PROCEDURE FOR CONTROLLING A LONG STATOR LINEAR MOTOR.
Patent Information
- Application Number
- IT502026000034402
- Authority / Receiving Office
- IT · IT
- Patent Type
- Patents
- Current Assignee / Owner
- Filing Date
- 2019-03-07
- Publication Date
- 2026-07-29
- Estimated Expiration
- 2039-03-07
AI Technical Summary
Long-stator linear motors face challenges in precise control and fault detection due to limited measurement capabilities, particularly when relying on a single measuring section, which restricts their ability to handle complex topologies and detect operational deviations or faults effectively.
The method involves overlapping measurement sections to generate redundant data, allowing for the determination of operating parameters by comparing first and second measured values, which can be used to approximate actual values and detect deviations, thereby enhancing control and fault detection through complementary and competitive sensor data fusion techniques.
This approach improves the precision of position and speed control, enables fault detection, and reduces the risk of system failure by utilizing redundant measurements, allowing for continuous operation even if one measuring section fails, and optimizes energy use by identifying uneven wear or misalignment.
Abstract
Description
[0001] The present invention relates to a method for controlling a long stator linear motor, wherein a first measured value is determined in a first measuring section and a second measured value is determined in a second measuring section along a transport path in a direction of movement.
[0002] A long-stator linear motor (LLM) comprises a plurality of adjacent electrical drive coils, which are fixedly arranged along a transport track on one, two, or more sides and form one or more stators. Furthermore, a number of excitation magnets, either permanent magnets, coils, or short-circuit windings, are arranged on each transport unit. The magnets are typically mounted on the transport unit on one, two, or more sides in the direction of movement so that they can interact with the drive coils of the stator. The long-stator linear motor can be designed as a synchronous machine, either self-excited or separately excited, or as an asynchronous machine. The interaction of the (electro)magnetic fields of the magnets and the drive coils generates a tractive force on the magnets of the transport unit, which in turn moves the transport unit in the direction of travel.This is achieved by controlling the individual drive coils to regulate the magnetic flux, which in turn influences the magnitude of the propulsive force. Long-stator linear motors are increasingly being used as replacements for conventional continuous conveyors or rotary-to-linear transmission units (e.g., rotary motor on conveyor belt, belt, chain, etc.) to meet the requirements of modern, flexible logistics units. A transport unit must, of course, be guided and held along the transport path in a suitable manner. In principle, any guide elements of the transport units can interact with guide elements of the transport path, using, for example, rollers, wheels, sliding elements, guide surfaces, etc. These guide elements can also be arranged on one, two, or more sides.To control the positions of the transport units on the stator, an actual position is required in addition to a target position. The device for detecting and specifying the actual position can be integrated into the long stator linear motor or implemented externally.
[0003] The position, speed, acceleration of transport units, or any other physical quantity along the entire transport route can be determined. This can be interpreted as the existence of a measurement section covering the entire transport route. A measurement section comprises one or more sensors for acquiring a measured value. Each measured value represents the actual value of a physical quantity. For example, the position of a transport unit within the measurement section can be determined as a measured value, thus representing the actual position as a physical quantity. This can be done directly by position sensors within the measurement section. Alternatively, position monitors can be used to determine the position based on other information such as voltages and currents.
[0004] If only one measuring section is provided, the transport path can only have a one-dimensional topology, i.e., a loop or a line. The transport path cannot then include any switches, as the measuring section would otherwise have to overlap itself. In contrast, the transport path of a long-stator linear motor can also be composed of multiple measuring sections. Typically, but not necessarily, each measuring section covers one transport segment or part of a transport segment. A transport segment is a modular section of the transport path and comprises a number of drive coils. The measuring sections are usually spaced apart from each other or arranged adjacent to each other along the transport path in the direction of movement.
[0005] To determine the global actual position of a transport unit, the measurement section where the transport unit is located can first be determined. Furthermore, the actual position of each section can be determined. The individual section actual positions are then fused to form a global actual position along the transport path. This allows the transport unit to be assigned a unique global actual position, for example, relative to a selected reference point. If the measurement sections together completely cover the transport path of the linear motor, a unique actual position relative to any reference point can be assigned to the transport units at any given time. This is called complementary sensor data fusion and is known, for example, from US 6,876,107 B2. In this process, the individual measurement sections complement each other and are seamlessly connected.
