Design of the Nano-Hexapod and associated Control Architectures - Summary

Let’s consider the block diagram shown in Figure 1 where the signals are:

• \(y\): the relative position of the sample with respect to the granite (the quantity to be controlled)
• \(d\): the disturbances affecting \(y\) (ground motion, stages’ vibrations)
• \(n\): the noise of the sensor measuring \(y\)
• \(r\): the reference signal, corresponding to the wanted \(y\)
• \(\epsilon = r - y\): the position error

The dynamical blocks are:

• \(G\): representing the dynamics from forces/torques applied by the nano-hexapod to the relative position sample/granite \(y\)
• \(G_d\): representing how the disturbances (e.g. ground motion) are affecting the relative position sample/granite \(y\)
• \(K\): representing the controller (to be designed)

Without the use of feedback (i.e. without the nano-hexapod), the disturbances will induce a sample motion error equal to:

\begin{equation} y = G_d d \label{eq:open_loop_error} \end{equation}

which is, in the case of the NASS out of the specifications (micro-meter range compare to the required \(\approx 10nm\)).

In the next section, is explained how the use of the feedback lowers the effect of the disturbances \(d\) on the sample motion error.

From the feedback diagram in Figure 1, the position error signal \(\epsilon = r - y\) can be written as a function of the reference signal \(r\), the disturbances \(d\) and the measurement noise \(n\): \[ \epsilon = \frac{1}{1 + GK} r + \frac{GK}{1 + GK} n - \frac{G_d}{1 + GK} d \]

It is common to define the following two transfer functions:

\begin{align} S &= \frac{1}{1 + GK} \\ T &= \frac{GK}{1 + GK} \end{align}

where \(S\) is called the sensitivity transfer function and \(T\) the transmissibility (or complementary sensitivity) transfer function.

And the position error can be rewritten as:

\begin{equation} \epsilon = S r + T n - G_d S d \label{eq:closed_loop_error} \end{equation}

From Eq. \eqref{eq:closed_loop_error} representing the closed-loop system behavior, it is seen that:

• the effect of disturbances \(d\) on \(\epsilon\) is multiplied by a factor \(S\) compared to the open-loop case
• the measurement noise \(n\) is injected and multiplied by a factor \(T\)

Ideally, it is desired to design the controller \(K\) such that:

• \(|S|\) is small to reduce the effect of disturbances
• \(|T|\) is small to limit the injection of sensor noise

From the definition of \(S\) and \(T\):

\begin{equation} S + T = \frac{1}{1 + GK} + \frac{GK}{1 + GK} = 1 \end{equation}

meaning that it is not possible to have \(|S|\) and \(|T|\) small at the same time.

There is therefore a trade-off between the disturbance rejection and the measurement noise filtering.

Typical shapes of \(|S|\) and \(|T|\) as a function of frequency are shown in Figure 2. It is shown that \(|S|\) and \(|T|\) exhibit different behaviors depending on the frequency band:

• At low frequency (inside the control bandwidth):
  • \(|S|\) can be made small and thus the effect of disturbances is reduced
  • \(|T| \approx 1\) and all the sensor noise is transmitted
• At high frequency (outside the control bandwidth):
  • \(|S| \approx 1\) and the feedback system does not reduce the effect of disturbances
  • \(|T|\) is small and thus the sensor noise is filtered
• Near the crossover frequency (between the two frequency bands):
  • The effect of disturbances is increased

As shown in the previous section, the effect of disturbances is reduced inside the control bandwidth.

Moreover, the slope of \(|S(j\omega)|\) is limited for stability reasons (not explained here), and therefore a large control bandwidth is required to obtain sufficient disturbance rejection at lower frequencies (where the disturbances have usually large effects).

The next important question is therefore what limits the attainable control bandwidth?

The main issue it that for stability reasons, the system dynamics must be known with only small uncertainty in the vicinity of the crossover frequency.

For mechanical systems, this generally means that the control bandwidth should take place before any appearing of flexible dynamics (right part of Figure 3).

This also means that any possible change in the system should have a small impact on the system dynamics in the vicinity of the crossover.

For the NASS, the possible changes in the system are:

• a modification of the payload mass and dynamics
• a change of experimental condition: spindle’s rotation speed, position of each micro-station’s stage
• a change in the micro-station dynamics (change of mechanical elements, aging effect, …)

The nano-hexapod and the control architecture have to be developed in such a way that the feedback system remains stable and exhibit acceptable performance for all these possible changes in the system.

This problem of robustness represent one of the main challenge for the design of the NASS.

The Power Spectral Density (PSD) \(S_{xx}(f)\) of the time domain signal \(x(t)\) is defined as the Fourier transform of the autocorrelation function: \[ S_{xx}(\omega) = \int_{-\infty}^{\infty} R_{xx}(\tau) e^{-j \omega \tau} d\tau \quad \frac{[\text{unit of } x]^2}{\text{Hz}} \]

The PSD \(S_{xx}(\omega)\) represents the distribution of the (average) signal power over frequency.

Thus, the total power in the signal can be obtained by integrating these infinitesimal contributions. The Root Mean Square (RMS) value of the signal \(x(t)\) is then:

\begin{equation} x_{\text{rms}} = \sqrt{\int_{0}^{\infty} S_{xx}(\omega) d\omega} \end{equation}

One can also integrate the infinitesimal power \(S_{xx}(\omega)d\omega\) over a finite frequency band to obtain the power of the signal \(x\) in that frequency band:

\begin{equation} P_{f_1,f_2} = \int_{f_1}^{f_2} S_{xx}(\omega) d\omega \quad [\text{unit of } x]^2 \end{equation}

The Cumulative Power Spectrum is the cumulative integral of the Power Spectral Density starting from \(0\ \text{Hz}\) with increasing frequency:

\begin{equation} CPS_{+_x}(f) = \int_0^f S_{xx}(\nu) d\nu \quad [\text{unit of } x]^2 \end{equation}

The Cumulative Power Spectrum \(CPS_{+}\) taken at frequency \(f\) thus represent the power in the signal in the frequency band \(0\) to \(f\).

