Last Review
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@ -52,10 +52,10 @@
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#+latex: \abstract{
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This paper investigates the use of Integral Force Feedback (IFF) for the active damping of rotating mechanical systems.
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Guaranteed stability, typical benefit of IFF, is lost as soon as the system is rotating due to gyroscopic effects.
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To overcome this issue, two modifications of the classical IFF control are proposed.
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To overcome this issue, two modifications of the classical IFF control scheme are proposed.
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The first consists of slightly modifying the control law while the second consists of adding springs in parallel with the force sensors.
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Conditions for stability and optimal parameters are derived.
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The results reveal that, despite their different implementations, both modified IFF control have almost identical damping authority on suspension modes.
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The results reveal that, despite their different implementations, both modified IFF control scheme have almost identical damping authority on suspension modes.
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#+latex: }
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* Introduction
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@ -105,7 +105,7 @@ To obtain the equations of motion for the system represented in Figure ref:fig:s
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\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i
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\end{equation}
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with $L = T - V$ the Lagrangian, $T$ the kinetic coenergy, $V$ the potential energy, $D$ the dissipation function, and $Q_i$ the generalized force associated with the generalized variable $\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}$.
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The equation of motion corresponding to the constant rotation in the $(\vec{i}_x, \vec{i}_y)$ is disregarded as the motion is considered to be imposed by the rotation stage.
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The equation of motion corresponding to the constant rotation in the $(\vec{i}_x, \vec{i}_y)$ plane is disregarded as the motion is considered to be imposed by the rotation stage.
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#+name: eq:energy_functions_lagrange
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\begin{equation}
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\begin{aligned}
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@ -123,18 +123,18 @@ Substituting equations eqref:eq:energy_functions_lagrange into eqref:eq:lagrangi
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\end{align}
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\end{subequations}
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The uniform rotation of the system induces two Gyroscopic effects as shown in Eq. eqref:eq:eom_coupled:
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The uniform rotation of the system induces two Gyroscopic effects as shown in eqref:eq:eom_coupled:
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- Centrifugal forces: that can been seen as added negative stiffness $- m \Omega^2$ along $\vec{i}_u$ and $\vec{i}_v$
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- Coriolis Forces: that couples the motion in the two orthogonal directions
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One can verify that without rotation ($\Omega = 0$) the system becomes equivalent as to two uncoupled one degree of freedom mass-spring-damper systems:
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#+name: eq:oem_no_rotation
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\begin{subequations}
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\begin{align}
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m \ddot{d}_u + c \dot{d}_u + k d_u &= F_u \\
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m \ddot{d}_v + c \dot{d}_v + k d_v &= F_v
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\end{align}
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\end{subequations}
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# One can verify that without rotation ($\Omega = 0$) the system becomes equivalent to two uncoupled one degree of freedom mass-spring-damper systems:
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# #+name: eq:oem_no_rotation
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# \begin{subequations}
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# \begin{align}
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# m \ddot{d}_u + c \dot{d}_u + k d_u &= F_u \\
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# m \ddot{d}_v + c \dot{d}_v + k d_v &= F_v
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# \end{align}
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# \end{subequations}
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#+latex: \par
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@ -199,7 +199,7 @@ In the rest of this study, rotational speeds smaller than the undamped natural f
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| <<fig:campbell_diagram_real>> Real Part | <<fig:campbell_diagram_imag>> Imaginary Part |
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Looking at the transfer function matrix $\bm{G}_d$ in Eq. eqref:eq:Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and the two off-diagonal (coupling) terms are opposite.
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The bode plot of these two distinct terms are shown in Figure ref:fig:plant_compare_rotating_speed for several rotational speeds $\Omega$.
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The bode plot of these two terms are shown in Figure ref:fig:plant_compare_rotating_speed for several rotational speeds $\Omega$.
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These plots confirm the expected behavior: the frequency of the two pairs of complex conjugate poles are further separated as $\Omega$ increases.
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For $\Omega > \omega_0$, the low frequency pair of complex conjugate poles $p_{-}$ becomes unstable.
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@ -292,7 +292,7 @@ The two IFF controllers $K_F$ consist of a pure integrator
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\end{equation}
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where $g$ is a scalar representing the gain of the controller.
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In order to see how the IFF affects the poles of the closed loop system, a Root Locus plot (Figure ref:fig:root_locus_pure_iff) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain $g$ varies from $0$ to $\infty$ for the two controllers simultaneously.
