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title = "Books"
author = ["Thomas Dehaeze"]
type = "book"
draft = false
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Here is the list of books I took note about.

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title = "Multivariable control systems: an engineering approach"
author = ["Thomas Dehaeze"]
draft = false
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Tags
: [Multivariable Control]({{< relref "multivariable_control" >}})
Reference
: <sup id="51789ae56fcd2284b8426153d90595c8"><a href="#albertos04_multiv_contr_system" title="Albertos \&amp; Antonio, Multivariable Control Systems: an Engineering Approach, Springer-Verlag (2004).">(Albertos \& Antonio, 2004)</a></sup>
Author(s)
: Albertos, P., & Antonio, S.
Year
: 2004
# Bibliography
<a id="albertos04_multiv_contr_system"></a>Albertos, P., & Antonio, S., *Multivariable control systems: an engineering approach* (2004), : Springer-Verlag. [↩](#51789ae56fcd2284b8426153d90595c8)

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title = "Modeling and control of vibration in mechanical systems"
author = ["Thomas Dehaeze"]
draft = false
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Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: <sup id="1d38bd128d92142dd456ab4e9bb4eb84"><a href="#du10_model_contr_vibrat_mechan_system" title="Chunling Du \&amp; Lihua Xie, Modeling and Control of Vibration in Mechanical Systems, CRC Press (2010).">(Chunling Du \& Lihua Xie, 2010)</a></sup>
Author(s)
: Du, C., & Xie, L.
Year
: 2010
Read Chapter 1 and 3.
# Bibliography
<a id="du10_model_contr_vibrat_mechan_system"></a>Du, C., & Xie, L., *Modeling and control of vibration in mechanical systems* (2010), : CRC Press. [↩](#1d38bd128d92142dd456ab4e9bb4eb84)

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title = "Multi-stage actuation systems and control"
author = ["Thomas Dehaeze"]
draft = false
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Tags
:
Reference
: <sup id="28e6aa60b8446943c921ff0d6578ac85"><a href="#du19_multi_actuat_system_contr" title="Chunling Du \&amp; Chee Khiang Pang, Multi-stage Actuation Systems and Control, CRC Press (2019).">(Chunling Du \& Chee Khiang Pang, 2019)</a></sup>
Author(s)
: Du, C., & Pang, C. K.
Year
: 2019
## Mechanical Actuation Systems {#mechanical-actuation-systems}
### Introduction {#introduction}
When high bandwidth, high position accuracy and long stroke are required simultaneously: dual-stage systems composed of a coarse (or primary) actuator and a fine actuator working together are used.
Popular choices for coarse actuator are:
- DC motor
- Voice coil motor (VCM)
- Permanent magnet stepper motor
- Permanent magnet linear synchronous motor
As fine actuators, most of the time piezoelectric actuator are used.
In order to overcome fine actuator stringent stroke limitation and increase control bandwidth, three-stage actuation systems are necessary in practical applications.
### Actuators {#actuators}
#### Primary Actuator {#primary-actuator}
Without loss of generality, the VCM actuator is used as the primary actuator.
When current passes through the coil, a force is produced which accelerates the actuator radially.
The produced force is a function of the current \\(i\_c\\):
\\[ f\_m = k\_t i\_c \\]
where \\(k\_t\\) is a linearized nominal value called the torque constant.
The resonance of the actuator is mainly due to the flexibility of the pivot bearing, arm, suspension.
Then the bandwidth of the control loop is low and the resonances are not a limiting factor of the control design, the actuator model can be considered as follows:
\\[ P\_v(s) = \frac{k\_{vcm}}{s^2} \\]
When the bandwidth is high, the actuator resonances have to be considered in the control design since the flexible resonance modes will reduce the system stability and affect the control performance. Then the actuator model becomes
\\[ P\_v(s) = \frac{k\_{vcm}}{s^2} P\_r(s) \\]
which includes the resonance model
\\[ P\_r(s) = \Pi\_{i=1}^{N} P\_{ri}(s) \\]
and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following forms
\begin{align\*}
P\_{ri}(s) &= \frac{\omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\\\
P\_{ri}(s) &= \frac{b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\\\
P\_{ri}(s) &= \frac{b\_{2i} s^2 + b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
\end{align\*}
#### Secondary Actuators {#secondary-actuators}
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org53c168b)) and the microactuator.
<a id="org53c168b"></a>
{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}}
There are three popular types of micro-actuators: electrostatic moving-slider microactuator, PZT slider-driven microactuator and thermal microactuator.
There characteristics are shown on table [1](#table--tab:microactuator).
<a id="table--tab:microactuator"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:microactuator">Table 1</a></span>:
Performance comparison of microactuators
</div>
| | Elect. | PZT | Thermal |
|-------------|-----------------------------------------------|-----------------------------------------------|----------------------------|
| TF | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{\tau s + 1}\\) |
| \\(\tau\\) | \\(<\SI{0.1}{ms}\\) | \\(<\SI{0.05}{ms}\\) | \\(>\SI{0.1}{ms}\\) |
| \\(omega\\) | \\(1-\SI{2}{kHz}\\) | \\(20-\SI{25}{kHz}\\) | \\(>\SI{15}{kHz}\\) |
### Single-Stage Actuation Systems {#single-stage-actuation-systems}
A typical closed-loop control system is shown on figure [2](#org5941a76), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
<a id="org5941a76"></a>
{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}}
### Dual-Stage Actuation Systems {#dual-stage-actuation-systems}
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="org8f04825"></a>
{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}}
### Three-Stage Actuation Systems {#three-stage-actuation-systems}
Due to the limited allowed stroke of the microactuator, the control bandwidth has to be restricted and that limits the dual-stage disturbance rejection capability.
A three-stage actuation system is therefore introduced to further increase the bandwidth.
Typically, a VCM actuator is used as the primary actuator, PZT milliactuator as the second stage actuator and a third actuator more collocated is used.
## High-Precision Positioning Control of Dual-Stage Actuation Systems {#high-precision-positioning-control-of-dual-stage-actuation-systems}
### Introduction {#introduction}
The sensitivity function of the closed-loop system has provided a straightforward view of its disturbance rejection capability.
It is demanded that the sensitivity function magnitude in the low-frequency range be sufficiently low, while its hump in high-frequency range stays low enough.
In view of this, the controller design for dual-stage actuation systems adopts a weighting function to shape the sensitivity function.
### Control Schemes {#control-schemes}
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#org03d53e8).
- \\(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\\).
- \\(d\_1\\) and \\(d\_2\\) denote internal disturbances
- \\(n\\) is the measurement noise
- \\(d\_u\\) stands for external vibration
<a id="org03d53e8"></a>
{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}}
The open-loop transfer function from \\(pes\\) to \\(y\\) is
\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) + P\_v(z) C\_v(z) \hat{P}\_p(z) C\_p(z) \\]
And the overall sensitivity function of the closed loop system from \\(r\\) to \\(pes\\) is
\\[ S(z) = \frac{1}{1 + G(z)} \\]
which is approximately
\\[ S(z) = \frac{1}{[1 + P\_p(z) C\_p(z)] [1 + P\_v(z)C\_v(z)]} \\]
since within a certain bandwidth
\\[ \hat{P}\_p(z) \approx P\_p(z) \\]
The sensitivity functions of the VCM loop and the secondary actuator loop are
\begin{equation}
S\_v(z) = \frac{1}{1 + P\_v(z) C\_v(z)}, \quad S\_p(z) = \frac{1}{1 + P\_p(z) C\_p(z)}
\end{equation}
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](#org37116a9).
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="org37116a9"></a>
{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}}
Because of the limited displacement range of the secondary actuator, the control efforts for the two actuators should be distributed properly when designing respective controllers to meet the required performance, make the actuators not conflict with each other, as well as prevent the saturation of the secondary actuator.
### 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](#org299b914) 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
\begin{equation}
\\|T\_{zw}\\|\_\infty < 1
\end{equation}
is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
<a id="org299b914"></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](#org361ec91) 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}
W(s) = \frac{\frac{1}{M}s^2 + 2\xi\omega\frac{1}{\sqrt{M}}s + \omega^2}{s^2 + 2\omega\sqrt{\epsilon}s + \omega^2\epsilon}
\end{equation}
where \\(\omega\\) is the desired bandwidth, \\(\epsilon\\) is used to determine the desired low frequency level of sensitivity magnitude and \\(\xi\\) is the damping ratio.
The controller can then be synthesis using the linear matrix inequality (LMI) approach.
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](#org60ad057), 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="org60ad057"></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](#org1d6afb9), 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="org1d6afb9"></a>
{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
A basic requirement of the dual-stage actuation control system is to make the individual primary and secondary loops stable.
It also required that the primary actuator path has a higher gain than the secondary actuator path at low frequency range and the secondary actuator path has a higher gain than the primary actuator path in high-frequency range.
These can be achieve by choosing appropriate weighting function for the controllers design.
### Conclusion {#conclusion}
The controller design has been discussed for high-precision positioning control of the dual-stage actuation systems.
The \\(\mathcal{H}\_\infty\\) loop shaping method has been applied and the design method has been presented.
With the weighting functions, the desired sensitivity function can achieved.
Such a design method can produce robust controllers with more disturbance rejection in the low frequency range and less disturbance amplification in the high-frequency range.
## Modeling and Control of a Three-Stage Actuation System {#modeling-and-control-of-a-three-stage-actuation-system}
### Introduction {#introduction}
In view of the additional bandwidth requirement which is limited by stroke constraint and saturation of secondary actuators, three-stage actuation systems are thereby proposed to meet the demand of a higher bandwidth.
In this section, a specific three-stage actuation system is presented and a controller strategy is proposed, which is based on a decoupled master-slave dual-stage control structure combined with a third stage actuation in parallel format.
### Actuator and Vibration Modeling {#actuator-and-vibration-modeling}
A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a PZT milliactuator as the second-stage actuator denoted by \\(P\_p(s)\\), and a thermal microactuator denoted by \\(P\_m(s)\\).
### Control Strategy and Controller Design {#control-strategy-and-controller-design}
Figure [9](#org0e50764) 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="org0e50764"></a>
{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}}
The open-loop transfer function of the three-stage actuation system is derived as
\begin{equation}
G(z) = G\_v(z) + G\_p(z) + G\_v(z) G\_p(z) + G\_m(z)
\end{equation}
with
\begin{align\*}
G\_v(z) &= P\_v(z) C\_v(z) \\\\\\
G\_p(z) &= P\_p(z) C\_p(z) \\\\\\
G\_m(z) &= P\_m(z) C\_m(z)
\end{align\*}
The overall sensitivity function is given by
\begin{equation}
S(z) = \frac{1}{1 + G(z)}
\end{equation}
The VCM actuator \\(P\_v(s)\\) works in a low bandwidth below \\(\SI{1}{kHz}\\).
The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandwidth up to \\(\SI{3}{kHz}\\).
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](#orgefe88f9).
<a id="orgefe88f9"></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](#orgd0c25f8).
<a id="orgd0c25f8"></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" >}}
### Performance Evaluation {#performance-evaluation}
External vibration from the system working environment is much higher than the internal disturbance, especially for ultra-high precision positioning systems.
In the presence of external vibration, the actuators control effort is dominantly determined by the external vibration.
But because the actuator input is constrained, the external vibration level has to be limited.
Otherwise, saturation will occur in the control loop and the control system performance will be degraded.
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](#org0e50764), 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
\begin{align\*}
u\_p &= C\_p e,\ y\_p = P\_p C\_p e \\\\\\
u\_m &= C\_m e,\ y\_m = P\_m C\_m e \\\\\\
u\_v &= C\_v ( 1 + \hat{P}\_pC\_p ) e,\ y\_v = P\_v ( u\_v + d\_1 )
\end{align\*}
The controller design for the microactuators with input constraints must take into account both external vibration requirements and actuators' stroke, based on which an appropriate bandwidth should be decided when designing the control system.
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](#org5bb499d)).
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](#orga34ddfe) and
\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
Denote the dual-stage open-loop transfer function as \\(G\_d\\)
\\[ G\_d(z) = G\_v(z) + G\_p(z) + G\_v(z) G\_p(z) \\]
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="org5bb499d"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}}
The control signals and the positions of the three actuators are
\begin{align\*}
u\_p &= C\_p(1 + \hat{P}\_m C\_m) e, \ y\_p = P\_p u\_p \\\\\\
u\_m &= C\_m e, \ y\_m = P\_m M\_m e \\\\\\
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](#org0a46272)).
<a id="org0a46272"></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" >}}
### Conclusion {#conclusion}
The relationship among the external vibration, the microactuator stroke, and the achievable control bandwidth has been discussed for being considered in the controller design.
The discussion suggests that in addition to the traditional wisdom of just increasing the resonant frequency, adding more stroke to the microactuator will give more freedom to the loop shaping for the control system design.
## Dual-Stage System Control Considering Secondary Actuator Stroke Limitation {#dual-stage-system-control-considering-secondary-actuator-stroke-limitation}
### Introduction {#introduction}
### More Freedom Loop Shaping for Microactuator Controller Design {#more-freedom-loop-shaping-for-microactuator-controller-design}
### Dual-Stage System Control Design for 5 kHz Bandwidth {#dual-stage-system-control-design-for-5-khz-bandwidth}
### Evaluation with the Consideration of External Vibration and Microactuator Stroke {#evaluation-with-the-consideration-of-external-vibration-and-microactuator-stroke}
### Conclusion {#conclusion}
## Saturation Control for Microactuators in Dual-Stage Actuation Systems {#saturation-control-for-microactuators-in-dual-stage-actuation-systems}
### Introduction {#introduction}
### Modeling and Feedback Control {#modeling-and-feedback-control}
### Anti-Windup Compensation Design {#anti-windup-compensation-design}
### Simulation and Experimental Results {#simulation-and-experimental-results}
### Conclusion {#conclusion}
## Time Delay and Sampling Rate Effect on Control Performance of Dual-Stage Actuation Systems {#time-delay-and-sampling-rate-effect-on-control-performance-of-dual-stage-actuation-systems}
### Introduction {#introduction}
### Modeling of Time Delay {#modeling-of-time-delay}
### Dual-Stage Actuation System Modeling with Time Delay for Controller Design {#dual-stage-actuation-system-modeling-with-time-delay-for-controller-design}
### Controller Design with Time Delay for the Dual-Stage Actuation Systems {#controller-design-with-time-delay-for-the-dual-stage-actuation-systems}
### Time Delay Effect on Dual-Stage System Control Performance {#time-delay-effect-on-dual-stage-system-control-performance}
### Sampling Rate Effect on Dual-Stage System Control Performance {#sampling-rate-effect-on-dual-stage-system-control-performance}
### Conclusion {#conclusion}
## PZT Hysteresis Modeling and Compensation {#pzt-hysteresis-modeling-and-compensation}
### Introduction {#introduction}
### Modeling of Hysteresis {#modeling-of-hysteresis}
#### PI Model {#pi-model}
#### GPI Model {#gpi-model}
#### Inverse GPI Model {#inverse-gpi-model}
### Application of GPI Model to a PZT-Actuated Structure {#application-of-gpi-model-to-a-pzt-actuated-structure}
#### Modeling of the Hysteresis in the PZT-Actuated Structure {#modeling-of-the-hysteresis-in-the-pzt-actuated-structure}
#### Hysteresis Compensator Design {#hysteresis-compensator-design}
#### Experimental Verification {#experimental-verification}
### Conclusion {#conclusion}
## Seeking Control of Dual-Stage Actuation Systems with Trajectory Optimization {#seeking-control-of-dual-stage-actuation-systems-with-trajectory-optimization}
### Introduction {#introduction}
### Current Profile of VCM Primary Actuator {#current-profile-of-vcm-primary-actuator}
#### PTOS Method {#ptos-method}
#### A General Form of VCM Current Profiles {#a-general-form-of-vcm-current-profiles}
### Control System Structure for the Dual-Stage Actuation System {#control-system-structure-for-the-dual-stage-actuation-system}
### Design of VCM Current Profile a[sub(v)] and Dual-Stage Reference Trajectory r[sub(d)] {#design-of-vcm-current-profile-a-sub--v--and-dual-stage-reference-trajectory-r-sub--d}
### Seeking within PZT Milliactuator Stroke {#seeking-within-pzt-milliactuator-stroke}
### Seeking over PZT Milliactuator Stroke {#seeking-over-pzt-milliactuator-stroke}
### Conclusion {#conclusion}
## High-Frequency Vibration Control Using PZT Active Damping {#high-frequency-vibration-control-using-pzt-active-damping}
### Introduction {#introduction}
### Singular Perturbation Method-Based Controller Design {#singular-perturbation-method-based-controller-design}
#### Singular Perturbation Control Topology {#singular-perturbation-control-topology}
#### Identification of Fast Dynamics Using PZT as a Sensor {#identification-of-fast-dynamics-using-pzt-as-a-sensor}
#### Design of Controllers {#design-of-controllers}
##### Fast Subsystem Estimator G[sub(v)][sup(\*)] {#fast-subsystem-estimator-g-sub--v--sup}
##### Fast Controller C[sub(v)] {#fast-controller-c-sub--v}
##### Slow Controller C[sub(v)] {#slow-controller-c-sub--v}
#### Simulation and Experimental Results {#simulation-and-experimental-results}
##### Frequency Responses {#frequency-responses}
##### Time Responses {#time-responses}
### H[sub(2)] Controller Design {#h-sub--2--controller-design}
### Design of C[sub(d)](z) with H[sub(2)] Method and Notch Filters {#design-of-c-sub--d----z--with-h-sub--2--method-and-notch-filters}
### Design of Mixed H[sub(2)]/H[sub(∞)] Controller C[sub(d)](z) {#design-of-mixed-h-sub--2--h-sub-----controller-c-sub--d----z}
### Application Results {#application-results}
#### System Modeling {#system-modeling}
#### H[sub(2)] Active Damping Control {#h-sub--2--active-damping-control}
#### Mixed H[sub(2)]/H[sub(∞)] Active Damping Control {#mixed-h-sub--2--h-sub-----active-damping-control}
#### Experimental Results {#experimental-results}
### Conclusion {#conclusion}
## Self-Sensing Actuation of Dual-Stage Systems {#self-sensing-actuation-of-dual-stage-systems}
### Introduction {#introduction}
### Estimation of PZT Secondary Actuator’s Displacement y[sub(p)][sup(\*)] {#estimation-of-pzt-secondary-actuator-s-displacement-y-sub--p--sup}
#### Self-Sensing Actuation and Bridge Circuit {#self-sensing-actuation-and-bridge-circuit}
#### PZT Displacement Estimation Circuit H[sub(B)] {#pzt-displacement-estimation-circuit-h-sub--b}
### Design of Controllers {#design-of-controllers}
#### VCM Controller and Controller C[sub(D)] {#vcm-controller-and-controller-c-sub--d}
#### PZT Controller {#pzt-controller}
### Performance Evaluation {#performance-evaluation}
#### Effectiveness of C[sub(D)] {#effectiveness-of-c-sub--d}
#### Position Errors {#position-errors}
### Conclusion {#conclusion}
## Modeling and Control of a MEMS Micro X–Y Stage Media Platform {#modeling-and-control-of-a-mems-micro-x-y-stage-media-platform}
### Introduction {#introduction}
### MEMS Micro X–Y Stage {#mems-micro-x-y-stage}
#### Design and Simulation of Micro X–Y Stage {#design-and-simulation-of-micro-x-y-stage}
##### Static {#static}
##### Dynamic {#dynamic}
#### Modeling of Micro X–Y Stage {#modeling-of-micro-x-y-stage}
#### Fabrication of the MEMS Micro X–Y Stage {#fabrication-of-the-mems-micro-x-y-stage}
### Capacitive Self-Sensing Actuation {#capacitive-self-sensing-actuation}
#### Design of CSSA Bridge Circuit {#design-of-cssa-bridge-circuit}
#### Experimental Verification {#experimental-verification}
### Robust Decoupling Controller Design {#robust-decoupling-controller-design}
#### Choice of Pre-Shaping Filters {#choice-of-pre-shaping-filters}
#### Controller Synthesis {#controller-synthesis}
#### Frequency Responses {#frequency-responses}
#### Time Responses {#time-responses}
#### Robustness Analysis {#robustness-analysis}
### Conclusion {#conclusion}
## Conclusions {#conclusions}
Many secondary actuators have been developed in addition to primary actuators in the field of mechanical actuation systems.
The aim is to provide high performance such as high precision and fast response.
Several types of secondary actuators have been introduced such as PZT milliactuator, electrostatic microactuator, PZT microactuator, and thermal microactuator.
Comparison of these secondary actuators has been made, and these secondary actuators have made dual and multi-stage actuation mechanisms possible.
Three-stage actuation systems have been proposed for the demand of wider bandwidth, to overcome the limitation by stroke constraint and saturation of secondary actuators.
After the characteristics of the three-stage systems have been developed and the models have been identified, the control strategy and algorithm have been developed to deal with vibrations and meet different requirements.
Particularly, for the three-stage actuation systems, the presented control strategies make it easy to further push the bandwidth and meet the performance requirement.
The control of the thermal microactuator based dual-stage system has been discussed in detail, including linearization and controller design method.
The developed advanced algorithms applied in the multi-stage systems include \\(\mathcal{H}\_\infty\\) loop shaping, anti-windup compensation, \\(\mathcal{H}\_2\\) control method,
and mixed \\(\mathcal{H}\_2/\mathcal{H}\_\infty\\) control method.
Typical problems of the milli and micro-actuators as the secondary actuators have been considered and appropriate solutions have been presented such as saturation compensation, hysteresis modeling and compensation, stroke limitation, and PZT self-sensing scheme.
Time delay and sampling rate effect on the control performance have been analyzed to help select appropriate sampling rate and design suitable controllers.
Specific usage of PZT elements has been produced for system performance improvement.
Using PZT elements as a sensor to deal with high-frequency vibration beyond the bandwidth has been proposed and systematic controller design methods have been developed.
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
<a id="du19_multi_actuat_system_contr"></a>Du, C., & Pang, C. K., *Multi-stage actuation systems and control* (2019), Boca Raton, FL: CRC Press. [↩](#28e6aa60b8446943c921ff0d6578ac85)