An alternative definition of the Cumulative Power Spectrum can be used where the PSD is integrated from \(f\) to \(\infty\):

\begin{equation} CPS_{-_x}(f) = \int_f^\infty S_{xx}(\nu) d\nu \quad [\text{unit of } x]^2 \end{equation}

And thus \(CPS_{-_x}(f)\) represents the power in the signal \(x\) for frequencies above \(f\).

The Cumulative Power Spectrum is generally shown as a function of frequency, and is used to identify the critical modes in a design, at which the effort should be targeted. It can also helps to determine at which frequencies the effect of disturbances must be reduced, and thus the approximate required control bandwidth.

A typical Cumulative Power Spectrum is shown in figure 4.

Let’s consider a signal \(u\) with a PSD \(S_{uu}\) going through a LTI system \(G(s)\) that outputs a signal \(y\) with a PSD (Figure 5).

The Power Spectral Density of the output signal \(y\) can be computed using:

\begin{equation} S_{yy}(\omega) = \left|G(j\omega)\right|^2 S_{uu}(\omega) \end{equation}

Let’s consider a signal \(y\) that is the sum of two uncorrelated signals \(u\) and \(v\) (Figure 6).

The PSD of \(y\) is equal to sum of the PSD and \(u\) and the PSD of \(v\) (can be easily shown from the definition of the PSD): \[ S_{yy} = S_{uu} + S_{vv} \]

Let’s consider the Feedback architecture in Figure 1 where the position error \(\epsilon\) is equal to: \[ \epsilon = S r + T n - G_d S d \]

Supposing that the signals \(r\), \(n\) and \(d\) are uncorrelated (which is a good approximation in our case), the PSD of \(\epsilon\) is equal to: \[ S_{\epsilon \epsilon}(\omega) = |S(j\omega)|^2 S_{rr}(\omega) + |T(j\omega)|^2 S_{nn}(\omega) + |G_d(j\omega) S(j\omega)|^2 S_{dd}(\omega) \]

And the RMS value of the residual motion can be computed using:

\begin{align*} \epsilon_\text{rms} &= \sqrt{ \int_0^\infty S_{\epsilon\epsilon}(\omega) d\omega} \\ &= \sqrt{ \int_0^\infty \Big( |S(j\omega)|^2 S_{rr}(\omega) + |T(j\omega)|^2 S_{nn}(\omega) + |G_d(j\omega) S(j\omega)|^2 S_{dd}(\omega) \Big) d\omega } \end{align*}

To estimate the PSD of the position error \(\epsilon\) and thus the RMS residual motion (in closed-loop), the following needs to be determined:

• The Power Spectral Densities of the signals affecting the system:
  • The disturbances \(S_{dd}\): this will be done in Section 3
  • The sensor noise \(S_{nn}\): this can be estimated from the sensor data-sheet
  • The wanted sample’s motion \(S_{rr}\): this is a deterministic signal that is chosen by the “user”. For a simple tomography experiment, the wanted sample’s motion can consider to be equal to \(0\) (the point of interest should stay on the focus X-ray)
• The dynamics of the complete system comprising the micro-station and the nano-hexapod: \(G\), \(G_d\). To do so, the dynamics of the micro-station (Section 2) should be identified and then included in a model (Section 4). Then a model of the nano-hexapod is merged with the micro-station model (Section 5)
• The controller \(K\) that will be designed in Section 6

The setup for the measurement of vibrations induced by rotation of the Spindle and Slip-ring is shown in Figure 15.

A geophone is fixed at the location of the sample and the motion is measured:

• without any rotation
• when rotating at 6rpm using only the slip-ring motor
• when rotating at 6rpm using the spindle motor synchronized with the slip-ring motor

The obtained Power Spectral Densities of the sample’s absolute velocity are shown in Figure 16.

It can be seen that when using the Slip-ring motor to rotate the sample, only a little increase of the motion is observed above 100Hz.

However, when rotating with the Spindle (normal functioning mode):

• a very sharp peak at 23Hz is observed. Its cause has not been identified yet
• a general large increase in motion above 30Hz

The same setup is used: a geophone is located at the sample’s location and another on the granite.

A 1Hz triangle motion with an amplitude of \(\pm 2.5mm\) is sent to the translation stage (Figure 17), and the absolute velocities of the sample and the granite are measured.

The time domain absolute vertical velocity of the sample and granite are shown in Figure 18. It is shown that quite large motion of the granite is induced by the translation stage scans. This could be a problem if this is shown to excite the metrology frame of the nano-focusing lens position stage.

The Amplitude Spectral Densities of the measured absolute velocities are shown in Figure 19. The ASD contains any peaks starting from 1Hz showing the large spectral content of the motion which is probably due to the triangular reference of the translation stage.

The sensibility to the spindle’s vibration for all the considered nano-hexapod stiffnesses (from \(10^3\,[N/m]\) to \(10^9\,[N/m]\)) is shown in Figure 30. It is shown that a softer nano-hexapod is better to filter out vertical vibrations of the spindle. More precisely, the nano-hexapod filters out the vibration starting at the first suspension mode of the payload on top of the nano-hexapod.

The same conclusion is made for vibrations of the translation stage.

The sensibility to ground motion in the Y and Z directions is shown in Figure 31. Above the suspension mode of the nano-hexapod, the norm of the transmissibility is close to one until the suspension mode of the granite. Thus, a stiff nano-hexapod (\(k>10^5\,[N/m]\)) is better for reducing the effect of ground motion at low frequency.

It will be suggested in Section 7.6 that using soft mounts for the granite can greatly lower the sensibility to ground motion.

Looking at the change of sensibility with the nano-hexapod’s stiffness helps understand the physics of the system. It however, does not permit to estimate the optimal stiffness that will lower the motion error due to disturbances.

To do so, the power spectral density of the disturbances should be taken into account, as the sensibility of one disturbance should be reduced only where the PSD of the considered disturbance is large compared to the other disturbances.

