**4. Modeling**

Two different models have been developed to permit the design of the three proposed control strategies:


#### **4.1. Four degrees of freedom model**

The system has been modeled by using four degrees of freedom describing the dynamics in *YZ* plane. Four flexural steel springs have been used to link the stage to the frame, four air springs are placed at the bottom of the frame, two actuators are working in series between the stage, and the frame and two geophones are used to measure the velocities of stage and frame respectively. As the axial stiffness of the flexural springs is very high, it can be assumed that there is no relative displacement between stage and frame along the vertical direction, which means that the relative displacement along the z axis between stage and frame are the same. Both stage and frame are assumed as moving about the frame mass center with the same rotating speed. The model reference frames are defined in Figure 2 (XY -plane view) and in Figure 13 (YZ -plane view).

**Figure 12.** YZ plane 4 dof kinematic relationships scheme.

The degrees of freedom of the model are:

528 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

Lead Lag approach, b) Feedforward control strategy.

**4.1. Four degrees of freedom model** 


**Figure 12.** YZ plane 4 dof kinematic relationships scheme.

2. Six degrees of freedom model used for the design of c) Modal controller.

in the range of ±30V.

**4. Modeling** 

control strategies:

stage performs the current control by means of an operational amplifier that is unity-gain stable with a bandwidth of 1.8MHz and it is internally protected against over-temperature conditions and current overloads. The second stage is a classical current amplifier with bipolar transistors in Darlington configuration to increase the current gain. The last stage provides the feedback signal to ensure the desired current in the load. The power supply is

Two different models have been developed to permit the design of the three proposed

1. Four degrees of freedom model used for the design of: a) a feedback controller with a

The system has been modeled by using four degrees of freedom describing the dynamics in *YZ* plane. Four flexural steel springs have been used to link the stage to the frame, four air springs are placed at the bottom of the frame, two actuators are working in series between the stage, and the frame and two geophones are used to measure the velocities of stage and frame respectively. As the axial stiffness of the flexural springs is very high, it can be assumed that there is no relative displacement between stage and frame along the vertical direction, which means that the relative displacement along the z axis between stage and frame are the same. Both stage and frame are assumed as moving about the frame mass center with the same rotating speed. The model reference frames are defined in Figure 2 (XY

$$\overline{X} = \begin{bmatrix} y\_{F'} \; ; & z\_{F'} \; \; \; \; \; \theta \; \; \; \; \; \; q\_S \end{bmatrix} \tag{6}$$

that indicate the displacement of the frame along *Y* -axis and *Z* -axis, the rotation of the frame (and stage) around the *X* -axis mass center and the stage displacement along its *Y* axis.

Referring to Figure 12, it is possible to obtain the formulation of the velocity of a generic point *S* of the stage:

$$\begin{aligned} \vec{V}\_S &= \vec{V}\_F + \dot{q}\_S \vec{j}\_S + \dot{\theta} \, \overrightarrow{FS\_0} \left[ \cos \alpha \, \vec{k}\_S - \sin \alpha \, \vec{j}\_S \right] = \\ \vec{y} &= \dot{y}\_F \vec{j}\_F + \dot{z}\_F \vec{k}\_F + \left( \dot{q}\_S - \dot{\theta} z\_{0S} \right) \vec{j}\_S + \dot{\theta} (y\_{0S} + q\_S) \vec{k}\_S \end{aligned} \tag{7}$$

The kinetic energy *T* of the system can be expressed as:

$$T = \frac{1}{2}m\_{\rm S}\vec{V}\_{\rm S}{}^2 + \frac{1}{2}J\_{\rm S}\dot{\theta}^2 + \frac{1}{2}m\_{\rm F}\vec{V}\_{\rm F}{}^2 + \frac{1}{2}J\_{\rm F}\dot{\theta}^2\tag{8}$$

Where *mS* and *JS* are the mass and the rotating inertia measured in the center of mass of the stage S, and *mF* and *JF* the mass and the rotating inertia measured in the center of mass of the frame *F* .

The potential energy *U* is obtained starting from the diagram reported in Figure 13.