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In order to see how the IFF affects the poles of the closed loop system, a Root Locus plot (Figure ref:fig:root_locus_pure_iff) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain $g$ varies from $0$ to $\infty$ for the two controllers $K_F$ simultaneously.
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As explained in cite:preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr, the closed-loop poles start at the open-loop poles (shown by $\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};$) for $g = 0$ and coincide with the transmission zeros (shown by $\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];$) as $g \to \infty$.
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The direction of increasing gain is indicated by arrows $\tikz[baseline=-0.6ex] \draw[-{Stealth[round]},line width=2pt] (0,0) -- (0.3,0);$.
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@ -307,7 +307,7 @@ This can be seen in the Root Locus plot (Figure ref:fig:root_locus_pure_iff) whe
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Physically, this can be explain like so: at low frequency, the loop gain is very large due to the pure integrators in $K_F$.
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The control system is thus canceling the spring forces which makes the suspended platform no able to hold the payload against centrifugal forces, hence the instability.
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In order to apply Decentralized IFF on rotating platforms, two solutions are proposed to deal with this instability problem.
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In order to apply decentralized IFF on rotating platforms, two solutions are proposed to deal with this instability problem.
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The first one consists of slightly modifying the control law (Section ref:sec:iff_hpf) while the second one consists of adding springs in parallel with the force sensors (Section ref:sec:iff_kp).
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* Integral Force Feedback with High Pass Filter
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@ -318,7 +318,7 @@ As was explained in the previous section, the instability comes in part from the
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In order to limit this low frequency controller gain, an high pass filter (HPF) can be added to the controller
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#+name: eq:IFF_LHF
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\begin{equation}
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\bm{K}_F(s) = \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix}, \quad K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}
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K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}
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\end{equation}
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This is equivalent to slightly shifting the controller pole to the left along the real axis.
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@ -363,14 +363,14 @@ Two parameters can be tuned for the modified controller eqref:eq:IFF_LHF: the ga
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The optimal values of $\omega_i$ and $g$ are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
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In order to visualize how $\omega_i$ does affect the attainable damping, the Root Loci for several $\omega_i$ are displayed in Figure ref:fig:root_locus_wi_modified_iff.
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It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes, the control gain $g$ may be limited to small values due to Eq. eqref:eq:gmax_iff_hpf.
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It is shown that even though small $\omega_i$ seem to allow more damping to be added to the suspension modes, the control gain $g$ may be limited to small values due to eqref:eq:gmax_iff_hpf.
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#+name: fig:root_locus_wi_modified_iff
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#+caption: Root Locus for several HPF cut-off frequencies $\omega_i$, $\Omega = 0.1 \omega_0$
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#+attr_latex: :scale 1
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[[file:figs/root_locus_wi_modified_iff.pdf]]
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In order to study this trade off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of the $\omega_i/\omega_0$.
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In order to study this trade off, the attainable closed-loop damping ratio $\xi_{\text{cl}}$ is computed as a function of $\omega_i/\omega_0$.
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The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also displayed and compared with the gain $g_{\text{max}}$ at which the system becomes unstable (Figure ref:fig:mod_iff_damping_wi).
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#+name: fig:mod_iff_damping_wi
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@ -381,7 +381,7 @@ The gain $g_{\text{opt}}$ at which this maximum damping is obtained is also disp
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Three regions can be observed:
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- $\omega_i/\omega_0 < 0.02$: the added damping is limited by the maximum allowed control gain $g_{\text{max}}$
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- $0.02 < \omega_i/\omega_0 < 0.2$: the attainable damping ratio is maximized and is reached for $g \approx 2$
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- $0.2 < \omega_i/\omega_0$: the added damping decreases as the $\omega_i/\omega_0$ increases
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- $0.2 < \omega_i/\omega_0$: the added damping decreases as $\omega_i/\omega_0$ increases
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* Integral Force Feedback with Parallel Springs
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<<sec:iff_kp>>
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@ -412,7 +412,7 @@ An example of such system is shown in Figure ref:fig:cedrat_xy25xs.
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#+latex: \par
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** Effect of the Parallel Stiffness on the Plant Dynamics :ignore:
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The forces $\begin{bmatrix}f_u, f_v\end{bmatrix}$ measured by the two force sensors represented in Figure ref:fig:system_parallel_springs are equal to
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The forces $\begin{bmatrix}f_u & f_v\end{bmatrix}$ measured by the two force sensors represented in Figure ref:fig:system_parallel_springs are equal to
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#+name: eq:measured_force_kp
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\begin{equation}
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\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
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@ -536,7 +536,7 @@ The two techniques are also compared with passive damping (Figure ref:fig:system
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\end{equation}
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Very similar results are obtained for the two proposed IFF modifications in terms of transmissibility (Figure ref:fig:comp_transmissibility) and compliance (Figure ref:fig:comp_compliance).
