METHOD FOR DETERMINING A DETECTION SENSITIVITY OF A ROTATION RATE SENSOR

Abstract

A method for determining a detection sensitivity of a rotation rate sensor, the rotation rate sensor including an oscillatory system. A first quadrature signal of the oscillatory system is determined in a first step. A controlled change of a transfer function of the oscillatory system takes place in a second step. A second quadrature signal of the oscillatory system is determined in a third step. The detection sensitivity is determined in a fourth step on the basis of the first and second quadrature signal. A method is also described for determining a detection sensitivity of a rotation rate sensor, the rotation rate sensor including one first oscillatory system and one second oscillatory system.

Description

CROSS REFERENCE

The present application claims the benefit under 35 U.S.C. § 119 of German Patent Application No. DE 102020203851.1 filed on Mar. 25, 2020, which is expressly incorporated herein by reference in its entirety.

FIELD

The present invention is directed to a method for determining a detection sensitivity of a rotation rate sensor.

BACKGROUND INFORMATION

Conventional rotation rate sensors are limited in their performance by adverse effects such as, for example, the drift of the detection sensitivity since, if not compensated or corrected for, this may lead to a false interpretation of the measured results. For this reason, rotation rate sensors frequently include compensation mechanisms, which counteract the drift of measured results.

Such a drift of sensitivity is dependent on a multitude of factors such as, for example, environmental influences such as temperature or atmospheric humidity or also on mechanical stresses, aging and many more. The actual physical cause of the drift is often due to a mechanical influence on the spring mass system (SMS) of the system of the sensor. Such a mechanical influence in turn results in a change of the amplitude response (amplification characteristic curve) and/or in the phase response (phase characteristic curve). Amplitude response and phase response are determined by the transfer function (also transfer function or system function) of the spring mass system, which characterizes the frequency-dependent response of the system, in that it indicates the correlation between the input signal (for example, drive signal) and the output signal (for example, the amplitude of the drive mode or detection mode, or of the associated electrical signal). The transfer function for various models of oscillatory systems is known in an analytical form. The following dependency of the frequency ω of the input signal, in particular, results for a damped oscillation of a system made up of a mass and a linear spring (the dimensional prefactor of the amplification G(ω) is omitted here):

$\begin{array}{cc}G\left(\omega \right)\sim \frac{1}{\sqrt{{\left(1-\frac{{\omega }^2}{{\omega }_0^2}\right)}^2+\frac{{\omega }^2}{Q^2·{\omega }_0^2}}}& \left(1\right)\\ \varphi \left(\omega \right)=\arctan \left(\frac{\frac{\omega }{Q·{\omega }_0}}{1-\frac{{\omega }^2}{{\omega }_0^2}}\right)& \left(2\right)\end{array}$

In this case, ω₀ refers to the inherent frequency (determined by the mass and the spring constant) and Q refers to the quality factor of the oscillatory system. In the event of a change, for example, of the spring constant (and thus the inherent frequency) of the system, the amplification characteristic curve and phase characteristic curve are influenced by the aforementioned effects in terms of their slope and in their position relative to the frequency axis. The profile of the characteristic curves may on the other hand also be influenced by a change in the quality of the system due to a temperature change. A change in the amplification or in the phase in turn influences the measured signal (cf. equation 3 further below), without this being based on an actual change of the rotation rate, so that here a measuring error results and the detection sensitivity is reduced. In order to identify a drift without resorting to a reference stimulus (i.e., a known external rotation rate) in the process, the quadrature (identified in the formulas as Quad) present in each real micromechanical system may be utilized. The quadrature produces a force effect independent of rotation rate Ω, or an additional contribution to the detection signal, which is phase shifted by 90° with respect to the useful signal generated by the rotation rate. Both signal contributions are, however, proportional to sensitivity G. Due to the phase shift, it is possible to separate quadrature signal s_Quad via demodulation from detection signal s_Ω.

s_Ω=G*Ω+G*Quad*ϕ  (3)

s_Quad=G*Quad  (4)

This fact is exploited in conventional methods for determining a sensitivity change. This is achieved by temporarily artificially amplifying by electrostatic means the quadrature of the oscillating detection apparatus via electrodes also integrated in the sensor. The output signals changed as a result are then compared during operation with a reference value, which has been established via a previous calibration. An indication about the sensitivity change may then be derived from the relationship of the output signal to the reference value. One disadvantage of such a method, however, is the additional design complexity associated therewith and the space requirement for the additional electrodes on the sensor. It is therefore desirable to obtain a measurement of the sensitivity drift without additional mechanical outlay. Moreover, with the conventional method described above, it is possible to determine only a temperature-dependent change in the amplification characteristic curve and phase characteristic curve of an oscillatory system. By contrast, a shift of the characteristic curve as a result of additional disruptive effects, such as mechanical stresses in the micromechanical structure that forms the sensor, may be ascertained to only a limited extent.

