Update Content - 2022-03-15
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title = "Active isolation and damping of vibrations via stewart platform"
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author = ["Thomas Dehaeze"]
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author = ["Dehaeze Thomas"]
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draft = true
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ref_author = "Hanieh, A. A."
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ref_year = 2003
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+++
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Tags
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: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}}), [Vibration Isolation]({{<relref "vibration_isolation.md#" >}}), [Active Damping]({{<relref "active_damping.md#" >}})
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: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Active Damping]({{< relref "active_damping.md" >}})
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Reference
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: ([Hanieh 2003](#orgf310fe8))
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: (<a href="#citeproc_bib_item_1">Hanieh 2003</a>)
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Author(s)
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: Hanieh, A. A.
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@@ -19,7 +19,8 @@ Year
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: 2003
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## Bibliography {#bibliography}
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<a id="orgf310fe8"></a>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.
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<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
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<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.</div>
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</div>
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@@ -1,16 +1,16 @@
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+++
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title = "Mechatronic design of a magnetically suspended rotating platform"
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author = ["Thomas Dehaeze"]
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author = ["Dehaeze Thomas"]
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draft = false
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ref_author = "Jabben, L."
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ref_year = 2007
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+++
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Tags
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: [Dynamic Error Budgeting]({{<relref "dynamic_error_budgeting.md#" >}})
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: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
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Reference
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: ([Jabben 2007](#org6250919))
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: (<a href="#citeproc_bib_item_1">Jabben 2007</a>)
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Author
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: Jabben, L.
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@@ -49,10 +49,9 @@ This approach allows frequency dependent error budgeting, which is why it is ref
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This noise can be modeled as a voltage source in series with the system impedance.
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The noise source has a PSD given by:
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\\[ S\_T(f) = 4 k T \text{Re}(Z(f)) \ [V^2/Hz] \\]
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with \\(k = 1.38 \cdot 10^{-23} \,[J/K]\\) the Boltzmann's constant, \\(T\\) the temperature [K] and \\(Z(f)\\) the frequency dependent impedance of the system.
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with \\(k = 1.38 \cdot 10^{-23} \\,[J/K]\\) the Boltzmann's constant, \\(T\\) the temperature [K] and \\(Z(f)\\) the frequency dependent impedance of the system.
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<div class="exampl">
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<div></div>
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A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of \\(0.13 \mu V\\) from zero up to one kHz.
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@@ -62,12 +61,11 @@ A kilo Ohm resistor at 20 degree Celsius will show a thermal noise of \\(0.13 \m
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Seen with junctions in a transistor.
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It has a white spectral density:
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\\[ S\_S = 2 q\_e i\_{dc} \ [A^2/Hz] \\]
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with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\, [C]\\)), \\(i\_{dc}\\) the average current [A].
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with \\(q\_e\\) the electronic charge (\\(1.6 \cdot 10^{-19}\\, [C]\\)), \\(i\_{dc}\\) the average current [A].
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<div class="exampl">
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<div></div>
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An averable current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\,[A]\\) from zero up to one kHz.
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An averable current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\\,[A]\\) from zero up to one kHz.
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</div>
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@@ -100,24 +98,23 @@ The corresponding PSD is white up to the Nyquist frequency:
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with \\(f\_N\\) the Nyquist frequency [Hz].
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<div class="exampl">
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<div></div>
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Let's take the example of a 16 bit ADC which has an electronic noise with a SNR of 80dB.
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Let's suppose the ADC is used to measure a position over a range of 1 mm.
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- ADC quantization noise: it has 16 bots over the 1 mm range.
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The standard diviation from the quantization is:
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\\[ \sigma\_{ADq} = \frac{1 \cdot 10^6/2^16}{\sqrt{12}} = 4.4\,[nm] \\]
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- ADC electronic noise: the RMS value of a sine that covers to full range is \\(\frac{0.5}{\sqrt{2}} = 0.354\,[mm]\\).
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- ADC quantization noise: it has 16 bits over the 1 mm range.
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The standard deviation from the quantization is:
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\\[ \sigma\_{ADq} = \frac{1 \cdot 10^6/2^{16}}{\sqrt{12}} = 4.4\\,[nm] \\]
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- ADC electronic noise: the RMS value of a sine that covers to full range is \\(\frac{0.5}{\sqrt{2}} = 0.354\\,[mm]\\).