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It is also confirmed that these two techniques can significantly damp the system's resonances.
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It is also confirmed that these two techniques can significantly damp the suspension modes.
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#+name: fig:comp_active_damping
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#+caption: Comparison of the two proposed Active Damping Techniques, $\Omega = 0.1 \omega_0$
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@ -551,7 +551,7 @@ The addition of the HPF or the use of the parallel stiffness permit to limit the
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* Conclusion
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<<sec:conclusion>>
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Due to gyroscopic effects, decentralized IFF with pure integrators was shown not to be stable when applied to rotating platforms.
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Due to gyroscopic effects, decentralized IFF with pure integrators was shown to be unstable when applied to rotating platforms.
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Two modifications of the classical IFF control have been proposed to overcome this issue.
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The first modification concerns the controller and consists of adding an high pass filter to the pure integrators.
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@ -560,7 +560,7 @@ This renders the closed loop system stable up to some value of the controller ga
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The second proposed modification concerns the mechanical system.
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Additional springs are added in parallel with the actuators and force sensors.
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It was shown that if the stiffness $k_p$ of the addition springs is larger than the negative stiffness $m \Omega^2$ induced by centrifugal forces, the classical decentralized IFF regains its unconditional stability property.
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It was shown that if the stiffness $k_p$ of the additional springs is larger than the negative stiffness $m \Omega^2$ induced by centrifugal forces, the classical decentralized IFF regains its unconditional stability property.
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While having very different implementations, both proposed modifications are very similar when it comes to the attainable damping and the obtained closed loop system behavior.
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BIN
paper/paper.pdf
BIN
paper/paper.pdf
Binary file not shown.
@ -1,4 +1,4 @@
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% Created 2020-07-08 mer. 18:07
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% Created 2020-07-09 jeu. 09:06
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% Intended LaTeX compiler: pdflatex
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\documentclass{ISMA_USD2020}
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\usepackage[utf8]{inputenc}
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@ -51,14 +51,14 @@
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\abstract{
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This paper investigates the use of Integral Force Feedback (IFF) for the active damping of rotating mechanical systems.
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Guaranteed stability, typical benefit of IFF, is lost as soon as the system is rotating due to gyroscopic effects.
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To overcome this issue, two modifications of the classical IFF control are proposed.
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To overcome this issue, two modifications of the classical IFF control scheme are proposed.
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The first consists of slightly modifying the control law while the second consists of adding springs in parallel with the force sensors.
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Conditions for stability and optimal parameters are derived.
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The results reveal that, despite their different implementations, both modified IFF control have almost identical damping authority on suspension modes.
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The results reveal that, despite their different implementations, both modified IFF control scheme have almost identical damping authority on suspension modes.
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}
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\section{Introduction}
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\label{sec:orgc580a8f}
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\label{sec:orgf2d9f1e}
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\label{sec:introduction}
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There is an increasing need to reduce the undesirable vibration of many sensitive equipment.
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A common method to reduce vibration is to mount the sensitive equipment on a suspended platform which attenuates the vibrations above the frequency of the suspension modes.
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@ -78,7 +78,7 @@ Section \ref{sec:iff_kp} proposes to add springs in parallel with the force sens
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Section \ref{sec:comparison} compares both proposed modifications to the classical IFF in terms of damping authority and closed-loop system behavior.
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\section{Dynamics of Rotating Platforms}
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\label{sec:orgf7cef1f}
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\label{sec:orgf97884c}
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\label{sec:dynamics}
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In order to study how the rotation does affect the use of IFF, a model of a suspended platform on top of a rotating stage is used.
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Figure \ref{fig:system} represents the model schematically which is the simplest in which gyroscopic forces can be studied.
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@ -104,7 +104,7 @@ To obtain the equations of motion for the system represented in Figure \ref{fig:
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\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) + \frac{\partial D}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = Q_i
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\end{equation}
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with \(L = T - V\) the Lagrangian, \(T\) the kinetic coenergy, \(V\) the potential energy, \(D\) the dissipation function, and \(Q_i\) the generalized force associated with the generalized variable \(\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}\).
