NASS - Effect of rotation

To obtain the equations of motion for the system represented in Figure fig:rotating_3dof_model_schematic, the Lagrangian equations are used:

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 \(\begin{bmatrix}q_1 & q_2\end{bmatrix} = \begin{bmatrix}d_u & d_v\end{bmatrix}\). The equation of motion corresponding to the constant rotation along \(\vec{i}_w\) is disregarded as this motion is considered to be imposed by the rotation stage.

Substituting equations eq:energy_functions_lagrange into equation eq:lagrangian_equations for both generalized coordinates gives two coupled differential equations eq:eom_coupled_1 and eq:eom_coupled_2.

The uniform rotation of the system induces two gyroscopic effects as shown in equation eq:eom_coupled:

• Centrifugal forces: that can been seen as an added negative stiffness \(- m \Omega^2\) along \(\vec{i}_u\) and \(\vec{i}_v\)
• 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 differential equations of motions eq: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 eq:Gd_mimo_tf is obtained. Its elements are shown in equation eq:Gd_indiv_el.

To simplify the analysis, the undamped natural frequency \(\omega_0\) and the damping ratio \(\xi\) are used as in equation eq:xi_and_omega.

The elements of transfer function matrix \(\mathbf{G}_d\) are now described by equation eq:Gd_w0_xi_k.

The poles of \(\mathbf{G}_d\) are the complex solutions \(p\) of equation eq:poles.

Supposing small damping (\(\xi \ll 1\)), two pairs of complex conjugate poles are obtained as shown in equation eq:pole_values.

The real and complex parts of these two pairs of complex conjugate poles are represented in Figure fig:rotating_campbell_diagram as a function of the rotational speed \(\Omega\). As the rotational speed increases, \(p_{+}\) goes to higher frequencies and \(p_{-}\) goes to lower frequencies. The system becomes unstable for \(\Omega > \omega_0\) as the real part of \(p_{-}\) is positive. Physically, the negative stiffness term \(-m\Omega^2\) induced by centrifugal forces exceeds the spring stiffness \(k\).

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 eq: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 plot of these two terms are shown in Figure fig:rotating_direct_coupling_bode_plot for several rotational speeds \(\Omega\). These plots confirm the expected behavior: the frequency 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.

In order to apply Integral Force Feedback, two force sensors are added in series with the actuators (Figure fig:rotating_3dof_model_schematic_iff). Two identical controllers \(K_F\) are then used to feedback each of the sensed force to its associated actuator:

The forces \(\begin{bmatrix}f_u & f_v\end{bmatrix}\) measured by the two force sensors represented in Figure fig:rotating_3dof_model_schematic_iff are described by equation eq:measured_force.

The transfer function matrix \(\mathbf{G}_{f}\) from actuator forces to measured forces in equation eq:Gf_mimo_tf can be obtained by inserting equation eq:Gd_w0_xi_k into equation eq:measured_force. Its elements are shown in equation eq:Gf_indiv_el.

The zeros of the diagonal terms of \(\mathbf{G}_f\) in equation eq:Gf_diag_tf are computed, and neglecting the damping for simplicity, two complex conjugated zeros \(z_{c}\) are obtained in equation eq:iff_zero_cc, and two real zeros \(z_{r}\) in equation eq:iff_zero_real.

It is interesting to see that the frequency of the pair of complex conjugate zeros \(z_c\) in equation eq:iff_zero_cc always lies between the frequency of the two pairs of complex conjugate poles \(p_{-}\) and \(p_{+}\) in equation eq:pole_values. This is what usually gives the unconditional stability of IFF when collocated force sensors are used.

However, for non-null rotational speeds, the two real zeros \(z_r\) in equation eq:iff_zero_real are inducing a non-minimum phase behavior. This can be seen in the Bode plot of the diagonal terms (Figure 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 eq: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 taking into account 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.

The transfer functions from actuator forces \([F_u,\ F_v]\) to the measured force sensors \([f_u,\ f_v]\) are identified for several rotating velocities and shown in Figure fig:rotating_iff_bode_plot_effect_rot.

