CN110487279A - A kind of paths planning method based on improvement A* algorithm - Google Patents
A kind of paths planning method based on improvement A* algorithm Download PDFInfo
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Abstract
The invention discloses a kind of based on the paths planning method for improving A* algorithm.There are more turning points in the path obtained for A* algorithm, causing path not is optimal problem, merges the crucial inflection point in path on the basis of A* algorithm first, this not only reduces the turning points in path, decrease cubic spline interpolation points, interpolation efficiency is improved, then according to the path node after merging inflection point, reaches smooth paths purpose using cubic spline interpolation, pass through improved algorithm, path length is shorter, whole smoother, more meets incomplete humanoid robot movement.
Description
Technical field
The invention belongs to field in intelligent robotics more particularly to a kind of paths planning methods based on improvement A* algorithm.
Background technique
With the continuous development of robot technology, more and more robots are used to various services, or even substitution
The work of people.Independent navigation is the key that robot is realized intelligent and played a role, and path planning be then robot from
The important composition of dynamic homing capability.Path planning problem is always in a hot research in intelligent mobile robot field
Hold.Path planning refers to that in priori map, mobile robot can obtain information using sensor according to ambient enviroment, from
It is dynamic to cook up the collisionless path from origin-to-destination.
There are many kinds of the classification of path planning algorithm.According to external environmental information whether it is known that global path rule can be divided into
Cost-effective method and local path planning algorithm；And according to the way of search of algorithm, blindness formula searching algorithm and heuristic can also be divided into
Searching algorithm.The search of blindness formula focuses on the process of search rather than search target, is usually associated with huge search space, causes
A large amount of memory sources and inefficiency are consumed, specifically there is breadth first algorithm, depth-priority-searching method and dijkstra's algorithm etc..
Heuristic search is during search, according to heuristic information relevant to problem, is unfolded to search for towards advantageous direction, can
To avoid many meaningless searching routes, the complexity for greatly reducing search range, reducing problem common are greedy algorithm
With A* algorithm.And A* algorithm is as a kind of heuristic search algorithm being widely used, combined dijkstra's algorithm and
The advantages of greedy algorithm, it can not only guarantee to find an optimal path, but also can make the direction of search definitely, to search for sky
Between smaller, search speed faster.
Summary of the invention
Goal of the invention: traditional A* algorithm in search process when extending neighboring node, only centered on present node to
One layer of external expansion, i.e. 8 neighbouring nodes of present node, the angle of the direction of motion of robot will be constrained to 45 degree at this time
Integral multiple, route is restricted, if according to 8 neighbors extended modes of traditional A* algorithm, in actual environment
The obtained final path of robot path planning may not be optimal.In view of the above problems, the present invention proposes that one kind is based on
The paths planning method of A* algorithm is improved, the path length of improved algorithmic rule is shorter, and path is more smooth, route searching
It is more efficient.
Technical solution: to achieve the purpose of the present invention, the technical scheme adopted by the invention is that: one kind is calculated based on A* is improved
The paths planning method of method, comprising the following steps:
Step 1: by environment representation locating for robot at grating map, and being searched out in grating map using A* algorithm
One initial path from starting point to target point；
Step 2: extracting all nodes in initial path, from the off, adjacent three sections are successively judged using area-method
Whether point is conllinear, to find all turning points on path, i.e. inflection point；
Step 3: from the off, being sequentially connected inflection point until target point using straight line, obtain updated path；Note is every
Section straight line is L (i, j), and i, j respectively indicate the beginning and end of straightway, i=0,1,2 ..., n-1, j=1,2 ..., n,
Middle n is inflection point number；
Step 4: updating whether the straightway in rear path passes through barrier by judgment step 3, reject redundancy inflection point, more
New route；
Step 5: rejecting the path after redundancy inflection point to step 4, be smoothed, obtained using cubic spline interpolation
The path that final planning is completed.
Further, in the step 4, as by environment representation locating for robot at grating map, it is (horizontal in fixed-direction
To or it is longitudinal) on whether to have obstacle between any two node in fixed step size searching route when, it is possible that missing inspection
The case where surveying barrier；The present invention judges whether straightway passes through barrier by the method for the dynamic select direction of search, when straight
When line segment L (i, j) does not pass through barrier, the redundancy inflection point between i and j is removed.