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The equation of motion corresponding to the constant rotation in the \((\vec{i}_x, \vec{i}_y)\) is disregarded as the motion is considered to be imposed by the rotation stage.
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The equation of motion corresponding to the constant rotation in the \((\vec{i}_x, \vec{i}_y)\) plane is disregarded as the motion is considered to be imposed by the rotation stage.
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\begin{equation}
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\label{eq:energy_functions_lagrange}
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\begin{aligned}
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@ -122,21 +122,12 @@ Substituting equations \eqref{eq:energy_functions_lagrange} into \eqref{eq:lagra
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\end{align}
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\end{subequations}
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The uniform rotation of the system induces two Gyroscopic effects as shown in Eq. \eqref{eq:eom_coupled}:
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The uniform rotation of the system induces two Gyroscopic effects as shown in \eqref{eq:eom_coupled}:
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\begin{itemize}
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\item Centrifugal forces: that can been seen as added negative stiffness \(- m \Omega^2\) along \(\vec{i}_u\) and \(\vec{i}_v\)
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\item Coriolis Forces: that couples the motion in the two orthogonal directions
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\end{itemize}
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One can verify that without rotation (\(\Omega = 0\)) the system becomes equivalent as to two uncoupled one degree of freedom mass-spring-damper systems:
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\begin{subequations}
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\label{eq:oem_no_rotation}
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\begin{align}
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m \ddot{d}_u + c \dot{d}_u + k d_u &= F_u \\
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m \ddot{d}_v + c \dot{d}_v + k d_v &= F_v
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\end{align}
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\end{subequations}
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\par
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To study the dynamics of the system, the differential equations of motions \eqref{eq:eom_coupled} are transformed in the Laplace domain and the \(2 \times 2\) transfer function matrix \(\bm{G}_d\) from \(\begin{bmatrix}F_u & F_v\end{bmatrix}\) to \(\begin{bmatrix}d_u & d_v\end{bmatrix}\) is obtained
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\begin{align}
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@ -205,7 +196,7 @@ In the rest of this study, rotational speeds smaller than the undamped natural f
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\end{figure}
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Looking at the transfer function matrix \(\bm{G}_d\) in Eq. \eqref{eq:Gd_w0_xi_k}, one can see that the two diagonal (direct) terms are equal and the two off-diagonal (coupling) terms are opposite.
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The bode plot of these two distinct terms are shown in Figure \ref{fig:plant_compare_rotating_speed} for several rotational speeds \(\Omega\).
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The bode plot of these two terms are shown in Figure \ref{fig:plant_compare_rotating_speed} for several rotational speeds \(\Omega\).
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These plots confirm the expected behavior: the frequency of the two pairs of complex conjugate poles are further separated as \(\Omega\) increases.
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For \(\Omega > \omega_0\), the low frequency pair of complex conjugate poles \(p_{-}\) becomes unstable.
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@ -225,7 +216,7 @@ For \(\Omega > \omega_0\), the low frequency pair of complex conjugate poles \(p
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\end{figure}
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\section{Decentralized Integral Force Feedback}
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\label{sec:orgcb8c9c7}
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\label{sec:orgf541d3f}
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\label{sec:iff}
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In order to apply IFF to the system, force sensors are added in series with the two actuators (Figure \ref{fig:system_iff}).
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As this study focuses on decentralized control, two identical controllers \(K_F\) are used to feedback each of the sensed force to its associated actuator and no attempt is made to counteract the interactions in the system.
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@ -302,7 +293,7 @@ The two IFF controllers \(K_F\) consist of a pure integrator
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\end{equation}
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where \(g\) is a scalar representing the gain of the controller.
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In order to see how the IFF affects the poles of the closed loop system, a Root Locus plot (Figure \ref{fig:root_locus_pure_iff}) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain \(g\) varies from \(0\) to \(\infty\) for the two controllers simultaneously.
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In order to see how the IFF affects the poles of the closed loop system, a Root Locus plot (Figure \ref{fig:root_locus_pure_iff}) is constructed as follows: the poles of the closed-loop system are drawn in the complex plane as the controller gain \(g\) varies from \(0\) to \(\infty\) for the two controllers \(K_F\) simultaneously.
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As explained in \cite{preumont08_trans_zeros_struc_contr_with,skogestad07_multiv_feedb_contr}, the closed-loop poles start at the open-loop poles (shown by \(\tikz[baseline=-0.6ex] \node[cross out, draw=black, minimum size=1ex, line width=2pt, inner sep=0pt, outer sep=0pt] at (0, 0){};\)) for \(g = 0\) and coincide with the transmission zeros (shown by \(\tikz[baseline=-0.6ex] \draw[line width=2pt, inner sep=0pt, outer sep=0pt] (0,0) circle[radius=3pt];\)) as \(g \to \infty\).