What is more important than comparing the sensitivity to disturbances, is to compare the resulting open-loop power spectral density of the sample’s position error with the change of the nano-hexapod’s stiffness. This is the dynamic noise budgeting.

From the Power Spectral Density of all the sources of disturbances identified in Section 3 is computed the Power Spectral Density of the vertical motion error for all the considered nano-hexapod stiffnesses (Figure 32).

It can be seen that the most important change is in the frequency range 30Hz to 300Hz where a stiffness smaller than \(10^5\,[N/m]\) greatly reduces the sample’s vibrations.

The most obvious change in the system is the change of payload.

In Figure 34 the dynamics is shown for payloads with a mass equal to 1kg, 20kg and 50kg (the resonance of the payload is fixed to 100Hz). On the left side, the change of dynamics is computed for a very soft nano-hexapod, while on the right side, it is computed for a very stiff nano-hexapod.

One can see that for the soft nano-hexapod:

• the first resonance (suspension mode of the nano-hexapod) is lowered with an increase of the sample’s mass. This is very logical as the first resonance corresponds to \(\omega = \sqrt{\frac{k_n}{m_n + m_s}}\) where \(k_n\) is the vertical nano-hexapod stiffness, \(m_n\) the mass of the nano-hexapod’s top platform, and \(m_s\) the sample’s mass
• the gain after the first resonance and up until the anti-resonance at 100Hz is changing with the sample’s mass. It is indeed equal to \(\frac{1}{(m_n + m_s) \omega^2}\)

For the stiff-nano-hexapod, the change of payload mass has very little effect (the vertical scale for the amplitude is quite small).

To minimize the uncertainty to the payload’s mass, the mass of the nano-hexapod’s top platform plus the metrology reflector should be maximized, and ideally close to the maximum payload’s mass. As the maximum payload’s mass is \(50\,kg\), this may however not be practical, and thus the control architecture must be developed to be robust to a change of the payload’s mass.

In Figure 35 is shown the effect of a change of payload dynamics. The mass of the payload is fixed and its resonance frequency is changing from 50Hz to 500Hz.

It can be seen (more easily for the soft nano-hexapod), that resonance of the payload produces an anti-resonance for the considered dynamics.

The dynamics for all the payloads (mass from 1kg to 50kg and first resonance from 50Hz to 500Hz) and all the considered nano-hexapod stiffnesses are display in Figure 36.

For nano-hexapod stiffnesses below \(10^6\,[N/m]\):

• the phase stays between 0 and -180deg which is a very nice property for control
• the dynamical change up until the resonance of the payload can be considered as a change of gain

For nano-hexapod stiffnesses above \(10^7\,[N/m]\):

• the dynamics is unchanged until the first resonance which is around 25Hz-35Hz
• above that frequency, the change of dynamics is quite chaotic (which this is due to the micro-station dynamics as shown in the next section) and it would be difficult to have a controller with high bandwidth which is robust to such change of dynamics

The micro-station dynamics is quite complex as was shown in Section 2, moreover, its dynamics can change due to:

• a change in some mechanical elements
• a change in the position of one stage. For instance, a large displacement of the micro-hexapod can change the micro-station compliance
• a change in a control loop of some of its stage

Thus, it would be much more robust if the plant dynamics were not depending on the micro-station dynamics. This as several other advantages:

• the control could be develop on top on another support and then added to the micro-station without changing the controller
• the nano-hexapod could be use on top of any other station much more easily

To identify the effect of the micro-station compliance on the system dynamics, the plant dynamics is identified in two different cases (Figure 37):

• with a micro-station considered as a solid body (solid curves)
• with the micro-station dynamics (dashed curves)

One can see that for nano-hexapod stiffnesses below \(10^6\,[N/m]\), the plant dynamics does not significantly changed due to the micro station dynamics (the solid and dashed curves are superimposed).

For nano-hexapod stiffnesses above \(10^7\,[N/m]\), the micro-station compliance appears in the plant dynamics starting at about 45Hz.

Let’s now consider the rotation of the Spindle.

The plant dynamics for spindle rotation speed varying from 0rpm up to 60rpm are identified and shown in Figure 38.

One can see that for nano-hexapods with a stiffness above \(10^5\,[N/m]\), the dynamics is mostly not changing with the spindle’s rotating speed.

For very soft nano-hexapods, the main resonance is split into two resonances and one anti-resonance that are all moving at a function of the rotating speed.

The change of dynamics is due to both centrifugal forces and Coriolis forces. This effect has been studied in details in this document.

Finally, let’s combined all the uncertainties and display the “spread” of the plant dynamics for all the nano-hexapod stiffnesses (Figure 39). This show how the dynamics evolves with the stiffness and how different effects enters the plant dynamics.

The Kinematic analysis of the Stewart platform can be divided into two problems: the inverse kinematics and the forward kinematics.

For inverse kinematic analysis, it is assumed that the position \({}^A\bm{P}\) and orientation of the moving platform \({}^A\bm{R}_B\) are given and the problem is to obtain the joint variables \(\bm{\mathcal{L}} = \left[ l_1, l_2, l_3, l_4, l_5, l_6 \right]^T\).

This problem can be easily solved, and the obtain joint variables are:

\begin{equation*} \begin{aligned} l_i = &\Big[ {}^A\bm{P}^T {}^A\bm{P} + {}^B\bm{b}_i^T {}^B\bm{b}_i + {}^A\bm{a}_i^T {}^A\bm{a}_i - 2 {}^A\bm{P}^T {}^A\bm{a}_i + \dots\\ &2 {}^A\bm{P}^T \left[{}^A\bm{R}_B {}^B\bm{b}_i\right] - 2 \left[{}^A\bm{R}_B {}^B\bm{b}_i\right]^T {}^A\bm{a}_i \Big]^{1/2} \end{aligned} \end{equation*}

If the position and orientation of the platform lie in the feasible workspace, the solution is unique. Otherwise, the solution gives complex numbers.

