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Use =bm= instead of =mathbf=

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Thomas Dehaeze 2025-04-18 18:04:43 +02:00
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@ -1917,7 +1917,7 @@ The transfer functions from $F$ to $L$ (i.e., control of the relative motion of
When the relative displacement of the nano-hexapod $L$ is controlled (dynamics shown in Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics), having a stiff nano-hexapod (i.e., with a suspension mode at higher frequency than the mode of the support) makes the dynamics less affected by the limited support compliance (Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_stiff).
This is why it is very common to have stiff piezoelectric stages fixed at the very top of positioning stages.
In such a case, the control of the piezoelectric stage using its integrated metrology (typically capacitive sensors) is quite simple as the plant is not much affected by the dynamics of the support on which it is fixed.
# TODO - Add references of such stations with piezo stages on top
# TODO - Add references of such stations with piezo stages on top, for instance [[cite:&schropp20_ptynam]]
If a soft nano-hexapod is used, the support dynamics appears in the dynamics between $F$ and $L$ (see Figure\nbsp{}ref:fig:uniaxial_effect_support_compliance_dynamics_soft) which will impact the control robustness and performance.
@ -2221,35 +2221,35 @@ The uniform rotation of the system induces two /gyroscopic effects/ as shown in
- /Coriolis forces/: that adds /coupling/ between the two orthogonal directions.
One can verify that without rotation ($\Omega = 0$), the system becomes equivalent to two /uncoupled/ one degree of freedom mass-spring-damper systems.
To study the dynamics of the system, the two differential equations of motions\nbsp{}eqref:eq:rotating_eom_coupled are converted into the Laplace domain and the $2 \times 2$ transfer function matrix $\mathbf{G}_d$ from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ in equation\nbsp{}eqref:eq:rotating_Gd_mimo_tf is obtained.
The four transfer functions in $\mathbf{G}_d$ are shown in equation\nbsp{}eqref:eq:rotating_Gd_indiv_el.
To study the dynamics of the system, the two differential equations of motions\nbsp{}eqref:eq:rotating_eom_coupled are converted into the Laplace domain and the $2 \times 2$ transfer function matrix $\bm{G}_d$ from $\begin{bmatrix}F_u & F_v\end{bmatrix}$ to $\begin{bmatrix}d_u & d_v\end{bmatrix}$ in equation\nbsp{}eqref:eq:rotating_Gd_mimo_tf is obtained.
The four transfer functions in $\bm{G}_d$ are shown in equation\nbsp{}eqref:eq:rotating_Gd_indiv_el.
\begin{equation}\label{eq:rotating_Gd_mimo_tf}
\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
\begin{subequations}\label{eq:rotating_Gd_indiv_el}
\begin{align}
\mathbf{G}_{d}(1,1) &= \mathbf{G}_{d}(2,2) = \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} \\
\mathbf{G}_{d}(1,2) &= -\mathbf{G}_{d}(2,1) = \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}
\bm{G}_{d}(1,1) &= \bm{G}_{d}(2,2) = \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} \\
\bm{G}_{d}(1,2) &= -\bm{G}_{d}(2,1) = \frac{2 m \Omega s}{\left( m s^2 + cs + k - m \Omega^2 \right)^2 + \left( 2 m \Omega s \right)^2}
\end{align}
\end{subequations}
To simplify the analysis, the undamped natural frequency $\omega_0$ and the damping ratio $\xi$ defined in\nbsp{}eqref:eq:rotating_xi_and_omega are used instead.
The elements of the transfer function matrix $\mathbf{G}_d$ are described by equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k.
The elements of the transfer function matrix $\bm{G}_d$ are described by equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k.
\begin{equation} \label{eq:rotating_xi_and_omega}
\omega_0 = \sqrt{\frac{k}{m}} \text{ in } \si{\radian\per\second}, \quad \xi = \frac{c}{2 \sqrt{k m}}
\end{equation}
\begin{subequations} \label{eq:rotating_Gd_w0_xi_k}
\begin{align}
\mathbf{G}_{d}(1,1) &= \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \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} \\
\mathbf{G}_{d}(1,2) &= \frac{\frac{1}{k} \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}
\bm{G}_{d}(1,1) &= \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \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} \\
\bm{G}_{d}(1,2) &= \frac{\frac{1}{k} \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}
\end{align}
\end{subequations}
**** System Poles: Campbell Diagram
The poles of $\mathbf{G}_d$ are the complex solutions $p$ of equation\nbsp{}eqref:eq:rotating_poles (i.e. the roots of its denominator).