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title = "Design, modeling and control of nanopositioning systems"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
Reference
: <sup id="1851788e0c4aa5b06afe3362c73ea5eb"><a href="#fleming14_desig_model_contr_nanop_system" title="Andrew Fleming \&amp; Kam Leang, Design, Modeling and Control of Nanopositioning Systems, Springer International Publishing (2014).">(Andrew Fleming \& Kam Leang, 2014)</a></sup>
Author(s)
: Fleming, A. J., & Leang, K. K.
Year
: 2014
# Bibliography
<a id="fleming14_desig_model_contr_nanop_system"></a>Fleming, A. J., & Leang, K. K., *Design, modeling and control of nanopositioning systems* (2014), : Springer International Publishing. [↩](#1851788e0c4aa5b06afe3362c73ea5eb)

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title = "The art of electronics - third edition"
author = ["Thomas Dehaeze"]
draft = false
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Tags
: [Reference Books]({{< relref "reference_books" >}}), [Electronics]({{< relref "electronics" >}})
Reference
: <sup id="b6e665e07855d8348f2f3aa70ec9b563"><a href="#horowitz15_art_of_elect_third_edition" title="Horowitz, The Art Of Electronics - Third Edition, Cambridge University Press (2015).">(Horowitz, 2015)</a></sup>
Author(s)
: Horowitz, P.
Year
: 2015
# Bibliography
<a id="horowitz15_art_of_elect_third_edition"></a>Horowitz, P., *The art of electronics - third edition* (2015), New York, NY, USA: Cambridge University Press. [↩](#b6e665e07855d8348f2f3aa70ec9b563)

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title = "Fundamental principles of engineering nanometrology"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Metrology]({{< relref "metrology" >}})
Reference
: <sup id="58bd6e601168ed1397ab2ec3cc3bab2d"><a href="#leach14_fundam_princ_engin_nanom" title="Richard Leach, Fundamental Principles of Engineering Nanometrology, Elsevier (2014).">(Richard Leach, 2014)</a></sup>
Author(s)
: Leach, R.
Year
: 2014
## Measurement of angles {#measurement-of-angles}
Unit:
- radian for plane angle
- steradian for solid angle
\\(1 rad \approx 55.3deg\\)
Instrument principles:
- subdivision: index tacle, angular gratings, polygons, ...
- ratio of two lengths: angular interferometers, sin cars, small angle generators, ...
- autocollimators with a flat mirror
### How that thing with two autocollimators can work? {#how-that-thing-with-two-autocollimators-can-work}
## Sources of error in displacement interferometry {#sources-of-error-in-displacement-interferometry}
Two error sources:
- error sources that are proportional to the displacement being measured \\(L\\): cumulative errors
- error sources that are independent of the displacement being measured: non-cumulative errors
### Thermal expansion of the metrology frame {#thermal-expansion-of-the-metrology-frame}
### Deadpath length {#deadpath-length}
Deadpath length, \\(d\\), is defined as the difference in distance in air between the reference and measurement reflectors and the beam splitter when the interferometer measurement is initiated.
Deadpath error occurs when there is a non-zero deadpath and environmental conditions change during a measurement.
### Cosine error {#cosine-error}
\\(\Delta l = l(1-\cos(\theta))\\)
For small angles: \\(\Delta l = \frac{l \theta^2}{2}\\)
The cosine error is then a second-order effect, contrary to the Abbe error which is a first order effect.
The second order nature means that cosine error quickly diminish as the alignment is improved.
## Latest advances in displacement interferometry {#latest-advances-in-displacement-interferometry}
Commercial interferometers
=> fused silica optics housed in Invar mounts
=> all the optical components are mounted to one central optic to reduce the susceptibility to thermal variations
One advantage that homodyme systems have over heterodyne systems is their ability to readily have the source fibre delivered to the interferometer.
### Spatially separated interferometers {#spatially-separated-interferometers}
It uses heterodyne interferometer and one quadrant photodiode.
By knowing the beam size and detector geometry, the measurement target's angle change can be determined by differencing matched pairs of measured phase from the quadrant photodiode while the displacement is determined from the average phase over the four quadrants.
## Angular interferometers {#angular-interferometers}
Determination of an angle by the ratio of two lengths.
The angular optics is used to create two parallel beam paths between the angular interferometer and the angular reflector.
The beam that illuminates the angular optics contains two frequencies, \\(f1\\) and \\(f2\\). A polarising beam splitter in the angular interferometer splits the frequencies that travel along separate paths.
The measurement of angles is then relative.
This type of angular interferometer is used to measure small angles (less than \\(10deg\\)).
# Bibliography
<a id="leach14_fundam_princ_engin_nanom"></a>Leach, R., *Fundamental principles of engineering nanometrology* (2014), : Elsevier. [↩](#58bd6e601168ed1397ab2ec3cc3bab2d)

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title = "Basics of precision engineering - 1st edition"
author = ["Thomas Dehaeze"]
draft = false
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Tags
: [Precision Engineering]({{< relref "precision_engineering" >}})
Reference
: <sup id="cc6e42420309d21c1aa596152d84cf8b"><a href="#leach18_basic_precis_engin_edition" title="Richard Leach \&amp; Stuart Smith, Basics of Precision Engineering - 1st Edition, CRC Press (2018).">(Richard Leach \& Stuart Smith, 2018)</a></sup>
Author(s)
: Leach, R., & Smith, S. T.
Year
: 2018
# Bibliography
<a id="leach18_basic_precis_engin_edition"></a>Leach, R., & Smith, S. T., *Basics of precision engineering - 1st edition* (2018), : CRC Press. [↩](#cc6e42420309d21c1aa596152d84cf8b)

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title = "The design of high performance mechatronics - 2nd revised edition"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Reference Books]({{< relref "reference_books" >}})
Reference
: <sup id="37468bbe5988cc7f4878fcd664d9cb7f"><a href="#schmidt14_desig_high_perfor_mechat_revis_edition" title="Schmidt, Schitter \&amp; Rankers, The Design of High Performance Mechatronics - 2nd Revised Edition, Ios Press (2014).">(Schmidt {\it et al.}, 2014)</a></sup>
Author(s)
: Schmidt, R. M., Schitter, G., & Rankers, A.
Year
: 2014
Section 2.2 Mechanics
> The core of a mechatronic system is its mechanical construction and in spite of many decade of excellent designs, optimizing the mechanical structure in strength, mass and endurance, the mechanical behavior will always remain the limiting factor of the performance of any mechatronic system.
Section 2.2.2 Force and Motion
> _Statics_ deals with the stress levels that are present in the mechanical system when (quasi-)static forces are exerted on it.
> It analyses the linear and non-linear strain effects that are caused by elastic and plastic deformation under these stress levels.
>
> _Dynamics_ deals with the behaviour of the mechanical system under changing forces, while often the effects are linearised and limited to strain levels well below any irreversible plastic deformation.
> 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.
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
<a id="schmidt14_desig_high_perfor_mechat_revis_edition"></a>Schmidt, R. M., Schitter, G., & Rankers, A., *The design of high performance mechatronics - 2nd revised edition* (2014), : Ios Press. [↩](#37468bbe5988cc7f4878fcd664d9cb7f)

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title = "Papers"
author = ["Thomas Dehaeze"]
type = "paper"
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Here is the list of papers I took note about.

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title = "Active structural vibration control: a review"
author = ["Thomas Dehaeze"]
draft = false
tags = ["tag1", "tag2"]
categories = ["cat1", "cat2"]
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Tags
:
Reference
: <sup id="279b5558de3a8131b329a9ba1a99e4f8"><a href="#alkhatib03_activ_struc_vibrat_contr" title="Rabih Alkhatib \&amp; Golnaraghi, Active Structural Vibration Control: a Review, {The Shock and Vibration Digest}, v(5), 367-383 (2003).">(Rabih Alkhatib \& Golnaraghi, 2003)</a></sup>
Author(s)
: Alkhatib, R., & Golnaraghi, M. F.
Year
: 2003
## Process of designing an active vibration control system {#process-of-designing-an-active-vibration-control-system}
1. Analyze the structure to be controled
2. Obtain an idealized mathematical model with FEM or experimental modal analysis
3. Reduce the model order is necessary
4. Analyze the resulting model: dynamics properties, types of disturbances, ...
5. Quantify sensors and actuators requirements. Decide on their types and location
6. Analyze the impact of the sensors and actuators on the overall dynamic characteristics
7. Specify performance criteria and stability tradeoffs
8. Device of the type of control algorythm to be employed and design a controller to meet the specifications
9. Simulate the resulting controlled system on a computer
10. If the controller does not meet the requirements, adjust the specifications or modify the type of controller
11. Choose hardware and software and integrate the components on a pilot plant
12. Formulate experiments and perform system identification and model updating
13. Implement controller and carry out system test to evaluate the performance
## Feedback control {#feedback-control}
### Active damping {#active-damping}
The objective is to reduce the resonance peaks of the closed loop transfer function.
\\[T(s) = \frac{G(s)H(s)}{1+G(s)H(s)}\\]
Then \\(T(s) \approx G(s)\\) except near the resonance peaks where the amplitude is reduced.
This method can be realized without a model of the structure with **guaranteed stability**, granted that the actuators and sensors are **collocated**.
### Model based feedback {#model-based-feedback}
Objective: keep a control variable (position, velocity, ...) to a desired value in spite of external disturbances \\(d(s)\\).
We have \\[\frac{y(s)}{d(s)} = \frac{1}{1+G(s)H(s)}\\] so we need large values of \\(G(s)H(s)\\) in the frequency range where the disturbance has considerable effect.
To do so, we need a mathematical model of the system, then the control bandwidth and effectiveness are restricted by the accuracy of the model.
Unmodeled structural dynamics may destabilize the system.
## Feedforward Control {#feedforward-control}
We need a signal that is correlated to the disturbance. Then feedforward can improve performance over simple feedback control.
An adaptive filter manipulates the signal correlated to the disturbance and the output is applied to the system by the actuator.
The filter coefficients are adapted in such a way that an error signal is minimized.
The idea is to generate a secondary disturbance, which destructively interferes with the effect of the primary distance at the location of the error sensor.
However, there is no guarantee that the global response is also reduced at other locations.
The method is considered to be a **local technique**, in contrast to feedback which is global.
Contrary to active damping which can only reduce the vibration near the resonance, **feedforward control can be effective for any frequency**.
The major restriction to the application of feedforward adaptive filtering is the accessibility of a reference signal correlated to the disturbance.
<a id="table--table:comparison-constrol-strat"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--table:comparison-constrol-strat">Table 1</a></span>:
Comparison of control strategies
</div>
| Type of control | Advantages | Disadvantages |
|--------------------------------|---------------------------------------------|-----------------------------------------------|
| Active Damping | Simple to implement | Effective only near resonance |
| | Does not required accurate model | |
| | Guaranteed stability (collocated) | |
| Model Based | Global method | Requires accurate model |
| | Attenuate all disturbance within bandwidth | Required low delay |
| | | Limited bandwidth |
| | | Spillover |
| Feedforward Adaptive filtering | No model is necessary | Error signal required |
| | Robust to change in plant transfer function | Local method: may amplify vibration elsewhere |
| | More effective for narrowband disturbance | Large amount of real-time computation |
## Controllability and Observability {#controllability-and-observability}
Controllability and Observability are two fundamental qualitave properties of dynamic systems.
A system is said to be **controllable** if every state vector can be transform to a desirate state in finite time by the application of unconstrained control inputs.
A system is said to be **observable** at time \\(t\_0\\) if for a state \\(z(t\_0)\\), there is a finite time \\(t\_1>t\_0\\) such that the knowledge of the input \\(u(t)\\) and output \\(y(t)\\) from \\(t\_0\\) to \\(t\_1\\) are sufficient to determine the state \\(z(t\_0)\\).
## Coordinate Coupling Control {#coordinate-coupling-control}
Coordinate coupling control (CCC) is an **energy-basded method**.
The idea is to **transfer the vibrations** from a low or undamped oscilatory system (the plant) to a damped system (the controller).
This can be implemented passively using tuned mass damper. But the key advantage of this technique is that one can replace the physical absorber with a computer model. The coupling terms can then be selected to maximise the energy transfer.
## Robust control {#robust-control}
Robust control concentrates on the **tradeoffs between performance and stability** in the presence of uncertainty in the system model as well as the exogenous inputs to which it is subjected.
Uncertainty can be divided into four types:
- parameter errors
- error in model order
- neglected disturbances
- 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](#org95c575a).
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="org95c575a"></a>
{{< figure src="/ox-hugo/alkhatib03_hinf_control.png" caption="Figure 1: Block diagram for robust control" >}}
The generalized plan \\(G\\) can be partitionned according to the input-output variables. And we have that the transfer function matrix from \\(d\\) to \\(z\\) is:
\\[ H\_{z/d} = G\_{z/d} + G\_{z/u} K (I - G\_{y/u} K)^{-1} G\_{y/d} \\]
This transfer function matrix contains measures of performance and stability robustness.
The objective of \\(\mathcal{H}\_\infty\\) control is to design an admissible control \\(u(s)=K(s)y(s)\\) such that \\(\\| H\_{z/d} \\|\_\infty\\) is minimum.
## Optimal Control {#optimal-control}
The control \\(u(t)\\) is designed to minimize a cost function \\(J\\), given the initial conditions \\(z(t\_0)\\) and \\(\dot{z}(t\_0)\\) subject to the constraint that:
\begin{align\*}
\dot{z} &= Az + Bu\\\\\\
y &= Cz
\end{align\*}
One such cost function appropriate to a vibration control is
\\[J = 1/2 \int\_{t\_0}^{t\_f} ( z^T A z + u^T R u ) dt\\]
Where \\(Q\\) and \\(R\\) and positive definite symmetric weighting matrices.
## State Observers (Estimators) {#state-observers--estimators}
It is not always possible to determine the entire state variables. There are usualy too many degrees of freedom and only limited measurements.
The state vector \\(z(t)\\) can be estimated independently of the control problem, and the resulting estimate \\(\hat{z}(t)\\) can be used.
## Intelligent Structure and Controller {#intelligent-structure-and-controller}
Intelligent structure would have the capability to:
- recognize the present dynamic state of its own structure and evaluate the functional performance of the structure
- identify functional descriptions of external and internal disturbances
- detect changes in structural properties and changes in external and internal disturbances
- predict possible future changes in structural properties
- make intelligent decisions regarding compensations for disturbances and adequately generale actuation forces
- learn from past performance to improve future actions
Two main methodologies:
- artificial neural networks
- fuzzy logic
## Adaptive Control {#adaptive-control}
Adaptive control is frequently used to control systems whose parameters are unknown, uncertain, or slowly varying.
The design of an adaptive controller involves several steps:
- selection of a controller structure with adjustable parameters
- selection of an adaptation law for adjusting those parameters
- selection of a performance index
- real-time evaluation of the performance with respect to some desired behavior
- real-time plant identification and model updating
- real-time adjustment of the controller parameters to bring the performance closer to the desired behavior
It essentially consists of a real-time system identification technique integrated with a control algorithm.
Two different methods
- **Direct method**: the controller parameters are adjusted directly based on the error between the measured and desired outputs.
- **Indirect method**: the computations are divided into two consecutive phases. First, the plant model is first estimated in real time. Second, the controller parameters are modified based on the most recent updated plant parameters.
## Active Control Effects on the System {#active-control-effects-on-the-system}
<a id="org7c357dd"></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](#org7c357dd), 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:
\\[ (m+g\_a)\ddot{x} + (c+g\_v)\dot{x} + (k+g\_d)x = f \\]
Depending of the type of signal used, the active control adds/substracts mass, damping and stiffness.
## Time Delays {#time-delays}
One of the limits to the performance of active control is the time delay in controllers and actuators. Time delay introduces phase shift, which deteriorates the controller performance or even causes instability in the system.
## Optimal Placement of Actuators {#optimal-placement-of-actuators}
The problem of optimizing the locations of the actuators can be more significant than the control law itself.
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
<a id="alkhatib03_activ_struc_vibrat_contr"></a>Alkhatib, R., & Golnaraghi, M. F., *Active structural vibration control: a review*, The Shock and Vibration Digest, *35(5)*, 367–383 (2003). http://dx.doi.org/10.1177/05831024030355002 [↩](#279b5558de3a8131b329a9ba1a99e4f8)