This means that from the wanted position of the nano-hexapod’s mobile platform with respect to the fixed platform (described by \({}^A\bm{P}\) and \({}^A\bm{R}_B\)) and for a specific geometry (position of the top joints \(^{B}\bm{b}\) and bottom joints \({}^A\bm{a}\)), the required motion of each leg can easily be determined.

In forward kinematic analysis, it is assumed that the vector of limb lengths \(\bm{L}\) is given and the problem is to find the position \({}^A\bm{P}\) and the orientation \({}^A\bm{R}_B\).

This is a difficult problem that requires to solve nonlinear equations.

However, as will be shown in the next section, approximate solution of the forward kinematic analysis can be obtained thanks to the Jacobian analysis.

The Jacobian matrix \(\bm{J}\) can be computed form the orientation of the legs (describes by the unit vectors \({}^A\hat{\bm{s}}_i\)) and the position of the top joints (described by the position vectors \({}^A\bm{b}_i\)) both expressed in the frame \(\{A\}\):

\begin{equation} \bm{J} = \begin{bmatrix} {\hat{\bm{s}}_1}^T & (\bm{b}_1 \times \hat{\bm{s}}_1)^T \\ {\hat{\bm{s}}_2}^T & (\bm{b}_2 \times \hat{\bm{s}}_2)^T \\ {\hat{\bm{s}}_3}^T & (\bm{b}_3 \times \hat{\bm{s}}_3)^T \\ {\hat{\bm{s}}_4}^T & (\bm{b}_4 \times \hat{\bm{s}}_4)^T \\ {\hat{\bm{s}}_5}^T & (\bm{b}_5 \times \hat{\bm{s}}_5)^T \\ {\hat{\bm{s}}_6}^T & (\bm{b}_6 \times \hat{\bm{s}}_6)^T \end{bmatrix} \end{equation}

It can be shown that the Jacobian matrix reveals the relation between the legs’ velocities to the moving platform linear and angular velocities:

\begin{equation} \dot{\bm{\mathcal{L}}} = \bm{J} \dot{\bm{\mathcal{X}}} \label{eq:jacobian_velocity} \end{equation}

with:

• \(\dot{\bm{\mathcal{L}}} = [ \dot{l}_1, \dot{l}_2, \dot{l}_3, \dot{l}_4, \dot{l}_5, \dot{l}_6 ]^T\): relative velocity of each strut
• \(\dot{\bm{X}} = [^A\bm{v}_p, {}^A\bm{\omega}]^T\): output twist vector of the mobile platform

For small legs motions \(\delta\bm{\mathcal{L}}\) and small mobile platform motion \(\delta \bm{\mathcal{X}}\), the following approximation can be computed from Eq. \eqref{eq:jacobian_velocity}:

\begin{equation} \delta\bm{\mathcal{L}} \approx \bm{J} \delta\bm{\mathcal{X}}, \quad \delta\bm{\mathcal{X}} \approx \bm{J}^{-1} \delta\bm{\mathcal{L}} \label{eq:jacobian_L} \end{equation}

Equations \eqref{eq:jacobian_L} can be used to approximate the forward and inverse kinematic problems for small displacements.

This approximation will be used to estimate the platform’s mobility from the legs’ stroke, or inversely, to estimate the required stroke of the legs from the wanted platform’s mobility.

Note that Eq. \eqref{eq:jacobian_L} is an approximation and is only valid for leg’s displacement less than \(1\%\) of the leg’s length which is the case for the nano-hexapod.

It can also be shown that the Jacobian matrix links the actuator forces to forces and moments acting on the moving platform:

\begin{equation} \bm{\mathcal{F}} = \bm{J}^T \bm{\tau}, \quad \bm{\tau} = \bm{J}^{-T} \bm{\mathcal{F}} \label{eq:jacobian_F} \end{equation}

with:

• \(\bm{\tau} = [\tau_1, \tau_2, \cdots, \tau_6]^T\): vector of actuator forces applied in each strut
• \(\bm{\mathcal{F}} = [\bm{f}, \bm{n}]^T\): external force/torque action on the mobile platform

And thus the Jacobian matrix can be used to compute the forces that should be applied by the Stewart platform’s legs from the wanted total forces/torques that is to be applied to the sample.

Linear transformations in Eq. \eqref{eq:jacobian_L} and \eqref{eq:jacobian_F} will be widely in the developed control architectures in Section 6.

For a specified geometry and actuator stroke, the mobility of the Stewart platform can be estimated thanks to the approximate forward kinematic analysis.

An example of the mobility considering only pure translations is shown in Figure 40.

From a wanted mobility and a specific geometry, the required actuator stroke can be estimated.

Suppose the following specifications on the mobility:

• x, y and z translations up to \(50\,\mu m\)
• x and y rotation up to \(30\,\mu rad\)
• no z rotation

The geometry is chosen arbitrary and corresponds to the wanted nano-hexapod size.

If only pure translations and pure rotations are considered, the required actuator stroke is \(76 \mu m\), whereas if combined translations and rotations are considered, the required actuator stroke is \(177 \mu m\).

This gives an idea of the relation between the mobility and the actuator stroke.

In order to determine the stiffness and compliance matrices of the Stewart platform, let’s model the actuators by a spring with a stiffness \(k_i\) in parallel with a force source \(\tau_i\).

