CN101261496A - Magnetic levitation flywheel high precision initiative vibration control system - Google Patents
Magnetic levitation flywheel high precision initiative vibration control system Download PDFInfo
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Abstract
A magnetic levitation flywheel high-precision active vibration control system comprises a displacement sensor, a current sensor, a magnetic bearing controller and a PWM modulation and a power amplifier, wherein, the magnetic bearing controller comprises stabilization of the controller, eccentric estimation, magnetic force compensation and action of a switch. The magnetic levitation flywheel high-precision active vibration control system introduces the eccentric estimation and the magnetic force compensation on the basis of the stable control and utilizes the unbalance vibration parameters of a flywheel to carry out the compensation of the unbalance amount and the displacement negative stiffness of the flywheel within the entire rotation speed range, thus realizing the control of the unbalance vibration of the flywheel within the entire rotation speed range and allowing the flywheel to be operated around a principal axis of inertia with high precision during the whole process of speed increasing and speed reducing.
Description
Technical field
The present invention relates to a kind of magnetic levitation flywheel high precision initiative vibration control system, be used for the inhibition and the flywheel high precision control of the rotor unbalance vibration of magnetically levitated flywheel.
Background technology
Flywheel is the main topworks of spacecraft attitude such as satellite control.Traditional mechanical bearing flywheel supports by ball bearing, the ball bearing of rigidity can be directly passed to spacecraft with the unbalanced moments of flywheel, the lubricated of ball bearing also can cause viscous moment, torque disturbance simultaneously, and static friction etc., thereby traditional mechanical bearing flywheel also is the main source of spacecraft attitude control system disturbing force or disturbing moment, and these disturbing forces or disturbing moment can reduce the spacecraft pointing accuracy.Along with the development of magnetic bearing technology, Magnetic Bearing Flywheel Technology research has been subjected to the generally attention of various countries.The unrivaled advantages of conventional mechanical bearings flywheel such as that magnetically levitated flywheel has is contactless, do not have friction, need not to lubricate, high precision, long-life are the desirable topworkies of high precision spacecraft attitude control.
Yet also there is certain problem in magnetically levitated flywheel in the application of high-precision attitude control, be that magnetically levitated flywheel still exists some vibration sources, disturbance that non-linear factors such as the flywheel fundamental frequency disturbance that causes as unbalance vibration, alignment error cause etc., if the vibration that these vibration sources are caused does not suppress, the precision of attitude control will be had a strong impact on, technical advantages such as the high precision of magnetically levitated flywheel also can't be embodied, and also can bring a series of problems such as power amplifier is saturated, power consumption increase simultaneously.Therefore how these vibrations are suppressed, become magnetically levitated flywheel and be used for spacecraft attitude and control one of subject matter to be solved.
The unbalance vibration of rotor can cause disturbances such as the fundamental frequency, two frequencys multiplication, frequency tripling of fly wheel system, is the main vibration source of magnetically levitated flywheel.Rotor exists uneven, is meant that the geometrical axis of rotor does not overlap with the principal axis of inertia, as shown in Figure 1.When flywheel works in high speed (more than the critical rotary speed) and since flywheel rotor from the centering effect, flywheel rotor can be tending towards rotate around the principal axis of inertia, at this moment the rotor geometrical axis can rotate around the principal axis of inertia.Because the principle of work of magnetically levitated flywheel is the geometrical axis that flywheel rotor is suspended in rotor, so magnetic bearing can apply active controlling force to flywheel rotor, because the principle of acting force and reacting force will have disturbing force to export this moment.And the High Accuracy Control of flywheel is meant the High Accuracy Control of flywheel output torque, comprises High Accuracy Control two parts of moment size and moment direction.Wherein the High Accuracy Control of output torque size is determined by the control accuracy of motor, and the High Accuracy Control of output torque direction is by the control accuracy decision of flywheel itself.So because the output of the disturbing force that the imbalance of rotor causes, must influence the direction of flywheel output torque, therefore how the unbalance vibration of rotor is suppressed, become magnetically levitated flywheel and be applied to high precision spacecraft attitude control institute problem demanding prompt solution.
