Finish first pass on all sections
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paper/paper.org
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@ -157,20 +157,15 @@ One can verify that without rotation ($\Omega = 0$) the system becomes equivalen
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\end{subequations}
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** Transfer Functions in the Laplace domain
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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 transfer function matrix from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ is obtained:
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#+name: eq:Gd_mimo_tf
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\begin{equation}
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\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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with $\bm{G}_d$ a $2 \times 2$ transfer function matrix
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#+name: eq:Gd_m_k_c
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\begin{equation}
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\bm{G}_{d} =
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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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\begin{bmatrix} d_u \\ d_v \end{bmatrix} &= \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix} \label{eq:Gd_mimo_tf} \\
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\bm{G}_{d} &=
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\begin{bmatrix}
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\frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} & \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} \\
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\frac{-2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} & \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}
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\end{bmatrix}
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\end{equation}
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\end{bmatrix} \label{eq:Gd_m_k_c}
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\end{align}
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# Change of variables
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To simply the analysis, the following change of variable is performed:
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@ -320,6 +315,7 @@ This low frequency gain can be explained as follows: a constant force induces a
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** Decentralized Integral Force Feedback with Pure Integrators
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The two IFF controllers $K_F$ are pure integrators
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#+NAME: eq:Kf_pure_int
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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}
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\end{equation}
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@ -428,12 +424,15 @@ It is shown that even tough small $\omega_i$ seems to allow more damping to be a
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* Integral Force Feedback with Parallel Springs
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<<sec:iff_kp>>
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** Stiffness in Parallel with the Force Sensor
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# Zeros = remove force sensor
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# We want to have stable zeros => add stiffnesses in parallel
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As was shown in the previous sections, the instability when using Decentralized IFF for rotating positioning platforms is due to Gyroscopic effects, more precisely to the negative stiffnesses induced by centrifugal forces.
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The idea in this section is to include additional springs in parallel with the force sensors to counteract the negative stiffness due to centrifugal forces.
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The idea in this section is to include additional springs in parallel with the force sensors to counteract this negative stiffness.
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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.
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Such system could consist of additional springs, or it could also be
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This could represent a system where ref:fig:cedrat_xy25xs.
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The use of such amplified piezoelectric actuator for IFF is discussed in cite:souleille18_concep_activ_mount_space_applic.
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#+attr_latex: :options [t]{0.55\linewidth}
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#+begin_minipage
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#+name: fig:system_parallel_springs
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@ -450,8 +449,8 @@ Such springs are schematically shown in Figure ref:fig:system_parallel_springs w
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[[file:figs/cedrat_xy25xs.png]]
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#+end_minipage
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# Sensed Force
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The forces measured by the force sensors are equal to:
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** Effect of the Parallel Stiffness on the Plant Dynamics
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The forces measured by the sensors 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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@ -459,33 +458,24 @@ The forces measured by the force sensors are equal to:
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\begin{bmatrix} d_u \\ d_v \end{bmatrix}
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\end{equation}
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# Maybe add the fact that this is equivalent to amplified piezo for instance
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# Add reference to cite:souleille18_concep_activ_mount_space_applic
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This could represent a system where ref:fig:cedrat_xy25xs.
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** Effect of the Parallel Stiffness on the Plant Dynamics
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We define an adimensional parameter $\alpha$, $0 \le \alpha < 1$, that describes the proportion of the stiffness in parallel with the actuator and force sensor:
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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
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\begin{subequations}
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\begin{align}
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k_p &= \alpha k \\
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k_a &= (1 - \alpha) k
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\end{align}
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\end{subequations}
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The overall stiffness $k$ stays constant:
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\begin{equation}
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k = k_a + k_p
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\end{equation}
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Note that the overall stiffness $k = k_a + k_p$ is kept constant.
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# Equations: sensed force
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The equations of motion are derived and transformed in the Laplace domain
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#+name: eq:Gk_mimo_tf
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\begin{equation}
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\begin{bmatrix} f_u \\ f_v \end{bmatrix} =
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\bm{G}_k
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\begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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with $\bm{G}_k$ a $2 \times 2$ transfer function matrix
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#+name: eq:Gk
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\begin{equation}
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\bm{G}_k =
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@ -497,94 +487,117 @@ The overall stiffness $k$ stays constant:
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# News terms with \alpha are added
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# w0 and xi are the same as before => only the zeros are changing and not the poles.