The stiffness of the actuator \(k_i\) links the applied (constant) actuator force \(\delta \tau_i\) and the corresponding small deflection \(\delta l_i\):

\begin{equation*} \tau_i = k_i \delta l_i, \quad i = 1,\ \dots,\ 6 \end{equation*}

Combining these 6 relations:

\begin{equation*} \bm{\tau} = \mathcal{K} \delta \bm{\mathcal{L}} \quad \mathcal{K} = \text{diag}\left[ k_1,\ \dots,\ k_6 \right] \end{equation*}

Equations \eqref{eq:jacobian_L} and \eqref{eq:jacobian_F} are used to obtain:

\begin{equation} \bm{\mathcal{F}} = \bm{K} \delta \bm{\mathcal{X}} \end{equation}

with the stiffness matrix

\begin{equation} \bm{K} = \bm{J}^T \mathcal{K} \bm{J} \label{eq:jacobian_K} \end{equation}

And the compliance matrix can be computed by taking the inverse of the Stiffness matrix:

\begin{equation*} \bm{C} = \bm{K}^{-1} = (\bm{J}^T \mathcal{K} \bm{J})^{-1} \end{equation*}

The compliance matrix of a manipulator shows the mapping of the moving platform wrench applied at \(\bm{O}_B\) to its small deflection by

\begin{equation*} \delta \bm{\mathcal{X}} = \bm{C} \cdot \bm{\mathcal{F}} \end{equation*}

Stiffness properties of the Stewart platform can then be estimated from the architecture (through the Jacobian matrix) and leg’s stiffness.

Equations \eqref{eq:jacobian_L}, \eqref{eq:jacobian_F} and \eqref{eq:jacobian_K} can be used to see how the maneuverability, the force authority and the stiffness of the Stewart platform are changing with a the geometry (position of the joints and orientation of the legs).

The effects of two changes in the manipulator’s geometry, namely the position and orientation of the legs, are summarized in Table 1. These results could have been easily deduced based on some mechanical principles, but thanks to the kinematic analysis, they can be quantified.

Table 1: Effect of a change in geometry on the manipulator’s stiffness, force authority and stroke

	legs pointing more vertically	legs further apart
Vertical stiffness	\(\nearrow\)	\(=\)
Horizontal stiffness	\(\searrow\)	\(=\)
Vertical rotation stiffness	\(\searrow\)	\(\nearrow\)
Horizontal rotation stiffness	\(\nearrow\)	\(\nearrow\)
Vertical force authority	\(\nearrow\)	\(=\)
Horizontal force authority	\(\searrow\)	\(=\)
Vertical torque authority	\(\searrow\)	\(\nearrow\)
Horizontal torque authority	\(\nearrow\)	\(\nearrow\)
Vertical stroke	\(\searrow\)	\(=\)
Horizontal stroke	\(\nearrow\)	\(=\)
Vertical rotation stroke	\(\nearrow\)	\(\searrow\)
Horizontal rotation stroke	\(\searrow\)	\(\searrow\)

Even tough Table 1 can be used to optimize the nano-hexapod’s geometry, the available space for the nano-hexapod is too small to obtain a significant impact on the manipulator’s stiffness and stroke.

A very popular choice of Stewart platform architecture in the scientific literature, especially for vibration isolation, is the Cubic architecture.

The cubic architecture is quite specific in the sense that the active struts are arranged in a mutually orthogonal configuration connecting the corners of a cube (Figure 41).

This architecture provides some advantages such as uniform stiffness and uniform mobility in all directions. It can also have very nice dynamical properties in specific conditions (center of mass of the payload located at the cube’s center) that are not met for the NASS.

The cubic configuration also puts much restriction on the position and orientation of the legs as there is only one design variable: the size of the cube.

[5] and [6] are good resources about the Cubic architecture for vibration isolation. Separate study of the cubic architecture is performed here.

Each of the nano-hexapod legs has a universal joint at one end and a spherical joint at the other end.

When only small stroke is required, flexible joints can be used: material is bend to achieve motion, rather than relying on sliding or rolling across two surfaces.

Example of flexible joints used for Stewart platforms are shown in Figures 42 and 43.

The flexible joints have few advantages compared to conventional joints such as the absence of wear, friction and backlash which allows extremely high-precision (predictable) motion.

The parasitic bending and torsional stiffness of these joints usually induce some limitation on the control performance.

This has been studied using the Simscape model (report available here), and conclusions on the required characteristics of the flexible joints are summarized below.

The bending and torsional stiffnesses somehow adds a parasitic stiffness in parallel with the struts. This is not found to be problematic for the control architecture that will be developed in Section 6 (it is however, if Integral Force Feedback is to be used, explained here).

The finite axial stiffness of the flexible joints can however be very problematic for control. Small values of the axial stiffness are shown to limit the achievable damping using Direct Velocity Feedback. The axial stiffness should therefore be maximized and taken into account in the model of the nano-hexapod.

For the identified optimal actuator stiffness \(k = 10^5\,[N/m]\), the flexible joint should have the following stiffness properties:

• Axial Stiffness: \(K_a > 10^7\,[N/m]\)
• Bending Stiffness: \(K_b < 50\,[Nm/rad]\)
• Torsion Stiffness: \(K_t < 50\,[Nm/rad]\)

These requirements are easily obtained in practice. For instance, the flexible joint used for the ID16 nano-hexapod have the following stiffness properties:

• Axial Stiffness: \(K_a = 6 \cdot 10^7\,[N/m]\)
• Bending Stiffness: \(K_b = 15\,[Nm/rad]\)
• Torsion Stiffness: \(K_t = 20\,[Nm/rad]\)

In order to test this modeling technique, some tests have been performed on a flexible piezoelectric stack actuator.

The APA95ML from Cedrat has been sketched into Ansys and the interface nodes chosen as shown in Figure 44. The top and bottom nodes are used as interface nodes to connect to other mechanical parts of the nano-hexapod. The ten interface nodes along the piezo stack are used to apply forces into the stack.

The reduced mass and stiffness matrices are exported using Ansys and imported into Matlab. The actuator is included in a Simscape model, and the dynamics from forces applied by the piezo stack to the vertical displacement of the amplified structure is identified using Simscape and compare with an harmonic response using Ansys (Figure 45).

A payload with a mass of 10kg is then added both in the Simscape model and in Ansys and the dynamic is identified again and compared (Figure 45). The dynamics obtained with Simscape and Ansys are very close to each other which validate the fact that we can interface the flexible element with other Simscape parts.

A test bench is planned to validate the presented modelling technique.

The DCM’s fast jack test bench will be slightly modified to integrate the APA95ML actuator (already available).