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The direction of increasing gain is indicated by arrows \(\tikz[baseline=-0.6ex] \draw[-{Stealth[round]},line width=2pt] (0,0) -- (0.3,0);\).
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@ -318,18 +309,18 @@ This can be seen in the Root Locus plot (Figure \ref{fig:root_locus_pure_iff}) w
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Physically, this can be explain like so: at low frequency, the loop gain is very large due to the pure integrators in \(K_F\).
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The control system is thus canceling the spring forces which makes the suspended platform no able to hold the payload against centrifugal forces, hence the instability.
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In order to apply Decentralized IFF on rotating platforms, two solutions are proposed to deal with this instability problem.
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In order to apply decentralized IFF on rotating platforms, two solutions are proposed to deal with this instability problem.
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The first one consists of slightly modifying the control law (Section \ref{sec:iff_hpf}) while the second one consists of adding springs in parallel with the force sensors (Section \ref{sec:iff_kp}).
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\section{Integral Force Feedback with High Pass Filter}
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\label{sec:org0b913ec}
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\label{sec:orgf53673d}
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\label{sec:iff_hpf}
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As was explained in the previous section, the instability comes in part from the high gain at low frequency caused by the pure integrators.
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In order to limit this low frequency controller gain, an high pass filter (HPF) can be added to the controller
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\begin{equation}
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\label{eq:IFF_LHF}
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\bm{K}_F(s) = \begin{bmatrix} K_F(s) & 0 \\ 0 & K_F(s) \end{bmatrix}, \quad K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}
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K_{F}(s) = g \cdot \frac{1}{s} \cdot \underbrace{\frac{s/\omega_i}{1 + s/\omega_i}}_{\text{HPF}} = g \cdot \frac{1}{s + \omega_i}
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\end{equation}
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This is equivalent to slightly shifting the controller pole to the left along the real axis.
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@ -368,7 +359,7 @@ Two parameters can be tuned for the modified controller \eqref{eq:IFF_LHF}: the
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The optimal values of \(\omega_i\) and \(g\) are here considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
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In order to visualize how \(\omega_i\) does affect the attainable damping, the Root Loci for several \(\omega_i\) are displayed in Figure \ref{fig:root_locus_wi_modified_iff}.
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It is shown that even though small \(\omega_i\) seem to allow more damping to be added to the suspension modes, the control gain \(g\) may be limited to small values due to Eq. \eqref{eq:gmax_iff_hpf}.
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It is shown that even though small \(\omega_i\) seem to allow more damping to be added to the suspension modes, the control gain \(g\) may be limited to small values due to \eqref{eq:gmax_iff_hpf}.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
@ -376,7 +367,7 @@ It is shown that even though small \(\omega_i\) seem to allow more damping to be
|
||||
\caption{\label{fig:root_locus_wi_modified_iff}Root Locus for several HPF cut-off frequencies \(\omega_i\), \(\Omega = 0.1 \omega_0\)}
|
||||
\end{figure}
|
||||
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||||
In order to study this trade off, the attainable closed-loop damping ratio \(\xi_{\text{cl}}\) is computed as a function of the \(\omega_i/\omega_0\).
|
||||
In order to study this trade off, the attainable closed-loop damping ratio \(\xi_{\text{cl}}\) is computed as a function of \(\omega_i/\omega_0\).
|
||||
The gain \(g_{\text{opt}}\) at which this maximum damping is obtained is also displayed and compared with the gain \(g_{\text{max}}\) at which the system becomes unstable (Figure \ref{fig:mod_iff_damping_wi}).
|
||||
|
||||
\begin{figure}[htbp]
|
||||
@ -389,11 +380,11 @@ Three regions can be observed:
|
||||
\begin{itemize}
|
||||
\item \(\omega_i/\omega_0 < 0.02\): the added damping is limited by the maximum allowed control gain \(g_{\text{max}}\)
|
||||
\item \(0.02 < \omega_i/\omega_0 < 0.2\): the attainable damping ratio is maximized and is reached for \(g \approx 2\)
|
||||
\item \(0.2 < \omega_i/\omega_0\): the added damping decreases as the \(\omega_i/\omega_0\) increases
|
||||
\item \(0.2 < \omega_i/\omega_0\): the added damping decreases as \(\omega_i/\omega_0\) increases
|
||||
\end{itemize}
|
||||
|
||||
\section{Integral Force Feedback with Parallel Springs}
|
||||
\label{sec:org082b3c2}
|
||||
\label{sec:org4c124af}
|
||||
\label{sec:iff_kp}
|
||||
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.
