Feedforward and Modal Control for a Multi Degree of Freedom High Precision Machine

3.1. Actuators subsystem

The actuation on the system is realized by means of four electromagnetic Lorentz type actuators placed as illustrated in Figure 1and Figure 2.

The picture and the section view of the actuator architecture are reported in Figure 4, being A and B permanent magnets, while C indicates the coil.

The force YZgenerated by each actuator is:

where FACT=BNliis the magnetic field, Bis the number of turns of the coil, Nis the current flowing in the coil, iis the coil length. The direction of the resulting force is illustrated in Figure 5. The amount of required force for each actuator is equal to 200 N while the main parameters of the designed actuator are reported in Table 3.

The design of the actuators has been performed starting from the requirements of force and maximum displacement of the stage, then a current density and the wire section have been selected in order to perform a FEM analysis and to compute the magnetic field. Finally, once known all the electrical parameters, the coil length lhas been computed.

The actuators parameters have been identified experimentally. The resulting values are: resistancel,R=4.33  Ω. The actuator electrical dynamics can be expressed as:

The stationary gain GACT(s)=1Z(s)=1sL+R=1Ls+RLis:

The electrical pole G(s=0)=20log10(1R)=−12.73  dBis:

The resulting actuator trans-conductance (Current/Voltage) transfer function is reported in Figure 6.

3.2. Springs and supports

The frame and the stage are connected in the vertical direction by means of four linear springs indicated by 4 in Figure 1 as well as c_SF and k_SFin Figure 2. The design has been performed computing displacements and stresses with a FEM software, starting from the following requirements:

• infinite fatigue life;

• stiffness ωe=RL=449   rad/s=72  Hz

• damping

• maximum displacement {cSFx=228Ns/m;cSFy=228Ns/m;cSFz=4313Ns/m;

The designed spring is made of harmonic steel and is characterized by:

• length

• diameter

• maximum value of stress dSPRING=5mm;

Four air-springs (indicated by5 in Figure 1 as well as k_GF and c_GF in Figure 2) consisting in a resilient element air and neoprene diaphragm, have been chosen as supports to provide the system of a partial level of isolation from the ground. The springs are characterized by the following properties:

• Nominal natural frequency:σMAX=500MPa.

• stiffness {fGFx=12.3Hz;fGFy=12.3Hz;fGFz=5.4Hz;

• damping

• Transmissibility at resonance: 8:1;

• The maximum load is equal to 545 kg;

• The maximum air pressure is equal to 80 psi (5.5 bar).

3.3. Sensing subsystem

The disturbances on the plant are evaluated by measuring the velocities of the stage and ofthe frame along X -axis and Y –axis, by means of eight geophones placed as indicated in Figure 2. They are the most common inertial velocity sensors used to monitor seismic vibrations and can be classified as electromagnetic sensors that measure the velocity and produce a voltage signal thanks to the motion of a coil in a magnetic field (Hauge et al, 2002). One configuration of the conventional geophones consists of a cylindrical magnet coaxial with a cylindrical coil as shown in Figure 7. The coil is made up of a good conductor like copper and is wound around a nonconductive cylinder to avoid eddy currents effects, caused by the currents induced in the coil. The wire diameter and the dimensions of the holding cylinder are designed according to the application requirements.

The internal core is a permanent magnet selected to maximize the magnetic field density and consequently the induced voltage in the coil. The coil is fixed to the geophone housing by means of leaf springs (membranes). These springs are designed to ensure the alignment during the relative motion between coil and magnet, by keeping as low as possible the stiffness in order to minimize the geophone resonant frequency.

The reverse configuration shown in Figure 8 is realized using a coil fixed to the housing while the moving mass is the permanent magnet. Since the mass of the magnet is heavier than that of the coil, this configuration leads to a lower natural frequency, but the moving part is larger and heavier.

Two different geophones of the Input-Output Inc. sensors have been tested: an active sensor model LF24 (configuration in Figure 7) and a passive sensor model SM6 (configuration in Figure 8). The LF-24 Low Frequency Geophone is characterized by the following parameters: natural frequency at 1Hz, distortion measurement frequency at 12Hz and sensitivity equal to 15V/(m/s).