Further, judge whether straightway passes through barrier by the method for the dynamic select direction of search, removal redundancy turns
Point, detailed process is as follows:
4-1, selecting step 3 update the seat that any two node in rear path is search starting point A and target point B, AB two o'clock
Mark is place grating map net center of a lattice, is denoted as (x respectively1, y1) and (x2, y2)；
4-2, the linear equation y=kx+b, k for calculating AB are straight slope, and b is the ordinate of straight line and longitudinal axis intersection point；
4-3 judges the abs (y of AB two o'clock2-y1) and abs (x2-x1) size, if abs (y2-y1) > abs (x2-x1), it uses
Horizon Search executes step 4-4；Otherwise, using longitudinal searching, step 4-5 is executed；
4-4, in a lateral direction whether to have barrier between fixed step size search two nodes of AB；If there is obstacle
Object, the inflection point between two nodes of AB cannot be rejected, and respective paths cannot update；If rejecting AB two sections without barrier
Inflection point between point updates respective paths；
4-5, in a longitudinal direction whether to have barrier between fixed step size search two nodes of AB；If there is obstacle
Object, the inflection point between two nodes of AB cannot be rejected, and respective paths cannot update；If rejecting AB two sections without barrier
Inflection point between point updates respective paths；
4-6 rejects redundancy inflection point, more new route according to all nodes in step 4-1~4-5 traverse path.
Further, whether the step 4-4 is in a lateral direction to have obstacle between fixed step size search two nodes of AB
Object, specific as follows:
All intersection points of longitudinal network ruling and line segment AB on grating map are denoted as a (1), a (2) ..., a by 4-4-a
(m), m is the number of intersection point；All intersection point ordinates are calculated using the linear equation of AB, can analyze by ordinate
Whether there are obstacles around to intersection point；
4-4-b is determined intersection point a (i), i=1,2 ..., number of grid of the m as common intersection；If intersection point a (i) is
The common intersection of four grids, enters step 4-4-c；If intersection point a (i) is the common intersection of two grids, 4- is entered step
4-d；
4-4-c judges the positive and negative of the slope k of line segment AB, if slope is positive, checks that adjoining three grids above line segment AB is
It is no to have barrier；If slope is negative, checks and adjoin whether three grids have barrier below line segment AB；
If there is barrier, the inflection point between two nodes of AB cannot be rejected, and respective paths cannot update, two nodes of AB
Between barrier search terminate；If i=i+1 goes to step 4-4-b without barrier, continue to judge next intersection point；
4-4-d judges whether the adjacent left and right grid of intersection point is barrier；If it is barrier, hinder between two nodes of AB
Object search is hindered to terminate；If not barrier, i=i+1 goes to step 4-4-b, continues to judge next intersection point；
4-4-e, after the completion of all intersection points traverse, barrier search terminates between two nodes of AB.
Further, whether the step 4-5 is in a longitudinal direction to have obstacle between fixed step size search two nodes of AB
Object, specific as follows:
All intersection points of transverse grid line and line segment AB on grating map are denoted as b (1), b (2) ..., b by 4-5-a
(m), m is the number of intersection point；All intersection point abscissas are calculated using the linear equation of AB, can analyze by abscissa
Whether there are obstacles around to intersection point；
4-5-b is determined intersection point b (j), j=1,2 ..., number of grid of the m as common intersection；If intersection point b (j) is
The common intersection of four grids, enters step 4-5-c；If intersection point b (j) is the common intersection of two grids, 4- is entered step
5-d；
4-5-c judges the positive and negative of the slope k of line segment AB, if slope is positive, checks that adjoining three grids above line segment AB is
It is no to have barrier；If slope is negative, checks and adjoin whether three grids have barrier below line segment AB；
If there is barrier, the inflection point between two nodes of AB cannot be rejected, and respective paths cannot update, two nodes of AB
Between barrier search terminate；If i=i+1 goes to step 4-5-b without barrier, continue to judge next intersection point；
4-5-d judges whether the adjacent left and right grid of intersection point is barrier, if it is barrier, hinders between two nodes of AB
Object search is hindered to terminate；If not barrier, i=i+1 goes to step 4-5-b, continues to judge next intersection point；
4-5-e, after the completion of all intersection points traverse, barrier search terminates between two nodes of AB.