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With a SNR of 80dB, the electronic noise from the ADC becomes:
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\\[ \sigma\_{ADn} = 35\,[nm] \\]
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\\[ \sigma\_{ADn} = 35\\,[nm] \\]
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Let's suppose the ADC is used to measure a sensor with an electronic noise having a standard deviation of \\(\sigma\_{sn} = 17\,[nm]\\).
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Let's suppose the ADC is used to measure a sensor with an electronic noise having a standard deviation of \\(\sigma\_{sn} = 17\\,[nm]\\).
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The PSD of this digitalized sensor noise is:
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\\[ \sigma\_s = \sqrt{\sigma\_{sn}^2 + \sigma\_{ADq}^2 + \sigma\_{ADn}^2} = 39\,[nm]\\]
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\\[ \sigma\_s = \sqrt{\sigma\_{sn}^2 + \sigma\_{ADq}^2 + \sigma\_{ADn}^2} = 39\\,[nm]\\]
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from which the PSD of the total sensor noise \\(S\_s\\) is calculated:
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\\[ S\_s = \frac{\sigma\_s^2}{f\_N} = 1.55\,[nm^2/Hz] \\]
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\\[ S\_s = \frac{\sigma\_s^2}{f\_N} = 1.55\\,[nm^2/Hz] \\]
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with \\(f\_N\\) is the Nyquist frequency of 1kHz.
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</div>
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@@ -132,9 +129,8 @@ To have a pressure difference, the body must have a certain minimum dimension, d
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For a body of typical dimensions of 100mm, only frequencies above 800 Hz have a significant disturbance contribution.
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<div class="exampl">
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<div></div>
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Consider a cube with a rib size of 100 mm located in a room with a sound level of 80dB, distributed between one and ten kHz, then the force disturbance PSD equal \\(2.2 \cdot 10^{-2}\,[N^2/Hz]\\)
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Consider a cube with a rib size of 100 mm located in a room with a sound level of 80dB, distributed between one and ten kHz, then the force disturbance PSD equal \\(2.2 \cdot 10^{-2}\\,[N^2/Hz]\\)
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</div>
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@@ -163,21 +159,21 @@ Three factors influence the performance:
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The DEB helps identifying which disturbance is the limiting factor, and it should be investigated if the controller can deal with this disturbance before re-designing the plant.
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The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
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In Figure [1](#orgcc56194), the generalized plant maps the disturbances to the performance channels.
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In Figure [1](#figure--fig:jabben07-general-plant), the generalized plant maps the disturbances to the performance channels.
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By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the variance of the performance channels is minimized.
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<a id="orgcc56194"></a>
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<a id="figure--fig:jabben07-general-plant"></a>
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{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="Figure 1: Control system with the generalized plant \\(G\\). The performance channels are stacked in \\(z\\), while the controller input is denoted with \\(y\\)" >}}
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{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="<span class=\"figure-number\">Figure 1: </span>Control system with the generalized plant \\(G\\). The performance channels are stacked in \\(z\\), while the controller input is denoted with \\(y\\)" >}}
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#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
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Since disturbances are generally not white, the system of Figure [1](#orgcc56194) needs to be augmented with so called **disturbance weighting filters**.
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Since disturbances are generally not white, the system of Figure [1](#figure--fig:jabben07-general-plant) needs to be augmented with so called **disturbance weighting filters**.
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A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
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This is illustrated in Figure [2](#org772dfb7) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
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This is illustrated in Figure [2](#figure--fig:jabben07-weighting-functions) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
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The generalized plant framework also allows to include **weighting filters for the performance channels**.
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This is useful for three reasons:
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@@ -186,9 +182,9 @@ This is useful for three reasons:
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- some performance channels may be of more importance than others
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- by using dynamic weighting filters, one can emphasize the performance in a certain frequency range
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<a id="org772dfb7"></a>
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<a id="figure--fig:jabben07-weighting-functions"></a>
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{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="Figure 2: Control system with the generalized plant \\(G\\) and weighting functions" >}}
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{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="<span class=\"figure-number\">Figure 2: </span>Control system with the generalized plant \\(G\\) and weighting functions" >}}
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The weighting filters should be stable transfer functions.
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@@ -209,13 +205,13 @@ By making the \\(\mathcal{H}\_2\\) norm of \\(V\_h\\) equal to the RMS-value of
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IF only the output \\(y\\) are considered in the performance channel \\(z\\), the resulting optimal controller might result in very large actuator signals.
