8 +3043,8 @@ Then, each PRF is, simply, a particular combination of the original FRFs, and th
On example of this form of pre-processing is shown on figure [19](#table--fig:PRF-numerical) for a numerically-simulation test data, and another in figure [20](#table--fig:PRF-measured) for the case of real measured test data.
The second plot [19](#org354c67b) helps to determine the true order of the system because the number of non-zero singular values is equal to this parameter.
The third plot [19](#org9b16f79) shows the genuine modes distinct from the computational modes.
The second plot [19](#org1966197) helps to determine the true order of the system because the number of non-zero singular values is equal to this parameter.
The third plot [19](#orgf7309a1) shows the genuine modes distinct from the computational modes.
<div class="important">
<div></div>
@ -3071,7 +3071,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
| ![](/ox-hugo/ewins00_PRF_numerical_FRF.png) | ![](/ox-hugo/ewins00_PRF_numerical_svd.png) | ![](/ox-hugo/ewins00_PRF_numerical_PRF.png) |
|---------------------------------------------|---------------------------------------------|---------------------------------------------|
| <a id="org7b84bb4"></a> FRF | <a id="org354c67b"></a> Singular Values | <a id="org9b16f79"></a> PRF |
| <a id="org85391fe"></a> FRF | <a id="org1966197"></a> Singular Values | <a id="orgf7309a1"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
<a id="table--fig:PRF-measured"></a>
@ -3082,7 +3082,7 @@ The two groups are usually separated by a clear gap (depending of the noise pres
| ![](/ox-hugo/ewins00_PRF_measured_FRF.png) | ![](/ox-hugo/ewins00_PRF_measured_svd.png) | ![](/ox-hugo/ewins00_PRF_measured_PRF.png) |
|--------------------------------------------|--------------------------------------------|--------------------------------------------|
| <a id="org9775a5d"></a> FRF | <a id="org109aae1"></a> Singular Values | <a id="orgcf6c0c3"></a> PRF |
| <a id="org410e27f"></a> FRF | <a id="org6b7d854"></a> Singular Values | <a id="orgc6a23d0"></a> PRF |
| width=\linewidth | width=\linewidth | width=\linewidth |
@ -3114,7 +3114,7 @@ The **Complex mode indicator function** (CMIF) is defined as
</div>
The actual mode indicator values are provided by the squares of the singular values and are usually plotted as a function of frequency in logarithmic form as shown in figure [27](#org1684572):
The actual mode indicator values are provided by the squares of the singular values and are usually plotted as a function of frequency in logarithmic form as shown in figure [27](#org08ac181):
- **Natural frequencies are indicated by large values of the first CMIF** (the highest of the singular values)
- **double or multiple modes by simultaneously large values of two or more CMIF**.
@ -3124,7 +3124,7 @@ Associated with the CMIF values at each natural frequency \\(\omega\_r\\) are tw
- the left singular vector \\(\\{U(\omega\_r)\\}\_1\\) which approximates the **mode shape** of that mode
- the right singular vector \\(\\{V(\omega\_r)\\}\_1\\) which represents the approximate **force pattern necessary to generate a response on that mode only**
<a id="org1684572"></a>
<a id="org08ac181"></a>
{{< figure src="/ox-hugo/ewins00_mifs.png" caption="Figure 27: Complex Mode Indicator Function (CMIF)" >}}
@ -3197,7 +3197,7 @@ In this method, it is assumed that close to one local mode, any effects due to t
This is a method which works adequately for structures whose FRF exhibit **well separated modes**.
This method is useful in obtaining initial estimates to the parameters.
The peak-picking method is applied as follows (illustrated on figure [28](#org00f64f0)):
The peak-picking method is applied as follows (illustrated on figure [28](#orgd1dacfd)):
1. First, **individual resonance peaks** are detected on the FRF plot and the maximum responses frequency \\(\omega\_r\\) is taken as the **natural frequency** of that mode
2. Second, the **local maximum value of the FRF** \\(|\hat{H}|\\) is noted and the **frequency bandwidth** of the function for a response level of \\(|\hat{H}|/\sqrt{2}\\) is determined.
@ -3219,7 +3219,7 @@ The peak-picking method is applied as follows (illustrated on figure [28](#org00
It must be noted that the estimates of both damping and modal constant depend heavily on the accuracy of the maximum FRF level \\(|\hat{H}|\\) which is difficult to measure with great accuracy, especially for lightly damped systems.
Only real modal constants and thus real modes can be deduced by this method.
<a id="org00f64f0"></a>
<a id="orgd1dacfd"></a>
{{< figure src="/ox-hugo/ewins00_peak_amplitude.png" caption="Figure 28: Peak Amplitude method of modal analysis" >}}
@ -3244,7 +3244,7 @@ In the case of a system assumed to have structural damping, the basic function w
\end{equation}
since the only effect of including the modal constant \\({}\_rA\_{jk}\\) is to scale the size of the circle by \\(|{}\_rA\_{jk}|\\) and to rotate it by \\(\angle {}\_rA\_{jk}\\).
A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#orga9233a4).
A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#org0c46692).
<a id="table--fig:modal-circle-figures"></a>
<div class="table-caption">
@ -3254,7 +3254,7 @@ A plot of the quantity \\(\alpha(\omega)\\) is given in figure [21](#orga9233a4)
| ![](/ox-hugo/ewins00_modal_circle.png) | ![](/ox-hugo/ewins00_modal_circle_bis.png) |
|----------------------------------------|--------------------------------------------------------------------|
| <a id="orga9233a4"></a> Properties | <a id="org0ad9bc7"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| <a id="org0c46692"></a> Properties | <a id="orga918af0"></a> \\(\omega\_b\\) and \\(\omega\_a\\) points |
| width=\linewidth | width=\linewidth |
For any frequency \\(\omega\\), we have the following relationship:
@ -3292,7 +3292,7 @@ It may also be seen that an **estimate of the damping** is provided by the sweep
\end{equation}
Suppose now we have two specific points on the circle, one corresponding to a frequency \\(\omega\_b\\) below the natural frequency and the other one \\(\omega\_a\\) above the natural frequency.
Referring to figure [21](#org0ad9bc7), we can write:
Referring to figure [21](#orga918af0), we can write:
\begin{equation}
\begin{aligned}
@ -3358,7 +3358,7 @@ The sequence is:
3. **Locate natural frequency, obtain damping estimate**.
The rate of sweep through the region is estimated numerically and the frequency at which it reaches the maximum is deduced.
At the same time, an estimate of the damping is derived using \eqref{eq:estimate_damping_sweep_rate}.
A typical example is shown on figure [29](#org751f6e5).
A typical example is shown on figure [29](#org96a13a2).
4. **Calculate multiple damping estimates, and scatter**.
A set of damping estimates using all possible combination of the selected data points are computed using \eqref{eq:estimate_damping}.
Then, we can choose the damping estimate to be the mean value.
@ -3368,7 +3368,7 @@ The sequence is:
5. **Determine modal constant modulus and argument**.
The magnitude and argument of the modal constant is determined from the diameter of the circle and from its orientation relative to the Real and Imaginary axis.
<a id="org751f6e5"></a>
<a id="org96a13a2"></a>
{{< figure src="/ox-hugo/ewins00_circle_fit_natural_frequency.png" caption="Figure 29: Location of natural frequency for a Circle-fit modal analysis" >}}
@ -3482,8 +3482,8 @@ We need to introduce the concept of **residual terms**, necessary in the modal a
The first occasion on which the residual problem is encountered is generally at the end of the analysis of a single FRF curve, such as by the repeated application of an SDOF curve-fit to each of the resonances in turn until all modes visible on the plot have been identified.
At this point, it is often desired to construct a theoretical curve (called "**regenerated**"), based on the modal parameters extracted from the measured data, and to overlay this on the original measured data to assess the success of the curve-fit process.
Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in figure [22](#orgf043f5e).
However, by the inclusion of two simple extra terms (the "**residuals**"), the modified regenerated curve is seen to correlate very well with the original experimental data as shown on figure [22](#org1a33e90).
Then the regenerated curve is compared with the original measurements, the result is often disappointing, as illustrated in figure [22](#org398d4d8).
However, by the inclusion of two simple extra terms (the "**residuals**"), the modified regenerated curve is seen to correlate very well with the original experimental data as shown on figure [22](#org8ee9d90).
<a id="table--fig:residual-modes"></a>
<div class="table-caption">
@ -3493,7 +3493,7 @@ However, by the inclusion of two simple extra terms (the "**residuals**"), the m
| ![](/ox-hugo/ewins00_residual_without.png) | ![](/ox-hugo/ewins00_residual_with.png) |
|--------------------------------------------|-----------------------------------------|
| <a id="orgf043f5e"></a> without residual | <a id="org1a33e90"></a> with residuals |
| <a id="org398d4d8"></a> without residual | <a id="org8ee9d90"></a> with residuals |
| width=\linewidth | width=\linewidth |
If we regenerate an FRF curve from the modal parameters we have extracted from the measured data, we shall use a formula of the type
@ -3522,9 +3522,9 @@ The three terms corresponds to:
2. the **high frequency modes** not identified
3. the **modes actually identified**
These three terms are illustrated on figure [30](#orgd0e499e).
These three terms are illustrated on figure [30](#org473ef14).
<a id="orgd0e499e"></a>
<a id="org473ef14"></a>
{{< figure src="/ox-hugo/ewins00_low_medium_high_modes.png" caption="Figure 30: Numerical simulation of contribution of low, medium and high frequency modes" >}}
@ -3818,7 +3818,7 @@ with
</div>
The composite function \\(HH(\omega)\\) can provide a useful means of determining a single (average) value for the natural frequency and damping factor for each mode where the individual functions would each indicate slightly different values.
As an example, a set of mobilities measured are shown individually in figure [23](#org2873c75) and their summation shown as a single composite curve in figure [23](#orge18611d).
As an example, a set of mobilities measured are shown individually in figure [23](#org1ee7063) and their summation shown as a single composite curve in figure [23](#orgfb3f6a3).
<a id="table--fig:composite"></a>
<div class="table-caption">
@ -3828,7 +3828,7 @@ As an example, a set of mobilities measured are shown individually in figure [23
| ![](/ox-hugo/ewins00_composite_raw.png) | ![](/ox-hugo/ewins00_composite_sum.png) |
|-------------------------------------------|-----------------------------------------|
| <a id="org2873c75"></a> Individual curves | <a id="orge18611d"></a> Composite curve |
| <a id="org1ee7063"></a> Individual curves | <a id="orgfb3f6a3"></a> Composite curve |
| width=\linewidth | width=\linewidth |
The global analysis methods have the disadvantages first, that the computation power required is high and second that there may be valid reasons why the various FRF curves exhibit slight differences in their characteristics and it may not always be appropriate to average them.
@ -4382,11 +4382,11 @@ There are basically two choices for the graphical display of a modal model:
##### Deflected shapes {#deflected-shapes}
A static display is often adequate for depicting relatively simple mode shapes.
Measured coordinates of the test structure are first linked as shown on figure [31](#org6eef557) (a).
Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on figure [31](#org6eef557) (b).
Measured coordinates of the test structure are first linked as shown on figure [31](#org0dcf72a) (a).
Then, the grid of measured coordinate points is redrawn on the same plot but this time displaced by an amount proportional to the corresponding element in the mode shape vector as shown on figure [31](#org0dcf72a) (b).
The elements in the vector are scaled according the normalization process used (usually mass-normalized), and their absolute magnitudes have no particular significance.
<a id="org6eef557"></a>
<a id="org0dcf72a"></a>
{{< figure src="/ox-hugo/ewins00_static_display.png" caption="Figure 31: Static display of modes shapes. (a) basic grid (b) single-frame deflection pattern (c) multiple-frame deflection pattern (d) complex mode (e) Argand diagram - quasi-real mode (f) Argand diagram - complex mode" >}}
@ -4395,16 +4395,16 @@ It is customary to select the largest eigenvector element and to scale the whole
##### Multiple frames {#multiple-frames}
If a series of deflection patterns that has been computed for a different instant of time are superimposed, we obtain a result as shown on figure [31](#org6eef557) (c).
If a series of deflection patterns that has been computed for a different instant of time are superimposed, we obtain a result as shown on figure [31](#org0dcf72a) (c).
Some indication of the motion of the structure can be obtained, and the points of zero motion (nodes) can be clearly identified.
It is also possible, in this format, to give some indication of the essence of complex modes, as shown in figure [31](#org6eef557) (d).
It is also possible, in this format, to give some indication of the essence of complex modes, as shown in figure [31](#org0dcf72a) (d).
Complex modes do not, in general, exhibit fixed nodal points.
##### Argand diagram plots {#argand-diagram-plots}
Another form of representation which is useful for complex modes is the representation of the individual complex elements of the eigenvectors on a polar plot, as shown in the examples of figure [31](#org6eef557) (e) and (f).
Another form of representation which is useful for complex modes is the representation of the individual complex elements of the eigenvectors on a polar plot, as shown in the examples of figure [31](#org0dcf72a) (e) and (f).
Although there is no attempt to show the physical deformation of the actual structure in this format, the complexity of the mode shape is graphically displayed.
@ -4427,11 +4427,11 @@ We then tend to interpret this as a motion which is purely in the x-direction wh
The second problem arises when the **grid of measurement points** that is chosen to display the mode shapes is **too coarse in relation to the complexity of the deformation patterns** that are to be displayed.
This can be illustrated using a very simple example: suppose that our test structure is a straight beam, and that we decide to use just three response measurements points.
If we consider the first six modes of the beam, whose mode shapes are sketched in figure [32](#org77a201c), then we see that with this few measurement points, modes 1 and 5 look the same as do modes 2, 4 and 6.