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title = "Guidelines for the selection of weighting functions for h-infinity control"
author = ["Thomas Dehaeze"]
draft = false
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Tags
: [H Infinity Control]({{< relref "h_infinity_control" >}})
Reference
: <sup id="5b41da575e27e6e86f1a1410a0170836"><a href="#bibel92_guidel_h" title="Bibel \&amp; Malyevac, Guidelines for the selection of weighting functions for H-infinity control, NAVAL SURFACE WARFARE CENTER DAHLGREN DIV VA, (1992).">(Bibel \& Malyevac, 1992)</a></sup>
Author(s)
: Bibel, J. E., & Malyevac, D. S.
Year
: 1992
## Properties of feedback control {#properties-of-feedback-control}
<a id="org82bead2"></a>
{{< figure src="/ox-hugo/bibel92_control_diag.png" caption="Figure 1: Control System Diagram" >}}
From the figure [1](#org82bead2), we have:
\begin{align\*}
y(s) &= T(s) r(s) + S(s) d(s) - T(s) n(s)\\\\\\
e(s) &= S(s) r(s) - S(s) d(s) - S(s) n(s)\\\\\\
u(s) &= S(s)K(s) r(s) - S(s)K(s) d(s) - S(s)K(s) n(s)
\end{align\*}
With the following definitions
- \\(L(s) = G(s)K(s)\\) is the **loop transfer matrix**
- \\(S(s) = [I+G(s)K(s)]^{-1}\\) is the **Sensitivity** function matrix
- \\(T(s) = [I+G(s)K(s)]^{-1}G(s)K(s)\\) is the **Transmissibility** function matrix
<div class="cbox">
<div></div>
\\[ S(s) + T(s) = 1 \\]
</div>
<div class="cbox">
<div></div>
- **Command following**: \\(S=0\\) and \\(T=1\\) => large gains
- **Disturbance rejection**: \\(S=0\\) => large gains
- **Sensor noise attenuation**: \\(T\\) small where the noise is concentrated
- **Control Sensitivity minimization**: \\(K S\\) small
- **Robustness to modeling errors**: \\(T\\) small in the frequency range of the expected model undertainties
</div>
## SISO tradeoff {#siso-tradeoff}
We want \\(S\\) small for command following and disturbance rejection.
We want \\(T\\) small to remain insensitive to sensor noise and modeling errors and to reduce control sensitivity.
However we cannot keep both \\(S\\) and \\(T\\) small as \\(S(s)+T(s)=1\\).
We must determine some **tradeoff** between the sensitivity and the complementary sensitivity functions.
Usually, reference signals and disturbances occur at low frequencies, while noise and modeling errors are concentrated at high frequencies. The tradeoff, in a SISO sense, is to make \\(|S(j\omega)|\\) small as low frequencies and \\(|T(j\omega)|\\) small at high frequencies.
## \\(H\_\infty\\) and weighting functions {#h-infty--and-weighting-functions}
<div class="cbox">
<div></div>
\\(\mathcal{H}\_\infty\\) control is a design technique with a state-space computation solution that utilizes frequency-dependent weighting functions to tune the controller's performance and robustness characteristics.
</div>
<a id="org71ea720"></a>
{{< figure src="/ox-hugo/bibel92_general_plant.png" caption="Figure 2: \\(\mathcal{H}\_\infty\\) control framework" >}}
New design framework (figure [2](#org71ea720)): \\(P(s)\\) is the **generalized plant** transfer function matrix:
- \\(w\\): exogenous inputs
- \\(z\\): regulated performance output
- \\(u\\): control inputs
- \\(y\\): measured output variables
The plant \\(P\\) has two inputs and two outputs, it can be decomposed into four sub-transfer function matrices:
\\[P = \begin{bmatrix}P\_{11} & P\_{12} \\ P\_{21} & P\_{22} \end{bmatrix}\\]
## Lower Linear Fractional Transformation {#lower-linear-fractional-transformation}
The transformation from the input \\(w\\) to the output \\(z\\), \\(T\_{zw}\\) is called the **Lower Linear Fractional Transformation** \\(F\_l (P, K)\\).
<div class="cbox">
<div></div>
\\[T\_{zw} = F\_l (P, K) = P\_{11} + P\_{12}K (I-P\_{22})^{-1} P\_{21}\\]
</div>
The \\(H\_\infty\\) control problem is to find a controller that minimizes \\(\\| T\_{zw} \\|\_\infty\\) over the space of all realizable controllers \\(K(s)\\) that stabilize the closed-loop system.
## 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](#org549c59f)).
<a id="org549c59f"></a>
{{< figure src="/ox-hugo/bibel92_hinf_weights.png" caption="Figure 3: Input and Output weights in \\(\mathcal{H}\_\infty\\) framework" >}}
The weights on the input and output variables are selected to reflect the spatial and **frequency dependence** of the respective signals and performance specifications.
These inputs and output weighting functions are defined as rational, stable and **minimum-phase transfer function** (no poles or zero in the right half plane).
## General Guidelines for Weight Selection: \\(W\_S\\) {#general-guidelines-for-weight-selection--w-s}
\\(W\_S\\) is selected to reflect the desired **performance characteristics**.
The sensitivity function \\(S\\) should have low gain at low frequency for good tracking performance and high gain at high frequencies to limit overshoot.
We have to select \\(W\_S\\) such that \\({W\_S}^-1\\) reflects the desired shape of \\(S\\).
<div class="cbox">
<div></div>
- **Low frequency gain**: set to the inverse of the desired steady state tracking error
- **High frequency gain**: set to limit overshoot (\\(0.1\\) to \\(0.5\\) is a good compromise between overshoot and response speed)
- **Crossover frequency**: chosen to limit the maximum closed-loop time constant (\\(\omega\_c \approx 1/\tau\\))
</div>
## General Guidelines for Weight Selection: \\(W\_T\\) {#general-guidelines-for-weight-selection--w-t}
We want \\(T\\) near unity for good tracking of reference and near zero for noise suppresion.
<div class="cbox">
<div></div>
A high pass weight is usualy used on \\(T\\) because the noise energy is mostly concentrated at high frequencies. It should have the following characteristics:
- The **crossover frequency** is chosen to **limit the closed-loop bandwidth**
- The **high frequency gain** is set high to proide **sensor noise rejection** and high frequency gain attenuation
</div>
When using both \\(W\_S\\) and \\(W\_T\\), it is important to make sure that the magnitude of theise weights at the crossover frequency is less that one to not violate \\(S+T=1\\).
## 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](#org379d5b1)).
<a id="org379d5b1"></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](#orgcc65489).
<a id="orgcc65489"></a>
{{< figure src="/ox-hugo/bibel92_weight_dynamics.png" caption="Figure 5: Example of unmodeled dynamics weight" >}}
## Inputs and Output weighting function {#inputs-and-output-weighting-function}
It is possible to **weight the control input and actuator rate**.
This is used to **prevent actuator saturation** and **limit amplification of sensor noise signals** on the control input signal.
Typically actuator input weights are constant over frequency and set at the inverse of the saturation limit.
## Order of the weighting functions {#order-of-the-weighting-functions}
**The order of the optimal controller is equal to the order of the nominal plant model plus the order of the weights**. The complexity of the controller is increase as the order of the weights increases.
**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
<a id="bibel92_guidel_h"></a>Bibel, J. E., & Malyevac, D. S., *Guidelines for the selection of weighting functions for h-infinity control* (1992). [↩](#5b41da575e27e6e86f1a1410a0170836)

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title = "Position control in lithographic equipment"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Multivariable Control]({{< relref "multivariable_control" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
Reference
: <sup id="6a014e3a2ee3e41d20bd0644654c56f0"><a href="#butler11_posit_contr_lithog_equip" title="Hans Butler, Position Control in Lithographic Equipment, {IEEE Control Systems}, v(5), 28-47 (2011).">(Hans Butler, 2011)</a></sup>
Author(s)
: Butler, H.
Year
: 2011
# Bibliography
<a id="butler11_posit_contr_lithog_equip"></a>Butler, H., *Position control in lithographic equipment*, IEEE Control Systems, *31(5)*, 28–47 (2011). http://dx.doi.org/10.1109/mcs.2011.941882 [↩](#6a014e3a2ee3e41d20bd0644654c56f0)

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title = "Identification and decoupling control of flexure jointed hexapods"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: <sup id="ba05ff213f8e5963d91559d95becfbdb"><a href="#chen00_ident_decoup_contr_flexur_joint_hexap" title="Yixin Chen \&amp; McInroy, Identification and Decoupling Control of Flexure Jointed Hexapods, nil, in in: {Proceedings 2000 ICRA. Millennium Conference. IEEE
International Conference on Robotics and Automation. Symposia
Proceedings (Cat. No.00CH37065)}, edited by (2000)">(Yixin Chen \& McInroy, 2000)</a></sup>
Author(s)
: Chen, Y., & McInroy, J.
Year
: 2000
# Bibliography
<a id="chen00_ident_decoup_contr_flexur_joint_hexap"></a>Chen, Y., & McInroy, J., *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) (pp. ) (2000). : . [↩](#ba05ff213f8e5963d91559d95becfbdb)

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title = "Review of active vibration isolation strategies"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: <sup id="2d69d483f210ca387ca8061596ec27ea"><a href="#collette11_review_activ_vibrat_isolat_strat" title="Christophe Collette, Stef Janssens \&amp; Kurt Artoos, Review of Active Vibration Isolation Strategies, {Recent Patents on Mechanical Engineeringe}, v(3), 212-219 (2011).">(Christophe Collette {\it et al.}, 2011)</a></sup>
Author(s)
: Collette, C., Janssens, S., & Artoos, K.
Year
: 2011
## Background and Motivations {#background-and-motivations}
### Passive Isolation Tradeoffs {#passive-isolation-tradeoffs}
<div class="cbox">
<div></div>
\\[ 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>
- \\(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.
In order to minimize the vibrations of a sensitive equipment, a general objective to design a good isolator is to minimize both \\(\abs{T\_{wx}}\\) and \\(\abs{T\_{Fx}}\\) in the frequency range of interest.
To decrease the amplitude of the overshoot at the resonance frequency, **damping** can be increased.
The price to pay is degradation of the isolation at high frequency (the roll off becomes \\(-1\\) instead of \\(-2\\)).
**First Trade-off**: Trade-off between damping and isolation.
To improve the transmissibility, the resonance frequency can be decreased.
However, the systems becomes more sensitive to external force \\(F\\) applied on the equipment.
**Second trade-off**: Trade-off between isolation and robustness to external force
### Active Isolation {#active-isolation}
We apply a feedback control.
The general expression of the force delivered by the actuator is \\(f = g\_a \ddot{x} + g\_v \dot{x} + g\_p x\\). \\(g\_a\\), \\(g\_v\\) and \\(g\_p\\) are constant gains.
<a id="table--table:active-isolation"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--table:active-isolation">Table 1</a></span>:
Active isolation techniques
</div>
| **Feedback Signal** | **Effect** | **Applications** |
|---------------------|------------------------------------------|------------------|
| Acceleration | Add virtual mass | Few |
| Velocity | Add virtual dashpot connected to the sky | Sky-Hook Damping |
| Position | Add virtual spring connected to the sky | Sky-Hook Spring |
## Practical Realizations {#practical-realizations}
## Sensor Limitations {#sensor-limitations}
## Conclusions {#conclusions}
<a id="orgef29aaf"></a>
{{< figure src="/ox-hugo/collette11_comp_isolation_strategies.png" caption="Figure 1: Comparison of Active Vibration Isolation Strategies" >}}
# Bibliography
<a id="collette11_review_activ_vibrat_isolat_strat"></a>Collette, C., Janssens, S., & Artoos, K., *Review of active vibration isolation strategies*, Recent Patents on Mechanical Engineeringe, *4(3)*, 212–219 (2011). http://dx.doi.org/10.2174/2212797611104030212 [↩](#2d69d483f210ca387ca8061596ec27ea)

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title = "Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Sensor Fusion]({{< relref "sensor_fusion" >}})
Reference
: <sup id="1223611da2f9b157af97503a4fec7631"><a href="#collette14_vibrat" title="Collette \&amp; Matichard, Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs, in in: {International Conference on Noise and Vibration Engineering
(ISMA2014)}, edited by (2014)">(Collette \& Matichard, 2014)</a></sup>
Author(s)
: Collette, C., & Matichard, F.
Year
: 2014
## Introduction {#introduction}
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
- Force sensor / collocated sensor to improve the robustness
## Different types of sensors {#different-types-of-sensors}
In this paper, three types of sensors are used. Their advantages and disadvantages are summarized table [1](#table--tab:sensors).
> Several types of sensors can be used for the feedback control of vibration isolation systems:
>
> - Feedback control based on **relative motion sensors** (inductive, capactive, ferromagnetic sensors...) typically permits to servo-position a system or platform relative to a reference (e.g. floor or support base), but does not provide isolation from the ground motion.
> - Feedback control based on **force sensors** typically lowers the effective natural frequency, and therefore increases the isolation, but sacrifices the systems compliance in doing so.
> - Feedback control based on **inertial sensors** (geophones, seismometers, accelerometers...) improves not only the vibration isolation but also the compliance. Inertial sensors are, however, AC coupled and noisy at low frequencies.
<a id="table--tab:sensors"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:sensors">Table 1</a></span>:
Types of sensors
</div>
| Sensors | Advantages | Disadvantages |
|------------------|----------------------------------|---------------------------------------|
| Relative motion | Servo-position | No isolation from gorund motion |
| Force sensors | Improve isolation | Increase compliance |
| Inertial sensors | Improve isolation and compliance | AC couple and noisy at high frequency |
## Inertial Control and sensor fusion configurations {#inertial-control-and-sensor-fusion-configurations}
For a simple 1DoF model, two fusion-sensor configuration are studied. The results are summarized Table [2](#table--tab:fusion-trade-off).
<a id="table--tab:fusion-trade-off"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:fusion-trade-off">Table 2</a></span>:
Sensor fusion configurations
</div>
| Low freq. sensor | High freq. sensor | Transmissibility | Compliance | Trade-off |
|------------------|-------------------|------------------|------------|----------------------------------------------------|
| Inertial | Force sensor | Unchanged | Degraded | Sensor noise filtering / compliance degradation |
| Inertial | Relative sensor | Degraded | Unchanged | Isolation in the bandwidth / amplification outside |
## Flexible structure {#flexible-structure}
Flexibility is added between the inertial sensor and the actuator.
Now the sensor and actuator are not collocated anymore and the system is unstable because there is no zero between the two poles.
We use sensor fusion to obtain stability at high frequency.
### Inertial and small accelerometer {#inertial-and-small-accelerometer}
The idea is to use a small accelerometer which is easier to locate near the actuator at high frequency.
However, it is important to verify that the noise introduced by the accelerometer does not degrades too much the isolation performance.
### Inertial and force sensor {#inertial-and-force-sensor}
Here the advantage is that the deformation mode is almost not present in the open-loop transfer function.
This simplifies the loop shaping of the controller.
### Inertial and relative sensor {#inertial-and-relative-sensor}
The relative sensor introduces coupling between both side of the actuator which induces degradation of the isolation at high frequency. However, the compliance remains unchanged at high frequency.
## Conclusion {#conclusion}
Fusion of inertial instruments with sensors collocated with the actuator permits to increase the feedback control bandwidth of active isolation systems.
Three types of sensors have been considered for the high frequency part of the fusion:
- The fusion with a **relative sensor** improves the stability but compromises the transmissibility. It can be of interested for stiff suspension where high frequency isolation can be sacrified to improve stability.
- The fusion with an **accelerometre** is used to increase the loop gain. However, as the accelerometer is not dual with the actuator, there is no guaranty stability when the isolation stage is mounted on a flexible support.
- 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
<a id="collette14_vibrat"></a>Collette, C., & Matichard, F., *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) (pp. ) (2014). : . [↩](#1223611da2f9b157af97503a4fec7631)

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title = "Sensor fusion methods for high performance active vibration isolation systems"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: <sup id="7772841a8f05142ec30f0f6daae20932"><a href="#collette15_sensor_fusion_method_high_perfor" title="Collette \&amp; Matichard, Sensor Fusion Methods for High Performance Active Vibration Isolation Systems, {Journal of Sound and Vibration}, v(nil), 1-21 (2015).">(Collette \& Matichard, 2015)</a></sup>
Author(s)
: Collette, C., & Matichard, F.
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 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:
- the high frequency sensor and the actuator are dual
- there exists a bandwidth where we can superimpose the open loop transfer functions obtained with the two sensors.
# Bibliography
<a id="collette15_sensor_fusion_method_high_perfor"></a>Collette, C., & Matichard, F., *Sensor fusion methods for high performance active vibration isolation systems*, Journal of Sound and Vibration, *342(nil)*, 1–21 (2015). http://dx.doi.org/10.1016/j.jsv.2015.01.006 [↩](#7772841a8f05142ec30f0f6daae20932)

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title = "The stewart platform manipulator: a review"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference
: <sup id="ad17e03f0fbbcc1a070557d7b5a0e1e1"><a href="#dasgupta00_stewar_platf_manip" title="Bhaskar Dasgupta \&amp; Mruthyunjaya, The Stewart Platform Manipulator: a Review, {Mechanism and Machine Theory}, v(1), 15-40 (2000).">(Bhaskar Dasgupta \& Mruthyunjaya, 2000)</a></sup>
Author(s)
: Dasgupta, B., & Mruthyunjaya, T.
Year
: 2000
<a id="table--tab:parallel-vs-serial-manipulators"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:parallel-vs-serial-manipulators">Table 1</a></span>:
Parallel VS serial manipulators
</div>
| | **Advantages** | **Disadvantages** |
|--------------|---------------------------|-----------------------|
| **Serial** | Manoeuverability | Poor precision |
| | Large workspace | Bends under high load |
| | | Vibrate at high speed |
| **Parallel** | High stiffness | Small workspace |
| | Good dynamic performances | |
| | Precise positioning | |
The generalized Stewart platforms consists of two rigid bodies (referred to as the base and the platoform) connected through six extensible legs, each with sherical joints at both ends.
# Bibliography
<a id="dasgupta00_stewar_platf_manip"></a>Dasgupta, B., & Mruthyunjaya, T., *The stewart platform manipulator: a review*, Mechanism and Machine Theory, *35(1)*, 15–40 (2000). http://dx.doi.org/10.1016/s0094-114x(99)00006-3 [↩](#ad17e03f0fbbcc1a070557d7b5a0e1e1)

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title = "A survey of control issues in nanopositioning"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
Reference
: <sup id="8ce53b8a612ce8ae3eb616cd1ed05630"><a href="#devasia07_survey_contr_issues_nanop" title="Devasia, Eleftheriou, Moheimani \&amp; SO Reza, A Survey of Control Issues in Nanopositioning, {IEEE Transactions on Control Systems Technology}, v(5), 802--823 (2007).">(Devasia {\it et al.}, 2007)</a></sup>
Author(s)
: Devasia, S., Eleftheriou, E., & Moheimani, S. R.
Year
: 2007
- Talks about Scanning Tunneling Microscope (STM) and Scanning Probe Microscope (SPM)
- 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="orga0f4b4e"></a>
{{< figure src="/ox-hugo/devasia07_piezoelectric_tradeoff.png" caption="Figure 1: Tradeoffs between bandwidth, precision and range" >}}
# Bibliography
<a id="devasia07_survey_contr_issues_nanop"></a>Devasia, S., Eleftheriou, E., & Moheimani, S. R., *A survey of control issues in nanopositioning*, IEEE Transactions on Control Systems Technology, *15(5)*, 802–823 (2007). [↩](#8ce53b8a612ce8ae3eb616cd1ed05630)

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title = "Nanopositioning system with force feedback for high-performance tracking and vibration control"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Force Sensors]({{< relref "force_sensors" >}})
Reference
: <sup id="c823f68dd2a72b9667a61b3c046b4731"><a href="#fleming10_nanop_system_with_force_feedb" title="Fleming, Nanopositioning System With Force Feedback for High-Performance Tracking and Vibration Control, {IEEE/ASME Transactions on Mechatronics}, v(3), 433-447 (2010).">(Fleming, 2010)</a></sup>
Author(s)
: Fleming, A.
Year
: 2010
Summary:
- The noise generated by a piezoelectric force sensor is much less than a capacitive sensor
- Dynamical model of a piezoelectric stack actuator and piezoelectric force sensor
- Noise of a piezoelectric force sensor
- IFF with a piezoelectric stack actuator and piezoelectric force sensor
- A force sensor is used as a displacement sensor below the frequency of the first zero
- Sensor fusion architecture with a capacitive sensor and a force sensor and using complementary filters
- Virtual sensor fusion architecture (called low-frequency bypass)
- Analog implementation of the control strategies to avoid quantization noise, finite resolution and sampling delay
## Model of a multi-layer monolithic piezoelectric stack actuator {#model-of-a-multi-layer-monolithic-piezoelectric-stack-actuator}
<a id="orgf7e4ab9"></a>
{{< figure src="/ox-hugo/fleming10_piezo_model.png" caption="Figure 1: Schematic of a multi-layer monolithic piezoelectric stack actuator model" >}}
The actuator experiences an internal stress in response to an applied voltage.
This stress is represented by the voltage dependent force \\(F\_a\\) and is related to free displacement by
\\[ \Delta L = \frac{F\_a}{k\_a} \\]
- \\(\Delta L\\) is the change in actuator length in [m]
- \\(k\_a\\) is the actuator stiffness in [N/m]
The developed force \\(F\_a\\) is related to the applied voltage by:
\\[ \Delta L = d\_{33} n V\_a \\]
- \\(d\_{33}\\) is the piezoelectric strain constant in [m/V]
- \\(n\\) is the number of layers
- \\(V\_a\\) is the applied voltage in [V]
Combining the two equations, we obtain:
\\[ F\_a = d\_{33} n k\_a V\_a \\]
The ratio of the developed force to applied voltage is \\(d\_{33} n k\_a\\) in [N/V].
We denote this constant by \\(g\_a\\) and:
\\[ F\_a = g\_a V\_a, \quad g\_a = d\_{33} n k\_a \\]
## Dynamics of a piezoelectric force sensor {#dynamics-of-a-piezoelectric-force-sensor}
Piezoelectric force sensors provide a high sensitivity and bandwidth with low noise at high frequencies.
If a **single wafer** of piezoelectric material is sandwiched between the actuator and platform:
\\[ D = d\_{33} T \\]
- \\(D\\) is the amount of generated charge per unit area in \\([C/m^2]\\)
- \\(T\\) is the stress in \\([N/m^2]\\)
- \\(d\_{33}\\) is the piezoelectric strain constant in \\([m/V] = [C/N]\\)
The generated charge is then
\\[ q = d\_{33} F\_s \\]
If an **n-layer** piezoelectric transducer is used as a force sensor, the generated charge is then:
\\[ q = n d\_{33} F\_s \\]
---
We can use a **charge amplifier** to measure the force \\(F\_s\\).
{{< figure src="/ox-hugo/fleming10_charge_ampl_piezo.png" caption="Figure 2: Electrical model of a piezoelectric force sensor is shown in gray. Developed charge \\(q\\) is proportional to the strain and hence the force experienced by the sensor. Op-amp charge amplifier produces an output voltage \\(V\_s\\) equal to \\(-q/C\_s\\)" >}}
The output voltage \\(V\_s\\) is equal to
\\[ V\_s = -\frac{q}{C\_s} = -\frac{n d\_{33}F\_s}{C\_s} \\]
that is, the scaling between the force and voltage is \\(-\frac{n d\_{33}F\_s}{C\_s}\ [V/N]\\) .
---
We can also use a voltage amplifier.
In that case, the generated charge is deposited on the transducer's internal capacitance.
The open-circuit voltage of a piezoelectric force sensor is:
\\[ V\_s = \frac{n d\_{33} F\_s}{C} \\]
- \\(C\\) is the transducer capacitance defined by \\(C = n \epsilon\_T A / h\\) in [F]
- \\(A\\) is the area in \\([m^2]\\)
- \\(h\\) is the layer thickness in [m]
- \\(\epsilon\_T\\) is the dielectric permittivity under a constant stress in \\([F/m]\\)
We obtain
\\[ V\_s = g\_s F\_s, \quad g\_s = \frac{n d\_{33}}{C} \\]
## Noise of a piezoelectric force sensor {#noise-of-a-piezoelectric-force-sensor}
As piezoelectric sensors have a capacitive source impedance, the sensor noise density \\(N\_{V\_s}(\omega)\\) is primarily due to current noise \\(i\_n\\) reacting the capacitive source impedance:
\\[ N\_{V\_s}(\omega) = i\_n \frac{1}{C \omega} \\]
- \\(N\_{V\_s}\\) is the measured noise in \\(V/\sqrt{\text{Hz}}\\)
- \\(i\_n\\) is the current noise in \\(A/\sqrt{\text{Hz}}\\)
- \\(C\\) is the capacitance of the piezoelectric in \\(F\\)
The current noise density of a general purpose LM833 FET-input op-amp is \\(0.5\ pA/\sqrt{\text{Hz}}\\).
The capacitance of a piezoelectric stack is typically between \\(1 \mu F\\) and \\(100 \mu F\\).
# Bibliography
<a id="fleming10_nanop_system_with_force_feedb"></a>Fleming, A., *Nanopositioning system with force feedback for high-performance tracking and vibration control*, IEEE/ASME Transactions on Mechatronics, *15(3)*, 433–447 (2010). http://dx.doi.org/10.1109/tmech.2009.2028422 [↩](#c823f68dd2a72b9667a61b3c046b4731)
## Backlinks {#backlinks}
- [Actuators]({{< relref "actuators" >}})
- [Force Sensors]({{< relref "force_sensors" >}})

