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title = "Vibrations and dynamic isotropy in hexapods-analytical studies"
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author = ["Dehaeze Thomas"]
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draft = false
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ref_author = "Afzali-Far, B."
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ref_year = 2016
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+++
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Tags
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: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Isotropy of Parallel Manipulator]({{< relref "isotropy_of_parallel_manipulator.md" >}})
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Reference
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: (<a href="#citeproc_bib_item_1">Afzali-Far 2016</a>)
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Author(s)
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: Afzali-Far, B.
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Year
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: 2016
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## Abstract {#abstract}
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> The present work was initiated based on an industrial demand for designing a **high-bandwidth** hexapod of an advanced large optical telescope.
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> In this dissertation, we have generalized this industrial problem to fully-parametric models of the hexapod vibrations as well as analytical studies on dynamic isotropy in parallel robots, which can be directly used in any hexapod applications.
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>
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> This work firstly establishes a comprehensive and fully parametric model for the vibrations in hexapods at symmetric configurations.
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> We have developed three models:
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>
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> - Cartesian-space formulation
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> - joint-space formulation
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> - refined model taking into account the inertia of the struts
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>
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> Kinematics of the hexapod are derived parametrically based on the Jacobian.
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> Inertia, stiffness and damping matrices are also parametrically formulated.
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> The eigenvectors and eigenfrequencies are then established in both the cartesian and joint spaces.
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> By introducing the inertia of the struts, despite the apparent symmetric geometry, the equivalent inertia matrix in the cartesian space turns out to be non-diagonal matrix.
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> In addition, the decoupled vibrations are analytically investigated where it is shown that the consideration of the strut inertia may lead to significant changes of the decoupling conditions.
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>
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> The problem of dynamic isotropy, as an optimal design solution for hexapods, is also addressed in this dissertation.
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> Dynamic isotropy is a condition in which all eigenfrequencies of a robot are equal.
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> This is a powerful tool in order to obtain dynamically optimized architectures for parallel robots.
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> We analytically present the conditions of dynamic isotropy in hexapods with and without the consideration of the strut inertia.
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## Introduction {#introduction}
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The design variables of a hexapod (i.e. geometry, stiffness, damping and inertia properties) can be optimized based upon the requirements on the modal behavior (i.e. eigenfrequencies and eigenvectors of the system).
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To do so, the following is performed parametrically:
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- parametric model
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- kinematics
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- linearized equations of motion
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- modal analysis
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The linearized equations of motion are identified by stiffness, damping and inertia matrices.
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These matrices can be expressed in terms of the **cartesian-space** or the **joint-space** coordinates.
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In the cartesian space, the stiffness matrix is a function of the flexibility of the struts as well as the geometrical variables.
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However, in the joint space, the stiffness matrix is not a function of geometrical variables.
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The inertia matrix is a function of inertia properties as well as the geometrical variables.
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Dynamic isotropy is an effective tool to avoid scattered eigenfrequencies in a system.
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In a dynamic isotropy condition, all the eigenfrequencies of a system are equal.
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Is is practically almost impossible to obtain dynamic isotropy based on the standard hexapod architecture.
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> Hence, due to the fact that the control bandwidth of a hexapod is mechanically restricted by its natural frequencies, the optimization of the natural frequencies is of great importance.
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## Parametric Modeling of Vibrations {#parametric-modeling-of-vibrations}
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## Analytical Studies on Dynamics Isotropy {#analytical-studies-on-dynamics-isotropy}
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<div class="definition">
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(complete) Dynamic isotropy is defined by:
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\begin{equation}
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M^{-1} K = \sigma I
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\end{equation}
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where \\(\sigma I\\) is a scaled identity matrix.
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This implies that the eigenfrequencies of the matrix \\(M^{-1} K\\) are all equal:
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\begin{equation}
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\omega\_1 = \dots = \omega\_6 = \sqrt{\sigma}
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\end{equation}
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</div>
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Dynamic isotropy for the Stewart platform leads to a series of restrictive conditions and a unique eigenfrequency:
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\begin{equation}
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\omega\_i = \sqrt{\frac{2k}{m\_p}}
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\end{equation}
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When considering inertia of the struts, conditions are becoming more complex.
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<a id="figure--fig:afzali-far16-isotropic-hexapod-example"></a>
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{{< figure src="/ox-hugo/afzali-far16_isotropic_hexapod_example.png" caption="<span class='figure-number'>Figure 1: </span>Architecture of the obtained dynamically isotropic hexapod" >}}
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<div class="definition">
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Static isotropy can be defined by:
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\begin{equation}
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K\_C = J^T K\_J J = \sigma I
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\end{equation}
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where \\(\sigma I\\) is a scaled identity matrix.
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</div>
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The isotropic constrain of the standard hexapod imposes special inertia of the top platform which may not be wanted in practice (\\(I\_{zz} = 4 I\_{yy} = 4 I\_{xx}\\)).
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A class of generalized Gough-Stewart platforms are proposed to eliminate the above constrains.
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[Figure 2](#figure--fig:afzali-far16-proposed-generalized-hexapod) shows a schematic of proposed generalized hexapod.
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<a id="figure--fig:afzali-far16-proposed-generalized-hexapod"></a>
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{{< figure src="/ox-hugo/afzali-far16_proposed_generalized_hexapod.png" caption="<span class='figure-number'>Figure 2: </span>Parametrization of the proposed generalized hexapod" >}}
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## Conclusions {#conclusions}
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<div class="sum">
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The main findings of this dissertation are:
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- Comprehensive and fully parametric model of the hexapod for symmetric configurations are established both in the Cartesian and joint space.
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- Inertia of the struts are taken into account to refine the model.
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- A novel approach in order to obtain dynamically isotropic hexapods is proposed.
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- A novel architecture of hexapod is introduced ([Figure 2](#figure--fig:afzali-far16-proposed-generalized-hexapod)) which is dynamically isotropic for a wide range of inertia properties.
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</div>
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## Bibliography {#bibliography}
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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>Afzali-Far, Behrouz. 2016. “Vibrations and Dynamic Isotropy in Hexapods-Analytical Studies.” Lund University.</div>
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</div>
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+++
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title = "Active damping of vibrations in high-precision motion systems"
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author = ["Dehaeze Thomas"]
|
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draft = false
|
||||
ref_author = "Babakhani, B."
|
||||
ref_year = 2012
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+++
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Tags
|
||||
: [Active Damping]({{< relref "active_damping.md" >}})
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||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Babakhani 2012</a>)
|
||||
|
||||
Author(s)
|
||||
: Babakhani, B.
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||||
|
||||
Year
|
||||
: 2012
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## Bibliography {#bibliography}
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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>Babakhani, Bayan. 2012. “Active Damping of Vibrations in High-Precision Motion Systems.” University of Twente. doi:<a href="https://doi.org/10.3990/1.9789036534642">10.3990/1.9789036534642</a>.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,27 @@
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+++
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||||
title = "Development of precision pointing controllers with and without vibration suppression for the NPS precision pointing hexapod"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
ref_author = "Bishop Jr, R. M."
|
||||
ref_year = 2002
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+++
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|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Bishop Jr 2002</a>)
|
||||
|
||||
Author(s)
|
||||
: Bishop Jr, R. M.
|
||||
|
||||
Year
|
||||
: 2002
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
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||||
|
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<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>Bishop Jr, Ronald M. 2002. “Development of Precision Pointing Controllers with and without Vibration Suppression for the NPS Precision Pointing Hexapod.” Naval Postgraduate School, Monterey, California.</div>
|
||||
</div>
|
||||
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||||
+++
|
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title = "Active isolation and damping of vibrations via stewart platform"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
ref_author = "Hanieh, A. A."
|
||||
ref_year = 2003
|
||||
+++
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|
||||
Tags
|
||||
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Active Damping]({{< relref "active_damping.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Abu Hanieh 2003</a>)
|
||||
|
||||
Author(s)
|
||||
: Hanieh, A. A.
|
||||
|
||||
Year
|
||||
: 2003
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||||
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||||
|
||||
## Bibliography {#bibliography}
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||||
|
||||
<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>Abu Hanieh, A. 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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+++
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||||
title = "Mechatronic design of a magnetically suspended rotating platform"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
ref_author = "Jabben, L."
|
||||
ref_year = 2007
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||||
+++
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|
||||
Tags
|
||||
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Jabben 2007</a>)
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||||
|
||||
Author
|
||||
: Jabben, L.
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||||
|
||||
Year
|
||||
: 2007
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||||
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||||
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||||
## Dynamic Error Budgeting {#dynamic-error-budgeting}
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||||
### Introduction {#introduction}
|
||||
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A large class of mechatronic machines have specifications based on their _standstill_ performance.
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The standstill performance is then limited by the (stochastic) disturbances action on the closed loop.
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|
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The difficulty in calculation with stochastic signals and Bode plots, is that, instead of calculating with the complex response at one frequency, the **area** over a frequency range should be taken into account.