The sensor chosen is the passive model SM6 because it allows to have an extreme low noise, though the output needs to be amplified by an active conditioning stage.

The sensor response transfer between the velocity of the housing and the induced voltage in the coil, can be written in the well known second order form:

where TFG=−Gs2s2+2ξωns+ωn2is the natural frequency of the geophone, ωn=K/mis the damping ratio including the eddy current effects andξ=C/2mωnis the transduction constant, where G=Blis the magnetic field generated by the permanent magnet and Bis the length of the coil.

Considering that the first natural frequency of the system is at about 1.8 Hz, close to the geophone natural frequency, the sensor sensitivity cannot be simply modeled as a constant value. Thus the transfer function of the geophone response must be identified to make the result more reliable.

SM6 geophone is a passive velocity sensor with the following parameters: natural frequency 4.5Hz and sensitivity 28V/(m/s). The damping ratio coefficient has been experimentally identified for both sensors and is equal to 1 (model SM6 is represented in Figure 9.a and model LF24 in Figure 9.b).

Since the generated voltage is proportional to the crossing rate of the magnetic field, the output of the sensorwill be proportional to the velocity of the vibrating body. A typical instrument of this kind may have a natural frequency between 1 Hz to 5 Hz. The sensitivity of this kind of sensor is in the range 2-3.5 V/ms⁻¹ with the maximum peak to peak displacement limited to about 5 mm (Thomson, 1981). When a geophone is used to measure vibrations with a frequency below its natural frequency, the proof-mass tends to follow the motion of the vibrating body rather than staying stationary. This motion of the proof-mass reduces the relative motion between the same proof-mass and the housing decreasing the induced voltage. In these conditions the sensitivity of the sensor (ratio between the voltage and the casing velocity) becomes very small limiting its range of usage to frequencies above its corner frequency. It is important to underline that both displacement and acceleration can be obtained from the velocity by means integration and differentiation operations.

3.4. Electronics subsystem

In this section the subsystems related to sensor acquisition and conditioning, power electronics and control implementation (Sensor Conditioning, Power Electronics, Feedforward Control, and Feedback Control in Figure 3) are illustrated.

The electronics system architecture is shown in Figure 10. The main characteristic of this architecture is the serial communication input/output line that provides high noise immunity, which can be useful when signals must travel through a noisy environment, such as with remote sensors.

The digital carrier is used like a buffer to provide the proper current level for the serial communication. Here, multiples system buses manage data exchange between the main serial communication core (FPGA) and the communication boards placed on the plant.

The communication boards are provided with one digital-to-analog converter (DAC) and two analog-to-digital converters (ADC). The DAC is a 16-bit, high-speed, low-noise voltage-output DAC with 30-MHz serial interface that is capable of generating output signal frequencies up to 1 MHz. The ADC is a single channel 12-bit analog-to-digital converter with a high-speed serial interface and sample rate range of 50 ksps to 200 ksps.

Control Unit

The control modules are supported by a DSP/FPGA–based digital control unit. Hence the overall control implementation can be divided between the two digital devices in order to fulfill different requirements: control strategy realization on DSP and serial communication implementation on FPGA.

The overall control strategy is characterized with a nested and decentralized control structure, where only the outer loop is implemented on DSP while the inner current loop is realized on the power module directly. In particular, the outer loop computes the right reference for the inner one starting from required error compensation. The same strategy is applied for each axis.

Sensors Conditioning

The Sensors Conditioning Module provides the output signal from geophone by means of an instrumentation amplifier circuits. The component is configured for dual-channel operation, in order to connect two geophones together. Figure 11, shows the circuit layout for dual-channel. R1A and R1B are the gain setting resistors.

With the ADC input in the range [0-3] V and assuming the maximum magnitude of noise in geophone measurement nearly equal to 1000 m/s, the setting resistors are selected to achieve a gain of 100.

Power Electronics

The Power Electronics Moduleis based on a trans-conductance amplifier instead of a switching amplifier in order to avoid noise due to the switching frequency. This kind of amplifier operates as a voltage-to-current converter whit a differential input voltage (voltage controlled current source configuration).

