WO2011055297A1

WO2011055297A1 - A method, of estimating the control parameters of an active-damping system, and corresponding system and computer-program product

Info

Publication number: WO2011055297A1
Application number: PCT/IB2010/054955
Authority: WO
Inventors: Gianpiero Negri; Gianluca Guadagno; Marco Palladino; Giuseppe Anzalone
Original assignee: Sintesi S.P.A.
Priority date: 2009-11-03
Filing date: 2010-11-02
Publication date: 2011-05-12
Also published as: ITTO20090845A1

Abstract

Α method for estimating the control parameters of a system for active damping of the vibrations (20) of a mechanical structure (30), wherein the damping system (20) is able to generate a force via rotation of at least three eccentric rotors. The method includes the steps of : - receiving a value indicative for the total force that the eccentric rotors must generate; and - estimating (206) the phases (PosVibr) and the angular velocities (VelVibr) to be applied to the eccentric rotors as a function of the value indicative for the total force.

Description

"A method of estimating the control parameters of an active-damping system, and corresponding system and computer-program product"

***

TEXT OF THE DESCRIPTION

Field of the invention

The present invention relates to systems and methods for active damping of vibrations.

The invention has been developed in particular in order to estimate the control parameters of an active- damping system applied to a mechanical structure, for example of a machine tool.

Description of the relevant art

In the sector of machine tools the need is particularly felt to reduce the vibrations of the tip of the tool, which might jeopardize the quality of the machining operations and reduce the mean time of use of the tool itself.

A possible index of evaluation of the vibrations is the stability lobe diagram, where appearing on the abscissae is the velocity of rotation of the spindle (in r.p.m.) and appearing on the ordinates is the depth of the pass (in millimetres) . .·

The analyses conducted on said graph enables highlighting of the stable (and unstable) regions for the machining operations on the machine (for example, a lathe or a milling machine) , which are distinguished by the boundary of the lobe.

Figure 1 shows an example of lobe diagram that represents a possible plot without control, NC, and two possible plots with active control of vibrations, AC1 and AC2.

In particular, in the top part of the curve, the machining operations are deemed unstable since the combination of the parameters depth of pass and spindle r.p.m. can trigger regenerative phenomena such as chatter, i.e., an unstable vibration, which jeopardizes the machining quality and can even cause damage to the tool and/or to the machine itself. This means that the intercept on the ordinate corresponding to the minimum of the lobes represents the maximum value of depth of pass before onset of chatter.

The adoption of an effective solution for damping the vibrations enables raising the minimum limit of depth of pass beyond which machining runs the risks of becoming unstable.

This entails the possibility of increasing the depth of pass, and, hence, the capacity of removal of stock by the machine, with a corresponding increase in productivity.

Known to the art are systems for active damping of vibrations for industrial machines, or in general for vibrating structures, which are based upon actuators of an inertial type. Usually, said inertial actuators are based upon the physical principle of generation of a force of reaction on a supporting structure, obtained by accelerating a suspended mass.

For example, the document No. US-A-2006/0157310 discloses an inertial actuator based upon a mass moved via an electromagnetic circuit.

A possible implementation of a device for damping the vibrations is based upon an inertial actuator, a sensor coupled to the actuator, and a simple controller that implements a DVF (Direct Velocity Feedback) algorithm.

A control scheme of this sort is referred to as LAC (Low-Authority Control) . The only parameter for the control system, apart from the position of the sensor/actuator pair, is the feedback gain.

Purpose and summary of the invention The inventors have noted that inertial actuators present a series of advantages as compared to other actuation technologies (for example, of the piezoelectric type) :

1) Non-invasiveness : with the use of said type of actuator no structural modification of the system the vibrations of which are to be damped is necessary. If the control is de-activated, the structure simply exhibits its original behaviour. As soon as the control is re-activated, the structural behaviour remains substantially the same, but the resonance peaks are usually damped.

2) Robustness and stability of the control: thanks to the sensor/actuator configuration, it is possible to obtain a control system that is more stable and robust.

3) The inertial damping is an "additive" process, in the sense that the higher the number of inertial actuators synchronised, the greater the damping obtained .

4 ) No mechanical connections are required: unlike classic actuators which require an anchorage point, inertial actuators are "hooked to the sky", i.e., they do not require any constraint to an external supporting structure .

5) No in-depth knowledge of the modal characteristics of the structure of which the modes are to be damped is required; consequently, no preliminary modal analysis of the structure itself is necessary.

6) Wide operative band: the actuator can damp all the modes sensed by the sensors coupled, and, unlike

"tuned" passive actuators, which cover a single frequency band, it presents an actuation band typically comprised between 25 and 2000 Hz.

7) Compactness: the dimensions of the actuator depend only upon the amplitude of the force that it is intended to generate, and are somewhat contained.

The inventors have, however, noted that the actuators based upon inertial technology available on the market present significant disadvantages. For example, said actuators can generate forces in just one direction and usually present a high ratio between weight and force generated.

The object of the invention is to provide a system for active damping of vibrations that will overcome the above drawbacks.

With a view to achieving the aforesaid purpose, the subject of the invention is a method for estimating the control parameters of a system for active damping of vibrations that presents the characteristics specified in the annexed Claim 1. The invention also regards the corresponding damping system, as well as a computer-program product, which can be loaded into the memory of at least one processor and comprises portions of software code, which are able to implement the steps of the method when the product is run on at least one processor. As used herein, the reference to such a computer-program product is understood as being equivalent to the reference to a computer-readable means containing instructions for control of the processing system so as to co-ordinate implementation of the method according to the invention. Reference to "at least one processor" is evidently meant to highlight the possibility for the present invention to be implemented in a modular and/or distributed form.

Further advantageous characteristics of the invention form the subject of the annexed dependent claims .