[0006] One objective of the present invention is to improve the control of the long stator linear motor.
[0007] This problem is solved according to the invention by having the first measuring section overlap with the second measuring section in the direction of movement in an overlap area, wherein the first measured value and the second measured value represent the same actual value of a physical quantity and wherein an operating parameter of the long stator linear motor is determined on the basis of an occurring deviation between the first measured value and the second measured value.
[0008] Due to the overlap of measurement sections, redundant measured values are generated in the overlap area. The respective measured value can be determined or observed in the corresponding measurement section by a sensor, or through the interaction of sensors within that section. It is determined whether and to what extent the first measured value of the first measurement section deviates from the second measured value of the second measurement section, whereby a tolerance can, of course, be provided. If a deviation occurs, an operating parameter is determined using this deviation. The first and / or second measured value can be used to determine the operating parameter. The first and second measured values themselves only need to represent the physical quantity. This does not mean that the measured values must directly represent the same physical quantity.For example, the first measurement can directly describe a current position, and the second measurement can describe a current from which the current position is then determined. Thus, the first measurement directly represents the current position as a physical quantity, and the second measurement indirectly represents the current position as a physical quantity.
[0009] Of course, the method according to the invention is not limited to two measured values, each originating from a corresponding measuring section. It is also possible to use one measured value from more than two overlapping measuring sections to determine the operating parameter, or even a plurality of measured values from two or more overlapping measuring sections.
[0010] The measuring sections can be located on opposite sides of the transport route.
[0011] While this results in an overlap area when viewed in the direction of movement, the measuring sections can certainly be spaced apart from each other in a transverse direction perpendicular to the direction of movement. Such arrangements are particularly common in long-stator linear motors with double combs. Double-comb long-stator linear motors are characterized by two drive sides arranged along the transport path, with one stator provided for each drive side. Drive coils are thus arranged on each side. Accordingly, excitation magnets are also provided on both sides of a transport unit, each interacting with the drive coils on one side.
[0012] The measuring sections can also be located on the same side of the transport route.
[0013] It is possible for the sensors of two measuring sections to be arranged overlapping in the direction of movement. However, the sensors belonging to the respective measuring sections are often not arranged overlappingly. It's important to note that the measuring sections refer to the sensors' field of view and not the physical dimensions of the sensors themselves. Generally speaking, overlapping measuring sections mean that the fields of view of the sensors overlap, and the sensors themselves can also overlap. This also applies, of course, to measuring sections located on opposite sides of the transport path.
[0014] Of course, along a transport route, multiple measuring sections can be arranged in such a way that a mixture of overlapping areas on opposite sides and on the same side of the transport route results. A third measurement from a third measuring section can also be compared with the first and second measurements, or further measurements from additional measuring sections, etc.
[0015] Advantageously, an approximation of the actual value is determined as an operating parameter.
[0016] This approximation can of course be performed over the entire overlap range for occurring actual values, using a first and a second measured value for each actual value.
[0017] Either the first or the second measured value can be selected as an approximation of the actual value.
[0018] This corresponds to a selective method, which enables a particularly rapid approximation of the actual value. Thus, even if one measuring section fails in the overlap area, the other measuring section can continue to provide measured values, thereby preventing a failure of the entire long-stator linear motor.
[0019] The selection of the first or second measurement can be based on a classification of the respective measurement and / or an expected accuracy of the respective measurement values.
[0020] The selection of the first or second measured value as an approximation of the actual value can be based on an existing classification of the measured value or the measurement section. For example, the measured value that can be assumed to be more accurate based on the classification can be selected.
[0021] It is also possible to select one of the measured values based on its accuracy. For example, it can be assumed that the accuracy decreases at the edge of a corresponding measurement section, thus allowing the position of the determined measurement relative to the measurement section to be factored into the selection. A decrease in measurement accuracy with increasing distance between the sensor and the object being measured can also be considered as a geometric factor during the selection process. For example, in the case of opposing measurement sections, the measured values of the section where a transport unit is closer can be selected.
[0022] The selection of a measurement value can also be made using learning algorithms, such as neural networks.
