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title = "Vibrations and dynamic isotropy in hexapods-analytical studies"
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author = ["Thomas Dehaeze"]
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draft = true
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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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: ([Afzali-Far 2016](#orga93b30a))
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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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<div></div>
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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="org64466c7"></a>
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{{< figure src="/ox-hugo/afzali-far16_isotropic_hexapod_example.png" caption="Figure 1: Architecture of the obtained dynamically isotropic hexapod" >}}
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<div class="definition">
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<div></div>
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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](#orgfab85fb) shows a schematic of proposed generalized hexapod.
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<a id="orgfab85fb"></a>
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{{< figure src="/ox-hugo/afzali-far16_proposed_generalized_hexapod.png" caption="Figure 2: Parametrization of the proposed generalized hexapod" >}}
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## Conclusions {#conclusions}
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<summary>
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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](#orgfab85fb)) which is dynamically isotropic for a wide range of inertia properties.
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</summary>
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## Bibliography {#bibliography}
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<a id="orga93b30a"></a>Afzali-Far, Behrouz. 2016. “Vibrations and Dynamic Isotropy in Hexapods-Analytical Studies.” Lund University.
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@@ -0,0 +1,23 @@
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+++
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title = "Active damping of vibrations in high-precision motion systems"
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author = ["Thomas Dehaeze"]
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draft = false
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+++
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Tags
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: [Active Damping]({{<relref "active_damping.md#" >}})
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Reference
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: ([Babakhani 2012](#org0b93bb2))
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Author(s)
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: Babakhani, B.
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Year
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: 2012
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## Bibliography {#bibliography}
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<a id="org0b93bb2"></a>Babakhani, Bayan. 2012. “Active Damping of Vibrations in High-Precision Motion Systems.” University of Twente. <https://doi.org/10.3990/1.9789036534642>.
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@@ -5,10 +5,10 @@ draft = false
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+++
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Tags
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: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
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: [Dynamic Error Budgeting]({{<relref "dynamic_error_budgeting.md#" >}})
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Reference
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: ([Monkhorst 2004](#org7671be3))
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: ([Monkhorst 2004](#orgb303aca))
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Author(s)
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: Monkhorst, W.
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@@ -74,10 +74,10 @@ The assumptions when applying DEB are:
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In practice, many disturbances will have a normal like distribution.
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### \\(\mathcal{H}\_2\\) control, maximizing performance {#mathcal-h-2--control-maximizing-performance}
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### \\(\mathcal{H}\_2\\) control, maximizing performance {#mathcal-h-2-control-maximizing-performance}
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#### The \\(\mathcal{H}\_2\\) norm and variance of the output {#the--mathcal-h-2--norm-and-variance-of-the-output}
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#### The \\(\mathcal{H}\_2\\) norm and variance of the output {#the-mathcal-h-2-norm-and-variance-of-the-output}
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The \\(\mathcal{H}\_2\\) norm is a norm defined on a system:
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\\[ \\|H\\|\_2^2 = \int\_{-\infty}^\infty |H(j2\pi f)|^2 df \\]
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@@ -85,7 +85,7 @@ The \\(\mathcal{H}\_2\\) norm is a norm defined on a system:
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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.
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#### The \\(\mathcal{H}\_2\\) control problem {#the--mathcal-h-2--control-problem}
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#### The \\(\mathcal{H}\_2\\) control problem {#the-mathcal-h-2-control-problem}
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Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_2\\) norm of the closed loop system \\(H\\):
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\\[ C\_{\mathcal{H}\_2} \in \arg \min\_C \\|H\\|\_2 \\]
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@@ -95,9 +95,9 @@ Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_
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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.
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This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#org16f42a7)).
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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](#orgdc82b09)).