**Figure 13.** *YZ* plane 4 dof model scheme.

The potential energy *U* is:

$$\begin{split} \mathbf{U} &= k\_{\mathrm{GFz}} \left( \mathbf{z}\_{\mathrm{F}} - \theta \mathbf{d}\_{1} - \mathbf{z}\_{\mathrm{G}} \right)^{2} + k\_{\mathrm{GFy}} (\mathbf{y}\_{\mathrm{F}} + \theta \boldsymbol{\hbar} - \mathbf{y}\_{\mathrm{G}})^{2} + k\_{\mathrm{GFz}} \left( \mathbf{z}\_{\mathrm{F}} + \theta \mathbf{d}\_{2} - \mathbf{z}\_{\mathrm{G}} \right)^{2} + \dots \\ &+ k\_{\mathrm{GFy}} (\mathbf{y}\_{\mathrm{F}} + \theta \mathbf{h} - \mathbf{y}\_{\mathrm{G}})^{2} + k\_{\mathrm{GFy}} \boldsymbol{\eta}\_{\mathrm{S}}^{2} \end{split} \tag{9}$$

where *yG* and *zG* are the displacement of the ground and *d1*, *d2*, and *h* the quantities reported in Figure 13.

Owing to the Rayleigh formulation, the damping of the system is governed by the following dissipation function:

$$\begin{split} \mathfrak{R} &= \mathbf{c}\_{\mathrm{GF}z} \left( \dot{\boldsymbol{z}}\_{F} - \dot{\boldsymbol{\theta}} \, d\_{1} - \dot{\boldsymbol{z}}\_{G} \right)^{2} + \mathbf{c}\_{\mathrm{GF}y} \left( \dot{\boldsymbol{y}}\_{F} + \dot{\boldsymbol{\theta}} \, \boldsymbol{h} - \dot{\boldsymbol{y}}\_{G} \right)^{2} + \mathbf{c}\_{\mathrm{GF}z} \left( \dot{\boldsymbol{z}}\_{F} + \dot{\boldsymbol{\theta}} \, d\_{2} - \dot{\boldsymbol{z}}\_{G} \right)^{2} + \dots \\ &+ \mathbf{c}\_{\mathrm{GF}y} \left( \dot{\boldsymbol{y}}\_{F} + \dot{\boldsymbol{\theta}} \, \boldsymbol{h} - \dot{\boldsymbol{y}}\_{G} \right)^{2} + \mathbf{c}\_{\mathrm{SF}y} \dot{\boldsymbol{q}}\_{S}^{2} \end{split} \tag{10}$$

where each damping term *<sup>i</sup> c* is obtained starting from the experimental identification of damping ratios *<sup>i</sup>* ς:

$$c\_i = \mathbb{Z}\varphi\_i \sqrt{k\_i m\_i} \tag{11}$$

The inputs of the system are: the force of the electromagnetic actuators *act F* , the force of the stage *SF* and the velocities from the ground in y direction *Gy v* and z direction *Gz v* . The output are the velocities *<sup>F</sup> v* of the frame and *Sv* of the stage measured with geophones sensors. Inputs and outputs are graphically represented in Figure 14.

**Figure 14.** YZ plane 4dof model scheme – input and output.

Using the Lagrange formulation is possible to write the equations of motion in the form:

$$\mathbf{M}\ddot{q} + \mathbf{C}\dot{q} + \mathbf{K}q + T\_G q\_G = \mathbf{T}\begin{bmatrix} F \end{bmatrix} \tag{12}$$

where

530 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

( )

( )

θ

+ +− +

ς:

*GFy F G SFy S*

**Figure 14.** YZ plane 4dof model scheme – input and output.

*c y hy c q* θ

θ

+ +− +

*GFy F G SFy S*

*k y hy k q* θ

2 2

2 2

sensors. Inputs and outputs are graphically represented in Figure 14.

( ) ( ) 2 2 <sup>2</sup> 1 2

*GFz F G GFy F G GFz F G*

where *yG* and *zG* are the displacement of the ground and *d1*, *d2*, and *h* the quantities reported

Owing to the Rayleigh formulation, the damping of the system is governed by the following

*GFz F G GFy F G GFz F G*

where each damping term *<sup>i</sup> c* is obtained starting from the experimental identification of

2 *i i ii c km* = ς

The inputs of the system are: the force of the electromagnetic actuators *act F* , the force of the stage *SF* and the velocities from the ground in y direction *Gy v* and z direction *Gz v* . The output are the velocities *<sup>F</sup> v* of the frame and *Sv* of the stage measured with geophones

*c z d z c y hy c z d z*

ℜ= − − + + − + + − +

( ) ( ) 2 2 <sup>2</sup> 1 2

= − − + +− + + − +

*Uk z d z k y hy k z d z*

( ) ...

