JP6444041B2 - Handwritten music symbol recognition apparatus and handwritten music symbol recognition program - Google Patents

Description

  The present invention relates to a handwritten music symbol recognition apparatus and a handwritten music symbol recognition program, and in particular, for recognizing handwritten symbols when a score is created by handwriting (including using a pen) on a tablet terminal (including a touch-compatible PC). The present invention relates to a technique suitable for use.

  A tablet terminal that supports handwriting is equipped with an API that acquires position information whenever a finger moves until the finger touches the screen and then leaves. A series of changes in position information is stored as a plurality of vectors, the features of finger movement are extracted from the vectors, and the drawn symbols are recognized.

The actual API operation acquires many minute changes in position information until the finger touches the screen and then leaves. Therefore, many small vectors are generated.
If it is as it is, it is difficult to grasp the characteristics of finger movement and the amount of calculation increases.

  In order to solve such a problem, Patent Document 1 proposes a linear component extraction method for extracting a linear component from input coordinate data.

JP-A-1-96770

The technique proposed in Patent Document 1 is a technique for making straight line data connecting start point coordinates and end point coordinates if a plurality of input coordinate data are within a predetermined linearization range. When it is determined that the input coordinate data is out of the linearization range while sequentially checking the linear component, the data before the coordinate data and the start point data are extracted and stored as the linear component. For this reason, there has been a problem that the amount of calculation required for recognizing the entire input symbol increases.
In view of the above-described problems, an object of the present invention is to reduce the amount of calculation required by compressing data when recognizing a symbol in which many vectors are generated by handwriting input.

The handwritten music symbol recognition apparatus according to the present invention includes a touch information acquisition unit that acquires position information every time the input unit moves until the input unit touches the screen and leaves, and a position obtained from the touch information acquisition unit. Stroke information storage means for creating vector information indicating movement of the input means from the information, storing a series of vectors from when the input means touches the screen until it completely leaves as one stroke information, and a series of 1 to 1 Includes object information storage means for storing one or more stroke information in which a plurality of vector information is stored, stroke feature quantity calculation means for calculating a feature quantity of the stroke from the vector information, and a plurality of vector information. Among them, data compression means for compressing the number of vectors by performing a collective process to convert a predetermined number of vectors into one vector It has a, and the summarizing processing obtains a straight line connecting the end point of the start point and the final vector of the first vector, as long as the distance to the straight line obtained from said end point of the vector is within the prescribed value, combine multiple vectors This is a process for generating one vector .
In addition, the handwritten music symbol recognition apparatus of the present invention is obtained from the touch information acquisition unit that acquires the position information every time the input unit moves until the input unit touches the screen and then leaves. Stroke information storage means for creating vector information indicating movement of the input means from the obtained position information, and storing a series of vectors as one stroke information from when the input means touches the screen until it completely leaves; Object information storage means for storing one or more stroke information in which one or more vector information is stored; stroke feature quantity calculation means for calculating a feature quantity of the stroke from the vector information; Data pressure that compresses the number of vectors by combining the number of vectors into one vector. Means for determining whether the vectors are adjacent to each other when the vectors are classified by direction, and in the case of vectors adjacent to each other, an upper limit angle and a lower limit of angles serving as threshold values thereof are determined. When the angle is calculated and both are within the range of the upper limit angle and the lower limit angle, these vectors are combined to generate one vector.
The handwritten music symbol recognition program according to the present invention includes a touch information acquisition process for acquiring position information every time the input unit moves until the input unit touches the screen and then leaves, and a position obtained from the touch information acquisition process. Stroke information storage processing for creating vector information indicating movement of the input means from the information, storing a series of vectors from when the input means touches the screen until it completely leaves as one stroke information; Is an object information storage process for storing one or more stroke information in which a plurality of vector information is stored, a stroke feature quantity calculation process for calculating a feature quantity of the stroke from the vector information, and a plurality of vector information. Among them, the data pressure that compresses the number of vectors by performing a batch process to convert a predetermined number of vectors into one vector A handwritten music symbol recognition program for executing a processing on a computer, and the summarizing processing obtains a straight line connecting the end point of the start point and the final vector of the first vector, the distance to the straight line obtained from said end point of the vector If it is within a prescribed value, it is a process for generating a single vector by combining a plurality of vectors .
Further, the handwritten music symbol recognition program of the present invention is obtained from the touch information acquisition process for acquiring the position information every time the input unit moves until the input unit touches the screen and then leaves, and the touch information acquisition process. Stroke information storage processing for creating a vector information indicating the movement of the input means from the obtained position information, and storing a series of vectors as one stroke information from when the input means touches the screen until it completely leaves; Object information storage processing for storing one or more stroke information in which one or more vector information is stored; stroke feature amount calculation processing for calculating a feature amount of the stroke from the vector information; Among the vectors, a predetermined number of vectors are combined into one vector, and the number of vectors is compressed. In the case of vectors adjacent to each other, it is determined whether the vectors are adjacent to each other when the vectors are classified by direction. The upper limit angle and the lower limit angle of the threshold angle are calculated, and when both are within the upper limit angle and the lower limit angle, the vectors are combined to generate one vector. .