All the annexed claims are to be understood as forming an integral part of the technical teaching provided herein in relation to the invention. According to the invention, the system for damping vibrations uses an actuation system constituted by a number of pairs of eccentric rotors via which it is possible to obtain a force of inertia linked to the eccentricity of the rotating masses.

In one embodiment, the action of the pairs of rotors is generated starting from a measurement signal coming from a sensor set in contact with a part of the mechanical structure the vibrations of which are to be damped, by means of an appropriate control strategy.

In one embodiment, according to the type of vibration measured by the structure, the system adaptively develops a force capable of opposing the vibration, thus damping it.

The inventors have noted that an actuation system constituted by a number of pairs of eccentric rotors presents various critical points. In general, a system of this sort is able to generate:

- magnitude of the force;

- frequency of the force;

- instantaneous phase of the force; and

- direction of the force.

Control of such a system is rather simple if the type of force to be damped (for example, vibrations induced by motors) includes only small variations of the parameters (in particular the frequency) around well-defined values.

However, in the sector of machine tools, the disturbance is characterized by a wide variability, both in temporal terms (transient disturbance) and in terms of frequency composition (multimodal disturbance) ; namely, the system for damping vibrations must generate forces that have a composite spectral content, i.e., multimodal, or are markedly non- stationary. In one embodiment, in order to solve this problem, a control method is used capable of compensating non- stationary and/or multimodal vibrational disturbance.

Brief description of the annexed drawings

The invention will now be described with reference to the annexed plates of drawings, which are provided purely by way of non-limiting example and in which:

- Figure 1 has already been described;

- Figures 2 to 5 show possible embodiments of the actuation system;

- Figures 6a and 6b show an example of installation of the actuation system on a machine tool;

- Figure 7 shows the reference system used for each single rotor; and

- Figures 8 and 9 are block diagrams that illustrate a possible embodiment of the control method used within the system for damping vibrations.

Detailed description of embodiments

In the ensuing description various specific details are illustrated, aimed at an in-depth understanding of the embodiments. The embodiments may be provided without one or more of the specific details, or with other methods, components, materials, etc. In other cases, known structures, materials, or operations are not shown or described in detail so that various aspects of the embodiments will not be obscured .

Reference to "an embodiment" or "one embodiment" in the framework of the present description indicates that a particular configuration, structure, or characteristic described in relation to the embodiment is comprised in at least one embodiment. Hence, phrases as "in an embodiment" or "in one embodiment" that may possibly be present in different points of the present description do not necessarily refer to one and the same embodiment. In addition, particular conformations, structures, or characteristics can be combined in an adequate way in one or more embodiments.

The references used herein are only provided for convenience and hence do not define the sphere of protection or the scope of the embodiments.

The idea underlying the solution described herein is the use of an actuation system constituted by at least three eccentric rotors via which it is possible to obtain a force of inertia linked to the eccentricity of the rotating masses.

In one embodiment, the action of the pairs of rotors is generated starting from a measurement signal coming from a sensor set in contact with the part of the mechanical structure the vibrations of which are to be damped, for example a machine tool.

In this way, according to the type of vibration measured by the structure, the system develops a force that able to oppose the vibration.

In one embodiment, the force generated is characterized in that it has a magnitude, instantaneous phase (and hence frequency) , and direction that can be controlled independently and "continuously", so as to follow appropriately and oppose the accelerations or forces that may arise during machining.

In one embodiment, the actuator used for damping is constituted by two pairs of eccentric coaxial counter-rotating rotors, the angular velocities of which can be controlled independently.

In one embodiment, the overall scheme of the damping system envisages the use of the following components and modules:

- a sensor or a number of sensors (preferably accelerometers or load cells) , set appropriately in contact with the mechanical structure the vibrations of which are to be damped;

- the actuation system, coupled with the sensor on the mechanical structure, constituted for example by four independent motors used for driving the eccentric masses in rotation;

- a regulator, appropriately calibrated for implementing the control strategy of the damping system; and

- a module, which implements an estimation method for decomposition of the acceleration signal measured and calculation of the references to be assigned to the individual rotors to obtain a force that will balance the force sensed by the accelerometer so as to oppose the vibrations present in the structure of the machine during machining.

Figure 2 shows a first embodiment of the actuation system 10.

In the embodiment considered, the system 10 comprises four eccentric rotors 100, which are fixed to the shafts of respective motors 102.

In the embodiment considered, the motors 102 are fixed to a supporting plate 104, thus enabling only a rotation of the eccentric rotors about the axes of the respective motors. For example, in the embodiment considered, the supporting plate 104 comprises four cylindrical openings 104a, i.e., one opening for each motor 102. The shafts of the motors 102 are inserted in the openings 104a during installation of the motors 102 on the plate 104, and the eccentric rotors 100 are subsequently fixed on the opposite side of the plate 104 to the shafts of the motors 102.

In the embodiment considered, also provided is a base plate 106, which includes four concave portions, i.e., one for each eccentric rotor 100. Said base plate 106 is fixed to the supporting plate in such a way as to close the eccentric rotors 100 within the plates 104 and 106.

Preferably, also provided are bearings 108, which are inserted in the concave portions of the base plate 106 so as to support the terminal portions of the shafts of the motors 100.

In the configuration shown in Figure 3, the force is generated in the plane of the bottom plate of the device, i.e., in the plane orthogonal to the axes of the rotors, which is preferably also the resting plane of the device itself.

Figure 3 shows a second configuration for the motors 102.

In the embodiment considered, two pairs of motors are mounted on opposite sides of the base plate 106.

In the embodiment considered, also provided are two supporting plates 104, each of which supports two motors 102.

In this case, the force is generated always in the plane perpendicular to the axis of the rotors, but their arrangement enables the actuator to be located so as to generate a component of force normal to the resting plane.

The actuation system hence exerts forces by using rotors unbalanced by eccentric masses since it is able to work in two directions.

