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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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: [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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