  ADVANTAGE OF THE INVENTION According to this invention, the amount of calculations required when recognizing the symbol by which many vectors were produced | generated by handwriting input can be reduced.

It is a block diagram which shows the structural example of the handwritten music symbol recognition apparatus which concerns on embodiment of this invention. It is a figure which shows the example which classify | categorizes a vector into nine types. It is a figure which shows the example of a change of the rotation angle from the said vector to the following vector. It is a figure which shows the example which collects the micro vector in 1st Embodiment, and converts it into one vector of a certain magnitude | size. It is a flowchart explaining the process sequence which compresses the amount of calculations in 1st Embodiment. It is a figure explaining the example put together into one vector, if the distance from the straight line which connects a starting point and an end point is less than a regulation value in a 2nd embodiment. It is a flowchart explaining the process sequence in 2nd Embodiment. In a 3rd embodiment, when a vector is grouped according to a direction and the vector of the same group continues, it is a figure showing an example which combines them into one. It is a flowchart explaining the process sequence in 3rd Embodiment. It is a figure which shows the example which puts together the vector of the threshold value vicinity in 4th Embodiment. It is a flowchart explaining the process sequence in 4th Embodiment. It is a flowchart explaining the process sequence which produces | generates the arrangement | sequence of the Index of the vector which should be put together from the vector arrangement | sequence in 4th Embodiment. It is a figure which shows the relationship between ID memorize | stored and an x-axis flag. It is a figure for determining the threshold angle of a vector according to an x-axis flag and temporary storage vector ID.

  First, an embodiment of a data compression method in a handwritten music symbol recognition apparatus according to an embodiment of the present invention will be described with reference to the drawings.

The handwritten music symbol recognition apparatus of the present invention will be described below with reference to the drawings.
FIG. 1 is a block diagram illustrating a schematic configuration of a handwritten music symbol recognition apparatus 10, and the handwritten music symbol recognition apparatus 10 includes a tablet terminal (including a touch-compatible PC).

  In FIG. 1, 11 is a bus, 12 is ROM, 13 is RAM, 14 is CPU, 15 is flash memory, 16 is a network interface, 17 is a touch display, 18 is an audio output device, and 19 is a MIDI output device.

A program for controlling the overall operation of the handwritten music symbol recognition apparatus 10 is stored in the flash memory 15, and is read out, expanded in the RAM 13, and executed by the CPU 14.
The touch information acquisition means includes the touch display 17, the RAM 13, the CPU 14, the flash memory 15, and the like. Each time the input means moves until the input means such as a finger or a touch pen moves away from touching the screen, the position information is obtained. get.

The stroke information storage means and the object information storage means are constituted by the RAM 13. The stroke information storage means stores a series of vectors as one stroke information from when the input means touches the screen until it completely leaves.
The object information storage means stores one or more stroke information in which a series of one or more vector information is stored.
The stroke feature amount calculating means, stroke recognizing means, and object recognizing means are composed of a RAM 13, a CPU 14, and a flash memory 15. The stroke feature quantity calculation means calculates the stroke feature quantity from the vector information indicating the movement of the input means.

The network interface 16 stores the recognized music symbol information as a file in other hardware, or reads a stored music symbol information file.
The audio output device 18 is provided in order to create music performance information from the recognized music symbols and output the sound.
The MIDI output device 19 is provided in order to create MIDI information from the recognized music symbols and output the MIDI information.