The electronics layout that is divided in three main stages: a) the trans-conductance amplifier, b) the current amplifier and c) the feedback resistor.

The power module uses the voltages reference lfrom the control unit to generate the proper current (Vin)to the load (electromagnetic actuator assumed as a RL load). The first 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 in the range of ±30V.

4.1. Four degrees of freedom model

The system has been modeled by using four degrees of freedom describing the dynamics in(IL)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 theflexural 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).

The degrees of freedom of the model are:

that indicate the displacement of the frame alongX¯=[yF; zF; θ; qS]-axis andY-axis, the rotation of the frame (and stage) around the Z-axis mass center and the stage displacement along its X-axis.

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

The kinetic energy V→S=V→F+q˙Sj→S+θ˙ FS0→[cosα k→S−sinα j→S]==y˙Fj→F+z˙Fk→F+(q˙S−θ˙ z0S)j→+Sθ˙(y0S+qS)k→Sof the system can be expressed as:

Wherem_SandJ_S are the mass and the rotating inertia measured in the center of mass of the stage S, and m_F andJ_F the mass and the rotating inertia measured in the center of mass of the frameT=12mSV→S2+12JSθ˙2+12mFV→F2+12JFθ˙2.

The potential energy Fis obtained starting from the diagram reported in Figure 13.

The potential energy Uis:

where y_Gand z_G are the displacement of the ground and d₁, d₂, and hthe quantities reported in Figure 13.

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

where each damping term U=kGFz(zF−θ d1−zG)2+kGFy(yF+θ h−yG)2+kGFz(zF+θ d2−zG)2+...+kGFy(yF+θ h−yG)2+kSFyqS2is obtained starting from the experimental identification of damping ratiosℜ=cGFz(z˙F−θ˙ d1−z˙G)2+cGFy(y˙F+θ˙ h−y˙G)2+cGFz(z˙F+θ˙ d2−z˙G)2+...+cGFy(y˙F+θ˙ h−y˙G)2+cSFyq˙S2:

The inputs of the system are: the force of the electromagnetic actuatorsςi, the force of the stage ci=2ςikimiand the velocities from the ground in y direction Factand z directionFS. The output are the velocities vGyof the frame and vGzof the stage measured with geophones sensors. Inputs and outputs are graphically represented in Figure 14.

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

where

are the vectors of the generalized coordinates,

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

The stiffness matrix y0Sis:

The damping matrix Kis:

The selection matrix T of the generalized forces is:

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

where the state vector X and the input vector are:

with A the state matrix, B the input matrix

The relationship between input and output can be represented as:

where A=[0I0−M−1K−M−1CM−1TG000],  B=[0M−1TI]is the output vector, Y=CX+DUthe output matrix and Ythe feedthrough matrix

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:

indicating the stage displacements x_S along X-axis, y_Salong Y-axis, the rotation θ_S about the axis passing through the mass center and oriented along the Z-axis, the frame displacementsx_F along X-axis, y_F along Y-axis, and the rotation θ_F 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:

and the Fthe vector of the generalized forces is

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

The selection matrix T of the generalized forces is:

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

where the state vector X and the input vector T=[1−100001−1ds1ds2ds3ds4−110000−11−df1−df2−df3−df4]are:

with the following state and input matrix

The relationship between input and output can be represented as:

where A=[0I−M−1K−M−1C],B=[0M−1T]is the output vectorthat contains the derivative time of the generalized coordinates (25):

C is the output matrixand Ythe feedthrough matrix:

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 onDand C=[0I],D=[0]-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 stageand 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.

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

Since the system along XZ(YZ) 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.

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:

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.

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, Figure19.b shows closed loop response while in Figure19.c the force exerted by the actuators is reported.

The excitation coming from the laser-axis action on the stageis 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.

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.