As was expected from the derived equations of motion:

• when \(0 < \Omega < \omega_0\): the low frequency gain is no longer zero and two (non-minimum phase) real zero appears at low frequency. The low frequency gain increases with \(\Omega\). A pair of (minimum phase) complex conjugate zeros appears between the two complex conjugate poles that are split further apart as \(\Omega\) increases.
• when \(\omega_0 < \Omega\): the low frequency pole becomes unstable.

The control diagram for decentralized Integral Force Feedback is shown in Figure fig:rotating_iff_diagram.

The decentralized IFF controller \(\bm{K}_F\) corresponds to a diagonal controller with integrators:

In order to see how the IFF controller affects the poles of the closed loop system, a Root Locus plot (Figure fig:rotating_root_locus_iff_pure_int) 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. As explained in (Preumont, De Marneffe, and Krenk 2008; Skogestad and Postlethwaite 2007), 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 \(g \to \infty\).

Physically, this can be explained like so: at low frequency, the loop gain is very large due to the pure integrator in \(K_{F}\) and the finite gain of the plant (Figure fig:rotating_iff_bode_plot_effect_rot). 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.

The Integral Force Feedback Controller is modified such that instead of using pure integrators, pseudo integrators (i.e. low pass filters) are used:

where \(\omega_i\) characterize down to which frequency the signal is integrated.

Let’s arbitrary choose the following control parameters:

%% Modified IFF - parameters
g = 2; % Controller gain
wi = 0.1; % HPF Cut-Off frequency [rad/s]

Kiff     = (g/s)*eye(2); % Pure IFF
Kiff_hpf = (g/(wi+s))*eye(2); % IFF with added HPF

The loop gains (\(K_F(s)\) times the direct dynamics \(f_u/F_u\)) with and without the added HPF are shown in Figure fig:rotating_iff_modified_loop_gain. The effect of the added HPF limits the low frequency gain to finite values as expected.

The Root Locus plots for the decentralized IFF with and without the HPF are displayed in Figure fig:rotating_iff_root_locus_hpf. With the added HPF, the poles of the closed loop system are shown to be stable up to some value of the gain \(g_\text{max}\) given by equation eq:gmax_iff_hpf.

It is interesting to note that \(g_{\text{max}}\) also corresponds to the controller gain at which the low frequency loop gain (Figure fig:rotating_iff_modified_loop_gain) reaches one.

Two parameters can be tuned for the modified controller in equation eq:iff_lhf: the gain \(g\) and the pole’s location \(\omega_i\). 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.

In order to visualize how \(\omega_i\) does affect the attainable damping, the Root Locus plots for several \(\omega_i\) are displayed in Figure fig:rotating_root_locus_iff_modified_effect_wi. 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 equation eq:gmax_iff_hpf.

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 fig:rotating_iff_hpf_optimal_gain).

Three regions can be observed:

• \(\omega_i/\omega_0 < 0.02\): the added damping is limited by the maximum allowed control gain \(g_{\text{max}}\)
• \(0.02 < \omega_i/\omega_0 < 0.2\): the attainable damping ratio is maximized and is reached for \(g \approx 2\)
• \(0.2 < \omega_i/\omega_0\): the added damping decreases as \(\omega_i/\omega_0\) increases.

Let’s choose \(\omega_i = 0.1 \cdot \omega_0\) and compute the damped plant. The undamped and damped plants are compared in Figure fig:rotating_iff_hpf_damped_plant in blue and red respectively. A well damped plant is indeed obtained.

However, the magnitude of the coupling term (\(d_v/F_u\)) is larger then IFF is applied.

In order to study how \(\omega_i\) affects the coupling of the damped plant, the closed-loop plant is identified for several \(\omega_i\). The direct and coupling terms of the plants are shown in Figure fig:rotating_iff_hpf_damped_plant_effect_wi_coupling (left) and the ratio between the two (i.e. the coupling ratio) is shown in Figure fig:rotating_iff_hpf_damped_plant_effect_wi_coupling (right).

The coupling ratio is decreasing as \(\omega_i\) increases. There is therefore a trade-off between achievable damping and coupling ratio for the choice of \(\omega_i\). The same trade-off can be seen between achievable damping and loss of compliance at low frequency (see Figure fig:rotating_iff_hpf_effect_wi_compliance).

The forces measured by the two force sensors represented in Figure fig:rotating_3dof_model_schematic_iff_parallel_springs are described by Eq. eq:measured_force_kp.

