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@@ -8,7 +8,7 @@ Tags
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: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}}), [Vibration Isolation]({{<relref "vibration_isolation.md#" >}}), [Cubic Architecture]({{<relref "cubic_architecture.md#" >}}), [Flexible Joints]({{<relref "flexible_joints.md#" >}}), [Multivariable Control]({{<relref "multivariable_control.md#" >}})
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Reference
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: ([Li 2001](#org7277b25))
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: ([Li 2001](#orgc147fe0))
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Author(s)
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: Li, X.
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@@ -22,15 +22,15 @@ Year
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### Flexure Jointed Hexapods {#flexure-jointed-hexapods}
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A general flexible jointed hexapod is shown in Figure [1](#org858f898).
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A general flexible jointed hexapod is shown in Figure [1](#orge84e431).
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<a id="org858f898"></a>
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<a id="orge84e431"></a>
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{{< figure src="/ox-hugo/li01_flexure_hexapod_model.png" caption="Figure 1: A flexure jointed hexapod. {P} is a cartesian coordinate frame located at, and rigidly attached to the payload's center of mass. {B} is the frame attached to the base, and {U} is a universal inertial frame of reference" >}}
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Flexure jointed hexapods have been developed to meet two needs illustrated in Figure [2](#orgda07839).
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Flexure jointed hexapods have been developed to meet two needs illustrated in Figure [2](#orga3eb26a).
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<a id="orgda07839"></a>
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<a id="orga3eb26a"></a>
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{{< figure src="/ox-hugo/li01_quet_dirty_box.png" caption="Figure 2: (left) Vibration machinery must be isolated from a precision bus. (right) A precision paylaod must be manipulated in the presence of base vibrations and/or exogenous forces." >}}
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@@ -41,12 +41,12 @@ On the other hand, the flexures add some complexity to the hexapod dynamics.
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Although the flexure joints do eliminate friction and backlash, they add spring dynamics and severely limit the workspace.
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Moreover, base and/or payload vibrations become significant contributors to the motion.
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The University of Wyoming hexapods (example in Figure [3](#orgccc775c)) are:
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The University of Wyoming hexapods (example in Figure [3](#org051e360)) are:
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- Cubic (mutually orthogonal)
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- Flexure Jointed
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<a id="orgccc775c"></a>
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<a id="org051e360"></a>
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{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 3: Flexure jointed Stewart platform used for analysis and control" >}}
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@@ -85,7 +85,7 @@ J = \begin{bmatrix}
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\end{bmatrix}
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\end{equation}
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where (see Figure [1](#org858f898)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
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where (see Figure [1](#orge84e431)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
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To make the dynamic model as simple as possible, the origin of {P} is located at the payload's center of mass.
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Thus all \\({}^Pp\_i\\) are found with respect to the center of mass.
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@@ -94,43 +94,96 @@ Thus all \\({}^Pp\_i\\) are found with respect to the center of mass.
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The dynamics of a flexure jointed hexapod can be written in joint space:
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\begin{equation}
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\begin{equation} \label{eq:hexapod\_eq\_motion}
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\begin{split}
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& \left( J^{-T} {}^B\_PR^P M\_x {}^B\_PR^T J^{-1} + M\_s \right) \ddot{l} + B \dot{l} + K (l - l\_r) = \\\\\\
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&\quad f\_m - \left( M\_s + J^{-T} {}^B\_PR^P M\_x {}^U\_PR^T J\_c J\_b^{-1} \right) \ddot{q}\_u + J^{-T} {}^U\_BR\_T(\mathcal{F}\_e + \mathcal{G} + \mathcal{C})
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& \left( J^{-T} \cdot {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T \cdot J^{-1} + M\_s \right) \ddot{l} + B \dot{l} + K (l - l\_r) = \\\\\\
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&\quad f\_m - \left( M\_s + J^{-T} \cdot {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + J^{-T} \cdot {}^U\_BR^T(\mathcal{F}\_e + \mathcal{G} + \mathcal{C})
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\end{split}
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\end{equation}
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where:
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### Test {#test}
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- \\({}^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
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- \\({}^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).