In order to be able to control independently magnitude, direction, and phase of the force generated, it is necessary to control precisely the motion of at least three rotors, preferably four rotors.

In one embodiment a modular structure of the actuator is envisaged. Said modular structure can enable arrangement of the pairs of rotors in at least two alternative configurations, which can be used for generating forces with different planes of lie. Figure 4 shows a possible embodiment of a single actuation device.

In the embodiment considered, the actuation device comprises an eccentric rotor 100, carried by the shaft of a motor 102. Also in this case, the motor 102 is fixed via appropriate fixing means to a supporting plate 104. For example, in the embodiment considered, the shaft of the motor 102 is inserted in an opening of the supporting plate 104 (not visible in Figure 4), and the motor 102 is fixed via screws and/or bolts 120 to the plate 104. Next, the eccentric rotor 100 is fixed on the opposite side of the plate 104 to the shaft of the motor 102.

In the embodiment considered, also provided is a base plate 106, which is fixed to the supporting plate 104 via appropriate fixing means in such a way as to close the rotor 100 within the plates 104 and 106. For example, in the embodiment considered, the base plate 106 is fixed to the supporting plate 104 via screws and/or bolts 122.

Also in this case, further bearings may be provided, for example, a bearing 108 so as to support the terminal portion of the shaft of the motor 102 (i.e., a bearing between the rotor 100 and the base plate 106) , and/or a bearing 110 so as to support the shaft of the motor 102 in the opening of the supporting plate 104 (i.e., a bearing between the motor 102 and the rotor 100) .

In the embodiment considered, a zero pin 112 is provided. Said pin can be useful during a procedure of "mechanical zeroing" for all the modules, to guarantee with a high precision the value of the initial phase (or angle) of each rotor.

In one embodiment, at least one sensor is provided (based, for example, upon optical technology) used during a zeroing procedure.

The actuation device also comprises a plurality of coupling means 130 (for example, centring and/or fixing holes) so as to couple the actuation devices to one another in such a way as to form more complex actuation systems .

Figures 5a and 5b show two views of a possible embodiment of an actuation system 10 comprising four modular actuation devices 10a, 10b, 10c and lOd.

In the embodiment considered, the actuation devices of Figure 4 are coupled in such a way as to create an actuation system that functions substantially as the actuation system shown in Figure 3.

In the embodiment considered, each actuation device, or the casing of the device, comprises at least one hole 130a perpendicular to the axis of the motor (for example, a through hole that traverses exclusively the supporting plate 104 or the base plate 106) and at least one hole 130b perpendicular both to the axis of the motor and to the axis of the hole 130a.

In the embodiment considered, the holes 130a and 130b are used for coupling two actuation devices, according to a mutual arrangement such that they are set alongside one another in a common plane perpendicular to the axis of the motors. For example, in the embodiment considered, the hole 130b is a threaded blind hole that enables fixing of two actuation devices via a screw 130c.

In the embodiment considered, the holes 130a and 130b together with the screws 130c enable coupling of the devices 10a and 10b and of the devices 10c and lOd.

In the embodiment considered, each actuation device lOa-lOd, or the casing of the device, also comprises at least one centring hole 130g parallel to the axis of the hole 130a and at least one centring hole 130h parallel to the axis of the hole 130b. In the embodiment considered, said centring holes 130g and 130h are blind holes, in which pins 130i can be inserted for optimizing coupling between two devices.

In the embodiment considered, each actuation device, or the casing of the device, also comprises at least two holes 130d and 130e with axes parallel to the axis of the motor.

In the embodiment considered, the holes 130d and 130e are used for coupling two actuation devices according to a mutual arrangement such that they are set on top of one another, in which the axes of the motors coincide. For example, in the embodiment considered the hole 130d is a through hole, and the hole 130e is a threaded blind hole that enables coupling of two actuation devices via screws 130f.

In the embodiment considered, the holes 130d and 130e, together with the screws 130f, enable coupling of the devices 10a and 10c and of the devices 10b and lOd.

A modular actuation device of the sort hence enables creation of different complex actuation systems that can develop forces in different planes. For example, the modular actuation devices can also be coupled to create a system that functions substantially as the actuation system shown in Figures 2.

However, the individual modular actuation devices can also be mounted in different points on the mechanical structure the vibrations of which are to be damped .

For example, Figures 5a and 5b show two views of an example of installation of four modular actuation devices lOa-lOd on a machine tool.

In the embodiment considered, the actuation devices are mounted directly on the operating element 34 of the machine tool 30 (i.e., on the element that carries the tool, for example the spindle) .

In the embodiment considered, the actuation devices are fixed in such a way that the eccentric rotors of the respective actuation devices lOa-lOd rotate in a common plane perpendicular to the axis of rotation 32 of the tool, in which the devices have substantially the same distance from the axis of rotation 32 of the tool.

Figure 7 shows the reference system used for each individual rotor, where the point P is the centre of mass of the respective eccentric mass 100, which is set at a distance r from the centre (i.e., the shaft of the respective motor 102) and has a mass m.

The position p of the mass can be described as p = r cos(S(t)) x + r sin(d(t)) y (1) where x = (1,0)^T and y = (0,1) ^T are the unit vectors associated to the cartesian axes in the plane, and (t) is the angle to be followed in a counterclockwise direction starting from 0°.

The velocity v of the mass can be described as v = p = -r sin(S(t)) co(t) x H- r cos(d(t)) co(t) y (2) where co(t) = S(t) is the angular velocity.

Finally, the acceleration a of the mass can be described as

a = v = - r cos(9-(t)) or(t) - r sin($(t)) d>(t) x +

(3)

- r sin($(t)) ccr(t) + r cos($(t)) <b(t) y

The dynamics of the accelerations generated in the plane is hence extremely non-linear in the angular variables, and the overall acceleration is the sum of the accelerations produced by the four rotors.