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|
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The **error budgeting** is often used to estimate how much each component contributes to the total error
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||||
|
||||
Since many of the disturbances have a stochastic nature, they can be modelled with their **Power Spectral Densities**.
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The PSD of the performance measure in the closed loop system is the weigted sum of PSDs of the contributions of each disturbance to the performance channel.
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||||
This approach allows frequency dependent error budgeting, which is why it is referred to as **Dynamic Error Budgeting**.
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### Common Mechatronics Disturbances {#common-mechatronics-disturbances}
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||||
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#### Ground vibrations {#ground-vibrations}
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|
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#### [Electronic Noise]({{< relref "electronic_noise.md" >}}) {#electronic-noise--electronic-noise-dot-md}
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**Thermal Noise** (or Johnson noise).
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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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<div class="exampl">
|
||||
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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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</div>
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**Shot Noise**.
|
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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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<div class="exampl">
|
||||
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A current of 1 A will introduce noise with a STD of \\(10 \cdot 10^{-9}\\,[A]\\) from zero up to one kHz.
|
||||
|
||||
</div>
|
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**Excess Noise** (or \\(1/f\\) noise).
|
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It results from fluctuating conductivity due to imperfect contact between two materials.
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The PSD of excess noise increases when the frequency decreases:
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\\[ S\_E = \frac{K\_f}{f^\alpha}\ [V^2/Hz] \\]
|
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where \\(K\_f\\) is dependent on the average voltage drop over the resistor and the index \\(\alpha\\) is usually between 0.8 and 1.4, and often set to unity for approximate calculation.
|
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|
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**Signal to Noise Ration**
|
||||
Electronic equipment does most often not come with detailed electric schemes, in which case the PSD should be determined from measurements.
|
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In the design phase however, one has to rely on information provided by specification sheets from the manufacturer.
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The noise performance of components like sensors, amplifiers, converters, etc., is often specified in terms of a **Signal to Noise Ratio** (SNR).
|
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**The SNR gives the ratio of the RMS value of a sine that covers the full range of the channel through which the signal is propagating over the RMS value of the electrical noise.**
|
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Usually, the SNR is specified up to a certain cut-off frequency.
|
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If no information on the colouring of the noise is available, then the corresponding **PSD can be assumed to be white up to the cut-off frequency** \\(f\_c\\):
|
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\\[ S\_{snr} = \frac{x\_{fr}^2}{8 f\_c C\_{snr}^2} \\]
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||||
with \\(x\_{fr}\\) the full range of \\(x\\), and \\(C\_{snr}\\) the SNR.
|
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|
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|
||||
#### AD and DA converters {#ad-and-da-converters}
|
||||
|
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ADC and DAC add quantization noise to the signal.
|
||||
The variance can be calculated to be:
|
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\\[ \sigma^2 = \frac{q^2}{12} \\]
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with \\(q\\) the quantization interval.
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|
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The corresponding PSD is white up to the Nyquist frequency:
|
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\\[ S\_Q = \frac{q^2}{12 f\_N} \\]
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with \\(f\_N\\) the Nyquist frequency [Hz].
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||||
|
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<div class="exampl">
|
||||
|
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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.
|
||||
|
||||
- ADC quantization noise: it has 16 bits over the 1 mm range.
|
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The standard deviation from the quantization is:
|
||||
\\[ \sigma\_{ADq} = \frac{1 \cdot 10^6/2^{16}}{\sqrt{12}} = 4.4\\,[nm] \\]
|
||||
- ADC electronic noise: the RMS value of a sine that covers to full range is \\(\frac{0.5}{\sqrt{2}} = 0.354\\,[mm]\\).
|
||||
With a SNR of 80dB, the electronic noise from the ADC becomes:
|
||||
\\[ \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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|
||||
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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from which the PSD of the total sensor noise \\(S\_s\\) is calculated:
|
||||
\\[ S\_s = \frac{\sigma\_s^2}{f\_N} = 1.55\\,[nm^2/Hz] \\]
|
||||
with \\(f\_N\\) is the Nyquist frequency of 1kHz.
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
#### Acoustic Noise {#acoustic-noise}
|
||||
|
||||
This can be a big error source in high precision machines, especially when the surface is big compare to the mass.
|
||||
|
||||
The disturbance force acting on a body, is the **difference of pressure between the front and the back times the surface**.
|
||||
To have a pressure difference, the body must have a certain minimum dimension, depending on the wave length of the sound.
|
||||
For a body of typical dimensions of 100mm, only frequencies above 800 Hz have a significant disturbance contribution.
|
||||
|
||||
<div class="exampl">
|
||||
|
||||
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]\\)
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
#### Brownian Noise {#brownian-noise}
|
||||
|
||||
This is due to thermal effects and it notable where a small mass needs positioning.
|
||||
|
||||
|
||||
#### Turbulence {#turbulence}
|
||||
|
||||
Rotation of the spindle introduces and air flow in which turbulence is cause by sharp angles on the rotor and stator.
|
||||
|
||||
|
||||
### Optimal Control {#optimal-control}
|
||||
|
||||
|
||||
#### The use of Optimal Control in DEB {#the-use-of-optimal-control-in-deb}
|
||||
|
||||
Three factors influence the performance:
|
||||
|
||||
- the disturbances: often a given value
|
||||
- the plant: can be costly to redesign
|
||||
- the controller
|
||||
|
||||
The DEB helps identifying which disturbance is the limiting factor, and it should be investigated if the controller can deal with this disturbance before re-designing the plant.
|
||||
|
||||
The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
|
||||
In [Figure 1](#figure--fig:jabben07-general-plant), the generalized plant maps the disturbances to the performance channels.
|
||||
By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the variance of the performance channels is minimized.
|
||||
|
||||
<a id="figure--fig:jabben07-general-plant"></a>
|
||||
|
||||
{{< 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\\)" >}}
|
||||
|
||||
|
||||
#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
|
||||
|
||||
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**.
|
||||
|
||||
A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
|
||||
|
||||
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\\) .
|
||||
|
||||
The generalized plant framework also allows to include **weighting filters for the performance channels**.
|
||||
This is useful for three reasons:
|
||||
|
||||
- the performance channels might have different dimensions, which require scaling in order to compare
|
||||
- some performance channels may be of more importance than others
|
||||
- by using dynamic weighting filters, one can emphasize the performance in a certain frequency range
|
||||
|
||||
<a id="figure--fig:jabben07-weighting-functions"></a>
|
||||
|
||||
{{< 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" >}}
|
||||
|
||||
The weighting filters should be stable transfer functions.
|
||||
|
||||
**Obtaining the weighting filters**:
|
||||
|
||||
If the PSD is given as a function \\(S\_x(j\omega)\\), the disturbance filter can be using **spectral factorization**:
|
||||
|
||||
> Given a positive even function \\(S\_x(f)\\) of finite area, find a minimum-phase stable function \\(L(s)\\), such that \\(|L(j2\pi f)|^2 = S(s)\\)
|
||||
|
||||
**Harmonic signals** can be approximately modeled by filtering white noise with a badly damped second order system, having a \\(+1\\) slope below the resonance frequency and a \\(-1\\) slope above the resonance frequency:
|
||||
\\[ V\_h = \frac{s}{s^2 + 2 \xi \omega\_h + \omega\_h^2} \\]
|
||||
with \\(\xi\\) the relative damping and \\(\omega\_h\\) the resonance frequency [rad/s].
|
||||
By making the \\(\mathcal{H}\_2\\) norm of \\(V\_h\\) equal to the RMS-value of the harmonic signal, the propagation of the disturbance to the performance channel can be well approximated.
|
||||
|
||||
|
||||
#### Balancing Control Effort vs Performance {#balancing-control-effort-vs-performance}
|
||||
|
||||
IF only the output \\(y\\) are considered in the performance channel \\(z\\), the resulting optimal controller might result in very large actuator signals.
|
||||
So, to obtain feasible controllers, the performance channel is a combination of controller output \\(u\\) and system output \\(y\\).
|
||||
By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
|
||||
\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
|
||||
|
||||
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in [Figure 3](#figure--fig:jabben07-pareto-curve-H2) is obtained.
|
||||
|
||||
<a id="figure--fig:jabben07-pareto-curve-H2"></a>
|
||||
|
||||
{{< 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\\)." >}}
|
||||
|
||||
|
||||
## Conclusion {#conclusion}
|
||||
|
||||
> Using the DEB analysis during the design helped to formulate the specifications of the several subcomponents, such as:
|
||||
>
|
||||
> - The target bandwidth of the decentralized closed loops, which is very important for the mechanical design, as mechanical resonances can severely limit the bandwidth.
|
||||
> This value was also used to specify the current loop bandwidth of the custom designed power amplifiers for the RTAs and other components such as sensors and filters.
|
||||
> - The target value of the stiffness of the actuators was derived at 1000 N/m.