The poles of $\bm{G}_d$ are the complex solutions $p$ of equation\nbsp{}eqref:eq:rotating_poles (i.e. the roots of its denominator).
\begin{equation}\label{eq:rotating_poles}
\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
@ -2291,7 +2291,7 @@ Physically, the negative stiffness term $-m\Omega^2$ induced by centrifugal forc
**** System Dynamics: Effect of rotation
The system dynamics from actuator forces $[F_u, F_v]$ to the relative motion $[d_u, d_v]$ is identified for several rotating velocities.
Looking at the transfer function matrix $\mathbf{G}_d$ in equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite.
Looking at the transfer function matrix $\bm{G}_d$ in equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k, one can see that the two diagonal (direct) terms are equal and that the two off-diagonal (coupling) terms are opposite.
The bode plots of these two terms are shown in Figure\nbsp{}ref:fig:rotating_bode_plot for several rotational speeds $\Omega$.
These plots confirm the expected behavior: the frequencies of the two pairs of complex conjugate poles are further separated as $\Omega$ increases.
For $\Omega > \omega_0$, the low-frequency pair of complex conjugate poles $p_{-}$ becomes unstable (shown be the 180 degrees phase lead instead of phase lag).
@ -2368,21 +2368,21 @@ The forces $\begin{bmatrix}f_u & f_v\end{bmatrix}$ measured by the two force sen
\begin{bmatrix} d_u \\ d_v \end{bmatrix}
\end{equation}
The transfer function matrix $\mathbf{G}_{f}$ from actuator forces to measured forces in equation\nbsp{}eqref:eq:rotating_Gf_mimo_tf can be obtained by inserting equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k into equation\nbsp{}eqref:eq:rotating_measured_force.
The transfer function matrix $\bm{G}_{f}$ from actuator forces to measured forces in equation\nbsp{}eqref:eq:rotating_Gf_mimo_tf can be obtained by inserting equation\nbsp{}eqref:eq:rotating_Gd_w0_xi_k into equation\nbsp{}eqref:eq:rotating_measured_force.
Its elements are shown in equation\nbsp{}eqref:eq:rotating_Gf.
\begin{equation}\label{eq:rotating_Gf_mimo_tf}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \mathbf{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix}
\begin{bmatrix} f_{u} \\ f_{v} \end{bmatrix} = \bm{G}_{f} \begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
\begin{subequations}\label{eq:rotating_Gf}
\begin{align}
\mathbf{G}_{f}(1,1) &= \mathbf{G}_{f}(2,2) = \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \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} \label{eq:rotating_Gf_diag_tf} \\
\mathbf{G}_{f}(1,2) &= -\mathbf{G}_{f}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \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} \label{eq:rotating_Gf_off_diag_tf}
\bm{G}_{f}(1,1) &= \bm{G}_{f}(2,2) = \frac{\left( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} \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} \label{eq:rotating_Gf_diag_tf} \\
\bm{G}_{f}(1,2) &= -\bm{G}_{f}(2,1) = \frac{- \left( 2 \xi \frac{s}{\omega_0} + 1 \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} \label{eq:rotating_Gf_off_diag_tf}
\end{align}
\end{subequations}
The zeros of the diagonal terms of $\mathbf{G}_f$ in equation\nbsp{}eqref:eq:rotating_Gf_diag_tf are computed, and neglecting the damping for simplicity, two complex conjugated zeros $z_{c}$ eqref:eq:rotating_iff_zero_cc, and two real zeros $z_{r}$ eqref:eq:rotating_iff_zero_real are obtained.
The zeros of the diagonal terms of $\bm{G}_f$ in equation\nbsp{}eqref:eq:rotating_Gf_diag_tf are computed, and neglecting the damping for simplicity, two complex conjugated zeros $z_{c}$ eqref:eq:rotating_iff_zero_cc, and two real zeros $z_{r}$ eqref:eq:rotating_iff_zero_real are obtained.
\begin{subequations}
\begin{align}
@ -2397,12 +2397,12 @@ This is what usually gives the unconditional stability of IFF when collocated fo
However, for non-null rotational speeds, the two real zeros $z_r$ in equation\nbsp{}eqref:eq:rotating_iff_zero_real are inducing a /non-minimum phase behavior/.