A method of assigning a vector ID according to this embodiment will be described.
In the present embodiment, the vectors are classified into nine types in order to simplify the processing.
As shown in FIG. 2, the slope a is obtained from the starting point (x1, y1) and the ending point (x2, y2) of the vector using the equation (1).
Then, the vector ID is determined in consideration of the positional relationship of x1, y1, x2, and y2. The lower side of the y-axis is “+”.

| a |> 2
y2 <y1: Lower → Up y2> y1: Lower → Up-2 ≦ a <−0.5
x2 <x1: lower left → upper right x2> x1: upper right → lower left
| a | ≦ 0.5
x2> x1: Left → Right x2 <x1: Right → Left 0.5 <a ≦ 2
x2> x1: upper left → lower right x2 <x1: lower right → upper left x1 = x2
y2 <y1: Bottom → Up y2> y1: Bottom → Up x1 = x2 and y1 = y2
It becomes a point.

<Description of each feature>
(Start point end point distance)
When this distance is smaller than the specified value, it is considered as a closed figure like a circle, and when it is larger, it is considered as a figure not closed like an arc line.
If the start point coordinates of the first vector are (x1, y1) and the end point coordinates of the final vector are (x2, y2), the start point / end point distance D _se is obtained by the following equation (2).

(Connection type)
It is used to judge whether the symbol is smooth or has corners.
It is determined from the ID of the vector and the ID of the next vector. In this embodiment, it is divided into an obtuse angle, a right angle, or an acute angle.

(Direction of rotation)
The point where the rotation direction has changed is important as a clue for identifying the symbol.
For example, in the first stroke of “f”, vectors are smoothly continuous like “S”, the start point and the end point are separated, but “S” has one point where the rotation direction changes. So you can distinguish. This is also determined from the ID of the vector and the ID of the next vector.

(rotation angle)
FIG. 3 shows a change example of the rotation angle from the vector to the next vector. From this total, it can be determined whether the symbol is a circle or a multiple circle.
If the start point of the vector is (x11, y11), the end point is (x12, y12), the start point of the next vector is (x21, y21), and the end point is (x22, y22), the angle Deg ( °) is obtained by the following formula.

(Left / Right direction and Up / Down direction)
It is determined whether the symbol is simply a line segment drawn from top to bottom or from left to right.
Obtained from the vector ID. If the vector ID is down → up, there is no left / right movement direction, and the up / down movement direction is up. If the vector ID is lower right → upper left, the left / right moving direction is left and the up / down moving direction is up.

The data compression method in the handwritten music symbol recognition apparatus of the present invention compresses the number of vectors by performing a collective process of converting a predetermined number of vectors into one vector among a plurality of vectors.
Hereinafter, a data compression method in the handwritten music symbol recognition apparatus of the present invention will be described in detail with reference to the drawings.

(First embodiment)
In the present embodiment, a vector conversion process for collectively converting minute vectors into a vector having a certain size will be described.
An example will be described in which ten minute vectors a1 to a10 shown in FIG. 4A are collectively converted into two vectors b1 and b2 having a certain size as shown in FIG. 4B.
For stroke data,
A vector array 1 for storing vector data generated from position information acquired from the API;
A vector array 2 for storing a small vector to be converted;
It has a vector array 3 that stores the converted vector.

The processing procedure of the present embodiment will be described with reference to the flowchart of FIG. The processing of this flowchart is realized by developing a program stored in the flash memory 15 in the RAM 13 and executing it by the CPU 14.
In S51, one vector is copied from the vector array 1 to the vector array 2.
In S52, a vector (test vector) having the start point of the first vector of the vector array 2 as the start point and the end point of the last vector of the vector array 2 as the end point is generated, and its size is obtained.
In S53, a specified size determination process for determining whether the size of the test vector is equal to or greater than a specified value is performed. If it is equal to or greater than the specified value, the process proceeds to S54. If it is less than the specified value, the process returns to S51, and the next vector is copied from the vector array 1 to the vector array 2.

  Further, a test vector having the start point of the first vector as the start point and the end point of the vector added this time as the end point is generated, and its size is obtained. The processes of S51 to S53 are repeatedly performed until the size exceeds the specified value.

In S54, the generated test vector is registered in the vector array 3, and the vector array 2 is cleared.
In S55, it is determined whether or not there is an unprocessed vector in the vector array 1. If there is no unprocessed vector, the process ends. If there is an unprocessed vector, the process returns to S51.

