CN104965519A - Bezier curve-based terminal guidance method with fall angle constraint - Google Patents

Abstract

A Bezier curve-based terminal guidance method with a fall angle constraint, which is disclosed by the invention, relates to a Bezier curve-based guidance method with a fall angel constraint which is used for aircraft guidance, and belongs to the technical field of aircraft guidance. The method includes the steps of: 1. building an aircraft particle dynamics kinematics equation; 2. performing kinematics trajectory planning based on a Bezier curve; and 3. solving angle of attack alpha<m> and angle of heel mu<m> guidance instructions based on an aircraft trajectory given based on the step 2, performing real-time feedback on the current condition of the aircraft, realizing on-line conducting of Bezier model-based trajectory replanning, and deducing an acceleration instruction of the next moment till full-course guidance is completed. The Bezier curve-based terminal guidance method with the fall angle constraint can adapt to a requirement of a precise strike under a wide-range fall angle constraint, can also ensure that the aircraft finally completes a strike task under the condition of a saturated controlled quantity, and has certain robustness to external interference and environmental uncertainty.

Description

The terminal guidance method that a kind of band angle of fall based on Bezier retrains

Technical field

The present invention relates to a kind of band angle of fall constraint method of guidance, particularly relate to a kind of band angle of fall based on Bezier for aircraft guidance constraint method of guidance, belong to technical field of air vehicle guidance.

Background technology

As the final tache that speed reentry aircrafts hits over the ground, high precision terminal guidance technology is related to the success or failure of whole aerial mission.In actual strike task, in order to obtain best damage effectiveness, terminal point constraint often has great importance.In recent years, Planning thought has also been used among Design of Guidance Law.By carrying out to flight path the demand that geometric programming can meet terminal position and angle, by carrying out planning the steering order that can directly obtain resolving to controlled quentity controlled variable, integration obtains corresponding flight path then.Utilize planning technology can considerably simplify the design process of Guidance Law.In addition, also its dependent variable can be introduced performance index and use restraint, more meet multiple constraint guidance requirement.Aircraft planning strategy has vital role in raising operational performance and viability.

Path planning, as a large class very important in planning strategy, obtains the extensive concern of relevant scholar.Path planning agreeing with under the different mission requirements of aircraft, for satisfied flight track cooked up by aircraft, can improve flight quality, thus effective success ratio improving aircraft and hit over the ground.Bezier is as the one of nurbs curve, because it has stronger geometric flexibility, simultaneously containing less moulding variable, for aircraft, geometry due to trajectory is in the nature a smooth curve, and Bezier almost can characterize all smooth curves, therefore Bezier has agreeing with property preferably when characterizing trajectory.

First the key concept of Bezier is introduced below.Bezier is the curve utilizing spatially one group of reference mark definition, and change of shape only depends on reference mark number and position.Definable N rank, N+1 reference mark Bezier, its expression-form is as follows:

$P\left(\mathrm{\&tau;}\right)=\underset{i=0}{\overset{n}{\mathrm{\&Sigma;}}}{P}_i{B}_{i,n}\left(\mathrm{\&tau;}\right),\mathrm{\&tau;}\&Element;\left[\mathrm{0,1}\right]---\left(1\right)$

In formula: P _i(0≤i≤n) is called as i-th reference mark coordinate of Bezier, connects P in turn _ican obtain the characteristic polygon of this Bezier, Bezier is uniquely determined by characteristic polygon.B _i,n(τ) be n Bernstein basis function, its expression-form is as follows:

$B_{i,n}\left(\mathrm{\&tau;}\right)=C_n^i{\mathrm{\&tau;}}^i{(1-\mathrm{\&tau;})}^{n-i},C_n^i=\frac{n!}{k!(n-k)!}---\left(2\right)$

Bezier has following characteristic:

Character 1: starting point, the terminal of the starting point of Bezier, terminal and corresponding characteristic polygon overlap.

Character 2: Bezier starting point is consistent with the trend on characteristic polygon Article 1 limit and the last item limit with the tangential direction of destination county.

Character 3: the point on Bezier all drops on by its reference mark P _iamong the convex closure formed.