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title = "Estimating the resolution of nanopositioning systems from frequency domain data"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
Reference
: <sup id="a1cc9b70316a7dda2f652efd146caf84"><a href="#fleming12_estim" title="Andrew Fleming, Estimating the resolution of nanopositioning systems from frequency domain data, nil, in in: {2012 IEEE International Conference on Robotics and
Automation}, edited by (2012)">(Andrew Fleming, 2012)</a></sup>
Author(s)
: Fleming, A. J.
Year
: 2012
# Bibliography
<a id="fleming12_estim"></a>Fleming, A. J., *Estimating the resolution of nanopositioning systems from frequency domain data*, In , 2012 IEEE International Conference on Robotics and Automation (pp. ) (2012). : . [↩](#a1cc9b70316a7dda2f652efd146caf84)

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title = "A review of nanometer resolution position sensors: operation and performance"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Position Sensors]({{< relref "position_sensors" >}})
Reference
: <sup id="3fb5b61524290e36d639a4fac65703d0"><a href="#fleming13_review_nanom_resol_posit_sensor" title="Andrew Fleming, A Review of Nanometer Resolution Position Sensors: Operation and Performance, {Sensors and Actuators A: Physical}, v(nil), 106-126 (2013).">(Andrew Fleming, 2013)</a></sup>
Author(s)
: Fleming, A. J.
Year
: 2013
- Define concise performance metric and provide expressions for errors sources (non-linearity, drift, noise)
- Review current position sensor technologies and compare their performance
## Sensor Characteristics {#sensor-characteristics}
### Calibration and nonlinearity {#calibration-and-nonlinearity}
Usually quoted as a percentage of the fill-scale range (FSR):
\begin{equation}
\text{mapping error (\%)} = \pm 100 \frac{\max{}|e\_m(v)|}{\text{FSR}}
\end{equation}
With \\(e\_m(v)\\) is the mapping error.
<a id="org64f54e9"></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." >}}
### Drift and Stability {#drift-and-stability}
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="org81ab6a9"></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." >}}
### Bandwidth {#bandwidth}
The bandwidth of a position sensor is the frequency at which the magnitude of the transfer function \\(P(s) = v(s)/x(s)\\) drops by \\(3\,dB\\).
Although the bandwidth specification is useful for predicting the resolution of sensor, it reveals very little about the measurement errors caused by sensor dynamics.
The frequency domain position error is
\begin{equation}
\begin{aligned}
e\_{bw}(s) &= x(s) - v(s) \\\\\\
&= x(s) (1 - P(s))
\end{aligned}
\end{equation}
If the actual position is a sinewave of peak amplitude \\(A = \text{FSR}/2\\):
\begin{equation}
\begin{aligned}
e\_{bw} &= \pm \frac{\text{FSR}}{2} |1 - P(s)| \\\\\\
&\approx \pm A n \frac{f}{f\_c}
\end{aligned}
\end{equation}
with \\(n\\) is the low pass filter order corresponding to the sensor dynamics and \\(f\_c\\) is the measurement bandwidth.
Thus, the sensor bandwidth must be significantly higher than the operating frequency if dynamic errors are to be avoided.
### Noise {#noise}
In addition to the actual position signal, all sensors produce some additive measurement noise.
In many types of sensor, the majority of noise arises from the thermal noise in resistors and the voltage and current noise in conditioning circuit transistors.
These noise processes can usually be approximated by a Gaussian random process.<br />
A Gaussian random process is usually described by its autocorrelation function or its Power Spectral Density.
The autocorrelation function of a random process \\(\mathcal{X}\\) is
\begin{equation}
R\_{\mathcal{X}}(\tau) = E[\mathcal{X}(t)\mathcal{X}(t + \tau)]
\end{equation}
where \\(E\\) is the expected value operator.
The variance of the process is equal to \\(R\_\mathcal{X}(0)\\) and is the expected value of the varying part squared:
\begin{equation}
\text{Var} \mathcal{X} = E \left[ (\mathcal{X} - E[\mathcal{X}])^2 \right]
\end{equation}
The standard deviation \\(\sigma\\) is the square root of the variance:
\begin{equation}
\sigma\_\mathcal{X} = \sqrt{\text{Var} \mathcal{X}}
\end{equation}
The standard deviation is also the Root Mean Square (RMS) value of a zero-mean random process.
The Power Spectral Density \\(S\_\mathcal{X}(f)\\) of a random process represents the distribution of power (or variance) across frequency \\(f\\).
For example, if the random process under consideration was measured in volts, the power spectral density would have the units of \\(V^2/\text{Hz}\\).
The Power Spectral Density can be obtained from the autocorrelation function from the Wiener-Khinchin relation:
\begin{equation}
S\_{\mathcal{X}} = 2 \mathcal{F}\\{ R\_\mathcal{X}(\tau) \\} = 2 \int\_{-\infty}^{\infty} R\_\mathcal{X}(\tau) e^{-2j\pi f \tau} d\tau
\end{equation}
If the power Spectral Density is known, the variance of the generating process can be found from the area under the curve:
\begin{equation}
\sigma\_\mathcal{X}^2 = E[\mathcal{X}^2(t)] = R\_\mathcal{X}(0) = \int\_0^\infty S\_\mathcal{X}(f) df
\end{equation}
Rather than plotting the frequency distribution of power, it is often convenient to plot the frequency distribution of the standard deviation, which is referred to as the spectral density.
It is related to the power spectral density by a square root:
\begin{equation}
\text{spectral density} = \sqrt{S\_\mathcal{X}(f)}
\end{equation}
The units of \\(\sqrt{S\_\mathcal{X}(f)}\\) are \\(\text{units}/\sqrt{Hz}\\).
The spectral density if preferred in the electronics literature as the RMS value of a noise process can be determined directly from the noise density and effective bandwidth.
### Resolution {#resolution}
The random noise of a position sensor causes an uncertainty in the measured position.
If the distance between two measured locations is smaller than the uncertainty, it is possible to mistake one point for the other.
To characterize the resolution, we use the probability that the measured value is within a certain error bound.
If the measurement noise is approximately Gaussian, the resolution can be quantified by the standard deviation \\(\sigma\\) (RMS value).
The empirical rule states that there is a \\(99.7\%\\) probability that a sample of a Gaussian random process lie within \\(\pm 3 \sigma\\).
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](#orgd8c6776)).
<a id="orgd8c6776"></a>
{{< figure src="/ox-hugo/fleming13_tradeoff_res_bandwidth.png" caption="Figure 3: The resolution versus banwidth of a position sensor." >}}
Many type of sensor have a limited full-scale-range (FSR) and tend to have an approximated proportional relationship between the resolution and range.
As a result, it is convenient to consider the ratio of resolution to the FSR, or equivalently, the dynamic range (DNR).
A convenient method for reporting this ratio is in parts-per-million (ppm):
\begin{equation}
\text{DNR}\_{\text{ppm}} = 10^6 \frac{\text{full scale range}}{6\sigma\text{-resolution}}
\end{equation}
## Comparison and summary {#comparison-and-summary}
<a id="table--tab:summary-position-sensors"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:summary-position-sensors">Table 1</a></span>:
Summary of position sensor characteristics. The dynamic range (DNR) and resolution are approximations based on a full-scale range of \(100\,\mu m\) and a first order bandwidth of \(1\,kHz\)
</div>
| Sensor Type | Range | DNR | Resolution | Max. BW | Accuracy |
|----------------|----------------------------------|---------|------------|----------|-----------|
| Metal foil | \\(10-500\,\mu m\\) | 230 ppm | 23 nm | 1-10 kHz | 1% FSR |
| Piezoresistive | \\(1-500\,\mu m\\) | 5 ppm | 0.5 nm | >100 kHz | 1% FSR |
| Capacitive | \\(10\,\mu m\\) to \\(10\,mm\\) | 24 ppm | 2.4 nm | 100 kHz | 0.1% FSR |
| Electrothermal | \\(10\,\mu m\\) to \\(1\,mm\\) | 100 ppm | 10 nm | 10 kHz | 1% FSR |
| Eddy current | \\(100\,\mu m\\) to \\(80\,mm\\) | 10 ppm | 1 nm | 40 kHz | 0.1% FSR |
| LVDT | \\(0.5-500\,mm\\) | 10 ppm | 5 nm | 1 kHz | 0.25% FSR |
| Interferometer | Meters | | 0.5 nm | >100kHz | 1 ppm FSR |
| Encoder | Meters | | 6 nm | >100kHz | 5 ppm FSR |
# Bibliography
<a id="fleming13_review_nanom_resol_posit_sensor"></a>Fleming, A. J., *A review of nanometer resolution position sensors: operation and performance*, Sensors and Actuators A: Physical, *190(nil)*, 106–126 (2013). http://dx.doi.org/10.1016/j.sna.2012.10.016 [↩](#3fb5b61524290e36d639a4fac65703d0)

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title = "Studies on stewart platform manipulator: a review"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference
: <sup id="cc10fe9545c7c381cc2b610e8f91a071"><a href="#furqan17_studies_stewar_platf_manip" title="Mohd Furqan, Mohd Suhaib \&amp; Nazeer Ahmad, Studies on Stewart Platform Manipulator: a Review, {Journal of Mechanical Science and Technology}, v(9), 4459-4470 (2017).">(Mohd Furqan {\it et al.}, 2017)</a></sup>
Author(s)
: Furqan, M., Suhaib, M., & Ahmad, N.
Year
: 2017
Lots of references.
# Bibliography
<a id="furqan17_studies_stewar_platf_manip"></a>Furqan, M., Suhaib, M., & Ahmad, N., *Studies on stewart platform manipulator: a review*, Journal of Mechanical Science and Technology, *31(9)*, 4459–4470 (2017). http://dx.doi.org/10.1007/s12206-017-0846-1 [↩](#cc10fe9545c7c381cc2b610e8f91a071)

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title = "Nanometre-cutting machine using a stewart-platform parallel mechanism"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: <sup id="bedab298599c84f60236313ebaad2714"><a href="#furutani04_nanom_cuttin_machin_using_stewar" title="Katsushi Furutani, Michio Suzuki \&amp; Ryusei Kudoh, Nanometre-Cutting Machine Using a Stewart-Platform Parallel Mechanism, {Measurement Science and Technology}, v(2), 467-474 (2004).">(Katsushi Furutani {\it et al.}, 2004)</a></sup>
Author(s)
: Furutani, K., Suzuki, M., & Kudoh, R.
Year
: 2004
- Lever mechanism to amplify the motion of piezoelectric stack actuators
- Use of flexure joints
- Eddy current displacement sensors for control (decentralized)
{{< figure src="/ox-hugo/furutani04_ctrl_arch.png" >}}
- Isotropic performance (cubic configuration even if not said so)
Possible sources of error:
- position error of the link ends in assembly => simulation of position error and it is not significant
- Inaccurate modelling of the links
- insufficient generative force
- unwanted deformation of the links
To minimize the errors, a calibration is done between the required leg length and the wanted platform pose.
Then, it is fitted with 4th order polynomial and included in the control architecture.
# Bibliography
<a id="furutani04_nanom_cuttin_machin_using_stewar"></a>Furutani, K., Suzuki, M., & Kudoh, R., *Nanometre-cutting machine using a stewart-platform parallel mechanism*, Measurement Science and Technology, *15(2)*, 467–474 (2004). http://dx.doi.org/10.1088/0957-0233/15/2/022 [↩](#bedab298599c84f60236313ebaad2714)

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title = "Measurement technologies for precision positioning"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Position Sensors]({{< relref "position_sensors" >}})
Reference
: <sup id="b820b918ced36901ea0ad4bf653202c6"><a href="#gao15_measur_techn_precis_posit" title="Gao, Kim, Bosse, Haitjema, , Chen, Lu, Knapp, Weckenmann, , Estler \&amp; Kunzmann, Measurement Technologies for Precision Positioning, {CIRP Annals}, v(2), 773-796 (2015).">(Gao {\it et al.}, 2015)</a></sup>
Author(s)
: Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …
Year
: 2015
# Bibliography
<a id="gao15_measur_techn_precis_posit"></a>Gao, W., Kim, S., Bosse, H., Haitjema, H., Chen, Y., Lu, X., Knapp, W., …, *Measurement technologies for precision positioning*, CIRP Annals, *64(2)*, 773–796 (2015). http://dx.doi.org/10.1016/j.cirp.2015.05.009 [↩](#b820b918ced36901ea0ad4bf653202c6)

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title = "Implementation challenges for multivariable control: what you did not learn in school!"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Multivariable Control]({{< relref "multivariable_control" >}})
Reference
: <sup id="07f63c751c1d9fcfe628178688f7ec24"><a href="#garg07_implem_chall_multiv_contr" title="Sanjay Garg, Implementation Challenges for Multivariable Control: What you did not learn in school!, nil, in in: {AIAA Guidance, Navigation and Control Conference and
Exhibit}, edited by (2007)">(Sanjay Garg, 2007)</a></sup>
Author(s)
: Garg, S.
Year
: 2007
Discusses:
- When to use multivariable control and when not to?
- Two major issues with implementing multivariable control: **gain scheduling** and **integrator wind up protection**
> Inline simple gain and phase margin measured for SISO, "robustness" determination of multivariable control requires complex analyses using **singular value techniques** and **Monte Carlo** simulations.
**When to use multivariable control**:
- System has high input/output coupling and not much separation between loop bandwidth
- System is complex with large number of states
- When sequential SISO loop closure will not meet performance requirements
Importance of having a mechanism to limit the control rate in the synthesis process.
The control rate should be weighted appropriately in order to not saturate the system and stay in the linearity regime.
- importance of scaling the plant prior to synthesis and also replacing pure integrators with slow poles
# Bibliography
<a id="garg07_implem_chall_multiv_contr"></a>Garg, S., *Implementation challenges for multivariable control: what you did not learn in school!*, In , AIAA Guidance, Navigation and Control Conference and Exhibit (pp. ) (2007). : . [↩](#07f63c751c1d9fcfe628178688f7ec24)

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title = "An intelligent control system for multiple degree-of-freedom vibration isolation"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: <sup id="76af0f5c88615842fa91864c8618fb58"><a class="reference-link" href="#geng95_intel_contr_system_multip_degree" title="Jason Geng, George Pan, Leonard Haynes, , Ben Wada \&amp; John Garba, An Intelligent Control System for Multiple Degree-Of-Freedom Vibration Isolation, {Journal of Intelligent Material Systems and Structures}, v(6), 787-800 (1995).">(Jason Geng {\it et al.}, 1995)</a></sup>
Author(s)
: Geng, Z. J., Pan, G. G., Haynes, L. S., Wada, B. K., & Garba, J. A.
Year
: 1995
<a id="org1384437"></a>
{{< figure src="/ox-hugo/geng95_control_structure.png" caption="Figure 1: Local force feedback and adaptive acceleration feedback for active isolation" >}}
# Bibliography
<a class="bibtex-entry" id="geng95_intel_contr_system_multip_degree"></a>Geng, Z. J., Pan, G. G., Haynes, L. S., Wada, B. K., & Garba, J. A., *An intelligent control system for multiple degree-of-freedom vibration isolation*, Journal of Intelligent Material Systems and Structures, *6(6)*, 787–800 (1995). http://dx.doi.org/10.1177/1045389x9500600607 [↩](#76af0f5c88615842fa91864c8618fb58)

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title = "Active isolation and damping of vibrations via stewart platform"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Active Damping]({{< relref "active_damping" >}})
Reference
: <sup id="10e535e895bdcd6b921bff33ef68cd81"><a href="#hanieh03_activ_stewar" title="@phdthesis{hanieh03_activ_stewar,
author = {Hanieh, Ahmed Abu},
school = {Universit{\'e} Libre de Bruxelles, Brussels, Belgium},
title = {Active isolation and damping of vibrations via Stewart
platform},
year = 2003,
tags = {parallel robot},
}">@phdthesis{hanieh03_activ_stewar,
author = {Hanieh, Ahmed Abu},
school = {Universit{\'e} Libre de Bruxelles, Brussels, Belgium},
title = {Active isolation and damping of vibrations via Stewart
platform},
year = 2003,
tags = {parallel robot},
}</a></sup>
Author(s)
: Hanieh, A. A.
Year
: 2003
# Bibliography
<a id="hanieh03_activ_stewar"></a>Hanieh, A. A., *Active isolation and damping of vibrations via stewart platform* (Doctoral dissertation) (2003). Universit{\'e} Libre de Bruxelles, Brussels, Belgium, . [↩](#10e535e895bdcd6b921bff33ef68cd81)

