CN109322221A - A method of it is linear using four Bezier curve segmented construction highway three-dimensional space - Google Patents
A method of it is linear using four Bezier curve segmented construction highway three-dimensional space Download PDFInfo
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- E—FIXED CONSTRUCTIONS
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Abstract
The invention discloses a kind of methods linear using four Bezier curve segmented construction highway three-dimensional space, comprising the following steps: S1, given Discrete control point sequence Pi(i=0,1 ..., n) determines corresponding segments interpolation parameter ti；S2, provide section [ti,ti+1] upper four Bezier curve ΓiAnalytic uniform formula；S3, calculate each Discrete control point PiEndpoint feature amount；S4, according to end-point condition construct m containing freedom degreeiOr ui,sI-th section of four Bezier curve Γi；S5, give freedom degree miOr ui,sAssignment obtains Γi, calculate ΓiOn curvature, the torsion mutation value at torsion and Discrete control point；S6, judge ΓiWhether Engineering constraint requirement is met；S7, judge whether the linear of all segmentations of construction complete；S8, output according to control point sequence PiThe space of segmented construction is linear.The present invention is by giving the uniform analytic expression formula of space curve between Discrete control point, and it is linear intuitively to construct the highway three-dimensional space met the requirements according to Discrete control point, and parsing accuracy is high, engineering practicability is good.
Description
Technical field
The present invention relates to road alignment design fields, and in particular to a kind of to use four Bezier curve segmented construction highways
The linear method of three-dimensional space.
Background technique
The action of road alignment design is that the restrictive conditions such as landform, atural object is overcome to determine an energy in route selection region
Guarantee the space curve of vehicle safety travel, the accurate and analytical expression of this space curve is to be asked.As highway alignment
Space curve need simultaneously meet running car comfort and stability Engineering constraint and geometric continuity constrain.Route selection
When, space curve control point sequence P can be obtained within the scope of route selection regional space according to condition on the spoti, control route trend.
Now the problem is that according to control point sequence Pi, seek the space curve Г met the requirements.According to space geometry modeling principle,
Curvature κ and torsion τ are the three-dimensional linear control parameters that space curve Г changes with arc length.Automobile is analyzed in model space geometric
Driving status, can determine corresponding to different designs speed and form is the Engineering constraint index of three-dimensional linear parameter, specific table
Up to the maximum curvature κ for space curvemax, maximum torsion τmaxWith torsion changing value Δ τ.
The target of geometric continuity constraint is to keep highway three-dimensional space linear and traval trace space geometry characteristic phase
Mutually matching.Vehicle track is a continuous space curve, is not in any break, wrong head or interruption, same point on track
Will not be there are two curvature and two curvature variations, therefore space curve need to meet that geometry G2 is continuous, i.e., curvature variation connects
It is continuous.Due to often needing to carry out the linear modification in part during Alignment Design, the exact analytic expression of space curve can also change therewith,
In the case where no analytic uniform form, adjustment linear each time is recalculated to all parameters, computationally intensive
And low efficiency.Therefore, even in the situation known to the boundary conditions such as Engineering constraint and geometric continuity constraint, it is also difficult to straight
It connects and effectively obtains required space curve Г.So that simplify space curve solution procedure, current road alignment design is adopted
With the flat vertical method for first separating recombinant, by be in the nature three-dimensional space curve highway alignment be split as horizontal curve and vertical curve into
Row design.Spatial parameter in Engineering constraint is reduced to plane parameter, using MINIMUM CURVE RADIUS etc. as alignment control index；
With the geometric continuity of horizontal curve geometric continuity approximate substitution space curve；By adjusting the combination of local horizontal and vertical alignment, reach
To the purpose for designing linear local directed complete set.The problem is that two dimension splits the obtained space of design, linear to will appear space several
The phenomenon that continuity decays, it means that highway alignment the lacking there are three-dimensional space geometrical property that two-dimensional design obtains
It falls into.
Road alignment design should revert to three-dimensional nature.Realize that the critical issue of three dimensional design is: if can provide
A kind of space curve with uniform analytic expression form improves solution efficiency, simultaneously because the resolution table of linear upper any section
It is identical up to form, therefore can be in control point sequence PiIn between any two adjacent control points according to constraint condition curve construction, point
Section design finally obtains highway alignment.Therefore, it is necessary to be directed to above situation, propose that a kind of segmented construction highway three-dimensional space is linear
Method.
Summary of the invention
It is a kind of using four Bezier curve segmentations the purpose of the present invention is in view of the above shortcomings of the prior art, providing
The linear method of highway three-dimensional space is constructed, the method can pass through the analytic uniform of space curve between given Discrete control point
Expression-form realizes the linear segment design of highway three-dimensional space, does not need two dimension and splits and directly obtain highway three-dimensional space
Linear accurate Analysis expression, while having evaded the linear defect in space geometry characteristic of existing two-dimension method design.