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<a id="org16f42a7"></a>
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<a id="orgdc82b09"></a>
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{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="Figure 1: The use of a weighting filter \\(V\_w(f)\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
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@@ -108,23 +108,23 @@ The PSD \\(S\_w(f)\\) of the weighted signal is:
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Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
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However, this can be avoided if the modelling of the disturbances is directly done in terms of weighting filters.
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Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#orgc49109b)).
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Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#org624c0f1)).
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<a id="orgc49109b"></a>
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<a id="org624c0f1"></a>
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{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="Figure 2: The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}}
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#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
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In this research, the outputs of the closed loop system (Figure [3](#org8b8bb94)) are:
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In this research, the outputs of the closed loop system (Figure [3](#org1993951)) are:
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- the performance (error) signal \\(e\\)
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- the controller output \\(u\\)
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In this way, the designer can analyze how much control effort is used to achieve the performance level at the performance output.
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<a id="org8b8bb94"></a>
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<a id="org1993951"></a>
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{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="Figure 3: The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}}
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@@ -151,4 +151,4 @@ Drawbacks however are, that no robustness guarantees can be given and that the o
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## Bibliography {#bibliography}
|
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|
||||
<a id="org7671be3"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
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<a id="orgb303aca"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
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@@ -5,10 +5,10 @@ draft = false
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+++
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Tags
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: [Finite Element Model]({{< relref "finite_element_model" >}})
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: [Finite Element Model]({{<relref "finite_element_model.md#" >}})
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Reference
|
||||
: ([Rankers 1998](#org98ff031))
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: ([Rankers 1998](#org2d6d98d))
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Author(s)
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: Rankers, A. M.
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@@ -22,20 +22,20 @@ Year
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Despite the fact, that mechanical vibrations in a servo device can be very complex and often involve the motion of many components of the system, there are three fundamental mechanisms that are often observed.
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These there basic dynamic phenomena can be indicated by:
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- _Actuator flexibility_: the mechanical system does not behave as one rigid body, due to flexibility between the location at which the servo force is applied and the actual point that needs to be positioned
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- _Guiding system flexibility_: the device usually rely on the guiding system to suppress motion in an undesired direction
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- _Limited mass and stiffness of the stationary machine part_: the reaction force that comes with the driving force will introduce a motion of the "stationary" part of the mechanical system
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- **Actuator flexibility**: the mechanical system does not behave as one rigid body, due to flexibility between the location at which the servo force is applied and the actual point that needs to be positioned
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- **Guiding system flexibility**: the device usually rely on the guiding system to suppress motion in an undesired direction
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- **Limited mass and stiffness of the stationary machine part**: the reaction force that comes with the driving force will introduce a motion of the "stationary" part of the mechanical system
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Whereas the first two phenomena mainly affect the stability of the control loop, the last phenomena manifests itself more often as a dynamic positional error in the set-point response.
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A tool that can be very useful in understanding the nature of more complex resonance phenomena and the underlying motion of the mechanical system, is "Modal Analysis".
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A tool that can be very useful in understanding the nature of more complex resonance phenomena and the underlying motion of the mechanical system, is **Modal Analysis**.
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Translating the mathematics of one single decoupled "modal" equation into a graphical representation, which includes all relevant data such as (effective) modal mass and stiffness plus the motion of each physical DoF, facilitates a better understanding of the modal concept.
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It enables a very intuitive link between the modal and the physical domain, and thus leads to a more creative use of "modal analysis" without the complications of the mathematical formalism.
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Dynamic phenomena of the mechanics in a servo positioning device can lead to stability problems of the control loop.
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Therefore it is important to investigate the frequency response (\\(x/F\\)), which characterizes the dynamics of the mechanical system, and especially the influence of mechanical resonances on it.
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Once the behavior of one individual mode is fully understood it is not so difficult to construct this frequency response and the interaction between the rigid-body motion of the device, and the dynamics of one additional mode.