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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.
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# Negative Stiffness due to rotation => the stiffness should be larger than that
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# For kp < negative stiffness => real zeros => non-minimum phase
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# For kp > negative stiffness => complex conjugate zeros => minimum phase
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The two real zeros $z_r$ eqref:eq:iff_zero_real that were inducing non-minimum phase behavior are transformed into complex conjugate zeros for
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\begin{equation}
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\begin{aligned}
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\alpha > \frac{\Omega^2}{{\omega_0}^2} \\
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\Leftrightarrow k_p > m \Omega^2
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\alpha &> \frac{\Omega^2}{{\omega_0}^2} \\
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\Leftrightarrow k_p &> m \Omega^2
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\end{aligned}
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\end{equation}
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If the added stiffness is higher than the maximum negative stiffness, then the poles of the IFF damped system will stay in the (stable) right half-plane.
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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.
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This is confirmed by the Bode plot in Figure ref:fig:plant_iff_kp.
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# while recovering the alternating poles and zeros near the imaginary axis.
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#+name: fig:plant_iff_kp
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#+caption: Bode Plot of $f_u/F_u$ without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
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#+attr_latex: :scale 1
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[[file:figs/plant_iff_kp.pdf]]
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# Location of the zeros as a function of kp => maybe to complex
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# Do we talk about siso zeros of mimo (transmission zeros)?
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# Try to show that we don't have anymore real zeros that was making the system non-minimum phase
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# Show that it is the case on the root locus
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# Root Locus plot
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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.
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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.
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#+name: fig:root_locus_iff_kp
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#+caption: Root Locus for IFF without parallel spring, with parallel springs with stiffness $k_p < m \Omega^2$ and $k_p > m \Omega^2$, $\Omega = 0.1 \omega_0$
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#+attr_latex: :scale 1
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[[file:figs/root_locus_iff_kp.pdf]]
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# For kp > m Omega => unconditionally stable
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** Optimal Parallel Stiffness
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# Attainable damping generally proportional to the distance between the poles and zeros (add reference, probably preumont)
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# The zero is the poles of the system without the force sensors => w0 = sqrt(kp/m) +/- Omega ?? => seems not true
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# Thus, small kp is wanted: kp close to m Omega^2 should give the optimal damping but is not acceptable for robustness reasons
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# Large Stiffness decreases the attainable damping
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Figure ref:fig:root_locus_iff_kps shows Root Loci plots for several parallel stiffnesses $k_p > m \Omega^2$.
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It is shown that large parallel stiffness $k_p$ reduces the attainable damping.
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This can be explained by the fact that as the parallel stiffnesses increases, the transmission zeros are closer to the poles.
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As explained in cite:preumont18_vibrat_contr_activ_struc_fourt_edition, the attainable damping is generally proportional to the distance between the poles and zeros.
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The frequency of the transmission zeros of the system are increasing with the fraction used as parallel stiffness $k_p$.
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# Example with kp = 5 m Omega
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For any $k_p > m \Omega^2$, the control gain $g$ can be tuned such that the maximum simultaneous damping is added to the resonances of the system as shown in Figure ref:fig:root_locus_opt_gain_iff_kp for $k_p = 5 m \Omega^2$.
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#+name: fig:root_locus_iff_kps_opt
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#+caption: Root Locus for IFF when parallel stiffness is used, $\Omega = 0.1 \omega_0$
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#+caption: Root Locus for IFF when parallel stiffness $k_p$ is added, $\Omega = 0.1 \omega_0$
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#+attr_latex: :environment subfigure :width 0.49\linewidth :align c
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| file:figs/root_locus_iff_kps.pdf | file:figs/root_locus_opt_gain_iff_kp.pdf |
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| <<fig:root_locus_iff_kps>> Three values of $k_p$ | <<fig:root_locus_opt_gain_iff_kp>> $k_p = 5 m \Omega^2$, optimal damping is shown |
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| file:figs/root_locus_iff_kps.pdf | file:figs/root_locus_opt_gain_iff_kp.pdf |
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| <<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 |
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* Comparison of the Proposed Active Damping Techniques for Rotating Positioning Stages
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* Comparison of the Proposed Modification to Decentralized Integral Force Feedback for Rotating Positioning Stages
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<<sec:comparison>>
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** Physical Comparison
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** Introduction :ignore:
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# Comparison in terms of modification to the system
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The two proposed modification to the decentralized IFF for rotating positioning stages are now compared.