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So, to obtain feasible controllers, the performance channel is a combination of controller output \\(u\\) and system output \\(y\\).
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By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
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\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
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\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
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By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#orgeab38dd) is obtained.
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By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#figure--fig:jabben07-pareto-curve-H2) is obtained.
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<a id="orgeab38dd"></a>
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<a id="figure--fig:jabben07-pareto-curve-H2"></a>
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{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="Figure 3: An illustration of a Pareto curve. Each point of the curve represents the performance obtained with an optimal controller. The curve is obtained by varying \\(\alpha\\) and calculating an \\(\mathcal{H}\_2\\) optimal controller for each \\(\alpha\\)." >}}
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{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="<span class=\"figure-number\">Figure 3: </span>An illustration of a Pareto curve. Each point of the curve represents the performance obtained with an optimal controller. The curve is obtained by varying \\(\alpha\\) and calculating an \\(\mathcal{H}\_2\\) optimal controller for each \\(\alpha\\)." >}}
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## Conclusion {#conclusion}
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@@ -237,8 +233,3 @@ By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha
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> To use the measured PSDs in an optimal control design, such as H2-control, the disturbances must be modelled using linear time invariant models with multiple white noise input.
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> To derive such models, spectral factorization is used.
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> It is recommended to investigate which methods for spectral factorization are currently available and numerically robust.
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## Bibliography {#bibliography}
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<a id="org6250919"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.
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@@ -1,16 +1,16 @@
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+++
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title = "Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods"
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author = ["Thomas Dehaeze"]
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author = ["Dehaeze Thomas"]
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draft = false
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ref_author = "Li, X."
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ref_year = 2001
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+++
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Tags
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: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}}), [Vibration Isolation]({{<relref "vibration_isolation.md#" >}}), [Cubic Architecture]({{<relref "cubic_architecture.md#" >}}), [Flexible Joints]({{<relref "flexible_joints.md#" >}}), [Multivariable Control]({{<relref "multivariable_control.md#" >}})
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: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Cubic Architecture]({{< relref "cubic_architecture.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}}), [Multivariable Control]({{< relref "multivariable_control.md" >}})
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Reference
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: ([Li 2001](#org8036ec7))
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: (<a href="#citeproc_bib_item_1">Li 2001</a>)
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Author(s)
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: Li, X.
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@@ -24,17 +24,17 @@ Year
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### Flexure Jointed Hexapods {#flexure-jointed-hexapods}
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A general flexible jointed hexapod is shown in Figure [1](#orgd9d105c).
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A general flexible jointed hexapod is shown in Figure [1](#figure--fig:li01-flexure-hexapod-model).
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<a id="orgd9d105c"></a>
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<a id="figure--fig:li01-flexure-hexapod-model"></a>
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{{< figure src="/ox-hugo/li01_flexure_hexapod_model.png" caption="Figure 1: A flexure jointed hexapod. {P} is a cartesian coordinate frame located at, and rigidly attached to the payload's center of mass. {B} is the frame attached to the base, and {U} is a universal inertial frame of reference" >}}
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{{< figure src="/ox-hugo/li01_flexure_hexapod_model.png" caption="<span class=\"figure-number\">Figure 1: </span>A flexure jointed hexapod. {P} is a cartesian coordinate frame located at, and rigidly attached to the payload's center of mass. {B} is the frame attached to the base, and {U} is a universal inertial frame of reference" >}}
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Flexure jointed hexapods have been developed to meet two needs illustrated in Figure [2](#orgaa02e76).
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Flexure jointed hexapods have been developed to meet two needs illustrated in Figure [2](#figure--fig:li01-quet-dirty-box).
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<a id="orgaa02e76"></a>
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<a id="figure--fig:li01-quet-dirty-box"></a>
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{{< figure src="/ox-hugo/li01_quet_dirty_box.png" caption="Figure 2: (left) Vibration machinery must be isolated from a precision bus. (right) A precision paylaod must be manipulated in the presence of base vibrations and/or exogenous forces." >}}
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{{< figure src="/ox-hugo/li01_quet_dirty_box.png" caption="<span class=\"figure-number\">Figure 2: </span>(left) Vibration machinery must be isolated from a precision bus. (right) A precision paylaod must be manipulated in the presence of base vibrations and/or exogenous forces." >}}
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Since only small movements are considered in flexure jointed hexapod, the Jacobian matrix, which relates changes in the Cartesian pose to changes in the strut lengths, can be considered constant.