Further, it in the step 5, to the path after rejecting redundancy inflection point, is carried out using cubic spline interpolation smooth
Processing, detailed process is as follows:
5-1, if there is n+1 back end in path, node coordinate is respectively (x0, y0), (x1, y1), (x2, y2) ...,
(xn, yn)；In each subinterval xi≤x≤xi+1In, create batten difference equation:
gi(x)=ai+bi(x-xi)+ci(x-xi)2+di(x-xi)3
Wherein, ai, bi, ci, diIndicate the coefficient of spline curve；
5-2, material calculation hi=xi+1-xi, i=0,1 ..., n-1；
5-3, by back end and end-point condition M0=0, Mn=0 brings following matrix equation into, obtains:
Wherein, Mi, i=0,1 ..., n indicate batten difference equation second differential value；
5-4, solution matrix equation obtain batten difference equation second differential value Mi, i=0,1 ..., n；
5-5 calculates the coefficient a of spline curvei, bi, ci, di, formula is as follows:
ai=yi
Wherein, i=0,1 ..., n-1；
5-6 solves the coefficient in every section of batten difference equation, the expression of every section of curve can be obtained.
The utility model has the advantages that compared with prior art, technical solution of the present invention has technical effect beneficial below:
Fusion proposed by the present invention improves the smooth track planning algorithm of A* algorithm and cubic spline interpolation relative to tradition
Cubic spline interpolation smooth paths algorithm, decreases interpolation point number, additionally reduces while reducing path turning point
Path total length.The introducing of cubic spline interpolation keeps whole path more smooth, so that robot be avoided to occur at turning
Anxious the case where accelerating and suddenly slowing down, keeps its forms of motion more coherent, be more in line with the dynamics Controlling of incomplete humanoid robot.
Detailed description of the invention
Fig. 1 is the overall flow figure of the method for the present invention；
Fig. 2 is three, section of the A* algorithmic rule schematic diagram of the embodiment of the present invention；
Fig. 3 is the merging key inflection point process schematic of the embodiment of the present invention；
Fig. 4 is the lateral traversal schematic diagram of the embodiment of the present invention；
Fig. 5 is longitudinal traversal schematic diagram of the embodiment of the present invention；
Fig. 6 is traditional A* algorithmic rule path simulation figure；
Fig. 7 is the improvement A* algorithmic rule path simulation figure of the embodiment of the present invention.
Specific embodiment
Further description of the technical solution of the present invention with reference to the accompanying drawings and examples.
It is of the present invention it is a kind of based on improve A* algorithm paths planning method, overall flow as shown in Figure 1, include with
Lower step:
Step 1: by environment representation locating for robot at grating map, and being searched out in grating map using A* algorithm
One initial path from starting point to target point.
Step 2: extracting all nodes in initial path, from the off, adjacent three sections are successively judged using area-method
Whether point is conllinear, to find all turning points on path, i.e. inflection point；It is specific as follows:
As shown in Fig. 2, A, B, C represent certain three adjacent node in the path that A* algorithmic rule goes out, the face triangle ABC is calculated
Product SABC, work as SABCWhen not being 0, A, B, 3 points of C not conllinear, and B point is the inflection point in path；
Triangle ABC area SABCCalculation formula it is as follows:
In formula, A point coordinate is (xA, yA), B point coordinate is (xB, yB), C point coordinate is (xC, yC)。
Step 3: from the off, being sequentially connected inflection point until target point using straight line, obtain updated path；Note is every
Section straight line is L (i, j), and i, j respectively indicate the beginning and end of straightway, i=0,1,2 ..., n-1, j=1,2 ..., n,
Middle n is inflection point number；
Step 4: updating whether the straightway in rear path passes through barrier by judgment step 3, reject redundancy inflection point, more
New route；
The present invention judges whether straightway passes through barrier by the method for the dynamic select direction of search, when straightway L (i,
When j) not passing through barrier, the redundancy inflection point between i and j is removed.Detailed process is as follows:
4-1, selecting step 3 update the seat that any two node in rear path is search starting point A and target point B, AB two o'clock
Mark is place grating map net center of a lattice, is denoted as (x respectively1, y1) and (x2, y2)。
4-2, the linear equation y=kx+b, k for calculating AB are straight slope, and b is the ordinate of straight line and longitudinal axis intersection point.
4-3 judges the abs (y of AB two o'clock2-y1) and abs (x2-x1) size, if abs (y2-y1) > abs (x2-x1), it uses
Horizon Search executes step 4-4；Otherwise, using longitudinal searching, step 4-5 is executed.