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Two modification to the decentralized IFF for rotating positioning stages have been proposed.
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The first modification concerns the controller.
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It consists of adding an high pass filter to $K_F$ eqref:eq:IFF_LHF.
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This allows the system to be stable for gains up to $g_\text{max}$ eqref:eq:gmax_iff_hpf.
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The second proposed modification concerns the mechanical system.
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If springs are added in parallel to the actuators and force sensors with a stiffness $k_p > m \Omega^2$, decentralized IFF can be applied with unconditional stability.
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** Attainable Damping
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These two methods are now compared in terms of added damping, closed-loop compliance and transmissibility.
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For the following comparisons, the high pass cut-off frequency is set to $\omega_i = 0.1 \omega_0$ and the parallel stiffness is $k_p = 5 m \Omega^2$.
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# Both techniques provides very good amount of damping
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** Comparison of the Attainable Damping
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Figure ref:fig:comp_root_locus shows to Root Locus plots for the two proposed IFF techniques.
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The maximum added damping is very similar for both techniques and are reached for $g_\text{opt} \approx 2$ in both cases.
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#+name: fig:comp_root_locus
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#+caption: Root Locus for the three proposed decentralized active damping techniques: IFF with HFP, IFF with parallel springs, and relative DVF, $\Omega = 0.1 \omega_0$
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#+caption: Root Locus for the two proposed modifications of decentralized IFF, $\Omega = 0.1 \omega_0$
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#+attr_latex: :scale 1
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[[file:figs/comp_root_locus.pdf]]
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** Transmissibility and Compliance
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** Comparison Transmissibility and Compliance
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The two proposed techniques are now compared in terms of closed-loop compliance and transmissibility.
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The compliance is defined as the transfer function from external forces applied to the payload to the displacement of the payload in an inertial frame.
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The transmissibility is the dynamics from the displacement of the rotating stage to the displacement of the payload.
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It is used to characterize how much vibration of the rotating stage is transmitted to the payload.
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The two techniques are also compared with passive damping (Figure ref:fig:system) with $c$ tuned to critically damp the resonance when $\Omega = 0$
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\begin{equation}
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c_\text{crit} = 2 \sqrt{k m}
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\end{equation}
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# IFF with HPF and IFF with kp give very similar results!
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Very similar results are obtained for both techniques as shown in Figures ref:fig:comp_compliance and ref:fig:comp_transmissibility.
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It is also confirmed that these techniques can significantly damp the system's resonances.
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# IFF degrades the compliance at low frequency (add reference)
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# Passive Damping degrades the transmissibility at high frequency
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# The roll-off is -1 instead of -2
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Compared to passive damping, the two techniques degrades the compliance at low frequency (Figure ref:fig:comp_compliance).
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They however do not degrades the transmissibility as high frequency as its the case with passive damping (Figure ref:fig:comp_transmissibility)
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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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#+attr_latex: :environment subfigure :width 0.45\linewidth :align c
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| file:figs/comp_compliance.pdf | file:figs/comp_transmissibility.pdf |
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| <<fig:comp_compliance>> Transmissibility | <<fig:comp_transmissibility>> Compliance |
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| file:figs/comp_compliance.pdf | file:figs/comp_transmissibility.pdf |
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| <<fig:comp_compliance>> Compliance | <<fig:comp_transmissibility>> Transmissibility |
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* Conclusion
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<<sec:conclusion>>
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MIMO approach to study the coupling effects?
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# MIMO approach to study the coupling effects?
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* Acknowledgment
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:PROPERTIES:
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:UNNUMBERED: t
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:END:
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This research benefited from a FRIA grant from the French Community of Belgium.