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Thus a static kinematic decoupling algorithm can be implemented for both vibration isolation and pointed controls on flexible jointed hexapods.
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@@ -43,14 +43,14 @@ On the other hand, the flexures add some complexity to the hexapod dynamics.
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Although the flexure joints do eliminate friction and backlash, they add spring dynamics and severely limit the workspace.
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Moreover, base and/or payload vibrations become significant contributors to the motion.
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The University of Wyoming hexapods (example in Figure [3](#orgf80b696)) are:
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The University of Wyoming hexapods (example in Figure [3](#figure--fig:li01-stewart-platform)) are:
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- Cubic (mutually orthogonal)
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- Flexure Jointed
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<a id="orgf80b696"></a>
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<a id="figure--fig:li01-stewart-platform"></a>
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{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 3: Flexure jointed Stewart platform used for analysis and control" >}}
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{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="<span class=\"figure-number\">Figure 3: </span>Flexure jointed Stewart platform used for analysis and control" >}}
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The objectives of the hexapods are:
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@@ -81,13 +81,13 @@ p\_x & p\_y & p\_z & \theta\_x & \theta\_y & \theta\_z
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\begin{equation}
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J = \begin{bmatrix}
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{}^B\hat{u}\_1^T & [({}^B\_PR^P p\_1) \times {}^B\hat{u}\_1]^T \\\\\\
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\vdots & \vdots \\\\\\
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{}^B\hat{u}\_1^T & [({}^B\_PR^P p\_1) \times {}^B\hat{u}\_1]^T \\\\
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\vdots & \vdots \\\\
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{}^B\hat{u}\_6^T & [({}^B\_PR^P p\_6) \times {}^B\hat{u}\_6]^T
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\end{bmatrix}
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\end{equation}
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where (see Figure [1](#orgd9d105c)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
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where (see Figure [1](#figure--fig:li01-flexure-hexapod-model)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
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To make the dynamic model as simple as possible, the origin of {P} is located at the payload's center of mass.
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Thus all \\({}^Pp\_i\\) are found with respect to the center of mass.
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@@ -98,7 +98,7 @@ The dynamics of a flexure jointed hexapod can be written in joint space:
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\begin{equation} \label{eq:hexapod\_eq\_motion}
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\begin{split}
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& \left( J^{-T} \cdot {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T \cdot J^{-1} + M\_s \right) \ddot{l} + B \dot{l} + K (l - l\_r) = \\\\\\
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& \left( J^{-T} \cdot {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T \cdot J^{-1} + M\_s \right) \ddot{l} + B \dot{l} + K (l - l\_r) = \\\\
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&\quad f\_m - \left( M\_s + J^{-T} \cdot {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + J^{-T} \cdot {}^U\_BR^T(\mathcal{F}\_e + \mathcal{G} + \mathcal{C})
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\end{split}
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\end{equation}
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@@ -131,20 +131,20 @@ Define a new input and a new output:
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u\_1 = J^T f\_m, \quad y = J^{-1} (l - l\_r)
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\end{equation}
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Equation \eqref{eq:hexapod_eq_motion} can be rewritten as:
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Equation <eq:hexapod_eq_motion> can be rewritten as:
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\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_1}
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\begin{split}
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& \left( {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T + J^T \cdot M\_s \cdot J \right) \cdot \ddot{y} + J^T \cdot B J \dot{y} + J^T \cdot K \cdot J y = \\\\\\
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& \left( {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T + J^T \cdot M\_s \cdot J \right) \cdot \ddot{y} + J^T \cdot B J \dot{y} + J^T \cdot K \cdot J y = \\\\
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&\quad u\_1 - \left( J^T \cdot M\_s + {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + {}^U\_BR^T\mathcal{F}\_e
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\end{split}
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\end{equation}
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|
||||
If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled (Figure [4](#org493f606)).
|
||||
If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled (Figure [4](#figure--fig:li01-decoupling-conf)).