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title = "Sensors and control of a space-based six-axis vibration isolation system"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
Reference
: <sup id="f9698a1741fe7492aa9b7b42c7724670"><a href="#hauge04_sensor_contr_space_based_six" title="Hauge \&amp; Campbell, Sensors and Control of a Space-Based Six-Axis Vibration Isolation System, {Journal of Sound and Vibration}, v(3-5), 913-931 (2004).">(Hauge \& Campbell, 2004)</a></sup>
Author(s)
: Hauge, G., & Campbell, M.
Year
: 2004
**Discusses**:
- Choice of sensors and control architecture
- Predictability and limitations of the system dynamics
- Two-Sensor control architecture
- Vibration isolation using a Stewart platform
- Experimental comparison of Force sensor and Inertial Sensor and associated control architecture for vibration isolation
<a id="org666133a"></a>
{{< figure src="/ox-hugo/hauge04_stewart_platform.png" caption="Figure 1: Hexapod for active vibration isolation" >}}
**Stewart platform** (Figure [1](#org666133a)):
- Low corner frequency
- Large actuator stroke (\\(\pm5mm\\))
- Sensors in each strut (Figure [2](#org4d96564)):
- three-axis load cell
- base and payload geophone in parallel with the struts
- LVDT
<a id="org4d96564"></a>
{{< figure src="/ox-hugo/hauge05_struts.png" caption="Figure 2: Strut" >}}
> Force sensors typically work well because they are not as sensitive to payload and base dynamics, but are limited in performance by a low-frequency zero pair resulting from the cross-axial stiffness.
**Performance Objective** (frequency domain metric):
- The transmissibility should be close to 1 between 0-1.5Hz
\\(-3dB < |T(\omega)| < 3db\\)
- The transmissibility should be below -20dB in the 5-20Hz range
\\(|T(\omega)| < -20db\\)
With \\(|T(\omega)|\\) is the Frobenius norm of the transmissibility matrix and is used to obtain a scalar performance metric.
**Challenge**:
- small frequency separation between the two requirements
**Robustness**:
- minimization of the transmissibility amplification (Bode's "pop") outside the performance region
**Model**:
- single strut axis as the cubic Stewart platform can be decomposed into 6 single-axis systems
<a id="org74432f8"></a>
{{< figure src="/ox-hugo/hauge04_strut_model.png" caption="Figure 3: Strut model" >}}
**Zero Pair when using a Force Sensor**:
- The frequency of the zero pair corresponds to the resonance frequency of the payload mass and the "parasitic" stiffness (sum of the cross-axial, suspension, wiring stiffnesses)
- This zero pair is usually not predictable nor repeatable
- In this Stewart platform, this zero pair uncertainty is due to the internal wiring of the struts
**Control**:
- Single-axis controllers => combine them into a full six-axis controller => evaluate the full controller in terms of stability and robustness
- Sensitivity weighted LQG controller (SWLQG) => address robustness in flexible dynamic systems
- Three type of controller:
- Force feedback (cell-based)
- Inertial feedback (geophone-based)
- Combined force/velocity feedback (load cell/geophone based)
> The use of multivariable and robust control on the full 6x6 hexapod does not improve performance over single-axis designs.
<a id="table--tab:hauge05-comp-load-cell-geophone"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:hauge05-comp-load-cell-geophone">Table 1</a></span>:
Typical characteristics of sensors used for isolation in hexapod systems
</div>
| | **Load cell** | **Geophone** |
|-----------------------------------------|---------------------------------|-------------------------------------|
| Type | Relative | Inertial |
| Relationship with voice coil | Collocated and Dual | Non-Collocated and non-Dual |
| Open loop transfer function | (+) Alternating poles/zeros | (-) Large phase drop |
| Limitation from low-frequency zero pair | (-) Yes | (+) No |
| Sensitive to payload/base dynamics | (+) No | (-) Yes |
| Best frequency range | High (low-freq zero limitation) | Low (high-freq toll-off limitation) |
**Ability of a sensor-actuator pair to improve performance**:
General system with input \\(u\\), performance \\(z\\), output \\(y\\) disturbance \\(u\\).
Given a sensor \\(u\\) and actuator \\(y\\) and a controller \\(u = -K(s) y\\), the closed loop disturbance to performance transfer function can be written as:
\\[ \left[ \frac{z}{w} \right]\_\text{CL} = \frac{G(s)\_{zw} + K(G(s)\_{zw} G(s)\_{yu} - G(s)\_{zu} G(s)\_{yw})}{1 + K G(s)\_{yu}} \\]
In order to obtain a significant performance improvement is to use a high gain controller, _provided_ the term \\(G(s)\_{zw} + K(G(s)\_{zw} G(s)\_{yu} - G(s)\_{zu} G(s)\_{yw})\\) is small.
We can compare the transfer function from \\(w\\) to \\(z\\) with and without a high gain controller.
And we find that for \\(u\\) and \\(y\\) to be an acceptable pair for high gain control:
\\[ \left| \frac{G(j\omega)\_{zw} G(j\omega)\_{yu} - G(j\omega)\_{zu} G(j\omega)\_{yw}}{K G(j\omega)\_{yu}} \right| \ll |G\_{zw}(j\omega)| \\]
**Controllers**:
**Force feedback**:
- Performance limited by the low frequency zero-pair
- It is desirable to separate the zero-pair and first most are separated by at least a decade in frequency
- This can be achieve by reducing the cross-axis stiffness
- If the low frequency zero pair is inverted, robustness is lost
- Thus, the force feedback controller should be designed to have combined performance and robustness at frequencies at least a decade above the zero pair
- The presented controller as a high pass filter at to reduce the gain below the zero-pair, a lag at low frequency to improve phase margin, and a low pass filter for roll off
**Inertial feedback**:
- Non-Collocated => multiple phase drops that limit the bandwidth of the controller
- Good performance, but the transmissibility "pops" due to low phase margin and thus this indicates robustness problems
**Combined force/velocity feedback**:
- Use the low frequency performance advantages of geophone sensor with the high robustness advantages of the load cell sensor
- A Single-Input-Multiple-Outputs (SIMO) controller is found using LQG
- The performance requirements are met
- Good robustness
<a id="orgca6905f"></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
<a id="hauge04_sensor_contr_space_based_six"></a>Hauge, G., & Campbell, M., *Sensors and control of a space-based six-axis vibration isolation system*, Journal of Sound and Vibration, *269(3-5)*, 913–931 (2004). http://dx.doi.org/10.1016/s0022-460x(03)00206-2 [↩](#f9698a1741fe7492aa9b7b42c7724670)

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title = "An instrument for 3d x-ray nano-imaging"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
Reference
: <sup id="66ab0e7602a1dedda963d7da60533b0d"><a href="#holler12_instr_x_ray_nano_imagin" title="Holler, Raabe, Diaz, Guizar-Sicairos, , Quitmann, Menzel \&amp; Bunk, An Instrument for 3d X-Ray Nano-Imaging, {Review of Scientific Instruments}, v(7), 073703 (2012).">(Holler {\it et al.}, 2012)</a></sup>
Author(s)
: Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., & Bunk, O.
Year
: 2012
Instrument similar to the NASS.
Obtain position stability of 10nm (standard deviation).
<a id="orgba4a339"></a>
{{< figure src="/ox-hugo/holler12_station.png" caption="Figure 1: Schematic of the tomography setup" >}}
- **Limited resolution due to instrumentation**:
The resolution of ptychographic tomography remains above 100nm due to instabilities and drifts of the scanning systems.
- **Need of a Metrology System**:
> To achieve positioning accuracy and stability in the nanometer range, one cannot rely on the position encoders built into individual positioning stages.
> A precise exteroceptive measurement of the relative position of the optical elements with respect to the sample is mandatory.
> Thus, thermal drifts and parasitic motions can be measured and compensated for.
- **Interferometer System Concept**:
The sample is aligned with the X-ray with the XYZ piezo stage.
As a result, the metrology sphere will be usually off center with respect to the rotation axis of the spindle.
That implies that the laser will not propagate back to the interferometer at all rotation angles.
A position sensitive detector (PSD) is used, it provides a measurement of the position of the sphere in the plane perpendicular to the laser.
The interferometer is positionned on top of a translation stage. The PSD information is used to close the loop so that the interferometer follows the displacement of the metrology sphere.
- **Feedback Loop**: Using the signals from the 2 interferometers, the loop is closed to compensate low frequency vibrations and thermal drifts.
# Bibliography
<a id="holler12_instr_x_ray_nano_imagin"></a>Holler, M., Raabe, J., Diaz, A., Guizar-Sicairos, M., Quitmann, C., Menzel, A., & Bunk, O., *An instrument for 3d x-ray nano-imaging*, Review of Scientific Instruments, *83(7)*, 073703 (2012). http://dx.doi.org/10.1063/1.4737624 [↩](#66ab0e7602a1dedda963d7da60533b0d)

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title = "Active damping based on decoupled collocated control"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Active Damping]({{< relref "active_damping" >}})
Reference
: <sup id="cc7836a555fe4dbae791e17008c29bfd"><a href="#holterman05_activ_dampin_based_decoup_colloc_contr" title="Holterman \&amp; deVries, Active Damping Based on Decoupled Collocated Control, {IEEE/ASME Transactions on Mechatronics}, v(2), 135-145 (2005).">(Holterman \& deVries, 2005)</a></sup>
Author(s)
: Holterman, J., & deVries, T.
Year
: 2005
# Bibliography
<a id="holterman05_activ_dampin_based_decoup_colloc_contr"></a>Holterman, J., & deVries, T., *Active damping based on decoupled collocated control*, IEEE/ASME Transactions on Mechatronics, *10(2)*, 135–145 (2005). http://dx.doi.org/10.1109/tmech.2005.844702 [↩](#cc7836a555fe4dbae791e17008c29bfd)

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title = "Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Actuators]({{< relref "actuators" >}})
Reference
: <sup id="aad53368e29e8a519e2f63857044fa46"><a href="#ito16_compar_class_high_precis_actuat" title="Shingo Ito \&amp; Georg Schitter, Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation, {IEEE/ASME Transactions on Mechatronics}, v(2), 1169-1178 (2016).">(Shingo Ito \& Georg Schitter, 2016)</a></sup>
Author(s)
: Ito, S., & Schitter, G.
Year
: 2016
## Classification of high-precision actuators {#classification-of-high-precision-actuators}
<div class="table-caption">
<span class="table-number">Table 1</span>:
Zero/Low and High stiffness actuators
</div>
| **Categories** | **Pros** | **Cons** |
|----------------|---------------------------|-----------------------------|
| Zero stiffness | No vibration transmission | Large and Heavy |
| Low stiffness | High vibration isolation | Typically for low load |
| High Stiffness | High control bandwidth | High vibration transmission |
## Time Delay of Piezoelectric Electronics {#time-delay-of-piezoelectric-electronics}
In this paper, the piezoelectric actuator/electronics adds a time delay which is much higher than the time delay added by the voice coil/electronics.
## Definition of low-stiffness and high-stiffness actuator {#definition-of-low-stiffness-and-high-stiffness-actuator}
- **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
{{< figure src="/ox-hugo/ito16_low_high_stiffness_actuators.png" caption="Figure 1: Definition of low-stiffness and high-stiffness actuator" >}}
## Low-Stiffness / High-Stiffness characteristics {#low-stiffness-high-stiffness-characteristics}
- The low stiffness actuators achieve smooth transition from active isolation to passive isolation.
- The high stiffness actuators can have a gap between the passive and active isolation vibration where the vibrations are amplified in a certain frequency band.
## Controller Design {#controller-design}
{{< figure src="/ox-hugo/ito16_transmissibility.png" caption="Figure 2: Obtained transmissibility" >}}
## Discussion {#discussion}
The stiffness requirement for low-stiffness actuators can be rephrased in the frequency domain as: "the cross-over frequency of the sensitivity function of the feedback system must be larger than \\(\sqrt{2} \omega\_r\\) with \\(\omega\_r\\) is the resonant frequency of the uncontrolled system".
In practice, this is difficult to achieve with piezoelectric actuators as their first resonant frequency \\(\omega\_r\\) is **too close to other resonant frequencies to ensure close-loop stability**.
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
<a id="ito16_compar_class_high_precis_actuat"></a>Ito, S., & Schitter, G., *Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation*, IEEE/ASME Transactions on Mechatronics, *21(2)*, 1169–1178 (2016). http://dx.doi.org/10.1109/tmech.2015.2478658 [↩](#aad53368e29e8a519e2f63857044fa46)
## Backlinks {#backlinks}
- [Actuators]({{< relref "actuators" >}})

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title = "Dynamic modeling and experimental analyses of stewart platform with flexible hinges"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: <sup id="ee917739f88877d6c2758c1c36565deb"><a href="#jiao18_dynam_model_exper_analy_stewar" title="Jian Jiao, Ying Wu, Kaiping Yu \&amp; Rui Zhao, Dynamic Modeling and Experimental Analyses of Stewart Platform With Flexible Hinges, {Journal of Vibration and Control}, v(1), 151-171 (2018).">(Jian Jiao {\it et al.}, 2018)</a></sup>
Author(s)
: Jiao, J., Wu, Y., Yu, K., & Zhao, R.
Year
: 2018
# Bibliography
<a id="jiao18_dynam_model_exper_analy_stewar"></a>Jiao, J., Wu, Y., Yu, K., & Zhao, R., *Dynamic modeling and experimental analyses of stewart platform with flexible hinges*, Journal of Vibration and Control, *25(1)*, 151–171 (2018). http://dx.doi.org/10.1177/1077546318772474 [↩](#ee917739f88877d6c2758c1c36565deb)

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title = "A new isotropic and decoupled 6-dof parallel manipulator"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference
: <sup id="17295cbc2858c65ecc60d51b450233e3"><a href="#legnani12_new_isotr_decoup_paral_manip" title="Legnani, Fassi, Giberti, Cinquemani, \&amp; Tosi, A New Isotropic and Decoupled 6-dof Parallel Manipulator, {Mechanism and Machine Theory}, v(nil), 64-81 (2012).">(Legnani {\it et al.}, 2012)</a></sup>
Author(s)
: Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D.
Year
: 2012
- Concepts of isotropy and decoupling for parallel manipulators
- **isotropy**: the kinetostatic properties (same applicable force, same possible velocity, same stiffness) are identical in all directions (e.g. cubic configuration for Stewart platform)
- **decoupling**: each DoF of the end effector can be controlled by a **single** actuator (not the case for the Stewart platform)
Example of generated isotropic manipulator (not decoupled).
<a id="orgd015b7e"></a>
{{< figure src="/ox-hugo/legnani12_isotropy_gen.png" caption="Figure 1: Location of the leg axes using an isotropy generator" >}}
<a id="orgb3cab58"></a>
{{< figure src="/ox-hugo/legnani12_generated_isotropy.png" caption="Figure 2: Isotropic configuration" >}}
# Bibliography
<a id="legnani12_new_isotr_decoup_paral_manip"></a>Legnani, G., Fassi, I., Giberti, H., Cinquemani, S., & Tosi, D., *A new isotropic and decoupled 6-dof parallel manipulator*, Mechanism and Machine Theory, *58(nil)*, 64–81 (2012). http://dx.doi.org/10.1016/j.mechmachtheory.2012.07.008 [↩](#17295cbc2858c65ecc60d51b450233e3)

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title = "Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods"
author = ["Thomas Dehaeze"]
draft = false
+++
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
: <sup id="f885df380638b868e509fbbf75912d1e"><a href="#li01_simul_fault_vibrat_isolat_point" title="@phdthesis{li01_simul_fault_vibrat_isolat_point,
author = {Li, Xiaochun},
school = {University of Wyoming},
title = {Simultaneous, Fault-tolerant Vibration Isolation and
Pointing Control of Flexure Jointed Hexapods},
year = 2001,
tags = {parallel robot},
}">@phdthesis{li01_simul_fault_vibrat_isolat_point,
author = {Li, Xiaochun},
school = {University of Wyoming},
title = {Simultaneous, Fault-tolerant Vibration Isolation and
Pointing Control of Flexure Jointed Hexapods},
year = 2001,
tags = {parallel robot},
}</a></sup>
Author(s)
: Li, X.
Year
: 2001
## Introduction {#introduction}
**Stewart Platform**:
- Cubic (mutually orthogonal)
- Flexure Joints => eliminate friction and backlash but add complexity to the dynamics
<a id="orgd72b050"></a>
{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 1: Flexure jointed Stewart platform used for analysis and control" >}}
**Goal**:
- Precise pointing in two axes (sub micro-radians)
- simultaneously, providing both passive and active vibration isolation in six axes
**Jacobian Analysis**:
\\[ \delta \mathcal{L} = J \delta \mathcal{X} \\]
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](#org2d875d1) or in Figure [3](#org3e247bd), 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="org2d875d1"></a>
{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 2: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
<a id="org3e247bd"></a>
{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 3: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
## Simultaneous Vibration Isolation and Pointing Control {#simultaneous-vibration-isolation-and-pointing-control}
Basic idea:
- acceleration feedback is used to provide high-frequency vibration isolation
- cartesian pointing feedback can be used to provide low-frequency pointing
The compensation is divided in frequency because:
- pointing sensors often have low bandwidth
- acceleration sensors often have a poor low frequency response
The control bandwidth is divided as follows:
- low-frequency disturbances as attenuated and tracking is accomplished by feedback from low bandwidth pointing sensors
- mid-frequency disturbances are attenuated by feedback from band-pass sensors like accelerometer or load cells
- high-frequency disturbances are attenuated by passive isolation techniques
### Vibration Isolation {#vibration-isolation}
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [4](#org3c42849).
<a id="org3c42849"></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](#org3c42849)
<a id="orga10e0a5"></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" >}}
Each compensator is designed using simple loop-shaping techniques.
The unity control bandwidth of the isolation loop is designed to be from **5Hz to 50Hz**.
> Despite a reasonably good match between the modeled and the measured transfer functions, the model based decoupling algorithm does not produce the expected decoupling.
> Only about 20 dB separation is achieve between the diagonal and off-diagonal responses.
### Pointing Control {#pointing-control}
A block diagram of the pointing control system is shown in Figure [6](#org3c3e6ad).
<a id="org3c3e6ad"></a>
{{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 6: Figure caption" >}}
The plant is decoupled into two independent SISO subsystems.
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](#orgc8fa614).
<a id="orgc8fa614"></a>
{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 7: Feedforward control" >}}
### Simultaneous Control {#simultaneous-control}
The simultaneous vibration isolation and pointing control is approached in two ways:
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](#org987b709) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
<a id="org987b709"></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](#orgb070c43).
<a id="orgb070c43"></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)" >}}
The effect of pointing control on the isolation plant has not much effect.
> The interaction between loops may affect the transfer functions of the **first** closed loop, and thus affect its relative stability.
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](#org0d64bc7) and [11](#orgb43f022))
As shown in Figure [10](#org0d64bc7), 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="org0d64bc7"></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](#orgb43f022)).
The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
<a id="orgb43f022"></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" >}}
> The isolation loop adds a second resonance peak at its high-frequency crossover in the pointing closed-loop transfer function, which may cause instability.
> Thus, it is recommended to design and implement the isolation control system first, and then identify the pointing plant with the isolation loop closed.
### Experimental results {#experimental-results}
Two hexapods are stacked (Figure [12](#org12b1e53)):
- the bottom hexapod is used to generate disturbances matching candidate applications
- the top hexapod provide simultaneous vibration isolation and pointing control
<a id="org12b1e53"></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](#org4b99c02).
<a id="org4b99c02"></a>
{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 13: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}}
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](#orged11c63) where the bandwidth of the isolation control is expanded to very low frequency.
<a id="orged11c63"></a>
{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 14: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
## Future research areas {#future-research-areas}
Proposed future research areas include:
- **Include base dynamics in the control**:
The base dynamics is here neglected since the movements of the base are very small.
The base dynamics could be measured by mounting accelerometers at the bottom of each strut or by using force sensors.
It then could be included in the feedforward path.
- **Robust control and MIMO design**
- **New decoupling method**:
The proposed decoupling algorithm do not produce the expected decoupling, despite a reasonably good match between the modeled and the measured transfer functions.
Incomplete decoupling increases the difficulty in designing the controller.
New decoupling methods are needed.
These methods must be static in order to be implemented practically on precision hexapods
- **Identification**:
Many advanced control methods require a more accurate model or identified plant.
A closed-loop identification method is propose to solve some problems with the current identification methods used.
- **Other possible sensors**:
Many sensors can be used to expand the utility of the Stewart platform:
- **3-axis load cells** to investigate the Coriolis and centripetal terms and new decoupling methods
- **LVDT** to provide differential position of the hexapod payload with respect to the base
- **Geophones** to provide payload and base velocity information
# Bibliography
<a id="li01_simul_fault_vibrat_isolat_point"></a>Li, X., *Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods* (Doctoral dissertation) (2001). University of Wyoming, . [↩](#f885df380638b868e509fbbf75912d1e)

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title = "Simultaneous vibration isolation and pointing control of flexure jointed hexapods"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: <sup id="e3df2691f750617c3995644d056d553a"><a href="#li01_simul_vibrat_isolat_point_contr" title="Xiaochun Li, Jerry Hamann \&amp; John McInroy, Simultaneous Vibration Isolation and Pointing Control of Flexure Jointed Hexapods, nil, in in: {Smart Structures and Materials 2001: Smart Structures and
Integrated Systems}, edited by (2001)">(Xiaochun Li {\it et al.}, 2001)</a></sup>
Author(s)
: Li, X., Hamann, J. C., & McInroy, J. E.
Year
: 2001
- 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
<a id="li01_simul_vibrat_isolat_point_contr"></a>Li, X., Hamann, J. C., & McInroy, J. E., *Simultaneous vibration isolation and pointing control of flexure jointed hexapods*, In , Smart Structures and Materials 2001: Smart Structures and Integrated Systems (pp. ) (2001). : . [↩](#e3df2691f750617c3995644d056d553a)

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title = "Advanced motion control for precision mechatronics: control, identification, and learning of complex systems"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Motion Control]({{< relref "motion_control" >}})
Reference
: <sup id="73fd325bd20a6ef8972145e535f38198"><a href="#oomen18_advan_motion_contr_precis_mechat" title="Tom Oomen, Advanced Motion Control for Precision Mechatronics: Control, Identification, and Learning of Complex Systems, {IEEJ Journal of Industry Applications}, v(2), 127-140 (2018).">(Tom Oomen, 2018)</a></sup>
Author(s)
: Oomen, T.
Year
: 2018
# Bibliography
<a id="oomen18_advan_motion_contr_precis_mechat"></a>Oomen, T., *Advanced motion control for precision mechatronics: control, identification, and learning of complex systems*, IEEJ Journal of Industry Applications, *7(2)*, 127–140 (2018). http://dx.doi.org/10.1541/ieejjia.7.127 [↩](#73fd325bd20a6ef8972145e535f38198)