4-4, in a lateral direction whether to have barrier between fixed step size search two nodes of AB；As shown in figure 4, black
Color grid representation barrier, white grid representation can indicate the line segment of two inflection points of any connection, stain table by region, straight line
Show the intersection point of straight line and grid；
If there is barrier, the inflection point between two nodes of AB cannot be rejected, and respective paths cannot update；If do not hindered
Hinder object, reject the inflection point between two nodes of AB, updates respective paths；
It is described to search between two nodes of AB whether have barrier in a lateral direction with fixed step size, the method is as follows:
All intersection points of longitudinal network ruling and line segment AB on grating map are denoted as a (1), a (2) ..., a by 4-4-a
(m), m is the number of intersection point；All intersection point ordinates are calculated using the linear equation of AB, can analyze by ordinate
Whether there are obstacles around to intersection point；
4-4-b is determined intersection point a (i), i=1,2 ..., number of grid of the m as common intersection；If intersection point a (i) is
The common intersection of four grids, enters step 4-4-c；If intersection point a (i) is the common intersection of two grids, 4- is entered step
4-d；
4-4-c judges the positive and negative of the slope k of line segment AB, if slope is positive, checks that adjoining three grids above line segment AB is
It is no to have barrier；If slope is negative, checks and adjoin whether three grids have barrier below line segment AB；
If there is barrier, the inflection point between two nodes of AB cannot be rejected, and respective paths cannot update, two nodes of AB
Between barrier search terminate；If i=i+1 goes to step 4-4-b without barrier, continue to judge next intersection point；
4-4-d judges whether the adjacent left and right grid of intersection point is barrier, if it is barrier, hinders between two nodes of AB
Object search is hindered to terminate；If not barrier, i=i+1 goes to step 4-4-b, continues to judge next intersection point；
4-4-e, after the completion of all intersection points traverse, barrier search terminates between two nodes of AB.
4-5, in a longitudinal direction whether to have barrier between fixed step size search two nodes of AB, as shown in Figure 5；Such as
Fruit has barrier, and the inflection point between two nodes of AB cannot be rejected, and respective paths cannot update；If rejected without barrier
Inflection point between two nodes of AB updates respective paths；
It is described to search between two nodes of AB whether have barrier in a longitudinal direction with fixed step size, the method is as follows:
All intersection points of transverse grid line and line segment AB on grating map are denoted as b (1), b (2) ..., b by 4-5-a
(m), m is the number of intersection point；All intersection point abscissas are calculated using the linear equation of AB, can analyze by abscissa
Whether there are obstacles around to intersection point；
4-5-b is determined intersection point b (j), j=1,2 ..., number of grid of the m as common intersection；If intersection point b (j) is
The common intersection of four grids, enters step 4-5-c；If intersection point b (j) is the common intersection of two grids, 4- is entered step
5-d；
4-5-c judges the positive and negative of the slope k of line segment AB, if slope is positive, checks that adjoining three grids above line segment AB is
It is no to have barrier；If slope is negative, checks and adjoin whether three grids have barrier below line segment AB；
If there is barrier, the inflection point between two nodes of AB cannot be rejected, and respective paths cannot update, two nodes of AB
Between barrier search terminate；If i=i+1 goes to step 4-5-b without barrier, continue to judge next intersection point；
4-5-d judges whether the adjacent left and right grid of intersection point is barrier, if it is barrier, hinders between two nodes of AB
Object search is hindered to terminate；If not barrier, i=i+1 goes to step 4-5-b, continues to judge next intersection point；
4-5-e, after the completion of all intersection points traverse, barrier search terminates between two nodes of AB.
4-6 rejects redundancy inflection point, more new route according to all nodes in step 4-1~4-5 traverse path.
As shown in figure 3, wherein A, B, C, D, E represent the adjacent path inflection point extracted.Since A point, AC is connected, when AD not
There is barrier, then B, two extra inflection points of C can be rejected, and more new route is A-D-E.There is barrier among when connecting AE,
Therefore D point cannot reject, path cannot update.Then again using D point as starting point, it is sequentially connected with subsequent inflection point and judges that path is
It is no to update, until terminal terminates.