In the magnetic levitation flywheel high precision control system that adopts at present, its unbalance vibration controller generally only adopts the amount of unbalance compensated part, the i.e. vibration that the flywheel rotor amount of unbalance is caused compensates, when flywheel rotor when the principal axis of inertia rotates, no longer amount of unbalance is partly applied active controlling force.But owing to the existence of displacement negative stiffness,, still have the vibration that causes than big displacement negative stiffness and spread out of, thereby the high precision active vibration that can't realize magnetically levitated flywheel is controlled in the actual magnetic bearing arrangement even the rotor unbalance value part is not applied active controlling force.
Summary of the invention
The objective of the invention is: overcome the deficiency that existing magnetically levitated flywheel control system exists aspect the control of flywheel high precision active vibration, particularly solve the vibration that unbalance vibration that amount of unbalance causes and displacement negative stiffness cause, a kind of magnetic levitation flywheel high precision initiative vibration control system is provided, realizes the high precision active vibration control of magnetically levitated flywheel.
Technical solution of the present invention is: a kind of magnetic levitation flywheel high precision unbalance vibration control system, its characteristics are to comprise: magnetic bearing controller, current sensor, power amplifier, PWM modulation and displacement transducer, current sensor links to each other with magnetic bearing controller with the magnetic bearing coil, is used for detecting magnetic bearing coil electric current and detected value is delivered to magnetic bearing controller; Power amplifier links to each other with the magnetic bearing coil with the PWM modulation, is used for exporting the amplification of the pwm signal after the modulation to the magnetic bearing coil, and flywheel rotor is applied ACTIVE CONTROL; The PWM modulation links to each other with power amplifier with magnetic bearing controller, is used for the controlled quentity controlled variable that magnetic bearing controller generates is carried out the PWM modulation to output to power amplifier; Displacement transducer links to each other with magnetic bearing controller with flywheel rotor, is used to detect the displacement signal of flywheel rotor and detected value is delivered to magnetic bearing controller; Magnetic bearing controller is made up of stability controller, magnetic compensation, eccentric estimation and the first effect switch, stability controller receives the output of displacement transducer, its output do with the output of current sensor and, do of the input of difference back with the output of magnetic compensation as the PWM modulation; The first effect switch is estimated to link to each other with eccentric with displacement transducer, is used to control and whether eccentric estimation is delivered in the output of displacement transducer; The eccentric output signal of estimating to receive the first effect switch, magnetic compensation is delivered in its output, and negative feedback simultaneously is to the output terminal of displacement transducer; The output that magnetic compensation is estimated with off-centre is as input, its output is stabilized delivers to the PWM modulation after controller output deducts with current sensor output sum, middle stability controller is used for flywheel is stablized control, eccentric size and the position of estimating to be used for determining the flywheel rotor amount of unbalance, and rotor unbalance value compensated to reduce the unbalance vibration of flywheel rotor, determine the size and the position of magnetic compensation simultaneously, magnetic compensation is used for the flywheel rotor suspension air gap caused unbalanced magnetic pull that fluctuates is compensated, to reduce the vibration of the flywheel rotor that the displacement negative stiffness causes, its size and position are definite by the off-centre estimation, and the first effect switch is used for determining the effect moment of eccentric estimation and magnetic compensation.
The compensation formula of described magnetic compensation (9) is:
Described eccentric estimation is made up of the second effect switch, the 3rd effect switch, open-loop compensation algorithm and adaptive notch filter, wherein adaptive notch filter links to each other with the second effect switch with the first effect switch, is used for the flywheel rotor position signalling of input is carried out adaptive notch; As input, its output negative feedback is used to control having or not of adaptive notch effect to the second effect switch to the flywheel rotor displacement signal with adaptive notch filter; The open-loop compensation algorithm is provided with its parameter by adaptive notch filter, export the 3rd effect switch to, being used for that flywheel rotor is carried out open-loop compensation the 3rd effect switch imports as it with the output of open-loop compensation algorithm, the output negative feedback is used to control having or not of open-loop compensation effect to the flywheel rotor displacement signal.
Described stability controller can be the Decentralized PID controller, also can be that Decentralized PID adds the intersection feedback controller, can also be that decoupling controller etc. can make magnetically levitated flywheel suspend and the stability controller of raising speed.
Principle of the present invention is: the stability controller among the present invention can be realized the stable control of magnetically levitated flywheel; On the stable basis of controlling, introduce imbalance compensation and the compensation of displacement negative stiffness to rotor unbalance value, utilize the amount of unbalance parameter that picks out, to carrying out the control of high precision active vibration in the whole range of speeds of flywheel, flywheel can high precision be turned round.