|
||||
|
||||
<a id="org493f606"></a>
|
||||
<a id="figure--fig:li01-decoupling-conf"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 4: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
|
||||
{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="<span class=\"figure-number\">Figure 4: </span>Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
|
||||
|
||||
Alternatively, a new set of inputs and outputs can be defined:
|
||||
|
||||
@@ -152,21 +152,20 @@ Alternatively, a new set of inputs and outputs can be defined:
|
||||
u\_2 = J^{-1} f\_m, \quad y = J^{-1} (l - l\_r)
|
||||
\end{equation}
|
||||
|
||||
And another decoupled plant is found (Figure [5](#orgbeff72d)):
|
||||
And another decoupled plant is found (Figure [5](#figure--fig:li01-decoupling-conf-bis)):
|
||||
|
||||
\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_2}
|
||||
\begin{split}
|
||||
& \left( J^{-1} \cdot J^{-T} \cdot {}^BM\_x + M\_s \right) \cdot \ddot{y} + B \dot{y} + K y = \\\\\\
|
||||
& \left( J^{-1} \cdot J^{-T} \cdot {}^BM\_x + M\_s \right) \cdot \ddot{y} + B \dot{y} + K y = \\\\
|
||||
&\quad u\_2 - J^{-1} \cdot J^{-T} \left( J^T \cdot M\_s + {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + {}^U\_BR^T\mathcal{F}\_e
|
||||
\end{split}
|
||||
\end{equation}
|
||||
|
||||
<a id="orgbeff72d"></a>
|
||||
<a id="figure--fig:li01-decoupling-conf-bis"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 5: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
|
||||
{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="<span class=\"figure-number\">Figure 5: </span>Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
These decoupling algorithms have two constraints:
|
||||
|
||||
@@ -201,17 +200,17 @@ The control bandwidth is divided as follows:
|
||||
|
||||
### Vibration Isolation {#vibration-isolation}
|
||||
|
||||
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [6](#orgd7c310d).
|
||||
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [6](#figure--fig:li01-vibration-isolation-control).
|
||||
|
||||
<a id="orgd7c310d"></a>
|
||||
<a id="figure--fig:li01-vibration-isolation-control"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 6: Vibration isolation control strategy" >}}
|
||||
{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="<span class=\"figure-number\">Figure 6: </span>Vibration isolation control strategy" >}}
|
||||
|
||||
One of the subsystem plant transfer function is shown in Figure [6](#orgd7c310d)
|
||||
One of the subsystem plant transfer function is shown in Figure [6](#figure--fig:li01-vibration-isolation-control)
|
||||
|
||||
<a id="org1d9e762"></a>
|
||||
<a id="figure--fig:li01-vibration-isolation-control"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="Figure 7: Plant transfer function of one of the SISO subsystem for Vibration Control" >}}
|
||||
{{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="<span class=\"figure-number\">Figure 7: </span>Plant transfer function of one of the SISO subsystem for Vibration Control" >}}
|
||||
|
||||
Each compensator is designed using simple loop-shaping techniques.
|
||||
A typical compensator consists of the following elements:
|
||||
@@ -225,7 +224,6 @@ A typical compensator consists of the following elements:
|
||||
The unity control bandwidth of the isolation loop is designed to be from **5Hz to 50Hz**, so the vibration isolation loop works as a band-pass filter.
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
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.
|
||||
@@ -233,7 +231,6 @@ Only about 20 dB separation is achieve between the diagonal and off-diagonal res
|
||||
</div>
|
||||
|
||||
<div class="note">
|
||||
<div></div>
|
||||
|
||||
Severe phase delay exists in the actual transfer function.
|
||||
This is due to the limited sample frequency and sensor bandwidth limitation.
|
||||
@@ -246,20 +243,20 @@ The reason is not explained.
|
||||
|
||||
### Pointing Control Techniques {#pointing-control-techniques}
|
||||
|
||||
A block diagram of the pointing control system is shown in Figure [8](#orge6a2624).
|
||||
A block diagram of the pointing control system is shown in Figure [8](#figure--fig:li01-pointing-control).
|
||||
|
||||
<a id="orge6a2624"></a>
|
||||
<a id="figure--fig:li01-pointing-control"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 8: Figure caption" >}}
|
||||
{{< figure src="/ox-hugo/li01_pointing_control.png" caption="<span class=\"figure-number\">Figure 8: </span>Figure caption" >}}
|
||||
|
||||
The plant is decoupled into two independent SISO subsystems.