If we consider the first six modes of the beam, whose mode shapes are sketched in figure [32](#org843940c), then we see that with this few measurement points, modes 1 and 5 look the same as do modes 2, 4 and 6.
All the higher modes will be indistinguishable from these first few.
This is a well known problem of **spatial aliasing**.
<a id="org77a201c"></a>
<a id="org843940c"></a>
{{< figure src="/ox-hugo/ewins00_beam_modes.png" caption="Figure 32: Misinterpretation of mode shapes by spatial aliasing" >}}
@ -4478,11 +4478,11 @@ However, it must be noted that there is an important **limitation to this proced
<div></div>
As an example, suppose that FRF data \\(H\_{11}\\) and \\(H\_{21}\\) are measured and analyzed in order to synthesize the FRF \\(H\_{22}\\) initially unmeasured.
The predict curve is compared with the measurements on figure [24](#orgf256093).
The predict curve is compared with the measurements on figure [24](#orga9d477f).
Clearly, the agreement is poor and would tend to indicate that the measurement/analysis process had not been successful.
However, the synthesized curve contained only those terms relating to the modes which had actually been studied from \\(H\_{11}\\) and \\(H\_{21}\\) and this set of modes did not include **all** the modes of the structure.
Thus, \\(H\_{22}\\) **omitted the influence of out-of-range modes**.
The inclusion of these two additional terms (obtained here only after measuring and analyzing \\(H\_{22}\\) itself) resulted in the greatly improved predicted vs measured comparison shown in figure [24](#org15312a1).
The inclusion of these two additional terms (obtained here only after measuring and analyzing \\(H\_{22}\\) itself) resulted in the greatly improved predicted vs measured comparison shown in figure [24](#orgc3d79ab).
</div>
@ -4494,7 +4494,7 @@ The inclusion of these two additional terms (obtained here only after measuring
| ![](/ox-hugo/ewins00_H22_without_residual.png) | ![](/ox-hugo/ewins00_H22_with_residual.png) |
|--------------------------------------------------------|-----------------------------------------------------------|
| <a id="orgf256093"></a> Using measured modal data only | <a id="org15312a1"></a> After inclusion of residual terms |
| <a id="orga9d477f"></a> Using measured modal data only | <a id="orgc3d79ab"></a> After inclusion of residual terms |
| width=\linewidth | width=\linewidth |
The appropriate expression for a "correct" response model, derived via a set of modal properties is thus
@ -4546,10 +4546,10 @@ If the **transmissibility** is measured during a modal test which has a single e
</div>
In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on figure [33](#orga7adcfa).
In general, the transmissibility **depends significantly on the excitation point** (\\({}\_iT\_{jk}(\omega) \neq {}\_qT\_{jk}(\omega)\\) where \\(q\\) is a different DOF than \\(i\\)) and it is shown on figure [33](#orgf71911f).
This may explain why transmissibilities are not widely used in modal analysis.
<a id="orga7adcfa"></a>
<a id="orgf71911f"></a>
{{< figure src="/ox-hugo/ewins00_transmissibility_plots.png" caption="Figure 33: Transmissibility plots" >}}
@ -4570,7 +4570,7 @@ The fact that the excitation force is not measured is responsible for the lack o
| ![](/ox-hugo/ewins00_conventional_modal_test_setup.png) | ![](/ox-hugo/ewins00_base_excitation_modal_setup.png) |
|---------------------------------------------------------|-------------------------------------------------------|
| <a id="orga22fea4"></a> Conventional modal test setup | <a id="orgba3a30c"></a> Base excitation setup |
| <a id="org8c4c8e9"></a> Conventional modal test setup | <a id="orge759ea9"></a> Base excitation setup |
| height=4cm | height=4cm |
@ -4611,6 +4611,7 @@ This is accomplished using the above equation in the form:
Because the rank of each pseudo matrix is less than its order, it cannot be inverted and so we are unable to construct stiffness or mass matrix from this approach.
## Bibliography {#bibliography}
<a id="org300069f"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.
<a id="org15876a9"></a>Ewins, DJ. 2000. _Modal Testing: Theory, Practice and Application_. _Research Studies Pre, 2nd Ed., ISBN-13_. Baldock, Hertfordshire, England Philadelphia, PA: Wiley-Blackwell.

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content/book/fleming14_desig_model_contr_nanop_system.md
View File

@ -9,7 +9,7 @@ Tags
Reference
: ([Fleming and Leang 2014](#org53722bc))
: ([Fleming and Leang 2014](#org6bfb955))
Author(s)
: Fleming, A. J., & Leang, K. K.
@ -821,11 +821,11 @@ Year
### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
<a id="orgaaa53eb"></a>
<a id="org1aabb30"></a>
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
Consider the electrical circuit shown in Figure [1](#orgaaa53eb) where a voltage source is connected to a piezoelectric actuator.
Consider the electrical circuit shown in Figure [1](#org1aabb30) where a voltage source is connected to a piezoelectric actuator.
The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
@ -946,6 +946,7 @@ The bandwidth limitations of standard piezoelectric drives were identified as:
### References {#references}
## Bibliography {#bibliography}
<a id="org53722bc"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.
<a id="org6bfb955"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.

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@ -8,7 +8,7 @@ Tags
: [Finite Element Model]({{< relref "finite_element_model" >}})
Reference
: ([Hatch 2000](#orgf661cf4))
: ([Hatch 2000](#org8ef052d))
Author(s)
: Hatch, M. R.
@ -21,14 +21,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction}
<a id="org1a34f33"></a>
<a id="orgf6309da"></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.
### Modal Analysis {#modal-analysis}
The diagram in Figure [1](#org3a8e4cc) shows the methodology for analyzing a lightly damped structure using normal modes.
The diagram in Figure [1](#orge443794) shows the methodology for analyzing a lightly damped structure using normal modes.
<div class="important">
<div></div>
@ -46,7 +46,7 @@ The steps are:
</div>
<a id="org3a8e4cc"></a>
<a id="orge443794"></a>
{{< 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></div>
Figure [2](#org0a7e8db) shows such process, the steps are:
Figure [2](#org5c37471) shows such process, the steps are:
- start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model)
@ -71,14 +71,14 @@ Figure [2](#org0a7e8db) shows such process, the steps are:
</div>
<a id="org0a7e8db"></a>
<a id="org5c37471"></a>
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}
### Notations {#notations}
Tables [3](#orgc82b5d8), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
Tables [3](#org9e923ac), [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>
<div class="table-caption">
@ -127,22 +127,22 @@ Tables [3](#orgc82b5d8), [2](#table--tab:notations-eigen-vectors-values) and [3]
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
<a id="orgea88319"></a>
<a id="orgcf960ed"></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.
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](#orgc82b5d8) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
Figure [3](#org9e923ac) 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\\).
<a id="orgc82b5d8"></a>
<a id="org9e923ac"></a>
{{< 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></div>
([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.
([Miu 1993](#org6d53cc3)) 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.
@ -151,12 +151,12 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis}
<a id="org79a4830"></a>
<a id="orgb8f9c89"></a>
## Modal Analysis {#modal-analysis}
<a id="orgb718e30"></a>
<a id="orgdd2d2c8"></a>
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}
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\\).
Let's consider the model shown in Figure [4](#org829b3b4) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
<a id="orgebf4457"></a>
<a id="org829b3b4"></a>
{{< 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{equation}
Virtual interpretation of the eigenvectors are shown in Figures [5](#org520a99d), [6](#org722a9ff) and [7](#org9e25b28).
Virtual interpretation of the eigenvectors are shown in Figures [5](#orgabe9314), [6](#org4283877) and [7](#orge77cc5c).
<a id="org520a99d"></a>
<a id="orgabe9314"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
<a id="org722a9ff"></a>
<a id="org4283877"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}
<a id="org9e25b28"></a>
<a id="orge77cc5c"></a>
{{< 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-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](#orgfbabf08).
This procedure is schematically shown in Figure [8](#org948bae0).
<a id="orgfbabf08"></a>
<a id="org948bae0"></a>
{{< 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}
<a id="org3f5ad6c"></a>
<a id="org4c868eb"></a>
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
- 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](#orge102983).
This will be applied to the model shown in Figure [9](#org4e1f260).
<a id="orge102983"></a>
<a id="org4e1f260"></a>
{{< 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>
Figure [10](#org3024448) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
Figure [10](#org87a6063) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
<a id="org3024448"></a>
<a id="org87a6063"></a>
{{< 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}
<a id="org6842a3c"></a>
<a id="org56b254f"></a>
### Introduction {#introduction}
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org2292476).
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org684a769).
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.
<a id="org2292476"></a>
<a id="org684a769"></a>
{{< 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}
<a id="orgc13e165"></a>
<a id="org1a30462"></a>
### 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}
<a id="org03aa6d1"></a>
<a id="org638024a"></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](#org143e4e8)).
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org084a5d0)).
### Actuator Description {#actuator-description}
<a id="org143e4e8"></a>
<a id="org084a5d0"></a>
{{< 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](#orgc294fc5)).
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](#org0cedd1b)).
<a id="orgc294fc5"></a>
<a id="org0cedd1b"></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" >}}
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
Figure [13](#orgc294fc5) shows the nodes used for the reduced Matlab model.
Figure [13](#org0cedd1b) 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 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}
<a id="org1f06bfa"></a>
<a id="orgcc9b585"></a>
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}
<a id="orgfc560f8"></a>
<a id="org85fa9f4"></a>
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org7003388)).
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#orgc1d7ce0)).
### Actuator Description {#actuator-description}
<a id="org7003388"></a>
<a id="orgc1d7ce0"></a>
{{< 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}
In Figure [15](#org472d510) are shown the principal nodes used for the model.
In Figure [15](#orgf58efee) are shown the principal nodes used for the model.
<a id="org472d510"></a>
<a id="orgf58efee"></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" >}}
@ -1352,11 +1352,11 @@ And we note:
G = zn * Gp;
```
<a id="orgfd8bb64"></a>
<a id="orgeaf5fed"></a>
{{< 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="org8f08d6b"></a>
<a id="orgca5d420"></a>
{{< 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}
Let's plot the frequency of the modes (Figure [18](#org6e52a4a)).
Let's plot the frequency of the modes (Figure [18](#org34eb51a)).
<a id="org6e52a4a"></a>
<a id="org34eb51a"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
<a id="org52d380e"></a>
<a id="orgfccb84f"></a>
{{< 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');
```
<a id="org873b074"></a>
<a id="org4280511"></a>
{{< 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
<a id="org295f621"></a>
<a id="org4a478d8"></a>
{{< 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);
```
<a id="org2867cfa"></a>
<a id="org016532c"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
@ -2137,6 +2137,6 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
## Bibliography {#bibliography}
<a id="orgf661cf4"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="org8ef052d"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="org849cfe4"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
<a id="org6d53cc3"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.

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@ -1,6 +1,8 @@
+++
title = "The art of electronics - third edition"
author = ["Thomas Dehaeze"]
description = "One of the best book in electronics. Cover most topics (both analog and digital)."
keywords = ["electronics"]
draft = false
+++
@ -8,7 +10,7 @@ Tags
: [Reference Books]({{< relref "reference_books" >}}), [Electronics]({{< relref "electronics" >}})
Reference
: ([Horowitz 2015](#orgfc7b505))
: ([Horowitz 2015](#org7d6347d))
Author(s)
: Horowitz, P.
@ -17,6 +19,7 @@ Year
: 2015
## Bibliography {#bibliography}
<a id="orgfc7b505"></a>Horowitz, Paul. 2015. _The Art of Electronics - Third Edition_. New York, NY, USA: Cambridge University Press.
<a id="org7d6347d"></a>Horowitz, Paul. 2015. _The Art of Electronics - Third Edition_. New York, NY, USA: Cambridge University Press.

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@ -8,7 +8,7 @@ Tags
: [Metrology]({{< relref "metrology" >}})
Reference
: ([Leach 2014](#org023e404))
: ([Leach 2014](#orgdf2e918))
Author(s)
: Leach, R.
@ -90,4 +90,4 @@ This type of angular interferometer is used to measure small angles (less than \
## Bibliography {#bibliography}
<a id="org023e404"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.
<a id="orgdf2e918"></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" >}})
Reference
: ([Leach and Smith 2018](#orgdc805b5))
: ([Leach and Smith 2018](#org4f15d94))
Author(s)
: Leach, R., & Smith, S. T.
@ -20,4 +20,4 @@ Year
## Bibliography {#bibliography}
<a id="orgdc805b5"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
<a id="org4f15d94"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.

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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" >}})
Reference
: ([Preumont 2018](#org29acb4a))
: ([Preumont 2018](#org6703487))
Author(s)
: Preumont, A.
@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback
#### Feedback {#feedback}
<a id="orga09f785"></a>
<a id="orge1596ba"></a>
{{< 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](#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 principle of feedback is represented on figure [1](#orge1596ba). 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.
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}
<a id="org57ee378"></a>
<a id="org8128933"></a>
{{< 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 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 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](#org8128933).
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}
<a id="org8ea735d"></a>
<a id="orgf360ea4"></a>
{{< 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](#org8ea735d).
The various steps of the design of a controlled structure are shown in figure [3](#orgf360ea4).
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}
From the block diagram of the control system (figure [4](#orga135390)):
From the block diagram of the control system (figure [4](#orgdf35e26)):
\begin{align\*}
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
\end{align\*}
<a id="orga135390"></a>
<a id="orgdf35e26"></a>
{{< 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\\).
\\(\sigma\_z(0)\\) is then the global RMS response.
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orge835b98).
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#org3807050).
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.
<a id="orge835b98"></a>
<a id="org3807050"></a>
{{< 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}
\end{equation}
<a id="orga21e5bb"></a>
<a id="org8a88959"></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\\)" >}}
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)).
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](#org8a88959)).
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 \\]
@ -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:
\\[ 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](#org4ad84e0)).