The principle of displacement negative stiffness compensation is as shown in Figure 2: the force diagram with a radial direction magnetic bearing place is an example, the O point is magnetic bearing stator center line position among the figure, also represent simultaneously the principal axis of inertia of flywheel rotor, promptly when the principal axis of inertia of flywheel rotor and magnetic bearing stator center line coincide, the geometrical axis of flywheel rotor will depart from magnetic bearing stator center line, at this moment the air gap between magnetic bearing stator and the rotor will fluctuate with frequency with rotating speed, because the existence of bias magnetic field, the magnetic bearing rotor can be subjected to the influence of magnetic bearing negative displacement rigidity (magnetic force/displacement coefficient), produce the unbalanced magnetic pull of rotation, so, must pass to corresponding electric current by giving the magnetic bearing coil, thereby balance out this part unbalanced magnetic pull.At this moment the magnetically levitated flywheel rotor will be under this dynamic balance, around its principal axis of inertia rotation, promptly do not have out-of-balance force and pass to the magnetic bearing stator, the magnetic bearing stator does not produce out-of-balance force yet and acts on flywheel rotor, has just realized the High Accuracy Control of magnetically levitated flywheel veritably yet.
The present invention's advantage compared with prior art is: the present invention is on stable control basis, introducing is to the imbalance compensation and the compensation of displacement negative stiffness of rotor unbalance value, having overcome existing magnetic levitation flywheel high precision control system can only compensate amount of unbalance, the shortcoming that still has the vibration that the displacement negative stiffness causes, thus realize the high precision active vibration control of magnetically levitated flywheel.
Description of drawings
The rotor geometrical axis and the principal axis of inertia concerned synoptic diagram when there was amount of unbalance in Fig. 1 for a kind of magnetically levitated flywheel rotor;
Fig. 2 is a kind of magnetic levitation flywheel high precision control system displacement negative stiffness compensation principle synoptic diagram of the present invention;
Fig. 3 is that a kind of magnetic levitation flywheel high precision unbalance control system of the present invention constitutes synoptic diagram;
Fig. 4 is the eccentric estimation of an a kind of magnetic levitation flywheel high precision unbalance vibration control system of the present invention structural representation;
Fig. 5 is the eccentric amount of unbalance algorithm for estimating process flow diagram of estimating of a kind of magnetic levitation flywheel high precision initiative vibration control system of the present invention;
Fig. 6 is the eccentric open-loop compensation algorithm flow chart of estimating of a kind of magnetic levitation flywheel high precision initiative vibration control system of the present invention;
Fig. 7 is a kind of magnetic levitation flywheel high precision initiative vibration control system adaptive notch filter schematic diagram of the present invention;
Fig. 8 is a kind of magnetic levitation flywheel high precision initiative vibration control system displacement negative stiffness backoff algorithm block diagram of the present invention;
Fig. 9 is a kind of magnetically levitated flywheel vibrational waveform without active vibration control;
Embodiment
As shown in Figure 3, the used magnetic levitation flywheel high precision initiative vibration control system of present embodiment comprises magnetic bearing controller 1, current sensor 3, power amplifier 6, PWM modulation 7 and displacement transducer 8.Current sensor 3 links to each other with magnetic bearing controller 1 with magnetic bearing coil 4, is used for detecting magnetic bearing coil 1 electric current and detected value is delivered to magnetic bearing controller 1; Power amplifier 6 and PWM modulation 7 links to each other with magnetic bearing coil 4, is used for exporting the amplification of the pwm signal after the modulation to magnetic bearing coil 4, and flywheel rotor is applied ACTIVE CONTROL; PWM modulation 7 links to each other with power amplifier 6 with magnetic bearing controller 1, is used for that the controlled quentity controlled variable that magnetic bearing controller 1 generates is carried out PWM and modulates to output to power amplifier 6 links; Displacement transducer 8 links to each other with magnetic bearing controller 1 with flywheel rotor 5, is used to detect the displacement signal of flywheel rotor 5 and detected value is delivered to magnetic bearing controller 1.Magnetic bearing controller 1 is made up of stability controller 2, magnetic compensation 