In one embodiment, the system for damping vibrations is made up of an accelerometric sensor (preferably a band-pass sensor) , a processing unit that implements both an estimation method and a numeric control, and the drives and rotors.

In one embodiment, the accelerometer measures the vibration that is to be damped, and the estimation method, on the basis of this measurement, generates the paths for the motors to be sent to the numeric control, which will close the control loops.

In one embodiment, to simplify the treatment; only an arbitrary force in the plane is generated with a band-pass characteristic.

Figure 8 represents a block diagram that describes an embodiment of the damping system 20 applied to a machine tool 30.

In the embodiment considered, in the course of machining an undesirable vibration is generated, which produces a measurable acceleration ACC on the structure of the machine.

Said acceleration ACC is measured by an accelerometric sensor 202 (for example, a biaxial accelerometer) and is passed to a control module 204.

In the embodiment considered, the control module 204 implements a control DVF, in which the measured acceleration (i.e.,. the value ACCmis) is compared with a reference value (which is preferably zero, since the system, in closed loop, has to damp the vibrations of the structure on which it is set) .

In the embodiment considered, the control module 204, starting from the comparison between the measured acceleration ACCmis and the reference, supplies at output an acceleration ACCxeg that the actuation system set downstream has to generate to counter the disturbance .

In one embodiment, the reference value used for closing the external control loop is considered zero. In this case, the system is used for opposing the vibrations of the mechanical structure on which it is set so as to minimize the deviation between ACCmis and the zero reference, that is in such a way as to render the total acceleration measured by the accelerometric sensor zero.

However, the device can carry out an arbitrary action, in the sense that the reference used can also be other than zero and/or variable in time. In this way, the actuation device can generate arbitrary paths (in frequency and amplitude) , which are useful, for example, if the device itself is set on a machine tool and the contribution that it has to generate must be such as to reduce the deviation between the path planned and the one effectively followed by the machine.

In the embodiment considered, said acceleration ACCreg is processed by an estimator 206, via which, instant by instant, the laws of motion of the individual eccentric rotors are determined such as to generate, globally, the required action.

The respective setpoints obtained via the estimator are supplied at input to the regulators of the motors of the actuation system 10 shown for example in Figures 3 and 4, and the motors, by appropriately rotating, generate a force F for damping the disturbance .

In the embodiment considered, thanks to the presence of (at least) four independent motors, it is possible to generate, by varying both their velocity VelVibr and the mutual position of the eccentric masses PosVibr, a force with magnitude, instantaneous frequency, and instantaneous direction that vary in a plane .

In fact, considering that each of the rotors is connected to an eccentric mass, and that the position and velocity of the equivalent eccentric mass can be controlled independently, each rotor can generate a force with component both centrifugal and tangential linked to rotation of the mass.

For example, the overall action of the system constituted by four rotors is, in each of the two orthogonal directions in which it takes place, the resultant of the projections of the forces generated by each individual rotor. The damping system 20 is hence able to:

a) impose an instantaneous frequency of the action generated by controlling the angular velocity of rotation;

b) impose the value of the magnitude of the force generated (comprised between 0 and the maximum value of force linked to the value of angular velocity and eccentricity) by controlling the angle of phase between the rotors of one pair; and

c) impose the direction of the action in the plane by controlling the phase angle between two pairs of rotors .

The characteristics of the force generated can vary adaptively, hence compensating a wide class of disturbance with characteristics of time variance and characterized by a modal content that is not necessarily monochromatic.

The system proposed herein is hence suited for compensating vibrations with characteristics that are non-stationary in time, with frequencies, amplitude, content in frequency that are variable within the operating band of the system.

The solution can also be generalized to the case of 2A7 rotors (with IV preferably even) both for the generation of forces in the plane and for the generation of forces in space. In one embodiment, the control module 204 and/or the estimator 206 are implemented within a processing unit, for example via portions of software code which can be executed on at least one microprocessor or computer.

In one embodiment, the estimator 206 is made up of two submodules:

- the first submodule 206a executes a double pseudo-integration of the acceleration signal measured ACCmis; and

- the second submodule 206b estimates phases PosVibr and angular velocities VelVibr to be assigned to the rotors.

In one embodiment, both steps are executed by purposely provided Kalman filters.

In one embodiment, the pseudo-integration exploits the fact that the measured acceleration ACCmis is a band-pass signal in order to prevent an uncontrollable drift in time of the integrated signals.

In one embodiment, the estimation of the phases

PosVibr and of the angular velocities VelVibr is carried out by reformulating - the problem in the complex domain exploiting the properties of phasors.

In one embodiment, to formulate the double pseudo- integrator, a zero-reference stationary Kalman filter is used. In fact, since the signal of the accelerometer is of a band-pass type, its stationary mean value is zero, as likewise its integrals.

In one embodiment, the signal coming from the accelerometer 202 is normalized with respect to r to obtain, for each rotor,

p = p / r = cos(d(t)) x + sin(S(t)) y (4) v = v / r = - sin(&(t)) co(t) i + cos(d(t)) co(t) y (5) a = a / r = cos(S(t)) co-(t) - sin(&(t)) d(t)J x +

(6) sin(S(t)) ar(t) + cos(&(t)) cb(t) y

where the total normalized quantities are the sum of those for each rotor.

In one embodiment, the normalization factor designated by r includes also possible gains for the control schemes upstream.

In one embodiment, concentrating upon just one channel (i.e., a single axis) of the acceleromete , the following system in the state space is applied in the submodule 206a:

X F · x_k + G · _k + w (7) y_k = H · x_k + v, (8) where :

- x_k e (with n = 2) is the state made the concatenation of normalized position

normalized velocit v :

(with 1 = 1) is the ■ input of the normalized acceleration:

(10)

- y_k e 9 " (with m = 1) is the measurement (which ero) , and

- the vectors w . e 9Γ and are the disturbance on the state and the noise on the measurements, respectively.