|
||||
> It was shown that the stiffness of a motor with back-iron is too much for the separated frame concept.
|
||||
> - The analysis pinpointed the most limiting component in the final design to be the Analogue-to-Digital Converter (ADC).
|
||||
|
||||
<!--quoteend-->
|
||||
|
||||
> In the DEB-framework there are three distinct factors which determine the performance.
|
||||
> These are the plant, the controller and the disturbances.
|
||||
> Synthesizing optimal controllers, such as H2-control, in the design helps to eliminate the controller out of the equation.
|
||||
> If the performance specifications are not met with an optimal controller, it is certain that a redesign of the system is required.
|
||||
> 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.
|
||||
> To derive such models, spectral factorization is used.
|
||||
> It is recommended to investigate which methods for spectral factorization are currently available and numerically robust.
|
||||
@@ -0,0 +1,406 @@
|
||||
+++
|
||||
title = "Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
ref_author = "Li, X."
|
||||
ref_year = 2001
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [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" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Li 2001</a>)
|
||||
|
||||
Author(s)
|
||||
: Li, X.
|
||||
|
||||
Year
|
||||
: 2001
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
|
||||
### Flexure Jointed Hexapods {#flexure-jointed-hexapods}
|
||||
|
||||
A general flexible jointed hexapod is shown in [Figure 1](#figure--fig:li01-flexure-hexapod-model).
|
||||
|
||||
<a id="figure--fig:li01-flexure-hexapod-model"></a>
|
||||
|
||||
{{< 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" >}}
|
||||
|
||||
Flexure jointed hexapods have been developed to meet two needs illustrated in [Figure 2](#figure--fig:li01-quet-dirty-box).
|
||||
|
||||
<a id="figure--fig:li01-quet-dirty-box"></a>
|
||||
|
||||
{{< 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." >}}
|
||||
|
||||
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.
|
||||
Thus a static kinematic decoupling algorithm can be implemented for both vibration isolation and pointed controls on flexible jointed hexapods.
|
||||
|
||||
On the other hand, the flexures add some complexity to the hexapod dynamics.
|
||||
Although the flexure joints do eliminate friction and backlash, they add spring dynamics and severely limit the workspace.
|
||||
Moreover, base and/or payload vibrations become significant contributors to the motion.
|
||||
|
||||
The University of Wyoming hexapods (example in [Figure 3](#figure--fig:li01-stewart-platform)) are:
|
||||
|
||||
- Cubic (mutually orthogonal)
|
||||
- Flexure Jointed
|
||||
|
||||
<a id="figure--fig:li01-stewart-platform"></a>
|
||||
|
||||
{{< 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" >}}
|
||||
|
||||
The objectives of the hexapods are:
|
||||
|
||||
- Precise pointing in two axes (sub micro-radians)
|
||||
- simultaneously, providing both passive and active vibration isolation in six axes
|
||||
|
||||
|
||||
### Jacobian matrix, Dynamic model, and decoupling algorithms {#jacobian-matrix-dynamic-model-and-decoupling-algorithms}
|
||||
|
||||
|
||||
#### Jacobian Matrix {#jacobian-matrix}
|
||||
|
||||
The Jacobian matrix \\(J\\) relates changes in the cartesian pose \\(\mathcal{X}\\) to changes in the strut lengths \\(l\\):
|
||||
|
||||
\begin{equation}
|
||||
\delta l = J \delta \mathcal{X}
|
||||
\end{equation}
|
||||
|
||||
where \\(\mathcal{X}\\) is a 6x1 vector of payload plate translations and rotations
|
||||
|
||||
\begin{equation}
|
||||
\mathcal{X} = \begin{bmatrix}
|
||||
p\_x & p\_y & p\_z & \theta\_x & \theta\_y & \theta\_z
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\\(J\\) is given by:
|
||||
|
||||
\begin{equation}
|
||||
J = \begin{bmatrix}
|
||||
{}^B\hat{u}\_1^T & [({}^B\_PR^P p\_1) \times {}^B\hat{u}\_1]^T \\\\
|
||||
\vdots & \vdots \\\\
|
||||
{}^B\hat{u}\_6^T & [({}^B\_PR^P p\_6) \times {}^B\hat{u}\_6]^T
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
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\\).
|
||||
To make the dynamic model as simple as possible, the origin of {P} is located at the payload's center of mass.
|
||||
Thus all \\({}^Pp\_i\\) are found with respect to the center of mass.
|
||||
|
||||
|
||||
#### Dynamic model of flexure jointed hexapods {#dynamic-model-of-flexure-jointed-hexapods}
|
||||
|
||||
The dynamics of a flexure jointed hexapod can be written in joint space:
|
||||
|
||||
\begin{equation} \label{eq:hexapod\_eq\_motion}
|
||||
\begin{split}
|
||||
& \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) = \\\\
|
||||
&\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})
|
||||
\end{split}
|
||||
\end{equation}
|
||||
|
||||
where:
|
||||
|
||||
- \\({}^PM\_x\\) is the 6x6 mass/inertia matrix of the payload, found with respect to the payload frame {P}, whose origin is at the hexapod payload's center of mass
|
||||
- \\({}^U\_BR\\) is the 6x6 rotation matrix from the base frame {B} to the inertial frame of reference {U} (it consists of two identical 3x3 rotation matrices forming a block diagonal 6x6 matrix).
|
||||
Similarly, \\({}^B\_PR\\) is the rotation matrix from the payload frame to the base frame, and \\({}^U\_PR = {}^U\_BR {}^B\_PR\\)
|
||||
- \\(J\\) is the 6x6 Jacobian matrix relating payload cartesian movements to strut length changes
|
||||
- \\(M\_s\\) is a diagonal 6x6 matrix containing the moving mass of each strut
|
||||
- \\(l\\) is the 6x1 vector of strut lengths
|
||||
- \\(B\\) and \\(K\\) are 6x6 diagonal matrices containing the damping and stiffness, respectively, of each strut
|
||||
- \\(l\_r\\) is the constant vector of relaxed strut lengths
|
||||
- \\(f\_m\\) is the vector of strut motor force
|
||||
- \\(J\_c\\) and \\(J\_b\\) are 6x6 Jacobian matrices capturing base motion
|
||||
- \\(\ddot{q}\_u\\) is a 6x1 vector of base acceleration along each strut
|
||||
- \\(\mathcal{F}\_r\\) is a vector of payload exogenous generalized forces
|
||||
- \\(\mathcal{C}\\) is a vector containing all the Coriolis and centripetal terms
|
||||
- \\(\mathcal{G}\\) is a vector containing all gravity terms
|
||||
|
||||
|
||||
#### Decoupling {#decoupling}
|
||||
|
||||
Two decoupling algorithms are proposed by combining static input-output transformations with hexapod geometric design.
|
||||
|
||||
Define a new input and a new output:
|
||||
|
||||
\begin{equation}
|
||||
u\_1 = J^T f\_m, \quad y = J^{-1} (l - l\_r)
|
||||
\end{equation}
|
||||
|
||||
Equation \ref{eq:hexapod\_eq\_motion} can be rewritten as:
|
||||
|
||||
\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_1}
|
||||
\begin{split}
|
||||
& \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 = \\\\
|
||||
&\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
|
||||
\end{split}
|
||||
\end{equation}
|
||||
|
||||
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="figure--fig:li01-decoupling-conf"></a>
|
||||
|
||||
{{< 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:
|
||||
|
||||
\begin{equation}
|
||||
u\_2 = J^{-1} f\_m, \quad y = J^{-1} (l - l\_r)
|
||||
\end{equation}
|
||||
|
||||
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 = \\\\
|
||||
&\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="figure--fig:li01-decoupling-conf-bis"></a>
|
||||
|
||||
{{< 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">
|
||||
|
||||
These decoupling algorithms have two constraints:
|
||||
|
||||
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.
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
## Simultaneous Vibration Isolation and Pointing Control {#simultaneous-vibration-isolation-and-pointing-control}
|
||||
|
||||
Many applications require simultaneous vibration isolation and precision pointing.
|
||||
|
||||
The basic idea to achieve such objective is to use:
|
||||
|
||||
- acceleration feedback to provide high-frequency vibration isolation
|
||||
- cartesian pointing feedback 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 are 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 7](#figure--fig:li01-vibration-isolation-control).
|
||||
|
||||
<a id="figure--fig:li01-vibration-isolation-control"></a>
|
||||
|
||||
{{< 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 7](#figure--fig:li01-vibration-isolation-control)
|
||||
|
||||
<a id="figure--fig:li01-vibration-isolation-control"></a>
|
||||
|
||||
{{< 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:
|
||||
|
||||
- first order lag-lead filter to provide adequate phase margin a the low frequency crossover
|
||||
- a second order lag-lead filter to increase the gain between crossovers and provide adequate phase margin at the high frequency crossover
|
||||
- a second order notch filter to cancel the mode at 150Hz
|
||||
- a second order low pass filter to provide steep roll-off and gain stabilize the plant at high frequency
|
||||
- a first order high pass filter to eliminate DC signals
|
||||
|
||||
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">
|
||||
|
||||
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.
|
||||
|
||||
</div>
|
||||
|
||||
<div class="note">
|
||||
|
||||
Severe phase delay exists in the actual transfer function.
|
||||
This is due to the limited sample frequency and sensor bandwidth limitation.
|
||||
|
||||
The zero at around 130Hz is non-minimum phase which limits the control bandwidth.
|
||||
The reason is not explained.