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* Bibliography :ignore:
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\bibliography{ref.bib}
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paper/paper.tex
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@ -1,4 +1,4 @@
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% Created 2020-07-01 mer. 14:20
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% Created 2020-07-01 mer. 16:58
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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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@ -37,13 +37,6 @@
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\usetikzlibrary{shapes.misc,arrows,arrows.meta}
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\date{}
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\title{Active Damping of Rotating Positioning Platforms using Force Feedback}
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\hypersetup{
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pdfauthor={},
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pdftitle={Active Damping of Rotating Positioning Platforms using Force Feedback},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 27.0.91 (Org mode 9.4)},
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pdflang={English}}
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\begin{document}
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\maketitle
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@ -53,7 +46,7 @@
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}
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\section{Introduction}
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\label{sec:org5531002}
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\label{sec:org5846940}
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\label{sec:introduction}
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Due to gyroscopic effects, the guaranteed robustness properties of Integral Force Feedback do not hold.
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Either the control architecture can be slightly modified or mechanical changes in the system can be performed.
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@ -61,12 +54,11 @@ This paper has been published
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The Matlab code that was use to obtain the results are available in \cite{dehaeze20_activ_dampin_rotat_posit_platf}.
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\section{Dynamics of Rotating Positioning Platforms}
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\label{sec:org35aebb5}
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\label{sec:orge0b9fb2}
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\label{sec:dynamics}
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\subsection{Model of a Rotating Positioning Platform}
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\label{sec:org61dc9aa}
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\label{sec:org906209e}
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In order to study how the rotation of a positioning platforms does affect the use of force feedback, a simple model of an X-Y positioning stage on top of a rotating stage is developed.
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The model is schematically represented in Figure \ref{fig:system} and forms the simplest system where gyroscopic forces can be studied.
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The rotating stage is supposed to be ideal, meaning it induces a perfect rotation \(\theta(t) = \Omega t\) where \(\Omega\) is the rotational speed in \(\si{\radian\per\second}\).
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@ -95,7 +87,7 @@ The position of the payload is represented by \((d_u, d_v)\) expressed in the ro
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\end{figure}
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\subsection{Equations of Motion}
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\label{sec:orgdd30881}
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\label{sec:org8561102}
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To obtain of equation of motion for the system represented in Figure \ref{fig:system}, the Lagrangian equations are used:
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\begin{equation}
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\label{eq:lagrangian_equations}
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@ -139,21 +131,16 @@ One can verify that without rotation (\(\Omega = 0\)) the system becomes equival
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\end{subequations}
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\subsection{Transfer Functions in the Laplace domain}
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\label{sec:org12fe22b}
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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 transfer function matrix 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{equation}
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\label{eq:Gd_mimo_tf}
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\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
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\end{equation}
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with \(\bm{G}_d\) a \(2 \times 2\) transfer function matrix
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\begin{equation}
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\label{eq:Gd_m_k_c}
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\bm{G}_{d} =
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\label{sec:org2ee1be9}
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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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\begin{bmatrix} d_u \\ d_v \end{bmatrix} &= \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix} \label{eq:Gd_mimo_tf} \\
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\bm{G}_{d} &=
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\begin{bmatrix}
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\frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} & \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} \\
|
||||
\frac{-2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2} & \frac{ms^2 + cs + k - m \Omega^2}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
\end{bmatrix} \label{eq:Gd_m_k_c}
|
||||
\end{align}
|
||||
|
||||
To simply the analysis, the following change of variable is performed:
|
||||
\begin{itemize}
|
||||
@ -173,11 +160,10 @@ The transfer function matrix \eqref{eq:Gd_m_k_c} becomes equal to
|
||||
\end{equation}
|
||||
|
||||
For all the numerical analysis in this study, \(\omega_0 = \SI{1}{\radian\per\second}\), \(k = \SI{1}{\newton\per\meter}\) and \(\xi = 0.025 = \SI{2.5}{\percent}\).
|
||||
|
||||
Even tough no system with such parameters will be encountered in practice, conclusions will be drawn relative to these parameters such that they can be generalized to any other parameter.
|
||||
|
||||
\subsection{System Dynamics and Campbell Diagram}
|
||||
\label{sec:org5558523}
|
||||
\label{sec:orga228a88}
|
||||
The poles of \(\bm{G}_d\) are the complex solutions \(p\) of
|
||||
\begin{equation}
|
||||
\left( \frac{p^2}{{\omega_0}^2} + 2 \xi \frac{p}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \right)^2 + \left( 2 \frac{\Omega}{\omega_0} \frac{p}{\omega_0} \right)^2 = 0
|
||||
@ -214,7 +200,6 @@ In the rest of this study, rotational speeds smaller than the undamped natural f
|
||||
|
||||
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.