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+++
title = "An exploration of active hard mount vibration isolation for precision equipment"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: <sup id="bcab548922e0e1ad6a2c310f63879596"><a href="#poel10_explor_activ_hard_mount_vibrat" title="@phdthesis{poel10_explor_activ_hard_mount_vibrat,
author = {van der Poel, Gerrit Wijnand},
doi = {10.3990/1.9789036530163},
isbn = {978-90-365-3016-3},
school = {University of Twente},
title = {An Exploration of Active Hard Mount Vibration Isolation for
Precision Equipment},
url = {https://doi.org/10.3990/1.9789036530163},
year = 2010,
year = 2010,
tags = {parallel robot},
}">@phdthesis{poel10_explor_activ_hard_mount_vibrat,
author = {van der Poel, Gerrit Wijnand},
doi = {10.3990/1.9789036530163},
isbn = {978-90-365-3016-3},
school = {University of Twente},
title = {An Exploration of Active Hard Mount Vibration Isolation for
Precision Equipment},
url = {https://doi.org/10.3990/1.9789036530163},
year = 2010,
year = 2010,
tags = {parallel robot},
}</a></sup>
Author(s)
: van der Poel, G. W.
Year
: 2010
# Bibliography
<a id="poel10_explor_activ_hard_mount_vibrat"></a>van der Poel, G. W., *An exploration of active hard mount vibration isolation for precision equipment* (Doctoral dissertation) (2010). University of Twente, . [↩](#bcab548922e0e1ad6a2c310f63879596)

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+++
title = "Force feedback versus acceleration feedback in active vibration isolation"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: <sup id="525e1e237b885f81fae3c25a3036ba6f"><a href="#preumont02_force_feedb_versus_accel_feedb" title="Preumont, Fran\ccois, Bossens, \&amp; Abu-Hanieh, Force Feedback Versus Acceleration Feedback in Active Vibration Isolation, {Journal of Sound and Vibration}, v(4), 605-613 (2002).">(Preumont {\it et al.}, 2002)</a></sup>
Author(s)
: Preumont, A., A. Francois, Bossens, F., & Abu-Hanieh, A.
Year
: 2002
Summary:
- Compares the force feedback and acceleration feedback for active damping
- The use of a force sensor always give alternating poles and zeros in the open-loop transfer function between for force actuator and the force sensor which **guarantees the stability of the closed loop**
- Acceleration feedback produces alternating poles and zeros only when the flexible structure is stiff compared to the isolation system
The force applied to a **rigid body** is proportional to its acceleration, thus sensing the total interface force gives a measured of the absolute acceleration of the solid body.
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](#org7b4f6ee)), the acceleration feedback gives larger damping on the higher mode.
- For heavy payload (Figure [2](#org361b58f)), the acceleration feedback do not give alternating poles and zeros and thus for high control gains, the system becomes unstable
<a id="org7b4f6ee"></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="org361b58f"></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" >}}
Guaranteed stability of the force feedback:
> If two arbitrary flexible, undamped structures are connected with a single-axis soft isolator with force feedback, the poles and zeros of the open-loop transfer function from the force actuator to the force sensor alternate on the imaginary axis.
The same is true for the transfer function from the force actuator to the relative displacement of the actuator.
> According to physical interpretation of the zeros, they represent the resonances of the subsystem constrained by the sensor and the actuator.
# Bibliography
<a id="preumont02_force_feedb_versus_accel_feedb"></a>Preumont, A., A. Fran\ccois, Bossens, F., & Abu-Hanieh, A., *Force feedback versus acceleration feedback in active vibration isolation*, Journal of Sound and Vibration, *257(4)*, 605–613 (2002). http://dx.doi.org/10.1006/jsvi.2002.5047 [↩](#525e1e237b885f81fae3c25a3036ba6f)

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title = "A six-axis single-stage active vibration isolator based on stewart platform"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: <sup id="8096d5b2df73551d836ef96b7ca7efa4"><a href="#preumont07_six_axis_singl_stage_activ" title="Preumont, Horodinca, Romanescu, de, Marneffe, Avraam, Deraemaeker, Bossens, \&amp; Abu Hanieh, A Six-Axis Single-Stage Active Vibration Isolator Based on Stewart Platform, {Journal of Sound and Vibration}, v(3-5), 644-661 (2007).">(Preumont {\it et al.}, 2007)</a></sup>
Author(s)
: Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …
Year
: 2007
Summary:
- **Cubic** Stewart platform (Figure [3](#org32a4f7c))
- Provides uniform control capability
- Uniform stiffness in all directions
- minimizes the cross-coupling among actuators and sensors of different legs
- Flexible joints (Figure [2](#orgf807976))
- 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](#org744bdc9))
- 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="org744bdc9"></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="orgf807976"></a>
{{< figure src="/ox-hugo/preumont07_flexible_joints.png" caption="Figure 2: Flexible joints used for the Stewart platform" >}}
<a id="org32a4f7c"></a>
{{< figure src="/ox-hugo/preumont07_stewart_platform.png" caption="Figure 3: Stewart platform" >}}
# Bibliography
<a id="preumont07_six_axis_singl_stage_activ"></a>Preumont, A., Horodinca, M., Romanescu, I., Marneffe, B. d., Avraam, M., Deraemaeker, A., Bossens, F., …, *A six-axis single-stage active vibration isolator based on stewart platform*, Journal of Sound and Vibration, *300(3-5)*, 644–661 (2007). http://dx.doi.org/10.1016/j.jsv.2006.07.050 [↩](#8096d5b2df73551d836ef96b7ca7efa4)

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title = "Advances in internal model control technique: a review and future prospects"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Complementary Filters]({{< relref "complementary_filters" >}})
Reference
: <sup id="14f767d8ba71d58fa8a3ec876628d750"><a href="#saxena12_advan_inter_model_contr_techn" title="Sahaj Saxena \&amp; YogeshV Hote, Advances in Internal Model Control Technique: a Review and Future Prospects, {IETE Technical Review}, v(6), 461 (2012).">(Sahaj Saxena \& YogeshV Hote, 2012)</a></sup>
Author(s)
: Saxena, S., & Hote, Y.
Year
: 2012
## Proposed Filter \\(F(s)\\) {#proposed-filter--fs}
\begin{align\*}
F(s) &= \frac{1}{(\lambda s + 1)^n} \\\\\\
F(s) &= \frac{n \lambda + 1}{(\lambda s + 1)^n}
\end{align\*}
## Internal Model Control {#internal-model-control}
Central concept in IMC: control can be acheive only if the control system involves, either implicitly or explicitly, some representation of the process to be controlled.
### Basic IMC structure {#basic-imc-structure}
IMC can be considered as a special case of classical feedback structure with plant \\(G(s)\\) and controller \\(C(s)\\).
The plan model \\(G\_M(s)\\) is added and substracted into the feedback path of feedback controller.
The structure can then be modified and we obtain a new controller \\(Q(s)\\).
IMC is related to the classical controller through:
\begin{align\*}
Q(s) = \frac{C(s)}{1+G\_M(s)C(s)} \\\\\\
C(s) = \frac{Q(s)}{1-G\_M(s)Q(s)}
\end{align\*}
Internal model control system is characterized by a control device consisting of the controller \\(Q(s)\\) and a predictive model \\(G\_M(s)\\) of the process (internal model).
The internal model loop uses the difference between the outputs of the process \\(G(s)\\) to be controlled and the internal model.
This difference \\(E(s)\\) represents the effect of disturbance and mismatch of the model.
### Features of IMC Structure {#features-of-imc-structure}
Three properties:
- **Dual stability**: assume that, if the plant model is perfect (\\(G\_M(s) = G(s)\\)) and disturbance is absent, the system becomes open-loop and the closed-loop stability is characterized by the stability of \\(G(s)\\) and \\(Q(s)\\)
- **Perfect control**: assume that, if the controller is equal to the model inverse (\\(Q(s) = G\_M^{-1}\\)) and \\(G(s) = G\_M(s)\\) with \\(G(s)\\) stable, then the system is perfectly controlled.
- **Zero Offset**: assume that, if the steady state gain of the controller is equal to the inverse of model gain, then offset free control is obtained for constant step of ramp type inputs and disturbances. As expected, the equivalent classical controller leads to integral action.
Issues:
- the plant model is never perfect
- inverting the model can cause instability
- control signal may have large magnitude
## Design procedure for IMC Compensator {#design-procedure-for-imc-compensator}
1. factorize the plant model as \\(G\_M(s) = G\_{M-}(s)G\_{M+}(s)\\) where \\(G\_{M-}(s)\\) is invertible and minimum phase and \\(G\_{M+}(s)\\) is non-invertible and contains all non-minimum phase elements (delays, RHP zeros). Then, the controller is the inverse of the invertible portion of the plant model: \\(Q\_1(s) = G\_{M-}^{-1}(s)\\).
2. Filter selection: to make the controller proper and robust against the plant-model mismatch, a low pass filter of the form \\(F(s) = \frac{n \lambda}{(\lambda s + 1)^n}\\) is augmented with the inverted model \\(Q\_1(s)\\): \\(Q(s) = Q\_1(s) F(s)\\). \\(\lambda\\) is a tuning parameter which has an inverse relationship with the speed of closed loop response, \\(n\\) is selected such that \\(Q(s)\\) becomes proper.
## Issues in IMC {#issues-in-imc}
### Filter selection and tuning guidelines {#filter-selection-and-tuning-guidelines}
## Some advantages and future prospects {#some-advantages-and-future-prospects}
## Conclusion {#conclusion}
The interesting feature regarding IMC is that the design scheme is identical to the open-loop control design procedure and the implementation of IMC results in a feedback system, thereby copying the disturbances and parameter uncertainties, while open-loop control is not.
# Bibliography
<a id="saxena12_advan_inter_model_contr_techn"></a>Saxena, S., & Hote, Y., *Advances in internal model control technique: a review and future prospects*, IETE Technical Review, *29(6)*, 461 (2012). http://dx.doi.org/10.4103/0256-4602.105001 [↩](#14f767d8ba71d58fa8a3ec876628d750)

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title = "Design for precision: current status and trends"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Precision Engineering]({{< relref "precision_engineering" >}})
Reference
: <sup id="89f7d8f4c31f79f83e3666017687f525"><a href="#schellekens98_desig_precis" title="Schellekens, Rosielle, Vermeulen, , Vermeulen, Wetzels \&amp; Pril, Design for Precision: Current Status and Trends, {Cirp Annals}, v(2), 557-586 (1998).">(Schellekens {\it et al.}, 1998)</a></sup>
Author(s)
: Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W.
Year
: 1998
# Bibliography
<a id="schellekens98_desig_precis"></a>Schellekens, P., Rosielle, N., Vermeulen, H., Vermeulen, M., Wetzels, S., & Pril, W., *Design for precision: current status and trends*, Cirp Annals, *(2)*, 557–586 (1998). http://dx.doi.org/10.1016/s0007-8506(07)63243-0 [↩](#89f7d8f4c31f79f83e3666017687f525)

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title = "Nanopositioning with multiple sensors: a case study in data storage"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}})
Reference
: <sup id="eb5a15a8c900d93de0b9bab520e1b6da"><a href="#sebastian12_nanop_with_multip_sensor" title="Abu Sebastian \&amp; Angeliki Pantazi, Nanopositioning With Multiple Sensors: a Case Study in Data Storage, {IEEE Transactions on Control Systems Technology}, v(2), 382-394 (2012).">(Abu Sebastian \& Angeliki Pantazi, 2012)</a></sup>
Author(s)
: Sebastian, A., & Pantazi, A.
Year
: 2012
# Bibliography
<a id="sebastian12_nanop_with_multip_sensor"></a>Sebastian, A., & Pantazi, A., *Nanopositioning with multiple sensors: a case study in data storage*, IEEE Transactions on Control Systems Technology, *20(2)*, 382–394 (2012). http://dx.doi.org/10.1109/tcst.2011.2177982 [↩](#eb5a15a8c900d93de0b9bab520e1b6da)

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title = "A soft 6-axis active vibration isolator"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: <sup id="a48f6708d087625a42ca2375407a2bc4"><a href="#spanos95_soft_activ_vibrat_isolat" title="Spanos, Rahman \&amp; Blackwood, A Soft 6-axis Active Vibration Isolator, nil, in in: {Proceedings of 1995 American Control Conference - ACC'95}, edited by (1995)">(Spanos {\it et al.}, 1995)</a></sup>
Author(s)
: Spanos, J., Rahman, Z., & Blackwood, G.
Year
: 1995
**Stewart Platform** (Figure [1](#org4317d08)):
- Voice Coil
- Flexible joints (cross-blades)
- Force Sensors
- Cubic Configuration
<a id="org4317d08"></a>
{{< figure src="/ox-hugo/spanos95_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
Total mass of the paylaod: 30kg
Center of gravity is 9cm above the geometry center of the mount (cube's center?).
Limitation of the **Decentralized Force Feedback**:
- high frequency pole due to internal resonances of the struts
- low frequency zero due to the rotational stiffness of the flexible joints
After redesign of the struts:
- high frequency pole at 4.7kHz
- 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="org67e505c"></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" >}}
The controller used consisted of:
- second order low pass filter to gain stabilize the plant at high frequencies and provide steep roll-off
- first order lead filter to provide adequate phase margin at the high frequency crossover
- 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](#orgf128817).
<a id="orgf128817"></a>
{{< figure src="/ox-hugo/spanos95_results.png" caption="Figure 3: Experimentally measured Frobenius norm of the 6-axis transmissibility" >}}
# Bibliography
<a id="spanos95_soft_activ_vibrat_isolat"></a>Spanos, J., Rahman, Z., & Blackwood, G., *A soft 6-axis active vibration isolator*, In , Proceedings of 1995 American Control Conference - ACC'95 (pp. ) (1995). : . [↩](#a48f6708d087625a42ca2375407a2bc4)

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title = "Interferometric characterization of rotation stages for x-ray nanotomography"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}}), [Positioning Stations]({{< relref "positioning_stations" >}})
Reference
: <sup id="abb1be5f48179255f7d8c45b1784bcf8"><a href="#stankevic17_inter_charac_rotat_stages_x_ray_nanot" title="Tomas Stankevic, Christer Engblom, Florent Langlois, , Filipe Alves, Alain Lestrade, Nicolas Jobert, , Gilles Cauchon, Ulrich Vogt \&amp; Stefan Kubsky, Interferometric Characterization of Rotation Stages for X-Ray Nanotomography, {Review of Scientific Instruments}, v(5), 053703 (2017).">(Tomas Stankevic {\it et al.}, 2017)</a></sup>
Author(s)
: Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …
Year
: 2017
- Similar Station than the NASS
- Similar Metrology with fiber based interferometers and cylindrical reference mirror
<a id="orgc1f98d0"></a>
{{< figure src="/ox-hugo/stankevic17_station.png" caption="Figure 1: Positioning Station" >}}
- **Thermal expansion**: Stabilized down to \\(5mK/h\\) using passive water flow through the baseplate below the sample stage and in the interferometry reference frame.
- **Controller**: Two Independant PID loops
- Repeatable errors => feedforward (Look Up Table)
- Non-repeatable errors => feedback
- Result: 40nm runout error
# Bibliography
<a id="stankevic17_inter_charac_rotat_stages_x_ray_nanot"></a>Stankevic, T., Engblom, C., Langlois, F., Alves, F., Lestrade, A., Jobert, N., Cauchon, G., …, *Interferometric characterization of rotation stages for x-ray nanotomography*, Review of Scientific Instruments, *88(5)*, 053703 (2017). http://dx.doi.org/10.1063/1.4983405 [↩](#abb1be5f48179255f7d8c45b1784bcf8)

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title = "Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}})
Reference
: <sup id="85f81ff678aabc195636437548e4234a"><a href="#tang18_decen_vibrat_contr_voice_coil" title="Jie Tang, Dengqing Cao \&amp; Tianhu Yu, 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}, v(1), 132-145 (2018).">(Jie Tang {\it et al.}, 2018)</a></sup>
Author(s)
: Tang, J., Cao, D., & Yu, T.
Year
: 2018
# Bibliography
<a id="tang18_decen_vibrat_contr_voice_coil"></a>Tang, J., Cao, D., & Yu, T., *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)*, 132–145 (2018). http://dx.doi.org/10.1177/0954406218756941 [↩](#85f81ff678aabc195636437548e4234a)

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title = "Sensor fusion for active vibration isolation in precision equipment"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Sensor Fusion]({{< relref "sensor_fusion" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: <sup id="ef30bc07c91e9d46a42198757dc610de"><a href="#tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip" title="Tjepkema, van Dijk \&amp; Soemers, Sensor Fusion for Active Vibration Isolation in Precision Equipment, {Journal of Sound and Vibration}, v(4), 735-749 (2012).">(Tjepkema {\it et al.}, 2012)</a></sup>
Author(s)
: Tjepkema, D., Dijk, J. v., & Soemers, H.
Year
: 2012
## Relative motion Control {#relative-motion-control}
Control law: \\(f = -G(x-w)\\)
\\[ \frac{x}{w} = \frac{k+G}{ms^2 + k+G} \\]
\\[ \frac{x}{F} = \frac{1}{ms^2 + k+G} \\]
## Force Control {#force-control}
Control law: \\(f = -G F\_a = -G \left(f-k(x-w)\right)\\)
\\[ \frac{x}{w} = \frac{k}{(1+G)ms^2 + k} \\]
\\[ \frac{x}{F} = \frac{1+G}{(1+G)ms^2 + k} \\]
## Inertial Control {#inertial-control}
Control law: \\(f = -Gx\\)
\\[ \frac{x}{w} = \frac{k}{ms^2 + k+G} \\]
\\[ \frac{x}{F} = \frac{1}{ms^2 + k+G} \\]
## Design constraints and control bandwidth {#design-constraints-and-control-bandwidth}
Heavier sensor => lower noise but it is harder to maintain collocation with the actuator => that limits the bandwidth.
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
<a id="tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip"></a>Tjepkema, D., Dijk, J. v., & Soemers, H., *Sensor fusion for active vibration isolation in precision equipment*, Journal of Sound and Vibration, *331(4)*, 735–749 (2012). http://dx.doi.org/10.1016/j.jsv.2011.09.022 [↩](#ef30bc07c91e9d46a42198757dc610de)

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title = "Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Nano Active Stabilization System]({{< relref "nano_active_stabilization_system" >}})
Reference
: <sup id="1bccbe15e35ed02229afbc6528c5057e"><a href="#wang12_autom_marker_full_field_hard" title="Jun Wang, Yu-chen Karen Chen, Qingxi Yuan, Andrei, Tkachuk, Can Erdonmez, Benjamin Hornberger, Michael \&amp; Feser, Automated Markerless Full Field Hard X-Ray Microscopic Tomography At Sub-50 Nm 3-dimension Spatial Resolution, {Applied Physics Letters}, v(14), 143107 (2012).">(Jun Wang {\it et al.}, 2012)</a></sup>
Author(s)
: Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M.
Year
: 2012
**Introduction of Markers**:
That limits the type of samples that is studied
There is a need for markerless nano-tomography
=> the key requirement is the precision and stability of the positioning stages.
**Passive rotational run-out error system**:
It uses calibrated metrology disc and capacitive sensors
# Bibliography
<a id="wang12_autom_marker_full_field_hard"></a>Wang, J., Chen, Y. K., Yuan, Q., Tkachuk, A., Erdonmez, C., Hornberger, B., & Feser, M., *Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution*, Applied Physics Letters, *100(14)*, 143107 (2012). http://dx.doi.org/10.1063/1.3701579 [↩](#1bccbe15e35ed02229afbc6528c5057e)

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title = "Investigation on active vibration isolation of a stewart platform with piezoelectric actuators"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}})
Reference
: <sup id="db95fac7cd46c14e2b4f38e8ca4158fe"><a href="#wang16_inves_activ_vibrat_isolat_stewar" title="Wang, Xie, Chen, Zhang \&amp; Zhiyi, Investigation on Active Vibration Isolation of a Stewart Platform With Piezoelectric Actuators, {Journal of Sound and Vibration}, v(), 1-19 (2016).">(Wang {\it et al.}, 2016)</a></sup>
Author(s)
: Wang, C., Xie, X., Chen, Y., & Zhang, Z.
Year
: 2016
**Model of the Stewart platform**:
- Struts are treated as flexible beams
- Payload and the base are treated as flexible plates
- The FRF synthesis method permits to derive FRFs of the Stewart platform
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="orgbc70494"></a>
{{< figure src="/ox-hugo/wang16_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
**Control**:
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](#org4b1fbd9)).
- the force sensor is mounted **between the base and the strut**
<a id="org4b1fbd9"></a>
{{< figure src="/ox-hugo/wang16_force_feedback.png" caption="Figure 2: Feedback of integrated forces in the platform" >}}
Sorts of HAC-LAC control:
- LAC: Decentralized integral force feedback
- HAC: Inertial control using accelerometers. Use of the Jacobian to decouple the motion and then Fx-LMS based adaptive control is used
**Experimental validation**:
- All 6 transfer function from actuator force to force sensors are almost the same (gain offset)
- Effectiveness of control methods are shown
# Bibliography
<a id="wang16_inves_activ_vibrat_isolat_stewar"></a>Wang, C., Xie, X., Chen, Y., & Zhang, Z., *Investigation on active vibration isolation of a stewart platform with piezoelectric actuators*, Journal of Sound and Vibration, *383()*, 1–19 (2016). http://dx.doi.org/10.1016/j.jsv.2016.07.021 [↩](#db95fac7cd46c14e2b4f38e8ca4158fe)