[0023] As mentioned, the actual value of a physical quantity is represented by measured values in both measurement sections. Since two measured values are available, a unique determination of the actual value is not possible. Therefore, the measured values are preferably processed to approximate the actual value. This allows the actual value to be determined with increased accuracy, as not just a single measured value from one measurement section, but rather measured values from two (or more) measurement sections serve as the basis. This is referred to as concurrent sensor data fusion. The actual value to be selected can be determined based on an existing classification of the measured value or the measurement section. The actual value can also be approximated, for example, by averaging the measured values.
[0024] Advantageously, however, the first and second measured values are each assigned a weighting factor, and the approximation of the actual value is determined as an operating parameter from the first and second measured values and the respective weighting factor.
[0025] Unlike a selective method, this approach does not choose a single measured value as an approximation of the actual value. Instead, both measured values are considered, with each being weighted accordingly. By applying a weighting factor to the respective measured values, the approximation of the actual value can be improved even further.
[0026] The weighting factor can also include a model factor, which is determined by the magnitude of the deviation of the measured value from a reference model.
[0027] The model factor can thus be determined based on a reference model, which can, for example, represent a physical behavior. For instance, the equations of motion of a transport unit can be used for modeling, and the deviation of the actual motion determined by the measured values can be incorporated into the model factor. The greater the deviation between a measured value and the model, the more likely that the measured value is incorrect. The model factor can then be chosen based on this. This can occur particularly with opposing measurement sections, as these may overlap in the direction of motion but still be separated laterally, which can very likely result in different measured values.
[0028] The weighting factor can include a geometry factor, which is determined by the position of the measured value in the measurement section.
[0029] For example, it can be assumed that the accuracy of the measured values decreases at the edge of a corresponding measurement section, allowing the position of the determined measurement value relative to the measurement section to be incorporated into the geometry factor. Measured values at the edge of the measurement section are thus weighted less than measured values in the center of the measurement section. A decrease in the accuracy of the measured values with increasing distance between the sensor and the object being measured can also be considered as a geometry factor. For example, in the case of opposing measurement sections, the measured values of the section where a transport unit is closer can be weighted more heavily.
[0030] The weighting factor can include a statistical factor, which is determined by a statistical distribution function.
[0031] The statistical factor can, for example, take into account a parameterized random distribution of the measurement signals, whereby the variance of the measured values can be estimated. This can be particularly useful when determining the actual position by assuming that the variance increases with increasing distance between the magnetic plate of the transport unit and the position sensor. This is especially effective in combination with a geometric weighting factor.
[0032] Furthermore, the weighting factors can also be used by machine learning algorithms, such as neural networks. Naturally, any combination of the aforementioned factors and methods can be used to determine the weighting functions.
[0033] One measurement position of a transport unit on the transport route can be determined as the first and second measurement values.
[0034] This allows the actual position to be represented as an actual value by the measured positions as measured values, thus improving the control of the transport units, since the actual position can be determined more accurately by taking into account the measured values of several measuring sections.
[0035] Similarly, the speed and / or acceleration of a transport unit along the transport route and / or the temperature and / or current can be determined as the first and second measured values.
[0036] The deviation between the first and second measured values can be used to determine the occurrence of a disturbance and / or a fault and / or wear on the long stator linear motor as an operating parameter.
[0037] If a malfunction, fault, or wear occurs, the first and second measured values may deviate from each other beyond a predefined tolerance. This deviation allows conclusions to be drawn about the malfunction, fault, or wear.
[0038] The measured values can be reliably acquired and / or reliably evaluated. "Reliability" can be defined according to a category in Table 10 of the standard DIN EN ISO 13849-1:2016-06, and thus, depending on the safety category, single-fault safety, double-fault safety, etc., can be provided.
[0039] An action can be triggered when a malfunction, fault, or wear is detected. This action could be an emergency stop, the output of a signal (e.g., acoustic or visual), the setting of a flag, etc.
[0040] Errors can include mechanical assembly errors, but also failures (e.g., of a measuring section due to a sensor error or magnetic plate loss, etc.). Incorrectly initialized parameters, such as an incorrect definition of a measuring section, can also be detected, resulting in a difference between the measured value of this section and that of an overlapping section.