|
||||
The bode plot of these two distinct terms are shown in Figure \ref{fig:plant_compare_rotating_speed} for several rotational speeds \(\Omega\).
|
||||
|
||||
It is confirmed that the two pairs of complex conjugate poles are further separated as \(\Omega\) increases.
|
||||
For \(\Omega > \omega_0\), the low frequency complex conjugate poles \(p_{-}\) becomes unstable.
|
||||
|
||||
@ -232,10 +217,10 @@ For \(\Omega > \omega_0\), the low frequency complex conjugate poles \(p_{-}\) b
|
||||
\end{figure}
|
||||
|
||||
\section{Decentralized Integral Force Feedback}
|
||||
\label{sec:orga42580e}
|
||||
\label{sec:orgd6c59cc}
|
||||
\label{sec:iff}
|
||||
\subsection{Force Sensors and Control Architecture}
|
||||
\label{sec:orgdd7381f}
|
||||
\label{sec:orgcc446a6}
|
||||
In order to apply Decentralized Integral Force Feedback to the system, force sensors are added in series with the two actuators (Figure \ref{fig:system_iff}).
|
||||
Two identical controllers \(K_F\) are added to feedback each of the sensed forces to its collocated actuator.
|
||||
The control diagram is shown in Figure \ref{fig:control_diagram_iff}.
|
||||
@ -255,7 +240,7 @@ The control diagram is shown in Figure \ref{fig:control_diagram_iff}.
|
||||
\end{minipage}
|
||||
|
||||
\subsection{Plant Dynamics}
|
||||
\label{sec:org7bc5629}
|
||||
\label{sec:org8c9a16a}
|
||||
The forces measured by the force sensors are equal to:
|
||||
\begin{equation}
|
||||
\label{eq:measured_force}
|
||||
@ -302,7 +287,6 @@ The low frequency gain of \(\bm{G}_f\) is no longer zero, and increases with the
|
||||
\end{equation}
|
||||
|
||||
This low frequency gain can be explained as follows: a constant force induces a small displacement of the mass, which then increases the centrifugal forces measured by the force sensors.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/plant_iff_compare_rotating_speed.pdf}
|
||||
@ -310,9 +294,10 @@ This low frequency gain can be explained as follows: a constant force induces a
|
||||
\end{figure}
|
||||
|
||||
\subsection{Decentralized Integral Force Feedback with Pure Integrators}
|
||||
\label{sec:orgbe4061b}
|
||||
\label{sec:org99fa3c9}
|
||||
The two IFF controllers \(K_F\) are pure integrators
|
||||
\begin{equation}
|
||||
\label{eq:Kf_pure_int}
|
||||
\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}
|
||||
\end{equation}
|
||||
where \(g\) is a scalar value representing the gain of the controller.
|
||||
@ -335,10 +320,10 @@ Two system modifications are proposed in the next sections to deal with this sta
|
||||
Either the control law can be change (Section \ref{sec:iff_hpf}) or the mechanical system slightly modified (Section \ref{sec:iff_kp}).
|
||||
|
||||
\section{Integral Force Feedback with High Pass Filters}
|
||||
\label{sec:orgd914f89}
|
||||
\label{sec:orgcdcaab1}
|
||||
\label{sec:iff_hpf}
|
||||
\subsection{Modification of the Control Low}
|
||||
\label{sec:org9cbf657}
|
||||
\label{sec:org9d35c60}
|
||||
In order to limit the low frequency controller gain, an high pass filter (HPF) can be added to the controller which becomes
|
||||
\begin{equation}
|
||||
\label{eq:IFF_LHF}
|
||||
@ -350,7 +335,7 @@ This modification of the IFF controller is typically done to avoid saturation as
|
||||
This is however not the case in this study as it will become in the next section.
|
||||
|
||||
\subsection{Feedback Analysis}
|
||||
\label{sec:org5571d5f}
|
||||
\label{sec:org54e6be5}
|
||||
The loop gains for an individual decentralized controller \(K_F(s)\) with and without the added HPF are shown in Figure \ref{fig:loop_gain_modified_iff}.
|
||||
The effect of the added HPF is a limitation of the low frequency gain.
|
||||
|
||||
@ -376,8 +361,9 @@ This gain also corresponds as to when the low frequency loop gain reaches one.