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title = "Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Flexible Joints]({{< relref "flexible_joints" >}}), [Cubic Architecture]({{< relref "cubic_architecture" >}})
Reference
: <sup id="d39b6222c8dd2baf188d677733c2826c"><a href="#yang19_dynam_model_decoup_contr_flexib" title="Yang, Wu, Chen, Kang, ShengZheng \&amp; Cheng, Dynamic Modeling and Decoupled Control of a Flexible Stewart Platform for Vibration Isolation, {Journal of Sound and Vibration}, v(), 398-412 (2019).">(Yang {\it et al.}, 2019)</a></sup>
Author(s)
: Yang, X., Wu, H., Chen, B., Kang, S., & Cheng, S.
Year
: 2019
**Discusses**:
- flexible-rigid model of Stewart platform
- the impact of joint stiffness is compensated using a displacement sensor and a force sensor
- then the MIMO system is decoupled in modal space and 6 SISO controllers are applied for vibration isolation using force sensors
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](#org936d8f9)):
- piezoelectric actuators
- flexible joints (Figure [2](#orgd8c916a))
- force sensors (used for vibration isolation)
- displacement sensors (used to decouple the dynamics)
- cubic (even though not said explicitly)
<a id="org936d8f9"></a>
{{< figure src="/ox-hugo/yang19_stewart_platform.png" caption="Figure 1: Stewart Platform" >}}
<a id="orgd8c916a"></a>
{{< figure src="/ox-hugo/yang19_flexible_joints.png" caption="Figure 2: Flexible Joints" >}}
The stiffness of the flexible joints (Figure [2](#orgd8c916a)) 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">
<span class="table-number"><a href="#table--tab:yang19-stiffness-flexible-joints">Table 1</a></span>:
Stiffness of flexible joints obtained by FEM
</div>
| \\(k\_{\theta u},\ k\_{\psi u}\\) | \\(72 Nm/rad\\) |
|-----------------------------------|-----------------|
| \\(k\_{\theta s}\\) | \\(51 Nm/rad\\) |
| \\(k\_{\psi s}\\) | \\(62 Nm/rad\\) |
| \\(k\_{\gamma s}\\) | \\(64 Nm/rad\\) |
**Dynamics**:
If the bending and torsional stiffness of the flexible joints are neglected:
\\[ M \ddot{x} + C \dot{x} + K x = J^T f \\]
- \\(M\\) is the mass matrix
- \\(C\\) is the damping matrix
- \\(K\\) is the stiffness matrix
- \\(x\\) is the generalized coordinates, representing the displacement and orientation of the payload plate
- \\(f\\) is the actuator forces
- \\(J\\) is the Jacobian matrix
In this paper, the parasitic bending stiffness of the flexible joints are considered:
\\[ M \ddot{x} + C \dot{x} + (K + K\_e) x = J^T f \\]
where \\(K\_e\\) is the stiffness matrix induced by the parasitic stiffness of the flexible joints.
Analytical expression for \\(K\_e\\) are derived in the paper.
**Controller Design**:
There is a strong coupling between the input forces and the state variables in the task space.
The traditional modal decoupled control strategy cannot work with the flexible Stewart platform because it is impossible to achieve simultaneous diagonalization of the mass, damped and stiffness matrices.
To make the six-dof system decoupled into six single-dof isolators, a controller based on the leg's force and position feedback is designed.
> The idea is to synthesize the control force that can compensate the parasitic bending and torsional torques of the flexible joints and simultaneously achieve diagonalization of the matrices \\(M\\), \\(C\\) and \\(K\\)
The force measured by the force sensors are:
\\[ y = f - k J x - c J \dot{x} \\]
The displacements measured by the position sensors are:
\\[ z = [\Delta l\_1\ \dots\ \Delta l\_6]^T \\]
Let's apply the feedback control based on both the force sensor and the position sensor:
\\[ f = -H(s) y + (1 + H(s)) K\_{el} z \\]
where \\(K\_{el} = J^{-T} K\_e J^T\\) is the stiffness matrix of the flexible joints expressed in joint space.
We thus obtain:
\\[ f = \frac{H(s)}{1 + H(s)} (k J x + c J \dot{x}) + J^{-T} K\_e x \\]
If we substitute \\(f\\) in the dynamic equation, we obtain that the parasitic stiffness effect of the flexible joints has been compensated by the actuation forces and the system can now be decoupled in modal space \\(x = \Phi u\\).
\\(\Phi\\) is the modal matrix selected such that \\(\Phi^T M \Phi = I\_6\\) and \\(k \Phi^T J^T J \Phi = \text{diag}(\omega\_1^2\ \dots\ \omega\_6^2)\\):
\\[ s^2 + \frac{1}{1 + H(s)} \frac{c \omega\_i^2}{k} s + \frac{1}{1 + H(s)} \omega\_i^2 = 0, \quad i = 1,\ \dots,\ 6 \\]
The six-dof system is now transformed into a six one-dof system where \\(H(s)\\) can be designed for control purpose.
In order to apply this control strategy:
- A force sensor and displacement sensor are need in each strut
- The joint stiffness has to be known
- 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](#orgeb7080e).
<a id="orgeb7080e"></a>
{{< figure src="/ox-hugo/yang19_control_arch.png" caption="Figure 3: Control Architecture used" >}}
\\(H(s)\\) is designed as a proportional plus integral compensator:
\\[ H(s) = k\_p + k\_i/s \\]
Substituting \\(H(s)\\) in the equation of motion gives that:
- an increase of \\(k\_i\\) increase the damping and thus suppress the resonance peaks
- an increase of \\(k\_p\\) lowers the resonance frequency and thus the bandwidth of vibration isolation is examped
**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](#org48c287d).
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="org48c287d"></a>
{{< figure src="/ox-hugo/yang19_results.png" caption="Figure 4: Frequency response of the acceleration ratio between the paylaod and excitation (Transmissibility)" >}}
> A model-based controller is then designed based on the leg’s force and position feedback.
> The position feedback compensates the effect of parasitic bending and torsional stiffness of the flexible joints.
> The force feedback makes the six-DOF MIMO system decoupled into six SISO subsystems in modal space, where the control gains can be designed and analyzed more effectively and conveniently.
> 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
<a id="yang19_dynam_model_decoup_contr_flexib"></a>Yang, X., Wu, H., Chen, B., Kang, S., & Cheng, S., *Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation*, Journal of Sound and Vibration, *439()*, 398–412 (2019). http://dx.doi.org/10.1016/j.jsv.2018.10.007 [↩](#d39b6222c8dd2baf188d677733c2826c)

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title = "Six dof active vibration control using stewart platform with non-cubic configuration"
author = ["Thomas Dehaeze"]
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Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
Reference
: <sup id="a457d4de462d2fe52a1bbb848182b554"><a href="#zhang11_six_dof" title="Zhen Zhang, J Liu, Jq Mao, Yx Guo \&amp; Yh Ma, Six DOF active vibration control using stewart platform with non-cubic configuration, nil, in in: {2011 6th IEEE Conference on Industrial Electronics and
Applications}, edited by (2011)">(Zhen Zhang {\it et al.}, 2011)</a></sup>
Author(s)
: Zhang, Z., Liu, J., Mao, J., Guo, Y., & Ma, Y.
Year
: 2011
- **Non-cubic** stewart platform
- **Flexible** joints
- Magnetostrictive actuators
- Strong coupled motions along different axes
- Non-cubic architecture => permits to have larger workspace which was required
- Structure parameters (radius of plates, length of struts) are determined by optimization of the condition number of the Jacobian matrix
- **Accelerometers** for active isolation
- Adaptive FIR filters for active isolation control
<a id="orge1b0233"></a>
{{< figure src="/ox-hugo/zhang11_platform.png" caption="Figure 1: Prototype of the non-cubic stewart platform" >}}
# Bibliography
<a id="zhang11_six_dof"></a>Zhang, Z., Liu, J., Mao, J., Guo, Y., & Ma, Y., *Six dof active vibration control using stewart platform with non-cubic configuration*, In , 2011 6th IEEE Conference on Industrial Electronics and Applications (pp. ) (2011). : . [↩](#a457d4de462d2fe52a1bbb848182b554)

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title = "Control Bootcamp"
author = ["Thomas Dehaeze"]
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Tags
:
<https://www.youtube.com/playlist?list=PLMrJAkhIeNNR20Mz-VpzgfQs5zrYi085m>
## Overview {#overview}
## Linear Systems {#linear-systems}
## Stability and Eigenvalues {#stability-and-eigenvalues}
## Linearizing Around a Fixed Point {#linearizing-around-a-fixed-point}
## Controllability {#controllability}
## Controllability, Reachability, and Eigenvalue Placement {#controllability-reachability-and-eigenvalue-placement}
## Controllability and the Discrete-Time Impulse Response {#controllability-and-the-discrete-time-impulse-response}
## Degrees of Controllability and Gramians {#degrees-of-controllability-and-gramians}
## Controllability and the PBH Test {#controllability-and-the-pbh-test}
## Cayley-Hamilton Theorem {#cayley-hamilton-theorem}
## Reachability and Controllability with Cayley-Hamilton {#reachability-and-controllability-with-cayley-hamilton}
## Inverted Pendulum on a Cart {#inverted-pendulum-on-a-cart}
## Eigenvalue Placement for the Inverted Pendulum on a Cart {#eigenvalue-placement-for-the-inverted-pendulum-on-a-cart}
## Linear Quadratic Regulator (LQR) Control for the Inverted Pendulum on a Cart {#linear-quadratic-regulator--lqr--control-for-the-inverted-pendulum-on-a-cart}
## Motivation for Full-State Estimation {#motivation-for-full-state-estimation}
## Observability {#observability}
## Full-State Estimation {#full-state-estimation}
## Kalman Filter {#kalman-filter}
## Observability Example in Matlab {#observability-example-in-matlab}
## Observability Example in Matlab (Part 2) {#observability-example-in-matlab--part-2}
## Kalman Filter Example in Matlab {#kalman-filter-example-in-matlab}
## Linear Quadratic Gaussian (LQG) {#linear-quadratic-gaussian--lqg}
## LQG Example in Matlab {#lqg-example-in-matlab}
## Introduction to Robust Control {#introduction-to-robust-control}
## Three Equivalent Representations of Linear Systems {#three-equivalent-representations-of-linear-systems}
## Example Frequency Response (Bode Plot) for Spring-Mass-Damper {#example-frequency-response--bode-plot--for-spring-mass-damper}
## Laplace Transforms and the Transfer Function {#laplace-transforms-and-the-transfer-function}
## Benefits of Feedback on Cruise Control Example {#benefits-of-feedback-on-cruise-control-example}
## Benefits of Feedback on Cruise Control Example (Part 2) {#benefits-of-feedback-on-cruise-control-example--part-2}
## Cruise Control Example with Proportional-Integral (PI) control {#cruise-control-example-with-proportional-integral--pi--control}
## Sensitivity and Complementary Sensitivity {#sensitivity-and-complementary-sensitivity}
## Sensitivity and Complementary Sensitivity (Part 2) {#sensitivity-and-complementary-sensitivity--part-2}
## Loop shaping {#loop-shaping}
## Loop Shaping Example for Cruise Control {#loop-shaping-example-for-cruise-control}
## Sensitivity and Robustness {#sensitivity-and-robustness}
## Limitations on Robustness {#limitations-on-robustness}
## Cautionary Tale About Inverting the Plant Dynamics {#cautionary-tale-about-inverting-the-plant-dynamics}
## Control systems with non-minimum phase dynamics {#control-systems-with-non-minimum-phase-dynamics}
<./biblio/references.bib>

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Tags
:
## Data-Driven Control {#data-driven-control}
### Overview {#overview}
#### Challenges {#challenges}
With modern control (LQR, LQG, H-Infinity), we work with linear system (or linearized systems) and we develop a control law that minimize some cost function.
Challenging systems where modern control is not efficient:
- Non-linear systems
- System with unknown dynamics
- High dimensional systems
- Limited measurements or control inputs
For these kinds of systems, data-driven control seems to be a good alternative.
#### What is control? {#what-is-control}
It's an optimization constrained by dynamics of the system.
The optimization is easy when the system is linear and the cost function is quadratic.
When the system is non-linear or when the cost function is non quadratic, the optimization becomes complicated (non closed form). Then the optimization should be rerun on the fly, which is what is done with MPC (model predictive control).
#### What is Machine-Learning? {#what-is-machine-learning}
Machine-learning is powerful non-linear optimization based on data.
#### Outline of this lecture {#outline-of-this-lecture}
##### Data Driven Models {#data-driven-models}
For the problem of unknown dynamics, we can use data driven models.
The goal is to collect data to generate of model of the system.
##### Machine Learning Control {#machine-learning-control}
When we use the control inputs, the system changes and the system model might be not valid anymore.
The idea is to use data driven machine learning directly to learn a good controller.
##### Sensor and actuator placement {#sensor-and-actuator-placement}
Use powerful optimization techniques from machine learning to learn what are good sensors and actuators.
### Linear System Identification {#linear-system-identification}
### The Goal of Balanced Model Reduction {#the-goal-of-balanced-model-reduction}
### Change of Variables in Control Systems {#change-of-variables-in-control-systems}
### Change of Variables in Control Systems (Correction) {#change-of-variables-in-control-systems--correction}
### Balancing Example {#balancing-example}
### Balancing Transformation {#balancing-transformation}
### Balanced Truncation {#balanced-truncation}
### Balanced Truncation Example {#balanced-truncation-example}
### Error Bounds for Balanced Truncation {#error-bounds-for-balanced-truncation}
### Balanced Proper Orthogonal Decomposition {#balanced-proper-orthogonal-decomposition}
### BPOD and Output Projection {#bpod-and-output-projection}
### Balanced Truncation and BPOD Example {#balanced-truncation-and-bpod-example}
### Eigensystem Realization Algorithm {#eigensystem-realization-algorithm}
### ERA and the Discrete-Time Impulse Response {#era-and-the-discrete-time-impulse-response}
### Eigensystem Realization Algorithm Procedure {#eigensystem-realization-algorithm-procedure}
### Balanced Models with ERA {#balanced-models-with-era}
### Observer Kalman Filter Identification {#observer-kalman-filter-identification}
### ERA\_OKID Example in Matlab {#era-okid-example-in-matlab}
## System Identification {#system-identification}
### Full-State Models with Control {#full-state-models-with-control}
### Regression Models {#regression-models}
### Dynamic Mode Decomposition with Control {#dynamic-mode-decomposition-with-control}
### DMD Control Example {#dmd-control-example}
### Koopman with Control {#koopman-with-control}
### Sparse Nonlinear Models with Control {#sparse-nonlinear-models-with-control}
### Model Predictive Control {#model-predictive-control}
### Sparse Identification of Nonlinear Dynamics for Model Predictive Control {#sparse-identification-of-nonlinear-dynamics-for-model-predictive-control}
## Machine Learning Control {#machine-learning-control}
### Overview {#overview}
### Genetic Algorithms {#genetic-algorithms}
### Tuning a PID Controller with Genetic Algorithms {#tuning-a-pid-controller-with-genetic-algorithms}
### Tuning a PID Controller with Genetic Algorithms (Part 2) {#tuning-a-pid-controller-with-genetic-algorithms--part-2}
### Genetic Programming {#genetic-programming}
### Genetic Programming Control {#genetic-programming-control}
## Extremum Seeking Control {#extremum-seeking-control}
### Introduction {#introduction}
### Matlab {#matlab}
### Simulink {#simulink}
### Challenging Example {#challenging-example}
### Applications {#applications}
<./biblio/references.bib>

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type = "zettels"
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Here is the list of subjects I took note about.

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Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Active isolation and damping of vibrations via stewart platform]({{< relref "hanieh03_activ_stewar" >}})
- [Active damping based on decoupled collocated control]({{< relref "holterman05_activ_dampin_based_decoup_colloc_contr" >}})

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:
## How to choose the correct actuator for my application? {#how-to-choose-the-correct-actuator-for-my-application}
For vibration isolation:
- In <sup id="aad53368e29e8a519e2f63857044fa46"><a href="#ito16_compar_class_high_precis_actuat" title="Shingo Ito \&amp; Georg Schitter, Comparison and Classification of High-Precision Actuators Based on Stiffness Influencing Vibration Isolation, {IEEE/ASME Transactions on Mechatronics}, v(2), 1169-1178 (2016).">(Shingo Ito \& Georg Schitter, 2016)</a></sup>, the effect of the actuator stiffness on the attainable vibration isolation is studied ([Notes]({{< relref "ito16_compar_class_high_precis_actuat" >}}))
## Piezoelectric {#piezoelectric}
| Suppliers | Links |
|--------------|------------------------------------------------------------------------------------|
| Cedrat | [link](http://www.cedrat-technologies.com/) |
| PI | [link](https://www.physikinstrumente.com/en/) |
| Piezo System | [link](https://www.piezosystem.com/products/piezo%5Factuators/stacktypeactuators/) |
| Noliac | [link](http://www.noliac.com/) |
A model of a multi-layer monolithic piezoelectric stack actuator is described in <sup id="c823f68dd2a72b9667a61b3c046b4731"><a href="#fleming10_nanop_system_with_force_feedb" title="Fleming, Nanopositioning System With Force Feedback for High-Performance Tracking and Vibration Control, {IEEE/ASME Transactions on Mechatronics}, v(3), 433-447 (2010).">(Fleming, 2010)</a></sup> ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
## Voice Coil {#voice-coil}
| Suppliers | Links |
|----------------------|----------------------------------------------|
| Geeplus | [link](https://www.geeplus.com/) |
| Maccon | [link](https://www.maccon.de/en.html) |
| TDS PP | [link](https://www.tds-pp.com/en/) |
| H2tech | [link](https://www.h2wtech.com/) |
| PBA Systems | [link](http://www.pbasystems.com.sg/) |
| Celera Motion | [link](https://www.celeramotion.com/) |
| Beikimco | [link](http://www.beikimco.com/) |
| Electromate | [link](https://www.electromate.com/) |
| Magnetic Innovations | [link](https://www.magneticinnovations.com/) |
## Shaker {#shaker}
| Suppliers | Links |
|--------------------|---------------------------------------------------------------|
| BKSV | [link](https://www.bksv.com/en/products/shakers-and-exciters) |
| Vibration Research | [link](https://vibrationresearch.com/shakers/) |
| Sentek Dynamics | [link](https://www.sentekdynamics.com/) |
<https://www.bksv.com/en/products/shakers-and-exciters/LDS-shaker-systems/permanent-magnet-shakers/V201>
## Brush-less DC Motor {#brush-less-dc-motor}
- <sup id="d2e68d39d09d7e8e71ff08a6ebd45400"><a href="#yedamale03_brush_dc_bldc_motor_fundam" title="Yedamale, Brushless Dc (BLDC) Motor Fundamentals, {Microchip Technology Inc}, v(), 3--15 (2003).">(Yedamale, 2003)</a></sup>
<https://www.electricaltechnology.org/2016/05/bldc-brushless-dc-motor-construction-working-principle.html>
# Bibliography
<a id="ito16_compar_class_high_precis_actuat"></a>Ito, S., & Schitter, G., *Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation*, IEEE/ASME Transactions on Mechatronics, *21(2)*, 1169–1178 (2016). http://dx.doi.org/10.1109/tmech.2015.2478658 [↩](#aad53368e29e8a519e2f63857044fa46)
<a id="fleming10_nanop_system_with_force_feedb"></a>Fleming, A., *Nanopositioning system with force feedback for high-performance tracking and vibration control*, IEEE/ASME Transactions on Mechatronics, *15(3)*, 433–447 (2010). http://dx.doi.org/10.1109/tmech.2009.2028422 [↩](#c823f68dd2a72b9667a61b3c046b4731)
<a id="yedamale03_brush_dc_bldc_motor_fundam"></a>Yedamale, P., *Brushless dc (bldc) motor fundamentals*, Microchip Technology Inc, *20()*, 3–15 (2003). [↩](#d2e68d39d09d7e8e71ff08a6ebd45400)
## Backlinks {#backlinks}
- [Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation]({{< relref "ito16_compar_class_high_precis_actuat" >}})

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draft = false
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Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Advances in internal model control technique: a review and future prospects]({{< relref "saxena12_advan_inter_model_contr_techn" >}})

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author = ["Thomas Dehaeze"]
draft = false
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Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Sensors and control of a space-based six-axis vibration isolation system]({{< relref "hauge04_sensor_contr_space_based_six" >}})
- [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})
- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})