[0041] Incorrect assembly of transport segments can lead to mispositioning of a transport unit along the transport route, disrupting higher-level processes, particularly the control of the transport units' positions and trajectories. This can result in discontinuous control signals, unstable control loops, overcurrent errors, trailing error interruptions, etc. It is also possible for control loops to partially cancel each other out, creating further unstable control loops and potentially increasing energy consumption. Such incorrect assembly of transport segments can be inferred from deviations in the measured values within the overlapping area, especially if the measurement sections each cover transport segments.
[0042] Similarly, disturbances such as environmental conditions (e.g., elevated temperature) can also be detected, as these affect the sensor readings of individual measurement sections. Furthermore, the failure of a measurement section or part of a measurement section (sensor failure, magnetic plate loss, etc.) can be detected.
[0043] A deviation in the measured values can also reveal, for example, wear on guide elements such as rollers, especially when measuring sections are positioned opposite each other. Different measured values can indicate a change in the distance between the moving parts and the respective measuring section. In the case of differing measured values from overlapping measuring sections, this allows conclusions to be drawn about uneven wear of the guide elements (e.g., rollers), loss of the magnetic plate, demagnetization of a magnetic plate, or a sensor malfunction (e.g., sensor drift).
[0044] It is also possible to determine on which side of the transport unit has a shorter distance to the transport track. This can occur, for example, due to one-sided sealing. This information can be used for targeted actuation of the drive coils on the side with the shorter distance, thus saving energy and reducing losses.
[0045] The present invention is described below with reference to the Figuren 1 bis 2b In more detail, the invention is explained, and exemplary, schematic, and non-restrictive embodiments are shown. This includes showing Fig. 1 a long stator linear motor, Fig. 2a two measuring sections on the same side of the transport path, Fig. 2b two measuring sections on opposite sides of the transport path.
[0046] Fig.1 Figure 2 represents a long-stator linear motor, where the stator of the long-stator linear motor 2 is exemplified as a closed transport section 20. A plurality of drive coils L are arranged one behind the other in the direction of movement r of a transport unit 1 on the transport section 20. In normal operation, each of these coils is energized by a coil current im, controlled by a control unit R, to generate a moving magnetic field. The coil current im through the respective drive coils L can differ fundamentally from one drive coil L to the next. The control unit R can be implemented as suitable hardware and / or as software running on suitable hardware. The drive coils L, arranged side by side in the direction of movement r, are mounted on a stationary support structure 3 (only indicated in the figures) on the transport section 20.Depending on the application and requirements, the transport route 20 can be of any shape and can include closed and / or open sections. The transport route 20 can lie in a plane or be arbitrarily routed in space.
[0047] A transport track 20 typically consists of several interconnected transport segments, each with a number of drive coils L. Switches can also be used to guide a transport unit 1 from a first transport segment 20 to a second transport segment.
[0048] A transport unit 1 must, of course, be guided and held along the transport track 20 in a suitable manner. Any guide elements of the transport unit 1 can interact with guide elements of the transport track 20, such as rollers, wheels, sliding elements, guide surfaces, etc. These guide elements can also be arranged on one side, two sides, or multiple sides in sections.
[0049] Two measuring sections 21 and 22 are arranged along the transport path 20 of the long-stator linear motor, with each measuring section 21 or 22 extending over a portion of the transport path 20. A measuring section 21 or 22 can extend over several consecutive transport segments or be limited to a single transport segment. Naturally, a measuring section 21 or 22 can also extend beyond a transport segment or be considered independently of transport segments. For this reason, the present description focuses on measuring sections 21 and 22, rather than transport segments. For clarity, the measuring sections 21 and 22 are shown in Fig. 1 not shown. Rather, part of transport route 20 is shown in the Fig. 2a und 2b considered, with overlapping measurement sections 21,22 shown.
[0050] A measuring section 21, 22 is designed to determine one or more measured values m1, m2, where each measured value m1, m2 represents an actual value X of a physical quantity G. The physical quantity G can be considered to be an actual position x and / or an actual velocity v and / or an actual acceleration a of a transport unit 1. Thus, each measured value m1, m2 represents a measured position, a measured velocity, or a measured acceleration, respectively, and therefore reflects an actual position x, an actual velocity v, or an actual acceleration a, whereby the two measured values m1, m2 do not necessarily have to directly represent the same physical quantity G, but merely reflect it.