Can find out that Bezier has larger geometric flexibility by above-mentioned introduction, simple structure, design parameter is few, can characterize complicated trajectory shape, meets the angle of fall constraint of-180deg ~ 0deg.Bezier is widely used in aircraft glide section and cruise section trajectory planning Bezier, and dive section is violent because aerodynamic parameter changes, and flight environment of vehicle is complicated and changeable.Often require higher to Guidance Law, proposition can adapt to the lower precision strike demand of angle of fall constraint on a large scale, and also can guarantee that when controlled quentity controlled variable is saturated aircraft finally completes strike task, and the terminal guidance method that interference and environmental uncertainty have a certain robustness is to external world very important.

Summary of the invention

The technical problem to be solved in the present invention is to provide the terminal guidance method that a kind of band angle of fall based on Bezier retrains, the lower precision strike demand of angle of fall constraint on a large scale can be adapted to, and also can guarantee that when controlled quentity controlled variable is saturated aircraft finally completes strike task, and interference and environmental uncertainty have certain robustness to external world.The described constraint of the angle of fall on a large scale refers to that angle of fall scope is-180 ° to 0 °.

The object of the invention is to be achieved through the following technical solutions:

The present invention discloses the terminal guidance method that a kind of band angle of fall based on Bezier retrains, and comprises the steps:

Step 1, ignores earth rotation, sets up aircraft particle dynamics kinematical equation:

${\stackrel{\&CenterDot;}{x}}_m=V_m\cos {\mathrm{\&gamma;}}_m\cos {\mathrm{\&chi;}}_m---\left(3\right)$

${\stackrel{\&CenterDot;}{y}}_m=V_m\sin {\mathrm{\&gamma;}}_m---\left(4\right)$

${\stackrel{\&CenterDot;}{z}}_m=-V_m\cos {\mathrm{\&gamma;}}_m\sin {\mathrm{\&chi;}}_m---\left(5\right)$

${\stackrel{\&CenterDot;}{V}}_m=-\frac{D_m}{m_m}-g\cos {\mathrm{\&gamma;}}_m---\left(6\right)$

${\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}_m=\frac{L_m\cos {\mathrm{\&mu;}}_m}{m_m{V}_m}-\frac{g\cos {\mathrm{\&gamma;}}_m}{V_m}---\left(7\right)$

${\stackrel{\&CenterDot;}{\mathrm{\&chi;}}}_m=-\frac{L_m\sin {\mathrm{\&mu;}}_m}{m_m{V}_m\cos {\mathrm{\&gamma;}}_m}---\left(8\right)$

Wherein: x _m, y _m, z _mfor the position coordinates of aircraft under inertial system; V _m, γ _m, χ _mbe respectively speed, trajectory tilt angle, trajectory deflection angle; G is acceleration of gravity; μ _mfor angle of heel; L _m, D _mbe respectively lift and resistance, wherein, s _reffor the area of reference of aircraft; ρ is atmospheric density; C _lm, C _dmbe respectively lift coefficient and resistance coefficient, lift coefficient C _lm, resistance coefficient C _dmabout angle of attack _mwith the function of Mach number Ma.

Step 2, carries out kinematic trajectory planning based on Bezier.

Current positional information and velocity reversal and terminal location and angle restriction determination aerial vehicle trajectory, respectively for fore-and-aft plane structure Bezier track, described track is required to be the Bezier on more than three rank.Preferably three rank Bezier structure Bezier tracks.

If given reference mark coordinate is followed successively by (x _a, y _a, z _a), (px _a, py _a, pz _a), (px _b, py _b, pz _b), (x _b, y _b, z _b), three rank bezier curve equations of aircraft coordinate can be obtained:

Here for convenience of representing, by controlling polygon starting point (x _a, y _a, z _a) and (x _b, y _b, z _b) be denoted as end points, (px _a, py _a, pz _a) and (px _b, py _b, pz _b) be still denoted as reference mark.

x _m＝a _xτ ³+b _xτ ²+c _xτ+d _x(9)

y _m＝a _yτ ³+b _yτ ²+c _yτ+d _y(10)

z _m＝a _zτ ³+b _zτ ²+c _zτ+d _z(11)

Wherein: τ ∈ [0,1] is intermediate variable, the multinomial coefficient in formula (9)-(11) can be determined by following formula (12)-(14):

d _x＝x _A

c _x＝3(px _A-x _A)

（12）

b _x＝3(px _B-px _A)-c _x

a _x＝x _B-x _A-c _x-b _x

d _y＝y _A

c _y＝3(py _A-y _A)

（13）

b _y＝3(py _B-py _A)-c _y

a _y＝y _B-y _A-c _y-b _y

d _z＝z _A

c _z＝3(pz _A-z _A)

（14）

b _z＝3(pz _B-pz _A)-c _z

a _z＝z _B-z _A-c _z-b _z

Step 3, the aerial vehicle trajectory provided based on step 2 solves angle of attack _m, angle of heel μ _mguidance command.