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
### Pointing Control Techniques {#pointing-control-techniques}
|
||||
|
||||
A block diagram of the pointing control system is shown in [Figure 8](#figure--fig:li01-pointing-control).
|
||||
|
||||
<a id="figure--fig:li01-pointing-control"></a>
|
||||
|
||||
{{< 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](#figure--fig:li01-transfer-function-angle) shows the measured transfer function of the \\(\theta\_x\\) axis.
|
||||
|
||||
<a id="figure--fig:li01-transfer-function-angle"></a>
|
||||
|
||||
{{< 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:
|
||||
|
||||
- a first order low pass filter to increase the low frequency loop gain and provide a slope of -20dB/decade for the magnitude curve at the crossover
|
||||
- two complex zeros with high \\(Q\\) to provide adequate phase margin at the crossover
|
||||
- a pole after the zeros to decrease the excess gain caused by these zeros
|
||||
- a second order notch filter to cancel the mode at 150Hz
|
||||
- a second order low pass filter to provide steep roll off and gain stabilize the plant at high frequency
|
||||
|
||||
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](#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="figure--fig:li01-feedforward-control"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="<span class='figure-number'>Figure 10: </span>Feedforward control" >}}
|
||||
|
||||
|
||||
### Simultaneous Control {#simultaneous-control}
|
||||
|
||||
The simultaneous vibration isolation and pointing control is approached in two ways:
|
||||
|
||||
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](#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="figure--fig:li01-parallel-control"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/li01_parallel_control.png" caption="<span class='figure-number'>Figure 11: </span>A parallel scheme" >}}
|
||||
|
||||
<div class="important">
|
||||
|
||||
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.
|
||||
|
||||
However, the interaction between loops may affect the transfer functions of the **first** closed loop, and thus affect its relative stability.
|
||||
|
||||
</div>
|
||||
|
||||
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 [Figure 12](#figure--fig:li01-closed-loop-pointing) and [Figure 13](#figure--fig:li01-closed-loop-vibration))
|
||||
|
||||
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="figure--fig:li01-closed-loop-pointing"></a>
|
||||
|
||||
{{< 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](#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="figure--fig:li01-closed-loop-vibration"></a>
|
||||
|
||||
{{< 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">
|
||||
|
||||
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.
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
### Experimental results {#experimental-results}
|
||||
|
||||
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="figure--fig:li01-test-bench"></a>
|
||||
|
||||
{{< 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](#figure--fig:li01-vibration-isolation-control-results).
|
||||
|
||||
<a id="figure--fig:li01-vibration-isolation-control-results"></a>
|
||||
|
||||
{{< 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](#figure--fig:li01-simultaneous-control-results) where the bandwidth of the isolation control is expanded to very low frequency.
|
||||
|
||||
<a id="figure--fig:li01-simultaneous-control-results"></a>
|
||||
|
||||
{{< 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">
|
||||
|
||||
A parallel control scheme is proposed in this chapters.
|
||||
This scheme is suitable for simultaneous vibration isolation and pointing control.
|
||||
Part of this scheme involves closing one loop first, then re-identifying and designing the new control before closed the other loop.
|
||||
|
||||
An investigation into the interaction between loops shows that the order of closing loops is not important.
|
||||
However, only two channels need to be re-designed or adjusted for the pointing loop if the isolation loop is closed first.
|
||||
Experiments show that this scheme takes advantage of the bandwidths of both pointing and vibration sensors, and provides vibration isolation and pointing controls over a broad band.
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
## Future research areas {#future-research-areas}
|
||||
|
||||
<div class="sum">
|
||||
|
||||
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
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
## 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>Li, X. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,166 @@
|
||||
+++
|
||||
title = "Dynamic error budgeting, a design approach"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
ref_author = "Monkhorst, W."
|
||||
ref_year = 2004
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Monkhorst 2004</a>)
|
||||
|
||||
Author(s)
|
||||
: Monkhorst, W.
|
||||
|
||||
Year
|
||||
: 2004
|
||||
|
||||
|
||||
## Introduction {#introduction}
|
||||
|
||||
The performance of a mechatronic system is generally defined by the error made, which is caused by the disturbances \\(d\\) that act on the system.
|
||||
In order to study how the disturbances \\(d\\) propagates to the error, frequency dependent models of the disturbances and subsystems must be used.
|
||||
Disturbances (which are stochastic) are modeled with their power spectral densities.
|
||||
The new design approach will be referred to as _Dynamic Error Budgeting_, where "dynamic" refers to the use of the frequency dependent models.
|
||||
|
||||
Challenge definition of this thesis:
|
||||
|
||||
> Develop a tool which enables the designer to account for stochastic disturbances during the design of a mechatronics system.
|
||||
|
||||
Develop tools should enable the designer to:
|
||||
|
||||
- Predict the final performance level of a system which is subject to stochastic disturbances
|
||||
- Gain insight in performance limiting factors of the system.
|
||||
This insight should enable the designer to point out critical system components/properties and to improve the performance of the system
|
||||
- Objectively compare the performance of different system designs
|
||||
|
||||
|
||||
## Dynamic Error Budgeting {#dynamic-error-budgeting}
|
||||
|
||||
|
||||
### Motivations {#motivations}
|
||||
|
||||
Main motivations are:
|
||||
|
||||
- **Cutting costs in the design phase**: if the error is not simulated during the design phase, the final performance level can only be found when a costly prototype is build and the performance can be measured physically. If the performance level is not met, the designer has to find out what component or disturbance causes the output to exceed the error budget and then redesign the system. If the error could be simulated beforehand however, changes can be made when the system is still in the design phase, cutting down the costs of the system.
|
||||
- **Speeding up the design process**: It can give a quick indication if a concept is feasible or not.
|
||||
Several concepts can be analyzed in a short period of time and the most promising concept can be chosen, speeding up the design process.
|
||||
- **Enhancing design insight**: If the performance specifications is not met, the designer wants to know which component or what system property is limiting the performance most.
|
||||
|
||||
|
||||
### DEB design process {#deb-design-process}
|
||||
|
||||
The DEB design process can be summarized as follows: choose a system concept and simulate the output error.
|
||||
If the total error is meets the performance specifications, the design is satisfying.
|
||||
If the error exceeds the specified budget, the designer has to change the system such that the specifications is met.
|
||||
|
||||
Step by step, the process is as follows:
|
||||
|
||||
- Design a concept system.
|
||||
- Model the concept system, such that the closed loop transfer functions can be determined.
|
||||
- Identify all significant disturbances.
|
||||
Model them with their _Power Spectral Density_
|
||||
- Define the performance outputs of the system and simulate the output error.
|
||||
Using the theory of _propagation_, the contribution of each disturbance to the output error can be analyzed and the critical disturbance can be pointed out.
|
||||
- Make changes to the system that are expected to improve the performance level, and simulate the output error again.
|
||||
Iterate until the error budget is meet.
|
||||
|
||||
|
||||
### Assumptions {#assumptions}
|
||||
|
||||
The assumptions when applying DEB are:
|
||||
|
||||
- The system can be accurately described with a **linear time invariant model**.
|
||||
This is usually the case as much effort is put in to make systems have a linear behavior and because feedback loops have a " linearizing" effect on the closed loop behavior.
|
||||
- The disturbances action on the system must be **stationary** (their statistical properties are not allowed to change over time).
|
||||
- The disturbances are **uncorrelated** with each other.
|
||||
This is more difficult to satisfy for MIMO systems and the designer must make sure that the separate disturbances all originate from separate independent sources.
|
||||
- The disturbance signals are modeled by their **Power Spectral Density**.
|
||||
This implies that only stochastic disturbances are allowed.
|
||||
For the deterministic part, other techniques can be used to determine their influence to the error.
|
||||
- The calculation method makes no assumption on the distribution of the distribution functions of the disturbances.
|
||||
In practice, many disturbances will have a normal like distribution.
|
||||
|
||||
|
||||
### \\(\mathcal{H}\_2\\) control, maximizing performance {#mathcal-h-2-control-maximizing-performance}
|
||||
|
||||
|
||||
#### The \\(\mathcal{H}\_2\\) norm and variance of the output {#the-mathcal-h-2-norm-and-variance-of-the-output}
|
||||
|
||||
The \\(\mathcal{H}\_2\\) norm is a norm defined on a system:
|
||||
\\[ \\|H\\|\_2^2 = \int\_{-\infty}^\infty |H(j2\pi f)|^2 df \\]
|
||||
|
||||
Stochastic interpretation of the \\(\mathcal{H}\_2\\) norm: the squared \\(\mathcal{H}\_2\\) norm can be interpreted as the output variance of a system with zero mean white noise input.