|
||||
The decoupling matrix consists of the columns of \\(J\\) corresponding to the pointing DoFs.
|
||||
|
||||
Figure [9](#org54b4cd4) shows the measured transfer function of the \\(\theta\_x\\) axis.
|
||||
Figure [9](#figure--fig:li01-transfer-function-angle) shows the measured transfer function of the \\(\theta\_x\\) axis.
|
||||
|
||||
<a id="org54b4cd4"></a>
|
||||
<a id="figure--fig:li01-transfer-function-angle"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_transfer_function_angle.png" caption="Figure 9: Experimentally measured plant transfer function of \\(\theta\_x/\theta\_{x\_d}\\)" >}}
|
||||
{{< figure src="/ox-hugo/li01_transfer_function_angle.png" caption="<span class=\"figure-number\">Figure 9: </span>Experimentally measured plant transfer function of \\(\theta\_x/\theta\_{x\_d}\\)" >}}
|
||||
|
||||
A typical compensator consists of the following elements:
|
||||
|
||||
@@ -271,13 +268,13 @@ A typical compensator consists of the following elements:
|
||||
|
||||
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 [10](#orga527171).
|
||||
A feedforward control is added as shown in Figure [10](#figure--fig:li01-feedforward-control).
|
||||
\\(C\_f\\) is the feedforward compensator which is a 2x2 diagonal matrix.
|
||||
Ideally, the feedforward compensator is an invert of the plant dynamics.
|
||||
|
||||
<a id="orga527171"></a>
|
||||
<a id="figure--fig:li01-feedforward-control"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 10: Feedforward control" >}}
|
||||
{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="<span class=\"figure-number\">Figure 10: </span>Feedforward control" >}}
|
||||
|
||||
|
||||
### Simultaneous Control {#simultaneous-control}
|
||||
@@ -287,14 +284,13 @@ The simultaneous vibration isolation and pointing control is approached in two w
|
||||
1. **Closing the vibration isolation loop first**: Design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators.
|
||||
2. **Closing the pointing loop first**: Reverse order.
|
||||
|
||||
Figure [11](#orge85d506) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
|
||||
Figure [11](#figure--fig:li01-parallel-control) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
|
||||
|
||||
<a id="orge85d506"></a>
|
||||
<a id="figure--fig:li01-parallel-control"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 11: A parallel scheme" >}}
|
||||
{{< figure src="/ox-hugo/li01_parallel_control.png" caption="<span class=\"figure-number\">Figure 11: </span>A parallel scheme" >}}
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
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.
|
||||
@@ -306,24 +302,23 @@ However, the interaction between loops may affect the transfer functions of the
|
||||
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 [12](#org1065b18) and [13](#orgba389c3))
|
||||
- Affect the closed loop transmission of the loop first closed (see Figures [12](#figure--fig:li01-closed-loop-pointing) and [13](#figure--fig:li01-closed-loop-vibration))
|
||||
|
||||
As shown in Figure [12](#org1065b18), the peak resonance of the pointing loop increase after the isolation loop is closed.
|
||||
As shown in Figure [12](#figure--fig:li01-closed-loop-pointing), 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="org1065b18"></a>
|
||||
<a id="figure--fig:li01-closed-loop-pointing"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="Figure 12: Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}}
|
||||
{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="<span class=\"figure-number\">Figure 12: </span>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 [13](#orgba389c3)).
|
||||
The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [13](#figure--fig:li01-closed-loop-vibration)).
|
||||
The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
|
||||
|
||||
<a id="orgba389c3"></a>
|
||||
<a id="figure--fig:li01-closed-loop-vibration"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="Figure 13: Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}}
|
||||
{{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="<span class=\"figure-number\">Figure 13: </span>Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}}
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
From the analysis above, it is hard to say which loop has more significant affect on the other loop, but 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.
|
||||
@@ -333,38 +328,37 @@ Thus, it is recommended to design and implement the isolation control system fir
|
||||
|
||||
### Experimental results {#experimental-results}
|
||||
|
||||
Two hexapods are stacked (Figure [14](#orgc3b1ba9)):
|
||||
Two hexapods are stacked (Figure [14](#figure--fig:li01-test-bench)):
|
||||
|
||||
- the bottom hexapod is used to generate disturbances matching candidate applications
|
||||
- the top hexapod provide simultaneous vibration isolation and pointing control
|
||||
|
||||
<a id="orgc3b1ba9"></a>
|
||||
<a id="figure--fig:li01-test-bench"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_test_bench.png" caption="Figure 14: Stacked Hexapods" >}}
|
||||
{{< figure src="/ox-hugo/li01_test_bench.png" caption="<span class=\"figure-number\">Figure 14: </span>Stacked Hexapods" >}}
|
||||
|
||||
First, the vibration isolation and pointing controls were implemented separately.