Step 3.1 adopts inverse dynamics to solve angle of attack _m, angle of heel μ _mguidance command.

After by Bezier determination aircraft movements track, utilize inverse dynamics theory to solve and guidance command, its normal acceleration a _ywith longitudinal acceleration a _zbe defined as: there is following expression-form:

$a_y=V_m^2\sin {\mathrm{\&gamma;}}_m{\mathrm{\&gamma;}}_m^{\&prime;}+g\cos {\mathrm{\&gamma;}}_m---\left(15\right)$

$a_z={-V}_m^2\sin {\mathrm{\&gamma;}}_m\cos {\mathrm{\&gamma;}}_m{\mathrm{\&chi;}}_m^{\&prime;}---\left(16\right)$

Wherein, ' represent y differentiate, γ ' _m, χ ' _mthere is following expression-form:

$\begin{array}{c}{\mathrm{\&gamma;}}_m^{\&prime;}=-{\sin }^2{\mathrm{\&gamma;}}_m(\cos {\mathrm{\&chi;}}_m{x}^{\&prime;\&prime;}-\sin {\mathrm{\&chi;}}_m{z}^{\&prime;\&prime;})\\ {\mathrm{\&chi;}}_m^{\&prime;}=-\tan {\mathrm{\&gamma;}}_m(\sin {\mathrm{\&chi;}}_m{x}_m^{\&prime;\&prime;}+\cos {\mathrm{\&chi;}}_m{z}_m^{\&prime;\&prime;})\end{array}---\left(17\right)$

Can see in conjunction with above-mentioned inverse dynamics analysis, aircraft acceleration instruction is not only relevant to aircraft current flight state, simultaneously also with shape (the range x of flight path _mwith horizontal journey z _mabout height y _mfirst order derivative and second derivative) relevant, therefore above-mentioned guidance problems is converted into trajectory planning problem in fact.

Try to achieve aircraft acceleration a _ywith a _zafter instruction, aircraft angle of heel μ _mcan be determined by following formula (18):

${\mathrm{\&mu;}}_m=\arctan \frac{a_z}{a_y}---\left(18\right)$

Defined by formula (18) and aircraft acceleration and can inquire into aircraft lift L _mwith lift coefficient C _lm, and then anti-release Aircraft Angle of Attack α _minstruction.

Step 3.2, carries out Real-time Feedback to aircraft the present situation, repeatedly repeats step 2 and step 3.1 and realizes the track weight-normality carried out online based on Bezier model and draw, inquire into the instruction of subsequent time acceleration, until complete omnidistance guidance.Guarantee that when controlled quentity controlled variable is saturated aircraft realizes the precision strike met under impact angle constraint, and interference and environmental uncertainty has certain robustness to external world.

Easily occur this situation saturated for the acute variation of aircraft dive section environment and controlled quentity controlled variable, now, aircraft will depart from the flight path planned in advance, and only simple employing is followed the tracks of fixation locus and be difficult to ensure attack precision.Aircraft current flight state is introduced close-loop feedback, pass through Real-time Feedback, take current state as initial conditions, repetition step 2 and step 3.1 realize the track weight-normality carried out online based on Bezier model and draw, inquire into the instruction of subsequent time acceleration, bring aircraft power kinematics model into, obtain new state of flight, repeatedly carry out this process, when aircraft is drawn because external disturbance and controlled quentity controlled variable get the full passing track weight-normality, this Guidance Law can correction of deviation gradually, finally completes omnidistance guidance.

Beneficial effect

1, the present invention adopts Bezier to carry out trajectory planning, enormously simplify calculated amount, has stronger engineering operability.Bezier has larger geometric flexibility, simple structure, and design parameter is few, can characterize complicated trajectory shape, can cover the requirement of the angle of fall on a large scale of-180 ° to 0 °, realizes hitting requirement to the full angle of fall of target.