This can be seen in the Bode plot of the diagonal terms (Figure\nbsp{}ref:fig:rotating_iff_bode_plot_effect_rot) where the low-frequency gain is no longer zero while the phase stays at $\SI{180}{\degree}$.
The low-frequency gain of $\mathbf{G}_f$ increases with the rotational speed $\Omega$ as shown in equation\nbsp{}eqref:eq:rotating_low_freq_gain_iff_plan.
The low-frequency gain of $\bm{G}_f$ increases with the rotational speed $\Omega$ as shown in equation\nbsp{}eqref:eq:rotating_low_freq_gain_iff_plan.
This can be explained as follows: a constant actuator force $F_u$ induces a small displacement of the mass $d_u = \frac{F_u}{k - m\Omega^2}$ (Hooke's law considering the negative stiffness induced by the rotation).
This small displacement then increases the centrifugal force $m\Omega^2d_u = \frac{\Omega^2}{{\omega_0}^2 - \Omega^2} F_u$ which is then measured by the force sensors.
\begin{equation}\label{eq:rotating_low_freq_gain_iff_plan}
\lim_{\omega \to 0} \left| \mathbf{G}_f (j\omega) \right| = \begin{bmatrix}
\lim_{\omega \to 0} \left| \bm{G}_f (j\omega) \right| = \begin{bmatrix}
\frac{\Omega^2}{{\omega_0}^2 - \Omega^2} & 0 \\
0 & \frac{\Omega^2}{{\omega_0}^2 - \Omega^2}
\end{bmatrix}
@ -2440,7 +2440,7 @@ The decentralized acrshort:iff controller $\bm{K}_F$ corresponds to a diagonal c
\begin{equation} \label{eq:rotating_Kf_pure_int}
\begin{aligned}
\mathbf{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\
\bm{K}_{F}(s) &= \begin{bmatrix} K_{F}(s) & 0 \\ 0 & K_{F}(s) \end{bmatrix} \\
K_{F}(s) &= g \cdot \frac{1}{s}
\end{aligned}
\end{equation}
@ -2579,23 +2579,23 @@ To keep the overall stiffness $k = k_a + k_p$ constant, thus not modifying the o
k_p = \alpha k, \quad k_a = (1 - \alpha) k
\end{equation}
After the equations of motion are derived and transformed in the Laplace domain, the transfer function matrix $\mathbf{G}_k$ in Eq.\nbsp{}eqref:eq:rotating_Gk_mimo_tf is computed.
After the equations of motion are derived and transformed in the Laplace domain, the transfer function matrix $\bm{G}_k$ in Eq.\nbsp{}eqref:eq:rotating_Gk_mimo_tf is computed.
Its elements are shown in Eqs.\nbsp{}eqref:eq:rotating_Gk_diag and eqref:eq:rotating_Gk_off_diag.
\begin{equation}\label{eq:rotating_Gk_mimo_tf}
\begin{bmatrix} f_u \\ f_v \end{bmatrix} =
\mathbf{G}_k
\bm{G}_k
\begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
\begin{subequations}\label{eq:rotating_Gk}
\begin{align}
\mathbf{G}_{k}(1,1) &= \mathbf{G}_{k}(2,2) = \frac{\big( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \big) \big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big) + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}{\big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big)^2 + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2} \label{eq:rotating_Gk_diag} \\
\mathbf{G}_{k}(1,2) &= -\mathbf{G}_{k}(2,1) = \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} \label{eq:rotating_Gk_off_diag}
\bm{G}_{k}(1,1) &= \bm{G}_{k}(2,2) = \frac{\big( \frac{s^2}{{\omega_0}^2} - \frac{\Omega^2}{{\omega_0}^2} + \alpha \big) \big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big) + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2}{\big( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \big)^2 + \big( 2 \frac{\Omega}{\omega_0} \frac{s}{\omega_0} \big)^2} \label{eq:rotating_Gk_diag} \\
\bm{G}_{k}(1,2) &= -\bm{G}_{k}(2,1) = \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} \label{eq:rotating_Gk_off_diag}
\end{align}
\end{subequations}
Comparing $\mathbf{G}_k$ in\nbsp{}eqref:eq:rotating_Gk with $\mathbf{G}_f$ in\nbsp{}eqref:eq:rotating_Gf shows that while the poles of the system remain the same, the zeros of the diagonal terms change.