The purpose of the present invention can be achieved through the following technical solutions:
A method of linear using four Bezier curve segmented construction highway three-dimensional space, the method includes following
Step:
S1: given Discrete control point Pi, wherein i=0,1,2 ... n, n indicate the number of Discrete control point, and determine corresponding point
Section interpolation parameter ti；
S2: provide section [ti,ti+1] upper i-th section of four Bezier curve ΓiAnalytic uniform formula；
S3: calculate each Discrete control point PiEndpoint feature amount；
Construct four Bezier curves, it is necessary to first obtain at first and last endpoint butt to unit vector ds、de, buckling vector
ks、keFor end-point condition；Therefore, full section highway three-dimensional space is linear need to provide each Discrete control point P for constructioniCharacteristic quantity
Butt is to unit vector diWith buckling vector kiAs end-point condition；
S4: Coefficient m containing freedom degree is constructed according to end-point conditioniOr ui,sI-th section of four Bezier curve ΓiUnified
Analysis formula, wherein miForOrWhen freedom degree coefficient, mi∈R；ui,sFor Li,s||Li||Li,eWhen freedom degree system
Number, ui,s∈R；LiFor head-end Pi,sThe osculating plane π at placei,sWith distal point Pi,eThe osculating plane π at placei,eIntersection；Li,s、
Li,eTangent line respectively at i-th section of four Bezier curve first and last endpoint；
Four Bezier curve ΓiBy Pi,s、Pi,1、Pi,2、Pi,3、Pi,eFive vertex are determining, wherein endpoint Pi,s、
Pi,eOn curve, with Discrete control point PiRelationship be Pi,s=Pi, Pi,e=Pi+1；By step S3Obtain Discrete control point Pi、
Pi+1And its characteristic quantity di、di+1、ki、ki+1, construct section [ti,ti+1] on four Bezier curve ΓiEnd-points interpolation condition
It determines therewith；According to endpoint Pi,s、Pi,e, endpoint butt is to unit vector di,s、di,e, wherein di,s=di,di,e=di+1, curvature
Vector ki,s、ki,e, wherein ki,s=ki,ki,e=ki+1,ki,s·di,s=ki,e·di,e=0, it is three remaining to solve controlling polygon
Vertex Pi,1、Pi,2、Pi,3Complete four Bezier curve ΓiConstruction；
S5: give freedom degree miOr ui,sAssignment obtains four Bezier curve Γi, calculate ΓiOn curvature, torsion and segmentation
Torsion mutation value at control point；
S6: judgment curves ΓiWhether Engineering constraint requirement is met；
It is such as unsatisfactory for, sets iteration step length, give freedom degree Coefficient miOr ui,s、ui,eAgain assignment, return step S5；If
Meet to get to i-th section of four Bezier curve；
S7: judge whether the linear of all segmentations of construction complete；
After obtaining i-th section of four Bezier curve, judge whether i is equal to n, if so, entering step S8；Otherwise i=i is enabled
+ 1, return step S5It is linear to construct next segmentation；When starting, i=0；
S8: output is according to control point sequence PiThe highway three-dimensional space of segmented construction is linear.
Further, the step S1In, interpolation parameter tiCalculation method specifically:
Firstly, obtaining interpolation parameter initial value according to amendment Chord Length Parameterization method:
And | Δ P-1|=| Δ Pn|=0
Wherein, | Δ Pi-1| for the chord length of (i-1)-th Discrete control o'clock to i-th of Discrete control point；giFor amendment chord length ginseng
The correction factor of numberization method；
Then, by normalized, standard parameter [t is obtained0,tn]=[0,1]:
Further, the step S2In, section [ti,ti+1] upper i-th section of four Bezier curve ΓiAnalytic uniform formula
Are as follows: Pi(s)=Bi,0(s)Pi,s+Bi,1(s)Pi,1+Bi,2(s)Pi,2+Bi,3(s)Pi,3+Bi,e(s)Pi,e, wherein Pi,s、Pi,eFor song
Line first and last endpoint, i.e. curve first and last Discrete control point, Pi,1、Pi,2With Pi,3For the other three vertex of curve controlled polygon；s∈[0,1]；For Bernstein basic function:
Further, the step S3In, Discrete control point butt is to unit vector diWith buckling vector kiCalculating pass through
The interpolation curve of Ai Er meter Te three times at structural segmentation control point realizes, three times Ai Er meter Te interpolation curve Hi(t) analytic expression
Are as follows:
Wherein, liFor Hi(t) Discrete control point P iniThe tangent vector at place；hi=ti+1-ti；
Discrete control point butt is to unit vector diCalculation formula are as follows:
Tangent vector liSolution equation are as follows:
Whereinhi=ti+1-ti；
Give a definition buckling vector k in space geometry Frenet frame { T, N, B } are as follows: to any point on any space curve L