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author = ["Thomas Dehaeze"]
draft = false
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Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [The art of electronics - third edition]({{< relref "horowitz15_art_of_elect_third_edition" >}})

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author = ["Thomas Dehaeze"]
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Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [A six-axis single-stage active vibration isolator based on stewart platform]({{< relref "preumont07_six_axis_singl_stage_activ" >}})
- [Nanometre-cutting machine using a stewart-platform parallel mechanism]({{< relref "furutani04_nanom_cuttin_machin_using_stewar" >}})
- [Dynamic modeling and experimental analyses of stewart platform with flexible hinges]({{< relref "jiao18_dynam_model_exper_analy_stewar" >}})
- [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})
- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})
- [Investigation on active vibration isolation of a stewart platform with piezoelectric actuators]({{< relref "wang16_inves_activ_vibrat_isolat_stewar" >}})
- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})

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Tags
:
## Suppliers {#suppliers}
| | |
|-----|---------------------------------------------------------------|
| PCB | [link](https://www.pcb.com/products/productfinder.aspx?tx=17) |
## 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 <sup id="c823f68dd2a72b9667a61b3c046b4731"><a href="#fleming10_nanop_system_with_force_feedb" title="Fleming, Nanopositioning System With Force Feedback for High-Performance Tracking and Vibration Control, {IEEE/ASME Transactions on Mechatronics}, v(3), 433-447 (2010).">(Fleming, 2010)</a></sup> ([Notes]({{< relref "fleming10_nanop_system_with_force_feedb" >}})).
# Bibliography
<a id="fleming10_nanop_system_with_force_feedb"></a>Fleming, A., *Nanopositioning system with force feedback for high-performance tracking and vibration control*, IEEE/ASME Transactions on Mechatronics, *15(3)*, 433–447 (2010). http://dx.doi.org/10.1109/tmech.2009.2028422 [↩](#c823f68dd2a72b9667a61b3c046b4731)

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title = "H Infinity Control"
author = ["Thomas Dehaeze"]
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Tags
:
## Nice Citations {#nice-citations}
From _Rosenbrock, H. H. (1974). Computer-Aided Control System Design, Academic Press, New York_:
> Solutions are constrained by so many requirements that it is virtually impossible to list them all.
> The designer finds himself threading a maze of such requirements, attempting to reconcile conflicting demands of cost, performance, easy maintenance, and so on.
> A good design usually has strong aesthetic appeal to those who are competent in the subject.
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Guidelines for the selection of weighting functions for h-infinity control]({{< relref "bibel92_guidel_h" >}})

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author = ["Thomas Dehaeze"]
draft = false
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Tags
: [Position Sensors]({{< relref "position_sensors" >}})
## Reviews {#reviews}
- <sup id="dd5109075933cf543c7eba0979c0ba50"><a href="#collette12_review" title="Collette, Janssens, Fernandez-Carmona, , Artoos, Guinchard, Hauviller \&amp; Preumont, Review: Inertial Sensors for Low-Frequency Seismic Vibration Measurement, {Bulletin of the Seismological Society of America}, v(4), 1289-1300 (2012).">(Collette {\it et al.}, 2012)</a></sup>
## Accelerometers {#accelerometers}
| | |
|--------------------|---------------------------------------------------------------|
| Micromega Dynamics | [link](https://micromega-dynamics.com/products/) |
| MMF | [link](https://www.mmf.de/seismic%5Faccelerometers.htm) |
| PCB | [link](https://www.pcb.com/products/productfinder.aspx?tx=14) |
Wireless Accelerometers
- <https://micromega-dynamics.com/products/recovib/miniature-vibration-recorder/>
<a id="orgdad9a09"></a>
{{< figure src="/ox-hugo/inertial_sensors_characteristics_accelerometers.png" caption="Figure 1: 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).\">(Collette {\it et al.}, 2011)</a></sup>" >}}
## Geophones {#geophones}
| | |
|----------|----------------------------------------------------------------|
| Sercel | [link](http://www.sercel.com/products/Pages/seismometers.aspx) |
| Wilcoxon | [link](https://wilcoxon.com/) |
<a id="org8c39d2f"></a>
{{< figure src="/ox-hugo/inertial_sensors_characteristics_geophone.png" caption="Figure 2: 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).\">(Collette {\it et al.}, 2011)</a></sup>" >}}
# Bibliography
<a id="collette12_review"></a>Collette, C., Janssens, S., Fernandez-Carmona, P., Artoos, K., Guinchard, M., Hauviller, C., & Preumont, A., *Review: inertial sensors for low-frequency seismic vibration measurement*, Bulletin of the Seismological Society of America, *102(4)*, 1289–1300 (2012). http://dx.doi.org/10.1785/0120110223 [↩](#dd5109075933cf543c7eba0979c0ba50)
<a id="collette11_review"></a>Collette, C., Artoos, K., Guinchard, M., Janssens, S., Carmona Fernandez, P., & Hauviller, C., *Review of sensors for low frequency seismic vibration measurement* (2011). [↩](#642a18d86de4e062c6afb0f5f20501c4)

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title = "Metrology"
author = ["Thomas Dehaeze"]
draft = false
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Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Fundamental principles of engineering nanometrology]({{< relref "leach14_fundam_princ_engin_nanom" >}})

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title = "Motion Control"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Advanced motion control for precision mechatronics: control, identification, and learning of complex systems]({{< relref "oomen18_advan_motion_contr_precis_mechat" >}})

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title = "Multivariable Control"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}})
- [Multivariable control systems: an engineering approach]({{< relref "albertos04_multiv_contr_system" >}})
- [Position control in lithographic equipment]({{< relref "butler11_posit_contr_lithog_equip" >}})
- [Implementation challenges for multivariable control: what you did not learn in school!]({{< relref "garg07_implem_chall_multiv_contr" >}})
- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})

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title = "Nano Active Stabilization System"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Interferometric characterization of rotation stages for x-ray nanotomography]({{< relref "stankevic17_inter_charac_rotat_stages_x_ray_nanot" >}})
- [Automated markerless full field hard x-ray microscopic tomography at sub-50 nm 3-dimension spatial resolution]({{< relref "wang12_autom_marker_full_field_hard" >}})
- [An instrument for 3d x-ray nano-imaging]({{< relref "holler12_instr_x_ray_nano_imagin" >}})

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title = "Position Sensors"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
## Reviews of position sensors {#reviews-of-position-sensors}
- <sup id="0b0b67de6dddc4d28031ab2d3b28cd3d"><a href="#collette12_compar" title="Collette, Janssens, Mokrani, Fueyo-Roza, L, Artoos, Esposito, Fernandez-Carmona, , Guinchard \&amp; Leuxe, Comparison of new absolute displacement sensors, in in: {International Conference on Noise and Vibration Engineering
(ISMA)}, edited by (2012)">(Collette {\it et al.}, 2012)</a></sup>
- <sup id="3fb5b61524290e36d639a4fac65703d0"><a href="#fleming13_review_nanom_resol_posit_sensor" title="Andrew Fleming, A Review of Nanometer Resolution Position Sensors: Operation and Performance, {Sensors and Actuators A: Physical}, v(nil), 106-126 (2013).">(Andrew Fleming, 2013)</a></sup>
## Relative Position Sensors {#relative-position-sensors}
<a id="orgf9f8137"></a>
{{< figure src="/ox-hugo/position_sensor_characteristics_relative_sensor.png" caption="Figure 1: Characteristics of relative measurement sensors <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).\">(Collette {\it et al.}, 2011)</a></sup>" >}}
<a id="org4ac51f1"></a>
{{< figure src="/ox-hugo/position_sensor_characteristics.png" caption="Figure 2: Position sensor characteristics <sup id=\"3fb5b61524290e36d639a4fac65703d0\"><a href=\"#fleming13_review_nanom_resol_posit_sensor\" title=\"Andrew Fleming, A Review of Nanometer Resolution Position Sensors: Operation and Performance, {Sensors and Actuators A: Physical}, v(nil), 106-126 (2013).\">(Andrew Fleming, 2013)</a></sup>" >}}
### Strain Gauge {#strain-gauge}
### Capacitive Sensor {#capacitive-sensor}
Description:
- <http://www.lionprecision.com/tech-library/technotes/cap-0020-sensor-theory.html>
- <https://www.lionprecision.com/comparing-capacitive-and-eddy-current-sensors>
| | |
|----------------|-------------------------------------------------------------------------------------------------|
| Micro Sense | [link](http://www.microsense.net/products-position-sensors.htm) |
| Micro-Epsilon | [link](https://www.micro-epsilon.com/displacement-position-sensors/capacitive-sensor/) |
| PI | [link](https://www.physikinstrumente.com/en/technology/sensor-technologies/capacitive-sensors/) |
| Unipulse | [link](https://www.unipulse.com/product/ps-ia/) |
| Lion-Precision | [link](https://www.lionprecision.com/products/capacitive-sensors) |
### Inductive Sensor (Eddy Current) {#inductive-sensor--eddy-current}
| | |
|----------------|------------------------------------------------------------------------------------------|
| Micro-Epsilon | [link](https://www.micro-epsilon.com/displacement-position-sensors/eddy-current-sensor/) |
| Lion Precision | [link](https://www.lionprecision.com/products/eddy-current-sensors) |
### Inductive Sensor (LVDT) {#inductive-sensor--lvdt}
| | |
|---------------|--------------------------------------------------------------------------------------------|
| Micro-Epsilon | [link](https://www.micro-epsilon.com/displacement-position-sensors/inductive-sensor-lvdt/) |
| Keyence | [link](https://www.keyence.eu/products/measure/contact-distance-lvdt/gt2/index.jsp) |
### Interferometers {#interferometers}
| | |
|----------|----------------------------------------------------------------------------------------------------------|
| Attocube | [link](http://www.attocube.com/) |
| Zygo | [link](https://www.zygo.com/?/met/markets/stageposition/zmi/) |
| Smaract | [link](https://www.smaract.com/interferometry) |
| Qutools | [link](https://www.qutools.com/qudis/) |
| Renishaw | [link](https://www.renishaw.com/en/fibre-optic-laser-encoder-products--6594) |
| Sios | [link](https://sios-de.com/products/length-measurement/laser-interferometer/) |
| Keysight | [link](https://www.keysight.com/en/pc-1000000393%3Aepsg%3Apgr/laser-heads?nid=-536900395.0&cc=FR&lc=fre) |
<div class="table-caption">
<span class="table-number">Table 1</span>:
Characteristics of Environmental Units
</div>
| | Temperature (\\(\pm\ ^oC\\)) | Pressure (\\(\pm\ hPa\\)) | Humidity \\(\pm\ \% RH\\) | Wavelength Accuracy (\\(\pm\ \text{ppm}\\)) |
|-----------|------------------------------|---------------------------|---------------------------|---------------------------------------------|
| Attocube | 0.1 | 1 | 2 | 0.5 |
| Renishaw | 0.2 | 1 | 6 | 1 |
| Picoscale | 0.2 | 2 | 2 | 1 |
Figure [3](#orgce716a4) is taken from
<sup id="7658b1219a4458a62ae8c6f51b767542"><a href="#jang17_compen_refrac_index_air_laser" title="Yoon-Soo Jang \&amp; Seung-Woo Kim, Compensation of the Refractive Index of Air in Laser Interferometer for Distance Measurement: a Review, {International Journal of Precision Engineering and
Manufacturing}, v(12), 1881-1890 (2017).">(Yoon-Soo Jang \& Seung-Woo Kim, 2017)</a></sup>.
<a id="orgce716a4"></a>
{{< figure src="/ox-hugo/position_sensor_interferometer_precision.png" caption="Figure 3: Expected precision of interferometer as a function of measured distance" >}}
### Fiber Optic Displacement Sensor {#fiber-optic-displacement-sensor}
| | |
|----------|----------------------------------------------------|
| Unipulse | [link](https://www.unipulse.com/product/atw200-2/) |
# Bibliography
<a id="collette12_compar"></a>Collette, C., Janssens, S., Mokrani, B., Fueyo-Roza, L., Artoos, K., Esposito, M., Fernandez-Carmona, P., …, *Comparison of new absolute displacement sensors*, In , International Conference on Noise and Vibration Engineering (ISMA) (pp. ) (2012). : . [↩](#0b0b67de6dddc4d28031ab2d3b28cd3d)
<a id="fleming13_review_nanom_resol_posit_sensor"></a>Fleming, A. J., *A review of nanometer resolution position sensors: operation and performance*, Sensors and Actuators A: Physical, *190(nil)*, 106–126 (2013). http://dx.doi.org/10.1016/j.sna.2012.10.016 [↩](#3fb5b61524290e36d639a4fac65703d0)
<a id="collette11_review"></a>Collette, C., Artoos, K., Guinchard, M., Janssens, S., Carmona Fernandez, P., & Hauviller, C., *Review of sensors for low frequency seismic vibration measurement* (2011). [↩](#642a18d86de4e062c6afb0f5f20501c4)
<a id="jang17_compen_refrac_index_air_laser"></a>Jang, Y., & Kim, S., *Compensation of the refractive index of air in laser interferometer for distance measurement: a review*, International Journal of Precision Engineering and Manufacturing, *18(12)*, 1881–1890 (2017). http://dx.doi.org/10.1007/s12541-017-0217-y [↩](#7658b1219a4458a62ae8c6f51b767542)
## Backlinks {#backlinks}
- [A review of nanometer resolution position sensors: operation and performance]({{< relref "fleming13_review_nanom_resol_posit_sensor" >}})
- [Measurement technologies for precision positioning]({{< relref "gao15_measur_techn_precis_posit" >}})

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title = "Positioning Stations"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Interferometric characterization of rotation stages for x-ray nanotomography]({{< relref "stankevic17_inter_charac_rotat_stages_x_ray_nanot" >}})
- [Position control in lithographic equipment]({{< relref "butler11_posit_contr_lithog_equip" >}})
- [An instrument for 3d x-ray nano-imaging]({{< relref "holler12_instr_x_ray_nano_imagin" >}})

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title = "Precision Engineering"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Design for precision: current status and trends]({{< relref "schellekens98_desig_precis" >}})
- [Basics of precision engineering - 1st edition]({{< relref "leach18_basic_precis_engin_edition" >}})

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title = "Reference Books"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Multivariable feedback control: analysis and design]({{< relref "skogestad07_multiv_feedb_contr" >}})
- [The design of high performance mechatronics - 2nd revised edition]({{< relref "schmidt14_desig_high_perfor_mechat_revis_edition" >}})
- [Parallel robots : mechanics and control]({{< relref "taghirad13_paral" >}})
- [Modal testing: theory, practice and application]({{< relref "ewins00_modal" >}})
- [The art of electronics - third edition]({{< relref "horowitz15_art_of_elect_third_edition" >}})
- [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}})

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title = "Sensor Fusion"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Sensor fusion for active vibration isolation in precision equipment]({{< relref "tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip" >}})
- [Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs]({{< relref "collette14_vibrat" >}})
- [Sensor fusion methods for high performance active vibration isolation systems]({{< relref "collette15_sensor_fusion_method_high_perfor" >}})
- [Nanopositioning system with force feedback for high-performance tracking and vibration control]({{< relref "fleming10_nanop_system_with_force_feedb" >}})
- [Nanopositioning with multiple sensors: a case study in data storage]({{< relref "sebastian12_nanop_with_multip_sensor" >}})

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title = "Stewart Platforms"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Six dof active vibration control using stewart platform with non-cubic configuration]({{< relref "zhang11_six_dof" >}})
- [Decentralized vibration control of a voice coil motor-based stewart parallel mechanism: simulation and experiments]({{< relref "tang18_decen_vibrat_contr_voice_coil" >}})
- [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})
- [Parallel robots : mechanics and control]({{< relref "taghirad13_paral" >}})
- [Investigation on active vibration isolation of a stewart platform with piezoelectric actuators]({{< relref "wang16_inves_activ_vibrat_isolat_stewar" >}})
- [Identification and decoupling control of flexure jointed hexapods]({{< relref "chen00_ident_decoup_contr_flexur_joint_hexap" >}})
- [The stewart platform manipulator: a review]({{< relref "dasgupta00_stewar_platf_manip" >}})
- [Modeling and control of vibration in mechanical systems]({{< relref "du10_model_contr_vibrat_mechan_system" >}})
- [Studies on stewart platform manipulator: a review]({{< relref "furqan17_studies_stewar_platf_manip" >}})
- [Nanometre-cutting machine using a stewart-platform parallel mechanism]({{< relref "furutani04_nanom_cuttin_machin_using_stewar" >}})
- [An intelligent control system for multiple degree-of-freedom vibration isolation]({{< relref "geng95_intel_contr_system_multip_degree" >}})
- [Active isolation and damping of vibrations via stewart platform]({{< relref "hanieh03_activ_stewar" >}})
- [Sensors and control of a space-based six-axis vibration isolation system]({{< relref "hauge04_sensor_contr_space_based_six" >}})
- [Dynamic modeling and experimental analyses of stewart platform with flexible hinges]({{< relref "jiao18_dynam_model_exper_analy_stewar" >}})
- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})
- [A new isotropic and decoupled 6-dof parallel manipulator]({{< relref "legnani12_new_isotr_decoup_paral_manip" >}})
- [Simultaneous vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_vibrat_isolat_point_contr" >}})
- [A six-axis single-stage active vibration isolator based on stewart platform]({{< relref "preumont07_six_axis_singl_stage_activ" >}})
- [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}})
- [A soft 6-axis active vibration isolator]({{< relref "spanos95_soft_activ_vibrat_isolat" >}})

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title = "System Identification"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Modal testing: theory, practice and application]({{< relref "ewins00_modal" >}})

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title = "Test File"
author = ["Thomas Dehaeze"]
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> This is a quote!
```matlab
a = 2;
figure;
```
<div class="important">
<div></div>
This is an important part of the text.
</div>
See Eq. [eq:test1](#eq:test1) and [eq:test2](#eq:test2).
\begin{equation}
a = 1
\end{equation}
\begin{equation}
a = 2 \label{eq:test2}
\end{equation}
Also look at [1](#org7280632) \eqref{eq:test2}.
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title = "Vibration Isolation"
author = ["Thomas Dehaeze"]
draft = false
+++
Tags
:
<./biblio/references.bib>
## Backlinks {#backlinks}
- [Six dof active vibration control using stewart platform with non-cubic configuration]({{< relref "zhang11_six_dof" >}})
- [Dynamic modeling and decoupled control of a flexible stewart platform for vibration isolation]({{< relref "yang19_dynam_model_decoup_contr_flexib" >}})
- [Investigation on active vibration isolation of a stewart platform with piezoelectric actuators]({{< relref "wang16_inves_activ_vibrat_isolat_stewar" >}})
- [Review of active vibration isolation strategies]({{< relref "collette11_review_activ_vibrat_isolat_strat" >}})
- [Vibration control of flexible structures using fusion of inertial sensors and hyper-stable actuator-sensor pairs]({{< relref "collette14_vibrat" >}})
- [Sensor fusion methods for high performance active vibration isolation systems]({{< relref "collette15_sensor_fusion_method_high_perfor" >}})
- [Modeling and control of vibration in mechanical systems]({{< relref "du10_model_contr_vibrat_mechan_system" >}})
- [An intelligent control system for multiple degree-of-freedom vibration isolation]({{< relref "geng95_intel_contr_system_multip_degree" >}})
- [Active isolation and damping of vibrations via stewart platform]({{< relref "hanieh03_activ_stewar" >}})
- [Sensors and control of a space-based six-axis vibration isolation system]({{< relref "hauge04_sensor_contr_space_based_six" >}})
- [Comparison and classification of high-precision actuators based on stiffness influencing vibration isolation]({{< relref "ito16_compar_class_high_precis_actuat" >}})
- [Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_fault_vibrat_isolat_point" >}})
- [Simultaneous vibration isolation and pointing control of flexure jointed hexapods]({{< relref "li01_simul_vibrat_isolat_point_contr" >}})
- [An exploration of active hard mount vibration isolation for precision equipment]({{< relref "poel10_explor_activ_hard_mount_vibrat" >}})
- [Force feedback versus acceleration feedback in active vibration isolation]({{< relref "preumont02_force_feedb_versus_accel_feedb" >}})
- [A six-axis single-stage active vibration isolator based on stewart platform]({{< relref "preumont07_six_axis_singl_stage_activ" >}})
- [Vibration Control of Active Structures - Fourth Edition]({{< relref "preumont18_vibrat_contr_activ_struc_fourt_edition" >}})
- [A soft 6-axis active vibration isolator]({{< relref "spanos95_soft_activ_vibrat_isolat" >}})
- [Sensor fusion for active vibration isolation in precision equipment]({{< relref "tjepkema12_sensor_fusion_activ_vibrat_isolat_precis_equip" >}})