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#orgbd3bc07)).
<a id="org4ad84e0"></a>
<a id="orgbd3bc07"></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" >}}
@ -457,9 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in
</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](#org0d5b542).
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](#org6c053d5).
<a id="org0d5b542"></a>
<a id="org6c053d5"></a>
{{< 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>
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.
By looking at figure [7](#orgbd3bc07), 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="org6f76f34"></a>
<a id="org63fc16c"></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" >}}
@ -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)}
\end{equation}
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}\\).
The corresponding Bode plot is represented in figure [9](#org63fc16c). 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.
@ -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.
The system consists of (see figure [10](#orgc872907)):
The system consists of (see figure [10](#org459d27b)):
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
- A coil which is free to move axially
<a id="orgc872907"></a>
<a id="org459d27b"></a>
{{< 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}
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 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](#orgc64500b)).
<a id="org4783db3"></a>
<a id="orgc64500b"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
@ -583,9 +583,9 @@ with:
</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](#org1a21332)).
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org1f0c996)).
<a id="org1a21332"></a>
<a id="org1f0c996"></a>
{{< 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\\).
<a id="orgb1e0d40"></a>
<a id="org26133de"></a>
{{< 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}
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#orgf2af0aa).
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#orgdcf2def).
<a id="orgf2af0aa"></a>
<a id="orgdcf2def"></a>
{{< 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.
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](#orgeb5cb84) can be used.
To do so, the bridge circuit as shown on figure [15](#orgd6dcc43) can be used.
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.
<a id="orgeb5cb84"></a>
<a id="orgd6dcc43"></a>
{{< 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 have the ability to respond significantly to stimuli of different physical nature.
Figure [16](#org1e5bcfc) lists various effects that are observed in materials in response to various inputs.
Figure [16](#org9608d58) lists various effects that are observed in materials in response to various inputs.
<a id="org1e5bcfc"></a>
<a id="org9608d58"></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" >}}
@ -761,7 +761,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
</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](#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:
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#orgf820772)) 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{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\\))
- \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
<a id="orgffdc1af"></a>
<a id="orgf820772"></a>
{{< 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}
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org890e9f3).
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org8b2066c).
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
\end{equation}
<a id="org890e9f3"></a>
<a id="org8b2066c"></a>
{{< 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}
Consider the system of figure [19](#org87aa6cd), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
Consider the system of figure [19](#orgc7393d7), 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}\\).
<a id="org87aa6cd"></a>
<a id="orgc7393d7"></a>
{{< 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
\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](#org12f8fb9)).
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](#orgff1c070)).
<a id="org12f8fb9"></a>
<a id="orgff1c070"></a>
{{< 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.
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#orgca40454).
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org288cbbb).
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:
@ -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 larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
<a id="orgca40454"></a>
<a id="org288cbbb"></a>
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
@ -1819,4 +1819,4 @@ This approach has the following advantages:
## Bibliography {#bibliography}
<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>.
<a id="org6703487"></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
: [Reference Books]({{< relref "reference_books" >}}), [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
Reference
: ([Schmidt, Schitter, and Rankers 2014](#orgaeaec45))
: ([Schmidt, Schitter, and Rankers 2014](#orgddac163))
Author(s)
: Schmidt, R. M., Schitter, G., & Rankers, A.
@ -29,7 +29,7 @@ Section 2.2.2 Force and Motion
> One should however be aware that another non-destructive source of non-linearity is found in a tried important field of mechanics, called _kinematics_.
> The relation between angles and positions is often non-linear in such a mechanism, because of the changing angles, and controlling these often requires special precautions to overcome the inherent non-linearities by linearisation around actual position and adapting the optimal settings of the controller to each position.
<a id="orgd7f3fab"></a>
<a id="org5a727b1"></a>
{{< figure src="/ox-hugo/schmidt14_high_low_freq_regions.png" caption="Figure 1: Stabiliby condition and robustness of a feedback controlled system. The desired shape of these curves guide the control design by optimising the lvels and sloppes of the amplitude Bode-plot at low and high frequencies for suppression of the disturbances and of the base Bode-plot in the cross-over frequency region. This is called **loop shaping design**" >}}
@ -42,6 +42,7 @@ Section 9.3: Mass Dilemma
> A reduced mass requires improved system dynamics that enable a higher control bandwidth to compensate for the increase sensitivity for external vibrations.
## Bibliography {#bibliography}
<a id="orgaeaec45"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press.
<a id="orgddac163"></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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@ -8,7 +8,7 @@ Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Reference Books]({{< relref "reference_books" >}})
Reference
: ([Taghirad 2013](#org128f66e))
: ([Taghirad 2013](#org5caa795))
Author(s)
: Taghirad, H.
@ -22,7 +22,7 @@ PDF version
## Introduction {#introduction}
<a id="orgff20cc6"></a>
<a id="org47dbe02"></a>
This book is intended to give some analysis and design tools for the increase number of engineers and researchers who are interested in the design and implementation of parallel robots.
A systematic approach is presented to analyze the kinematics, dynamics and control of parallel robots.
@ -47,14 +47,14 @@ The control of parallel robots is elaborated in the last two chapters, in which
## Motion Representation {#motion-representation}
<a id="orgb57b1f9"></a>
<a id="org50588c0"></a>
### Spatial Motion Representation {#spatial-motion-representation}
Six independent parameters are sufficient to fully describe the spatial location of a rigid body.
Consider a rigid body in a spatial motion as represented in Figure [1](#org5836a85).
Consider a rigid body in a spatial motion as represented in Figure [1](#org25b870d).
Let us define:
- A **fixed reference coordinate system** \\((x, y, z)\\) denoted by frame \\(\\{\bm{A}\\}\\) whose origin is located at point \\(O\_A\\)
@ -62,7 +62,7 @@ Let us define:
The absolute position of point \\(P\\) of the rigid body can be constructed from the relative position of that point with respect to the moving frame \\(\\{\bm{B}\\}\\), and the **position and orientation** of the moving frame \\(\\{\bm{B}\\}\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
<a id="org5836a85"></a>
<a id="org25b870d"></a>
{{< figure src="/ox-hugo/taghirad13_rigid_body_motion.png" caption="Figure 1: Representation of a rigid body spatial motion" >}}
@ -87,7 +87,7 @@ It can be **represented in several different ways**: the rotation matrix, the sc
##### Rotation Matrix {#rotation-matrix}
We consider a rigid body that has been exposed to a pure rotation.
Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#org808f5cb)).
Its orientation has changed from a state represented by frame \\(\\{\bm{A}\\}\\) to its current orientation represented by frame \\(\\{\bm{B}\\}\\) (Figure [2](#orgd6c2a37)).
A \\(3 \times 3\\) rotation matrix \\({}^A\bm{R}\_B\\) is defined by
@ -109,7 +109,7 @@ in which \\({}^A\hat{\bm{x}}\_B, {}^A\hat{\bm{y}}\_B\\) and \\({}^A\hat{\bm{z}}\
The nine elements of the rotation matrix can be simply represented as the projections of the Cartesian unit vectors of frame \\(\\{\bm{B}\\}\\) on the unit vectors of frame \\(\\{\bm{A}\\}\\).
<a id="org808f5cb"></a>
<a id="orgd6c2a37"></a>
{{< figure src="/ox-hugo/taghirad13_rotation_matrix.png" caption="Figure 2: Pure rotation of a rigid body" >}}
@ -135,7 +135,7 @@ The term screw axis for this axis of rotation has the benefit that a general mot
The screw axis representation has the benefit of **using only four parameters** to describe a pure rotation.
These parameters are the angle of rotation \\(\theta\\) and the axis of rotation which is a unit vector \\({}^A\hat{\bm{s}} = [s\_x, s\_y, s\_z]^T\\).
<a id="orgaafaac9"></a>
<a id="orge51f311"></a>
{{< figure src="/ox-hugo/taghirad13_screw_axis_representation.png" caption="Figure 3: Pure rotation about a screw axis" >}}
@ -161,7 +161,7 @@ Three other types of Euler angles are consider with respect to a moving frame: t
The pitch, roll and yaw angles are defined for a moving object in space as the rotations along the lateral, longitudinal and vertical axes attached to the moving object.
<a id="org7158ffe"></a>
<a id="orgb51325b"></a>
{{< figure src="/ox-hugo/taghirad13_pitch-roll-yaw.png" caption="Figure 4: Definition of pitch, roll and yaw angles on an air plain" >}}
@ -364,10 +364,10 @@ There exist transformations to from screw displacement notation to the transform
##### Consecutive transformations {#consecutive-transformations}
Let us consider the motion of a rigid body described at three locations (Figure [5](#orgda91190)).
Let us consider the motion of a rigid body described at three locations (Figure [5](#org2fa078f)).
Frame \\(\\{\bm{A}\\}\\) represents the initial location, frame \\(\\{\bm{B}\\}\\) is an intermediate location, and frame \\(\\{\bm{C}\\}\\) represents the rigid body at its final location.
<a id="orgda91190"></a>
<a id="org2fa078f"></a>
{{< figure src="/ox-hugo/taghirad13_consecutive_transformations.png" caption="Figure 5: Motion of a rigid body represented at three locations by frame \\(\\{\bm{A}\\}\\), \\(\\{\bm{B}\\}\\) and \\(\\{\bm{C}\\}\\)" >}}
@ -430,7 +430,7 @@ Hence, the **inverse of the transformation matrix** can be obtain by
## Kinematics {#kinematics}
<a id="org205ff51"></a>
<a id="orgc37d0be"></a>
### Introduction {#introduction}
@ -537,11 +537,11 @@ The position of the point \\(O\_B\\) of the moving platform is described by the
\end{bmatrix}
\end{equation}
<a id="org3a7000d"></a>
<a id="org2b43912"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_schematic.png" caption="Figure 6: Geometry of a Stewart-Gough platform" >}}
The geometry of the manipulator is shown Figure [6](#org3a7000d).
The geometry of the manipulator is shown Figure [6](#org2b43912).
#### Inverse Kinematics {#inverse-kinematics}
@ -590,7 +590,7 @@ The complexity of the problem depends widely on the manipulator architecture and
## Jacobian: Velocities and Static Forces {#jacobian-velocities-and-static-forces}
<a id="org5675f4f"></a>
<a id="org52d5bda"></a>
### Introduction {#introduction}
@ -685,9 +685,9 @@ The matrix \\(\bm{\Omega}^\times\\) denotes a **skew-symmetric matrix** defined
\end{bmatrix}}
\end{equation}
Now consider the general motion of a rigid body shown in Figure [7](#org48af385), in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and **the problem is to find the absolute velocity** of point \\(P\\) with respect to a fixed frame \\(\\{\bm{A}\\}\\).
Now consider the general motion of a rigid body shown in Figure [7](#orgfd75b99), in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and **the problem is to find the absolute velocity** of point \\(P\\) with respect to a fixed frame \\(\\{\bm{A}\\}\\).
<a id="org48af385"></a>
<a id="orgfd75b99"></a>
{{< figure src="/ox-hugo/taghirad13_general_motion.png" caption="Figure 7: Instantaneous velocity of a point \\(P\\) with respect to a moving frame \\(\\{\bm{B}\\}\\)" >}}
@ -949,9 +949,9 @@ We obtain that the **Jacobian matrix** constructs the **transformation needed to
#### Static Forces of the Stewart-Gough Platform {#static-forces-of-the-stewart-gough-platform}
As shown in Figure [8](#orga0c6cd5), the twist of moving platform is described by a 6D vector \\(\dot{\bm{\mathcal{X}}} = \left[ {}^A\bm{v}\_P \ {}^A\bm{\omega} \right]^T\\), in which \\({}^A\bm{v}\_P\\) is the velocity of point \\(O\_B\\), and \\({}^A\bm{\omega}\\) is the angular velocity of moving platform.<br />
As shown in Figure [8](#org5bff8ab), the twist of moving platform is described by a 6D vector \\(\dot{\bm{\mathcal{X}}} = \left[ {}^A\bm{v}\_P \ {}^A\bm{\omega} \right]^T\\), in which \\({}^A\bm{v}\_P\\) is the velocity of point \\(O\_B\\), and \\({}^A\bm{\omega}\\) is the angular velocity of moving platform.<br />
<a id="orga0c6cd5"></a>
<a id="org5bff8ab"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_static_forces.png" caption="Figure 8: Free-body diagram of forces and moments action on the moving platform and each limb of the Stewart-Gough platform" >}}
@ -1108,9 +1108,9 @@ in which \\(\sigma\_{\text{min}}\\) and \\(\sigma\_{\text{max}}\\) are the small
#### Stiffness Analysis of the Stewart-Gough Platform {#stiffness-analysis-of-the-stewart-gough-platform}
In this section, we restrict our analysis to a 3-6 structure (Figure [9](#org1aa505c)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
In this section, we restrict our analysis to a 3-6 structure (Figure [9](#orge167af1)) in which there exist six distinct attachment points \\(A\_i\\) on the fixed base and three moving attachment point \\(B\_i\\).
<a id="org1aa505c"></a>
<a id="orge167af1"></a>
{{< figure src="/ox-hugo/taghirad13_stewart36.png" caption="Figure 9: Schematic of a 3-6 Stewart-Gough platform" >}}
@ -1140,7 +1140,7 @@ The largest axis of the stiffness transformation hyper-ellipsoid is given by thi
## Dynamics {#dynamics}
<a id="orga872c92"></a>
<a id="org3802547"></a>
### Introduction {#introduction}
@ -1239,7 +1239,7 @@ Linear acceleration of a point \\(P\\) can be easily determined by time derivati
Note that this is correct only if the derivative is taken with respect to a **fixed** frame.<br />
Now consider the general motion of a rigid body, in which a moving frame \\(\\{\bm{B}\\}\\) is attached to the rigid body and the problem is to find the absolute acceleration of point \\(P\\) with respect to the fixed frame \\(\\{\bm{A}\\}\\).