There is buckling vector k, size is identical as the space of points curvature, and direction is identical as the main method arrow direction B,It is asking
Solve tangent vector liAfterwards, Ai Er meter Te interpolation curve Hi(t) it determines therewith, according to definition, buckling vector k at Discrete control pointiMeter
Formula are as follows:
Further, the step S4In, i-th section of four Bezier curve ΓiThe calculating on middle endpoint and remaining three vertex
Method
Specifically:
Wherein, li,s、li,eThe side length modulus of the controlling polygon of respectively four times Bezier curves
di,s、di,eRespectively four Bezier curves correspond to first and last endpoint Pi,s、Pi,eUnit tangent vector, and di,s=di, di,e=
di+1；
Wherein, ki,s、ki,eRespectively four Bezier curves correspond to endpoint Pi,s、Pi,eBuckling vector, and ki,s=ki、
ki,e=ki+1；CoefficientCoefficientLi,s、Li,eRespectively four Bezier curve endpoint Pi,s、
Pi,eThe tangent line at place；LiFor endpoint Pi,sThe osculating plane π at placei,sWith endpoint Pi,eThe osculating plane π at placei,eIntersection；miForOrWhen freedom degree coefficient, mi∈R；ui,sFor Li,s||Li||Li,eWhen freedom degree coefficient, ui,s∈R；Its
In, mi∈ M={ mi∈R:mi(ki,e·di,s)>0,(Pi,e-mi,sdi,s,di,e,ki,e)-mi(ki,s·di,e)|ki,e|2>0}.；ui,s
In straight line ui,s(ki,s,ki,e,di,s)+ui,e(ki,s,ki,e,di,e)=(ki,s,ki,e,Pi,e-li,sdi,s-li,edi,e) on ui,eFrom
By changing, ui,e∈R。
The side length modulus l of the controlling polygon of four Bezier curvesi,s, li,eIt is calculated by following formula:
Further, the step S5In, in conjunction with i-th section of four Bezier curve ΓiAnalytic uniform formula, according to differential
Geometrical principle, curve ΓiCurvature κi(s), torsion τi(s) and Discrete control point at torsion mutation value Δ τiBy following formula
It calculates:
Further, the step S6In, give engineering constraints κmax、τmaxWith Δ τmax, i-th section of four Bezier
Curve ΓiWhether meet Engineering constraint requirement to be differentiated by following formula:
Wherein, κmax、τmaxWith Δ τmaxThe curvature maximum threshold value that is respectively given by Engineering constraint, torsion maximum value threshold
Value and torsion change rate maximum threshold, | κi(s)|maxFor curve ΓiUpper curvature maximum；|τi(s)|maxFor curve ΓiOn scratch
Rate maximum value.
Compared with the prior art, the invention has the following advantages and beneficial effects:
1, the present invention can satisfy the linear various constraint conditions of highway three-dimensional space, according to Discrete control point intuitively structure
The space curve met the requirements is produced, the process of road alignment design is simplified, has evaded the space of existing two-dimensional design method
Geometrical property defect.
2, the present invention has excellent adaptive performance, can be adjusted by the way that reasonable freedom degree parameter is arranged in Discrete control point
Interpolation space curve shape and position, adaptation to the ground, atural object restrictive condition make the linear more section of the highway three-dimensional space constructed
It is more reasonable to learn.
3, The present invention gives the analytic uniform form of space curve, the highway three-dimensional space that each segmented construction obtains is linear
With accurate uniform analytic expression form, in the case where end-point condition is fixed, the curve adjustment being arbitrarily segmented will not be drawn
The parameter of curve variation for playing other segmentations, there is good engineering practicability.
4, the linear method adaptive performance of segmented construction highway three-dimensional space provided by the invention is excellent, parses accuracy
Height, engineering practicability are good, have highly application value.
Detailed description of the invention
Fig. 1 is the present invention method flow diagram linear using four Bezier curve segmented construction highway three-dimensional space.
Fig. 2 is section [t of the present inventioni,ti+1] on four Bezier curve ΓiSchematic diagram.
Fig. 3 is the schematic diagram of present invention segmentation four Bezier curve endpoints splicing.
Fig. 4 is the linear schematic diagram in space completed using the method for the invention segmented construction.
Fig. 5 is the linear result figure in space that specific configuration of the embodiment of the present invention is completed.
Specific embodiment
Present invention will now be described in further detail with reference to the embodiments and the accompanying drawings, but embodiments of the present invention are unlimited
In this.