Step 5: rejecting the path after redundancy inflection point to step 4, be smoothed, obtained using cubic spline interpolation
The path that final planning is completed.Detailed process is as follows:
5-1, if there is n+1 back end in path, node coordinate is respectively (x0, y0), (x1, y1), (x2, y2) ...,
(xn, yn)；In each subinterval xi≤x≤xi+1In, create batten difference equation:
gi(x)=ai+bi(x-xi)+ci(x-xi)2+di(x-xi)3
Wherein, ai, bi, ci, diIndicate the coefficient of spline curve；
5-2, material calculation hi=xi+1-xi, i=0,1 ..., n-1；
5-3, by back end and end-point condition M0=0, Mn=0 brings following matrix equation into, obtains:
Wherein, Mi, i=0,1 ..., n indicate batten difference equation second differential value；
5-4, solution matrix equation obtain batten difference equation second differential value Mi, i=0,1 ..., n；
5-5 calculates the coefficient a of spline curvei, bi, ci, di, formula is as follows:
ai=yi
Wherein, i=0,1 ..., n-1；
5-6 solves the coefficient in every section of batten difference equation, the expression of every section of curve can be obtained.
Fig. 6 is the planning path figure of traditional A* algorithm, and Fig. 7 is after A* algorithm improvement of the embodiment of the present invention and by smooth place
Planning path figure after reason, S indicate starting point, and E indicates target point.As shown in Figure 7, after merging crucial inflection point, the turnover in path
Point number significantly reduces, and in addition to the redundancy inflection point of traditional A* algorithm, while also reducing path total length, path is gradually smooth.
But but there is many spikes at path turning, cubic spline interpolation is re-introduced into keep path smooth enough, to allow machine
Device people keeps continuous in turning velocity and acceleration.
Claims (6)
1. a kind of based on the paths planning method for improving A* algorithm, it is characterised in that: method includes the following steps:
Step 1: by environment representation locating for robot at grating map, and one is searched out in grating map using A* algorithm
Initial path from starting point to target point；
Step 2: extracting all nodes in initial path and successively judge that adjacent three nodes are using area-method from the off
It is no conllinear, to find all turning points on path, i.e. inflection point；
Step 3: from the off, being sequentially connected inflection point until target point using straight line, obtain updated path；Remember every section it is straight
Line is L (i, j), and i, j respectively indicate the beginning and end of straightway, i=0,1,2 ..., n-1, j=1,2 ..., n, wherein n
For inflection point number；
Step 4: updating whether the straightway in rear path passes through barrier by judgment step 3, reject redundancy inflection point, update road
Diameter；
Step 5: rejecting the path after redundancy inflection point to step 4, be smoothed using cubic spline interpolation, obtained final
Plan the path completed.
2. according to claim 1 a kind of based on the paths planning method for improving A* algorithm, it is characterised in that: the step
In 4, judge whether straightway passes through barrier by the method for the dynamic select direction of search, when straightway L (i, j) does not pass through barrier
When hindering object, the redundancy inflection point between i and j is removed.
3. according to claim 2 a kind of based on the paths planning method for improving A* algorithm, it is characterised in that: pass through dynamic
The method of the selection direction of search judges whether straightway passes through barrier, removes redundancy inflection point, detailed process is as follows:
4-1, it is equal for the coordinate of search starting point A and target point B, AB two o'clock that selecting step 3 updates any two node in rear path
For place grating map net center of a lattice, it is denoted as (x respectively1,y1) and (x2,y2)；
4-2, the linear equation y=kx+b, k for calculating AB are straight slope, and b is the ordinate of straight line and longitudinal axis intersection point；
4-3 judges the abs (y of AB two o'clock2-y1) and abs (x2-x1) size, if abs (y2-y1) > abs (x2-x1), using transverse direction
Search executes step 4-4；Otherwise, using longitudinal searching, step 4-5 is executed；
4-4, in a lateral direction whether to have barrier between fixed step size search two nodes of AB；If there is barrier, AB
Inflection point between two nodes is not rejected, and respective paths do not update；If rejecting turning between two nodes of AB without barrier
Point updates respective paths；
4-5, in a longitudinal direction whether to have barrier between fixed step size search two nodes of AB；If there is barrier, AB
Inflection point between two nodes is not rejected, and respective paths do not update；If rejecting turning between two nodes of AB without barrier
Point updates respective paths；
4-6 rejects redundancy inflection point, more new route according to all nodes in step 4-1~4-5 traverse path.