Comparing $\bm{G}_k$ in\nbsp{}eqref:eq:rotating_Gk with $\bm{G}_f$ in\nbsp{}eqref:eq:rotating_Gf shows that while the poles of the system remain the same, the zeros of the diagonal terms change.
The two real zeros $z_r$ in\nbsp{}eqref:eq:rotating_iff_zero_real that were inducing a non-minimum phase behavior are transformed into two complex conjugate zeros if the condition in\nbsp{}eqref:eq:rotating_kp_cond_cc_zeros holds.
Thus, if the added /parallel stiffness/ $k_p$ is higher than the /negative stiffness/ induced by centrifugal forces $m \Omega^2$, the dynamics from the actuator to its collocated force sensor will show /minimum phase behavior/.
@ -2716,13 +2716,13 @@ Let's note $\bm{G}_d$ the transfer function between actuator forces and measured
The elements of $\bm{G}_d$ were derived in Section\nbsp{}ref:sec:rotating_system_description are shown in\nbsp{}eqref:eq:rotating_rdc_plant_elements.
\begin{equation}\label{eq:rotating_rdc_plant_matrix}
\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \mathbf{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
\begin{bmatrix} d_u \\ d_v \end{bmatrix} = \bm{G}_d \begin{bmatrix} F_u \\ F_v \end{bmatrix}
\end{equation}
\begin{subequations}\label{eq:rotating_rdc_plant_elements}
\begin{align}
\mathbf{G}_{d}(1,1) &= \mathbf{G}_{d}(2,2) = \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \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} \\
\mathbf{G}_{d}(1,2) &= -\mathbf{G}_{d}(2,1) = \frac{\frac{1}{k} \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}
\bm{G}_{d}(1,1) &= \bm{G}_{d}(2,2) = \frac{\frac{1}{k} \left( \frac{s^2}{{\omega_0}^2} + 2 \xi \frac{s}{\omega_0} + 1 - \frac{{\Omega}^2}{{\omega_0}^2} \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} \\
\bm{G}_{d}(1,2) &= -\bm{G}_{d}(2,1) = \frac{\frac{1}{k} \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}
\end{align}
\end{subequations}
@ -3265,10 +3265,10 @@ If these local feedback controls were turned OFF, this would have resulted in ve
The top part representing the active stabilization stage was disassembled as the active stabilization stage will be added in the multi-body model afterwards.
To perform the modal analysis from the measured responses, the $n \times n$ frequency response function matrix $\mathbf{H}$ needs to be measured, where $n$ is the considered number of degrees of freedom.
To perform the modal analysis from the measured responses, the $n \times n$ frequency response function matrix $\bm{H}$ needs to be measured, where $n$ is the considered number of degrees of freedom.
The $H_{jk}$ element of this acrfull:frf matrix corresponds to the frequency response function from a force $F_k$ applied at acrfull:dof $k$ to the displacement of the structure $X_j$ at acrshort:dof $j$.
Measuring this acrshort:frf matrix is time consuming as it requires to make $n \times n$ measurements.
However, due to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /point measurement/ ($H_{jj}$), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix $\mathbf{H}$ [[cite:&ewins00_modal chapt. 5.2]].
However, due to the principle of reciprocity ($H_{jk} = H_{kj}$) and using the /point measurement/ ($H_{jj}$), it is possible to reconstruct the full matrix by measuring only one column or one line of the matrix $\bm{H}$ [[cite:&ewins00_modal chapt. 5.2]].
Therefore, a minimum set of $n$ frequency response functions is required.
This can be done either by measuring the response $X_{j}$ at a fixed acrshort:dof $j$ while applying forces $F_{i}$ at all $n$ considered acrshort:dof, or by applying a force $F_{k}$ at a fixed acrshort:dof $k$ and measuring the response $X_{i}$ for all $n$ acrshort:dof.