[0051] If an actual position is determined as a physical quantity G, this can be done with respect to a reference point, where the reference point can be a measurement section 21, 22, a transport segment, or any other point in space. Other physical quantities G, such as a prevailing force, a flowing current, a prevailing temperature, etc., can also be represented by the measured values m1, m2. From this, a physical quantity G such as an actual position x can be calculated, which can also be done via an observer.
[0052] A first measurement m1 can directly represent a physical quantity G, e.g., the current position. This means that the first measurement m1 represents the current position itself. A second measurement m2, on the other hand, can represent a different physical quantity, e.g., an electric current, from which the current position is derived as a physical quantity G. Thus, the first measurement m1 describes the physical quantity G directly, and the second measurement m2 describes the physical quantity G indirectly. Both measurements m1 and m2, however, represent the physical quantity G.
[0053] Magnetic field sensors, such as Hall sensors or magnetoresistive sensors, can be used as sensors. However, other physical measurement principles can also be employed, such as optical sensors, capacitive sensors, inductive sensors, etc. Current sensors, which determine the coil current through a drive coil L, can also be used. As is known, the normal force and / or propulsive force acting on a transport unit 1 can be determined from the coil current. A temperature sensor can also be used.
[0054] It is in Fig. 2a A first and a second measuring section 21, 22 are shown as examples. According to the invention, at least two measuring sections 21, 22 have an overlap area B in the direction of movement r, i.e., along the transport path 20. The measuring sections 21, 22 overlapping in an overlap area B can be arranged as shown in Fig. 2a on the same side of transport route 20, or also on opposite sides of transport route 20, as in Fig. 2b depicted.
[0055] In both cases shown, a first measured value m1 is determined in the overlap area B of the first measurement section 21, and a second measured value m2 is determined in the overlap area B of the second measurement section 22. Both measured values m1 and m2 represent the same actual value X of a physical quantity G. For example, the actual position x of a transport unit 1 can be represented as the actual value X by the first measured value m1 of the first measurement section 21. Similarly, the actual position x of the transport unit 1 can also be represented as the actual value X by the second measured value m2 of the second measurement section 22, i.e., as the second measured actual position.
[0056] It is in the Fig. 2a, b Only one actual value X is shown; of course, other and / or further actual values X can also be determined in the overlap area B, whereby measured values m1, m2, which represent the other / further actual values X, are determined in each case.
[0057] If the first measured value m1 and the second measured value m2 differ, an operating parameter P of the long stator linear motor 2 can be determined from the deviation of the first measured value m1 from the second measured value m2, whereby the operating parameter P is given as a function of the measured values P=f(m1, m2).
[0058] This takes place in the Fig.2a, 2b For example, this can take place in a processing unit V, but it can also be done, for example, in the control unit R or another unit already present on the long stator linear motor 1. The operating parameter P can also be output and / or processed, for example, to control the transport units 1.
[0059] An approximation of the actual value X can be determined from the deviation of the first and second measured values m1, m2 as operating parameter P. This can be done by transforming the measurement sections 21, 22 into a common coordinate system. The measured values m1, m2 can be averaged or each assigned a weighting factor f1, f2, resulting in the operating parameter P as a function of the measured values m1, m2 and their respective weighting factors f1, f2: P = f(m1, f2 ; m2, f2). An approximation of the actual value X can be determined as the operating parameter P. A weighting factor f1, f2 for a measurement section 21, 22 can be initially fixed and / or adjusted over time.
[0060] The respective measurement section 21, 22 can also contain areas of varying measurement accuracy, whereby the measurement accuracy can change discretely and / or continuously across a measurement section 21, 22 or a part thereof. Likewise, the measurement accuracy of a measurement section 21, 22 can change over time and / or depending on other influences, such as temperature, contamination, and / or sensor aging, etc. Thus, the respective weighting factor f1, f2 can include a geometry factor, which is determined by the position of the measured value m1, m2 within the measurement section 21, 22. For example, the geometry factor can incorporate factors such as accuracy dependent on the position within the measurement section 21, 22, the distance to the object being measured, the temperature, magnetic stray fields, etc.If the accuracy of the measured values m1, m2 decreases towards the edge of the measuring section 21, 22, the geometry factor can be used as a function of the distance to the center of the measuring section 21, 22.
[0061] Of course, a weighting factor f1, f2 can vary depending on the position of the measured value in relation to the measuring section 21, 22, which can also be achieved by a geometry factor.