( ) ...

(11)

θθ

(9)

(10)

θθ

The potential energy *U* is:

in Figure 13.

dissipation function:

damping ratios *<sup>i</sup>*

$$q = \begin{pmatrix} y\_F & z\_F & \mathcal{O} & q\_S \end{pmatrix}^T \tag{13}$$
 
$$q\_n$$

$$\boldsymbol{q}\_{G} = \begin{pmatrix} \boldsymbol{y}\_{G} & \boldsymbol{z}\_{G} \end{pmatrix}^{T}$$

are the vectors of the generalized coordinates,

$$F = \begin{pmatrix} v\_{Gy} & v\_{Gz} & F\_S & F\_{act} \end{pmatrix}^T \tag{14}$$

is the vector of the generalized forces and M is the mass matrix

$$M = \begin{bmatrix} m\_{tot} & 0 & -m\_S z\_{0S} & m\_S \\ 0 & m\_{tot} & m\_S y\_{0S} & 0 \\ -m\_S z\_{0S} & m\_S y\_{0S} & J\_{tot} & -m\_S z\_{0S} \\ m\_S & 0 & -m\_S z\_{0S} & m\_S \end{bmatrix} \tag{15}$$

with *m mm tot S F* = + , 2 2 0 0 ( ) *tot S F S S S J J J my z* =++ + . 0*<sup>S</sup> y* , <sup>0</sup>*<sup>S</sup> z* are the initial position of the stage. The matrix is symmetric and not diagonal because it takes into account the coupling between the stage and the frame dynamics.

The stiffness matrix *K* is:

$$K = \begin{vmatrix} 4k\_{\rm GFy} & 0 & 4k\_{\rm GFy}h & 0\\ 0 & 4k\_{\rm GFz} & -2k\_{\rm GFz}d\_1 + 2k\_{\rm GFz}d\_2 & 0\\ 4k\_{\rm GFy}h & -2k\_{\rm GFz}d\_1 + 2k\_{\rm GFz}d\_2 & 2k\_{\rm GFz}d\_1^2 + 4k\_{\rm GFy}h^2 + 2k\_{\rm GFz}d\_2^2 & 0\\ 0 & 0 & 0 & 4k\_{\rm Fy} \end{vmatrix} \tag{16}$$

The damping matrix *C* is:

$$\mathbf{C} = \begin{vmatrix} 4c\_{GFy} & 0 & 4c\_{GFy}h & 0 \\ 0 & 4c\_{GFz} & -2c\_{GFz}d\_1 + 2c\_{GFz}d\_2 & 0 \\ 4c\_{GFy}h & -2c\_{GFz}d\_1 + 2c\_{GFz}d\_2 & 2c\_{GFz}d\_1^2 + 4c\_{GFy}h^2 + 2c\_{GFz}d\_2^2 & 0 \\ 0 & 0 & 0 & 4c\_{GFy} \end{vmatrix} \tag{17}$$

The selection matrix T of the generalized forces is:

$$T = \begin{bmatrix} -4c\_{GFy} & 0 & 1 & 0 \\ 0 & -4c\_{GFz} & 0 & 0 \\ 4c\_{GFy}h & 2c\_{GFz}d\_1 - 2c\_{GFz}d\_2 & 0 & 0 \\ 0 & 0 & 1 & -1 \end{bmatrix} \tag{18}$$

$$T\_G = \begin{bmatrix} -4k\_{GFy} & 0\\ 0 & 4k\_{Gzy} \\ 4k\_{GFy}h & 2k\_{GFz}d\_1 - 2k\_{GFz}d\_2 \\ 0 & 0 \end{bmatrix}$$

In the state space formulation the equations of motion of the system can rewritten as:

$$
\dot{X} = AX + B\mathcal{U} \tag{19}
$$

where the state vector X and the input vector are:

$$\mathbf{X} = \begin{Bmatrix} \mathbf{q} & \dot{\mathbf{q}} & \mathbf{q}\_G \end{Bmatrix}^T, \mathbf{J} \mathbf{I} = \begin{Bmatrix} \mathbf{v}\_{\text{Gy}} & \mathbf{v}\_{\text{Gz}} & F\_{\text{S}} & F\_{\text{act}} \end{Bmatrix}^T \tag{20}$$

with A the state matrix, B the input matrix

$$A = \begin{bmatrix} 0 & I & 0 \\ -M^{-1}K & -M^{-1}C & M^{-1}T\_G \\ 0 & 0 & 0 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 \\ M^{-1}T \\ I \end{bmatrix} \tag{21}$$