|
||||
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [15](#org933bc12).
|
||||
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [15](#figure--fig:li01-vibration-isolation-control-results).
|
||||
|
||||
<a id="org933bc12"></a>
|
||||
<a id="figure--fig:li01-vibration-isolation-control-results"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 15: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}}
|
||||
{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="<span class=\"figure-number\">Figure 15: </span>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 [16](#org3618406) where the bandwidth of the isolation control is expanded to very low frequency.
|
||||
The results of simultaneous control is shown in Figure [16](#figure--fig:li01-simultaneous-control-results) where the bandwidth of the isolation control is expanded to very low frequency.
|
||||
|
||||
<a id="org3618406"></a>
|
||||
<a id="figure--fig:li01-simultaneous-control-results"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 16: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
|
||||
{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="<span class=\"figure-number\">Figure 16: </span>Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
|
||||
|
||||
|
||||
### Summary and Conclusion {#summary-and-conclusion}
|
||||
|
||||
<div class="sum">
|
||||
<div></div>
|
||||
|
||||
A parallel control scheme is proposed in this chapters.
|
||||
This scheme is suitable for simultaneous vibration isolation and pointing control.
|
||||
@@ -380,7 +374,6 @@ Experiments show that this scheme takes advantage of the bandwidths of both poin
|
||||
## Future research areas {#future-research-areas}
|
||||
|
||||
<div class="sum">
|
||||
<div></div>
|
||||
|
||||
Proposed future research areas include:
|
||||
|
||||
@@ -406,7 +399,8 @@ Proposed future research areas include:
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<a id="org8036ec7"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.</div>
|
||||
</div>
|
||||
|
||||
@@ -7,10 +7,10 @@ ref_year = 2004
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Dynamic Error Budgeting]({{<relref "dynamic_error_budgeting.md#" >}})
|
||||
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
|
||||
|
||||
Reference
|
||||
: <monkhorst04_dynam_error_budget>
|
||||
: (<a href="#citeproc_bib_item_1">Monkhorst 2004</a>)
|
||||
|
||||
Author(s)
|
||||
: Monkhorst, W.
|
||||
@@ -106,11 +106,11 @@ Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_
|
||||
|
||||
In order to synthesize an \\(\mathcal{H}\_2\\) controller that will minimize the output error, the total system including disturbances needs to be modeled as a system with zero mean white noise inputs.
|
||||
|
||||
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#orgfce1d5b)).
|
||||
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#figure--fig:monkhorst04-weighting-filter)).
|
||||
|
||||
<a id="orgfce1d5b"></a>
|
||||
<a id="figure--fig:monkhorst04-weighting-filter"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="Figure 1: The use of a weighting filter \\(V\_w(f)\\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
|
||||
{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="<span class=\"figure-number\">Figure 1: </span>The use of a weighting filter \\(V\_w(f)\\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
|
||||
|
||||
The white noise input \\(w(t)\\) is dimensionless, and when the weighting filter has units [SI], the resulting weighted signal \\(\bar{w}(t)\\) has units [SI].
|
||||
The PSD \\(S\_w(f)\\) of the weighted signal is:
|
||||
@@ -119,25 +119,25 @@ The PSD \\(S\_w(f)\\) of the weighted signal is:
|
||||
Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
|
||||
However, this can be avoided if the modeling of the disturbances is directly done in terms of weighting filters.
|
||||
|
||||
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#orgd937879)).
|
||||
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#figure--fig:monkhorst04-general-weighted-plant)).
|
||||
|
||||
<a id="orgd937879"></a>
|
||||
<a id="figure--fig:monkhorst04-general-weighted-plant"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="Figure 2: The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}}
|
||||
{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="<span class=\"figure-number\">Figure 2: </span>The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}}
|
||||
|
||||
|
||||
#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
|
||||
|
||||
In this research, the outputs of the closed loop system (Figure [3](#orgf4dc585)) are:
|
||||
In this research, the outputs of the closed loop system (Figure [3](#figure--fig:monkhorst04-closed-loop-H2)) are:
|
||||
|
||||
- the performance (error) signal \\(e\\)
|
||||
- the controller output \\(u\\)
|
||||
|
||||
In this way, the designer can analyze how much control effort is used to achieve the performance level at the performance output.