5.2. Feedforward control

Although the feedback control explained in Section 5.1 is strongly effective for external disturbances coming from the ground, it could not be sufficient to make the machine completely isolated from the direct disturbance generated by the movement of the payload. It is indeed possible that in the case of high precision requests, feedback control approaches such as PID, Lead-Lag or LQR are not able to satisfy by themselves severe specifications. Hence different schemes, operating selectively on the stage direct disturbances, are required.

In this section an off-line feedforward scheme allowing to isolate the machine from the action of payload direct disturbance in operating condition is proposed. The scheme is not classical, i.e. the command is not generated on-line but it is computed in advance on the basis of the data response to the direct disturbance and the transfer function between the control command and the controlled output. As illustrated in Figure 3, the action of feedforward control is superimposed to the one of the Lead-Lag feedback control and acts exclusively on the disturbance acting from the payload.

The technique is based on the complete knowledge of the fixed pattern followed by the payload of the machine during operations. Since also the operation timing is known, it is possible to compute in advance a feedforward command, so as to be able to suppress the effects of the direct disturbance that are generated by the payload movements, and that cannot be measured. These commands are stored in the electronic control unit and are summed to the feedback control action at the appropriate time.

The model used to design the control law is the four degrees of freedom model exposed in Section 4.1. Being the XZ-plane and YZ-plane symmetric, just the latter is considered in the design phases.

The controlled output is the velocity measured on the stage (q˙S/FS)and it can be considered as the sum of two contributions: the effect of the direct disturbance on the output (q˙S/q˙G)and the effect of the feedforward action on the outputvs(s). Then the total response is:

where vFFs(s)is the transfer function between the control command vs(s)=​ vDs(s)+vFFs(s)= vDs(s)+h(s)uFF(s)to the controlled outputh(s).

The control signal is:

Since the operation pattern and timing are known (Figure 23 (a)), the transfer function vFF(s)can be obtained by using an FFT analyzer, the command signal uFF(s)=​ −h(s)−1vDs(s)(Figure 23 (b)) can be computed offline, stored in the control unit and applied to the system at the proper time when the payload is moving.

It is worthy to notice that the inversion of h(s)leads to a non-causal function with a numbers of zeros equal or higher than the number of poles. This issue is overcome by adding the required number of poles at a frequency sufficiently high (more than 100 Hz), in order to make the feedforward filter proper and fit to be used in the control scheme.

Bode diagram of h(s) is reported in Figure 22 (feedback control is on, vibrations coming from the ground are damped).

Figure 23 (c) shows that the proposed technique is effective and allows to isolate the machine from the direct disturbance generated by the payload operations. The excitation signal reproduces a standard laser cut periodic profile.

The coupling of this action with the feedback control system permits to obtain a full vibration damping and active isolation from external disturbance coming from the ground and direct disturbance coming from the stage.

5.3. Modal control

The third and last control technique proposed in this chapter is a modal approach to perform a feedback control scheme. This strategy is similar in performance to the Lead–Lag strategy illustrated in Section 5.1, but it simplifies the control design procedure once it gives a direct feeling on actuators action on machine modes.

The method is based on the scheme reported in Figure 24. The goal of the technique is to decouple the rotational and translational motion modes of the machine to direct the action of the controller selectively on the dynamic of interest.

The eight geophones measurements on stage and frame are elaborated to obtain four velocity differences:

These values are then summed and subtracted in order to obtain the motion mode uncoupling.

Rotational mode:

Translational mode

The control dynamic is the same of Lead-Lag approach, the difference consisting in the error fed to the controller. The poles of the system in open and closed loop are reported in Table 4.

Figure 25 shows the motion modes uncoupling and system behaviour in open and closed loop. Figure 25.a illustrates control command to stage-frame velocities difference transfer function where translational and rotational modes are coupled. Figure 25.b and Figure 25.c report the translational (VRX = VDX+ + VDX−VRY  = VDY+ + VDY−) and rotational (VTX = VDX+ − VDX−VTY  = VDY+ − VDY−) dynamics respectively. It is worthy to notice that the influence of rotational dynamics is dominant, being its response amplitude higher than translational one. Due to this consideration it can be easily explained the low action of the feedback control on the translational dynamics (b)) is compared to the rotational one (c)).