The rigid body performs a general motion, which is a combination of a translation, denoted by the velocity vector \\({}^A\bm{v}\_{O\_B}\\), and an instantaneous angular rotation denoted by \\(\bm{\Omega}\\) (see Figure [7](#org48af385)).
The rigid body performs a general motion, which is a combination of a translation, denoted by the velocity vector \\({}^A\bm{v}\_{O\_B}\\), and an instantaneous angular rotation denoted by \\(\bm{\Omega}\\) (see Figure [7](#orgfd75b99)).
To determine acceleration of point \\(P\\), we start with the relation between absolute and relative velocities of point \\(P\\):
\begin{equation}
@ -1272,7 +1272,7 @@ For the case where \\(P\\) is a point embedded in the rigid body, \\({}^B\bm{v}\
In this section, the properties of mass, namely **center of mass**, **moments of inertia** and its characteristics and the required transformations are described.
<a id="orgc259b1e"></a>
<a id="org7d8eb1c"></a>
{{< figure src="/ox-hugo/taghirad13_mass_property_rigid_body.png" caption="Figure 10: Mass properties of a rigid body" >}}
@ -1364,7 +1364,7 @@ On the other hand, if the reference frame \\(\\{B\\}\\) has **pure rotation** wi
##### Linear Momentum {#linear-momentum}
Linear momentum of a material body, shown in Figure [11](#orgd93c822), with respect to a reference frame \\(\\{\bm{A}\\}\\) is defined as
Linear momentum of a material body, shown in Figure [11](#orgf0e919a), with respect to a reference frame \\(\\{\bm{A}\\}\\) is defined as
\begin{equation}
{}^A\bm{G} = \int\_V \frac{d\bm{p}}{dt} \rho dV
@ -1386,14 +1386,14 @@ in which \\({}^A\bm{v}\_C\\) denotes the velocity of the center of mass with res
This result implies that the **total linear momentum** of differential masses is equal to the linear momentum of a **point mass** \\(m\\) located at the **center of mass**.
This highlights the important of the center of mass in dynamic formulation of rigid bodies.
<a id="orgd93c822"></a>
<a id="orgf0e919a"></a>
{{< figure src="/ox-hugo/taghirad13_angular_momentum_rigid_body.png" caption="Figure 11: The components of the angular momentum of a rigid body about \\(A\\)" >}}
##### Angular Momentum {#angular-momentum}
Consider the solid body represented in Figure [11](#orgd93c822).
Consider the solid body represented in Figure [11](#orgf0e919a).
Angular momentum of the differential masses \\(\rho dV\\) about a reference point \\(A\\), expressed in the reference frame \\(\\{\bm{A}\\}\\) is defined as
\\[ {}^A\bm{H} = \int\_V \left(\bm{p} \times \frac{d\bm{p}}{dt} \right) \rho dV \\]
in which \\(d\bm{p}/dt\\) denotes the velocity of differential mass with respect to the reference frame \\(\\{\bm{A}\\}\\).
@ -1523,7 +1523,7 @@ With \\(\bm{v}\_{b\_{i}}\\) an **intermediate variable** corresponding to the ve
\bm{v}\_{b\_{i}} = \bm{v}\_{p} + \bm{\omega} \times \bm{b}\_{i}
\end{equation}
As illustrated in Figure [12](#orgd839e6c), the piston-cylinder structure of the limbs is decomposed into two separate parts, the masses of which are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
As illustrated in Figure [12](#org8ad224c), the piston-cylinder structure of the limbs is decomposed into two separate parts, the masses of which are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
The position vector of these two center of masses can be determined by the following equations:
\begin{align}
@ -1531,7 +1531,7 @@ The position vector of these two center of masses can be determined by the follo
\bm{p}\_{i\_2} &= \bm{a}\_{i} + ( l\_i - c\_{i\_2}) \hat{\bm{s}}\_{i}
\end{align}
<a id="orgd839e6c"></a>
<a id="org8ad224c"></a>
{{< figure src="/ox-hugo/taghirad13_free_body_diagram_stewart.png" caption="Figure 12: Free-body diagram of the limbs and the moving platform of a general Stewart-Gough manipulator" >}}
@ -1558,7 +1558,7 @@ We assume that each limb consists of two parts, the cylinder and the piston, whe
We also assume that the centers of masses of the cylinder and the piston are located at a distance of \\(c\_{i\_1}\\) and \\(c\_{i\_2}\\) above their foot points, and their masses are denoted by \\(m\_{i\_1}\\) and \\(m\_{i\_2}\\).
Moreover, consider that the pistons are symmetric about their axes, and their centers of masses lie at their midlengths.
The free-body diagrams of the limbs and the moving platforms is given in Figure [12](#orgd839e6c).
The free-body diagrams of the limbs and the moving platforms is given in Figure [12](#org8ad224c).
The reaction forces at fixed points \\(A\_i\\) are denoted by \\(\bm{f}\_{a\_i}\\), the internal force at moving points \\(B\_i\\) are dentoed by \\(\bm{f}\_{b\_i}\\), and the internal forces and moments between cylinders and pistons are denoted by \\(\bm{f}\_{c\_i}\\) and \\(\bm{M\_{c\_i}}\\) respectively.
Assume that the only existing external disturbance wrench is applied on the moving platform and is denoted by \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\).
@ -1586,7 +1586,7 @@ in which \\(m\_{c\_e}\\) is defined as
##### Dynamic Formulation of the Moving Platform {#dynamic-formulation-of-the-moving-platform}
Assume that the **moving platform center of mass is located at the center point** \\(P\\) and it has a mass \\(m\\) and moment of inertia \\({}^A\bm{I}\_{P}\\).
Furthermore, consider that gravitational force and external disturbance wrench are applied on the moving platform, \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\) as depicted in Figure [12](#orgd839e6c).
Furthermore, consider that gravitational force and external disturbance wrench are applied on the moving platform, \\(\bm{\mathcal{F}}\_d = [\bm{F}\_d, \bm{n}\_d]^T\\) as depicted in Figure [12](#org8ad224c).
The Newton-Euler formulation of the moving platform is as follows:
@ -1745,9 +1745,9 @@ in which
##### Forward Dynamics Simulations {#forward-dynamics-simulations}
As shown in Figure [13](#org7b2216f), it is **assumed that actuator forces and external disturbance wrench applied to the manipulator are given and the resulting trajectory of the moving platform is to be determined**.
As shown in Figure [13](#org59a1fc3), it is **assumed that actuator forces and external disturbance wrench applied to the manipulator are given and the resulting trajectory of the moving platform is to be determined**.
<a id="org7b2216f"></a>
<a id="org59a1fc3"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_forward_dynamics.png" caption="Figure 13: Flowchart of forward dynamics implementation sequence" >}}
@ -1758,7 +1758,7 @@ The closed-form dynamic formulation of the Stewart-Gough platform corresponds to
In inverse dynamics simulations, it is assumed that the **trajectory of the manipulator is given**, and the **actuator forces required to generate such trajectories are to be determined**.
As illustrated in Figure [14](#org3acba0f), inverse dynamic formulation is implemented in the following sequence.
As illustrated in Figure [14](#orgd3aaf90), inverse dynamic formulation is implemented in the following sequence.
The first step is trajectory generation for the manipulator moving platform.
Many different algorithms are developed for a smooth trajectory generation.
For such a trajectory, \\(\bm{\mathcal{X}}\_{d}(t)\\) and the time derivatives \\(\dot{\bm{\mathcal{X}}}\_{d}(t)\\), \\(\ddot{\bm{\mathcal{X}}}\_{d}(t)\\) are known.
@ -1780,7 +1780,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
\bm{\tau} = \bm{J}^{-T} \left( \bm{M}(\bm{\mathcal{X}})\ddot{\bm{\mathcal{X}}} + \bm{C}(\bm{\mathcal{X}}, \dot{\bm{\mathcal{X}}})\dot{\bm{\mathcal{X}}} + \bm{G}(\bm{\mathcal{X}}) - \bm{\mathcal{F}}\_d \right)
\end{equation}
<a id="org3acba0f"></a>
<a id="orgd3aaf90"></a>
{{< figure src="/ox-hugo/taghirad13_stewart_inverse_dynamics.png" caption="Figure 14: Flowchart of inverse dynamics implementation sequence" >}}
@ -1805,7 +1805,7 @@ Therefore, actuator forces \\(\bm{\tau}\\) are computed in the simulation from
## Motion Control {#motion-control}
<a id="org5c662d7"></a>
<a id="org65878c4"></a>
### Introduction {#introduction}
@ -1826,7 +1826,7 @@ However, using advanced techniques in nonlinear and MIMO control permits to over
### Controller Topology {#controller-topology}
<a id="org00b9812"></a>
<a id="org694cde5"></a>
<div class="important">
<div></div>
@ -1871,11 +1871,11 @@ In general, the desired motion of the moving platform may be represented by the
To perform such motion in closed loop, it is necessary to **measure the output motion** \\(\bm{\mathcal{X}}\\) of the manipulator by an instrumentation system.
Such instrumentation usually consists of two subsystems: the first subsystem may use accurate accelerometers, or global positioning systems to calculate the position of a point on the moving platform; and a second subsystem may use inertial or laser gyros to determine orientation of the moving platform.<br />
Figure [15](#org868a832) shows the general topology of a motion controller using direct measurement of the motion variable \\(\bm{\mathcal{X}}\\), as feedback in the closed-loop system.
Figure [15](#org6edb728) shows the general topology of a motion controller using direct measurement of the motion variable \\(\bm{\mathcal{X}}\\), as feedback in the closed-loop system.
In such a structure, the measured position and orientation of the manipulator is compared to its desired value to generate the **motion error vector** \\(\bm{e}\_\mathcal{X}\\).
The controller uses this error information to generate suitable commands for the actuators to minimize the tracking error.<br />
<a id="org868a832"></a>
<a id="org6edb728"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback.png" caption="Figure 15: The general topology of motion feedback control: motion variable \\(\bm{\mathcal{X}}\\) is measured" >}}
@ -1883,9 +1883,9 @@ However, it is usually much **easier to measure the active joint variable** rath
The relation between the **joint variable** \\(\bm{q}\\) and **motion variable** of the moving platform \\(\bm{\mathcal{X}}\\) is dealt with the **forward and inverse kinematics**.
The relation between the **differential motion variables** \\(\dot{\bm{q}}\\) and \\(\dot{\bm{\mathcal{X}}}\\) is studied through the **Jacobian analysis**.<br />
It is then possible to use the forward kinematic analysis to calculate \\(\bm{\mathcal{X}}\\) from the measured joint variables \\(\bm{q}\\), and one may use the control topology depicted in Figure [16](#org373a3c7) to implement such a controller.
It is then possible to use the forward kinematic analysis to calculate \\(\bm{\mathcal{X}}\\) from the measured joint variables \\(\bm{q}\\), and one may use the control topology depicted in Figure [16](#orga6b318d) to implement such a controller.
<a id="org373a3c7"></a>
<a id="orga6b318d"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_bis.png" caption="Figure 16: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured" >}}
@ -1894,26 +1894,26 @@ As described earlier, this is a **complex task** for parallel manipulators.
It is even more complex when a solution has to be found in real time.<br />
However, as shown herein before, the inverse kinematic analysis of parallel manipulators is much easier to carry out.
To overcome the implementation problem of the control topology in Figure [16](#org373a3c7), another control topology is usually implemented for parallel manipulators.
To overcome the implementation problem of the control topology in Figure [16](#orga6b318d), another control topology is usually implemented for parallel manipulators.
In this topology, depicted in Figure [17](#org1ae20e4), the desired motion trajectory of the robot \\(\bm{\mathcal{X}}\_d\\) is used in an **inverse kinematic analysis** to find the corresponding desired values for joint variable \\(\bm{q}\_d\\).
In this topology, depicted in Figure [17](#orgf913d55), the desired motion trajectory of the robot \\(\bm{\mathcal{X}}\_d\\) is used in an **inverse kinematic analysis** to find the corresponding desired values for joint variable \\(\bm{q}\_d\\).
Hence, the controller is designed based on the **joint space error** \\(\bm{e}\_q\\).
<a id="org1ae20e4"></a>
<a id="orgf913d55"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_ter.png" caption="Figure 17: The general topology of motion feedback control: the active joint variable \\(\bm{q}\\) is measured, and the inverse kinematic analysis is used" >}}
Therefore, the **structure and characteristics** of the controller in this topology is totally **different** from that given in the first two topologies.
The **input and output** of the controller depicted in Figure [17](#org1ae20e4) are **both in the joint space**.
The **input and output** of the controller depicted in Figure [17](#orgf913d55) are **both in the joint space**.
However, this is not the case in the previous topologies where the input to the controller is the motion error in task space, while its output is in the joint space.
For the topology in Figure [17](#org1ae20e4), **independent controllers** for each joint may be suitable.<br />
For the topology in Figure [17](#orgf913d55), **independent controllers** for each joint may be suitable.<br />
To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [18](#org08e1864).
To generate a **direct input to output relation in the task space**, consider the topology depicted in Figure [18](#orgac91dcd).
A force distribution block is added which maps the generated wrench in the task space \\(\bm{\mathcal{F}}\\), to its corresponding actuator forces/torque \\(\bm{\tau}\\).
<a id="org08e1864"></a>
<a id="orgac91dcd"></a>
{{< figure src="/ox-hugo/taghirad13_general_topology_motion_feedback_quater.png" caption="Figure 18: The general topology of motion feedback control in task space: the motion variable \\(\bm{\mathcal{X}}\\) is measured, and the controller output generates wrench in task space" >}}
@ -1923,16 +1923,16 @@ For a fully parallel manipulator such as the Stewart-Gough platform, this mappin
### Motion Control in Task Space {#motion-control-in-task-space}
<a id="org36cede4"></a>
<a id="orgcf26437"></a>
#### Decentralized PD Control {#decentralized-pd-control}
In the control structure in Figure [19](#org59c122f), a number of linear PD controllers are used in a feedback structure on each error component.