Embodiment:
Present embodiments provide a kind of method linear using four Bezier curve segmented construction highway three-dimensional space, institute
Method is stated by the uniform analytic expression form of space curve between given Discrete control point, realizes linear point of highway three-dimensional space
Section design, the linear accurate Analysis expression of highway three-dimensional space can be directly obtained, evade line obtained by existing two-dimension method by having
The characteristics of space geometry characteristic defective of shape.Flow chart is as shown in Figure 1, comprising the following steps:
S1: given Discrete control point sequence Pi(i=0,1 ..., n) determines corresponding segments interpolation parameter ti；
Given Discrete control point sequence Pi, i=0,1 ..., n, according to amendment Chord Length Parameterization method to Discrete control point sequence
Interpolation parameter processing is carried out, determines the interpolation parameter t at corresponding orderly control pointi, so that [t0,tn]=[0,1].Specific steps
It is as follows:
Firstly, obtaining interpolation parameter initial value according to amendment Chord Length Parameterization method::
And | Δ P-1|=| Δ Pn|=0
Wherein: | Δ pi-1| for the chord length of (i-1)-th Discrete control o'clock to i-th of Discrete control point；miFor amendment chord length ginseng
The correction factor of numberization method；
Then, by normalization, standardization parameter [t is obtained0,tn]=[0,1]:
S2: provide section [ti,ti+1] upper four Bezier curve ΓiAnalytic uniform formula；
It is illustrated in figure 2 section [t of the present inventioni,ti+1] on four Bezier curve ΓiSchematic diagram, according to Bezier song
The definition of line, ΓiIt is section [ti,ti+1] on four Bezier curve Pi(t) (i=0,1 ..., n), Pi,s、Pi,eFor curve
First and last endpoint (Discrete control point), Pi,1、Pi,2With Pi,3For the other three vertex of curve controlled polygon, then Pi(t) general
Form are as follows:
Wherein,It is Bernstein basic function；
To make, curve indicates more general and form is succinct, orderS ∈ [0,1], then four Bezier curves
ΓiGeneral type are as follows:
Pi(s)=Bi,0(s)Pi,s+Bi,1(s)Pi,1+Bi,2(s)Pi,2+Bi,3(s)Pi,3+Bi,e(s)Pi,e；
Wherein, Bernstein basic function is
S3: calculate each Discrete control point PiEndpoint feature amount；
Construct four Bezier curves, it is necessary to first obtain at first and last endpoint butt to unit vector ds、de, buckling vector
ks、keFor end-point condition；Therefore, full section highway three-dimensional space is linear need to provide each Discrete control point P for constructioni(i=0,
1 ..., n) characteristic quantity butt to unit vector diWith buckling vector kiAs end-point condition.
It is illustrated in figure 3 the schematic diagram of present invention segmentation four Bezier curve endpoints splicing, needs to meet at endpoint several
G required by what continuity constraint2Continuously, i.e., second order parameter is continuous.By Differential Geometry correlation theory, Frenet frame T,
N, B } T is unit tangent vector at any point, N is unit pair method arrow, B is unit method arrow on down space curve L, provide buckling vector
The definition of k:The second order parameter condition of continuity is are as follows:
di,s=di,di,e=di+1；
ki,s=ki,ki,e=ki+1,ki,s·di,s=ki,e·di,e=0.
Wherein, di,s、di,eFor four Bezier curve ΓiEndpoint butt is to unit vector；ki,s、ki,eFor four Bezier
Curve ΓiEndpoint buckling vector.
In order to meet the condition, Discrete control point list cuts direction vector diWith buckling vector kiCalculating pass through structural segmentation
G at control point2Continuous Ai Er meter Te interpolation curve H three timesi(t) method is realized.To make curve have part adjustable
Property, to any Discrete control point Pi, only choose five point P closed onj(j=i-2, i-1 ..., i+2) it is used for diCalculating, in
Between put lead arrow and averaged can obtain.The Ai Er meter Te three times (Hermite) of construction is led according to Discrete control point coordinate and single order
Interpolation curve Hi(t) expression formula are as follows:
Wherein: liFor Discrete control point PiThe tangent vector at place；hi=ti+1-ti。
Second order parameter is continuous at Discrete control point, i.e. Hi"(ti)=Hi+1"(ti), its continuity equation can be obtained are as follows:
λili-1+2li+μili+1=Ci, i=1,2 ..., n-1
Wherein: It is by three
Point (ti-1,Pi-1)、(ti,Pi)、(ti+1,Pi+1) interpolation curve in t=tiThe single order at place leads arrow, is equal in ti-1、ti、ti+1Place
Single order lead the weighted average of arrow.
Continuity equation, which is one, (n+1) a unknown quantity li(i=0,1,2 ..., (n-1) rank equation group n), addition
Natural boundary conditions: L0"(t0)=Ln"(tn)=0, hereafter, there are two additional conditions are as follows:
2l0+μ0l1=C0
λnln-1+2ln=Cn
Therefore, Discrete control point PiTangent vector l at (i=0,1,2 ..., n)i(i=0,1,2 ..., n) it can be by following
Complete continuity equation acquires:
Wherein, for left side matrix other than main triangle element, remaining element is 0, therefore the process of solving equations is stable
's.
diFor liUnit vector, can be calculated by following formula:
Solving liAfterwards, Ai Er meter Te (Hermite) interpolation curve Hi(t) it determines, while is also determined on curve therewith
The buckling vector k of every biti(i=0,1 ..., n).According to interpolation curve Hi(t) the buckling vector k acquirediCalculating formula are as follows:
S4: m containing freedom degree is constructed according to end-point conditioniOr ui,sI-th section of four Bezier curve Γi；
Four Bezier curve ΓiBy Pi,s、Pi,1、Pi,2、Pi,3、Pi,eFive vertex are determining, wherein endpoint Pi,s、Pi,e?
On curve, with Discrete control point PiRelationship be Pi,s=Pi, Pi,e=Pi+1.By S3Obtain Discrete control point Pi、Pi+1And its feature
Measure di、di+1、ki、ki+1, construct section [ti,ti+1] four Bezier curve ΓiEnd-points interpolation condition determine therewith.According to
Endpoint Pi,s、Pi,e, endpoint butt is to unit vector di,s、di,e(di,s=di,di,e=di+1), buckling vector ki,s、ki,e(ki,s=
ki,ki,e=ki+1,ki,s·di,s=ki,e·di,e=0) Γ, is completediConstruction problem be to solve controlling polygon remaining three
Vertex Pi,1、Pi,2、Pi,3。
Solve three vertex P of controlling polygon residuei,1、Pi,2、Pi,3, detailed process are as follows:
Clearly following lemma is needed first:
Lemma 1: a, b, x ∈ R are set3, | a |=1, then equation a × x=b can be solved when ab=0, and equation institute
There is the form of solution are as follows: x=ma+b × a；
Lemma 2: { e is seti, i=1,2 ..., n } it is Rn,a∈RnIn base vector, then a=0 is equivalent to aei=0, i=
1,2,...,n。
With Pi,s、Pi,1、Pi,2、Pi,3、Pi,eFor four Bezier curve controlling polygons on vertex, there is vector Δ Pi,j:
Wherein: li,jFor the side length modulus of controlling polygon.
Then have:
Meanwhile being had according to the definition of buckling vector and four Bezier curve properties:
According to lemma 1 it is found that there are ui,sWith ui,e, so that
Finally, by the continuous constraint on Bezier curve controlling polygon vertex, Δ Pi,s、ΔPi,1、ΔPi,2With Δ Pi,3It must
Following conditions must be met:
ΔPi,s+ΔPi,1+ΔPi,2+ΔPi,3=Pi,e-Pi,s (5)
The condition that the construction problem of i.e. four times Bezier curves can solve are as follows:
In fact, working as ui,s、ui,e、li,s、li,eMeet formula (6), P can be found out according to formula (1), formula (3) and formula (4)i,1、
Pi,2、Pi,3, to provide the solution of four Bezier curve construction problems.
Next, with ui,s、ui,e、li,s、li,eFor object, discuss from the angle of geometry to the solution of problem.
If πi,s、πi,eWith Li,s、Li,eOsculating plane and tangent line at respectively four Bezier curve endpoints indicate are as follows:
Wherein x indicates the dynamic point on corresponding track.
To four Bezier curve ΓiObvious hasPi,2∈πi,s∩πi,e.If LiFor πi,sWith πi,eFriendship
Line, LiDirection be ki,s×ki,e(ki,s×ki,e≠0).In order to find LiOn a little to determine Li, need to be according to Li,s、Li,eWith
LiParallel situation is divided into two groups of situations and discusses:
(1)Or
Consider firstSituation, i.e.,And di,s·ki,e≠0.Enabling p is Li,sWith LiIntersection point,
Obvious p ∈ Li,s, p ∈ πi,e, then there is Coefficient mi,s∈ R makes p=mi,sdi,s, (p-Pi,e)·ki,e=0.Therefore it can obtain:
Notice LiDirection be ki,s×ki,e, and Pi,2∈Li, then freedom degree m is certainly existediSo that:
Pi,2=mi,sdi,s+miki,s×ki,e,mi∈R (7)
Thus formula convolution (2) can obtain:
By lemma 2 and formula (8), can obtain:
It is available by the above process: whenAnd di,s·ki,eWhen ≠ 0, what four Bezier curves constructed
The solution of problem can be combined by formula (7) and (9) to be provided, and there are one degree of freedom m for solutioni:
mi∈ M={ mi∈R:mi(ki,e·di,s)>0,(Pi,e-mi,sdi,s,di,e,ki,e)-mi(ki,s·di,e)|ki,e|2>
0}.