4. according to claim 3 a kind of based on the paths planning method for improving A* algorithm, it is characterised in that: the step
4-4 is specific as follows in a lateral direction whether to have barrier between fixed step size search two nodes of AB:
All intersection points of longitudinal network ruling and line segment AB on grating map are denoted as a (1), a (2) ..., a (m), m by 4-4-a
For the number of intersection point；All intersection point ordinates are calculated using the linear equation of AB, analyze to obtain the draconitic revolution by ordinate
Enclose that whether there are obstacles；
4-4-b is determined intersection point a (i), i=1,2 ..., number of grid of the m as common intersection；If intersection point a (i) is four
The common intersection of grid, enters step 4-4-c；If intersection point a (i) is the common intersection of two grids, 4-4-d is entered step；
4-4-c judges the positive and negative of the slope k of line segment AB, if slope is positive, checks and adjoins whether three grids have above line segment AB
Barrier；If slope is negative, checks and adjoin whether three grids have barrier below line segment AB；
If there is barrier, the inflection point between two nodes of AB is not rejected, and respective paths do not update, barrier between two nodes of AB
Search terminates；If i=i+1 goes to step 4-4-b without barrier, continue to judge next intersection point；
4-4-d judges whether the adjacent left and right grid of intersection point is barrier；If it is barrier, barrier between two nodes of AB
Search terminates；If not barrier, i=i+1 goes to step 4-4-b, continues to judge next intersection point；
4-4-e, after the completion of all intersection points traverse, barrier search terminates between two nodes of AB.
5. according to claim 3 a kind of based on the paths planning method for improving A* algorithm, it is characterised in that: the step
4-5 is specific as follows in a longitudinal direction whether to have barrier between fixed step size search two nodes of AB:
All intersection points of transverse grid line and line segment AB on grating map are denoted as b (1), b (2) ..., b (m), m by 4-5-a
For the number of intersection point；All intersection point abscissas are calculated using the linear equation of AB, analyze to obtain the draconitic revolution by abscissa
Enclose that whether there are obstacles；
4-5-b is determined intersection point b (j), j=1,2 ..., number of grid of the m as common intersection；If intersection point b (j) is four
The common intersection of grid, enters step 4-5-c；If intersection point b (j) is the common intersection of two grids, 4-5-d is entered step；
4-5-c judges the positive and negative of the slope k of line segment AB, if slope is positive, checks and adjoins whether three grids have above line segment AB
Barrier；If slope is negative, checks and adjoin whether three grids have barrier below line segment AB；
If there is barrier, the inflection point between two nodes of AB is not rejected, and respective paths do not update, barrier between two nodes of AB
Search terminates；If i=i+1 goes to step 4-5-b without barrier, continue to judge next intersection point；
4-5-d judges whether the adjacent left and right grid of intersection point is barrier, if it is barrier, barrier between two nodes of AB
Search terminates；If not barrier, i=i+1 goes to step 4-5-b, continues to judge next intersection point；
4-5-e, after the completion of all intersection points traverse, barrier search terminates between two nodes of AB.
6. -5 any a kind of paths planning method based on improvement A* algorithm according to claim 1, it is characterised in that: institute
It states in step 5, to the path after rejecting redundancy inflection point, is smoothed using cubic spline interpolation, detailed process is as follows:
5-1, if there is n+1 back end in path, node coordinate is respectively (x0,y0),(x1,y1),(x2,y2),...,(xn,yn)；
In each subinterval xi≤x≤xi+1In, create batten difference equation:
gi(x)=ai+bi(x-xi)+ci(x-xi)2+di(x-xi)3
Wherein, ai,bi,ci,diIndicate the coefficient of spline curve；
5-2, material calculation hi=xi+1-xi, i=0,1 ..., n-1；
5-3, by back end and end-point condition M0=0, Mn=0 brings following matrix equation into, obtains:
Wherein, Mi, i=0,1 ..., n indicates batten difference equation second differential value；
5-4, solution matrix equation obtain batten difference equation second differential value Mi, i=0,1 ..., n；
5-5 calculates the coefficient a of spline curvei,bi,ci,di, formula is as follows:
ai=yi
Wherein, i=0,1 ..., n-1；
5-6 solves the coefficient in every section of batten difference equation, the expression of every section of curve can be obtained.
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