In the control structure in Figure [19](#orgdb03b09), a number of linear PD controllers are used in a feedback structure on each error component.
The decentralized controller consists of **six disjoint linear controllers** acting on each error component \\(\bm{e}\_x = [e\_x,\ e\_y,\ e\_z,\ e\_{\theta\_x},\ e\_{\theta\_y},\ e\_{\theta\_z}]\\).
The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), in which \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(6 \times 6\\) **diagonal matrices** denoting the derivative and proportional controller gains for each error term.
<a id="org59c122f"></a>
<a id="orgdb03b09"></a>
{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_task_space.png" caption="Figure 19: Decentralized PD controller implemented in task space" >}}
@ -1951,11 +1951,11 @@ The controller gains are generally tuned experimentally based on physical realiz
#### Feed Forward Control {#feed-forward-control}
A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the decentralized PD controller structure as depicted in Figure [20](#org48b2525).
A feedforward wrench denoted by \\(\bm{\mathcal{F}}\_{ff}\\) may be added to the decentralized PD controller structure as depicted in Figure [20](#orgf1d1d54).
This term is generated from the dynamic model of the manipulator in the task space, represented in a closed form by the following equation:
\\[ \bm{\mathcal{F}}\_{ff} = \bm{\hat{M}}(\bm{\mathcal{X}}\_d)\ddot{\bm{\mathcal{X}}}\_d + \bm{\hat{C}}(\bm{\mathcal{X}}\_d, \dot{\bm{\mathcal{X}}}\_d)\dot{\bm{\mathcal{X}}}\_d + \bm{\hat{G}}(\bm{\mathcal{X}}\_d) \\]
<a id="org48b2525"></a>
<a id="orgf1d1d54"></a>
{{< figure src="/ox-hugo/taghirad13_feedforward_control_task_space.png" caption="Figure 20: Feed forward wrench added to the decentralized PD controller in task space" >}}
@ -2011,14 +2011,14 @@ By this means, **nonlinear and coupling behavior of the robotic manipulator is s
</div>
General structure of IDC applied to a parallel manipulator is depicted in Figure [21](#org8872c95).
General structure of IDC applied to a parallel manipulator is depicted in Figure [21](#org32f9766).
A corrective wrench \\(\bm{\mathcal{F}}\_{fl}\\) is added in a **feedback structure** to the closed-loop system, which is calculated from the Coriolis and centrifugal matrix and gravity vector of the manipulator dynamic formulation.
Furthermore, mass matrix is added in the forward path in addition to the desired trajectory acceleration \\(\ddot{\bm{\mathcal{X}}}\_d\\).
As for the feedforward control, the **dynamics and kinematic parameters of the robot are needed**, and in practice estimates of these matrices are used.<br />
<a id="org8872c95"></a>
<a id="org32f9766"></a>
{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_task_space.png" caption="Figure 21: General configuration of inverse dynamics control implemented in task space" >}}
@ -2138,14 +2138,14 @@ in which
\\[ \bm{\eta} = \bm{M}^{-1} \left( \tilde{\bm{M}} \bm{a}\_r + \tilde{\bm{C}} \dot{\bm{\mathcal{X}}} + \tilde{\bm{G}} \right) \\]
is a measure of modeling uncertainty.
<a id="org3cb90cc"></a>
<a id="org88907b2"></a>
{{< figure src="/ox-hugo/taghirad13_robust_inverse_dynamics_task_space.png" caption="Figure 22: General configuration of robust inverse dynamics control implemented in the task space" >}}
#### Adaptive Inverse Dynamics Control {#adaptive-inverse-dynamics-control}
<a id="org235b439"></a>
<a id="org0c70404"></a>
{{< figure src="/ox-hugo/taghirad13_adaptative_inverse_control_task_space.png" caption="Figure 23: General configuration of adaptative inverse dynamics control implemented in task space" >}}
@ -2158,7 +2158,7 @@ If this measurement is available without any doubt, such topologies are among th
However, as explained in Section , in many practical situations measurement of the motion variable \\(\bm{\mathcal{X}}\\) is difficult or expensive, and usually just the active joint variables \\(\bm{q}\\) are measured.
In such cases, the controllers developed in the joint space may be recommended for practical implementation.<br />
To generate a direct input to output relation in the joint space, consider the topology depicted in Figure [16](#org373a3c7).
To generate a direct input to output relation in the joint space, consider the topology depicted in Figure [16](#orga6b318d).
In this topology, the controller input is the joint variable error vector \\(\bm{e}\_q = \bm{q}\_d - \bm{q}\\), and the controller output is directly the actuator force vector \\(\bm{\tau}\\), and hence there exists a **one-to-one correspondence between the controller input to its output**.<br />
The general form of dynamic formulation of parallel robot is usually given in the task space.
@ -2217,7 +2217,7 @@ Furthermore, the main dynamic matrices are all functions of the motion variable
Hence, in practice, to find the dynamic matrices represented in the joint space, **forward kinematics** should be solved to find the motion variable \\(\bm{\mathcal{X}}\\) for any given joint motion vector \\(\bm{q}\\).<br />
Since in parallel robots the forward kinematic analysis is computationally intensive, there exist inherent difficulties in finding the dynamic matrices in the joint space as an explicit function of \\(\bm{q}\\).
In this case it is possible to solve forward kinematics in an online manner, it is recommended to use the control topology depicted in [16](#org373a3c7), and implement control law design in the task space.<br />
In this case it is possible to solve forward kinematics in an online manner, it is recommended to use the control topology depicted in [16](#orga6b318d), and implement control law design in the task space.<br />
However, one implementable alternative to calculate the dynamic matrices represented in the joint space is to use the **desired motion trajectory** \\(\bm{\mathcal{X}}\_d\\) instead of the true value of motion vector \\(\bm{\mathcal{X}}\\) in the calculations.
This approximation significantly reduces the computational cost, with the penalty of having mismatch between the estimated values of these matrices to their true values.
@ -2226,11 +2226,11 @@ This approximation significantly reduces the computational cost, with the penalt
#### Decentralized PD Control {#decentralized-pd-control}
The first control strategy introduced in the joint space consists of the simplest form of feedback control in such manipulators.
In this control structure, depicted in Figure [24](#orgecf2422), a number of PD controllers are used in a feedback structure on each error component.
In this control structure, depicted in Figure [24](#orge46fe49), a number of PD controllers are used in a feedback structure on each error component.
The PD controller is denoted by \\(\bm{K}\_d s + \bm{K}\_p\\), where \\(\bm{K}\_d\\) and \\(\bm{K}\_p\\) are \\(n \times n\\) **diagonal** matrices denoting the derivative and proportional controller gains, respectively.<br />
<a id="orgecf2422"></a>
<a id="orge46fe49"></a>
{{< figure src="/ox-hugo/taghirad13_decentralized_pd_control_joint_space.png" caption="Figure 24: Decentralized PD controller implemented in joint space" >}}
@ -2250,9 +2250,9 @@ To remedy these shortcomings, some modifications have been proposed to this stru
#### Feedforward Control {#feedforward-control}
The tracking performance of the simple PD controller implemented in the joint space is usually not sufficient at different configurations.
To improve the tracking performance, a feedforward actuator force denoted by \\(\bm{\tau}\_{ff}\\) may be added to the structure of the controller as depicted in Figure [25](#orgdbbacb2).
To improve the tracking performance, a feedforward actuator force denoted by \\(\bm{\tau}\_{ff}\\) may be added to the structure of the controller as depicted in Figure [25](#orgc35f9a0).
<a id="orgdbbacb2"></a>
<a id="orgc35f9a0"></a>
{{< figure src="/ox-hugo/taghirad13_feedforward_pd_control_joint_space.png" caption="Figure 25: Feed forward actuator force added to the decentralized PD controller in joint space" >}}
@ -2293,14 +2293,14 @@ By this means, the **nonlinear and coupling characteristics** of robotic manipul
</div>
The general structure of inverse dynamics control applied to a parallel manipulator in the joint space is depicted in Figure [26](#org4f32038).
The general structure of inverse dynamics control applied to a parallel manipulator in the joint space is depicted in Figure [26](#org63ab6b7).
A corrective torque \\(\bm{\tau}\_{fl}\\) is added in a **feedback** structure to the closed-loop system, which is calculated from the Coriolis and Centrifugal matrix, and the gravity vector of the manipulator dynamic formulation in the joint space.
Furthermore, the mass matrix is acting in the **forward path**, in addition to the desired trajectory acceleration \\(\ddot{\bm{q}}\_q\\).
Note that to generate this term, the **dynamic formulation** of the robot, and its **kinematic and dynamic parameters are needed**.
In practice, exact knowledge of dynamic matrices are not available, and there estimates are used.<br />
<a id="org4f32038"></a>
<a id="org63ab6b7"></a>
{{< figure src="/ox-hugo/taghirad13_inverse_dynamics_control_joint_space.png" caption="Figure 26: General configuration of inverse dynamics control implemented in joint space" >}}
@ -2574,7 +2574,7 @@ Hence, it is recommended to design and implement controllers in the task space,
## Force Control {#force-control}
<a id="orgf5b2cda"></a>
<a id="org2ffeff1"></a>
### Introduction {#introduction}
@ -2625,12 +2625,12 @@ However, note that the motion control of the robot when the robot is in interact
To follow **two objectives** with different properties in one control system, usually a **hierarchy** of two feedback loops is used in practice.
This kind of control topology is called **cascade control**, which is used when there are **several measurements and one prime control variable**.
Cascade control is implemented by **nesting** the control loops, as shown in Figure [27](#org0896015).
Cascade control is implemented by **nesting** the control loops, as shown in Figure [27](#org0555e95).
The output control loop is called the **primary loop**, while the inner loop is called the secondary loop and is used to fulfill a secondary objective in the closed-loop system.
</div>
<a id="org0896015"></a>
<a id="org0555e95"></a>
{{< figure src="/ox-hugo/taghirad13_cascade_control.png" caption="Figure 27: Block diagram of a closed-loop system with cascade control" >}}
@ -2658,32 +2658,32 @@ Consider the force control schemes, in which **force tracking is the prime objec
In such a case, it is advised that the outer loop of cascade control structure is constructed by wrench feedback, while the inner loop is based on position feedback.
Since different types of measurement units may be used in parallel robots, different control topologies may be constructed to implement such a cascade structure.<br />
Consider first the cascade control topology shown in Figure [28](#orgae903ae) in which the measured variables are both in the **task space**.
Consider first the cascade control topology shown in Figure [28](#orgf4cb78e) in which the measured variables are both in the **task space**.
The inner loop is constructed by position feedback while the outer loop is based on force feedback.
As seen in Figure [28](#orgae903ae), the force controller block is fed to the motion controller, and this might be seen as the **generated desired motion trajectory for the inner loop**.
As seen in Figure [28](#orgf4cb78e), the force controller block is fed to the motion controller, and this might be seen as the **generated desired motion trajectory for the inner loop**.
The output of motion controller is also designed in the task space, and to convert it to implementable actuator force \\(\bm{\tau}\\), the force distribution block is considered in this topology.<br />
<a id="orgae903ae"></a>
<a id="orgf4cb78e"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop.png" caption="Figure 28: Cascade topology of force feedback control: position in inner loop and force in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
Other alternatives for force control topology may be suggested based on the variations of position and force measurements.
If the force is measured in the joint space, the topology suggested in Figure [29](#org499cb0c) can be used.
If the force is measured in the joint space, the topology suggested in Figure [29](#org04f708f) can be used.
In this topology, the measured actuator force vector \\(\bm{\tau}\\) is mapped into its corresponding wrench in the task space by the Jacobian transpose mapping \\(\bm{\mathcal{F}} = \bm{J}^T \bm{\tau}\\).<br />
<a id="org499cb0c"></a>
<a id="org04f708f"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau.png" caption="Figure 29: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
Consider the case where the force and motion variables are both measured in the **joint space**.
Figure [30](#org05be049) suggests the force control topology in the joint space, in which the inner loop is based on measured motion variable in the joint space, and the outer loop uses the measured actuator force vector.
Figure [30](#org4f8b19a) suggests the force control topology in the joint space, in which the inner loop is based on measured motion variable in the joint space, and the outer loop uses the measured actuator force vector.
In this topology, it is advised that the force controller is designed in the **task** space, and the Jacobian transpose mapping is used to project the measured actuator force vector into its corresponding wrench in the task space.
However, as the inner loop is constructed in the joint space, the desired motion variable \\(\bm{\mathcal{X}}\_d\\) is mapped into joint space using **inverse kinematic** solution.
Therefore, the structure and characteristics of the position controller in this topology is totally different from that given in the first two topologies.<br />
<a id="org05be049"></a>
<a id="org4f8b19a"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_outer_loop_tau_q.png" caption="Figure 30: Cascade topology of force feedback control: position in inner loop and force in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
@ -2695,30 +2695,30 @@ In such a case, force tracking is not the primary objective, and it is advised t
Since different type of measurement units may be used in parallel robots, different control topologies may be constructed to implement such cascade controllers.<br />
Figure [31](#org9e05714) illustrates the cascade control topology for the system in which the measured variables are both in the task space (\\(\bm{\mathcal{F}}\\) and \\(\bm{\mathcal{X}}\\)).
Figure [31](#org31cfb98) illustrates the cascade control topology for the system in which the measured variables are both in the task space (\\(\bm{\mathcal{F}}\\) and \\(\bm{\mathcal{X}}\\)).
The inner loop is loop is constructed by force feedback while the outer loop is based on position feedback.