|
||||
|
||||
|
||||
#### The \\(\mathcal{H}\_2\\) control problem {#the-mathcal-h-2-control-problem}
|
||||
|
||||
Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_2\\) norm of the closed loop system \\(H\\):
|
||||
\\[ C\_{\mathcal{H}\_2} \in \arg \min\_C \\|H\\|\_2 \\]
|
||||
|
||||
|
||||
#### Using weighting filters to model disturbances {#using-weighting-filters-to-model-disturbances}
|
||||
|
||||
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](#figure--fig:monkhorst04-weighting-filter)).
|
||||
|
||||
<a id="figure--fig:monkhorst04-weighting-filter"></a>
|
||||
|
||||
{{< 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:
|
||||
\\[ |S\_w(f)| = V\_w(j 2 \pi f) V\_w^T(-j 2 \pi f) \\]
|
||||
|
||||
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](#figure--fig:monkhorst04-general-weighted-plant)).
|
||||
|
||||
<a id="figure--fig:monkhorst04-general-weighted-plant"></a>
|
||||
|
||||
{{< 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](#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="figure--fig:monkhorst04-closed-loop-H2"></a>
|
||||
|
||||
{{< 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.
|
||||
When varying \\(\alpha\\), one can plot the amount of control effort at one axis and the achieve performance on the other axis.
|
||||
The resulting points lie on the so-called **Pareto curve**.
|
||||
|
||||
|
||||
## Conclusions {#conclusions}
|
||||
|
||||
\\(\mathcal{H}\_2\\) control strategy is an extension of the DEB approach.
|
||||
It offers the designer the opportunity to optimize over the degree of freedom given by the controller, enabling the designer to predict the maximum achievable performance level of a system concept.
|
||||
Using this technique, the designer is able to objectively compare the performance potential of different system concepts.
|
||||
|
||||
The accuracy of the predicted performance by DEB with respect to the measured results can be improved by using higher order models of the disturbances.
|
||||
Increasing of order of the disturbance model might even allow modelling of harmonic disturbances by using inverse notches.
|
||||
To achieve the highest degree of prediction accuracy, it is recommended to use to actual measured disturbance spectra in the simulations.
|
||||
|
||||
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, W. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,26 @@
|
||||
+++
|
||||
title = "An exploration of active hard mount vibration isolation for precision equipment"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
ref_author = "van der Poel, G. W."
|
||||
ref_year = 2010
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Van der Poel 2010</a>)
|
||||
|
||||
Author(s)
|
||||
: van der Poel, G. W.
|
||||
|
||||
Year
|
||||
: 2010
|
||||
|
||||
|
||||
## 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>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
@@ -0,0 +1,27 @@
|
||||
+++
|
||||
title = "Modeling and robust adaptive tracking control of a planar precision positioning system"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
ref_author = "Treichel, K."
|
||||
ref_year = 2017
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Treichel 2017</a>)
|
||||
|
||||
Author(s)
|
||||
: Treichel, K.
|
||||
|
||||
Year
|
||||
: 2017
|
||||
|
||||
|
||||
## 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>Treichel, Kai. 2017. “Modeling and Robust Adaptive Tracking Control of a Planar Precision Positioning System.” Fakultät für Informatik und Automatisierung der Technischen Universität Ilmenau.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,443 @@
|
||||
+++
|
||||
title = "Robust mass damper design for bandwidth increase of motion stages"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Verbaan 2015</a>)
|
||||
|
||||
Author(s)
|
||||
: Verbaan, C.
|
||||
|
||||
Year
|
||||
: 2015
|
||||
|
||||
> This thesis addresses the challenge to increase the modal damping of the bandwidth limiting resonances of motions stages.
|
||||
> This modal damping increase is realized by adding passive elements, called robust tuned mass dampers, at specific stage locations.
|
||||
>
|
||||
> [...]
|
||||
>
|
||||
> The damper parameters that have to be determined are mass, stiffness, and damping.
|
||||
> The optimal parameters are obtained by executing optimization algorithm.
|
||||
>
|
||||
> The first motion stage design is optimized based on an open-loop criterion for modal damping increase between 1 and 4kHz.
|
||||
> Experimental validation shows that a suppression factor of over 24dB is obtained.
|
||||
|
||||
|
||||
## Robust Mass Damper and broad banded damping {#robust-mass-damper-and-broad-banded-damping}
|
||||
|
||||
> In high tech motion systems, the finite stiffness of mechanical components results in natural frequencies which limit the bandwidth of the control system.
|
||||
> This is usually counteracted by increasing the controller complexity by adding notch filters.
|
||||
> The height of the non-rigid body modes in the frequency response function and the amount of damping significantly affect the achievable bandwidth.
|
||||
> This chapter described a method to add damping to the flexible behavior of a motion stage, by using robust mass dampers which are mass-spring-damper systems with an **over-critical** damping value.
|
||||
> This high damping results in robust dynamic behavior with respect to stiffness and damping variations for both the motion stage and the damper mechanisms.
|
||||
> The main result is a significant increase in modal damping over a broad band of resonance frequencies.
|
||||
|
||||
|
||||
### Tuned mass damper {#tuned-mass-damper}
|
||||
|
||||
The effectiveness of the TMD is related to the mass ratio between \\(m\\) and \\(M\\).
|
||||
To obtain a substantial suppression factor in combination with a relatively small increase in mass, the mass ratio is usually determined to be approximately 5 to 10% of the main structural mass.
|
||||
The undamped natural frequency of the TMD has to be tuned close to the targeted natural frequency of the main structure.
|
||||
|
||||
A drawback of the TMD is the relatively **large sensitivity of the suppression factor for variations in stiffness and damping values**.
|
||||
This sensitivity also holds for natural frequency variations of the main structure.
|
||||
|
||||
<a id="figure--fig:verbaan15-tmd-principle"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_tmd_principle.png" caption="<span class='figure-number'>Figure 1: </span>TMD principle" >}}
|
||||
|
||||
|
||||
### Damper design and validation {#damper-design-and-validation}
|
||||
|
||||
This damper is designed and tested to prove that it is possible to create dampers with over-critical damping values and with natural frequencies that are high enough to be useful.
|
||||
The spring and damper are assumed to behave linearly.
|
||||
In addition, the vibration amplitudes of high-tech positioning tables are small, which allows for assuming linear system theory.
|
||||
These small vibration amplitudes lead to small damper strokes.
|
||||
Therefore **flexures** can be used to provide for the guidance of the moving mass.
|
||||
The dimensions of the flexures determine the spring stiffness and therefore the natural frequency of the TMD.
|
||||
An additional advantage of flexures is the lack of hysteresis, which **enables the damper to work even if the damper strokes are very small**.
|
||||
|
||||
The dampers are intended to act purely in z-direction.
|
||||
The natural frequency in this direction is determined at 1250Hz and the natural frequency in the other directions should be as high as possible.
|
||||
|
||||
<a id="figure--fig:verbaan15-tmd-modes"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_tmd_modes.png" caption="<span class='figure-number'>Figure 2: </span>Natural frequency of the TMD. First natural frequency at 1250Hz and the second at 8100Hz." >}}
|
||||
|
||||
The second challenge is to create a damping mechanism with a high damping coefficient in a relatively small volume.
|
||||
The damper is designed to be **passive**.
|
||||
This guarantees stability of the damper system itself and preserves from increasing complexity.
|
||||
As damping concept, a **viscous fuild damper** is chosen due to the following properties:
|
||||
|
||||
- the linear time independent behavior
|
||||
- the ability to create an extremely large damping constant in a small volume
|
||||
- separation of stiffness and damping
|
||||
- the supreme damping properties of fuilds with respect to other damping materials
|
||||
|
||||
The guild applied is Rocol Kilopoise 0868 and is chosen based on the extremely high viscosity of 220 Pas.
|
||||
|
||||
In order to measure the damping the measurement bench shown in [Figure 3](#figure--fig:verbaan15-tmd-mech-system) is used.
|
||||
The measured FRF are shown in [Figure 4](#figure--fig:verbaan15-obtained-damping-bench).
|
||||
The measurement clearly shows that the damper mechanism is over-critically damped.
|
||||
|
||||
<a id="figure--fig:verbaan15-tmd-mech-system"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_tmd_mech_system.png" caption="<span class='figure-number'>Figure 3: </span>Damper test setup to measure the damping characteristics" >}}
|
||||
|
||||
<a id="figure--fig:verbaan15-obtained-damping-bench"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_obtained_damping_bench.png" caption="<span class='figure-number'>Figure 4: </span>Obtained damping results" >}}
|
||||
|
||||
|
||||
## Linear viscoelastic characterisation of an ultra-high viscosity fluid {#linear-viscoelastic-characterisation-of-an-ultra-high-viscosity-fluid}
|
||||
|
||||
> This chapter presents the use of a state of the art damper for high precision motion stages as a sliding plate rheometer for measuring linear viscoelastic properties in the frequency range of 10Hz to 10kHz.