The relationship between input and output can be represented as:

$$Y = CX + DUI\tag{22}$$

where *Y* is the output vector, *C* the output matrix and *D* the feedthrough matrix

$$\mathbf{Y} = \begin{bmatrix} \upsilon\_S & \upsilon\_F \end{bmatrix}^T, \mathbf{C} = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 0 & -z\_{\text{geS}} & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & -z\_{\text{geF}} & 1 & 0 & 0 \end{bmatrix}, \mathbf{D} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix} \tag{23}$$

#### **4.2. Six degrees of freedom model**

As well as the dynamics on the YZ plane described in the previous section, it has been developed a six degrees of freedom model of system dynamics on the XY plane. In this case, the degrees of freedom of the model are:

$$\overline{X} = \begin{bmatrix} \mathbf{x}\_{\mathcal{S}}; & y\_{\mathcal{S}}; & \theta\_{\mathcal{S}}; & \mathbf{x}\_{\mathcal{F}}; & y\_{\mathcal{F}}; & \theta\_{\mathcal{F}} \end{bmatrix} \tag{24}$$

indicating the stage displacements *xS* along X-axis, *yS* along Y-axis, the rotation θ*<sup>S</sup>* about the axis passing through the mass center and oriented along the Z-axis, the frame displacements *xF* along X-axis, *yF* along Y-axis, and the rotation θ*<sup>F</sup>*about the axis passing through the mass center oriented along the Z-axis. Stage and frame degrees of freedom, inputs, and geometric properties are illustrated in Figure 15 and 16.

Resorting to the Lagrange formulation as reported in (12), the *q* vector of the generalized coordinates is:

$$\boldsymbol{\eta} = \begin{pmatrix} \boldsymbol{\chi}\_{\mathcal{S}} & \boldsymbol{y}\_{\mathcal{S}} & \boldsymbol{\theta}\_{\mathcal{S}} & \boldsymbol{\chi}\_{\mathcal{F}} & \boldsymbol{y}\_{\mathcal{F}} & \boldsymbol{\theta}\_{\mathcal{F}} \end{pmatrix}^{\mathrm{T}} \tag{25}$$

and the *F* the vector of the generalized forces is

532 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

*G*

*T*

where the state vector X and the input vector are:

with A the state matrix, B the input matrix

{ }

*S F*

**4.2. Six degrees of freedom model** 

case, the degrees of freedom of the model are:

*T*

*GFy*

*c*

1 2

1 2

*X qqq U v v F F <sup>G</sup> Gy Gz S act* = = (20)

*X AX BU* = + (19)

*I*

*Y CX DU* = + (22)

*Gzy*

*k*

(18)

(21)

(23)

*GFz*

4 0 10 0 4 00 4 2 2 00 0 0 11

*c*

*GFy GFz GFz*

4 0 0 4 42 2 0 0

*GFy GFz GFz*

{ } , { } *<sup>T</sup> <sup>T</sup>*

1 11 1 0 00

− −− − =− − <sup>=</sup>

*A MK MC MT B MT*

*G*

000010 100 <sup>0000</sup> , , 000010 100 0000

*z*

*z* <sup>−</sup> = = <sup>=</sup> <sup>−</sup>

As well as the dynamics on the YZ plane described in the previous section, it has been developed a six degrees of freedom model of system dynamics on the XY plane. In this

> ;;;; ; *Xx y x y SSS F FF* = θ

*geoF*

 θ(24)

,

*kh kd kd* <sup>−</sup> <sup>=</sup> <sup>−</sup>

*GFy*

In the state space formulation the equations of motion of the system can rewritten as:

.

0 00

where *Y* is the output vector, *C* the output matrix and *D* the feedthrough matrix

*T geoS*

*Yvv C D*

The relationship between input and output can be represented as:

*I*

*k*

*ch cd cd* <sup>−</sup> <sup>−</sup> <sup>=</sup> <sup>−</sup> <sup>−</sup>

$$F = \begin{pmatrix} F\_{X+} & F\_{X-} & F\_{Y+} & F\_{Y-} \end{pmatrix}^T \tag{26}$$

it is possible to obtain the corresponding mass matrix M, stiffness matrix K and damping matrix C (not reported due to its excessive size).