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This leads to four interaction patterns:
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This leads to **four interaction patterns**:
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- -2 slope / zero / pole / -2 slope
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- -2 slope / pole / zero / -2 slope
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@@ -43,14 +43,13 @@ This leads to four interaction patterns:
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- -2 slope / pole / -2 slope (non-minimum phase and rarely occurring)
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It is not possible to judge the potential destabilizing effect of each of the typical characteristics without considering the frequency of the resonance in relation to the envisaged bandwidth of the control loop.
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The phase plot of a typical open loop frequency response of a PID controlled positioning device without mechanical resonances can be divided into three frequency ranges:
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The phase plot of a typical open loop frequency response of a PID controlled positioning device without mechanical resonances can be divided into three frequency ranges (supposing the plant model is just a mass line):
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- at low frequencies, the phase lies below -180 deg due to integrator action of the controller
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- at medium frequency (centered by the bandwidth frequency), the phase lies above -180 deg due to the differential action of the controller, which is necessary in order to achieve a stable position control-loop
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- at high frequencies, the phase eventually drops again below -180 deg due to additional low-pass filtering
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The potential destabilizing effect of each of the three typical characteristics can be judged in relation to the frequency range.
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Whether instability occurs depends very strongly on the resonance amplitude and damping of the additional mode.
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The potential **destabilizing effect** of each of the three typical characteristics can be judged in relation to the frequency range:
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- A -2 slope / zero / pole / -2 slope characteristics leads to a phase lead and is therefore potentially destabilizing in the low-frequency and high frequency regions.
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In the medium frequency region it adds an extra phase leads to the already existing margin, which does not harm the stability.
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@@ -58,15 +57,16 @@ Whether instability occurs depends very strongly on the resonance amplitude and
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It is potentially destabilizing in the medium-frequency range and is harmless in the low and high frequency ranges.
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- The -2 slope / poles / -4 slope behavior always has a devastating effect on the stability of the loop if located in the low or medium frequency range.
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Whether instability occurs depends very strongly on the resonance amplitude and damping of the additional mode.
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On the basis of these considerations, it is possible to give design guidelines for servo positioning devices.
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The subject of machine dynamics and its interaction with the control system plays a dominant role in fast and accurate positioning devices, so it is vital to consider these issues during the entire design process.
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Modeling and simulation can be adequate tools for that purpose; however, two conditions are crucial to the success:
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**Modeling and simulation** can be adequate tools for that purpose; however, two conditions are crucial to the success:
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|
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- usefulness of results
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- speed
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The analysis process has usually a top-down structure.
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The analysis process has usually a **top-down structure**.
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Starting with very elementary simulation models to support the selection of the proper concept, these models should become more refined, just like the product or machine under development.
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In various project throughout the past years, a three-step modeling approach has evolved, in which the following phases can be distinguished:
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@@ -163,13 +163,13 @@ The basic questions that are addressed in this thesis are:
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### Basic Control Aspects {#basic-control-aspects}
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A block diagram representation of a typical servo-system is shown in Figure [1](#orgfda2012).
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A block diagram representation of a typical servo-system is shown in Figure [1](#orgf3f4585).
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The main task of the system is achieve a desired positional relation between two or more components of the system.
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Therefore, a sensor measures the position which is then compared to the desired value, and the resulting error is used to generate correcting forces.
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||||
In most systems, the "actual output" (e.g. position of end-effector) cannot be measured directly, and the feedback will therefore be based on a "measured output" (e.g. encoder signal at the motor).
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It is important to realize that these two outputs can differ, first due to resilience in the mechanical system, and second because of geometrical imperfections in the mechanical transmission between motor and end-effector.
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<a id="orgfda2012"></a>
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<a id="orgf3f4585"></a>
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{{< figure src="/ox-hugo/rankers98_basic_el_mech_servo.png" caption="Figure 1: Basic elements of mechanical servo system" >}}
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@@ -180,10 +180,10 @@ The correction force \\(F\\) is defined by:
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F = k\_p \epsilon + k\_d \dot{\epsilon} + k\_i \int \epsilon dt
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\end{equation}
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It is illustrative to see that basically the proportional and derivative part of such a position control loop is very similar to a mechanical spring and damper that connect two points (Figure [2](#org2d694a7)).