2, the present invention has stronger robustness to external disturbance and uncertainty, and can successfully manage the controlled quentity controlled variable saturated phenomenon in guidance process.

Accompanying drawing explanation

Fig. 1 (a) is constrained to the Bezier Parameters design of acute angle for last angle;

Fig. 1 (b) is constrained to the Bezier Parameters design at obtuse angle for last angle;

Fig. 1 (c) is constrained to the Bezier Parameters design at right angle for last angle;

Fig. 2 is the terminal guidance method process flow diagram that a kind of band angle of fall based on Bezier of the present invention retrains;

Fig. 3 is the geometric locus of two Bezier splicings;

Fig. 4 is Selecting parameter schematic diagram in Bezier splicing strategy;

Fig. 5 (a) is for two-dimensional case, the flight path curve under the constraint of the different angle of falls;

Fig. 5 (b) is for two-dimensional case, the trajectory tilt angle change curve under the constraint of the different angle of falls;

Fig. 5 (c) is for two-dimensional case, the speed change curves under the constraint of the different angle of falls;

Fig. 5 (d) is for two-dimensional case, the angle of attack variation curve under the constraint of the different angle of falls;

Fig. 6 (a) is for two-dimensional case, the flight path curve under different initial angle;

Fig. 6 (b) is for two-dimensional case, the trajectory tilt angle change curve under different initial angle;

Fig. 6 (c) is for two-dimensional case, the angle of attack variation curve under different initial angle;

Fig. 7 (a) is for two-dimensional case, the flight path curve under the large angle of fall (-160 ° ~-180 °) constraint;

Fig. 7 (b) is for two-dimensional case, the trajectory tilt angle change curve under the large angle of fall (-160 ° ~-180 °) constraint;

Fig. 7 (c) is for two-dimensional case, the angle of attack variation curve under the large angle of fall (-160 ° ~-180 °) constraint;

Fig. 8 (a) in three dimensions for the flight path curve under the constraint of the different angle of falls;

Fig. 8 (b) in three dimensions for the trajectory tilt angle change curve under the constraint of the different angle of falls;

Fig. 8 (c) in three dimensions for the trajectory deflection angle change curve under the constraint of the different angle of falls;

Fig. 8 (d) in three dimensions for the angle of attack variation curve under the constraint of the different angle of falls;

Fig. 8 (e) in three dimensions for the angle of heel change curve under the constraint of the different angle of falls;

Fig. 9 (a) is the flight path curve under initial situation different in three dimensions;

Fig. 9 (b) is the trajectory tilt angle change curve under initial situation different in three dimensions;

Fig. 9 (c) is the trajectory deflection angle change curve under initial situation different in three dimensions;

Fig. 9 (d) is the angle of attack variation curve under initial situation different in three dimensions;

Fig. 9 (e) is the angle of heel change curve under initial situation different in three dimensions.

Embodiment

In order to better objects and advantages of the present invention are described, below in conjunction with accompanying drawing and example, technical scheme is described in further details.

Embodiment 1: the present embodiment is for two-dimensional space, and the angle of fall provided in fore-and-aft plane constrains in the guidance example within the scope of-150 ° ~ 0 °.

Step 1, ignores earth rotation, sets up aircraft particle dynamics kinematical equation in two-dimensional space as shown in formula (19)-(22).

${\stackrel{\&CenterDot;}{x}}_m=V_m\cos {\mathrm{\&gamma;}}_m---\left(19\right)$

${\stackrel{\&CenterDot;}{y}}_m=V_m\sin {\mathrm{\&gamma;}}_m---\left(20\right)$

${\stackrel{\&CenterDot;}{V}}_m=-\frac{D_m}{m_m}-g\cos {\mathrm{\&gamma;}}_m---\left(21\right)$

${\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}_m=\frac{L_m}{m_m{V}_m}-\frac{g\cos {\mathrm{\&gamma;}}_m}{V_m}---\left(22\right)$

Step 2, carries out kinematic trajectory planning based on Bezier.

Current positional information and velocity reversal and terminal location and angle restriction determination aerial vehicle trajectory, respectively for fore-and-aft plane structure Bezier track, described track is required to be the Bezier on more than three rank.Preferably three rank Bezier structure Bezier tracks.