The idea is to identify the transfer functions from forces applied by the stack actuator to the vertical displacement as what done both using Simscape and Ansys.

This test bench requires very little work and will permit to gain much confident on the modelling technique used as well as on the dynamics of amplified piezoelectric actuators.

During all the mechanical design of the nano-hexapod, it is planned to use the presented modelling technique to ensure that no parasitic modes will be problematic for the control part.

More specifically, it is wanted that both the flexible joints and the amplified piezoelectric actuators do not introduce parasitic modes in the dynamics to be controlled up to 200Hz.

This flexible modeling technique is thus a very important element during the mechanical design of the nano-hexapod.

To see why Integral Force Feedback should not be applied to damp the nano-hexapod’s modes, a simple model of a rotating positioning platform integration force sensors has been developed (described in details here).

The platform main resonance frequency is \(\omega_0\) and the rotation speed is \(\omega\).

Root Locus plots for Integral Force Feedback and Direct Velocity Feedback are shown in Figure 3. These plots show the evolution of the system’s poles in the complex plane as a function of the control gain.

Table 3: Variation of the Root Locus for DVF and IFF in presence of rotation. \(\omega\) is the spindle rotation speed, and \(\omega_0\) is the resonance frequency of the considered rotating system.

Direct Velocity Feedback	Integral Force Feedback

To understand what the root locus means, consider Figure 47 where two resonant systems are compared:

• The first one (represented in blue) is undamped. The corresponding poles of this system is located on the imaginary axis.
• The second one (represented in red) has some added damping which can be easily see by the reduction of the amplitude near the resonance. On the complex plane, this is manifested by the being “moved” such that the angle \(\phi\) it makes with the imaginary axis is increase.

As a matter of fact, the damping ratio of a system resonance can be approximated by \(\xi \approx \sin \phi\).

To damp a system, it is thus wanted to move the poles to the left part of the complex plane. A pole with a positive real part corresponds to an unstable system, and thus the right part of the complex should be avoided.

Coming back to the Root Locus in Figure 3, it can be seen that:

• For Direct Velocity Feedback:
  • The system’s poles are staying in the left half plane which means guaranteed stability
  • Arbitrary damping can be added to the system’s resonances
• For Integral Force Feedback:
  • For non-null rotation speed, and whatever the control gain is, a pole is located in the right part of the complex plane, showing that the closed-loop system is unstable
  • Limited damping can be added to the system

Similar observations are made using the Simscape model of the NASS, and this shows why Direct Velocity Feedback is the most suitable active damping technique for the NASS.

Relative motion sensors are included in each of the nano-hexapod’s leg and a decentralized direct velocity feedback control architecture is applied (Figure 48).

The signals shown in Figure 48 are:

• \(\bm{\tau}\): Actuator forces applied in each leg
• \(\bm{\tau}_m\): Force sensor signal located in each leg (not used here)
• \(\bm{\mathcal{X}}\): Measurement of the payload position with respect to the granite by the metrology system (used for the HAC part)
• \(d\bm{\mathcal{L}}\): Measurement of the (small) relative motion of each leg

\(\bm{K}_{\text{DVF}}\) is a diagonal controller with derivative action. This control architecture is equivalent as to have six independent control loops from the relative motion sensor of one leg to the actuator of the same leg (hence the term decentralized). The force applied in each leg being proportional to the relative velocity of the associated leg (thanks to the derivative action), this is equivalent as adding damping in each of the legs.

The dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) for three payload masses is shown in Figure 49. It is shown that for all the payload masses, the dynamics shows an alternation of poles and zeros which makes the direct velocity feedback loop robust.

This is confirmed by the Root Locus in Figure 50 where all the poles are staying in the left half plane. Moreover, it is seen that arbitrary damping can be applied to the nano-hexapod’s modes.

The DVF gain is here chosen in such a way that the suspension modes of the nano-hexapod are critically damped whatever the sample mass.

This may not be the optimal choice as will be further explained.

One objective of the active damping technique is to lower the sensibility to disturbances which are shown in Figure 51 without active damping (solid) and with the use of DVF (dashed).

The Direct Velocity Feedback control lowers the sensibility to disturbances in the vicinity of the nano-hexapod resonance but increases the sensibility at higher frequencies.

A smaller control gain could probably limit the increase of the sensibility at higher frequencies while still providing sufficient sensibility reduction near the nano-hexapod resonances. Further optimization of the gain should then be performed.

Another control objective for the LAC is to render the plant dynamics simpler to control for the High Authority Controller.

The plant dynamics before (solid curves) and after (dashed curves) the Low Authority Control implementation are compared in Figure 52. It is clear that the use of the DVF reduces the dynamical spread of the plant dynamics between 5Hz and 100Hz. This will make the primary controller more robust and easier to develop.

Let’s consider the two HAC-LAC control architectures shown in Figures 53 and 54 where an outer control loop is added to the already damped plant.

To do so, the block Compute Pos. Error is used to compute the position error \(\bm{\epsilon}_{\mathcal{X}_n}\) of the sample with respect to the nano-hexapod’s base platform from the actual measurement of the sample’s pose \(\bm{\mathcal{X}}\) and the wanted pose \(\bm{r}_\mathcal{X}\). The computation done in such block was briefly explained in Section 4.3.