|
||||
> This design is flexure based to minimize parasitic nonlinear forces.
|
||||
> Design and the damping mechanism are elaborated and a model is presented that describes the dynamic behavior.
|
||||
|
||||
The damper shown in [Figure 5](#figure--fig:verbaan15-damper-parts) can be used as a sliding plate rheometer to measure the linear viscoelastic properties of ultra-high viscosity fluids in the frequency range 10Hz to 10kHz.
|
||||
|
||||
<a id="figure--fig:verbaan15-damper-parts"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_damper_parts.png" caption="<span class='figure-number'>Figure 5: </span>Damper parts" >}}
|
||||
|
||||
The full damper assembly consists of a mass, mounted on two springs and a damper in parallel configuration.
|
||||
The mass can make small strokes in the x-direction and is fixed in all other directions.
|
||||
The spring is a double leaf spring guide.
|
||||
The space between the lead springs is used to accommodate for the damping mechanism.
|
||||
|
||||
<a id="figure--fig:verbaan15-tmd-slot-fin-parts"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_tmd_slot_fin_parts.png" caption="<span class='figure-number'>Figure 6: </span>Exploded view of the damper parts" >}}
|
||||
|
||||
A high-viscosity fluid is applied to create a velocity dependent force.
|
||||
For this purpose, the sliding plate principle is used which induces a **shear flow**: the fluid is placed between two slot plates and a fin is positioned between these two plates ([Figure 7](#figure--fig:verbaan15single-double-fin)).
|
||||
A **flexible encapsulation** is used to hold the fluid between find and slot part.
|
||||
|
||||
To study different damping values with the same fluid, two damper designs with different geometries are used (see [Figure 7](#figure--fig:verbaan15single-double-fin)).
|
||||
|
||||
<a id="figure--fig:verbaan15single-double-fin"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15single_double_fin.png" caption="<span class='figure-number'>Figure 7: </span>Cross-sectional views of the two different damping mechanims. The single fin (left) and double fin (right)." >}}
|
||||
|
||||
To excite the damper mass, a voice coil is mounted to the hardware.
|
||||
The damper position is measured with a laser vibrometer.
|
||||
|
||||
A sliding plate damper for high frequencies introduces side effects:
|
||||
|
||||
1. geometry related effects
|
||||
2. frequency dependent effects
|
||||
|
||||
A first geometrical effect is due to the **finite length of the plates**.
|
||||
The ratio length/gap here is more than 100 which makes this effect negligible.
|
||||
A second geometrical effect is due to the difficulty to get the **plates parallel to each other**, especially with the normal forces acting on the moving fin, induced by the flow.
|
||||
This design counteracts this problem in two-ways: the damper part is **symmetric**, which means that the fin normal forces cancel each other.
|
||||
In addition, the double leaf spring mechanism has a **very high lateral stiffness**, which minimizes lateral displacements.
|
||||
A third geometrical effect is pumping of the fluid, which appears in the case of closed ends and introduces a flow opposite to the fin velocity, and therefore introduces a parasitic damping force.
|
||||
This problem is avoided by letting the gaps' ends open.
|
||||
The **fin is shorted than the slot** to maintain the same damping area over the damper stroke.
|
||||
|
||||
These effects all arise at low frequencies, at which the flow can be assumed homogeneous.
|
||||
The ratio between inertial and viscous effects determines up to which frequency the flow can be assumed homogeneous:
|
||||
|
||||
\begin{equation}
|
||||
t\_c = \frac{10 \rho h^2}{\eta}
|
||||
\end{equation}
|
||||
|
||||
in which \\(\rho\\) describes the fluid density in \\(kg/m^3\\), \\(\eta\\) the dynamic viscosity in \\(Pa s\\) and \\(h\\) the gap width in \\(m\\).
|
||||
Dimensions are provided in [Table 1](#table--tab:single-fin-parameters).
|
||||
This estimate results in a frequency above 100kHz.
|
||||
It shows that high fluid viscosities and small gap widths enable high frequencies without losing homogeneous flow conditions.
|
||||
|
||||
<a id="table--tab:single-fin-parameters"></a>
|
||||
<div class="table-caption">
|
||||
<span class="table-number"><a href="#table--tab:single-fin-parameters">Table 1</a>:</span>
|
||||
Parameters for the single fin design
|
||||
</div>
|
||||
|
||||
| Dimension | Value [mm] |
|
||||
|----------------|------------|
|
||||
| Length \\(l\\) | 16 |
|
||||
| Width \\(w\\) | 8.5 |
|
||||
| Gap \\(h\\) | 0.12 |
|
||||
|
||||
**Conclusion**:
|
||||
A design of a sliding plate damper that can be used to characterize fluid behavior of high viscosity fluids in the frequency range between 10Hz and 10kHz.
|
||||
The drawbacks of standard sliding plate devices are taken care off by the mechanical design.
|
||||
The flexure mechanism very precisely determines the position of the fin with respect to the slot part.
|
||||
A three mode Maxwell model can accurately describe the behavior.
|
||||
|
||||
|
||||
## Damping optimization of a complex motion stage {#damping-optimization-of-a-complex-motion-stage}
|
||||
|
||||
|
||||
### Stage and damper dynamic models {#stage-and-damper-dynamic-models}
|
||||
|
||||
This chapter presents the results of a robust mass damper implementation on a complex motion stage with realistic natural frequencies to increase the modal damping of flexible modes.
|
||||
A design approach is presented which results in parameter values for the dampers to improve the modal damping over a specified frequency range.
|
||||
|
||||
[Figure 8](#figure--fig:verbaan15-stage-undamped) shows a collocated FRF of the stage's corner.
|
||||
The goal is to increase the modal damping of modes 7, 9, 10/11 and 13.
|
||||
|
||||
<a id="figure--fig:verbaan15-stage-undamped"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_stage_undamped.png" caption="<span class='figure-number'>Figure 8: </span>FRF at the stage corner in the z-direction, undamped" >}}
|
||||
|
||||
The transfer function \\(T\_i(s)\\) is defined as the contribution of the a single mode \\(i\\) in an input/output transfer function:
|
||||
|
||||
\begin{equation}
|
||||
T\_i(s) = \frac{\phi\_i^{\text{act}} \phi\_i^{\text{sen}}}{s^2 + 2 \xi \omega\_i s + \omega\_i^2} = \frac{1}{m\_i s^2 + c\_i s + k\_i}
|
||||
\end{equation}
|
||||
|
||||
With \\(\phi\_i^{\text{act}}\\) and \\(\phi\_i^{\text{sen}}\\) the modal factors of the actuator and sensor.
|
||||
|
||||
From this equation, it appears that the modal mass of a mode in a certain transfer function equals:
|
||||
|
||||
\begin{equation}
|
||||
m\_i = \frac{1}{\phi\_i^{\text{act}} \phi\_i^{\text{sen}}}
|
||||
\end{equation}
|
||||
|
||||
This equation shows that a certain mode's modal mass depends on the locations of the actuator and sensor.
|
||||
Since a TMD can be seen as a local control loop, the actuator and sensor location are equal.
|
||||
This results in the following equation for the apparent modal mass for mode \\(i\\) at the TMD location:
|
||||
|
||||
\begin{equation}
|
||||
m\_i = \frac{}{(\phi\_i^{\text{TMD}})^2}
|
||||
\end{equation}
|
||||
|
||||
It is known from literature that the efficiency of a TMD depends on the **mass ratio** of the TMD and the mode that has to be damped.
|
||||
It follows that the efficiency of a TMD to damp a certain resonance depends on the position of the damper on the stage in a quadratic sense.
|
||||
The TMD has to be located at the maximum displacement of the mode(s) to be damped.
|
||||
|
||||
The damper configuration consists of an inertial mass \\(m\\), a transnational flexible guide designed as a double leaf spring mechanism with total stiffness \\(c\\) and a part that creates the damping force with damping constant \\(d\\) (model shown in [Figure 9](#figure--fig:verbaan15-maxwell-fluid-model)).
|
||||
The velocity dependent damper force is the result of two parameters:
|
||||
|
||||
- the fluid's mechanical properties
|
||||
- the damper geometry
|
||||
|
||||
The fluid model is presented in [Figure 10](#figure--fig:verbaan15-fluid-lve-model).
|
||||
This figure shows the viscous and elastic properties of the fluid as a function of the frequency.
|
||||
The damper principle is chosen to be a parallel plate damper based on the shear principle with the viscous fluid in between the two parallel plates.
|
||||
In case of a velocity difference between these plates, a velocity gradient is created in the fluid causing a specific force per unit of area, which, multiplied by the effective area submerged in the fluid, leads to a damping force.