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
|
||||
\subsection{Optimal Control Parameters}
|
||||
\label{sec:org5313add}
|
||||
\label{sec:org9dc1d7f}
|
||||
|
||||
Two parameters can be tuned for the controller \eqref{eq:IFF_LHF}: the gain \(g\) and the location of the pole \(\omega_i\).
|
||||
The optimal values of \(\omega_i\) and \(g\) are considered as the values for which the damping of all the closed-loop poles are simultaneously maximized.
|
||||
@ -398,14 +384,19 @@ It is shown that even tough small \(\omega_i\) seems to allow more damping to be
|
||||
\end{figure}
|
||||
|
||||
\section{Integral Force Feedback with Parallel Springs}
|
||||
\label{sec:org9bf4afa}
|
||||
\label{sec:orgfb6b0e8}
|
||||
\label{sec:iff_kp}
|
||||
\subsection{Stiffness in Parallel with the Force Sensor}
|
||||
\label{sec:org7253a28}
|
||||
\label{sec:org4af80eb}
|
||||
|
||||
As was shown in the previous sections, the instability when using Decentralized IFF for rotating positioning platforms is due to Gyroscopic effects, more precisely to the negative stiffnesses induced by centrifugal forces.
|
||||
The idea in this section is to include additional springs in parallel with the force sensors to counteract the negative stiffness due to centrifugal forces.
|
||||
The idea in this section is to include additional springs in parallel with the force sensors to counteract this negative stiffness.
|
||||
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.
|
||||
|
||||
Such system could consist of additional springs, or it could also be
|
||||
This could represent a system where \ref{fig:cedrat_xy25xs}.
|
||||
The use of such amplified piezoelectric actuator for IFF is discussed in \cite{souleille18_concep_activ_mount_space_applic}.
|
||||
|
||||
\begin{minipage}[t]{0.55\linewidth}
|
||||
\begin{center}
|
||||
\includegraphics[scale=1]{figs/system_parallel_springs.pdf}
|
||||
@ -416,12 +407,13 @@ Such springs are schematically shown in Figure \ref{fig:system_parallel_springs}
|
||||
\begin{minipage}[t]{0.40\linewidth}
|
||||
\begin{center}
|
||||
\includegraphics[width=\linewidth]{figs/cedrat_xy25xs.png}
|
||||
\captionof{figure}{\label{fig:cedrat_xy25xs}XY Piezoelectric Stage (XY25XS from Cedrat-Technology)}
|
||||
\captionof{figure}{\label{fig:cedrat_xy25xs}XY Piezoelectric Stage (XY25XS from Cedrat Technology)}
|
||||
\end{center}
|
||||
\end{minipage}
|
||||
|
||||
|
||||
The forces measured by the force sensors are equal to:
|
||||
\subsection{Effect of the Parallel Stiffness on the Plant Dynamics}
|
||||
\label{sec:org47b3420}
|
||||
The forces measured by the sensors are equal to
|
||||
\begin{equation}
|
||||
\label{eq:measured_force_kp}
|
||||
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} =
|
||||
@ -429,31 +421,23 @@ The forces measured by the force sensors are equal to:
|
||||
\begin{bmatrix} d_u \\ d_v \end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
This could represent a system where
|
||||
|
||||
\subsection{Effect of the Parallel Stiffness on the Plant Dynamics}
|
||||
\label{sec:orgf4cd5ba}
|
||||
|
||||
We define an adimensional parameter \(\alpha\), \(0 \le \alpha < 1\), that describes the proportion of the stiffness in parallel with the actuator and force sensor:
|
||||
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{subequations}
|
||||
\begin{align}
|
||||
k_p &= \alpha k \\
|
||||
k_a &= (1 - \alpha) k
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
Note that the overall stiffness \(k = k_a + k_p\) is kept constant.