In order to keep the overall stiffness \(k = k_a + k_p\) constant, thus not modifying the open-loop poles as \(k_p\) is changed, a scalar parameter \(\alpha\) (\(0 \le \alpha < 1\)) is defined to describe the fraction of the total stiffness in parallel with the actuator and force sensor as in Eq. eq:kp_alpha.

After the equations of motion derived and transformed in the Laplace domain, the transfer function matrix \(\mathbf{G}_k\) in Eq. eq:Gk_mimo_tf is computed. Its elements are shown in Eq. eq:Gk_diag and eq:Gk_off_diag.

Comparing \(\mathbf{G}_k\) in Eq. eq:Gk with \(\mathbf{G}_f\) in Eq. 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 \(z_r\) in Eq. eq:iff_zero_real that were inducing a non-minimum phase behavior are transformed into two complex conjugate zeros if the condition in Eq. eq:kp_cond_cc_zeros holds.

The IFF plant (transfer function from \([F_u, F_v]\) to \([f_u, f_v]\)) is identified in three different cases:

• without parallel stiffness \(k_p = 0\)
• with a small parallel stiffness \(k_p < m \Omega^2\)
• with a large parallel stiffness \(k_p > m \Omega^2\)

The Bode plots of the obtained dynamics are shown in Figure fig:rotating_iff_effect_kp. One can see that for \(k_p > m \Omega^2\), the two real zeros with \(k_p < m \Omega^2\) are transformed into two complex conjugate zeros and the systems shows alternating complex conjugate poles and zeros.

Figure fig:rotating_iff_kp_root_locus shows the Root Locus plots for \(k_p = 0\), \(k_p < m \Omega^2\) and \(k_p > m \Omega^2\) when \(K_F\) is a pure integrator as in Eq. 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 are bounded on the (stable) left half-plane, and hence the unconditional stability of IFF is recovered.

Even though the parallel stiffness \(k_p\) has no impact on the open-loop poles (as the overall stiffness \(k\) is kept constant), it has a large impact on the transmission zeros. Moreover, as the attainable damping is generally proportional to the distance between poles and zeros (Preumont 2018), the parallel stiffness \(k_p\) is foreseen to have some impact on the attainable damping.

To study this effect, Root Locus plots for several parallel stiffnesses \(k_p > m \Omega^2\) are shown in Figure fig:rotating_iff_kp_root_locus_effect_kp. The frequencies of the transmission zeros of the system are increasing with an increase of the parallel stiffness \(k_p\) (thus getting closer to the poles) and the associated attainable damping is reduced.

This is confirmed by the Figure fig:rotating_iff_kp_optimal_gain where the attainable closed-loop damping ratio \(\xi_{\text{cl}}\) and the associated optimal control gain \(g_\text{opt}\) are computed as a function of the parallel stiffness.

Let’s choose a parallel stiffness equal to \(k_p = 2 m \Omega^2\) and compute the damped plant. The damped and undamped transfer functions from \(F_u\) to \(d_u\) are compared in Figure fig:rotating_iff_kp_damped_plant.

Even though the two resonances are well damped, the IFF changes the low frequency behavior of the plant which is usually not wanted. This is due to the fact that “pure” integrators are used, and that the low frequency loop gains becomes large below some frequency.

In order to lower the low frequency gain, an high pass filter is added to the IFF controller (which is equivalent as shifting the controller pole to the left in the complex plane):

Let’s see how the high pass filter impacts the attainable damping. The controller gain \(g\) is kept constant while \(\omega_i\) is changed, and the minimum damping ratio of the damped plant is computed. The obtained damping ratio as a function of \(\omega_i/\omega_0\) (where \(\omega_0\) is the resonance of the system without rotation) is shown in Figure fig:rotating_iff_kp_added_hpf_effect_damping.

Let’s choose \(\omega_i = 0.1 \cdot \omega_0\) and compute the damped plant again. The Bode plots of the undamped, damped with “pure” IFF, and with added high pass filters are shown in Figure fig:rotating_iff_kp_added_hpf_damped_plant. The added high pass filter gives almost the same damping properties while giving acceptable low frequency behavior.