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Similarly, \\({}^B\_PR\\) is the rotation matrix from the payload frame to the base frame, and \\({}^U\_PR = {}^U\_BR {}^B\_PR\\)
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- \\(J\\) is the 6x6 Jacobian matrix relating payload cartesian movements to strut length changes
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- \\(M\_s\\) is a diagonal 6x6 matrix containing the moving mass of each strut
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- \\(l\\) is the 6x1 vector of strut lengths
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- \\(B\\) and \\(K\\) are 6x6 diagonal matrices containing the damping and stiffness, respectively, of each strut
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- \\(l\_r\\) is the constant vector of relaxed strut lengths
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- \\(f\_m\\) is the vector of strut motor force
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- \\(J\_c\\) and \\(J\_b\\) are 6x6 Jacobian matrices capturing base motion
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- \\(\ddot{q}\_u\\) is a 6x1 vector of base acceleration along each strut
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- \\(\mathcal{F}\_r\\) is a vector of payload exogenous generalized forces
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- \\(\mathcal{C}\\) is a vector containing all the Coriolis and centripetal terms
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- \\(\mathcal{G}\\) is a vector containing all gravity terms
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**Jacobian Analysis**:
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\\[ \delta \mathcal{L} = J \delta \mathcal{X} \\]
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The origin of \\(\\{P\\}\\) is taken as the center of mass of the payload.
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**Decoupling**:
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If we refine the (force) inputs and (displacement) outputs as shown in Figure [4](#org7721136) or in Figure [5](#orgdc42940), we obtain a decoupled plant provided that:
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#### Decoupling {#decoupling}
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Two decoupling algorithms are proposed by combining static input-output transformations with hexapod geometric design.
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Define a new input and a new output:
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\begin{equation}
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u\_1 = J^T f\_m, \quad y = J^{-1} (l - l\_r)
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\end{equation}
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Equation \eqref{eq:hexapod_eq_motion} can be rewritten as:
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\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_1}
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\begin{split}
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& \left( {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR^T + J^T \cdot M\_s \cdot J \right) \cdot \ddot{y} + J^T \cdot B J \dot{y} + J^T \cdot K \cdot J y = \\\\\\
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&\quad u\_1 - \left( J^T \cdot M\_s + {}^B\_PR \cdot {}^PM\_x \cdot {}^U\_PR^T \cdot J\_c \cdot J\_b^{-1} \right) \ddot{q}\_u + {}^U\_BR^T\mathcal{F}\_e
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\end{split}
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\end{equation}
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If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled (Figure [4](#org8deb4db)).
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<a id="org8deb4db"></a>
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{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 4: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
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Alternatively, a new set of inputs and outputs can be defined:
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\begin{equation}
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u\_2 = J^{-1} f\_m, \quad y = J^{-1} (l - l\_r)
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\end{equation}
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And another decoupled plant is found (Figure [5](#org7a23a21)):
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\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_2}
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\begin{split}
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& \left( J^{-1} \cdot J^{-T} \cdot {}^BM\_x + M\_s \right) \cdot \ddot{y} + B \dot{y} + K y = \\\\\\
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&\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
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\end{split}
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\end{equation}
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<a id="org7a23a21"></a>
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{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 5: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
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<div class="important">
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<div></div>
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These decoupling algorithms have two constraints:
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1. the payload mass/inertia matrix must be diagonal (the CoM is coincident with the origin of frame \\(\\{P\\}\\))
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2. the geometry of the hexapod and the attachment of the payload to the hexapod must be carefully chosen
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> 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.
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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.
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<a id="org7721136"></a>
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{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 4: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
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<a id="orgdc42940"></a>
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{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 5: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
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</div>
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## Simultaneous Vibration Isolation and Pointing Control {#simultaneous-vibration-isolation-and-pointing-control}
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Basic idea:
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Many applications require simultaneous vibration isolation and precision pointing.
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- acceleration feedback is used to provide high-frequency vibration isolation
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- cartesian pointing feedback can be used to provide low-frequency pointing
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The basic idea to achieve such objective is to use:
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- acceleration feedback to provide high-frequency vibration isolation
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- cartesian pointing feedback to provide low-frequency pointing
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The compensation is divided in frequency because:
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@@ -139,115 +192,158 @@ The compensation is divided in frequency because:
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The control bandwidth is divided as follows:
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- low-frequency disturbances as attenuated and tracking is accomplished by feedback from low bandwidth pointing sensors
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- low-frequency disturbances are attenuated and tracking is accomplished by feedback from low bandwidth pointing sensors
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- mid-frequency disturbances are attenuated by feedback from band-pass sensors like accelerometer or load cells
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- high-frequency disturbances are attenuated by passive isolation techniques
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### Vibration Isolation {#vibration-isolation}
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The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [6](#org0dd19dc).