|
||||
|
||||
The overall stiffness \(k\) stays constant:
|
||||
\begin{equation}
|
||||
k = k_a + k_p
|
||||
\end{equation}
|
||||
|
||||
The equations of motion are derived and transformed in the Laplace domain
|
||||
\begin{equation}
|
||||
\label{eq:Gk_mimo_tf}
|
||||
\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
|
||||
\begin{equation}
|
||||
\label{eq:Gk}
|
||||
\bm{G}_k =
|
||||
@ -463,22 +447,27 @@ The overall stiffness \(k\) stays constant:
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
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 for
|
||||
\begin{equation}
|
||||
\begin{aligned}
|
||||
\alpha > \frac{\Omega^2}{{\omega_0}^2} \\
|
||||
\Leftrightarrow k_p > m \Omega^2
|
||||
\alpha &> \frac{\Omega^2}{{\omega_0}^2} \\
|
||||
\Leftrightarrow k_p &> m \Omega^2
|
||||
\end{aligned}
|
||||
\end{equation}
|
||||
|
||||
If the added stiffness is higher than the maximum negative stiffness, then the poles of the IFF damped system will stay in the (stable) right half-plane.
|
||||
|
||||
|
||||
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}.
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/plant_iff_kp.pdf}
|
||||
\caption{\label{fig:plant_iff_kp}Bode Plot of \(f_u/F_u\) without parallel spring, with parallel springs with stiffness \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\), \(\Omega = 0.1 \omega_0\)}
|
||||
\end{figure}
|
||||
|
||||
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.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/root_locus_iff_kp.pdf}
|
||||
@ -486,60 +475,94 @@ If the added stiffness is higher than the maximum negative stiffness, then the p
|
||||
\end{figure}
|
||||
|
||||
\subsection{Optimal Parallel Stiffness}
|
||||
\label{sec:orgeadc68d}
|
||||
\label{sec:orgbfbcf95}
|
||||
Figure \ref{fig:root_locus_iff_kps} shows Root Loci plots for several parallel stiffnesses \(k_p > m \Omega^2\).
|
||||
It is shown that large parallel stiffness \(k_p\) reduces the attainable damping.
|
||||
This can be explained by the fact that as the parallel stiffnesses increases, the transmission zeros are closer to the poles.
|
||||
As explained in \cite{preumont18_vibrat_contr_activ_struc_fourt_edition}, the attainable damping is generally proportional to the distance between the poles and zeros.
|
||||
The frequency of the transmission zeros of the system are increasing with the fraction used as parallel stiffness \(k_p\).
|
||||
|
||||
For any \(k_p > m \Omega^2\), the control gain \(g\) can be tuned such that the maximum simultaneous damping is added to the resonances of the system as shown in Figure \ref{fig:root_locus_opt_gain_iff_kp} for \(k_p = 5 m \Omega^2\).
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}[c]{0.49\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/root_locus_iff_kps.pdf}
|
||||
\caption{\label{fig:root_locus_iff_kps} Three values of \(k_p\)}
|
||||
\caption{\label{fig:root_locus_iff_kps} Comparison of three parallel stiffnesses \(k_p\)}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}[c]{0.49\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/root_locus_opt_gain_iff_kp.pdf}
|
||||
\caption{\label{fig:root_locus_opt_gain_iff_kp} \(k_p = 5 m \Omega^2\), optimal damping is shown}
|
||||
\caption{\label{fig:root_locus_opt_gain_iff_kp} \(k_p = 5 m \Omega^2\), optimal damping \(\xi_\text{opt}\) is shown}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:root_locus_iff_kps_opt}Root Locus for IFF when parallel stiffness is used, \(\Omega = 0.1 \omega_0\)}
|
||||
\caption{\label{fig:root_locus_iff_kps_opt}Root Locus for IFF when parallel stiffness \(k_p\) is added, \(\Omega = 0.1 \omega_0\)}
|
||||
\centering
|
||||
\end{figure}
|
||||
|
||||
\section{Comparison of the Proposed Active Damping Techniques for Rotating Positioning Stages}
|
||||
\label{sec:org21311c2}
|
||||
\section{Comparison of the Proposed Modification to Decentralized Integral Force Feedback for Rotating Positioning Stages}
|
||||
\label{sec:org25de90d}
|
||||
\label{sec:comparison}
|
||||
\subsection{Physical Comparison}
|
||||
\label{sec:orgff9353a}
|
||||
The two proposed modification to the decentralized IFF for rotating positioning stages are now compared.
|
||||
Two modification to the decentralized IFF for rotating positioning stages have been proposed.
|
||||
|
||||
The first modification concerns the controller.
|
||||
It consists of adding an high pass filter to \(K_F\) \eqref{eq:IFF_LHF}.