SUMMARY

It is an object of the present invention to provide a possibility of obtaining pieces of information about a potential drift of the detection sensitivity without the disadvantages described above occurring in the process.

The method according to an example embodiment of the present invention may have the advantage that it is possible to dispense with an artificial influencing of the quadrature of the oscillatory system and, in particular, with the additional electrodes required for this purpose. The method allows for an efficient way of establishing a change of sensitivity of an oscillatory system (also referred to hereinafter as spring mass system or abbreviated SMS). The SMS may, in particular, be the detection apparatus of a rotation rate sensor. In this case, the detection of this change is independent of a potential change of the quadrature and is also virtually temperature-independent. In accordance with an example embodiment of the present invention, the method allows for the determination of the position of the transfer function (TF) or the position of the operating point on the amplification characteristic curve or phase characteristic curve determined by the transfer function. In this way, it is possible to determine not only the detection sensitivity, but also the phase change (regardless of the temperature influence). This provides, in particular, the possibility of establishing a correction factor or a correction factor function or correction factor table, which may be utilized during the operation of the sensor for compensating for the drift of sensitivity and phase. Based on the measured results determined in the method, it is possible to determine a temperature-dependent absolute phase difference and a temperature-independent relative phase difference, which may serve as a further indicator for additional influence factors.

Each of the aforementioned possibilities may be implemented, in particular, in the case of the detection apparatus, without the additional electrodes conventionally used. They are also independent of an applied rotation rate and may thus be used during the operation of the sensor without further restrictions. It is also conceivable to use additional electrodes for influencing the SMS and to employ them complementarily to the method according to the present invention, in order to improve the accuracy of potential compensations and in this way to contribute to a rotation rate-independent absolute determination of the sensitivity.

The features of the present invention are based on initiating a controlled change of the transfer function (i.e., of the amplitude response and/or phase response) and at a fixed drive frequency to ascertain the relative change of the amplification or the phase shift for determining the position of the transfer function in relation to the frequency axis. At a predefined drive frequency, the detection signal allows for conclusions to be drawn about the actual amplification only with knowledge of the applied rotation rate. According to equation 3, the detection signal contains (in addition to the quadrature component determined by demodulation) amplification G and rotation rate Ω in the form of a product, so that based on the detection signal alone, no separation of the two influences is possible. In the method according to the present invention on the other hand, quadrature component s_Quad is determined for at least two different amplifications (caused by the controlled change of the transfer function), by which sufficient information is available in order to determine the true amplification.

With Equation 4, the relationship v_i of the quadrature signal s_Quad,i,Δ=s_Quad,i+Δs_Quad belonging to the changed transfer function (variables belonging to the changed transfer function are identified below with an additional index Δ and differences by a Δ placed in front of the variable) and of quadrature signal s_Quad,i belonging to the unchanged transfer function results in:

$\begin{array}{cc}{v}_i=\frac{s_{\mathrm{Quad},i,\Delta }}{s_{\mathrm{Quad},i}}=\frac{\left(G_i+\Delta G_i\right)}{G_i·\mathrm{Quad}}=\frac{G_i+\Delta G_i}{G_i}& \left(5\right)\end{array}$

As is apparent from the equation, the relationship thus established is independent of the quadrature. For the output signal of the rotation rate,

s_Ω,i,Δ=(G_i+ΔG_i)Ω(ϕ_i+Δϕ_i)  (6)

is applicable after the change of the transfer function.