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The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [6](#org0dc1d11).
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<a id="org0dd19dc"></a>
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<a id="org0dc1d11"></a>
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{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 6: Figure caption" >}}
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{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 6: Vibration isolation control strategy" >}}
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One of the subsystem plant transfer function is shown in Figure [6](#org0dd19dc)
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One of the subsystem plant transfer function is shown in Figure [6](#org0dc1d11)
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<a id="org6a21353"></a>
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<a id="orgcd4b06b"></a>
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{{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="Figure 7: Plant transfer function of one of the SISO subsystem for Vibration Control" >}}
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Each compensator is designed using simple loop-shaping techniques.
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A typical compensator consists of the following elements:
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The unity control bandwidth of the isolation loop is designed to be from **5Hz to 50Hz**.
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- first order lag-lead filter to provide adequate phase margin a the low frequency crossover
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- a second order lag-lead filter to increase the gain between crossovers and provide adequate phase margin at the high frequency crossover
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- a second order notch filter to cancel the mode at 150Hz
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- a second order low pass filter to provide steep roll-off and gain stabilize the plant at high frequency
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- a first order high pass filter to eliminate DC signals
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> Despite a reasonably good match between the modeled and the measured transfer functions, the model based decoupling algorithm does not produce the expected decoupling.
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> Only about 20 dB separation is achieve between the diagonal and off-diagonal responses.
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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.
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<div class="important">
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<div></div>
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Despite a reasonably good match between the modeled and the measured transfer functions, the model based decoupling algorithm does not produce the expected decoupling.
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Only about 20 dB separation is achieve between the diagonal and off-diagonal responses.
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</div>
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<div class="note">
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<div></div>
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Severe phase delay exists in the actual transfer function.
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This is due to the limited sample frequency and sensor bandwidth limitation.
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The zero at around 130Hz is non-minimum phase which limits the control bandwidth.
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The reason is not explained.
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</div>
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### Pointing Control {#pointing-control}
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### Pointing Control Techniques {#pointing-control-techniques}
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A block diagram of the pointing control system is shown in Figure [8](#orgb338488).
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A block diagram of the pointing control system is shown in Figure [8](#orgec13571).
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<a id="orgb338488"></a>
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<a id="orgec13571"></a>
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{{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 8: Figure caption" >}}
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The plant is decoupled into two independent SISO subsystems.
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The compensators are design with inverse-dynamics methods.
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The decoupling matrix consists of the columns of \\(J\\) corresponding to the pointing DoFs.
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Figure [9](#org23ec3f5) shows the measured transfer function of the \\(\theta\_x\\) axis.
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<a id="org23ec3f5"></a>
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{{< figure src="/ox-hugo/li01_transfer_function_angle.png" caption="Figure 9: Experimentally measured plant transfer function of \\(\theta\_x/\theta\_{x\_d}\\)" >}}
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A typical compensator consists of the following elements:
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- 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
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- two complex zeros with high \\(Q\\) to provide adequate phase margin at the crossover
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- a pole after the zeros to decrease the excess gain caused by these zeros
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- a second order notch filter to cancel the mode at 150Hz
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- a second order low pass filter to provide steep roll off and gain stabilize the plant at high frequency
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The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**.
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A feedforward control is added as shown in Figure [9](#orgb372596).
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A feedforward control is added as shown in Figure [10](#org68adfa5).
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\\(C\_f\\) is the feedforward compensator which is a 2x2 diagonal matrix.
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Ideally, the feedforward compensator is an invert of the plant dynamics.
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<a id="orgb372596"></a>
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<a id="org68adfa5"></a>
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{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 9: Feedforward control" >}}
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{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 10: Feedforward control" >}}
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### Simultaneous Control {#simultaneous-control}
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The simultaneous vibration isolation and pointing control is approached in two ways:
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1. design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators
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2. the reverse design order
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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.
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2. **Closing the pointing loop first**: Reverse order.
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Figure [10](#orgbafcf4b) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
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Figure [11](#orgedfc92b) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
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<a id="orgbafcf4b"></a>
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<a id="orgedfc92b"></a>
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{{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 10: A parallel scheme" >}}
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{{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 11: A parallel scheme" >}}
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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.
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<div class="important">
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<div></div>
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The effect of the isolation loop on the pointing loop is large around the natural frequency of the plant as shown in Figure [11](#org2a20ab8).