As freedom degree miWhen changing in its value range M, corresponding change can also occur for the shape of four Bezier curves,
Therefore its control parameter that can be used as curve shape uses, in highway space curve design process, the freedom degree m that can pass throughi
Iteration chooses best value automatically, and the curvature value of curve is made to meet the requirement of binding target system.
ForThe case where, the problem of four Bezier curves construction, can be provided by following formula:
Pi,2=mi,edi,e+miki,e×ki,s,mi∈R (10)
Wherein:
mi∈ M={ mi∈R:mi(ki,s·di,e)>0,(Pi,s-mi,edi,e,di,s,ki,s)-mi(ki,e·di,s)|ki,s|2>
0}.。
(2)Li,s||Li||Li,e
There is d in this situationi,s||ki,s×ki,e||di,e, i.e. di,s·ki,e=di,e·ki,s=0.Formula can be enabled at this time
(6) respectively with ki,s、ki,e、ki,s×ki,eMake inner product, can obtain:
ui,s(ki,s,ki,e,di,s)+ui,e(ki,s,ki,e,di,e)=(ki,s,ki,e,Pi,e-li,sdi,s-li,edi,e) (13)
Therefore, in ki,s×ki,e≠ 0 and di,s·ki,e=di,e·ki,sIn the case where, four times Bezier construction problem has solution
Condition be and if only if (Pi,e·ki,e)(ki,s,ki,e,di,s) < 0 and (Pi,e·ki,s)(ki,s,ki,e,di,e)<0.At this point, li,s
With li,eIt is uniquely determined by formula (12), ui,sWith ui,eIt can freely change on the straight line determined by formula (13), equally be used as
Bezier curve shape control parameter can realize that curvature of curve meets the design of highway space curve about by the automatic Iterative of the two
The requirement of beam index system.
In summary situation can provide i-th section of four Bezier curve ΓiIn three vertex of residue in addition to endpoint
Calculation method are as follows:
Wherein: li,s, li,eFor the side length modulus of the controlling polygon of four Bezier curvesdi,s,
di,eEndpoint P is corresponded to for four Bezier curvesi,s, Pi,eUnit tangent vector, and di,s=di, di,e=di+1；
To avoid computing repeatedly, whenWithIt is preferential to use when existing simultaneouslyThe case where,Wherein: ki,s, ki,eFor four Bezier curves
Corresponding Pi,s, Pi,eBuckling vector, and ki,s=ki, ki,e=ki+1；CoefficientCoefficient
Li,s、Li,eRespectively four Bezier curve endpoint Pi,s, Pi,eThe tangent line at place；LiFor endpoint Pi,s, Pi,eThe osculating plane at place
πi,s、πi,eIntersection；miAnd ui,sRespectively correspond to the freedom degree in situation.
li,s, li,eIt is calculated by following formula:
S5: give freedom degree miOr ui,sAssignment obtains Γi, calculate ΓiOn curvature, the torsion at torsion and Discrete control point
Mutation value；
In conjunction with S2In the general type of i-th section of four Bezier curve analytic expression that provides, it is bent according to differential geometry principle
Line ΓiCurvature κi(s), torsion τi(s) and Discrete control point at torsion mutation value Δ τiIt is calculated by following formula:
S6: judge ΓiWhether Engineering constraint requirement is met；
It is such as unsatisfactory for, sets freedom degree miOr ui,s, ui,eIteration step length, again assignment, return step S5, if it is satisfied, i.e.
Obtain i-th section of four Bezier curve；
In given Engineering constraint κmax、τmaxWith Δ τmaxCondition, i-th section of four Bezier curve whether meet the requirements by
Following formula differentiates:
Wherein, | κi(s)|maxFor curve ΓiUpper curvature maximum；|τi(s)|maxFor curve ΓiUpper torsion maximum value.
S7: judge whether the linear of all segmentations of construction complete；
After obtaining i-th section (i=0,1 ..., n) four Bezier curves, judge whether i is equal to n, if so, into S8,
Otherwise i=i+1 is enabled, S is returned5It is linear to construct next segmentation.When starting, i=0.
S8: output is according to PiThe three-dimensional space of segmented construction is linear.It is illustrated in figure 4 the sky that segmented construction of the present invention is completed
Between linear schematic diagram.
Highway three can be carried out after given Discrete control point and engineering constraints in practical applications using the present invention
The linear tectonic sieving of dimension space.
