Review - Section 5

paper/paper.org

@@ -386,8 +386,7 @@ Three regions can be observed:
 * Integral Force Feedback with Parallel Springs
 <<sec:iff_kp>>
 ** Stiffness in Parallel with the Force Sensor                       :ignore:
-As was explained in section ref:sec:iff_pure_int, the instability when using decentralized IFF for rotating platforms is due to Gyroscopic effects and, more precisely, due to the negative stiffness induced by centrifugal forces.
-In this section additional springs in parallel with the force sensors are added to counteract this negative stiffness.
+In this section additional springs in parallel with the force sensors are added to counteract the negative stiffness induced by the rotation.
 Such springs are schematically shown in Figure ref:fig:system_parallel_springs where $k_a$ is the stiffness of the actuator and $k_p$ the stiffness in parallel with the actuator and force sensor.
 
 Amplified piezoelectric stack actuators can also be used for such purpose where a part of the piezoelectric stack is used as an actuator while the rest is used as a force sensor cite:souleille18_concep_activ_mount_space_applic.
@@ -421,27 +420,22 @@ The forces $\begin{bmatrix}f_u, f_v\end{bmatrix}$ measured by the two force sens
   \begin{bmatrix} d_u \\ d_v \end{bmatrix}
 \end{equation}
 
-In order to keep the overall stiffness $k = k_a + k_p$ constant, a scalar parameter $\alpha$ ($0 \le \alpha < 1$) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor
+In order to keep the overall stiffness $k = k_a + k_p$ constant, thus not modifying the open-loop poles as $k_p$ is changed, a scalar parameter $\alpha$ ($0 \le \alpha < 1$) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor
 \begin{equation}
   k_p = \alpha k, \quad k_a = (1 - \alpha) k
 \end{equation}
 
 The equations of motion are derived and transformed in the Laplace domain
-#+name: eq:Gk_mimo_tf
-\begin{equation}
-\begin{bmatrix} f_u \\ f_v \end{bmatrix} =
+\begin{align}
+\begin{bmatrix} f_u \\ f_v \end{bmatrix} &=
 \bm{G}_k
-\begin{bmatrix} F_u \\ F_v \end{bmatrix}
-\end{equation}
-with $\bm{G}_k$ a $2 \times 2$ transfer function matrix
-#+name: eq:Gk
-\begin{equation}
-\bm{G}_k =
+\begin{bmatrix} F_u \\ F_v \end{bmatrix} \label{eq:Gk_mimo_tf} \\
+\bm{G}_k &=
 \begin{bmatrix}
   \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} & \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} \\
   \frac{\left( 2 \xi \frac{s}{\omega_0} + 1 - \alpha \right) \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2} & \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \right) \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right) + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}{\left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \right)^2}
-\end{bmatrix}
-\end{equation}
+\end{bmatrix} \label{eq:Gk}
+\end{align}
 
 Comparing $\bm{G}_k$ eqref:eq:Gk with $\bm{G}_f$ eqref:eq:Gf shows that while the poles of the system are kept the same, the zeros of the diagonal terms have changed.
 The two real zeros $z_r$ eqref:eq:iff_zero_real that were inducing non-minimum phase behavior are transformed into complex conjugate zeros if the following condition hold
@@ -451,7 +445,7 @@ The two real zeros $z_r$ eqref:eq:iff_zero_real that were inducing non-minimum p
 \end{equation}
 
 Thus, if the added parallel stiffness $k_p$ is higher than the negative stiffness induced by centrifugal forces $m \Omega^2$, the direct dynamics from actuator to force sensor will show minimum phase behavior.
-This is confirmed by the Bode plot in Figure ref:fig:plant_iff_kp.
+This is confirmed by the Bode plot of the direct dynamics in Figure ref:fig:plant_iff_kp.
 
 Figure ref:fig:root_locus_iff_kp shows Root Loci plots for $k_p = 0$, $k_p < m \Omega^2$ and $k_p > m \Omega^2$ when $K_F$ is a pure integrator eqref:eq:Kf_pure_int.
 It is shown that if the added stiffness is higher than the maximum negative stiffness, the poles of the closed-loop system stay in the (stable) right half-plane, and hence the unconditional stability of IFF is recovered.
@@ -482,8 +476,7 @@ To study this effect, Root Locus plots for several parallel stiffnesses $k_p > m
 The frequencies of the transmission zeros of the system are increasing with the parallel stiffness $k_p$ and the associated attainable damping is reduced.
 Therefore, even though the parallel stiffness $k_p$ should be larger than $m \Omega^2$ for stability reasons, it should not be taken too high as this would limit the attainable bandwidth.
 
-For any $k_p > m \Omega^2$, the control gain $g$ can be tuned such that the maximum simultaneous damping $\xi_\text{opt}$ is added to the resonances of the system.
-# An example is shown in Figure ref:fig:root_locus_opt_gain_iff_kp for $k_p = 5 m \Omega^2$ where the damping $\xi_{\text{opt}} \approx 0.83$ is obtained for a control gain $g_\text{opt} \approx 2 \omega_0$.
+This is confirmed in Figure ref:fig:mod_iff_damping_kp where the attainable closed-loop damping ratio $\xi_{\text{cl}}$ and the associated control gain $g_\text{opt}$ are computed as a function of $\alpha$.
 
 #+attr_latex: :options [t]{0.48\linewidth}
 #+begin_minipage
@@ -501,12 +494,6 @@ For any $k_p > m \Omega^2$, the control gain $g$ can be tuned such that the maxi
 [[file:figs/mod_iff_damping_kp.pdf]]
 #+end_minipage
 
-# #+name: fig:root_locus_iff_kps_opt
-# #+caption: Root Locus for IFF when parallel stiffness $k_p$ is added, $\Omega = 0.1 \omega_0$
-# #+attr_latex: :environment subfigure :width 0.45\linewidth :align c
-# | file:figs/root_locus_iff_kps.pdf                                          | file:figs/mod_iff_damping_kp.pdf                                                                   |
-# | <<fig:root_locus_iff_kps>> Comparison of three parallel stiffnesses $k_p$ | <<fig:root_locus_opt_gain_iff_kp>> $k_p = 5 m \Omega^2$, optimal damping $\xi_\text{opt}$ is shown |
-
 * Comparison and Discussion
 <<sec:comparison>>
 ** Introduction                                                      :ignore: