Master-slave teleoperation motion trajectory real-time alignment method

Abstract

The application belongs to the technical field of master-slave remote operation, and provides a master-slave remote operation motion trajectory real-time alignment method, which comprises the following steps: acquiring master end remote operation motion trajectory points and slave end execution motion trajectory points, obtaining a speed norm, a first-order difference sequence and a second-order difference sequence; constructing a cosine similarity matrix based on the first-order difference sequence and the second-order difference sequence respectively, and obtaining a fused similarity matrix; performing a primary path search, and expanding the search to determine a local sub-matrix; according to an amplitude relationship, selecting a corresponding difference order to construct a normalized difference sequence, obtaining a normalized difference distance matrix, updating a corresponding local sub-matrix of a current cost matrix and obtaining a new cost matrix; performing a secondary path search to obtain a final alignment path, and completing master-slave remote operation motion trajectory real-time alignment. The application uses morphological information and change trends to real-time calculate a stable alignment path under the condition that master-slave motion trajectories are similar but there is an uncertain offset.

Description

Technical Field

[0001] This invention relates to the field of master-slave teleoperation control technology, and specifically to a method for real-time alignment of master-slave teleoperation motion trajectories. Background Technology

[0002] In a classic master-slave teleoperation system, the slave device follows the master device in performing the same movements. Because the movement, stopping, acceleration, and deceleration of both devices are synchronized, the master-slave teleoperation system exhibits excellent motion-following characteristics, specifically, the master and slave motion trajectories are similar, providing a good user experience. However, when autonomous movement of the slave device is introduced through shared control, the slave's movement will no longer be identical to the master's. The slave may exhibit uncertain offsets such as leading or lagging behind the master. To ensure motion following and provide a good user experience, additional motion control constraints are required. The key is real-time alignment of the master and slave motion trajectories and estimation of time offsets. Directly comparing position or velocity errors at the same sampling time can easily misjudge scale differences as following errors. This makes it difficult to real-time align similar motion trajectories in master-slave teleoperation, and consequently, it is impossible to calculate the time offset between trajectories and assess their similarity in real time.

[0003] Dynamic Time Warping (DTW) enables nonlinear alignment of time series. Traditional standard DTW typically targets known complete sequences and employs boundary constraints that force alignment from the start to the end of each sequence. In real-time master-slave teleoperation scenarios, the master and slave motion trajectories grow point-by-point with each sampling process, making it impossible to obtain the complete trajectory in advance. The alignment matrix also needs to dynamically expand as new data arrives. Furthermore, when the slave's motion leads or lags the master's, the start points of the two sequences may be aligned at the current moment, but the end sampling points may not correspond. Forcing end alignment may introduce incorrect matching relationships. Therefore, a real-time alignment method that allows the alignment path endpoint to be dynamically determined is needed. Summary of the Invention

[0004] To overcome the limitations of existing technologies, this invention provides a real-time alignment method for master-slave teleoperation motion trajectories. This method uses the velocity norm curve as the feature of the master-slave teleoperation motion trajectory, constructs a multi-order curve shape similarity matrix through first-order and second-order differences, forms a fusion similarity matrix through matrix fusion, updates the cost matrix through local submatrices and normalized difference distance matrices, and finally obtains the real-time master-slave alignment path based on path search starting from the starting point.

[0005] The real-time alignment scheme for master-slave teleoperation motion trajectories in this application includes: acquiring the current sampling points of the master-end 3D position trajectory and the slave-end 3D position trajectory in real time, forming a real-time trajectory sequence according to the sampling time sequence, and calculating the velocity norm curve from the trajectory sequence; performing multi-order difference calculations, including first-order and second-order differences, on the real-time velocity norm curve; constructing cosine similarity matrices of each order of difference; conditionally fusing the multi-order cosine similarity matrices of difference; performing an initial path search on the fused similarity matrix to obtain the current alignment endpoint; expanding the search to include local submatrices containing the endpoint; constructing a normalized difference distance matrix and updating the cost matrix according to the difference order selection rule; performing a final path search on the updated cost matrix and outputting the alignment path and time deviation of the master-slave teleoperation motion trajectory.

[0006] The steps for real-time alignment of motion trajectories are as follows:

[0007] S1. Obtain the master-end teleoperation motion trajectory points and the slave-end execution motion trajectory points, and append them to the master-end motion trajectory sequence and the slave-end motion trajectory sequence according to the sampling time sequence. Obtain the velocity norm based on the master-end or slave-end position, and obtain the first-order difference sequence and the second-order difference sequence based on the velocity norm.

[0008] S2. Construct cosine similarity matrices based on first-order and second-order difference sequences respectively, and obtain the fused similarity matrix;

[0009] S3. Using the fused similarity matrix as the cost matrix, perform the initial path search starting from the starting point to obtain the current aligned path endpoint, and determine the local submatrix through extended search.

[0010] S4. Based on the magnitude relationship between the first-order and second-order differences at the starting point of the local submatrix, select the corresponding difference order to construct a normalized difference sequence, and then obtain the normalized difference distance matrix. Use the normalized difference distance matrix to update the corresponding local submatrix of the current cost matrix and obtain a new cost matrix.

[0011] S5. Perform path search again on the updated cost matrix to obtain the final alignment path and complete the real-time alignment of the master-slave teleoperation trajectory.

[0012] Furthermore, the calculation process and expressions for the velocity norm, first-order difference sequence, and second-order difference sequence in step S1 are as follows:

[0013]

[0014]

[0015]

[0016]

[0017]

[0018] In the formula, The position of the master or slave end. For speed, For the velocity norm, It is a first-order difference sequence. It is a second-order difference sequence. The sampling time is , where k is the sequence index of the trajectory; the sequence index ranges from 0 to N-1. This represents the number of sampling points.

[0019] Furthermore, the cosine similarity matrix and the fusion similarity matrix in step S2 are expressed as follows:

[0020]

[0021]

[0022]

[0023] In the formula, These are the sequence indices of the master and slave trajectories, respectively. The first-order cosine similarity matrix, It is a second-order cosine similarity matrix. To merge the similarity matrices, To prevent extremely small positive numbers with a denominator of zero.

[0024] Furthermore, in step S3, the fused similarity matrix is ​​used as the cost matrix, and an initial path search is performed starting from the starting point to obtain the aligned path endpoint. The process of determining the local submatrix through extended search is as follows:

[0025]

[0026] In the formula, The sequence index of the master trajectory. For the sequence index of the slave trajectory, For the sequence index of the alignment path, This represents the cost matrix.

[0027] Furthermore, in step S4, based on the magnitude relationship between the first-order and second-order differences at the starting point of the local submatrix, the corresponding difference order is selected to construct a normalized difference sequence, and then the normalized difference distance matrix is ​​obtained. The process of updating the corresponding local submatrix of the current cost matrix with the normalized difference distance matrix to obtain a new cost matrix is ​​as follows:

[0028] Based on the magnitude and order of the first and second differences at the starting point of the local submatrix, if the magnitude of the first difference at the starting point is greater than or equal to the magnitude of the second difference, then the first difference sequence within the local submatrix is... as the difference sequence within the local submatrix Otherwise, the second-order difference sequence within the local submatrix will be used. as the difference sequence within the local submatrix Determine the difference sequence within the local submatrix. Then, the difference sequence within the local submatrix is ​​normalized by dividing it by the difference value at the starting point of the local submatrix, resulting in the normalized difference sequence. ;

[0029]

[0030]

[0031] Where m is the sequence index of the master trajectory within the local submatrix, n is the sequence index of the slave trajectory within the local submatrix, r is the sequence index of the master trajectory at the starting point of the local submatrix, and c is the sequence index of the slave trajectory at the starting point of the local submatrix.

[0032] Sequence index; To prevent extremely small positive numbers with a denominator of zero;

[0033] The difference sequence within the local submatrix of the master end. For the difference sequence within the local submatrix of the slave end, The normalized difference sequence within the local submatrix of the master end. For the normalized difference sequence within the local submatrix of the slave end;

[0034]

[0035] In the formula, This is the normalized distance matrix;

[0036] The positions corresponding to the local submatrices in the cost matrix are normalized using the distance matrix. The replacement is performed to obtain the updated cost matrix. .

[0037] Furthermore, in step S5, path search is performed again on the updated cost matrix to obtain the final aligned path.

[0038] The way to express it is:

[0039]

[0040] in Let l be the sequence index of the master-slave trajectory at the l-th alignment point. The sequence index is the alignment path, and L is the endpoint of the alignment path. After aligning the original real-time master and slave trajectories according to the alignment path P, the aligned master motion trajectory and slave motion trajectory are obtained.

[0041] Furthermore, it also includes performing path search again on the updated cost matrix, and outputting the time difference that represents whether the master and slave follow ahead or behind.

[0042]

[0043] Indicates the time difference between master and slave. Indicates the slave end time of the alignment path endpoint. Indicates the master-end time of the alignment path endpoint. When When positive, it indicates that the final alignment path will match from the end of the line to a relatively later time position; when... When negative, it indicates a match from the end of the line to a relatively earlier time position.

[0044] The advantages of this invention compared to the prior art are:

[0045] This invention provides a master-slave teleoperation trajectory alignment method that utilizes morphological information and change trends, and can progressively align shapes based on real-time acquired existing motion trajectories. This allows for real-time calculation of stable alignment paths and time offsets even when master-slave trajectories are similar but exhibit uncertain offsets. The main advantages are as follows: First, this invention utilizes both first-order change trends and second-order bending trends through multi-order differencing, enabling the construction of a cosine similarity matrix at the local morphological level, thus improving the reliability of trajectory alignment under nonlinear time offset conditions. Second, this invention uses a normalized difference distance matrix for local cost updates, enhancing the continuity and stability of the real-time alignment path. Third, this invention can update incrementally based on sampling points, achieving real-time trajectory alignment. It outputs the master-slave time difference at each sampling moment, which can be used as input for master-slave teleoperation synchronization control compensation, follow-up quality evaluation, and motion segment alignment. This invention can be applied to scenarios such as master-slave teleoperated robots, motion segment matching, online time offset estimation, and motion trajectory similarity evaluation. Attached Figure Description

[0046] Figure 1 This is a flowchart of the real-time alignment method for master-slave teleoperation motion trajectories according to the present invention;

[0047] Figure 2 This is a visualization of the first-order difference cosine similarity matrix of the embodiment;

[0048] Figure 3This is a visualization of the second-order difference cosine similarity matrix of the embodiment;

[0049] Figure 4 This is a visualization of the fusion similarity matrix of the embodiment;

[0050] Figure 5 This is a schematic diagram of the alignment path and the updated cost matrix in an embodiment;

[0051] Figure 6 The graphs show the velocity norm curve, real-time alignment result, and time difference estimation curve of the embodiment. Detailed Implementation

[0052] The embodiments of the technical solutions of this application will now be described in detail with reference to the accompanying drawings. Unless otherwise stated, the technical or scientific terms used in this application have the ordinary meanings understood by those skilled in the art.

[0053] Example 1 Figure 1 This paper presents the design flow of a real-time alignment method for master-slave teleoperation motion trajectories based on multi-order difference cosine similarity. The method includes the following steps:

[0054] Step 1: Real-time acquisition of master-slave teleoperation motion trajectories and calculation of multi-order difference sequences

[0055] At the current sampling moment, the teleoperation trajectory points of the master end and the execution trajectory points of the slave end are acquired in real time and appended to the master end and slave end motion trajectory sequences according to the sampling time sequence. Then, the corresponding velocity norms are calculated based on the velocities of the master end and the slave end, respectively, forming the master end real-time feature sequence and the slave end real-time feature sequence that are updated with the sampling point increment. Finally, the first-order difference sequence and the second-order difference sequence are calculated based on the real-time feature sequences.

[0056] In a practical master-slave teleoperation system, the motion trajectories of the master and slave devices are acquired in real time by sensors, the master hand device, the robot controller, or a simulation environment at sampling times. To describe the real-time data stream, assume that the currently accumulated trajectories acquired by the master and slave devices are both position trajectories, and the number of sampling points is... The resulting trajectory sequence has an index range of 0 to N-1, and the sampling time is... ,in , where k is the sequence index of the trajectory. Sampling time The main end position is From the end position is .

[0057] First, calculate the velocity based on the positions of the master and slave ends, and then calculate the corresponding velocity norm:

[0058]

[0059]

[0060]

[0061] Then, based on the velocity norm, the first-order difference sequence and the second-order difference sequence are calculated respectively:

[0062]

[0063]

[0064] In the formula, The position of the master or slave end. For speed, For the velocity norm, It is a first-order difference sequence. It is a second-order difference sequence.

[0065] Step 2, Matrix Construction and Fusion: Calculation of multi-order difference cosine similarity matrix and fused similarity matrix.

[0066] Cosine similarity matrices are constructed based on first-order difference sequences and second-order difference sequences respectively. Based on the second-order difference cosine similarity matrix, the first-order difference cosine similarity is superimposed when the second-order difference cosine similarity satisfies the preset fusion conditions to obtain the fused similarity matrix.

[0067] Based on first-order difference sequences and second-order difference sequences Construct first-order and second-order cosine similarity matrices respectively:

[0068]

[0069]

[0070] Based on the second-order difference-cosine similarity matrix, when the second-order difference-cosine similarity satisfies the preset fusion conditions, the first-order difference-cosine similarity is superimposed to obtain the fused similarity matrix:

[0071]

[0072] In the formula, Sequence indexes for master and slave trajectories. The first-order cosine similarity matrix, It is a second-order cosine similarity matrix. To merge the similarity matrices, To prevent extremely small positive numbers with a denominator of zero.

[0073] Step 3, Initial Path Search: Based on DTW, search for the initial path and expand to determine the local submatrix.

[0074] Use the fusion similarity matrix at the current time as the cost matrix. The initial path search begins from the starting point. The current aligned path endpoint is obtained. A local submatrix is ​​determined through an expanded search. The path starts from the top left corner of the matrix, and the current path point is [insert path point here]. The next path point is selected from the three candidate points: the bottom right, the right-hand side, and the bottom right. The one with the lowest cost value is chosen.

[0075]

[0076] In the formula, The sequence index of the master trajectory. For the sequence index of the slave trajectory, The sequence index for the alignment path.

[0077] The path stops when it reaches the last row or column, yielding the endpoint sequence index of the initially aligned path. Using this endpoint sequence index as the target element, boundary expansion begins. The expansion extends outwards along the target row and column, ensuring that the corresponding value within the expanded region is less than 1, thus obtaining a local submatrix. .

[0078] Step 4, Local Cost Update: Calculation of the normalized difference distance matrix within the local submatrix and local update of the cost matrix.

[0079] Based on the magnitude relationship between the first-order and second-order differences at the starting point of the local submatrix, the corresponding difference order is selected to construct a normalized difference distance matrix, and the corresponding local submatrix of the current cost matrix is ​​updated with the normalized difference distance matrix to obtain a new cost matrix.

[0080] In local submatrix Within, the following processing is performed on both the master and slave ends respectively: based on the starting point of the submatrix The magnitudes of the first and second differences are determined. If the magnitude of the first difference at the starting point is greater than or equal to the magnitude of the second difference, then the first difference sequence within the local submatrix is... as the difference sequence within the local submatrix Otherwise, the second-order difference sequence within the local submatrix will be used. as the difference sequence within the local submatrix .

[0081] Determine the difference sequence within a local submatrix Then, the difference sequence within the local submatrix is ​​normalized by dividing it by the difference value at the starting point of the local submatrix, resulting in the normalized difference sequence. :

[0082]

[0083]

[0084] in, This is the sequence index of the main end trajectory within the local submatrix. This is the sequence index of the trailing trajectory within the local submatrix. This is the sequence index of the main trajectory at the starting point of the local submatrix. This is the sequence index of the main trajectory at the starting point of the local submatrix.

[0085] Difference sequences within local submatrices It can be divided into and , The difference sequence within the local submatrix of the master end. It is the difference sequence within the local submatrix of the slave end.

[0086] Normalized difference sequence within a local submatrix It can be divided into and , The normalized difference sequence within the local submatrix of the master end. It is the normalized difference sequence within the local submatrix of the slave end.

[0087] Calculate the normalized distance matrix within the local submatrix based on the normalized difference sequence data of the master and slave ends in the local submatrix. :

[0088]

[0089] Cost matrix Local submatrices in The corresponding position is represented by the normalized distance matrix. The substitution is performed to obtain the final cost matrix. .

[0090] Initial time, cost matrix As the fusion similarity matrix progresses, the cost matrix... It will be continuously updated to achieve the updated cost matrix. Each local submatrix in the matrix will be a normalized distance matrix.

[0091] Step 5: Final alignment path and time difference output

[0092] Perform path search again on the updated cost matrix, and output the master-end aligned index sequence, the slave-end aligned index sequence, and the time difference representing whether the master or slave is ahead or behind.

[0093] The updated cost matrix Then, perform the path search again starting from the origin to obtain the final aligned path. ,in Let l be the sequence index of the master-slave trajectory at the l-th alignment point. Here, L is the sequence index of the alignment path, and L is the endpoint of the alignment path. After aligning the original real-time curves according to the alignment path P, the aligned master-end motion trajectory and slave-end motion trajectory can be obtained.

[0094] The master-slave time difference is obtained by subtracting the master time from the slave time at the end of the alignment path:

[0095]

[0096] in, Indicates the time difference between master and slave. Indicates the slave end time of the alignment path endpoint. Indicates the endpoint of the alignment path

[0097] The main moment.

[0098] when When positive, it indicates that the final alignment path will match from the end of the line to a relatively later time position; when... A negative value indicates a match from the end point to a relatively earlier time position. This time difference can be used as input for master-slave teleoperation synchronization and similarity analysis or subsequent control compensation.

[0099] Example 2 Figure 2 and Figure 3 The visualizations show the effects of the first-order difference cosine similarity matrix calculated in real time based on the master and slave motion trajectories. Figure 4 The visualization of the fused similarity matrix is ​​shown. The first-order and second-order difference cosine similarity matrices contain their respective trajectory features. By merging them into a fused similarity matrix, feature point fusion is achieved, and local submatrices segmented by feature points are obtained, which are the multiple small rectangles in the visualization of the fused similarity matrix.

[0100] Figure 5 The final cost matrix and the alignment path obtained from the search are shown. The submatrices in the cost matrix are replaced by normalized distance matrices, and the values ​​in the submatrices represent the similarity between sequence pairs; the smaller the value, the higher the similarity. The final alignment path is the sequence alignment relationship with the highest similarity.

[0101] Figure 6The "Master-Slave Velocity Norm Original Curve and Sequence Alignment Relationship" diagram illustrates the alignment relationship between the master-slave velocity norm trajectory curves based on the final alignment path (the red line represents the master-end velocity norm, the green dashed line represents the slave-end velocity norm, and the gray line represents the master-slave alignment relationship). Based on this alignment relationship, the slave-end curve is aligned along the time axis, resulting in the "Master-Slave Velocity Norm Alignment Curve" diagram (the red line represents the master-end velocity norm, and the green dashed line represents the aligned slave-end velocity norm). The slave-end curve is aligned with a portion of the master-end curve's motion segment with the highest similarity. Master-slave start-point alignment indicates that the master and slave begin moving synchronously; however, the master-slave ends are not aligned, and the difference in the time axis corresponding to their ends represents the time offset of the master-slave movement. The "Real-Time Estimated Master-Slave Time Difference" diagram (the purple line represents the time difference) shows the estimated time difference calculated in real-time during the master-slave movement.

[0102] As can be seen from the above embodiments, this method can align similar master-slave motion trajectories in real time and stably, and output the time offset of master-slave motion.

[0103] This invention uses specific embodiments to illustrate the principles and implementation methods of this application. The above description of the embodiments is only for the purpose of helping to understand this application and its core ideas. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this application. Therefore, the content of this specification should not be construed as a limitation of this invention.

Claims

1. A method for real-time alignment of master-slave teleoperation motion trajectories, characterized in that: The method includes the following steps: S1. Obtain the master-end teleoperation motion trajectory points and the slave-end execution motion trajectory points, and append them to the master-end motion trajectory sequence and the slave-end motion trajectory sequence according to the sampling time sequence. Obtain the velocity norm based on the master-end or slave-end position, and obtain the first-order difference sequence and the second-order difference sequence based on the velocity norm. S2. Construct cosine similarity matrices based on first-order and second-order difference sequences respectively, and obtain the fused similarity matrix; S3. Using the fused similarity matrix as the cost matrix, perform the initial path search starting from the starting point to obtain the current aligned path endpoint, and determine the local submatrix through extended search. S4. Based on the magnitude relationship between the first-order and second-order differences at the starting point of the local submatrix, select the corresponding difference order to construct a normalized difference sequence, and then obtain the normalized difference distance matrix. Use the normalized difference distance matrix to update the corresponding local submatrix of the current cost matrix and obtain a new cost matrix. S5. Perform path search again on the updated cost matrix to obtain the final alignment path and complete the real-time alignment of the master-slave teleoperation trajectory.