@ -3441,7 +3441,7 @@ After all measurements are conducted, a $n \times p \times q$ acrlongpl:frf matr
For each frequency point $\omega_{i}$, a 2D complex matrix is obtained that links the 3 force inputs to the 69 output accelerations\nbsp{}eqref:eq:modal_frf_matrix_raw.
\begin{equation}\label{eq:modal_frf_matrix_raw}
\mathbf{H}(\omega_i) = \begin{bmatrix}
\bm{H}(\omega_i) = \begin{bmatrix}
\frac{D_{1_x}}{F_x}(\omega_i) & \frac{D_{1_x}}{F_y}(\omega_i) & \frac{D_{1_x}}{F_z}(\omega_i) \\
\frac{D_{1_y}}{F_x}(\omega_i) & \frac{D_{1_y}}{F_y}(\omega_i) & \frac{D_{1_y}}{F_z}(\omega_i) \\
\frac{D_{1_z}}{F_x}(\omega_i) & \frac{D_{1_z}}{F_y}(\omega_i) & \frac{D_{1_z}}{F_z}(\omega_i) \\
@ -3526,10 +3526,10 @@ The position of each accelerometer with respect to the center of mass of the cor
| Spindle | $0$ | $0$ | $-580\,\text{mm}$ |
| Hexapod | $-4\,\text{mm}$ | $6\,\text{mm}$ | $-319\,\text{mm}$ |
Using\nbsp{}eqref:eq:modal_cart_to_acc, the frequency response matrix $\mathbf{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\mathbf{H}$.
Using\nbsp{}eqref:eq:modal_cart_to_acc, the frequency response matrix $\bm{H}_\text{CoM}$ eqref:eq:modal_frf_matrix_com expressing the response at the center of mass of each solid body $D_i$ ($i$ from $1$ to $6$ for the $6$ considered solid bodies) can be computed from the initial acrshort:frf matrix $\bm{H}$.
\begin{equation}\label{eq:modal_frf_matrix_com}
\mathbf{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
\bm{H}_\text{CoM}(\omega_i) = \begin{bmatrix}
\frac{D_{1,T_x}}{F_x}(\omega_i) & \frac{D_{1,T_x}}{F_y}(\omega_i) & \frac{D_{1,T_x}}{F_z}(\omega_i) \\
\frac{D_{1,T_y}}{F_x}(\omega_i) & \frac{D_{1,T_y}}{F_y}(\omega_i) & \frac{D_{1,T_y}}{F_z}(\omega_i) \\
\frac{D_{1,T_z}}{F_x}(\omega_i) & \frac{D_{1,T_z}}{F_y}(\omega_i) & \frac{D_{1,T_z}}{F_z}(\omega_i) \\
@ -3545,8 +3545,8 @@ Using\nbsp{}eqref:eq:modal_cart_to_acc, the frequency response matrix $\mathbf{H
**** Verification of solid body assumption
<<ssec:modal_solid_body_assumption>>
From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\mathbf{H}_{\text{CoM}}$), and using equation\nbsp{}eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered location.
In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements $\mathbf{H}$.
From the response of one solid body expressed by its 6 acrshortpl:dof (i.e. from $\bm{H}_{\text{CoM}}$), and using equation\nbsp{}eqref:eq:modal_cart_to_acc, it is possible to compute the response of the same solid body at any considered location.
In particular, the responses at the locations of the four accelerometers can be computed and compared with the original measurements $\bm{H}$.
This is what is done here to check whether the solid body assumption is correct in the frequency band of interest.
The comparison is made for the 4 accelerometers fixed on the micro-hexapod (Figure\nbsp{}ref:fig:modal_comp_acc_solid_body_frf).
@ -3690,27 +3690,27 @@ The eigenvalues $s_r$ and $s_r^*$ can then be computed from equation\nbsp{}eqref
**** Verification of the modal model validity
<<ssec:modal_model_validity>>
To check the validity of the modal model, the complete $n \times n$ acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ is first synthesized from the modal parameters.
Then, the elements of this acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ that were already measured can be compared to the measured acrshort:frf matrix $\mathbf{H}$.
To check the validity of the modal model, the complete $n \times n$ acrshort:frf matrix $\bm{H}_{\text{syn}}$ is first synthesized from the modal parameters.