|
||||
This allows the system to be stable for gains up to \(g_\text{max}\) \eqref{eq:gmax_iff_hpf}.
|
||||
|
||||
The second proposed modification concerns the mechanical system.
|
||||
If springs are added in parallel to the actuators and force sensors with a stiffness \(k_p > m \Omega^2\), decentralized IFF can be applied with unconditional stability.
|
||||
|
||||
\subsection{Attainable Damping}
|
||||
\label{sec:org43e21e8}
|
||||
These two methods are now compared in terms of added damping, closed-loop compliance and transmissibility.
|
||||
For the following comparisons, the high pass cut-off frequency is set to \(\omega_i = 0.1 \omega_0\) and the parallel stiffness is \(k_p = 5 m \Omega^2\).
|
||||
\subsection{Comparison of the Attainable Damping}
|
||||
\label{sec:orgd307a58}
|
||||
Figure \ref{fig:comp_root_locus} shows to Root Locus plots for the two proposed IFF techniques.
|
||||
The maximum added damping is very similar for both techniques and are reached for \(g_\text{opt} \approx 2\) in both cases.
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\centering
|
||||
\includegraphics[scale=1]{figs/comp_root_locus.pdf}
|
||||
\caption{\label{fig:comp_root_locus}Root Locus for the three proposed decentralized active damping techniques: IFF with HFP, IFF with parallel springs, and relative DVF, \(\Omega = 0.1 \omega_0\)}
|
||||
\caption{\label{fig:comp_root_locus}Root Locus for the two proposed modifications of decentralized IFF, \(\Omega = 0.1 \omega_0\)}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Transmissibility and Compliance}
|
||||
\label{sec:org4cd8990}
|
||||
\subsection{Comparison Transmissibility and Compliance}
|
||||
\label{sec:orgb48201a}
|
||||
The two proposed techniques are now compared in terms of closed-loop compliance and transmissibility.
|
||||
|
||||
The compliance is defined as the transfer function from external forces applied to the payload to the displacement of the payload in an inertial frame.
|
||||
The transmissibility is the dynamics from the displacement of the rotating stage to the displacement of the payload.
|
||||
It is used to characterize how much vibration of the rotating stage is transmitted to the payload.
|
||||
|
||||
The two techniques are also compared with passive damping (Figure \ref{fig:system}) with \(c\) tuned to critically damp the resonance when \(\Omega = 0\)
|
||||
\begin{equation}
|
||||
c_\text{crit} = 2 \sqrt{k m}
|
||||
\end{equation}
|
||||
|
||||
Very similar results are obtained for both techniques as shown in Figures \ref{fig:comp_compliance} and \ref{fig:comp_transmissibility}.
|
||||
It is also confirmed that these techniques can significantly damp the system's resonances.
|
||||
|
||||
Compared to passive damping, the two techniques degrades the compliance at low frequency (Figure \ref{fig:comp_compliance}).
|
||||
They however do not degrades the transmissibility as high frequency as its the case with passive damping (Figure \ref{fig:comp_transmissibility})
|
||||
|
||||
\begin{figure}[htbp]
|
||||
\begin{subfigure}[c]{0.45\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/comp_compliance.pdf}
|
||||
\caption{\label{fig:comp_compliance} Transmissibility}
|
||||
\caption{\label{fig:comp_compliance} Compliance}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}[c]{0.45\linewidth}
|
||||
\includegraphics[width=\linewidth]{figs/comp_transmissibility.pdf}
|
||||
\caption{\label{fig:comp_transmissibility} Compliance}
|
||||
\caption{\label{fig:comp_transmissibility} Transmissibility}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:comp_active_damping}Comparison of the two proposed Active Damping Techniques, \(\Omega = 0.1 \omega_0\)}
|
||||
\centering
|
||||
\end{figure}
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:org2c33141}
|
||||
\label{sec:orgf8ee171}
|
||||
\label{sec:conclusion}
|
||||
|
||||
MIMO approach to study the coupling effects?
|
||||
|
||||
\section*{Acknowledgment}
|
||||
\label{sec:org33821c1}
|
||||
\label{sec:org66fd2b5}
|
||||
This research benefited from a FRIA grant from the French Community of Belgium.
|
||||
|
||||
\bibliography{ref.bib}
|
||||
\end{document}
|
||||
|
||||
Loading…
Reference in New Issue
Block a user