The two proposed control architectures are very similar in the sense that their outcome is a multi-input multi-output controller \(\bm{K}\). The difference between the two architectures relies in the way the controllers are designed:

• For the architecture shown in Figure 53:
  • The controller \(\bm{K}_\mathcal{X}\) is designed in the task space: from the position/orientation error \(\bm{\epsilon}_{\mathcal{X}_n}\), it generates a force/torque \(\bm{\mathcal{F}}\) to be applied to sample
  • The forces/torques are then further converted to actuators forces \(\bm{\tau}^\prime\) with the use of the Jacobian matrix \(\bm{J}^{-T}\)
  • The full controller is \(\bm{K} = \bm{J}^{-T} \bm{K}_\mathcal{X}\)
• For the architecture shown in Figure 54:
  • The sample’s position error \(\bm{\epsilon}_{\mathcal{X}_n}\) is first converted to the corresponding length errors of the six nano-hexapod’s legs \(\bm{\epsilon}_\mathcal{L}\) with the approximate inverse kinematics using the Jacobian matrix \(\bm{J}\)
  • The controller \(\bm{K}_\mathcal{L}\) then computes the actuator forces \(\bm{\tau}^\prime\) such that each of the legs have the wanted displacement
  • The full controller is \(\bm{K} = \bm{K}_\mathcal{L} \bm{J}\)

The choice of whether the controller should be designed in the leg space or in the task space does however makes some differences, that can be seen by looking at the dynamics to be controlled:

• Typical dynamics from \(\bm{\mathcal{F}}\) to \(\bm{\epsilon}_{\mathcal{X}_n}\) is shown in Figure 55:
  • The suspension modes of the Stewart platform are separated, and the direct (diagonal) dynamical terms are different
  • The coupling is very small except for the dynamics from \(\mathcal{F}_{x,y}\) to \(R_{y,x}\) and from \(\mathcal{M}_{x,y}\) to \(D_{y,x}\) which is due to an non-diagonal stiffness and mass matrices
• Typical dynamics from \(\bm{\tau}\) to \(\bm{\epsilon}_\mathcal{L}\) is shown in Figure 56:
  • The dynamics from \(\tau_i\) to \(d\mathcal{L}_i\) are all identical and contains all the Stewart platform modes, and thus only one controller has to be designed
  • The coupling is small at low frequency, quite high near the suspension modes of the Stewart platform and then small again at high frequency

The differences of a control in the leg space and in the task space are summarized in Table 4.

Table 4: Comparison of a control in the leg space and in the task space

	Control in the leg space	Control in the task space
Plant Meaning	\(\delta\mathcal{L}_i/\tau_i\)	\(\delta\mathcal{X}_i/\mathcal{F}_i\)
Obtained Decoupling	Decoupled at DC	Dynamical decoupling except few terms
Diagonal Elements	Identical with all the Resonances	Different, resonances are cancel out
Mechanical Architecture	Architecture Independent	Better with Cubic Architecture
Advantages	One controller to be designed	Possible to have different controllers for each
		direction as the requirements are not the same

Both control architecture have been applied and the control in the leg space appears to be simpler to apply and have good robustness properties.

An alternative that could increase the control performance and robustness would be to design the full multi-input multi-outputs controller \(\bm{K}\) in one step using optimal and robust control synthesis techniques such as the \(\mathcal{H}_\infty\) loop shaping.

The plant dynamics from \(\tau_i\) to \(\epsilon_{\mathcal{L}_i}\) for each of the six legs and for the three payload’s masses is shown in Figure 57. The dynamical spread is kept reasonably small thanks to both the optimal nano-hexapod design and the Low Authority Controller.

The diagonal controller \(\bm{K}_\mathcal{L}\) is then tuned in such a way that the control bandwidth is around 100Hz and such that enough stability margins are obtained for all the payload’s masses. The obtained loop gain is shown in Figure 58.

The sensibility to disturbance after the use of HAC-LAC control is shown in Figure 59. The change of sensibility is very typical for feedback system:

• the sensibility is reduced within the controller bandwidth (here \(\approx 100\,[Hz]\))
• the sensibility is increase around the crossover frequency and mostly unchanged outside of the control bandwidth

The large increase at around 250Hz when using a mass of either 1kg or 10kg is probably caused by insufficient stability margins.

A simulation of a tomography is performed with the optimal nano-hexapod and the HAC-LAC architecture implemented. The results of this simulation are compared to the simulation performed in Section 4.4 without the nano-hexapod. All the disturbances are included such as ground motion, spindle and translation stage vibrations.

The Power Spectral Density of the sample’s position error is plotted in Figure 60 and the Cumulative Amplitude Spectrum is shown in Figure 61. The top three plots corresponds to the X, Y and Z translations and the bottom three plots corresponds to the X,Y and Z rotations.

Several observations can be made:

• The sample’s vibrations are reduced within the control bandwidth as was expected
• The obtained performances for all the three considered masses are very similar. This is an indication of the good system’s robustness
• From the Cumulative Amplitude Spectrum (Figure 61), it can be seen that Z motion is reduced down to \(\approx 30\,nm\,[rms]\) and the Y motion down to \(\approx 25\,nm\,[rms]\)

• An increase in the rotational vibrations is observed. This is due to the fact that:

  1. no perturbations inducing rotations were included in the simulation and thus the rotational vibrations are very small
  2. the high authority control induces some coupling between translations and rotations. It then introduces some rotational motion due to vibrations in translation

  This increase in rotation is still very small and is not foreseen to be a problem

• The vertical rotation plot is meaningless as the spindle rotation was considered to be perfect and no attempt was made to compensate these vibrations by the nano-hexapod

The time domain sample’s vibrations are shown in Figure 62. The use of the nano-hexapod combined with the HAC-LAC architecture is shown to considerably reduce the sample’s vibrations.

An animation of the experiment is shown in Figure 63 and it can be seen that the actual sample’s position is more closely following the ideal position compared to the simulation of the micro-station alone in Figure 26 (same scale was used for both animations).

Let’s consider that the micro-hexapod introduces a 10mm offset on the sample’s position such that the X-ray is focus on an interesting part of the sample.

The sample’s mass is 1kg and the spindle’s rotation speed is 60rpm.

The control objective is to keep the point of interest on the focused X-ray.

An animation showing the simulation is shown in Figure 64.

One can see that the forces applied by the actuator are fluctuating around a constant value (Figure 65). This is because the controller generates the actuator forces such that they counteracts the disturbances affecting the sample’s position. The disturbance causing this constant force is the centrifugal force induced by the spindle’s rotation which is a constant force in the frame of the nano-hexapod (provided the rotation speed is constant), directed radially outwards the rotation spindle’s axis, and is equal to \(F = m r \omega^2 \approx 12 \cdot 0.01 \cdot (2\pi)^2 \approx 5\,[N]\).