Then, the elements of this acrshort:frf matrix $\bm{H}_{\text{syn}}$ that were already measured can be compared to the measured acrshort:frf matrix $\bm{H}$.
In order to synthesize the full acrshort:frf matrix, the eigenvectors $\phi_r$ are first organized in matrix from as shown in equation\nbsp{}eqref:eq:modal_eigvector_matrix.
\begin{equation}\label{eq:modal_eigvector_matrix}
\Phi = \begin{bmatrix}
\bm{\Phi} = \begin{bmatrix}
& & & & &\\
\phi_1 & \dots & \phi_N & \phi_1^* & \dots & \phi_N^* \\
& & & & &
\end{bmatrix}_{n \times 2m}
\end{equation}
The full acrshort:frf matrix $\mathbf{H}_{\text{syn}}$ can be obtained using\nbsp{}eqref:eq:modal_synthesized_frf.
The full acrshort:frf matrix $\bm{H}_{\text{syn}}$ can be obtained using\nbsp{}eqref:eq:modal_synthesized_frf.
\begin{equation}\label{eq:modal_synthesized_frf}
[\mathbf{H}_{\text{syn}}(\omega)]_{n\times n} = [\Phi]_{n\times2m} [\mathbf{H}_{\text{mod}}(\omega)]_{2m\times2m} [\Phi]_{2m\times n}^{\intercal}
\left[\bm{H}_{\text{syn}}(\omega)\right]_{n\times n} = \left[\bm{\Phi}\right]_{n\times2m} \left[\bm{H}_{\text{mod}}(\omega)\right]_{2m\times2m} \left[\bm{\Phi}\right]_{2m\times n}^{\intercal}
\end{equation}
With $\mathbf{H}_{\text{mod}}(\omega)$ a diagonal matrix representing the response of the different modes\nbsp{}eqref:eq:modal_modal_resp.
With $\bm{H}_{\text{mod}}(\omega)$ a diagonal matrix representing the response of the different modes\nbsp{}eqref:eq:modal_modal_resp.
\begin{equation}\label{eq:modal_modal_resp}
\mathbf{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m}
\bm{H}_{\text{mod}}(\omega) = \text{diag}\left(\frac{1}{a_1 (j\omega - s_1)},\ \dots,\ \frac{1}{a_m (j\omega - s_m)}, \frac{1}{a_1^* (j\omega - s_1^*)},\ \dots,\ \frac{1}{a_m^* (j\omega - s_m^*)} \right)_{2m\times 2m}
\end{equation}
A comparison between original measured frequency response functions and synthesized ones from the modal model is presented in Figure\nbsp{}ref:fig:modal_comp_acc_frf_modal.
@ -6454,7 +6454,7 @@ However, theoretical frameworks for evaluating flexible joint contribution to th
\bm{K} = \bm{J}^{\intercal} \bm{\mathcal{K}} \bm{J}
\end{equation}
It is assumed that the stiffness of all struts is the same: $\bm{\mathcal{K}} = k \cdot \mathbf{I}_6$.
It is assumed that the stiffness of all struts is the same: $\bm{\mathcal{K}} = k \cdot \bm{I}_6$.
In that case, the obtained stiffness matrix linearly depends on the strut stiffness $k$, and is structured as shown in equation\nbsp{}eqref:eq:detail_kinematics_stiffness_matrix_simplified.
\begin{equation}\label{eq:detail_kinematics_stiffness_matrix_simplified}
@ -12107,11 +12107,11 @@ A modal analysis of the nano-hexapod is first performed in Section\nbsp{}ref:sse
The results of the modal analysis will be useful to better understand the measured dynamics from actuators to sensors.
A block diagram of the (open-loop) system is shown in Figure\nbsp{}ref:fig:test_nhexa_nano_hexapod_signals.
The frequency response functions from controlled signals $\mathbf{u}$ to the force sensors voltages $\mathbf{V}_s$ and to the encoders measured displacements $\mathbf{d}_e$ are experimentally identified in Section\nbsp{}ref:ssec:test_nhexa_identification.