In one embodiment, the matrices F e , G e 9r^lxJ, H e ¾^m^of the system are

F = (11) v° ¹

H = (l 0) (13)

In one embodiment, it is assumed that w_k « N(0, Q) is a gaussian noise with covariance Q e 3V" , and v_A. « N(0, R) is a gaussian noise with covariance

R e m^m'^m .

In the embodiment considered, the double pseudo- integrator provided by means of a zero-reference Kalman filter forces the normalized position and the normalized velocity to have a zero stationary mean value, but for a variability linked to the uncertainty of measurement having covariance R e ¾^{J,r m} _

n one embodiment, said covariance matrices are:

R = (N / 2)^" / 12 (15) where N is the number of rotors and At is the sampling pitch .

In the embodiment considered, the matrix Q depends upon the measurement uncertainty of the accelerometer

In the embodiment considered, the value determined for R results from the consideration that the sum of the N normalized positions can be conservatively assumed as a random variable evenly distributed in the interval [-N/2 , Λ7/2 ] .

In one embodiment, a discrete Kalman filter is used, which exploits the fact that y is 0.

In one embodiment, the state vector is then calculated via the equation

χ_λ- = A χ_Λ__! + G a _k (16) where the matrix A is defined as: A = F(I — K · H) (17) where I is an identity matrix of appropriate size and K e y\^m'n is the Kalman gain, as follows:

K = PH^r(HPH^T + R)^_1 (18) and P e W^'*" is the covariance matrix of the error of estimation of the state.

In one embodiment the Kalman filter is applied in its complete formulation. Since the aforesaid Kalman filter is rapidly convergent, in one embodiment, the matrices A and G are pre-calculated to save computational time, accepting a slight deterioration in the quality of the estimate.

For example, in one embodiment, for the matrix P the stable solution of the discrete Riccati equation associated to the Kalman filter is used:

P = dare F^r,H^T,Q,R) (19) where the function X = dare{A,B ,Q,R) of Matlab© calculates the solution of the discrete Riccati equation :

A^rXA - X - A^rXB(B^TXB + R)^-1(B^:rXA + Q) = 0 (20)

In one embodiment, substantially the 'same structure is used on both channels of the accelerometer .

Figure 9 shows the block diagram of a possible implementation of Eq. (16) for one of the (at least) two channels of the accelerometer.

In the embodiment considered, the previous state vector Xk-i is multiplied by the matrix A in a block 2060, and the normalized acceleration a is multiplied by the matrix G in a block 2062.

The results of the multiplications 2060 and 2062 are added in a block 2064 to yield the new state vector x_kr i.e., the current position p_k and velocity ^. Finally, the state vector x_k is saved for an instant of time in a block 2066 to yield the previous state vector x^-i for the next instant of time.

The results for each axis of the accelerometer are supplied to a complex estimator, the purpose of which is to derive phases and angular velocities of the rotors starting from the estimation of the sum of the N normalized positions obtained in the preceding step.

In fact, the inventors have noted that thanks to the normalization it is possible to treat the sum of the N normalized positions along the axes x and y as if it were the sum of N phasors .

In one embodiment, said estimator is formulated in the complex field to render more convenient decomposition of the signal of "desired" acceleration in the references to be assigned to the four rotors in order for the overall action exerted to correspond effectively to the desired acceleration.

In one embodiment, for each rotor, the following first-order autoregressive (AR) linear model is assumed for the evolution of the an le of phase:

= ®i,_k (2D where i is the index of the rotor and k is the time index.

In the embodiment considered, the following parameters are defined:

a_iik = e^j0'' (22) β;_,, = e-¹⁸- (23) β^* = e^~j -^l: (24)

In the embodiment considered, the presence of the complex-conjugate phasor β is necessary in order to write the observation equation, as will be clarified hereinafter .

In one embodiment, the equations that describe the time evolution of the state for each rotor are the following :

In the embodiment considered, an evolution with constant angular velocity is hence assumed, for reasons of simplicity.

Given the definitions, we can write

(28) s_iik = sin 9_iiJt (29) ω ,. = log (a_iiA.) / j t (30)

At

where the function log indicates the natural- logarithmic function.

In one embodiment, the state of the estimator is made up of the concatenation of the phasor parameters for each of the rotors, whereas the measurements coming from the estimation of the double pseudo-integrator are the estimated normalized position^' and velocity, respectivel in the directions x and y,

In one embodiment', in view of the not-linearity of the equations, the matrices F_k and H_k are calculated as the Jacobians of the transformations. Given that each rotor is independent, the matrix F_k is a block diagonal matrix, whereas the matrix H_k is the horizontal concatenation of independent blocks

F,l,k 0 0 0

0 0

F„ = (32)

0 0

0 O O F,

where F and Hi _k are the i-th block of the matrices

In one embodiment, the state covariance P of the previous estimator is used as uncertainty on the measurement y .

In one embodiment, the uncertainty of the complex state is set to (<¾_ma. At^~)^~ for just the states representing the angular velocity, where co_raa., is the maximum angular acceleration allowed by the rotor.

However, both of the uncertainties can be optimized for the specific application.

In one embodiment an extended Kalman filter (EKF) is used.

An EKF is a classic Kalman filter in which the equations representing the evolution of the state and the equations representing the measurement are, generally, non-linear. Since the evolution in time of the state-covariance matrix P depends upon the matrix F and the Kalman gain K depends upon the state-covariance matrix P and the observation matrix H (see the equations of the double pseudo-integrator) , it is necessary to obtain a linearized form of the aforesaid matrices at each instant in time k. In general, if

x._k+1 = f (x_A.) + w_t (36) y, = h(x_t) + n, (37) w, « N(0, Q,) (38) n_i: « N(0, R,) (39) is a generic non-linear system, the equations of an EKF for each instant k are the following:

Xk + 1 = f (x, (41)

= ^F^ + ^Q _K (4:

x,_+I = x;₊₁ + K_k(y_k - h(x-₊₁)) (45)

P_A-_{+ I} = ^p ₊i - K ,H,P _{+ 1} (46)

In one embodiment, downstream of the operations of updating of the state, a normalization thereof is carried out.