[0062] The weighting factor f1, f2 can also include a statistical factor determined by a statistical distribution function. If the probability distributions of the individual measurement sections 21, 22 are known, independent of each other, normally distributed, and have the same mean, then a maximum likelihood estimator using weighted least squares can be employed. The variance on a measurement section 21, 22 can be a function of both time and position on that section.
[0063] Similarly, model factors can be incorporated into the weighting factors f1 and f2. The Kalman filter serves as an example of a model-based estimator. Assumptions about the probability distribution of the measured values m1 and m2 can also be made when designing a Kalman filter.
[0064] Alternatively, the first or second measurement m1, m2 itself, or an average of the first or second measurement m1, m2, could be selected as an approximation of the actual value X. The information mentioned above regarding weighting factors, which is incorporated into the statistical and / or geometric factors, can likewise be used to select a measurement m1, m2 as an approximation of the actual value X.
[0065] The deviation of the first and second measured values m1, m2 can be used to determine the occurrence of a disturbance, fault, and / or wear on the long stator linear motor 2 as an operating parameter P. This is possible if the measured values m1, m2 of overlapping measuring sections 21, 22 differ from each other due to the disturbance, fault, or wear. Conversely, this allows conclusions to be drawn about the disturbance, fault, or wear. For example, the type of disturbance, fault, or wear can be inferred based on the magnitude of the deviation. Changes in environmental conditions, such as an increased temperature, can also be considered disturbances.
Claims
1. Method for controlling a long stator linear motor (2), wherein a first measured value (m1) and a second measured value (m2) are determined along a transport section (20) in a direction of movement (r) in a first measuring section (21) and in a second measuring section (22), characterized by the fact that the first measuring section (21) in the direction of movement (r) overlaps with the second measuring section (22) in an overlap area (B), that the first measured value (m1) and the second measured value (m2) represent the same actual value (X) of a physical quantity (G), and that An operating parameter (P) of the long stator linear motor (2) is determined based on a deviation occurring between the first measured value (m1) and the second measured value (m2).
2. Method according to claim 1, characterized by the fact that the measuring sections (21, 22) are provided on opposite sides of the transport route (20).
3. Method according to claim 1 or 2, characterized by the fact thatthe measuring sections (21, 22) are provided on the same side of the transport route (20).
4. Method according to any one of claims 1 to 3, characterized by the fact that an approximation of the actual value (X) is determined as an operating parameter (P).
5. Method according to claim 4, characterized by the fact that The first or the second measured value (m1, m2) is selected as an approximation of the actual value (X).
6. Method according to claim 4, characterized by the fact that The selection of the first or second measured value (m1, m2) is based on a classification of the respective measured values.
7. Method according to claim 5 or 6, characterized by the fact that The selection of the first or second measurement (m1, m2) is based on the expected accuracy of the respective measurements.
8. Method according to claim 4, characterized by the fact that The first and second measured values (m1, m2) are each assigned a weighting factor (f1, f2), and thatThe approximation of the actual value (X) as the operating parameter (P) is determined from the first and second measured values (m1, m2) and the respective weighting factor (f2, f2).
9. Method according to claim 8, characterized by the fact that The weighting factor (f1, f2) includes a model factor which is determined by the magnitude of a deviation of the associated measured value (m1, m2) from a reference model.
10. Method according to one of claims 8 or 9, characterized by the fact that the weighting factor (f1, f2) includes a geometry factor which is determined by the position of the respective measured value (m1, m2) in the associated measurement section (21, 22).
11. Method according to any one of claims 8 to 10, characterized by the fact that The weighting factor (f1, f2) includes a statistical factor which is determined by a statistical distribution function.
12. Method according to any one of claims 1 to 11, characterized by the fact thatthe occurrence of a disturbance and / or a fault and / or wear on the long stator linear motor (2) is determined as an operating parameter (P).
13. Method according to any one of claims 1 to 12, characterized by the fact that In each case, a position of a transport unit (1) on the transport route (20) is determined as the first and second measured value (m1, m2).
14. Method according to any one of claims 1 to 12, characterized by the fact that A speed and / or an acceleration of a transport unit (1) on the transport route (20) is determined as the first and second measured values (m1, m2).
15. Method according to any one of claims 1 to 12, characterized by the fact that Each measurement determines a temperature and / or a current as the first and second measured value (m1, m2).