**Figure 15.** XY Plane 6 dof model scheme: stage degrees of freedom and inputs.

**Figure 16.** XY Plane 6 dof model scheme: stage degrees of freedom and inputs.

The selection matrix T of the generalized forces is:

$$T = \begin{bmatrix} 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 \\ d\_{s1} & d\_{s2} & d\_{s3} & d\_{s4} \\ -1 & 1 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ -d\_{f1} & -d\_{f2} & -d\_{f3} & -d\_{f4} \end{bmatrix} \tag{27}$$

Similarly in the state space formulation the equations of motion of the system can rewritten as:

$$
\dot{X} = AX + B\mathcal{U} \tag{28}
$$

where the state vector X and the input vector *U* are:

$$\mathbf{X} = \begin{bmatrix} q & \dot{q} \end{bmatrix}^T, \mathbf{U} = \begin{bmatrix} F \end{bmatrix}^T \tag{29}$$

with the following state and input matrix

534 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

**Figure 16.** XY Plane 6 dof model scheme: stage degrees of freedom and inputs.

*T*

1234

(2*7*)

*X AX BU* = + (28)

{ } {} , *T T X qq U F* = = (29)

*ssss*

*dddd*

11 0 0 0 0 11

1 10 0 001 1

<sup>−</sup> <sup>−</sup>

1234

*ffff*

*dddd*

Similarly in the state space formulation the equations of motion of the system can rewritten

.

<sup>=</sup> <sup>−</sup> <sup>−</sup> −−−−

The selection matrix T of the generalized forces is:

where the state vector X and the input vector *U* are:

as:

$$A = \begin{bmatrix} 0 & I \\ -M^{-1}K & -M^{-1}C \end{bmatrix} \\ B = \begin{bmatrix} 0 \\ M^{-1}T \end{bmatrix} \tag{30}$$

The relationship between input and output can be represented as:

$$Y = CX + DUI\tag{31}$$

where *Y* is the output vector that contains the derivative time of the generalized coordinates (25):

$$Y = \begin{bmatrix} \dot{q} \end{bmatrix}^T \tag{32}$$

C is the output matrix and *D* the feedthrough matrix:

$$C = \begin{bmatrix} 0 & I \end{bmatrix}, D = \begin{bmatrix} 0 \end{bmatrix} \tag{33}$$

#### **5. Control design & results**

In this section three different control strategies to damp vibration and isolate the machine are proposed: a) Feedback control by the use of a Lead-Lag technique, b) Feedforward control and c) Modal control. The experimental validation has been carried out just for the first strategy as proof of the correctness of the modeling approach. Feedforward and modal controls are validated numerically.

#### **5.1. Feedback control**

The control action is designed to achieve two main goals: active isolation of the payload from the ground disturbances and vibration damping during the machine work processes. These two actions allow to operate on the stage without external disturbances. Dynamics on *XZ* and *YZ* -planes are considered the same and decoupled so the control laws along the two planes are equivalent.

Furthermore, from the control point of view, the adopted model is oversized with respect to the control requirements if the goal is the isolation of the stage. As a matter of fact, in this case a two degrees of freedom model is sufficient while if also the dynamics of the frame is required to be controlled, then a 4 dof model is necessary.

The considered system can be regarded as intrinsically stable due to the presence of mechanical stiffness between the stage and the frame, which allows to obtain a negative real part for all the eigenvalues of the system.

Root loci of the system in open and closed loop configurations are reported in Figure 17.

**Figure 17.** Root loci of open loop (a) and closed loop (b) configurations (Circles: zeros; Crosses: poles).


Poles and zeros of the system are reported in Table 4.

536 Smart Actuation and Sensing Systems – Recent Advances and Future Challenges

**Figure 17.** Root loci of open loop (a) and closed loop (b) configurations (Circles: zeros; Crosses: poles).

**Table 4.** Poles and zeros of the system

Since the system along *YZ* ( *XZ* ) presents one actuation point and a couple of sensors (frame and stage velocities), a solution with a SISO control strategy is not feasible. A simplest solution to this problem considers the difference between the measured velocities as the feedback signal, so the system can be assumed as SISO and the control design becomes simpler.

Figure 18 shows that the system dynamics has a peak at 1.8 Hz related to the stage and higher modes related to the interaction of the stage with the frame and the ground at 10 Hz and beyond.

**Figure 18.** Vibration damping action. Transfer function from the actuator force to the difference of frame and stage velocities ( ( ) *S F ACT qqF* − ). Open-loop vs Closed-loop. Solid line: experimental; Dashed line: numerical.