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The transfer function matrix for the pointing loop after the vibration isolation is closed is still decoupled.
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The same happens when closing the pointing loop first and looking at the transfer function matrix of the vibration isolation.
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<a id="org2a20ab8"></a>
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However, the interaction between loops may affect the transfer functions of the **first** closed loop, and thus affect its relative stability.
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{{< figure src="/ox-hugo/li01_effect_isolation_loop_closed.png" caption="Figure 11: \\(\theta\_x/\theta\_{x\_d}\\) transfer function with the isolation loop closed (simulation)" >}}
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The effect of pointing control on the isolation plant has not much effect.
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> The interaction between loops may affect the transfer functions of the **first** closed loop, and thus affect its relative stability.
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</div>
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The dynamic interaction effect:
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- only happens in the unity bandwidth of the loop transmission of the first closed loop.
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- affect the closed loop transmission of the loop first closed (see Figures [12](#orgc137ea3) and [13](#orgc06274a))
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- Only happens in the unity bandwidth of the loop transmission of the first closed loop.
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- Affect the closed loop transmission of the loop first closed (see Figures [12](#orgfc5ad76) and [13](#org8dcf497))
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As shown in Figure [12](#orgc137ea3), the peak resonance of the pointing loop increase after the isolation loop is closed.
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As shown in Figure [12](#orgfc5ad76), the peak resonance of the pointing loop increase after the isolation loop is closed.
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The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness.
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<a id="orgc137ea3"></a>
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<a id="orgfc5ad76"></a>
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{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="Figure 12: Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}}
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The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [13](#orgc06274a)).
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The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [13](#org8dcf497)).
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The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
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<a id="orgc06274a"></a>
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<a id="org8dcf497"></a>
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{{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="Figure 13: Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}}
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> The isolation loop adds a second resonance peak at its high-frequency crossover in the pointing closed-loop transfer function, which may cause instability.
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> Thus, it is recommended to design and implement the isolation control system first, and then identify the pointing plant with the isolation loop closed.
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<div class="important">
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<div></div>
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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.
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Thus, it is recommended to design and implement the isolation control system first, and then identify the pointing plant with the isolation loop closed.
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</div>
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### Experimental results {#experimental-results}
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Two hexapods are stacked (Figure [14](#org2a11277)):
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Two hexapods are stacked (Figure [14](#org66cdd5c)):
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- the bottom hexapod is used to generate disturbances matching candidate applications
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- the top hexapod provide simultaneous vibration isolation and pointing control
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<a id="org2a11277"></a>
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<a id="org66cdd5c"></a>
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{{< figure src="/ox-hugo/li01_test_bench.png" caption="Figure 14: Stacked Hexapods" >}}
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Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [15](#org5933a45).
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First, the vibration isolation and pointing controls were implemented separately.
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Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [15](#org3b66ca1).
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<a id="org5933a45"></a>
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<a id="org3b66ca1"></a>
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{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 15: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}}
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@@ -256,15 +352,34 @@ The simultaneous control is of dual use:
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- it provide simultaneous pointing and isolation control
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- 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
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The results of simultaneous control is shown in Figure [16](#org996a848) where the bandwidth of the isolation control is expanded to very low frequency.
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The results of simultaneous control is shown in Figure [16](#orgb25318f) where the bandwidth of the isolation control is expanded to very low frequency.
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<a id="org996a848"></a>
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<a id="orgb25318f"></a>
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{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 16: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
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### Summary and Conclusion {#summary-and-conclusion}
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<div class="sum">
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<div></div>
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A parallel control scheme is proposed in this chapters.
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This scheme is suitable for simultaneous vibration isolation and pointing control.
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Part of this scheme involves closing one loop first, then re-identifying and designing the new control before closed the other loop.
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An investigation into the interaction between loops shows that the order of closing loops is not important.
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However, only two channels need to be re-designed or adjusted for the pointing loop if the isolation loop is closed first.
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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.
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</div>
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## Future research areas {#future-research-areas}
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<div class="sum">
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<div></div>
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Proposed future research areas include:
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- **Include base dynamics in the control**:
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@@ -286,8 +401,10 @@ Proposed future research areas include:
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- **LVDT** to provide differential position of the hexapod payload with respect to the base
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- **Geophones** to provide payload and base velocity information
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</div>
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## Bibliography {#bibliography}
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<a id="org7277b25"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.
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<a id="orgc147fe0"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.
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