The determination of the amplification on the basis of the relationship v_i may be described as exemplified by a controlled horizontal shift (in particular, by a corresponding shift of the inherent frequency) of the amplification characteristic curve and phase characteristic curve provided by equations 1 and 2 (cf. FIG. 3): Since the amplification according to equation 1 increases strictly monotonically at frequencies below the inherent frequency and drops strictly monotonically above the inherent frequency, it is possible to establish a range in which relationship v_i allows for a clear conclusion to be drawn about the position of the operating point. A similar approach may be carried out on the basis of the phase characteristic curve, i.e., by deducing the position of the characteristic curve from the mechanical phase change. If an unknown external influence (for example, mechanical stresses) then acts on the SMS of the drive or of the detection unit and causes a change of inherent frequency ω₀→ω_i, then the relationship v₀→v_i also changes. FIG. 4 graphically illustrates this process in a more general form.

The above explanations that describe in detail the present invention as exemplified by a controlled horizontal shift may be transferred to the general case, in which a largely arbitrary, reproducible and controlled influence of the transfer function of an oscillatory spring mass system may be utilized for ascertaining the operating point on the amplification characteristic curves and phase characteristic curves provided by the transfer function, and thus ultimately for determining the sensitivity change as a result of external influences.

Beyond the application of the shift of the inherent frequency as a compensation method (whether in a mechanical or electrically induced variant), this method may also be utilized for further functions or in further application alternatives.

According to one preferred specific embodiment of the present invention, a controlled change of an inherent frequency of the oscillatory system takes place and/or a controlled change of the quality factor of the oscillatory system takes place in the second step. Inherent frequency ω₀ and the quality Q according to equations 1 and 2 determine the shape and position of the amplification characteristic curve and phase characteristic curve. The quality in this case determines essentially the width of the curve, i.e., the drop of the flanks to the left and right of the maximum, whereas the inherent frequency determines the position of the characteristic curves relative to the frequency axis. Both variables may thus be used in the method according to the present invention to change the transfer function in a controlled and reversible manner and enable a determination of the sensitivity drift via the above-described measurement of the quadrature signal.

According to one particularly preferred specific embodiment of the present invention, the change of the inherent frequency takes place via a controlled change of the spring constant of the oscillatory system. The spring constant, together with the mass of the oscillating body, determines the inherent frequency of the oscillatory system, so that the increase or reduction of the spring constant results directly in a shift of the inherent frequency which, as explained above, may be utilized for a determination of the sensitivity drift.

According to one preferred specific embodiment of the present invention, further controlled changes of the transfer function take place in a fifth step following the third step and preceding the fourth step, and further quadrature signals of the rotation rate sensor are determined, the detection sensitivity being determined in the fourth step on the basis of the first, second, and further quadrature signals. In this way, it is possible to advantageously improve the accuracy of the ascertained sensitivity drift via a repetition or multiple repetition of the second and third step of the method according to the present invention. In the simplest case, the sensitivity drift may, for example, be ascertained with each repetition and a suitable selection may be made or an average value may be formed from the values for the drift thus ascertained.

According to one preferred specific embodiment of the present invention, a compensation variable for a detection signal is determined in a fifth step following the fourth step on the basis of the detection sensitivity determined in the fourth step. In order to be able to utilize the relationship v_i for compensating, a matrix or function may be established in advance (under initial conditions), which links the changed relationship v_i to a compensation factor CF. This may be achieved, for example, by measuring the relationship v_i across a particular range and by determining the associated correction factor for each of these values. In the simplest case, the correction factor may be established by setting the quadrature signal belonging to the shifted inherent frequency in relationship to the reference signal:

$\begin{array}{cc}{\mathrm{CF}}_{G,i}=\frac{G_0{Q}_0}{G_i{Q}_0}=\frac{G_0}{G_i}& \left(7\right)\end{array}$

FIG. 6 graphically illustrates this process in a more general form. Since this measurement takes place under initial conditions, the quadrature is not subject to any changes, as a result of which the correction factor is independent of potential changes of the quadrature. The calculated correction factor may then be used in a further step to the (previously quadrature-compensated) output signal of the demodulated detection channel:

$\begin{array}{cc}{s}_{\Omega ,i}{\mathrm{CF}}_{G,i}=\Omega ·G_i·\frac{G_0}{G_i}=\Omega ·G_0=s_{\Omega ,0}& \left(8\right)\end{array}$

Thus, a result is achieved as it would have been measured under initial conditions without a measurement of the quadrature or of the quadrature change being necessary. FIG. 5 graphically illustrates this process in a more general form. A correction factor for the mechanical phase change may be similarly calculated and used.