If given reference mark coordinate is followed successively by (x _a, y _a), (px _a, py _a), (px _b, py _b), (x _b, y _b), wherein for convenience of representing, by controlling polygon starting point (x _a, y _a) and (x _b, y _b) be denoted as end points, (px _a, py _a) and (px _b, py _b) be still denoted as reference mark.

Aircraft range x _m, height y _mconcrete form as shown in formula (9)-(10), polynomial parameters is determined by formula (12)-(13).

In the construction process of Bezier, in order to the selection of reduced parameter, define new variables Bezier parameter.Provide the choosing method of Bezier and parameter in xoy plane below.

1)-90°<γ _f<0°

$k_1=\frac{p{x}_A-x_A}{x_B-x_A},k_2=\frac{p{x}_B-x_A}{x_B-x_A}---\left(23\right)$

0≤k ₁≤k ₂≤1 (24)

When realizing following the trail of strike task, if desired trajectory inclination value is less, as shown in Fig. 1 (a), reference mark horizontal ordinate is all positioned at closed interval [x _a, x _b] on, in formula, inequality ensure that flatness and the accessibility of curve.

2)-150°<γ _f<-90°

Along with the change of the task of strike, for dealing task head-on blows, will realize larger impingement angle constraint at collision moment, the Parameters design provided in this up-to-date style (23)-(24) is no longer applicable, adopt following parameter selection method, as shown in figure (1) b:

$\begin{array}{c}{k}_1=\frac{p{x}_A-x_A}{p{x}_i-x_A},k_2=\frac{p{x}_B-x_A}{p{x}_B-x_A}\\ 0\&le;k_1,k_2\&le;1\end{array}---\left(25\right)$

Wherein: (px _i, py _i) be the intersecting point coordinate of Bezier starting point place's tangent line and End of Curve place tangent line.

$p{x}_i=\frac{y_A-y_B-x_A\tan {\mathrm{\&gamma;}}_0+x_B\tan {\mathrm{\&gamma;}}_f}{\tan {\mathrm{\&gamma;}}_f-\tan {\mathrm{\&gamma;}}_0}---\left(26\right)$

$p{y}_i=\frac{y_A\tan {\mathrm{\&gamma;}}_f-\tan {\mathrm{\&gamma;}}_0[x_A-y_B-x_A\tan {\mathrm{\&gamma;}}_0+x_B\tan {\mathrm{\&gamma;}}_f)]}{\tan {\mathrm{\&gamma;}}_f-\tan {\mathrm{\&gamma;}}_0}---\left(27\right)$

After Bezier parameter is determined, corresponding reference mark coordinate can be obtained:

$\left\{\begin{array}{c}p{x}_A=k_1(p{x}_i-x_A)+x_A\\ p{x}_B=k_2(x_B-p{x}_i)+p{x}_i\\ p{y}_A=\tan {\mathrm{\&gamma;}}_0(p{x}_A-x_A)+y_A\\ p{y}_B=\tan {\mathrm{\&gamma;}}_f(p{x}_B-x_B)+y_B\end{array}\right.---\left(28\right)$

When expecting that the angle of fall is-90 °, when continuing to adopt said method structure Bezier parameter, the reference mark coordinate that formula (26)-Shi (28) tries to achieve there will be unusual, by distortion, now Fig. 1 (b) is deformed into Fig. 1 (c), corresponding reference mark coordinate can be obtained as follows:

$\begin{array}{c}{k}_1=\frac{p{x}_A-x_A}{p{x}_i-x_A}\\ k_2=\frac{p{y}_B-p{y}_i}{y_B-p{y}_i}\\ 0\&le;k_1,k_2\&le;1\end{array}---\left(29\right)$

$\left\{\begin{array}{c}p{x}_A=k_1(p{x}_i-x_A)+x_A\\ p{x}_B=x_B\\ p{y}_A=\tan {\mathrm{\&gamma;}}_0(p{x}_A-x_A)+y_A\\ p{y}_B=k_2(y_B-p{y}_i)+p{y}_i\end{array}\right.---\left(30\right)$

Here it is noted that Bezier parametric configuration method that formula (25) provides is for-90 ° of < γ _fthe partial picture of <0 ° is applicable equally, but when expecting trajectory tilt angle γ _fsatisfy condition with angle of sight λ | γ _f| <| λ | time, the tangent line at Bezier end points place cannot intersect, and formula (25) cannot be suitable for.