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It is illustrative to see that basically the proportional and derivative part of such a position control loop is very similar to a mechanical spring and damper that connect two points (Figure [2](#org7066514)).
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If \\(c\\) and \\(d\\) represent the constant mechanical stiffness and damping between points \\(A\\) and \\(B\\), and a reference position profile \\(h(t)\\) is applied at \\(A\\), then an opposing force \\(F\\) is generated as soon as the position \\(x\\) and speed \\(\dot{x}\\) of point \\(B\\) does not correspond to \\(h(t)\\) and \\(\dot{h}(t)\\).
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<a id="org2d694a7"></a>
|
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<a id="org7066514"></a>
|
||||
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||||
{{< figure src="/ox-hugo/rankers98_basic_elastic_struct.png" caption="Figure 2: Basic Elastic Structure" >}}
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||||
@@ -199,9 +199,9 @@ These properties are very essential since they introduce the issue of **servo st
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An important aspect of a feedback controller is the fact that control forces can only result from an error signal.
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||||
Thus any desired set-point profile first leads to a position error before the corresponding driving forces are generated.
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||||
Most modern servo-systems have not only a feedback section, but also a feedforward section, as indicated in Figure [3](#org0c46a44).
|
||||
Most modern servo-systems have not only a feedback section, but also a feedforward section, as indicated in Figure [3](#orgd3dd201).
|
||||
|
||||
<a id="org0c46a44"></a>
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||||
<a id="orgd3dd201"></a>
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||||
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||||
{{< figure src="/ox-hugo/rankers98_feedforward_example.png" caption="Figure 3: Mechanical servo system with feedback and feedforward control" >}}
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||||
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||||
@@ -246,9 +246,9 @@ Basically, machine dynamics can have two deterioration effects in mechanical ser
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||||
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||||
#### Actuator Flexibility {#actuator-flexibility}
|
||||
|
||||
The basic characteristics of what is called "actuator flexibility" is the fact that in the frequency range of interest (usually \\(0-10\times \text{bandwidth}\\)) the driven system no longer behaves as one rigid body (Figure [4](#org765d5f4)) due to compliance between the motor and the load.
|
||||
The basic characteristics of what is called "actuator flexibility" is the fact that in the frequency range of interest (usually \\(0-10\times \text{bandwidth}\\)) the driven system no longer behaves as one rigid body (Figure [4](#orgac87759)) due to compliance between the motor and the load.
|
||||
|
||||
<a id="org765d5f4"></a>
|
||||
<a id="orgac87759"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_actuator_flexibility.png" caption="Figure 4: Actuator Flexibility" >}}
|
||||
|
||||
@@ -258,9 +258,9 @@ The basic characteristics of what is called "actuator flexibility" is the fact t
|
||||
The second category of dynamic phenomena results from the limited stiffness of the guiding system in combination with the fact the the device is driven in such a way that it has to rely on the guiding system to suppress motion in an undesired direction (in case of a linear direct drive system this occurs if the driving force is not applied at the center of gravity).
|
||||
|
||||
In general, a rigid actuator possesses six degrees of freedom, five of which need to be suppressed by the guiding system in order to leave one mobile degree of freedom.
|
||||
In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#org387cdc7)).
|
||||
In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#org7ada0d8)).
|
||||
|
||||
<a id="org387cdc7"></a>
|
||||
<a id="org7ada0d8"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_guiding_flexibility_planar.png" caption="Figure 5: Planar actuator with guiding system flexibility" >}}
|
||||
|
||||
@@ -280,14 +280,14 @@ The last category of dynamic phenomena results from the limited mass and stiffne
|
||||
In contrast to many textbooks on mechanics and machine dynamics, it is good practice always to look at the combination of driving force on the moving part, and reaction force on the stationary part, of a positioning device.