A further example embodiment of the present invention is a further method. Whereas the first method is based on a controlled change of the transfer function of one oscillatory system, two oscillatory systems of the same sensor are used in this variant, which differ in a defined manner with respect to their transfer functions. Relationship v_i accordingly from the quadrature signals of both systems, i.e., in equation 5, is the variable s_Quad,i provided by the quadrature signal of the first oscillatory system and the variable s_Quad,i,Δ provided by the quadrature signal of the second oscillatory system. The observations explained above are, however, directly transferable.

According to one preferred specific embodiment of the present invention, a quality factor of the first oscillatory system is identical to a quality factor of the second oscillatory system, a mass of the first oscillatory system differing from a mass of the second oscillatory system and/or a spring constant of the first oscillatory system differing from a spring constant of the second oscillatory system. In this specific embodiment, the difference between the transfer functions of the two systems is generated either by the mass, the spring constant and/or by the quality factor. The mass in this case, together with the spring constant, determines the inherent frequency, so that by enlarging or reducing the mass, a further possibility results for shifting the inherent frequencies of the two systems with respect to one another.

According to one preferred specific embodiment of the present invention, a compensation variable for a detection signal occurs in a fifth step following the fourth step on the basis of the detection sensitivity determined in the fourth step. The compensation variable is determined in a manner similar to that described above for the case of a single oscillatory system.

The following described specific embodiments may be understood as variants both of the first method, i.e., change of the transfer function of one oscillatory system, as well as of the second method, i.e., two oscillatory systems having different transfer functions.

According to one preferred specific embodiment of the present invention, the rotation rate sensor includes a register that includes a plurality of value pairs of the detection sensitivity and of the compensation variable, the compensation variable occurring via a selection of a value from the register, or the compensation variable is determined by an analytical, in particular, linear correlation between the detection sensitivity and the compensation variable. Thus, a linear correlation between the detection signal relationship v and the compensation factor CF may be, in particular, even approximately assumed as a function of the selection of the operating point. In this way, a factor, instead of a table or the like may be ascertained, so that CF may be calculated by multiplying v by this factor.

According to one preferred specific embodiment of the method according to the present invention, a first detection signal is determined in the first step and a second detection signal is determined in the second step, a temperature effect on a mechanical phase of the oscillatory system being ascertained in a sixth step following the fourth step on the basis of the first and second detection signal and of the first and second detection signal. The following correlation for the absolute phase change may be calculated based on equations 3 and 4 and on a shift of the detection modes:

$\begin{array}{cc}\frac{s_{\Omega ,0}}{s_{\mathrm{Quad},0}}-\frac{s_{\Omega ,i}}{s_{\mathrm{Quad},i}}=\left(\frac{\Omega }{\mathrm{Quad}}+\varphi \right)-\left(\frac{\Omega }{\mathrm{Quad}}+\varphi +\Delta \varphi \right)=\mathrm{\Delta \varphi }& \left(9\right)\end{array}$

Since, in the case of an inherent frequency shift, the same SMS is examined and a relationship is formed, the determination of the horizontal position on the amplification characteristic curve with the method according to the present invention is virtually independent of temperature influences. However, thermal disruptive influences are also not identified or compensated for in this way. The calculation of the phase change caused by the shift of the inherent frequency is an absolute value, however, and therefore a function of the temperature. Thus, the absolute phase change in one variant of the method according to the present invention may serve as an indicator for a temperature influence.

A further subject matter of the present invention is a rotation rate sensor, including an oscillatory system and a control unit, the rotation rate sensor, in particular, the control unit being configured to carry out a method(s) in accordance with an example embodiment of the present invention. A further subject matter is a rotation rate sensor including a first and a second oscillatory system and a control unit, the rotation rate sensor, in particular, the control unit being configured to carry out a method in accordance with an example embodiment of the present invention, or being configured to carry out a method variant in accordance with the present invention, and to carry out a method in accordance with another method variant of the present invention. The specific embodiments of the method according to the present invention described further above are each transferable directly to specific embodiments of the rotation rate sensor according to the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic representation of an oscillatory system, which is formed by a flexibly mounted mass and may be stimulated by an electrode arrangement to oscillate.