9, eccentric estimation 10, effect switch 11, stability controller 2 receives the output of displacement transducer 8, its output do with the output of current sensor 3 and, do of the input of difference back with the output of magnetic compensation 9 as PWM modulation 7; The first effect switch 11 estimates that with eccentric 10 link to each other with displacement transducer 8, is used to control and whether eccentric estimation 10 is delivered in the output of displacement transducer 8; The eccentric output signal of estimating 10 receptions, the first effect switch 11, magnetic compensation 9 is delivered in its output, and negative feedback simultaneously is to the output terminal of displacement transducer 8; Magnetic compensation 9 is estimated 10 output as input with off-centre, and its output is stabilized to be delivered to PWM after controller 2 outputs deduct with current sensor 3 output sums and modulate 7; Middle stability controller 3 is used for flywheel is stablized control; Eccentric estimation 10 is used for determining the size and the position of flywheel rotor amount of unbalance, and rotor unbalance value is compensated to reduce the unbalance vibration of flywheel rotor, determines the size and the position of magnetic compensation 9 simultaneously; Magnetic compensation 9 is used for the flywheel rotor suspension air gap caused unbalanced magnetic pull that fluctuates is compensated, and to reduce the vibration of the flywheel rotor that the displacement negative stiffness causes, its size and position estimate that by off-centre 10 is definite; The first effect switch 11 is used for determining the effect moment of eccentric estimation 10 and magnetic compensation 9.
As shown in Figure 4, eccentric estimation 10 is made up of the second effect switch 12, the 3rd effect switch 13, open-loop compensation algorithm 14 and adaptive notch filter 15, wherein adaptive notch filter 15 links to each other with the second effect switch 12 with the first effect switch 11, is used for the flywheel rotor position signalling of input is carried out adaptive notch; As input, its output negative feedback is used to control having or not of adaptive notch effect to the second effect switch 12 to the flywheel rotor displacement signal with adaptive notch filter 15; Open-loop compensation algorithm 14 is provided with its parameter by adaptive notch filter 15, export the 3rd effect switch 13 to, being used for that flywheel rotor 5 is carried out open-loop compensation the 3rd effect switch 13 imports as it with the output of open-loop compensation algorithm 14, the output negative feedback is used to control having or not of open-loop compensation effect to the flywheel rotor displacement signal.
The centre frequency of adaptive notch filter 15 is set according to the flywheel rotating speed, when the flywheel rotating speed reaches critical rotary speed when above, extract and feed back in the displacement signal by the amount of unbalance in 15 pairs of flywheel rotor displacement signals of adaptive notch filter, thereby realization is to the closed loop compensation of the amount of unbalance of flywheel rotor 5; After obtaining stable compensation rate, the size of noting adaptive notch filter 15 compensation rate when each position of flywheel is characterizing the size and the position of the amount of unbalance of rotor respectively to set up a two-dimentional numerical table.
As shown in Figure 5, the eccentric amount of unbalance algorithm for estimating structure of estimating is as follows: after the flywheel rotating speed reaches more than the critical rotary speed, the closed second effect switch 12, disconnect the 3rd effect switch 13 (present embodiment passes through software design), adaptive notch filter 15 extracts the amount of unbalance signal from the displacement signal of flywheel rotor 5, by acting on after the negative feedback in flywheel rotor 5 displacement signals, the unbalance vibration of flywheel rotor is carried out closed loop compensation.When flywheel rotor after the stable rotation of the principal axis of inertia, obtain the two-dimentional numerical table of stable sign amount of unbalance, disconnect the second effect switch 12, closed the 3rd effect switch 13 (present embodiment employing software design), the size and the position of the rotor unbalance value that obtains according to closed loop compensation, and the current position signal of the flywheel rotor that provides in conjunction with flywheel position identification device, the flywheel rotor unbalance vibration is carried out open-loop compensation, thereby realize the active vibration control (this for (2) to rotor unbalance value compensate to reduce the unbalance vibration of flywheel rotor) of flywheel in the whole range of speeds, make flywheel at whole liter, can both the high precision running in the reduction of speed process.