  By performing the processing of S51 to S55, minute vectors can be collected and converted into a vector of a certain size. As a result, data compression can be performed and the characteristics of the symbols can be easily grasped.

(Second Embodiment)
In the present embodiment, a process for correcting fluctuation of a straight line vector will be described.
When humans draw straight lines with their fingers, the lines may fluctuate. If the process of absorbing this fluctuation and making it a single straight line is not performed, what should be correctly determined as a straight line may be erroneously determined as an arc line.

In this embodiment, as shown in FIG. 6 (a), if the distance from the straight line connecting the start point and the end point is within a specified value, they are combined into one vector as shown in FIG. 6 (b). Hereinafter, the processing procedure of this embodiment will be described with reference to the flowchart of FIG. The processing of this flowchart is realized by developing a program stored in the flash memory 15 in the RAM 13 and executing it by the CPU 14.
In S71, a straight line equation connecting the start point of the first vector and the end point of the last vector is obtained.
In S72, a perpendicular is drawn from the end point of the vector to the straight line obtained in S71, and the distance between the intersection and the end point of the vector is obtained.

In S73, a deviation width specified value determination process is performed to determine whether or not the distance obtained in S72 is within a specified value. If it is outside the specified value, the process ends. If it is within the specified value, the process proceeds to S74.
In S74, it is determined whether there is an unprocessed vector. If there is an unprocessed vector, the process returns to S72 to process the next vector, and if there is no unprocessed vector, the process proceeds to S75.

  In S75, a vector having the start point of the first vector in the vector array as the start point and the end point of the last vector as the end point is generated. Then, all the vectors in the vector array are deleted, and the generated vector is stored in the vector array.

  By performing such processing, when a plurality of minute vectors are input, the distance from the straight line connecting the start point of the first vector and the end point of the final vector is within the specified value in the plurality of minute vectors. For example, they can be combined into one vector. As a result, data compression can be performed and the characteristics of the symbols can be easily grasped.

(Third embodiment)
In the present embodiment, a case will be described in which vectors are grouped by direction, and when vectors of the same group are continuous, they are combined into one.
As shown in FIG. 8A, the vectors are grouped into nine groups based on the gradient of the vectors.

  In the present embodiment, grouping means nine IDs of point, bottom → top, bottom left → top right, left → right, top left → bottom, top → bottom, top right → bottom left, right → left, bottom right → top left. Of these, one ID is given to the vector. The method of giving the ID is as described above with reference to FIG.

As shown in FIG. 8B, description will be made by taking as an example a stroke composed of six vectors b1 to b6.
The first three vectors b1, b2, b3 have IDs from left to right. In this case, as shown in FIG. 8C, a vector c1 having the start point of the first vector b1 as the start point and the end point of the third vector b3 as the end point is generated and replaced with these three vectors. Further, the fourth to sixth vectors b4, b5, and b6 have IDs of up → down. Also in this case, a vector c2 having the start point of the fourth vector b4 as the start point and the end point of the sixth vector b6 as the end point is generated and replaced with these three vectors.

A processing procedure for replacing a vector will be described with reference to the flowchart of FIG. The processing of this flowchart is realized by developing a program stored in the flash memory 15 in the RAM 13 and executing it by the CPU 14.
In S91, the nth vector is copied as a reference vector. The initial value of n is 1.
In S92, it is compared whether or not the vector IDs of the reference vector and the (n + m) th vector are the same. The initial value of m is 1. If the IDs are the same, the process proceeds to S93, and if they are different, the process proceeds to S94.

In S93, the end point of the reference vector is changed to the end point of the (n + m) th vector, the vector ID of the reference vector is recalculated, and m is incremented by “+1”. Thereafter, the process proceeds to S95.
In S94, the reference vector is stored in the processed vector array. Let n = n + m. The process proceeds to S96.

In S95, it is determined whether or not all vectors have been processed. If finished, the process is completed. If all the vectors have not been processed, the process returns to S93 to perform the above-described processing.
In S96, it is determined whether or not all vectors have been processed. If finished, the process is completed. If all the vectors have not been processed, the process returns to S91 to perform the above-described processing.