FIG. 2 illustrates the transfer function of an oscillatory system via a representation of the amplification characteristic curves of the drive and detection oscillation and of the phase characteristic curve of the detection oscillation.

FIG. 3 shows the flowchart of an initializing process, in which a plurality of values of the compensation variable is determined.

FIG. 4 shows the flowchart of a compensation process, in which a compensation variable for the detection signal is determined.

FIG. 5 shows the flowchart for determining the relationship v_i.

DETAILED DESCRIPTION OF EXAMPLE EMBODIMENTS

FIG. 1 is a schematic illustration of an oscillatory system 1 as it is used in rotation rate sensors. Oscillatory system 1 includes here a seismic mass 2, which is flexibly coupled via a spring arrangement 3 to the sensor substrate and may be stimulated by an electrostatic alternating field to oscillate (drive oscillation or drive mode). One common option for determining the rotation rate is that in addition to the drive mode, oscillatory mass 2 includes a further oscillation mode, which extends, for example, along a direction perpendicular to the drive oscillation and serves as a detection mode. A rotation rate present at the sensor effectuates a coupling of drive mode and detection mode so that the amplitude of the detection mode (which is a function of the strength of the coupling and thus of the present rotation rate) may be utilized for measuring the rotation rate. In addition to this coupling, real sensors also include (usually undesirable) couplings of the two modes, which are generally referred to as quadrature effects and result in an excitation of the detection mode regardless of the rotation rate. The measured signal contains both components; the quadrature signal, however, may be separated by demodulation from the useful signal. The measured signal may be detected as indicated in the figure by the change associated with the detection oscillation of the capacitance formed by mass 2 and a measuring electrode 4 fixed relative to the substrate. Via a bias voltage 5 of the measuring electrode, it is also possible to adapt the spring constant of the system and thus to achieve a shift of the inherent frequency. This represents one simple possibility for effectuating the controlled change of the transfer function according to the present invention, since the transfer function is shifted toward higher or lower frequencies when the inherent frequency is increased or reduced.

FIG. 2 is a representation of the amplification curve and phase curve generated by shifting the inherent frequency at a drive frequency, which operates with a phase-locked loop, PPL, in the global maximum. Shown is amplification characteristic curve 10 of the drive mode (above), and two amplification characteristic curves 11, 12 shifted with respect to one another of the detection mode (center) and two phase characteristic curves 13, 14 of the detection mode (below) shifted with respect to one another are shown. The frequency in each case is plotted on horizontal axis 6. Vertical axis 7 of the upper graphic corresponds to amplification G_drive, vertical axis 8 of the middle graphic corresponds to amplification G_sense, and vertical axis 9 of the lower graphic corresponds to phase ϕ_sense. The vertical plotting is logarithmic in each case.

One specific embodiment of the method according to the present invention is based on the approach of comparing the amplification or phase of an oscillatory system 1 at different inherent frequencies. FIG. 2 represents graphically the transfer functions provided by equations 1 and 2 for a sensor made up of a drive-SMS (above) and a detection SMS (middle and below) via amplification characteristic curve and phase characteristic curve 10, 11, 12, 13, 14. The detection SMS in this example has been influenced by an increase in the spring constant, as a result of which inherent frequency ω₀ of the system, and thus characteristic curves 11, 13 are shifted in their totality toward higher frequencies (indicated by arrow 15). The difference in the curves is easily possible by evaluating the associated quadrature signals. As is apparent from equation 5, the relationship of the quadrature signals corresponds directly to the relationship v_i of the amplifications, i.e., to values 16 and 16′ of amplification characteristic curves 11 and 12 at the drive frequency. The drive frequency is identified in the figure by vertical line 35 and corresponds to the frequency at which the drive mode experiences maximum amplification G_drive. The flanks of characteristic curve 11 (and those of its shifted counterpart 12) exhibit a strictly monotonically ascending profile left of the maximum and a strictly monotonically descending profile to the right thereof. Conclusions about the operating point in the ranges in which both characteristic curves are monotonic may be clearly drawn from the value of the relationship v_i at a known shift of the inherent frequency. This fact is utilized by the method according to the present invention in order to determine from the relationship the true amplification.