In one embodiment, also the effects of saturation on the angular velocity are considered prior to renormalization .

In one embodiment, to obtain the estimate of the phases and of the angular velocities of the rotors, the following calculation is carried out on the state vector:

co_{i A} = arg (a_{i A}. /At) (47)

&_lik = atan2[¾(- j(p. _k - ^ , δ ,_,, + J (48) where the symbol indicates extraction of the real part to prevent minor numeric uncertainties from possibly falsifying the calculation of the arctangent.

In one embodiment, as an alternative to the Kalman filter that functions as pseudo-integrator, in the submodule 206a a digital input filter of a band-pass type that behaves as integrator in the band of interest is used for actuation. For example, in one embodiment, said filter is a cascade of a band-pass filter and a set of integrators .

In addition, other estimators may also be used in the submodule 206b.

For example, in one embodiment, the submodule 206b estimates the control parameters of the motors only as a function of the pseudo-velocity v. For example, in this case, the submodule 206b can integrate the acceleration ACCreg in the band of interest even just once.

In the embodiment considered, an EKF can be used, which operates on the aforesaid pseudo-velocity signal.

For example, in one embodiment, a model of constant-velocity evolution is used for the phases:

(49)

- [Θ, af_k = x_k is the state of the model; - Θ is a [4x1] vector of the phases θ_χ , where the phases Q_i correspond to the phases i¾ described previously, with i = 1, . . . , N;

- ω is a [4x1] vector of the angular velocities

CO; ;

- I is a [4x4] identity matrix;

- At is the sampling period; and

- v is a [4x1] vector of gaussian noise with zero mean and covariance Q.

In one embodiment, the covariance Q is the following [8x8] matrix:

0 0

(50)

0 Ισ^:

where σ depends upon the maximum acceleration that the motors have available.

In one embodiment, the model of pseudo-velocity observation is regulated by the following non-linear law

where is a gaussian-noise vector with zero mean and covariance R.

For example, in one embodiment, the covariance R is a [2x2] matrix linked to the uncertainty of the accelerometer and to the processing to which the acceleration signal has been subjected to obtain velocity .

In the embodiment considered, in order to guarantee the coherence between estimated the phase and angular velocity, the step of prediction of the Kalman algorithm is hence carried out on the entire state vector, and the correction step is carried out on just the velocity part, moreover controlling that the variation of ω between one instant and the next can be implemented with the maximum angular acceleration that the motors have available. The updated estimate of the phase can be derived by integration of the new angular velocity .

For example, in the embodiment considered, after prediction of the state vector and calculation of the Kalman gain, the term of correction of the angular velocity is calculated by applying the following equation

d ω = z (52) where :

- z is the innovation vector, given by the difference between the observation and its reconstruction obtained by means of the predicted state, namely: sin(^. )

z = y - (53) cos(#;)

i=l

- K_w is formed by the rows of the Kalman gain corresponding to the part of angular velocity of the state .

If - d CD_ma.. < d ω < d Q_ma„ , then ω is directly corrected by means of the relation ω = ω^" + d ω ; otherwise, d ω is saturated at its maximum value, after which the correction is made.

In one embodiment, a further control is carried out on the value of ω , since the motors cannot turn beyond a certain velocity. Also in this case, in the case where the maximum limit is exceeded saturation is effected.

In the embodiment considered, for implementation of the extended Kalman filter the non-linear model and measurement functions are linearized. In particular, in the embodiment considered, the model equations are linear, whereas the equations that describe the observations need to be linearized - by means of the calculation of the Jacobian - around the prediction ( x- ) of the current state.

In one embodiment, in the extended Kalman filter described above, it is envisaged to introduce a fictitious observation with the aim of directing the rotation of the motors towards a particular frequency considered of interest. For example, said indication could be obtained from a preliminary analysis of the data of a machining operation or from an experimental modal analysis of the structure of which the vibrations are to attenuated.

In the embodiment considered, the model of evolution of the state does not change with respect to the one defined previously, whereas added to the vector of the observations are " fictitious measurements " co_r calculated as a function of the frequency of interest. In particular, the following fictitious-measurement vector is defined in way that two motors will turn in one direction and two in the o osite direction:

In the embodiment considered, the new vector of the observations to be used in the steps of correction of the extended Kalman filter is hence - a>_iik sin (0 _i:

= l

Ί

∑ ∞i,A- cos (9_J(J

i=l

+ W, (55)

- ω ,.

- co_r

Said vector is characterized by a covariance matrix R - which is diagonal and of size [6x6] - where the first two elements are calculated taking into account the uncertainty of the accelerometer and the presence of the input filter, whereas the variance of the fictitious measurements can be chosen according to how much it is desired for the effective rotation of the motors to differ from the one suggested. The coherence between the estimated phase and angular velocity is guaranteed with the technique described previously .

As an alternative to the versions described previously, which seek to obtain separately the four steps to be imparted on the motors to reconstruct the control action, it is also possible to formulate a filter that operates on a pseudo-position signal and attempts to evaluate characteristic parameters thereof useful for reconstruction of the control action.

For example, in one embodiment, the signal ACCreg is processed by means of an input filter 206a, which has a double-integrator behaviour.

In this case, the signal y of observation of the pseudo-position can be described, for example, by the followin relation:

where, in relation to the control action, ψ defines 30 T/IB2010/054955

the direction, φ the amplitude, and Θ the frequency thereof .