The feedback controller is focused on damping the mode related to the stage by adding on the loop a lead-lag compensator.

The two actions can be expressed as:

$$\begin{aligned} \mathbf{C}\_{LAG} &= \frac{\mathbf{s} + \mathbf{z}\_{LAG}}{\mathbf{s} + p\_{LAG}}\\ \mathbf{C}\_{LEAD} &= \frac{\mathbf{s} + \mathbf{z}\_{LEAD}}{\mathbf{s} + p\_{LEAD}} \end{aligned} \tag{34}$$

The *CLAG* action is used to improve the transient response at low frequency, while the *CLEAD* is useful to increase the stability margin of the closed-loop system.

Therefore the resulting Lag-Lead action allows to compensate the critical phase behavior of the geophones and furthermore guarantees a quick damping action with good levels of stability margins.

The experimental tests have been performed to validate the two control actions. Figure 18 shows the numerical and experimnental frequency response function in open loop and closed loop, obtained from the actuator force to the velocity measured on the stage. The force acts both on the stage and the frame, the dynamics of both the subsystems are visible. The vibration damping effect of the control action is validated on the stage mode (1.8 Hz peak) and the good correspondence shown between the simulated and experimental response is useful to validate the modeling approach.

and beyond.

Dashed line: numerical.

stability margins.

the loop a lead-lag compensator.

The two actions can be expressed as:

Figure 18 shows that the system dynamics has a peak at 1.8 Hz related to the stage and higher modes related to the interaction of the stage with the frame and the ground at 10 Hz

**Figure 18.** Vibration damping action. Transfer function from the actuator force to the difference of frame and stage velocities ( ( ) *S F ACT qqF* − ). Open-loop vs Closed-loop. Solid line: experimental;

*LAG*

*s z <sup>C</sup> s p*

+ = +

*s z <sup>C</sup>*

*LEAD*

*CLEAD* is useful to increase the stability margin of the closed-loop system.

response is useful to validate the modeling approach.

The feedback controller is focused on damping the mode related to the stage by adding on

*LAG*

*LAG LEAD*

*s p*

+ = +

The *CLAG* action is used to improve the transient response at low frequency, while the

Therefore the resulting Lag-Lead action allows to compensate the critical phase behavior of the geophones and furthermore guarantees a quick damping action with good levels of

The experimental tests have been performed to validate the two control actions. Figure 18 shows the numerical and experimnental frequency response function in open loop and closed loop, obtained from the actuator force to the velocity measured on the stage. The force acts both on the stage and the frame, the dynamics of both the subsystems are visible. The vibration damping effect of the control action is validated on the stage mode (1.8 Hz peak) and the good correspondence shown between the simulated and experimental

*LEAD*

(34)

**Figure 19.** Impulse time response, force from the actuator and velocity measured on the stage. Openloop (a), Closed-loop (b), Force exerted by the actuators. Solid line: experimental results. Dashed line: numerical results.

A further demonstration of the correctness of the damping action is the velocity time response reported in Figure 19. In this case the system is excited with an impulse from the actuator and the velocity is measured on the stage. Numerical and experimental responses are superimposed to provide a further validation of the model (the position time response is not reported since the machine is not provided with displacement sensors and hence this validation could not be possible to performed). Figure 19.a shows open loop response, Figure 19.b shows closed loop response while in Figure 19.c the force exerted by the actuators is reported.

The excitation coming from the laser-axis action on the stage is controlled in an effective way as shown in Figure 20 where the numerical transfer function between a force impulse on the stage and the related measured velocity is reported.

**Figure 20.** Vibration damping action. Transfer function from a force applied on the stage to the velocity measured on the stage ( ) *S S q F* . Numerical response. Solid line: closed-loop; Dashed line: Open-loop.

**Figure 21.** Active isolation action. Transfer function from a simulated ground velocity to the velocity measured on the stage ( ) *S G q q* . Numerical response. Solid line: Open loop configuration. Dashed line: Closed loop configuration.

The active isolation action is verified by simulating the excitation coming from the ground. The experimental test in this case has not been performed since in reality it is difficult to excite the machine from the ground in a controlled and effective way. Nevertheless the model is reliable as proved in Figure 14 and the obtained results can be assumed as a good validation of the control action.

Figure 21 illustrates that the closed loop system is capable to reject the disturbances coming from the ground in an effective way.