By this means, when the manipulator is not in contact with a stiff environment, position tracking is guaranteed through the primary controller.
However, when there is interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure controls the force-motion relation.
This configuration may be seen as if the **outer loop generates a desired force trajectory for the inner loop**.<br />
<a id="org9e05714"></a>
<a id="org31cfb98"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_F.png" caption="Figure 31: Cascade topology of force feedback control: force in inner loop and position in outer loop. Moving platform wrench \\(\bm{\mathcal{F}}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured in the task space" >}}
Other alternatives for control topology may be suggested based on the variations of position and force measurements.
If the force is measured in the joint space, control topology shown in Figure [32](#org62ccc96) can be used.
If the force is measured in the joint space, control topology shown in Figure [32](#org09698d7) can be used.
In such case, the Jacobian transpose is used to map the actuator force to its corresponding wrench in the task space.<br />
<a id="org62ccc96"></a>
<a id="org09698d7"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau.png" caption="Figure 32: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and motion variable \\(\bm{\mathcal{X}}\\) are measured" >}}
If the force and motion variables are both measured in the **joint** space, the control topology shown in Figure [33](#org389242f) is suggested.
If the force and motion variables are both measured in the **joint** space, the control topology shown in Figure [33](#orga67daf8) is suggested.
The inner loop is based on the measured actuator force vector in the joint space \\(\bm{\tau}\\), and the outer loop is based on the measured actuated joint position vector \\(\bm{q}\\).
In this topology, the desired motion in the task space is mapped into the joint space using **inverse kinematic** solution, and **both the position and force feedback controllers are designed in the joint space**.
Thus, independent controllers for each joint may be suitable for this topology.
<a id="org389242f"></a>
<a id="orga67daf8"></a>
{{< figure src="/ox-hugo/taghira13_cascade_force_inner_loop_tau_q.png" caption="Figure 33: Cascade topology of force feedback control: force in inner loop and position in outer loop. Actuator forces \\(\bm{\tau}\\) and joint motion variable \\(\bm{q}\\) are measured in the joint space" >}}
@ -2737,7 +2737,7 @@ Thus, independent controllers for each joint may be suitable for this topology.
### Direct Force Control {#direct-force-control}
<a id="org59c6ea9"></a>
<a id="org52d002e"></a>
{{< figure src="/ox-hugo/taghira13_direct_force_control.png" caption="Figure 34: Direct force control scheme, force feedback in the outer loop and motion feedback in the inner loop" >}}
@ -2799,7 +2799,7 @@ Nevertheless, note that Laplace transform is only applicable for **linear time i
<div class="exampl">
<div></div>
Consider an RLC circuit depicted in Figure [35](#orgd749458).
Consider an RLC circuit depicted in Figure [35](#org772f157).
The differential equation relating voltage \\(v\\) to the current \\(i\\) is given by
\\[ v = L\frac{di}{dt} + Ri + \int\_0^t \frac{1}{C} i(\tau)d\tau \\]
in which \\(L\\) denote the inductance, \\(R\\) the resistance and \\(C\\) the capacitance.
@ -2815,7 +2815,7 @@ The impedance of the system may be found from the Laplace transform of the above
<div class="exampl">
<div></div>
Consider the mass-spring-damper system depicted in Figure [35](#orgd749458).
Consider the mass-spring-damper system depicted in Figure [35](#org772f157).
The governing dynamic formulation for this system is given by
\\[ m \ddot{x} + c \dot{x} + k x = f \\]
in which \\(m\\) denote the body mass, \\(c\\) the damper viscous coefficient and \\(k\\) the spring stiffness.
@ -2828,7 +2828,7 @@ The impedance of the system may be found from the Laplace transform of the above
</div>
<a id="orgd749458"></a>
<a id="org772f157"></a>
{{< figure src="/ox-hugo/taghirad13_impedance_control_rlc.png" caption="Figure 35: Analogy of electrical impedance in (a) an electrical RLC circuit to (b) a mechanical mass-spring-damper system" >}}
@ -2846,7 +2846,7 @@ An impedance \\(\bm{Z}(s)\\) is called
</div>
Hence, for the mechanical system represented in Figure [35](#orgd749458):
Hence, for the mechanical system represented in Figure [35](#org772f157):
- mass represents inductive impedance
- viscous friction represents resistive impedance
@ -2879,24 +2879,24 @@ In the impedance control scheme, **regulation of the motion-force dynamic relati
Therefore, when the manipulator is not in contact with a stiff environment, position tracking is guaranteed by a primary controller.
However, when there is an interacting wrench \\(\bm{\mathcal{F}}\_e\\) applied to the moving platform, this structure may be designed to control the force-motion dynamic relation.<br />
As a possible impedance control scheme, consider the closed-loop system depicted in Figure [36](#org4dc07b4), in which the position feedback is considered in the outer loop, while force feedback is used in the inner loop.
As a possible impedance control scheme, consider the closed-loop system depicted in Figure [36](#org6c87ae0), in which the position feedback is considered in the outer loop, while force feedback is used in the inner loop.
This structure is advised when a desired impedance relation between the force and motion variables is required that consists of desired inductive, resistive, and capacitive impedances.
As shown in Figure [36](#org4dc07b4), the motion-tracking error is directly determined from motion measurement by \\(\bm{e}\_x = \bm{\mathcal{X}}\_d - \bm{\mathcal{X}}\\) in the outer loop and the motion controller is designed to satisfy the required impedance.
As shown in Figure [36](#org6c87ae0), the motion-tracking error is directly determined from motion measurement by \\(\bm{e}\_x = \bm{\mathcal{X}}\_d - \bm{\mathcal{X}}\\) in the outer loop and the motion controller is designed to satisfy the required impedance.
Moreover, direct force-tracking objective is not assigned in this control scheme, and therefore the desired force trajectory \\(\bm{\mathcal{F}}\_d\\) is absent in this scheme.
However, an auxiliary force trajectory \\(\bm{\mathcal{F}}\_a\\) is generated from the motion control law and is used as the reference for the force tracking.
By this means, no prescribed force trajectory is tracked, while the **motion control scheme would advise a force trajectory for the robot to ensure the desired impedance regulation**.<br />
<a id="org4dc07b4"></a>
<a id="org6c87ae0"></a>
{{< figure src="/ox-hugo/taghira13_impedance_control.png" caption="Figure 36: Impedance control scheme; motion feedback in the outer loop and force feedback in the inner loop" >}}
The required wrench \\(\bm{\mathcal{F}}\\) in the impedance control scheme, is based on inverse dynamics control and consists of three main parts.
In the inner loop, the force control scheme is based on a feedback linearization part in addition to a mass matrix adjustment, while in the outer loop usually a linear motion controller is considered based on the desired impedance requirements.
Although many different impedance structures may be considered as the basis of the control law, in Figure [36](#org4dc07b4), a linear impedance relation between the force and motion variables is generated that consists of desired inductive \\(\bm{M}\_d\\), resistive \\(\bm{C}\_d\\) and capacitive impedances \\(\bm{K}\_d\\).<br />
Although many different impedance structures may be considered as the basis of the control law, in Figure [36](#org6c87ae0), a linear impedance relation between the force and motion variables is generated that consists of desired inductive \\(\bm{M}\_d\\), resistive \\(\bm{C}\_d\\) and capacitive impedances \\(\bm{K}\_d\\).<br />
According to Figure [36](#org4dc07b4), the controller output wrench \\(\bm{\mathcal{F}}\\), applied to the manipulator may be formulated as
According to Figure [36](#org6c87ae0), the controller output wrench \\(\bm{\mathcal{F}}\\), applied to the manipulator may be formulated as
\\[ \bm{\mathcal{F}} = \hat{\bm{M}} \bm{M}\_d^{-1} \bm{e}\_F + \bm{\mathcal{F}}\_{fl} \\]
with:
@ -2923,6 +2923,7 @@ The impedance control scheme is very popular in practice, wherein tuning the for
However, note that for a good performance, an accurate model of the system is required, and the obtained force and motion dynamics are not robust to modeling uncertainty.
## Bibliography {#bibliography}
<a id="org128f66e"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.
<a id="org5caa795"></a>Taghirad, Hamid. 2013. _Parallel Robots : Mechanics and Control_. Boca Raton, FL: CRC Press.

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@ -161,21 +161,21 @@ Three factors influence the performance:
The DEB helps identifying which disturbance is the limiting factor, and it should be investigated if the controller can deal with this disturbance before re-designing the plant.
The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
In Figure [1](#org7b34df5), the generalized plant maps the disturbances to the performance channels.
In Figure [1](#orgbf22b5e), the generalized plant maps the disturbances to the performance channels.
By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the variance of the performance channels is minimized.
<a id="org7b34df5"></a>
<a id="orgbf22b5e"></a>
{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="Figure 1: Control system with the generalized plant \\(G\\). The performance channels are stacked in \\(z\\), while the controller input is denoted with \\(y\\)" >}}
#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
Since disturbances are generally not white, the system of Figure [1](#org7b34df5) needs to be augmented with so called **disturbance weighting filters**.
Since disturbances are generally not white, the system of Figure [1](#orgbf22b5e) needs to be augmented with so called **disturbance weighting filters**.
A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
This is illustrated in Figure [2](#org5013433) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
This is illustrated in Figure [2](#org27e9aeb) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
The generalized plant framework also allows to include **weighting filters for the performance channels**.
This is useful for three reasons:
@ -184,7 +184,7 @@ This is useful for three reasons:
- some performance channels may be of more importance than others
- by using dynamic weighting filters, one can emphasize the performance in a certain frequency range
<a id="org5013433"></a>
<a id="org27e9aeb"></a>
{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="Figure 2: Control system with the generalized plant \\(G\\) and weighting functions" >}}
@ -209,9 +209,9 @@ So, to obtain feasible controllers, the performance channel is a combination of
By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#org47370f3) is obtained.
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#org5ae58f0) is obtained.
<a id="org47370f3"></a>
<a id="org5ae58f0"></a>
{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="Figure 3: An illustration of a Pareto curve. Each point of the curve represents the performance obtained with an optimal controller. The curve is obtained by varying \\(\alpha\\) and calculating an \\(\mathcal{H}\_2\\) optimal controller for each \\(\alpha\\)." >}}

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@ -8,7 +8,7 @@ Tags
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
Reference
: ([Monkhorst 2004](#org5371372))
: ([Monkhorst 2004](#org7671be3))
Author(s)
: 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.
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)).
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](#org16f42a7)).
<a id="orgbc6c1df"></a>
<a id="org16f42a7"></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)\\)." >}}
@ -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_.
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](#org5bbc30c)).
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#orgc49109b)).
<a id="org5bbc30c"></a>
<a id="orgc49109b"></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\\)" >}}
#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
In this research, the outputs of the closed loop system (Figure [3](#orgff3035c)) are:
In this research, the outputs of the closed loop system (Figure [3](#org8b8bb94)) are:
- the performance (error) signal \\(e\\)
- 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.
<a id="orgff3035c"></a>
<a id="org8b8bb94"></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." >}}
@ -151,4 +151,4 @@ Drawbacks however are, that no robustness guarantees can be given and that the o
## Bibliography {#bibliography}
<a id="org5371372"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
<a id="org7671be3"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.

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## Control systems with non-minimum phase dynamics {#control-systems-with-non-minimum-phase-dynamics}
<./biblio/references.bib>

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@ -29,5 +29,3 @@ Tags
| Manufacturer | links | Country |
|--------------|----------------------------------|---------|
| ACE | [link](https://www.ace-ace.com/) | Germany |
<./biblio/references.bib>

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@ -7,14 +7,14 @@ draft = false
Tags
: [Complementary Filters]({{< relref "complementary_filters" >}})
([Beijen et al. 2019](#orgc359149))
([Beijen et al. 2019](#orgc6f7554))
([Beijen 2018](#org585205d)) (section 6.3.1)
([Beijen 2018](#org8e8fef4)) (section 6.3.1)
## Bibliography {#bibliography}
<a id="org585205d"></a>Beijen, MA. 2018. “Disturbance Feedforward Control for Vibration Isolation Systems: Analysis, Design, and Implementation.” Technische Universiteit Eindhoven.
<a id="org8e8fef4"></a>Beijen, MA. 2018. “Disturbance Feedforward Control for Vibration Isolation Systems: Analysis, Design, and Implementation.” Technische Universiteit Eindhoven.
<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>.
<a id="orgc6f7554"></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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@ -17,12 +17,12 @@ Links to specific actuators:
For vibration isolation:
- 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" >}}))
- In ([Ito and Schitter 2016](#orga71edd4)), 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}
- ([Yedamale 2003](#org1638958))
- ([Yedamale 2003](#org0ac1a74))
<https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html>
@ -30,6 +30,6 @@ For vibration isolation:
## Bibliography {#bibliography}
<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="orga71edd4"></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="org1638958"></a>Yedamale, Padmaraja. 2003. “Brushless Dc (BLDC) Motor Fundamentals.” _Microchip Technology Inc_ 20:3–15.
<a id="org0ac1a74"></a>Yedamale, Padmaraja. 2003. “Brushless Dc (BLDC) Motor Fundamentals.” _Microchip Technology Inc_ 20:3–15.

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@ -1,6 +1,7 @@
+++
title = "Analog to Digital Converters"
author = ["Thomas Dehaeze"]
keywords = ["electronics"]
draft = false
+++
@ -12,7 +13,7 @@ Tags
<https://dewesoft.com/daq/types-of-adc-converters>
- Delta Sigma ([Baker 2011](#orgb22f10b))
- Delta Sigma ([Baker 2011](#org60f0e22))
- Successive Approximation
@ -31,9 +32,9 @@ Let's suppose that the ADC is ideal and the only noise comes from the quantizati
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).
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](#org57805de)).