In the embodiment considered, these three parameters constitute the references to be assigned to the motors to obtain a correct actuation. For example, the references for the four motors can be determined as follows :

θ_λ = ψ + φ + Θ

θ,, = ψ - φ + θ

(57) θ₃ = ψ + φ - θ

θ₄ = ψ - φ - θ

In one embodiment, in order to describe the evolution of ψ, φ , and Θ a constant-velocity model is used defined b the relations

where

[θ, φ, ψ, θ, φ, ψ] = x_k is the state of the model;

- I and 0 are [3x3] matrices; and

- v_k is a gaussian-noise vector with zero mean and covariance Q.

For example, in one embodiment, the covariance Q is a 6x6] matrix with

where σ depends upon the maximum acceleration that the motors have available. The inventors have noted that this choice is justified by the fact that Θ must be the parameter with the greatest variability, whereas the other must be less variable.

In one embodiment, to carry out the estimate of the state, instead of using an EKF, the submodule 206b uses an unscented Kalman filter (UKF) , which, as against a higher computational complexity, enables a better evaluation of the references. In particular, a UKF is a Kalman filter for non-linear problems, which uses an unscented transformation to obtain an accurate propagation of the mean value and covariance. An unscented transformation is a deterministic sampling technique via which it is possible to select a set of state vectors, referred to as "sigma points", the sample mean and covariance of, which are equal to the mean of the state ( x ) and to the covariance matrix of the estimation error ( P ) . These sigma points are propagated through the non-linear functions and then used for recovering the mean and covariance of the estimate. With this technique a filter is obtained that, as compared to the EKF, produces a more accurate estimate of the actual mean and covariance.

In general,

y, = h(K_k) + n, w„~ N(0, Q_k)

is a generic non-linear system, where x is the state vector, y is the vector of the observations, and Q and R are the respective covariance matrices.

If x is the estimated state and P the covariance matrix of the estimation error, the equations of a UKF for the initialization k = 0 are (61) p - E (62)

Instead, in order to carry out the predictions for each instant k > 0, sigma points are selected for updating the state from the instant k - 1 to the instant k

= x^ + x^U) , for i = l, 2, ... , 2n

xⁱⁿ⁺ⁱ' = -(-Jn-P ,I, for i = 1, 2, ... , n (63) Next, the propagation of the sigma points through the non-l near state function is calculated as follows:

and an a pri ori estimate is made of the state x^~ and an a pri ori estimate is made of the covariance of the error .

For example, in the embodiment considered, the mean of the sigma points propagated as a pri ori estimate is used:

* = — ∑ (65) P- - (i

- *M^] - xj + Q,__x (66)

Said a pri ori estimates are then corrected and the propagation of the sigma points through the non-linear measurement function is calculated.

For example, it is possible to select a new set of sigma points or else re-use the sigma points obtained previously:

ΥΪ = h{k?) (67) Next, the prediction of the observation y_k and the estimate of the covariance of the measurement P predicted at step k are calculated.

For example, in the embodiment considered, the mean of the sigma points propagated is used:

pr = ^ 2n _i∑₌₁ (y"' - ΫΜ' - y + ** <⁶⁹>

On the basis of the a priori estimate of the state ~. and the prediction of the observation y ^. it is then possible to calculate the estimate of the covariance between the prediction of the state and the prediction of the measurement P ._r :

In the embodiment considered, said estimation of the covariance between the prediction of the state and the prediction of the measurement P. is used to update the a pri ori estimates described previously and to estimate the state x,. and the covariance of the error at the instant

In one embodiment, the version of the UKF described previously is modified. In the first place, if n is the number of states, 2n + 1 sigma points are used, instead of 2n, which are generated by weighting the covariance P by a factor Γ = fn + λ , rather than n^~ . The value of λ is obtained by choosing another three parameters (in general set at 1, 2, and 0, respectively) and, with these, it is possible to obtain values via which a weighted mean of the sigma points can be carried out.

Ιη· addition, in order to take into account the limits on the maximum and on the minimum velocity of rotation of the motors, it has been chosen to constrain the sigma points by applying the projection method with the least-sguares approach. In particular, in the embodiment considered, a set of relations between the estimated angular velocities is defined as follows:

x_v = Dx (74) where D(i) is the i-th row of the matrix D :

x,.(i) = D(i)x = [000110]x = θ + φ (75) and for each sigma point it is verified that it falls in the limits d(i) that it is desired to impose:

x_v(i) ≤ d(i) (76)

In one embodiment, in the case where a limit is exceeded, the exceeding value is saturated

x_v{i) = d(i) (77) and the state is modified accordingly

x = D^_1x_v (77)

As mentioned previously, an actuation system constituted by a number of pairs of eccentric rotors presents various critical points. In general, a system of this sort is able to generate:

- magnitude of the force;

- frequency of the force;

- instantaneous phase of the force; and

- direction of the force.

The control of a system of this sort is rather simple if the type of force to be damped (for example, vibrations induced by motors) includes only small variations of the parameters (in particular the frequency) around well-defined values, or forces are not generated that have a composite spectral content, i.e., are multimodal, or markedly non-stationary, which are, on the other hand, typical in the machine-tool sectors .

In effect, in the majority of the machining operations carried out by machine tools, the limiting vibrations are linked to the presence of a certain number of "main" modes, in which the majority of the undesired vibration energy is concentrated. In particular, the first limiting mode (i.e., the mode characterized by highest energy) is considered as the main cause of constraint in the possibility of increasing the capacity of removal of stock by the machine and/or as the main cause of an unacceptable surface finish of the workpiece.

Instead, the estimators presented herein are able to estimate the phases and the angular velocities to be assigned to a number N ≥ 3 of unbalanced rotors to cause them to release a desired force with a magnitude, phase, and direction (in the plane) that vary independently .

In fact, the solutions described herein present the characteristic of supplying references to the motors for generation of a multimodal signal, even in the case where a further observation is introduced, the purpose of which is to guide the evolution of the system towards states with fixed velocity, or else in the case where the estimated parameters are referred to a sinusoidal action with well-defined amplitude, frequency, and direction.