According to the above method, engineering constraints when given design speed is 80km/h are (κmax=0.0025, τmax=
0.0001, Δ τ=0.0039), with one group of control point P0(0,100,100)、P1(550,200,150)、P2(1000,0,75)、P3
(1435,-200,125)、P4(2000,250,110) it is Discrete control point, constructs one by four sections of four Beizer curve matchings
Made of space interpolation curve see Fig. 5, be illustrated in figure 5 the linear result figure in space of the present embodiment construction complete.Each segmentation four
Secondary Beizer curve maximum curvature κi(s)maxWith torsion τi(s)maxAnd torsion Sudden Changing Rate Δ τ at each section of curve tie pointiIt calculates
Value is shown in Table 1.
Table 1
By data in table 1 it is found that respectively the maximum curvature of four Beizer curves of segmentation expires with tie point torsion Sudden Changing Rate
The requirement of the given Engineering constraint of foot.
As can be seen that the method that the present invention is linear using four Beizer curve segmentation construction highway three-dimensional space, can make
The space curve finally obtained is G2Requirement that is continuous and meeting Engineering constraint, thus this method theoretically for have it is abundant
Feasibility, solve the defect of existing method to a certain extent.
The above, only the invention patent preferred embodiment, but the scope of protection of the patent of the present invention is not limited to
This, anyone skilled in the art is in the range disclosed in the invention patent, according to the present invention the skill of patent
Art scheme and its patent of invention design are subject to equivalent substitution or change, belong to the scope of protection of the patent of the present invention.
Claims (8)
1. a kind of method linear using four Bezier curve segmented construction highway three-dimensional space, which is characterized in that the side
Method the following steps are included:
S1: given Discrete control point Pi, wherein i=0,1,2 ... n, n indicate the number of Discrete control point, and determine that corresponding segments are inserted
Value parameter ti；
S2: provide section [ti,ti+1] upper i-th section of four Bezier curve ΓiAnalytic uniform formula；
S3: calculate each Discrete control point PiEndpoint feature amount, provide each Discrete control point PiCharacteristic quantity butt to unit
Vector diWith buckling vector kiAs end-point condition；
S4: Coefficient m containing freedom degree is constructed according to end-point conditioniOr ui,sI-th section of four Bezier curve ΓiAnalytic uniform
Formula, wherein miForOrWhen freedom degree coefficient, mi∈R；ui,sFor Li,s||Li||Li,eWhen freedom degree system
Number, ui,s∈R；LiFor head-end Pi,sThe osculating plane π at placei,sWith distal point Pi,eThe osculating plane π at placei,eIntersection；Li,s、
Li,eTangent line respectively at i-th section of four Bezier curve first and last endpoint；
Four Bezier curve ΓiBy Pi,s、Pi,1、Pi,2、Pi,3、Pi,eFive vertex are determining, wherein endpoint Pi,s、Pi,eIn song
On line, with Discrete control point PiRelationship be Pi,s=Pi, Pi,e=Pi+1；By step S3Obtain Discrete control point Pi、Pi+1And its it is special
Sign amount di、di+1、ki、ki+1, construct section [ti,ti+1] on four Bezier curve ΓiEnd-points interpolation condition determine therewith；
According to endpoint Pi,s、Pi,e, endpoint butt is to unit vector di,s、di,e, wherein di,s=di,di,e=di+1, buckling vector ki,s、
ki,e, wherein ki,s=ki,ki,e=ki+1,ki,s·di,s=ki,e·di,e=0, solve three vertex P of controlling polygon residuei,1、
Pi,2、Pi,3Complete four Bezier curve ΓiConstruction；
S5: give freedom degree Coefficient miOr ui,sAssignment obtains four Bezier curve Γi, calculate ΓiOn curvature, torsion and segmentation
Torsion mutation value at control point；
S6: judgment curves ΓiWhether Engineering constraint requirement is met；
It is such as unsatisfactory for, sets iteration step length, give freedom degree Coefficient miOr ui,s、ui,eAgain assignment, return step S5；If it is satisfied,
Obtain i-th section of four Bezier curve；
S7: judge whether the linear of all segmentations of construction complete；
After obtaining i-th section of four Bezier curve, judge whether i is equal to n, if so, entering step S8；Otherwise i=i+1 is enabled, is returned
Return step S5It is linear to construct next segmentation；When starting, i=0；
S8: output is according to control point sequence PiThe highway three-dimensional space of segmented construction is linear.