|
||||
|
||||
<a id="orgf4dc585"></a>
|
||||
<a id="figure--fig:monkhorst04-closed-loop-H2"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="Figure 3: The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}}
|
||||
{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="<span class=\"figure-number\">Figure 3: </span>The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}}
|
||||
|
||||
The resulting problem is a multi-objective control problem: while constraining the variance of the controller output \\(u\\), the variance of the performance channel should be minimized.
|
||||
This problem can be solved by scaling the controller output \\(u\\) with a factor \\(\alpha\\) during the \\(\mathcal{H}\_2\\) synthesis.
|
||||
@@ -157,3 +157,10 @@ To achieve the highest degree of prediction accuracy, it is recommended to use t
|
||||
|
||||
When an \\(\mathcal{H}\_2\\) controller is synthesized for a particular system, it can give the control designer useful hints about how to control the system best for optimal performance.
|
||||
Drawbacks however are, that no robustness guarantees can be given and that the order of the \\(\mathcal{H}\_2\\) controller will generally be too high for implementation.
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.</div>
|
||||
</div>
|
||||
|
||||
@@ -1,16 +1,16 @@
|
||||
+++
|
||||
title = "An exploration of active hard mount vibration isolation for precision equipment"
|
||||
author = ["Thomas Dehaeze"]
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
ref_author = "van der Poel, G. W."
|
||||
ref_year = 2010
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Vibration Isolation]({{<relref "vibration_isolation.md#" >}})
|
||||
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: ([Poel 2010](#org4dd001c))
|
||||
: (<a href="#citeproc_bib_item_1">Van der Poel 2010</a>)
|
||||
|
||||
Author(s)
|
||||
: van der Poel, G. W.
|
||||
@@ -19,7 +19,8 @@ Year
|
||||
: 2010
|
||||
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<a id="org4dd001c"></a>Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. <https://doi.org/10.3990/1.9789036530163>.
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. doi:<a href="https://doi.org/10.3990/1.9789036530163">10.3990/1.9789036530163</a>.</div>
|
||||
</div>
|
||||
|
||||
File diff suppressed because it is too large
Load Diff
@@ -1,16 +1,16 @@
|
||||
+++
|
||||
title = "Element and system design for active and passive vibration isolation"
|
||||
author = ["Thomas Dehaeze"]
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
ref_author = "Zuo, L."
|
||||
ref_year = 2004
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Vibration Isolation]({{<relref "vibration_isolation.md#" >}})
|
||||
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: ([Zuo 2004](#org05cd1c8))
|
||||
: (<a href="#citeproc_bib_item_1">Zuo 2004</a>)
|
||||
|
||||
Author(s)
|
||||
: Zuo, L.
|
||||
@@ -28,24 +28,25 @@ Year
|
||||
> They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback.
|
||||
> Note that reaction force actuators can also work with soft mounts or hard mounts.
|
||||
|
||||
<a id="orgdaec88b"></a>
|
||||
<a id="figure--fig:zuo04-piezo-spring-series"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}}
|
||||
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="<span class=\"figure-number\">Figure 1: </span>PZT actuator and spring in series" >}}
|
||||
|
||||
<a id="org84417be"></a>
|
||||
<a id="figure--fig:zuo04-voice-coil-spring-parallel"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}}
|
||||
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="<span class=\"figure-number\">Figure 2: </span>Voice coil actuator and spring in parallel" >}}
|
||||
|
||||
<a id="orge3c9205"></a>
|
||||
<a id="figure--fig:zuo04-piezo-plant"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}}
|
||||
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="<span class=\"figure-number\">Figure 3: </span>Transmission from PZT voltage to geophone output" >}}
|
||||
|
||||
<a id="orge26e6a6"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}}
|
||||
<a id="figure--fig:zuo04-voice-coil-plant"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="<span class=\"figure-number\">Figure 4: </span>Transmission from voice coil voltage to geophone output" >}}
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<a id="org05cd1c8"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.</div>
|
||||
</div>
|
||||
|
||||
Reference in New Issue
Block a user