Let’s note \(\bm{G}_d\) the transfer function between actuator forces and measured relative motion in parallel with the actuators:

With:

Neglecting the damping for simplicity (\(\xi \ll 1\)), the direct terms have two complex conjugate zeros:

Which are between the two pairs of complex conjugate poles at:

Therefore, for \(\Omega < \sqrt{k/m}\) (i.e. stable system), the transfer functions for Relative Damping Control have alternating complex conjugate poles and zeros.

The transfer functions from \([F_u,\ F_v]\) to \([d_u,\ d_v]\) is identified and shown in Figure fig:rotating_rdc_plant_effect_rot for several rotating velocities.

In order to see if large damping can be added with Relative Damping Control, the root locus is computed (Figure fig:rotating_rdc_root_locus). The closed-loop system is unconditionally stable and the poles can be damped as much as wanted.

Let’s select a reasonable “Relative Damping Control” gain, and compute the closed-loop damped system. The open-loop and damped plants are compared in Figure fig:rotating_rdc_damped_plant.

The rotating aspect does not add any complexity for the use of Relative Damping Control. It does not increase the low frequency coupling as compared to Integral Force Feedback.

Figure fig:rotating_comp_techniques_root_locus shows the Root Locus plots for the two proposed IFF modifications as well as for relative damping control. While the two pairs of complex conjugate open-loop poles are identical for both IFF modifications, the transmission zeros are not. This means that the closed-loop behavior of both systems will differ when large control gains are used.

One can observe that the closed loop poles corresponding to the system with added springs (in red) are bounded to the left half plane implying unconditional stability. This is not the case for the system where the controller is augmented with an HPF (in blue).

It is interesting to note that the maximum added damping is very similar for both techniques.

The actively damped plants are computed for the three techniques and compared in Figure fig:rotating_comp_techniques_dampled_plants.

The proposed active damping techniques are now compared in terms of closed-loop transmissibility and compliance.

The transmissibility is here defined as the transfer function from a displacement of the rotating stage along \(\vec{i}_x\) to the displacement of the payload along the same direction. It is used to characterize how much vibration is transmitted through the suspended platform to the payload.

The compliance describes the displacement response of the payload to external forces applied to it. This is a useful metric when disturbances are directly applied to the payload. It is here defined as the transfer function from external forces applied on the payload along \(\vec{i}_x\) to the displacement of the payload along the same direction.

Very similar results are obtained for the two proposed IFF modifications in terms of transmissibility and compliance (Figure fig:rotating_comp_techniques_transmissibility_compliance).

For the NASS, the maximum rotating velocity is \(\Omega = \SI[parse-numbers=false]{2\pi}{\radian\per\s}\) for a suspended mass on top of the nano-hexapod’s actuators equal to \(m_n + m_s = \SI{16}{\kilo\gram}\). The parallel stiffness corresponding to the centrifugal forces is \(m \Omega^2 \approx \SI{0.6}{\newton\per\mm}\).

The transfer functions from nano-hexapod actuator force \(F_u\) to the displacement of the nano-hexapod in the same direction \(d_u\) as well as in the orthogonal direction \(d_v\) (coupling) are shown in Figure fig:rotating_nano_hexapod_dynamics for all three considered nano-hexapod stiffnesses.

Let’s apply Integral Force Feedback with an added High Pass Filter to the three nano-hexapods.

First, let’s find the parameters of the IFF controller that yield best simultaneous damping. The results are shown in Figure fig:rotating_iff_hpf_nass_optimal_gain. The added damping for the soft nano-hexapod is quite low and is limited by the maximum usable gain.

The IFF parameters are chosen as follow:

• for \(k_n = \SI{0.01}{\N\per\mu\m}\): \(\omega_i\) is chosen such that the maximum damping is achieved while the gain is less than half of the maximum gain at which the system is unstable. This is done to have some control robustness.
• for \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\): the largest \(\omega_i\) is chosen such that obtained damping is \(\SI{95}{\percent}\) of the maximum achievable damping. Large \(\omega_i\) is chosen here to limit the loss of compliance and the increase of coupling at low frequency as was shown in Section sec:rotating_iff_pseudo_int.

The obtained IFF parameters and the achievable damping are summarized in Table tab:iff_hpf_opt_iff_hpf_params_nass.

The Root Locus for all three nano-hexapods are shown in Figure fig:rotating_root_locus_iff_hpf_nass with included optimal chosen gains.