|
||||
When doing so, one has to consider what the effect of the reaction force on the systems performance will be.
|
||||
In the discussion of the previous two dynamic phenomena, the stationary part of the machine was assumed to be infinitely stiff and heavy, and therefore the effect of the reaction force was negligible.
|
||||
However, in general the stationary part is neither infinitely heavy, nor is it connected to its environment with infinite stiffness, so the stationary part will exhibit a resonance that is excited by the reaction forces (Figure [6](#orgb3f73d2)).
|
||||
However, in general the stationary part is neither infinitely heavy, nor is it connected to its environment with infinite stiffness, so the stationary part will exhibit a resonance that is excited by the reaction forces (Figure [6](#org60fe278)).
|
||||
|
||||
<a id="orgb3f73d2"></a>
|
||||
<a id="org60fe278"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_limited_m_k_stationary_machine_part.png" caption="Figure 6: Limited Mass and Stiffness of Stationary Machine Part" >}}
|
||||
|
||||
|
||||
## [Modal Decomposition]({{< relref "modal_decomposition" >}}) {#modal-decomposition--modal-decomposition-dot-md}
|
||||
## [Modal Decomposition]({{<relref "modal_decomposition.md#" >}}) {#modal-decomposition--modal-decomposition-dot-md}
|
||||
|
||||
|
||||
### Mathematics of Modal Decomposition {#mathematics-of-modal-decomposition}
|
||||
@@ -451,9 +451,9 @@ The overall transfer function can be found by summation of the individual modal
|
||||
|
||||
### Basic Characteristics of Mechanical FRF {#basic-characteristics-of-mechanical-frf}
|
||||
|
||||
Consider the position control loop of Figure [7](#orgeee8a5d).
|
||||
Consider the position control loop of Figure [7](#org96447ac).
|
||||
|
||||
<a id="orgeee8a5d"></a>
|
||||
<a id="org96447ac"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_mechanical_servo_system.png" caption="Figure 7: Mechanical position servo-system" >}}
|
||||
|
||||
@@ -463,7 +463,7 @@ In the ideal situation the mechanical system behaves as one rigid body with mass
|
||||
\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{m s^2}
|
||||
\end{equation}
|
||||
|
||||
<a id="org2ffb1b1"></a>
|
||||
<a id="orgda68028"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_ideal_bode_nyquist.png" caption="Figure 8: FRF of an ideal system with no resonances" >}}
|
||||
|
||||
@@ -487,11 +487,11 @@ which simplifies equation \eqref{eq:effect_one_mode} to:
|
||||
\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{\alpha}{m s^2 + m \omega\_i^2}
|
||||
\end{equation}
|
||||
|
||||
<a id="orgc9ecb0c"></a>
|
||||
<a id="orgdfb8041"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_frf_effect_alpha.png" caption="Figure 9: Contribution of rigid-body motion and modal dynamics to the amplitude and phase of FRF for various values of \\(\alpha\\)" >}}
|
||||
|
||||
<a id="org5180ee3"></a>
|
||||
<a id="org080f036"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/rankers98_final_frf_alpha.png" caption="Figure 10: Bode diagram of final FRF (\\(x\_{\text{servo}}/F\_{\text{servo}}\\)) for six values of \\(\alpha\\)" >}}
|
||||
|
||||
@@ -551,6 +551,7 @@ It has static solution capacity, and the frequency of the highest fixed-interfac
|
||||
Through the enormous performance drive in mechatronics systems, much has been learned in the past years about the influence of machine dynamics in servo positioning-devices.
|
||||
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<a id="org98ff031"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.
|
||||
<a id="org2d6d98d"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.
|
||||
|
||||
Reference in New Issue
Block a user