|
||||
|
||||
The damping can be expressed with a geometrical damping factor (GDF) in meters:
|
||||
|
||||
\begin{equation}
|
||||
\text{GDF} = \frac{A}{h} = \frac{2 n l w}{h}
|
||||
\end{equation}
|
||||
|
||||
with \\(A\\) the total area of the damper fins, \\(n\\) is the number of fins, \\(l\\) is the fin length, \\(w\\) is the fin width and \\(h\\) is the effective gap width in which the fluid is applied.
|
||||
|
||||
This GDF, combined with the fluid properties in Pas and Pa, lead to a spring stiffness in N/m and a damping constant in N/(m/s).
|
||||
|
||||
In general, larger suppression factors can be obtained with larger TMD masses.
|
||||
In the example, the modal mass is 3.5kg and the damper mass is 110g (useful inertial mass of 65g).
|
||||
|
||||
<a id="figure--fig:verbaan15-maxwell-fluid-model"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_maxwell_fluid_model.png" caption="<span class='figure-number'>Figure 9: </span>Damper model with multi-mode Maxwell fluid model included" >}}
|
||||
|
||||
<a id="figure--fig:verbaan15-fluid-lve-model"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_fluid_lve_model.png" caption="<span class='figure-number'>Figure 10: </span>Storage and loss modulus of the 3 Maxwell mode LVE fluid model" >}}
|
||||
|
||||
|
||||
### TMD and RMD optimisation {#tmd-and-rmd-optimisation}
|
||||
|
||||
An algorithm is used to optimize the damping and is used in two cases:
|
||||
|
||||
- a small banded optimisation which includes a single resonance.
|
||||
This results in a **tuned mass damper** optimal design
|
||||
- a broad banded optimization which includes a range of resonances.
|
||||
This results in a **robust mass damper** optimal design
|
||||
|
||||
The algorithm is first used to calculate the optimal parameters to suppress a **single** resonance frequency.
|
||||
The result is shown in [Figure 11](#figure--fig:verbaan15-tmd-optimization) and shows **Tuned Mass Damper** behavior.
|
||||
|
||||
For this single frequency, stiffness and damping values can be calculated by hand.
|
||||
|
||||
<a id="figure--fig:verbaan15-tmd-optimization"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_tmd_optimization.png" caption="<span class='figure-number'>Figure 11: </span>Result of the optimization procedure. The cost function is specified between 1kHz and 2kHz. This implies that the first mode is suppressed by the damper." >}}
|
||||
|
||||
To obtain broad banded damping, the cost function is redefined between 1 and 4kHz.
|
||||
[Figure 12](#figure--fig:verbaan15-broadbanded-damping-results) presents the resulting bode diagram.
|
||||
|
||||
<a id="figure--fig:verbaan15-broadbanded-damping-results"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_broadbanded_damping_results.png" caption="<span class='figure-number'>Figure 12: </span>Result of the optimization procedure with the cost function specified between 1 and 4kHz. The result is a range of resonances that are suppressed by the dampers." >}}
|
||||
|
||||
Results of optimizations for increasing damper mass, in the range from 10 to 250g per damper are shown in [Figure 13](#figure--fig:verbaan15-results-fct-mass).
|
||||
|
||||
<a id="figure--fig:verbaan15-results-fct-mass"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_results_fct_mass.png" caption="<span class='figure-number'>Figure 13: </span>Optimal damper parameters as a function of the damper mass. The upper graph shows the suppression factor in dB, the second graph shows the natural frequency of the damper in Hz and the lower graph shows the geometrical damping factor in m." >}}
|
||||
|
||||
|
||||
### Damper Design and Validation {#damper-design-and-validation}
|
||||
|
||||
A damper mechanism is design which contains the following properties:
|
||||
|
||||
- a moving mass \\(m\_d = 65\\,g\\)
|
||||
- a mounting mass \\(m\_m = 45\\,g\\)
|
||||
- a natural frequency \\(\omega\_0 = 1270\\,Hz\\)
|
||||
- other natural frequencies as high as possible
|
||||
- a geometrical damping factor of 14.3m
|
||||
- an encapsulation to contain the fluid
|
||||
|
||||
[Figure 14](#figure--fig:verbaan15-RMD-mechanical-parts) shows an exploded view of the RMD design.
|
||||
The mechanism part is monolithically designed and consists of:
|
||||
|
||||
1. a mounting side
|
||||
2. leaf spring pair
|
||||
3. the damper side
|
||||
|
||||
The fluid is surrounded by a flexible encapsulation, which prevents it from running out.
|
||||
|
||||
<a id="figure--fig:verbaan15-RMD-mechanical-parts"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_RMD_mechanical_parts.png" caption="<span class='figure-number'>Figure 14: </span>Exploded view of the robust mass damper design with different parts indicated" >}}
|
||||
|
||||
<a id="figure--fig:verbaan15-RMD-design-modes"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_RMD_design_modes.png" caption="<span class='figure-number'>Figure 15: </span>Four lowest natural frequencies and corresponding mode shapes of the RMD while mounted to a stage corner" >}}
|
||||
|
||||
<a id="figure--fig:verbaan15-tmd-side-front-views"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_tmd_side_front_views.png" caption="<span class='figure-number'>Figure 16: </span>A side view and a front view of the fin and slot parts" >}}
|
||||
|
||||
| Dimension | Value | Unit |
|
||||
|-------------|-------|------|
|
||||
| Length fin | 17 | mm |
|
||||
| Height fins | 4 | mm |
|
||||
| Gap width | 50 | um |
|
||||
| GDF | 14 | m |
|
||||
|
||||
<a id="figure--fig:verbaan15-damped-undamped-frf"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/verbaan15_damped_undamped_frf.png" caption="<span class='figure-number'>Figure 17: </span>Measured undamped and damped FRF" >}}
|
||||
|
||||
|
||||
### Conclusion {#conclusion}
|
||||
|
||||
This chapter shows an approach to add damping to a range of resonances of a motion stage by adding robust mass dampers.
|
||||
Analysis is performed to calculate the damping increase beforehand, and experiments are conducted to validate the behavior of both the damper and the stage with dampers added.
|
||||
|
||||
The broadbanded solution shows a resonance suppression of at least 24.3dB between 1kHz and 4kHz.
|
||||
The overall mass increase is less than 2%.
|
||||
|
||||
The robustness, as one of the most important properties of the RMD, is proven: the suppression factor is well predictable despite different errors and estimations:
|
||||
|
||||
- stage model errors (the natural frequencies resulting from the FEM are an overestimation of the real frequencies)
|
||||
- fluid model errors
|
||||
- a simplified 1DoF model is applied as a damper model
|
||||
- production tolerances for the dampers
|
||||
|
||||
Tuned mass dampers are well known in literature.
|
||||
The equations are proven to calculate the optimal suppression factor, natural frequency and damping ratio.
|
||||
In these equations, the damper behavior is assumed to be purely viscous.
|
||||
We shows that larger suppression factors are possible by using visco-elastic fluids as damping medium.
|
||||
Although this effect is relatively small for single resonance suppression, it is larger for broadbanded suppression.
|
||||
The damper benefits from the frequency dependent stiffness of the fluid.
|
||||
|
||||
|
||||
## Conclusion {#conclusion}
|
||||
|
||||
In this thesis, the opportunities to increase the performance of high-tech motion systems are investigated by increasing the modal damping of non-rigid body resonances by introducing robust mass dampers (RMD), which provides damping over a broad frequency band.
|
||||
A combination of techniques is applied to improve the performance of motion stages in a systematical way, including mechanical design, dynamic modeling, material characterization and optimization procedures.
|
||||
Theoretical improvement factors are calculated and experimental validation is provided to support the theory.
|
||||
The main conclusions of the previous chapters are summarized and listed by subject.
|
||||
|
||||
|
||||
### Robust Mass Dampers {#robust-mass-dampers}
|
||||
|
||||
Robust mass dampers have proven to be able to provide **broad banded damping**.
|
||||
In addition, **robust behavior** is proven in case of parameter variations of both the motion stage and/or the parameters of the RMDs.
|
||||
This property explicitly underlines the suitability of RMDs to improve the behavior of motion stages that are operated in closed-loop conditions: parameter sensitive designs will result in a performance decrease and might eventually lead to destabilization of the closed-loop system.
|
||||
|
||||
The RMDs in this thesis are **passive and stand-alone devices**.
|
||||
Advantages of these types of devices are
|
||||
|
||||
1. the stabilizing behavior due to the principle of energy dissipation.
|
||||
2. The stand-alone property implies that no connection between any structural part and the motion stage is created, and no signal or power cables are needed which prevents the introduction of disturbance forces.
|
||||
3. The damper design by application of LVE behavior enables larger suppression factors than purely viscous fluid behavior.
|
||||
|
||||
At least in case of motion stages with a relatively large length-height ratio it appears that an overall mass contribution by the RMDs of 2 % of the stage mass is sufficient to improve the stage performance significantly.