  By performing such processing, when a plurality of minute vectors are input, grouping is performed based on the gradient of the vectors, and when the same group of vectors continues, these vectors are combined into one vector. Can be summarized. As a result, data compression can be performed and the characteristics of the symbols can be easily grasped.

(Fourth embodiment)
In the present embodiment, an example will be described in which vectors in the vicinity of the inclination threshold are collected.
When vectors are classified by direction, classification is performed based on the gradient of the vectors. At this time, if an attempt is made to draw a straight line that is close to the threshold of inclination, a vector that is given an ID that exceeds the threshold and another vector that is below the threshold is given due to camera shake or the like. May become different IDs, and the vectors may not be collected by the method of the third embodiment described above.

In order to solve such problems, in the present embodiment, vectors near the threshold of inclination are collected.
Vector slope is
If it is less than -2.0, the vector ID will be down → up,
In the case of -2.0 to -0.5, the vector ID is from lower left to upper right,
In the case of -0.5 or more, it is assumed that the vector ID changes from left to right (see FIG. 2).

An example of this case will be described with reference to FIG.
As shown to (a), the vector a1 and the vector a2 which are near inclination-2.0 are put together into one as shown to the vector a3.
As shown in (b), when vectors near the threshold of inclination are collected, the vectors b1, b2, and b3 become one vector b7, and the vectors b4, b5, and b6 become one vector b8.

Next, the processing procedure of this embodiment will be described with reference to the flowchart of FIG. The processing of this flowchart is realized by developing a program stored in the flash memory 15 in the RAM 13 and executing it by the CPU 14.
In S111, an array of vector indices to be collected is generated from the vector array. Details of this processing will be described later with reference to the flowchart of FIG.
In S112, it is determined whether or not there is an Index array in the list created in S111. If Index is empty, the process ends. If there is an Index, the process proceeds to S113.

In S113, a summary vector is generated with the start point of the first Index vector in the Index array as the start point and the end point of the last vector in the Index array as the end point.
In S114, the Index vector of the Index array is deleted from the vector array.
In S115, a summary vector is inserted at the position of the first index in the vector array. Thereafter, the process returns to S111, and the vector array to be collected is generated.

  As described above, when vectors near the tilt threshold are collected, when a straight line close to the tilt threshold is written, due to camera shake or the like, one vector is an ID that exceeds the threshold, and another vector is less than the threshold. It is possible to solve the problem of giving an ID.

  Next, the contents of the process of generating an index array of vectors to be collected from the vector array performed in S111 will be described with reference to the flowchart of FIG. The processing of the flowchart in FIG. 12 is realized by developing a program stored in the flash memory 15 in the RAM 13 and executing it by the CPU 14.

In S121, the IDs (directions) of the vector n and the vector n + 1 are checked to determine whether or not they are adjacent to each other. If the vectors are adjacent to each other, the ID of vector n or vector n + 1 is stored as a temporary storage vector ID according to the combination, and x indicates whether the threshold of the adjacent vector ID is an inclination close to the x-axis. Set the axis flag.
The stored ID and x-axis flag are as shown in FIG.

In S122, processing is executed according to the temporary storage vector ID and the previous temporary storage vector ID.
If the temporary storage vector ID is not directionless, the process proceeds to S123 (case 1).
If the temporary storage vector ID is no direction and the previous temporary storage vector ID is also no direction, the process returns to S121 (case 2).
If the temporary storage vector ID is non-directional and the previous temporary storage vector ID is other than non-directional, the process ends (Case 3).

In S123, the previous temporary storage vector ID and the current temporary storage vector ID are compared. If they are the same, the process ends. If they are different, the process proceeds to S124.
In S124, the threshold angle of the vector is determined according to the x-axis flag and the temporary storage vector ID. This is determined as shown in FIG. Further, the angles of the vector n and the vector n + 1 are obtained, and the process proceeds to S125.

In S125, if both the angles of vectors n and n + 1 are within the threshold, the index of vector n + 1 is registered in the index array of the summary vector. If n = 0, the index of vector n is registered, then the index of n + 1 is registered, and the process proceeds to S126.
If the angle of the vector n or n + 1 is outside the threshold, the temporary storage vector ID is changed to no direction and the process proceeds to S126.

In S126, the current and previous temporary storage vector IDs are checked. If the previous temporary storage vector ID is not directionless but is different from the current temporary storage vector ID (Case 1), the process ends. Otherwise (Case 2), the process proceeds to S127.
In S127, the current temporary storage vector ID is substituted for the previous temporary storage vector ID, and the process returns to S121.