2. the method linear using four Bezier curve segmented construction highway three-dimensional space according to claim 1,
It is characterized in that, the step S1In, interpolation parameter tiCalculation method specifically:
Firstly, obtaining interpolation parameter initial value according to amendment Chord Length Parameterization method:
I=1,2 ..., n and | Δ P-1|=| Δ Pn|=0
Wherein, | Δ Pi-1| for the chord length of (i-1)-th Discrete control o'clock to i-th of Discrete control point；giTo correct Chord Length Parameterization
The correction factor of method；
Then, by normalized, standard parameter [t is obtained0,tn]=[0,1]:
3. the method linear using four Bezier curve segmented construction highway three-dimensional space according to claim 1,
It is characterized in that, the step S2In, section [ti,ti+1] upper i-th section of four Bezier curve ΓiAnalytic uniform formula are as follows: Pi(s)
=Bi,0(s)Pi,s+Bi,1(s)Pi,1+Bi,2(s)Pi,2+Bi,3(s)Pi,3+Bi,e(s)Pi,e, wherein Pi,s、Pi,eFor curve first and last end
Point, i.e. curve first and last Discrete control point, Pi,1、Pi,2With Pi,3For the other three vertex of curve controlled polygon；I=0,1 ..., n is Bernstein basic function:
4. the method linear using four Bezier curve segmented construction highway three-dimensional space according to claim 1,
It is characterized in that, the step S3In, Discrete control point butt is to unit vector diWith buckling vector kiCalculating pass through structural segmentation
The interpolation curve of Ai Er meter Te three times at control point realizes, three times Ai Er meter Te interpolation curve Hi(t) analytic expression are as follows:
Wherein, liFor Hi(t) Discrete control point P iniThe tangent vector at place；hi=ti+1-ti；
Discrete control point butt is to unit vector diCalculation formula are as follows:
(i=0,1,2 ..., n)
Tangent vector liSolution equation are as follows:
Whereinhi=ti+1-ti；
Solving tangent vector liAfterwards, Ai Er meter Te interpolation curve Hi(t) it determines therewith, buckling vector k at Discrete control pointiMeter
Formula are as follows:
(i=0,1,2 ..., n).
5. the method linear using four Bezier curve segmented construction highway three-dimensional space according to claim 1,
It is characterized in that, the step S4In, i-th section of four Bezier curve ΓiThe calculation method on middle endpoint and remaining three vertex is specific
Are as follows:
Wherein, li,s、li,eThe side length modulus of the controlling polygon of respectively four times Bezier curvesdi,s、
di,eRespectively four Bezier curves correspond to first and last endpoint Pi,s、Pi,eUnit tangent vector, and di,s=di, di,e=di+1；
Wherein, ki,s、ki,eRespectively four Bezier curves correspond to endpoint Pi,s、Pi,eBuckling vector, and ki,s=ki、ki,e=
ki+1；CoefficientCoefficientLi,s、Li,eRespectively four Bezier curve endpoint Pi,s、Pi,ePlace
Tangent line；LiFor endpoint Pi,sThe osculating plane π at placei,sWith endpoint Pi,eThe osculating plane π at placei,eIntersection；miForOrWhen freedom degree coefficient, mi∈R；ui,sFor Li,s||Li||Li,eWhen freedom degree coefficient, ui,s∈R；Wherein, mi∈ M=
{mi∈R:mi(ki,e·di,s)>0,(Pi,e-mi,sdi,s,di,e,ki,e)-mi(ki,s·di,e)|ki,e|2>0}.；ui,sIn straight line ui,s
(ki,s,ki,e,di,s)+ui,e(ki,s,ki,e,di,e)=(ki,s,ki,e,Pi,e-li,sdi,s-li,edi,e) on ui,eFreely change,
ui,e∈R。
6. the method linear using four Bezier curve segmented construction highway three-dimensional space according to claim 5,
It is characterized in that, the side length modulus l of the controlling polygon of four Bezier curvesi,s, li,eIt is calculated by following formula:
7. the method linear using four Bezier curve segmented construction highway three-dimensional space according to claim 3,
It is characterized in that, the step S5In, in conjunction with i-th section of four Bezier curve ΓiAnalytic uniform formula, according to differential geometry principle,
Curve ΓiCurvature κi(s), torsion τi(s) and Discrete control point at torsion mutation value Δ τiIt is calculated by following formula:
8. the method linear using four Bezier curve segmented construction highway three-dimensional space according to claim 1,
It is characterized in that, the step S6In, give engineering constraints κmax、τmaxWith Δ τmax, i-th section of four Bezier curve ΓiIt is
It is no to meet Engineering constraint requirement and differentiated by following formula:
Wherein, κmax、τmaxWith Δ τmaxThe curvature maximum threshold value that is respectively given by Engineering constraint, torsion maximum threshold and
Torsion change rate maximum threshold, | κi(s)|maxFor curve ΓiUpper curvature maximum；|τi(s)|maxFor curve ΓiUpper torsion is most
Big value.
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