For each considered nano-hexapod stiffness, the parallel stiffness \(k_p\) is varied from \(k_{p,\text{min}} = m\Omega^2\) (the minimum stiffness to have unconditional stability) to \(k_{p,\text{max}} = k_n\) (the total nano-hexapod stiffness). In order to keep the overall stiffness constant, the actuator stiffness \(k_a\) is decreased when \(k_p\) is increased:

With \(k_n\) the total nano-hexapod stiffness.

An high pass filter is also added to limit the low frequency gain. The cut-off frequency \(\omega_i\) is chosen to be one tenth of the system resonance:

The achievable maximum simultaneous damping of all the modes is computed as a function of the parallel stiffnesses. The comparison for the nano-hexapod stiffnesses is done in Figure fig:rotating_iff_kp_nass_optimal_gain. It is shown that the soft nano-hexapod cannot yield good damping. For the two stiff options, the achievable damping starts to significantly decrease when the parallel stiffness is one tenth of the total stiffness \(k_p = k_n/10\).

Let’s choose \(k_p = \SI{e3}{\newton\per\m}\), \(k_p = \SI{e4}{\newton\per\m}\) and \(k_p = \SI{e6}{\newton\per\m}\) for the three considered nano-hexapods respectively based on Figure fig:rotating_iff_kp_nass_optimal_gain.

The corresponding optimal controller gains are shown in Table tab:iff_kp_opt_iff_kp_params_nass.

The root locus for the three nano-hexapod with parallel stiffnesses are shown in Figure fig:rotating_root_locus_iff_kp_nass.

For each considered nano-hexapod stiffness, relative damping control is applied and the achievable damping ratio as a function of the controller gain is shown in Figure fig:rotating_rdc_optimal_gain.

The gain is chosen is chosen such that 99% of modal damping is obtained. The root locus for all three nano-hexapod stiffnesses are shown in Figure fig:rotating_root_locus_rdc_nass.

Let’s now compare the obtained damped plants for the three active damping techniques applied on the three nano-hexapod stiffnesses (Figure fig:rotating_nass_damped_plant_comp).

In order to be a bit closer to the NASS application, the 2DoF nano-hexapod (modelled as shown in Figure fig:rotating_3dof_model_schematic) is now located on top of a model of the micro-station including (see Figure fig:rotating_nass_model for a 3D view):

• the floor whose motion is imposed
• a 2DoF granite (\(k_{g,x} = k_{g,y} = \SI{950}{\N\per\mu\m}\), \(m_g = \SI{2500}{\kg}\))
• a 2DoF \(T_y\) stage (\(k_{t,x} = k_{t,y} = \SI{520}{\N\per\mu\m}\), \(m_g = \SI{600}{\kg}\))
• a spindle (vertical rotation) stage whose rotation is imposed (\(m_s = \SI{600}{\kg}\))
• a 2DoF micro-hexapod (\(k_{h,x} = k_{h,y} = \SI{61}{\N\per\mu\m}\), \(m_g = \SI{15}{\kg}\))

A payload is rigidly fixed to the nano-hexapod and the \(x,y\) motion of the payload is measured with respect to the granite.

The dynamics of the undamped and damped plants are identified. The active damping parameters used are the optimal ones previously identified (i.e. for the rotating nano-hexapod fixed on a rigid platform).

The undamped and damped plants are shown in Figure fig:rotating_nass_plant_comp_stiffness. Three nano-hexapod velocities are shown (from left to right): \(k_n = \SI{0.01}{\N\per\mu\m}\), \(k_n = \SI{1}{\N\per\mu\m}\) and \(k_n = \SI{100}{\N\per\mu\m}\). The direct terms are shown by the solid curves while the coupling terms are shown by the shaded ones.

To confirm that the coupling is smaller when the stiffness of the nano-hexapod is increase, the coupling ratio for the three nano-hexapod stiffnesses are shown in Figure fig:rotating_nass_plant_coupling_comp.

The effect of three disturbances are considered:

• Floor motion (Figure fig:rotating_nass_effect_floor_motion)
• Micro-Station vibrations (Figure fig:rotating_nass_effect_stage_vibration)
• Direct force applied on the payload (Figure fig:rotating_nass_effect_direct_forces)