Update Content - 2024-12-17

content/phdthesis/li01_simul_fault_vibrat_isolat_point.md

@@ -24,13 +24,13 @@ Year
 
 ### Flexure Jointed Hexapods {#flexure-jointed-hexapods}
 
-A general flexible jointed hexapod is shown in [1](#figure--fig:li01-flexure-hexapod-model).
+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 [2](#figure--fig:li01-quet-dirty-box).
+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>
 
@@ -43,7 +43,7 @@ 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 [3](#figure--fig:li01-stewart-platform)) are:
+The University of Wyoming hexapods (example in [Figure 3](#figure--fig:li01-stewart-platform)) are:
 
 -   Cubic (mutually orthogonal)
 -   Flexure Jointed
@@ -87,7 +87,7 @@ J = \begin{bmatrix}
 \end{bmatrix}
 \end{equation}
 
-where (see [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\\).
+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.
 
@@ -140,7 +140,7 @@ Equation <eq:hexapod_eq_motion> can be rewritten as:
 \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 ([4](#figure--fig:li01-decoupling-conf)).
+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>
 
@@ -152,7 +152,7 @@ Alternatively, a new set of inputs and outputs can be defined:
 u\_2 = J^{-1} f\_m, \quad y = J^{-1} (l - l\_r)
 \end{equation}
 
-And another decoupled plant is found ([5](#figure--fig:li01-decoupling-conf-bis)):
+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}
@@ -200,13 +200,13 @@ The control bandwidth is divided as follows:
 
 ### Vibration Isolation {#vibration-isolation}
 
-The system is decoupled into six independent SISO subsystems using the architecture shown in [6](#figure--fig:li01-vibration-isolation-control).
+The system is decoupled into six independent SISO subsystems using the architecture shown in [Figure 6](#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 [6](#figure--fig:li01-vibration-isolation-control)
+One of the subsystem plant transfer function is shown in [Figure 6](#figure--fig:li01-vibration-isolation-control)
 
 <a id="figure--fig:li01-vibration-isolation-control"></a>
 
@@ -243,7 +243,7 @@ The reason is not explained.
 
 ### Pointing Control Techniques {#pointing-control-techniques}
 
-A block diagram of the pointing control system is shown in [8](#figure--fig:li01-pointing-control).
+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>
 
@@ -252,7 +252,7 @@ A block diagram of the pointing control system is shown in [8](#figure--fig:li01
 The plant is decoupled into two independent SISO subsystems.
 The decoupling matrix consists of the columns of \\(J\\) corresponding to the pointing DoFs.
 
-[9](#figure--fig:li01-transfer-function-angle) shows the measured transfer function of the \\(\theta\_x\\) axis.
+[Figure 9](#figure--fig:li01-transfer-function-angle) shows the measured transfer function of the \\(\theta\_x\\) axis.
 
 <a id="figure--fig:li01-transfer-function-angle"></a>
 
@@ -268,7 +268,7 @@ A typical compensator consists of the following elements:
 
 The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**.
 
-A feedforward control is added as shown in [10](#figure--fig:li01-feedforward-control).
+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.
 
@@ -284,7 +284,7 @@ The simultaneous vibration isolation and pointing control is approached in two w
 1.  **Closing the vibration isolation loop first**: Design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators.
 2.  **Closing the pointing loop first**: Reverse order.
 
-[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.
+[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>
 
@@ -302,16 +302,16 @@ However, the interaction between loops may affect the transfer functions of the
 The dynamic interaction effect:
 
 -   Only happens in the unity bandwidth of the loop transmission of the first closed loop.
--   Affect the closed loop transmission of the loop first closed (see [12](#figure--fig:li01-closed-loop-pointing) and [13](#figure--fig:li01-closed-loop-vibration))
+-   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 [12](#figure--fig:li01-closed-loop-pointing), the peak resonance of the pointing loop increase after the isolation loop is closed.
+As shown in [Figure 12](#figure--fig:li01-closed-loop-pointing), the peak resonance of the pointing loop increase after the isolation loop is closed.
 The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness.
 
 <a id="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 ([13](#figure--fig:li01-closed-loop-vibration)).
+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>
@@ -328,7 +328,7 @@ Thus, it is recommended to design and implement the isolation control system fir
 
 ### Experimental results {#experimental-results}
 
-Two hexapods are stacked ([14](#figure--fig:li01-test-bench)):
+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
@@ -338,7 +338,7 @@ Two hexapods are stacked ([14](#figure--fig:li01-test-bench)):
 {{< 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 [15](#figure--fig:li01-vibration-isolation-control-results).
+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>
 
@@ -349,7 +349,7 @@ 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 [16](#figure--fig:li01-simultaneous-control-results) where the bandwidth of the isolation control is expanded to very low frequency.
+The results of simultaneous control is shown in [Figure 16](#figure--fig:li01-simultaneous-control-results) where the bandwidth of the isolation control is expanded to very low frequency.
 
 <a id="figure--fig:li01-simultaneous-control-results"></a>