The frequency response functions from controlled signals $\bm{u}$ to the force sensors voltages $\bm{V}_s$ and to the encoders measured displacements $\bm{d}_e$ are experimentally identified in Section\nbsp{}ref:ssec:test_nhexa_identification.
The effect of the payload mass on the dynamics is discussed in Section\nbsp{}ref:ssec:test_nhexa_added_mass.
#+name: fig:test_nhexa_nano_hexapod_signals
#+caption: Block diagram of the studied system. The command signal generated by the speedgoat is $\mathbf{u}$, and the measured dignals are $\mathbf{d}_{e}$ and $\mathbf{V}_s$. Units are indicated in square brackets.
#+caption: Block diagram of the studied system. The command signal generated by the speedgoat is $\bm{u}$, and the measured dignals are $\bm{d}_{e}$ and $\bm{V}_s$. Units are indicated in square brackets.
#+attr_latex: :width 0.9\linewidth
[[file:figs/test_nhexa_nano_hexapod_signals.png]]
@ -12168,7 +12168,7 @@ These modes are summarized in Table\nbsp{}ref:tab:test_nhexa_hexa_modal_modes_li
The dynamics of the nano-hexapod from the six command signals ($u_1$ to $u_6$) to the six measured displacement by the encoders ($d_{e1}$ to $d_{e6}$) and to the six force sensors ($V_{s1}$ to $V_{s6}$) were identified by generating low-pass filtered white noise for each command signal, one by one.
The $6 \times 6$ FRF matrix from $\mathbf{u}$ ot $\mathbf{d}_e$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_de.
The $6 \times 6$ FRF matrix from $\bm{u}$ ot $\bm{d}_e$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_de.
The diagonal terms are displayed using colored lines, and all the 30 off-diagonal terms are displayed by gray lines.
All six diagonal terms are well superimposed up to at least $1\,kHz$, indicating good manufacturing and mounting uniformity.
@ -12185,17 +12185,17 @@ Up to at least 1kHz, an alternating pole/zero pattern is observed, which makes t
This would not have occurred if the encoders were fixed to the struts.
#+name: fig:test_nhexa_identified_frf_de
#+caption: Measured FRF for the transfer function from $\mathbf{u}$ to $\mathbf{d}_e$. The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the gray lines.
#+caption: Measured FRF for the transfer function from $\bm{u}$ to $\bm{d}_e$. The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the gray lines.
#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_identified_frf_de.png]]
Similarly, the $6 \times 6$ FRF matrix from $\mathbf{u}$ to $\mathbf{V}_s$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_Vs.
Similarly, the $6 \times 6$ FRF matrix from $\bm{u}$ to $\bm{V}_s$ is shown in Figure\nbsp{}ref:fig:test_nhexa_identified_frf_Vs.
Alternating poles and zeros can be observed up to at least 2kHz, which is a necessary characteristics for applying decentralized IFF.
Similar to what was observed for the encoder outputs, all the "diagonal" terms are well superimposed, indicating that the same controller can be applied to all the struts.
The first flexible mode of the struts as 235Hz has large amplitude, and therefore, it should be possible to add some damping to this mode using IFF.
#+name: fig:test_nhexa_identified_frf_Vs
#+caption: Measured FRF for the transfer function from $\mathbf{u}$ to $\mathbf{V}_s$. The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.
#+caption: Measured FRF for the transfer function from $\bm{u}$ to $\bm{V}_s$. The 6 diagonal terms are the colored lines (all superimposed), and the 30 off-diagonal terms are the shaded black lines.
#+attr_latex: :scale 0.8
[[file:figs/test_nhexa_identified_frf_Vs.png]]
@ -12271,7 +12271,7 @@ This is checked in Section\nbsp{}ref:ssec:test_nhexa_comp_model_masses.
The multi-body model of the nano-hexapod was first configured with 4-DoF flexible joints, 2-DoF APA, and rigid top and bottom plates.
The stiffness values of the flexible joints were chosen based on the values estimated using the test bench and on the FEM.
The parameters of the APA model were determined from the test bench of the APA.
The $6 \times 6$ transfer function matrices from $\mathbf{u}$ to $\mathbf{d}_e$ and from $\mathbf{u}$ to $\mathbf{V}_s$ are then extracted from the multi-body model.