As shown in Figure 6, the process flow diagram of open-loop compensation algorithm 14 is as follows in the eccentric estimation 10: at first carry out the compensation of closed loop unbalance vibration by 15 pairs of flywheel rotors of adaptive notch filter 5, size and position with the amount of unbalance that obtains flywheel rotor 5, and generation characterizes the two-dimentional numerical table of amount of unbalance size and position, and, carry out the unbalance vibration compensation in conjunction with the current location of flywheel rotor.
As shown in Figure 7, the adaptive notch filter 15 that present embodiment adopts is based on the orthogonality principle of orthogonal function, with the centre frequency of flywheel rotating speed as adaptive notch filter, with frequency is just to change frequently, cosine value is made integration after multiplying each other with the displacement signal of flywheel rotor respectively, it is same frequently just with rotating speed to be used for obtaining the flywheel rotor displacement signal, the coefficient of cosine component, be just to change frequently with frequency respectively again with this two coefficient then, cosine signal is done multiplication, to obtain rotor displacement signal intermediate frequency rate serves as just to change frequently, 9 cosine signal components, again this two component is done and, the final flywheel rotor displacement signal medium frequency that obtains is commentaries on classics component value frequently, and feeds back in the rotor displacement signal.The flywheel rotating speed be multiply by doubly (n=1 of n, 2,3), can realize to rotor displacement signal intermediate frequency rate being that n doubly changes (n=1,2 frequently with centre frequency as adaptive notch filter, 3) extraction of component, the n that is extracted is doubly changeed the component negative feedback of frequency (n=1,2,3) to the rotor displacement signal, realization is to rotor n times of amount of unbalance (n=1,2,3) closed loop compensation is in order to obtain to characterize n times of amount of unbalance (n=1,2,3) size and the two-dimentional numerical table of position according to this two-dimentional numerical table and the flywheel actual position signal that provided in conjunction with flywheel position identification device, can be realized n is doubly changeed frequency (n=1,2,3) open loop unbalance vibration compensation.
As shown in Figure 8, described have the algorithm of magnetic compensation 9 to be: will measure and displacement negative stiffness matrix with frequency in AX, BX, AY, the BY channel bit shifting signal
Do multiplication, gained is the corresponding displacement negative stiffness of each passage compensation rate.In the formula, K
eBe displacement negative stiffness compensation matrix, K
iCharacterize magnetic bearing current stiffness matrix, k
iBe magnetic bearing current stiffness value, K
xCharacterize magnetic bearing displacement rigidity matrix, k
xBe magnetic bearing displacement rigidity value, T
x, T
sBe sensor coordinates and magnetic bearing transformation of coordinates matrix, a, b are two ends magnetic bearing span radially, a
s, b
sBe two end sensor spans radially.
For the effect of magnetic levitation flywheel high precision active vibration control is described, Fig. 9 a, Fig. 9 b have provided the flywheel rotor vibrational waveform that adds active vibration control when rotating speed is 3600 rev/mins and do not add active vibration control and have compared.Fig. 9 b is when rotating speed is 3600 rev/mins, the vibrational waveform of flywheel rotor when not adding active vibration control, and by Fig. 9 b as seen, the amplitude of radial vibration signal is bigger, and the amplitude of once per revolution vibration is higher (3.79dBV); Fig. 9 a is when 3600 rev/mins of rotating speeds, add the control of high precision active vibration after, the vibrational waveform of flywheel rotor, by Fig. 9 a as seen, the amplitude of radial vibration signal has very high attenuation than Fig. 9 b, the spectrum value of once per revolution vibration decays to-18dBV about.Fig. 9 a compares with Fig. 9 b, the spectrum value of the once per revolution vibration in the flywheel rotor vibrational waveform by-3.79dB decay to-18dB about, decay to original 1/5th (14dB=0.1995), the unbalance vibration of flywheel rotor is significantly suppressed, and has reached the effect of high precision active vibration control.
The content that is not described in detail in the instructions of the present invention belongs to this area professional and technical personnel's known prior art.