2. The real-time alignment method for master-slave teleoperation motion trajectories according to claim 1, characterized in that: The calculation process and expressions for the velocity norm, first-order difference sequence, and second-order difference sequence in S1 are as follows: In the formula, The position of the master or slave end. For speed, For the velocity norm, It is a first-order difference sequence. It is a second-order difference sequence. The sampling time is , where k is the sequence index of the trajectory; the sequence index ranges from 0 to N-1. This represents the number of sampling points.

3. The real-time alignment method for master-slave teleoperation motion trajectories according to claim 1, characterized in that: The cosine similarity matrix and the fused similarity matrix in S2 are expressed as follows: In the formula, These are the sequence indices of the master and slave trajectories, respectively. The first-order cosine similarity matrix, It is a second-order cosine similarity matrix. To merge the similarity matrices, To prevent extremely small positive numbers with a denominator of zero.

4. The real-time alignment method for master-slave teleoperation motion trajectories according to claim 1, characterized in that: In S3, the fused similarity matrix is ​​used as the cost matrix. An initial path search is performed starting from the starting point to obtain the current aligned path endpoint. The process of determining the local submatrix through extended search is as follows: In the formula, The sequence index of the master trajectory. For the sequence index of the slave trajectory, For the sequence index of the alignment path, This represents the cost matrix.

5. The real-time alignment method for master-slave teleoperation motion trajectories according to claim 1, characterized in that: In S4, the process of selecting the corresponding difference order to construct the normalized difference sequence based on the magnitude relationship between the first and second difference at the starting point of the local submatrix is ​​as follows: Based on the magnitude and order of the first and second differences at the starting point of the local submatrix, if the magnitude of the first difference at the starting point is greater than or equal to the magnitude of the second difference, then the first difference sequence within the local submatrix is... as the difference sequence within the local submatrix Otherwise, the second-order difference sequence within the local submatrix will be used. as the difference sequence within the local submatrix Determine the difference sequence within the local submatrix. Then, the difference sequence is normalized by dividing the difference sequence within the local submatrix by the difference value at the starting point of the local submatrix, resulting in the normalized difference sequence. .

6. The real-time alignment method for master-slave teleoperation motion trajectories according to claim 5, characterized in that: The process of obtaining the normalized difference distance matrix in S4 and updating the corresponding local submatrices of the current cost matrix with the normalized difference distance matrix to obtain the new cost matrix is ​​as follows: in, This is the sequence index of the main end trajectory within the local submatrix. This is the sequence index of the trailing trajectory within the local submatrix. This is the sequence index of the main trajectory at the starting point of the local submatrix. This is the sequence index of the main trajectory at the starting point of the local submatrix; To prevent extremely small positive numbers with a denominator of zero; The difference sequence within the local submatrix of the master end. For the difference sequence within the local submatrix of the slave end, The normalized difference sequence within the local submatrix of the master end. For the normalized difference sequence within the local submatrix of the slave end; In the formula, This is the normalized distance matrix; The positions corresponding to the local submatrices in the cost matrix are normalized using the distance matrix. The replacement is performed to obtain the updated cost matrix. .

7. The real-time alignment method for master-slave teleoperation motion trajectories according to claim 1, characterized in that: In S5, a path search is performed again on the updated cost matrix, and the final aligned path is expressed as follows: in Let l be the sequence index of the master-slave trajectory at the l-th alignment point. L is the sequence index of the alignment path, where L is the pair. The endpoint of the alignment path is reached by aligning the original real-time master and slave trajectories according to the alignment path P, thus obtaining the aligned master and slave motion trajectories.

8. The real-time alignment method for master-slave teleoperation motion trajectories according to claim 1, characterized in that: It also includes S6, which performs path search again on the updated cost matrix and outputs the time difference representing whether the master and slave follow ahead or behind. Indicates the time difference between master and slave. Indicates the slave end time of the alignment path endpoint. Indicates the master end time of the alignment path endpoint.

9. The real-time alignment method for master-slave teleoperation motion trajectories according to claim 8, characterized in that: when When positive, it indicates that the final alignment path will match from the end of the line to a relatively later time position; when... When negative, it indicates a match from the end of the line to a relatively earlier time position.

Images