(Other embodiments)
The present invention can also be realized by executing the following processing. That is, software (computer program) that implements the functions of the above-described embodiments is supplied to a system or apparatus via a network or various computer-readable storage media. Then, the computer (or CPU, MPU, etc.) of the system or apparatus reads out and executes the program.

10 Handwritten Music Symbol Recognition Device 11 Bus 12 ROM
13 RAM
14 CPU
15 Flash memory 16 Network interface 17 Touch display 18 Audio output device 19 MIDI output device

Claims (4)

Touch information acquisition means for acquiring position information each time the input means moves until the input means touches the screen and then leaves,
A stroke that creates vector information indicating the movement of the input means from the position information obtained from the touch information acquisition means, and stores a series of vectors as one stroke information from when the input means touches the screen until it completely leaves. Information storage means;
Object information storage means for storing one or more stroke information in which a series of one or more vector information is stored;
Stroke feature value calculating means for calculating a feature value of the stroke from the vector information;
Data compression means for compressing the number of vectors by performing a summarizing process to convert a predetermined number of vectors into one vector among the plurality of vectors;
The summarization process is to obtain a straight line connecting the start point of the first vector and the end point of the final vector, and if the distance from the end point of the vector to the calculated straight line is within a specified value, a plurality of vectors are combined into one vector. handwritten music symbol recognition device you being a process of generating. Touch information acquisition means for acquiring position information each time the input means moves until the input means touches the screen and then leaves,
A stroke that creates vector information indicating the movement of the input means from the position information obtained from the touch information acquisition means, and stores a series of vectors as one stroke information from when the input means touches the screen until it completely leaves. Information storage means;
Object information storage means for storing one or more stroke information in which a series of one or more vector information is stored;
Stroke feature value calculating means for calculating a feature value of the stroke from the vector information;
Data compression means for compressing the number of vectors by performing a summarizing process to convert a predetermined number of vectors into one vector among the plurality of vectors;
In the summary processing, when vectors are classified by direction, it is determined whether or not the vectors are adjacent to each other. If the vectors are adjacent to each other, the upper limit angle and the lower limit angle of the threshold values are calculated. If it is within the range between the upper angle and lower angles, handwritten music symbol recognition device characterized in that collectively their vectors is a process of generating one vector. Touch information acquisition processing for acquiring the position information every time the input means moves until the input means touches the screen and then leaves,
A stroke that creates vector information indicating the movement of the input means from the position information obtained from the touch information acquisition process, and stores a series of vectors from the touch of the input means until the input means completely leaves as one stroke information Information storage processing;
An object information storage process for storing one or more stroke information in which a series of one or more vector information is stored;
A stroke feature amount calculation process for calculating a feature amount of the stroke from the vector information;
A handwritten music symbol recognition program that performs a summarizing process to convert a predetermined number of vectors into one vector from the plurality of vectors, and causes a computer to execute a data compression process for compressing the number of vectors .
The summarization process is to obtain a straight line connecting the start point of the first vector and the end point of the final vector, and if the distance from the end point of the vector to the calculated straight line is within a specified value, a plurality of vectors are combined into one vector. A program for recognizing handwritten music symbols, which is a process for generating Touch information acquisition processing for acquiring the position information every time the input means moves until the input means touches the screen and then leaves,
A stroke that creates vector information indicating the movement of the input means from the position information obtained from the touch information acquisition process, and stores a series of vectors from the touch of the input means until the input means completely leaves as one stroke information Information storage processing;
An object information storage process for storing one or more stroke information in which a series of one or more vector information is stored;
A stroke feature amount calculation process for calculating a feature amount of the stroke from the vector information;
A handwritten music symbol recognition program that performs a summarizing process to convert a predetermined number of vectors into one vector from the plurality of vectors, and causes a computer to execute a data compression process for compressing the number of vectors .
In the summary processing, when vectors are classified by direction, it is determined whether or not the vectors are adjacent to each other. If the vectors are adjacent to each other, the upper limit angle and the lower limit angle of the threshold values are calculated. A handwritten music symbol recognition program, which is a process of generating one vector by combining the vectors when the angle is within the range of the upper limit angle and the lower limit angle .

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