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](#orgee08810)).
<a id="org57805de"></a>
<a id="orgee08810"></a>
{{< figure src="/ox-hugo/probability_density_function_adc.png" caption="Figure 1: Probability density function \\(p(e)\\) of the ADC error \\(e\\)" >}}
@ -88,4 +89,4 @@ The quantization is:
## Bibliography {#bibliography}
<a id="orgb22f10b"></a>Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” _Analog Applications_ 7.
<a id="org60f0e22"></a>Baker, Bonnie. 2011. “How Delta-Sigma Adcs Work, Part.” _Analog Applications_ 7.

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Tags
:
<a id="org3e05411"></a>
<a id="org67aca6e"></a>
{{< figure src="/ox-hugo/bipolar_transistor_basic_circuits.svg" caption="Figure 1: 5 basic circuits using the bipolar transistor" >}}
<./biblio/references.bib>

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## Software {#software}
- [WireViz](https://github.com/formatc1702/WireViz) is a nice software to easily document cables and wiring harnesses
<./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}
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)))
Two basic circuits of charge amplifiers are shown in Figure [1](#org7d016e2) (taken from ([Fleming 2010](#org467f88f))) and Figure [2](#orgb83f736) (taken from ([Schmidt, Schitter, and Rankers 2014](#org80f2485)))
<a id="org45de288"></a>
<a id="org7d016e2"></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\\)" >}}
<a id="org8955723"></a>
<a id="orgb83f736"></a>
{{< 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 gain of the charge amplified (Figure [1](#org45de288)) is equal to:
The gain of the charge amplified (Figure [1](#org7d016e2)) is equal to:
\\[ \frac{V\_s}{q} = \frac{-1}{C\_s} \\]
@ -50,6 +50,6 @@ The gain of the charge amplified (Figure [1](#org45de288)) is equal to:
## Bibliography {#bibliography}
<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="org467f88f"></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="orgf9a1421"></a>Schmidt, R Munnig, Georg Schitter, and Adrian Rankers. 2014. _The Design of High Performance Mechatronics - 2nd Revised Edition_. Ios Press.
<a id="org80f2485"></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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## Collocated/Dual actuator and sensor {#collocated-dual-actuator-and-sensor}
According to ([Preumont 2018](#org5d050e8)):
According to ([Preumont 2018](#orgbf8f4c5)):
> A **collocated** control system is a control system where the actuator and the sensor are attached to the same degree of freedom.
>
@ -19,9 +19,9 @@ According to ([Preumont 2018](#org5d050e8)):
## Nearly Collocated Actuator Sensor Pair {#nearly-collocated-actuator-sensor-pair}
From Figure [1](#org0d605b2), it is clear that at some frequency / for some mode, the actuator and the sensor will not be collocated anymore (here starting with mode 3).
From Figure [1](#org5d460f9), it is clear that at some frequency / for some mode, the actuator and the sensor will not be collocated anymore (here starting with mode 3).
<a id="org0d605b2"></a>
<a id="org5d460f9"></a>
{{< figure src="/ox-hugo/preumont18_nearly_collocated_schematic.png" caption="Figure 1: Mode shapes for a uniform beam. \\(u\\) and \\(y\\) are not collocated actuator and sensor" >}}
@ -38,6 +38,7 @@ Of course, this will reduce the sensibility.
- [ ] What happens is small pieces of actuators are mixed with small pieces of sensors?
## Bibliography {#bibliography}
<a id="org5d050e8"></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="orgbf8f4c5"></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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## 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](#org0c35169)).
The shaping of complementary filters can be done using the \\(\mathcal{H}\_\infty\\) synthesis ([Dehaeze, Vermat, and Christophe 2019](#org066e272)).
## Bibliography {#bibliography}
<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>.
<a id="org066e272"></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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## BNC {#bnc}
BNC connectors can have an impedance of 50Ohms or 75Ohms as shown in Figure [1](#orgfe209b2).
BNC connectors can have an impedance of 50Ohms or 75Ohms as shown in Figure [1](#orgd1b23d3).
<a id="orgfe209b2"></a>
<a id="orgd1b23d3"></a>
{{< figure src="/ox-hugo/bnc_50_75_ohms.jpg" caption="Figure 1: 75Ohms and 50Ohms BNC connectors" >}}
<./biblio/references.bib>

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@ -13,7 +13,7 @@ Tags
## Special Properties {#special-properties}
Cubic Stewart Platforms can be decoupled provided that (from ([Chen and McInroy 2000](#org0969434)))
Cubic Stewart Platforms can be decoupled provided that (from ([Chen and McInroy 2000](#org2ea9cff)))
> 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.
@ -22,4 +22,4 @@ Cubic Stewart Platforms can be decoupled provided that (from ([Chen and McInroy
## Bibliography {#bibliography}
<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>.
<a id="org2ea9cff"></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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title = "Decoupled Control"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Multivariable Control]({{< relref "multivariable_control" >}})

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:
<./biblio/references.bib>

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draft = false
+++
Backlinks:
- [Dynamic error budgeting, a design approach]({{< relref "monkhorst04_dynam_error_budget" >}})
- [Systems and Signals Norms]({{< relref "norms" >}})
- [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}})
- [The design of high performance mechatronics - 2nd revised edition]({{< relref "schmidt14_desig_high_perfor_mechat_revis_edition" >}})
- [Mechatronic design of a magnetically suspended rotating platform]({{< relref "jabben07_mechat" >}})
Tags
:
A good introduction to Dynamic Error Budgeting is given in ([Monkhorst 2004](#orgce880aa)).
A good introduction to Dynamic Error Budgeting is given in ([Monkhorst 2004](#orgda61e4e)).
## Step by Step process {#step-by-step-process}
Taken from ([Monkhorst 2004](#orgce880aa)): ([Notes]({{< relref "monkhorst04_dynam_error_budget" >}}))
Taken from ([Monkhorst 2004](#orgda61e4e)): ([Notes]({{< relref "monkhorst04_dynam_error_budget" >}}))
> Step by step, the process is as follows:
>
@ -34,6 +26,7 @@ Taken from ([Monkhorst 2004](#orgce880aa)): ([Notes]({{< relref "monkhorst04_dyn
> Iterate until the error budget is meet.
## Bibliography {#bibliography}
<a id="orgce880aa"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
<a id="orgda61e4e"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.

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| [Kaman](https://www.kamansensors.com/product/smt-9700/) | USA |
| [Keyence](https://www.keyence.com/ss/products/measure/measurement%5Flibrary/type/inductive/) | USA |
| [Althen](https://www.althensensors.com/sensors/linear-position-sensors/eddy-current-sensors/) | Netherlands |
<./biblio/references.bib>

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@ -29,14 +29,14 @@ With:
- \\(\omega\_0 = \frac{1}{R\sqrt{C\_1 C\_2}}\\)
- \\(\xi = \frac{C\_2}{C\_1}\\)
<a id="org21a1d35"></a>
<a id="orgb2c3453"></a>
{{< figure src="/ox-hugo/elec_active_second_order_low_pass_filter.png" caption="Figure 1: Second Order Low Pass Filter" >}}
## High Pass Filter {#high-pass-filter}
Same as [1](#org21a1d35) but by exchanging R1 with C1 and R2 with C2
Same as [1](#orgb2c3453) but by exchanging R1 with C1 and R2 with C2
\begin{equation}
\frac{V\_o}{V\_i}(s) = \frac{R^2 C\_1 C\_2 s^2}{R^2 C\_1 C\_2 s^2 + 2 R C\_2 s + 1}
@ -46,5 +46,3 @@ With:
- \\(\omega\_0 = \frac{1}{R\sqrt{C\_1 C\_2}}\\)
- \\(\xi = \frac{C\_2}{C\_1}\\)
<./biblio/references.bib>

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@ -16,29 +16,27 @@ TODOS:
## First Order Low Pass Filter {#first-order-low-pass-filter}
<a id="orgf718550"></a>
<a id="org1c6b488"></a>
{{< figure src="/ox-hugo/elec_passive_first_order_low_pass_filter.png" caption="Figure 1: First Order Low Pass Filter using an RC circuit" >}}
## First Order High Pass Filter {#first-order-high-pass-filter}
<a id="orgc9b929d"></a>
<a id="orgecf7617"></a>
{{< figure src="/ox-hugo/elec_passive_first_order_high_pass_filter.png" caption="Figure 2: First Order High Pass Filter using an RC circuit" >}}
## Second Order Low Pass Filter {#second-order-low-pass-filter}
<a id="orgb56edb0"></a>
<a id="orgcfc4c15"></a>
{{< figure src="/ox-hugo/elec_passive_second_order_low_pass_filter.png" caption="Figure 3: Second Order Low Pass Filter using an RLC circuit" >}}
## Second Order High Pass Filter {#second-order-high-pass-filter}
<a id="org1bcacc5"></a>
<a id="org0b32ffe"></a>
{{< figure src="/ox-hugo/elec_passive_second_order_high_pass_filter.png" caption="Figure 4: Second Order High Pass Filter using an RLC circuit" >}}
<./biblio/references.bib>

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draft = false
+++
Backlinks:
- [Charge Amplifiers]({{< relref "charge_amplifiers" >}})
- [Signal Conditioner]({{< relref "signal_conditioner" >}})
- [Analog to Digital Converters]({{< relref "analog_to_digital_converters" >}})
- [Transconductance Amplifiers]({{< relref "transconductance_amplifiers" >}})
- [Digital to Analog Converters]({{< relref "digital_to_analog_converters" >}})
- [The art of electronics - third edition]({{< relref "horowitz15_art_of_elect_third_edition" >}})
- [Signal to Noise Ratio]({{< relref "signal_to_noise_ratio" >}})
- [Voltage Amplifier]({{< relref "voltage_amplifier" >}})
- [Transimpedance Amplifiers]({{< relref "transimpedance_amplifiers" >}})
Tags
:
<./biblio/references.bib>

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draft = false
+++
Backlinks:
- [Vibration Simulation using Matlab and ANSYS]({{< relref "hatch00_vibrat_matlab_ansys" >}})
Tags
:
@ -16,17 +12,18 @@ Tags
Some resources:
- ([Hatch 2000](#org4303bb7)) ([Notes]({{< relref "hatch00_vibrat_matlab_ansys" >}}))
- ([Khot and Yelve 2011](#org31239e2))
- ([Kovarac et al. 2015](#org304a9dd))
- ([Hatch 2000](#orgddee845)) ([Notes]({{< relref "hatch00_vibrat_matlab_ansys" >}}))
- ([Khot and Yelve 2011](#orgb0a5955))
- ([Kovarac et al. 2015](#org7660da4))
The idea is to extract reduced state space model from Ansys into Matlab.
## Bibliography {#bibliography}
<a id="org4303bb7"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="orgddee845"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="org31239e2"></a>Khot, SM, and Nitesh P Yelve. 2011. “Modeling and Response Analysis of Dynamic Systems by Using ANSYS and MATLAB.” _Journal of Vibration and Control_ 17 (6). SAGE Publications Sage UK: London, England:953–58.
<a id="orgb0a5955"></a>Khot, SM, and Nitesh P Yelve. 2011. “Modeling and Response Analysis of Dynamic Systems by Using ANSYS and MATLAB.” _Journal of Vibration and Control_ 17 (6). SAGE Publications Sage UK: London, England:953–58.
<a id="org304a9dd"></a>Kovarac, A, M Zeljkovic, C Mladjenovic, and A Zivkovic. 2015. “Create SISO State Space Model of Main Spindle from ANSYS Model.” In _12th International Scientific Conference, Novi Sad, Serbia_, 37–41.
<a id="org7660da4"></a>Kovarac, A, M Zeljkovic, C Mladjenovic, and A Zivkovic. 2015. “Create SISO State Space Model of Main Spindle from ANSYS Model.” In _12th International Scientific Conference, Novi Sad, Serbia_, 37–41.

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@ -12,16 +12,16 @@ Tags
Books:
- ([Lobontiu 2002](#orgf96bd1c))
- ([Henein 2003](#org77c1a30))
- ([Smith 2005](#orgdf03b02))
- ([Soemers 2011](#orgc441221))
- ([Cosandier 2017](#orgc637f07))
- ([Lobontiu 2002](#org0e711a7))
- ([Henein 2003](#org4fb65e1))
- ([Smith 2005](#orgbf46163))
- ([Soemers 2011](#orgf482067))
- ([Cosandier 2017](#orgf099485))
## Flexure Joints for Stewart Platforms: {#flexure-joints-for-stewart-platforms}
From ([Chen and McInroy 2000](#org26c43a0)):
From ([Chen and McInroy 2000](#org14378b5)):
> 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.
@ -31,14 +31,14 @@ From ([Chen and McInroy 2000](#org26c43a0)):
## Bibliography {#bibliography}
<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="org14378b5"></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="orgc637f07"></a>Cosandier, Florent. 2017. _Flexure Mechanism Design_. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press.
<a id="orgf099485"></a>Cosandier, Florent. 2017. _Flexure Mechanism Design_. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press.
<a id="org77c1a30"></a>Henein, Simon. 2003. _Conception Des Guidages Flexibles_. Lausanne, Suisse: Presses polytechniques et universitaires romandes.
<a id="org4fb65e1"></a>Henein, Simon. 2003. _Conception Des Guidages Flexibles_. Lausanne, Suisse: Presses polytechniques et universitaires romandes.
<a id="orgf96bd1c"></a>Lobontiu, Nicolae. 2002. _Compliant Mechanisms: Design of Flexure Hinges_. CRC press.
<a id="org0e711a7"></a>Lobontiu, Nicolae. 2002. _Compliant Mechanisms: Design of Flexure Hinges_. CRC press.
<a id="orgdf03b02"></a>Smith, Stuart T. 2005. _Foundations of Ultra-Precision Mechanism Design_. Vol. 2. CRC Press.