Claims (5)
1, a kind of magnetic levitation flywheel high precision unbalance vibration control system, it is characterized in that: comprising: magnetic bearing controller (1), current sensor (3), power amplifier (6), PWM modulate (7) and displacement transducer (8), current sensor (3) links to each other with magnetic bearing controller (1) with magnetic bearing coil (4), is used for detecting magnetic bearing coil (4) electric current and detected value is delivered to magnetic bearing controller (1); Power amplifier (6) links to each other with magnetic bearing coil (4) with PWM modulation (7), is used for exporting the amplification of the pwm signal after the modulation to magnetic bearing coil (4), and flywheel rotor is applied ACTIVE CONTROL; PWM modulation (7) links to each other with power amplifier (6) with magnetic bearing controller (1), is used for the controlled quentity controlled variable that magnetic bearing controller (1) generates is carried out the PWM modulation to output to power amplifier (6); Displacement transducer (8) links to each other with magnetic bearing controller (1) with flywheel rotor (5), is used to detect the displacement signal of flywheel rotor (5) and detected value is delivered to magnetic bearing controller (1); Magnetic bearing controller (1) is made up of stability controller (2), magnetic compensation (9), eccentric estimation (10) and the first effect switch (11), stability controller (2) receives the output of displacement transducer (8), the output of the output of stability controller (2) and current sensor (3) do and, do of the input of difference back with the output of magnetic compensation (9) as PWM modulation (7); The first effect switch (11) estimates that with eccentric (10) link to each other with displacement transducer (8), is used for control and whether eccentric estimate (10) is delivered in the output of displacement transducer (8); The eccentric output signal of estimating (10) reception first effect switch (11), magnetic compensation (9) is delivered in its output, and negative feedback simultaneously is to the output terminal of displacement transducer (8); The output that magnetic compensation (9) is estimated (10) with off-centre is as input, and its output is stabilized to be delivered to PWM after controller (2) output deducts with current sensor (3) output sum and modulate (7); Wherein stability controller (2) is used for flywheel is stablized control, eccentric estimation (10) is used for determining the size and the position of flywheel rotor amount of unbalance, and rotor unbalance value compensated to reduce the unbalance vibration of flywheel rotor, determine the size and the position of magnetic compensation (9) simultaneously, magnetic compensation (9) is used for the flywheel rotor suspension air gap caused unbalanced magnetic pull that fluctuates is compensated, to reduce the vibration of the flywheel rotor that the displacement negative stiffness causes, its size and position estimate that by off-centre (10) are definite, and the first effect switch (11) is used for determining the eccentric effect moment of estimating (10) and magnetic compensation (9).
2, magnetic levitation flywheel high precision initiative vibration control system according to claim 1 is characterized in that: the compensation formula of described magnetic compensation (9) is:
In the formula, K
eBe displacement negative stiffness compensation matrix, K
iCharacterize magnetic bearing current stiffness matrix, k
iBe magnetic bearing current stiffness value, K
xCharacterize magnetic bearing displacement rigidity matrix, k
xBe magnetic bearing displacement rigidity value, T
x, T
sBe sensor coordinates and magnetic bearing transformation of coordinates matrix, a, b are two ends magnetic bearing span radially, a
s, b
sBe two end sensor spans radially.
3, magnetic levitation flywheel high precision initiative vibration control system according to claim 1, it is characterized in that: described eccentric estimation (10) is made up of the second effect switch (12), the 3rd effect switch (13), open-loop compensation algorithm (14) and adaptive notch filter (15), wherein adaptive notch filter (15) links to each other with the second effect switch (12) with the first effect switch (11), is used for the flywheel rotor position signalling of input is carried out adaptive notch; As input, its output negative feedback is used to control having or not of adaptive notch effect to the second effect switch (12) to the flywheel rotor displacement signal with adaptive notch filter (15); Open-loop compensation algorithm (14) is provided with its parameter by adaptive notch filter (15), export the 3rd effect switch (13) to, being used for that flywheel rotor (5) is carried out open-loop compensation the 3rd effect switch (13) imports as it with the output of open-loop compensation algorithm (14), the output negative feedback is used to control having or not of open-loop compensation effect to the flywheel rotor displacement signal.
4, magnetic levitation flywheel high precision initiative vibration control system according to claim 1, it is characterized in that: described eccentric estimate (10), promptly comprise with changeing the estimation and the compensation of amount of unbalance frequently, also comprise the estimation and the compensation of two frequencys multiplication, frequency tripling amount of unbalance.
5, magnetic levitation flywheel high precision initiative vibration control system according to claim 1, it is characterized in that: described stability controller (2) is the Decentralized PID controller, or Decentralized PID adds the intersection feedback controller, or decoupling controller, maybe can make the also stability controller of raising speed of magnetically levitated flywheel suspension.
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