FIG. 3 shows the sequence of an initialization of the sensor, in which the associated values of compensation variable CF_i(v_i) are collected for various values of relationship v_i. For the sake of simplicity, the sequences are described here and in the following based on the specific implementation with the aid of the shift of the inherent frequency, a transfer to the general case of a controlled change of the transfer function being readily possible, however. The processes are carried out, in particular, in an automated manner by a control unit of the sensor.

The initialization is started in block 17 and it is initially established in block 18 whether the desired number of measuring points is achieved. Initially, no measured values are present, so that in block 20 the inherent frequency of the SMS is shifted and the quadrature signal is subsequently measured in block 21 and the relationship v_i is determined from the measured value and from the value of the quadrature signal belonging to the unshifted inherent frequency. Correction factor CF_i=v_i/v is determined in block 22 and stored in block 23 as assignment CF_i(v_i), for example, in a register. Sequence 18, 20, 21, 22, 23 is repeated until it is determined in block 18 that the desired number of measured points is achieved and the process is ended in block 19.

FIG. 4 shows the sequence of a compensation of a sensitivity change or phase change due to disruptive influences. Once the process has been started in 24, the relationship v_i is determined in 25, and in 26 the associated correction factor CF_i(v_i) is retrieved from 28 (CF_i(v_i) may be present here, for example, as a table or also as a function). The process is ended in 27.

FIG. 5 shows the sequence of the determination of the relationship After start 29, quadrature signal s_Quad,i is measured in 30. The inherent frequency of the SMS is shifted in step 31 and associated quadrature signal s_Quad,i,Δ is measured in subsequent step 32. Relationship v_i=s_Quad,i,Δ/s_Quad,i is then calculated in 33 and the sequence is ended in 34.

Claims

1. A method for determining a detection sensitivity of a rotation rate sensor, the rotation rate sensor including an oscillatory system, the method comprising:

in a first step, determining a first quadrature signal of the oscillatory system;

in a second step, performing a controlled change of a transfer function of the oscillatory system;

in a third step, determining a second quadrature signal of the oscillatory system; and

in a fourth step, determining the detection sensitivity based on the first quadrature signal and the second quadrature signal.

2. The method as recited in claim 1, wherein, in the second step, a controlled change of an inherent frequency of the oscillatory system takes place and/or a controlled change of a quality factor of the oscillatory system takes place.

3. The method as recited in claim 2, wherein the change of the inherent frequency takes place via a controlled change of a spring constant of the oscillatory system.

4. The method as recited in claim 1, wherein further controlled changes of the transfer function take place and further quadrature signals of the rotation rate sensor are determined in a fifth step following the third step and preceding the fourth step, the detection sensitivity being determined in the fourth step based on the first, second and further quadrature signals.

5. The method as recited in claim 1, wherein a compensation variable for a detection signal is determined in a fifth step following the fourth step based on the detection sensitivity determined in the fourth step.

6. A method for determining a detection sensitivity of a rotation rate sensor, the rotation rate sensor including one first oscillatory system and one second oscillatory system, a transfer function of the first oscillatory system differing from a transfer function of the second oscillatory system, the method comprising:

in a first step, determining a first quadrature signal of the first oscillatory system;

in a second step, determining a second quadrature signal of the second oscillatory system; and

in a fourth step, determining the detection sensitivity based on the first quadrature signal and the second quadrature signal.

7. The method as recited in claim 6, wherein a quality factor of the first oscillatory system is identical to a quality factor of the second oscillatory system, and/or a mass of the first oscillatory system differs from a mass of the second oscillatory system and/or a spring constant of the first oscillatory system differs from a spring constant of the second oscillatory system.

8. The method as recited in claim 6, wherein a compensation variable for a detection signal is determined in a fifth step following the fourth step based on the detection sensitivity determined in the fourth step.

9. The method as recited in claim 5, wherein: (i) the rotation rate sensor includes a register that includes a plurality of value pairs of the detection sensitivity and of the compensation variable, the compensation variable being determined via selection of a value from the register, or (ii) the compensation variable is determined via an analytical, linear correlation between the detection sensitivity and the compensation variable.

10. The method as recited in claim 1, wherein a first detection signal is determined in the first step and a second detection signal is determined in the second step, a temperature effect on a mechanical phase of the oscillatory system being ascertained in a sixth step following the fourth step based on the first detection signal and the second detection signal.