In fact, the action required of the motors is such as to cause tuning thereof at a well-defined frequency (which generally can be the frequency of the first limiting mode) , being characterized, however, by the presence of modulations in amplitude and frequency such as to obtain an action with wider spectrum in order to:

- intervene on the modes at a frequency that is a multiple of that of the first limiting mode; and

- intervene on further independent modes, which, albeit characterized by lower energy, can contribute to the undesired effect on the workpiece or to the limitation of the capacity of removal of stock by the machine .

The inventors have noted that the initial states of the various estimators are very important. Preferably, the accelerometer will have to read initially a zero acceleration, and the initial phase and angular velocity of the rotors will have to be known with a high level of precision.

The inventors have also noted that all the parameters require calibrations of a small amount for the specific application.

However, the estimation method is particularly stable, even starting or passing through conditions of singularity of kinematic representation and even in conditions of saturation.

The inventors have also noted that the system is suited for implementation of a self-tuning or self- calibration method.

In fact, with the system installed on a target machine tool, starting from the signals measured by the accelerometer included in the system in response to appropriate input forces generated by the solution itself, the system is able to estimate the parameters present within the estimation methods discussed above (principally, variances or covariances of the state matrix) so as to optimize their own performance on the machine for which the system is used.

In one embodiment, the system implements also a self-diagnostics procedure. For example, the system can detect autonomously its own malfunctioning, for example starting from appropriate measurements of the quantities such as acceleration or current of the motors .

In general, the system described herein is able to generate, in a controlled way, forces and momenta. In the scheme so far discussed, however, the arm of the resultant force with respect to the plane of action on which the force is developed has been reduced to a minimum to enable generation above all of forces. However, the same actuation system, with a slightly different scheme (where, for example, the distance between the resultant force and the plane of action is increased) may also be used as generator of torques.

Of course, without prejudice to the principle of the invention, the details of construction and the embodiments may vary widely with respect to what is described and illustrated herein purely by way of example, without thereby departing from the scope of the present invention, as defined by ensuing claims.

Claims

1. A method for estimating the control parameters of a system for active damping of the vibrations (20) of a mechanical structure (30), wherein said damping system (20) is able to generate a force via rotation of at least three eccentric rotors (100), wherein said method includes the steps of:

- receiving a value (a^") that is indicative for the total force that said at least three eccentric rotors (100) must generate; and

- estimating (206) the phases {PosVibr, Q) and the angular velocities (VeJVij r, ω) to be applied to said at least three rotors (100) as a function of said value (a ) indicative for the total force.

2. The method according to Claim 1, wherein said value (a ) indicative for the total force is an acceleration that said at least three eccentric rotors (100) must generate.

3. The method according to Claim 2, wherein said acceleration (a ) that said at least three eccentric rotors (100) must generate is determined (204) as a function of a value of measured acceleration (ACC is) .

4. The method according to Claim 2 or Claim 3, wherein said estimation (206) of the phases ( PosVijbr, θ ) and angular velocities {VelVibr, ω ) to be applied to said at least three rotors (100) includes:

- estimating (206a) a value indicative for a total position (p) and a value indicative for a total velocity (v ) as a function of said acceleration ( a ) ; and

- estimating (206b) said phases (PosVijbr, θ ) and said angular velocities {VelVibr, o ) as a function of said value indicative for a total position (p) and said value indicative for a total velocity ( v ) .

5. The method according to Claim 4, wherein said value indicative for a total position (p) and said value indicative for a total velocity ( ) are estimated (206a) via a double pseudo-integration of said acceleration (a ) .

.

6 . The method according to Claim .5, wherein said double pseudo-integration of said acceleration (a ) is carried out via a Kalman filter.

7. The method according to any one of Claims 4 to 6, wherein a value indicative for a total position (p) and a value indicative for a total velocity ( ) are estimated for each axis of acceleration.

8 . The method according to any one of Claims 4 to

7, wherein said phases (PosVibr,Q) and said angular velocities {VelVibr, ω ) are estimated (206b) via a complex first-order autoregressive linear model.

9. The method according to any one of Claims 4 to

8, wherein said phases ( PosVibr, 9- ) and said angular velocities (VelVibr, ω ) are estimated (206b) via an extended Kalman filter.

10. A system for active damping of the vibrations (20) of a mechanical structure (30), comprising:

- an inertial actuation system (10) for generating a force via rotation of at least three eccentric rotors (100);

- a sensor (202) for detecting a value (a ) indicative for the total force that said at least three eccentric rotors (100) must generate; and

- a control module configured for implementing the estimation method according to any one of Claims 1 to

9.

11. The system according to Claim 10, wherein said control module is configured for carrying out a self- calibration procedure, which comprises the steps of:

- generating at least one force via said at least three eccentric rotors (100) ; and

- estimating at least one variance and/or covariance used within said control module as a function of the value ( a ) detected via said sensor (202) .

12. The system according to Claim 10 or Claim 11, wherein said sensor (202) is an acceleration sensor.

13. The system according to any one of Claims 10 to 12, wherein said mechanical structure (30) is a machine tool.

14. An actuation device (lOa-lOd) that can be used in a system for damping vibrations (20) according to any one of Claims 10 to 13, wherein said actuation device (lOa-lOd) comprises a motor (102), an eccentric rotor (100) coupled to said motor (102), and a casing (104, 106), wherein said casing (104, 106) comprises at least one first hole (130a) with axis perpendicular to the axis of said motor (102), at least one second hole with axis perpendicular both to the axis of said motor (102) and to the axis of said first hole (130a) , and at least two holes (130d, 130e) with axes parallel to the axis of said motor (102) for coupling · a plurality of said actuation devices (lOa-lOd) .

15. A computer-program product that can be loaded into the memory of at least one processor and comprises portions of software code for implementing the method according to any one of Claims 1 to 9.