|
||||
This is proven by experiments.
|
||||
|
||||
|
||||
### Influence on stage dynamics {#influence-on-stage-dynamics}
|
||||
|
||||
The relatively high modal damping of the RMDs prevents for visible effects in the rigid body mass line of the frequency response functions.
|
||||
In other directions, the natural frequencies of the RMDs can be designed above 6 kHz for dampers of 65 g.
|
||||
This is usually high enough to prevent for detrimental properties in the direction of motion
|
||||
|
||||
|
||||
### RMD locations {#rmd-locations}
|
||||
|
||||
The **location of an RMD on the mechanical stage is a significant factor in the performance increase factor**.
|
||||
The effectiveness of the RMD to improve the modal damping factor scales quadratically with the stage displacement at the damper location.
|
||||
Therefore, if the limiting natural frequencies are determined, **the locations with large displacements for the corresponding mode shapes have to be found**.
|
||||
In case of more than one resonance this might be a weighted criterion for the different modes.
|
||||
This approach is applicable for both open- loop and closed-loop performance criteria.
|
||||
|
||||
|
||||
### The fluid model {#the-fluid-model}
|
||||
|
||||
A **linear visco-elastic fluid model** is derived from measurements and applied in the optimization formulations.
|
||||
The results show that the model quality is good enough to predict the system’s damped behavior quite accurately.
|
||||
|
||||
|
||||
### Open-loop modal damping improvement {#open-loop-modal-damping-improvement}
|
||||
|
||||
The principle of **broad banded damping** is well applicable for practical cases: the intended damping range was 1-4 kHz.
|
||||
In addition, a damping increase is visible up to 6 kHz.
|
||||
This frequency range abundantly covers the range in which performance limiting flexibilities usually arise in motion stage designs.
|
||||
An optimization criterion in terms of resonance suppression is applied and works well: this criterion inherently only optimizes the visible resonances at the actuator and sensor location.
|
||||
The choice which resonances should be suppressed, therefore, is specified in the cost function by the frequency response function.
|
||||
Robustness of the solution and broad banded effect in practical cases is proven by the experimental validation.
|
||||
The calculated suppression factor compares well to the measured ones.
|
||||
The suppression factor amounts approximately 24 dB between 1 and 4 kHz, which indicates a modal damping increase factor of 16.
|
||||
|
||||
|
||||
### Closed-loop performance increase {#closed-loop-performance-increase}
|
||||
|
||||
The principle of closed-loop performance increase is formulated in an optimization formulation which accurately estimates the bandwidth improvement factor.
|
||||
The optimization formulation is non-convex, however, a hybrid optimization procedure is able to solve this specific problem in a limited amount of time.
|
||||
In addition to the improvements in the intended control loops, other control loops often benefit from the damping increase.
|
||||
|
||||
|
||||
### Advantages in analysis {#advantages-in-analysis}
|
||||
|
||||
A more general observation regarding the analyses method is presented.
|
||||
The approach with separate RMDs is an efficient approach which contains two large advantages: It enables to continue with the current applied mechanical design approach for high natural frequencies and increase the modal damping afterwards.
|
||||
This enables to still apply the materials with high specific stiffness and low damping.
|
||||
|
||||
In the analysis phase the advantages are enormous:
|
||||
|
||||
1. Undamped natural frequencies and mode shapes can be calculated and are valid for the low damped stage’s mechanical design.
|
||||
These algorithms are very efficient and large models can be solved.
|
||||
2. State space models can be created which contain the complexity of the FEM model and can be validated by calculating the responses by means of superposition of the undamped modes in the FEM software.
|
||||
3. RMDs can be added at specific locations.
|
||||
This results in non-proportional damping and complex mode shapes, which are correctly calculated by the state space model.
|
||||
4. This enables to apply optimization algorithms and compare different RMDs very quickly.
|
||||
|
||||
The complete model including dampers can be solved in FEM, however, this approach contains serious drawbacks:
|
||||
|
||||
1. The mode shapes change from real normal modes to complex modes due to the damping at specific locations.
|
||||
This implies that complex solvers have to be applied.
|
||||
These solvers are much more time consuming than the solvers for real natural modes.
|
||||
2. The frequency response functions can be calculated using fully harmonic solvers.
|
||||
This results in the most accurate solution because the model is not truncated as in case of a state space model with a limited number of modes.
|
||||
However, this algorithm solves the complete model for every frequency point in the frequency response function and, therefore, this approach is extremely time-consuming.
|
||||
3. Therefore, in this approach the ability to implement different RMD parameters and execute optimization algorithms practically vanishes due to the limitations listed above.
|
||||
|
||||
<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>Verbaan, C.A.M. 2015. “Robust mass damper design for bandwidth increase of motion stages.” Mechanical Engineering; Technische Universiteit Eindhoven.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,27 @@
|
||||
+++
|
||||
title = "Dynamic modeling, experimental identification, and active vibration control design of a smart parallel manipulator."
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = true
|
||||
ref_author = "Wang, X."
|
||||
ref_year = 2007
|
||||
+++
|
||||
|
||||
Tags
|
||||
:
|
||||
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Wang 2007</a>)
|
||||
|
||||
Author(s)
|
||||
: Wang, X.
|
||||
|
||||
Year
|
||||
: 2007
|
||||
|
||||
|
||||
## 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>Wang, Xiaoyun. 2007. “Dynamic Modeling, Experimental Identification, and Active Vibration Control Design of a Smart Parallel Manipulator.” University of Toronto.</div>
|
||||
</div>
|
||||
@@ -0,0 +1,78 @@
|
||||
+++
|
||||
title = "Element and system design for active and passive vibration isolation"
|
||||
author = ["Dehaeze Thomas"]
|
||||
draft = false
|
||||
ref_author = "Zuo, L."
|
||||
ref_year = 2004
|
||||
+++
|
||||
|
||||
Tags
|
||||
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Eddy Current Damping]({{< relref "eddy_current_damping.md" >}})
|
||||
|
||||
Reference
|
||||
: (<a href="#citeproc_bib_item_1">Zuo 2004</a>)
|
||||
|
||||
Author(s)
|
||||
: Zuo, L.
|
||||
|
||||
Year
|
||||
: 2004
|
||||
|
||||
> Vibration isolation systems can have various system architectures.
|
||||
> When we configure an active isolation system, we can use compliant actuators (such as voice coils) or stiff actuators (such as PZT stacks).
|
||||
> We also need to consider how to **combine the active actuation with passive elements**: we can place the actuator in parallel or in series with the passive elements.
|
||||
> Most of the isolation systems fall into the category of soft active mounts, in which a compliant actuator is placed in parallel with a spring.
|
||||
> A second category is **hard active mounts**, in which the **payload mass is directly mounted to a stiff actuator**.
|
||||
> Soft active mounts generally have advantages for better passive performance; hard active mounts are favored for payload disturbance rejection, but combination with passive stages is required due to the lack of isolation performance out of the control bandwidth.
|
||||
> Beard, von Flotow and Schubert proposed another type of hard mount, wherein **a stiff PZT actuator is placed in series with a spring** stiffer than the top passive stage.
|
||||
> 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.
|
||||
|
||||
|
||||
## Passive Vibration Isolation {#passive-vibration-isolation}
|
||||
|
||||
|
||||
### The Role of damping and its practical constructions {#the-role-of-damping-and-its-practical-constructions}
|
||||
|
||||
|
||||
#### Viscous damping {#viscous-damping}
|
||||
|
||||
|
||||
#### Eddy-current damper {#eddy-current-damper}
|
||||
|
||||
<a id="figure--fig:zuo04-eddy-current-magnets"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zuo04_eddy_current_magnets.png" caption="<span class='figure-number'>Figure 1: </span>(left) Magnetic field and conductor plates assemblies, (right) magnet arrays" >}}
|
||||
|
||||
<a id="figure--fig:zuo04-eddy-current-setup"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zuo04_eddy_current_setup.png" caption="<span class='figure-number'>Figure 2: </span>Single DoF system damped by eddy current damper" >}}
|
||||
|
||||
|
||||
## Elements and configurations for active vibration systems {#elements-and-configurations-for-active-vibration-systems}
|
||||
|
||||
|
||||
### System architectures {#system-architectures}
|
||||
|
||||
<a id="figure--fig:zuo04-piezo-spring-series"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="<span class='figure-number'>Figure 3: </span>PZT actuator and spring in series" >}}
|
||||
|
||||
<a id="figure--fig:zuo04-voice-coil-spring-parallel"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="<span class='figure-number'>Figure 4: </span>Voice coil actuator and spring in parallel" >}}
|
||||
|
||||
<a id="figure--fig:zuo04-piezo-plant"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="<span class='figure-number'>Figure 5: </span>Transmission from PZT 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 6: </span>Transmission from voice coil voltage to geophone output" >}}
|
||||
|
||||
|
||||
## 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>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