The $6 \times 6$ transfer function matrices from $\bm{u}$ to $\bm{d}_e$ and from $\bm{u}$ to $\bm{V}_s$ are then extracted from the multi-body model.
First, is it evaluated how well the models matches the "direct" terms of the measured FRF matrix.
To do so, the diagonal terms of the extracted transfer function matrices are compared with the measured FRF in Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_diag.
@ -12301,9 +12301,9 @@ At higher frequencies, no resonances can be observed in the model, as the top pl
<<ssec:test_nhexa_comp_model_coupling>>
Another desired feature of the model is that it effectively represents coupling in the system, as this is often the limiting factor for the control of MIMO systems.
Instead of comparing the full 36 elements of the $6 \times 6$ FFR matrix from $\mathbf{u}$ to $\mathbf{d}_e$, only the first "column" is compared (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all), which corresponds to the transfer function from the command $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$.
Instead of comparing the full 36 elements of the $6 \times 6$ FFR matrix from $\bm{u}$ to $\bm{d}_e$, only the first "column" is compared (Figure\nbsp{}ref:fig:test_nhexa_comp_simscape_de_all), which corresponds to the transfer function from the command $u_1$ to the six measured encoder displacements $d_{e1}$ to $d_{e6}$.
It can be seen that the coupling in the model matches the measurements well up to the first un-modeled flexible mode at 237Hz.
Similar results are observed for all other coupling terms and for the transfer function from $\mathbf{u}$ to $\mathbf{V}_s$.
Similar results are observed for all other coupling terms and for the transfer function from $\bm{u}$ to $\bm{V}_s$.
#+name: fig:test_nhexa_comp_simscape_de_all
#+caption: Comparison of the measured (in blue) and modeled (in red) frequency transfer functions from the first control signal $u_1$ to the six encoders $d_{e1}$ to $d_{e6}$. The APA are here modeled with a 2-DoF mass-spring-damper system.
@ -12380,7 +12380,7 @@ The frequency response functions from the six DAC voltages $\bm{u}$ to the six f
This indicates that it is possible to implement decentralized Integral Force Feedback in a robust manner.
The developed multi-body model of the nano-hexapod was found to accurately represents the suspension modes of the Nano-Hexapod (Section\nbsp{}ref:sec:test_nhexa_model).
Both FRF matrices from $\mathbf{u}$ to $\mathbf{V}_s$ and from $\mathbf{u}$ to $\mathbf{d}_e$ are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective.
Both FRF matrices from $\bm{u}$ to $\bm{V}_s$ and from $\bm{u}$ to $\bm{d}_e$ are well matching with the measurements, even when considering coupling (i.e. off-diagonal) terms, which are very important from a control perspective.
At frequencies above the suspension modes, the Nano-Hexapod model became inaccurate because the flexible modes were not modeled.
It was found that modeling the APA300ML using a /super-element/ allows to model the internal resonances of the struts.
The same can be done with the top platform and the encoder supports; however, the model order would be higher and may become unpractical for simulation.
@ -12755,7 +12755,7 @@ Therefore, the model can be used for model-based control if necessary.
It is interesting to note that the anti-resonances in the force sensor plant now appear as minimum-phase, as the model predicts (Figure\nbsp{}ref:fig:test_id31_comp_simscape_iff_diag_masses).
#+name: fig:test_id31_picture_masses
#+caption: The four tested payload conditions. (\subref{fig:test_id31_picture_mass_m0}) without payload. (\subref{fig:test_id31_picture_mass_m1}) with $13\,\text{kg}$ payload. (\subref{fig:test_id31_picture_mass_m2}) with $26\,\text{kg}$ payload. (\subref{fig:test_id31_picture_mass_m3}) with $39\,\text{kg}$ payload.
#+caption: The four tested payload conditions: (\subref{fig:test_id31_picture_mass_m0}) no payload, (\subref{fig:test_id31_picture_mass_m1}) $13\,\text{kg}$ payload, (\subref{fig:test_id31_picture_mass_m2}) $26\,\text{kg}$ payload, (\subref{fig:test_id31_picture_mass_m3}) $39\,\text{kg}$ payload.
#+attr_latex: :options [htbp]
#+begin_figure
#+attr_latex: :caption \subcaption{\label{fig:test_id31_picture_mass_m0}$m=0\,\text{kg}$}