|
||||
@ -417,7 +408,7 @@ An example of such system is shown in Figure \ref{fig:cedrat_xy25xs}.
|
||||
\end{minipage}
|
||||
|
||||
\par
|
||||
The forces \(\begin{bmatrix}f_u, f_v\end{bmatrix}\) measured by the two force sensors represented in Figure \ref{fig:system_parallel_springs} are equal to
|
||||
The forces \(\begin{bmatrix}f_u & f_v\end{bmatrix}\) measured by the two force sensors represented in Figure \ref{fig:system_parallel_springs} are equal to
|
||||
\begin{equation}
|
||||
\label{eq:measured_force_kp}
|
||||
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
|
||||
@ -494,7 +485,7 @@ This is confirmed in Figure \ref{fig:mod_iff_damping_kp} where the attainable cl
|
||||
\end{minipage}
|
||||
|
||||
\section{Comparison and Discussion}
|
||||
\label{sec:org1f46ad4}
|
||||
\label{sec:org537f1b3}
|
||||
\label{sec:comparison}
|
||||
Two modifications to adapt the IFF control strategy to rotating platforms have been proposed in Sections \ref{sec:iff_hpf} and \ref{sec:iff_kp}.
|
||||
These two methods are now compared in terms of added damping, closed-loop compliance and transmissibility.
|
||||
@ -532,7 +523,7 @@ The two techniques are also compared with passive damping (Figure \ref{fig:syste
|
||||
\end{equation}
|
||||
|
||||
Very similar results are obtained for the two proposed IFF modifications in terms of transmissibility (Figure \ref{fig:comp_transmissibility}) and compliance (Figure \ref{fig:comp_compliance}).
|
||||
It is also confirmed that these two techniques can significantly damp the system's resonances.
|
||||
It is also confirmed that these two techniques can significantly damp the suspension modes.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}[c]{0.49\linewidth}
|
||||
@ -554,10 +545,10 @@ On can see in Figure \ref{fig:comp_transmissibility} that the problem of the deg
|
||||
The addition of the HPF or the use of the parallel stiffness permit to limit the degradation of the compliance as compared with classical IFF (Figure \ref{fig:comp_compliance}).
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:org071db57}
|
||||
\label{sec:orga805aaa}
|
||||
\label{sec:conclusion}
|
||||
|
||||
Due to gyroscopic effects, decentralized IFF with pure integrators was shown not to be stable when applied to rotating platforms.
|
||||
Due to gyroscopic effects, decentralized IFF with pure integrators was shown to be unstable when applied to rotating platforms.
|
||||
Two modifications of the classical IFF control have been proposed to overcome this issue.
|
||||
|
||||
The first modification concerns the controller and consists of adding an high pass filter to the pure integrators.
|
||||
@ -566,7 +557,7 @@ This renders the closed loop system stable up to some value of the controller ga
|
||||
|
||||
The second proposed modification concerns the mechanical system.
|
||||
Additional springs are added in parallel with the actuators and force sensors.
|
||||
It was shown that if the stiffness \(k_p\) of the addition springs is larger than the negative stiffness \(m \Omega^2\) induced by centrifugal forces, the classical decentralized IFF regains its unconditional stability property.
|
||||
It was shown that if the stiffness \(k_p\) of the additional springs is larger than the negative stiffness \(m \Omega^2\) induced by centrifugal forces, the classical decentralized IFF regains its unconditional stability property.
|
||||
|
||||
While having very different implementations, both proposed modifications are very similar when it comes to the attainable damping and the obtained closed loop system behavior.
|
||||
|
||||
@ -575,7 +566,7 @@ Future work will focus on the experimental validation of the proposed active dam
|
||||
The Matlab code that was used for this study is available under a MIT License and archived in Zenodo \cite{dehaeze20_activ_dampin_rotat_posit_platf}.
|
||||
|
||||
\section*{Acknowledgment}
|
||||
\label{sec:orgd8daf24}
|
||||
\label{sec:orge39bf3f}
|
||||
This research benefited from a FRIA grant from the French Community of Belgium.
|
||||
|
||||
\bibliography{ref.bib}
|
||||
|
||||
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Reference in New Issue
Block a user