<a id="orgbf46163"></a>Smith, Stuart T. 2005. _Foundations of Ultra-Precision Mechanism Design_. Vol. 2. CRC Press.
<a id="orgc441221"></a>Soemers, Herman. 2011. _Design Principles for Precision Mechanisms_. T-Pointprint.
<a id="orgf482067"></a>Soemers, Herman. 2011. _Design Principles for Precision Mechanisms_. T-Pointprint.

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## Materials {#materials}
- ([Smith 2000](#org48aca5d))
- ([Lobontiu 2002](#org45b1d4f))
- ([Henein 2003](#org516fc89))
- ([Cosandier 2017](#orgb511c9e))
- ([Smith 2000](#org903194d))
- ([Lobontiu 2002](#org353b748))
- ([Henein 2003](#org26cb408))
- ([Cosandier 2017](#org684f025))
## Bibliography {#bibliography}
<a id="orgb511c9e"></a>Cosandier, Florent. 2017. _Flexure Mechanism Design_. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press.
<a id="org684f025"></a>Cosandier, Florent. 2017. _Flexure Mechanism Design_. Boca Raton, FL Lausanne, Switzerland: Distributed by CRC Press, 2017EOFL Press.
<a id="org516fc89"></a>Henein, Simon. 2003. _Conception Des Guidages Flexibles_. Lausanne, Suisse: Presses polytechniques et universitaires romandes.
<a id="org26cb408"></a>Henein, Simon. 2003. _Conception Des Guidages Flexibles_. Lausanne, Suisse: Presses polytechniques et universitaires romandes.
<a id="org45b1d4f"></a>Lobontiu, Nicolae. 2002. _Compliant Mechanisms: Design of Flexure Hinges_. CRC press.
<a id="org353b748"></a>Lobontiu, Nicolae. 2002. _Compliant Mechanisms: Design of Flexure Hinges_. CRC press.
<a id="org48aca5d"></a>Smith, Stuart T. 2000. _Flexures: Elements of Elastic Mechanisms_. Crc Press.
<a id="org903194d"></a>Smith, Stuart T. 2000. _Flexures: Elements of Elastic Mechanisms_. Crc Press.

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@ -17,9 +17,9 @@ There are two main technique for force sensors:
The choice between the two is usually based on whether the measurement is static (strain gauge) or dynamics (piezoelectric).
Main differences between the two are shown in Figure [1](#org921c881).
Main differences between the two are shown in Figure [1](#orgd4cde6e).
<a id="org921c881"></a>
<a id="orgd4cde6e"></a>
{{< figure src="/ox-hugo/force_sensor_piezo_vs_strain_gauge.png" caption="Figure 1: Piezoelectric Force sensor VS Strain Gauge Force sensor" >}}
@ -29,7 +29,7 @@ Main differences between the two are shown in Figure [1](#org921c881).
### Dynamics and Noise of a piezoelectric force sensor {#dynamics-and-noise-of-a-piezoelectric-force-sensor}
An analysis the dynamics and noise of a piezoelectric force sensor is done in ([Fleming 2010](#org26fffc0)) ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
An analysis the dynamics and noise of a piezoelectric force sensor is done in ([Fleming 2010](#org6f75dec)) ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
### Manufacturers {#manufacturers}
@ -75,6 +75,7 @@ However, if a charge conditioner is used, the signal will be doubled.
| [Althen](https://www.althensensors.com/sensors/weighing-sensors-load-cells/) | Netherlands |
## Bibliography {#bibliography}
<a id="org26fffc0"></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="org6f75dec"></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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@ -15,16 +15,16 @@ The documentation for the toolbox is accessible [here](https://fomcon.net/fomcon
Here are the parameters that are used to define the wanted properties of the fractional model:
```matlab
wb = 2*pi*0.1; % Lowest frequency bound
wh = 2*pi*1e3; % Highest frequency bound
n = 8; % Approximation order
r = 0.5; % Wanted slope, The corresponding phase will be pi*r
wb = 2*pi*0.1; % Lowest frequency bound
wh = 2*pi*1e3; % Highest frequency bound
n = 8; % Approximation order
r = 0.5; % Wanted slope, The corresponding phase will be pi*r
```
Then, to create an approximation of a fractional-order operator \\(s^r\\) of order \\(n\\) which is valid in the frequency range \\([\omega\_b\, \omega\_h]\\), the `oustafod` function can be used:
```matlab
G = oustafod(r,n,wb,wh);
G = oustafod(r,n,wb,wh);
```
```text
@ -37,10 +37,8 @@ G =
Continuous-time transfer function.
```
Few examples of different slopes are shown in Figure [1](#orgaa7c066).
Few examples of different slopes are shown in Figure [1](#org9241d6d).
<a id="orgaa7c066"></a>
<a id="org9241d6d"></a>
{{< figure src="/ox-hugo/approximate_deriv_int.png" caption="Figure 1: Example of fractional approximations" >}}
<./biblio/references.bib>

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draft = false
+++
Backlinks:
- [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}})
- [Control of spacecraft and aircraft]({{< relref "bryson93_contr_spacec_aircr" >}})
Tags
:
High-Authority Control/Low-Authority Control
From ([Preumont 2018](#org2917245)):
From ([Preumont 2018](#org4171546)):
> The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [1](#org9ce3153). 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:
> The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [1](#org5a821d8). 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:
>
> - The active damping extends outside the bandwidth of the HAC and reduces the settling time of the modes which are outsite the bandwidth
> - 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)
<a id="org9ce3153"></a>
<a id="org5a821d8"></a>
{{< figure src="/ox-hugo/hac_lac_control_architecture.png" caption="Figure 1: HAC-LAC Control Architecture" >}}
Nice papers:
- ([Williams and Antsaklis 1989](#orge6af6e6))
- ([Aubrun 1980](#org05dd00f))
- ([Williams and Antsaklis 1989](#orgb65b217))
- ([Aubrun 1980](#org9a935c0))
## Bibliography {#bibliography}
<a id="org05dd00f"></a>Aubrun, J.N. 1980. “Theory of the Control of Structures by Low-Authority Controllers.” _Journal of Guidance and Control_ 3 (5):444–51. <https://doi.org/10.2514/3.56019>.
<a id="org9a935c0"></a>Aubrun, J.N. 1980. “Theory of the Control of Structures by Low-Authority Controllers.” _Journal of Guidance and Control_ 3 (5):444–51. <https://doi.org/10.2514/3.56019>.
<a id="org2917245"></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="org4171546"></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="orge6af6e6"></a>Williams, T.W.C., and P.J. Antsaklis. 1989. “Limitations of Vibration Suppression in Flexible Space Structures.” In _Proceedings of the 28th IEEE Conference on Decision and Control_, nil. <https://doi.org/10.1109/cdc.1989.70563>.
<a id="orgb65b217"></a>Williams, T.W.C., and P.J. Antsaklis. 1989. “Limitations of Vibration Suppression in Flexible Space Structures.” In _Proceedings of the 28th IEEE Conference on Decision and Control_, nil. <https://doi.org/10.1109/cdc.1989.70563>.

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## 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](#orge266e77))
- Collette, C. et al., Comparison of new absolute displacement sensors ([Collette, Janssens, Mokrani, et al. 2012](#orga0b31ea))
- Collette, C. et al., Review: inertial sensors for low-frequency seismic vibration measurement ([Collette, Janssens, Fernandez-Carmona, et al. 2012](#orga092f9a))
- Collette, C. et al., Comparison of new absolute displacement sensors ([Collette, Janssens, Mokrani, et al. 2012](#orgef1075b))
<a id="orgaa8be44"></a>
<a id="org9a5fa73"></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" >}}
@ -35,7 +35,7 @@ Wireless Accelerometers
- <https://micromega-dynamics.com/products/recovib/miniature-vibration-recorder/>
<a id="org47441e2"></a>
<a id="org1693047"></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>" >}}
@ -52,7 +52,7 @@ Wireless Accelerometers
| [Guralp](https://www.guralp.com/products/surface) | UK |
| [Nanometric](https://www.nanometrics.ca/products/seismometers) | Canada |
<a id="orga5e26ab"></a>
<a id="org6d70737"></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>" >}}
@ -60,6 +60,6 @@ Wireless Accelerometers
## Bibliography {#bibliography}
<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="orga092f9a"></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="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)_.
<a id="orgef1075b"></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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Tags
: [Active Damping]({{< relref "active_damping" >}})
<./biblio/references.bib>

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## 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](#org1b86993))).
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](#org90df4b2))).
<a id="table--tab:index-air"></a>
<div class="table-caption">
@ -59,16 +59,16 @@ Typical characteristics of commercial environmental units are shown in Table [2]
## Interferometer Precision {#interferometer-precision}
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))).
Figure [1](#org195a5db) 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](#org4c766f1))).
<a id="org3490ef0"></a>
<a id="org195a5db"></a>
{{< 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 error in laser interferometry are well described in ([Ducourtieux 2018](#org588696d)).
Sources of error in laser interferometry are well described in ([Ducourtieux 2018](#org08e49c8)).
It includes:
@ -78,10 +78,10 @@ It includes:
- Pressure: \\(K\_P \approx 0.27 ppm hPa^{-1}\\)
- Humidity: \\(K\_{HR} \approx 0.01 ppm \% RH^{-1}\\)
- These errors can partially be compensated using an environmental unit.
- Air turbulence (Figure [2](#orgceb0667))
- Air turbulence (Figure [2](#org7f738e4))
- Non linearity
<a id="orgceb0667"></a>
<a id="org7f738e4"></a>
{{< figure src="/ox-hugo/interferometers_air_turbulence.png" caption="Figure 2: Effect of air turbulences on measurement stability" >}}
@ -89,8 +89,8 @@ It includes:
## Bibliography {#bibliography}
<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="org08e49c8"></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="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="org4c766f1"></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="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.
<a id="org90df4b2"></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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@ -32,7 +32,7 @@ Tags
>
> 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](#orge62ce0f))
From ([Shaw and Srinivasan 1990](#org82fbcc5))
> 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.
@ -52,4 +52,4 @@ From <https://dsp.stackexchange.com/a/30999>
## Bibliography {#bibliography}
<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.
<a id="org82fbcc5"></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/linear_variable_differential_transformers.md
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@ -15,5 +15,3 @@ Tags
| [Micro-Epsilon](https://www.micro-epsilon.com/displacement-position-sensors/inductive-sensor-lvdt/) | Germany |
| [Keyence](https://www.keyence.eu/products/measure/contact-distance-lvdt/gt2/index.jsp) | USA |
| [Althen](https://www.althensensors.com/sensors/linear-position-sensors/lvdt-sensors/) | Netherlands |
<./biblio/references.bib>

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content/zettels/mass_spring_damper_systems.md
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@ -10,7 +10,7 @@ Tags
## Actuated Mass Spring Damper System {#actuated-mass-spring-damper-system}
Let's consider Figure [1](#orga358a0b) where:
Let's consider Figure [1](#orgbf5f22b) where:
- \\(m\\) is the mass in [kg]
- \\(ḱ\\) is the spring stiffness in [N/m]
@ -20,7 +20,7 @@ Let's consider Figure [1](#orga358a0b) where:
- \\(w\\) is ground motion
- \\(x\\) is the absolute mass motion
<a id="orga358a0b"></a>
<a id="orgbf5f22b"></a>
{{< figure src="/ox-hugo/mass_spring_damper_system.png" caption="Figure 1: Mass Spring Damper System" >}}

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content/zettels/matlab.md
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@ -12,11 +12,11 @@ Tags
Books:
- ([Higham 2017](#org68f863c))
- ([Attaway 2018](#org3441bfb))
- ([OverFlow 2018](#org8e0ff2b))
- ([Johnson 2010](#org019531d))
- ([Hahn and Valentine 2016](#orgbeacac3))
- ([Higham 2017](#org28e00d3))
- ([Attaway 2018](#org46f9de5))
- ([OverFlow 2018](#orgfac8ed6))
- ([Johnson 2010](#org9e4fa10))
- ([Hahn and Valentine 2016](#org56f31fb))
## Useful Commands {#useful-commands}
@ -108,12 +108,12 @@ Nice functions:
## Bibliography {#bibliography}
<a id="org3441bfb"></a>Attaway, Stormy. 2018. _MATLAB : a Practical Introduction to Programming and Problem Solving_. Amsterdam: Butterworth-Heinemann.
<a id="org46f9de5"></a>Attaway, Stormy. 2018. _MATLAB : a Practical Introduction to Programming and Problem Solving_. Amsterdam: Butterworth-Heinemann.
<a id="orgbeacac3"></a>Hahn, Brian, and Daniel T Valentine. 2016. _Essential MATLAB for Engineers and Scientists_. Academic Press.
<a id="org56f31fb"></a>Hahn, Brian, and Daniel T Valentine. 2016. _Essential MATLAB for Engineers and Scientists_. Academic Press.
<a id="org68f863c"></a>Higham, Desmond. 2017. _MATLAB Guide_. Philadelphia: Society for Industrial and Applied Mathematics.
<a id="org28e00d3"></a>Higham, Desmond. 2017. _MATLAB Guide_. Philadelphia: Society for Industrial and Applied Mathematics.
<a id="org019531d"></a>Johnson, Richard K. 2010. _The Elements of MATLAB Style_. Cambridge University Press.
<a id="org9e4fa10"></a>Johnson, Richard K. 2010. _The Elements of MATLAB Style_. Cambridge University Press.
<a id="org8e0ff2b"></a>OverFlow, Stack. 2018. _MATLAB Notes for Professionals_. GoalKicker.com.
<a id="orgfac8ed6"></a>OverFlow, Stack. 2018. _MATLAB Notes for Professionals_. GoalKicker.com.

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