The relative motions of the nano-hexapod’s legs is shown in Figure 66 and are in the micro-meter range.

Finally, the position/orientation error of the sample is shown in Figure 67. The root mean square value of the x-y-z error motions is around \(30\,nm\) which is very similar than for the “simple” tomography experiment.

In this simulation:

• the sample has a mass of 1kg
• the spindle rotation speed is set to 60rpm
• scans with the translation stage are performed simultaneously with a triangular shape having a period of 4s and an amplitude of 5mm

The obtained sample’s motion during the simulation is shown in Figure 68.

The forces applied by the nano-hexapod’s are shown in Figure 69. Peak values of the forces are appearing when the translation stage changes the direction of the scan.

The relative motions of the nano-hexapod’s legs is shown in Figure 70 and are again in the micro-meter range.

The time domain position/orientation error of the sample is shown in Figure 71. The RMS value of the x-y-z position error is again \(\approx 30\,nm\).

The wanted dimension of the nano-hexapod are shown in Figure 72:

• Diameter of the bottom platform: 300mm
• Diameter of the top platform: 200mm
• Maximum Height: 90mm

The limiting height might be quite problematic for the integration of the flexible joints, the actuators and sensors.

Flexible joints are located at each end of the six struts. These flexible joints should have the following properties:

• High Axial Stiffness: \(K_a > 10^7\,[N/m]\)
• Small Bending Stiffness: \(K_b < 50\,[Nm/rad]\)
• Small Torsion Stiffness: \(K_t < 50\,[Nm/rad]\)

The required angular stroke has not been estimated in this study. It is however simple to do so as the angular motion of each joint can easily be measured in the multi-body model used to perform the simulations. Typical angular stroke for such flexible joints is expected.

The axial stiffness of the struts (between two flexible joints) should be equal to \(\approx 10^5 - 10^6\,[N/m]\).

If voice coils are used, this corresponds to the axial stiffness of the membrane guiding the moving part of the voice coil.

Based on simulations:

• Continuous Force: \(\pm 5\,[N]\) (due to centrifugal forces)
• “Variable” Force: \(\pm 1\,[N]\)

If static deflection is to be compensated by the actuator, \(\approx 100\,[N]\) of continuous force is required for each actuator.

Based on simulations, the required actuator stroke seems to be \(\pm 5\,[\mu m]\).

This however does not take into account two error types that will have to be compensated by the nano-hexapod:

• the static positioning errors of all the micro-station’s stages. These errors have been measured by Hans-Peter, and are in the order of tens of \(\mu m\) and tens of \(\mu rad\).
• the tracking errors of the translation stage and tilt stage. This is probably more difficult to estimate. However, by limiting the acceleration of these stages, we may limit the dynamic tracking errors to acceptable levels

Some security margin should be taken as if the nano-hexapod has not enough stroke to compensated the above errors, the system will not be able to compensate all the vibrations.

Thus, an actuator stroke of \(\pm 50 \mu m\) would be quite safe.

Note that a piezoelectric stack have a maximum strain of \(0.1\%\). A piezo stack with a stroke of \(\pm 50\,[\mu m]\) will have a length size of \(\approx 100\,[mm]\) making it difficult to integrate in the nano-hexapod.

Some amplified piezoelectric actuators that fulfill the requirements are listed in Table 5. The actuators that seems the most suited will be modeled using a FE software and integrated into the Simscape model as explained in Section 5.5. Simulation will be performed with the chosen actuator to make sure that the obtained performance is acceptable.

Table 5: List of some amplified piezoelectric actuators that could be used for the nano-hexapod

	Specification	APA150M	APA400MML	APA300ML	FPA-0500E-P-1036	FPA-0300E-S-0536	FPA-0150E-S-0518
Stroke	\(> 100\, [\mu m]\)	187	368	304	432	240	132
Stiffness	\(0.1-1\, [N/\mu m]\)	0.7	0.55	1.8	0.87	0.58	0.8
Resolution	\(< 2\, [nm]\)	2	4	3
Blocked Force	\(> 100\, [N]\)	127	201	546	376	139	106
Height	\(< 50\, [mm]\)	22	24	30	27	16	15
Price					2300$	1400$	890$

A relative displacement sensor must be included in each of the nano-hexapod’s legs as explained in Section 6.

The sensors must as the following properties:

• High bandwidth \(> 1\,[kHz]\)
• Fine resolution \(< 10\,[nm]\)
• Range of \(> 100\,[\mu m]\)

Note that the sensor signal will have to pass through the Slip-Ring. This adds few constrains:

• The sensor signal must be immune to some “electrical noise” that could be induced by the slip-ring
• Limited number of slip-ring channels should be required for the six sensors

Several sensor technology could be used for the nano-hexapod. Characteristics of those sensors are shown in Table 6.

Table 6: Characteristics of relative measurement sensors collette11_review

Technology	Frequency	Resolution	Range	T Range
LVDT	DC-200 Hz	10 nm rms	1-10 mm	-50,100 °C
Eddy current	5 kHz	0.1-100 nm rms	0.5-55 mm	-50,100 °C
Capacitive	DC-100 kHz	0.05-50 nm rms	50 nm - 1 cm	-40,100 °C
Interferometer	300 kHz	0.1 nm rms	10 cm	-250,100 °C
Encoder	DC-1 MHz	1 nm rms	7-27 mm	0,40 °C

Two sensor technologies seem the most adapted to the nano-hexapod: the eddy current sensor and the capacitive sensor. They both exhibit few nanometers of resolution over a stroke of \(100\,[\mu m]\) and a sufficient bandwidth.

Cedrat proposes to integrate Eddy Current Sensors in their amplified piezoelectric actuator as shown in Figure 73. An alternative could be to use the capacitive sensors such as the very compact D-100 proposed by PI.

As explained in section 5.4 the orientation of the legs and position of the joints are very much constrained by the limited height of the nano-hexapod.