Step 3, the aerial vehicle trajectory provided based on step 2 solves angle of attack _mguidance command.

Step 3.1 adopts inverse dynamics to solve angle of attack _mguidance command.

Because aircraft dive section flying height is generally monotone decreasing, rule of thumb, employing highly replaces t _go, building model as independent variable can Simplified analysis, more meets engineering practice simultaneously, decreases by t _goestimate the error introduced.Aircraft particle power and kinematical equation are expressed as with height y _mequation for independent variable:

$x_m^{\&prime;}=\frac{d{x}_m}{d{y}_m}=\cot {\mathrm{\&gamma;}}_m---\left(31\right)$

$V_m^{\&prime;}=\frac{d{V}_m}{d{y}_m}=-\frac{D_m+m_{m}g\sin {\mathrm{\&gamma;}}_m}{m_m{V}_m\sin {\mathrm{\&gamma;}}_m}---\left(32\right)$

${\mathrm{\&gamma;}}_m^{\&prime;}=\frac{d{\mathrm{\&gamma;}}_m}{d{y}_m}=\frac{a_y-g\cos {\mathrm{\&gamma;}}_m}{{V_m}^2\sin {\mathrm{\&gamma;}}_m}---\left(33\right)$

In formula: ' represent y _mdifferentiate, a _yrepresent longitudinal acceleration, they have following expression-form:

$a_y=\frac{L_m}{m}---\left(34\right)$

By formula (33) can its acceleration expression formula of reverse as follows:

$a_y=V_m^2\sin {\mathrm{\&gamma;}}_m{\mathrm{\&gamma;}}_m^{\&prime;}+g\cos {\mathrm{\&gamma;}}_m---\left(35\right)$

Namely acceleration instruction can by instant position, speed, angle and γ ' _mexpression obtains, and utilizes inverse dynamics theoretical, continues differentiate then can obtain γ ' to formula (31) _mexpression formula:

γ′ _m＝-sin ²γ _mx″ (36)

Convolution (35)-(36), aircraft guidance instruction and aerial vehicle trajectory are closely related, and therefore above-mentioned guidance problems are converted into trajectory planning problem, can solve obtain aircraft acceleration instruction a in conjunction with formula (18) _y, and then instead can inquire into obtain Aircraft Angle of Attack α _m.In formula (31)-(33) ' all represent height y _mdifferentiate, and Bezier expression formula is all expressed as the function of intermediate variable τ, utilizes compound function derivation law to carry out change and arranges:

$x^{\&prime;}=\frac{\mathrm{dx}/\mathrm{d\&tau;}}{\mathrm{dy}/\mathrm{d\&tau;}}---\left(37\right)$

$x^{\&prime;\&prime;}=\frac{d^{2}x/d{\mathrm{\&tau;}}^2-(\mathrm{dx}/\mathrm{d\&tau;})(d^{2}y/d{\mathrm{\&tau;}}^2)/\mathrm{dy}/\mathrm{d\&tau;}}{{(\mathrm{dy}/\mathrm{d\&tau;})}^2}---\left(38\right)$

Step 3.2, carries out Real-time Feedback to aircraft the present situation, repeatedly repeats step 2 and step 3.1 and realizes the track weight-normality carried out online based on Bezier model and draw, inquire into the instruction of subsequent time acceleration, until complete omnidistance guidance.Guarantee that when controlled quentity controlled variable is saturated aircraft realizes the precision strike met under impact angle constraint, and interference and environmental uncertainty has certain robustness to external world.

Easily occur this situation saturated for the acute variation of aircraft dive section environment and controlled quentity controlled variable, now, aircraft will depart from the flight path planned in advance, and only simple employing is followed the tracks of fixation locus and be difficult to ensure attack precision.Aircraft current flight state is introduced close-loop feedback, pass through Real-time Feedback, take current state as initial conditions, repetition step 2 and step 3.1 realize the track weight-normality carried out online based on Bezier model and draw, inquire into the instruction of subsequent time acceleration, bring aircraft power kinematics model into, obtain new state of flight, repeatedly carry out this process, when aircraft is drawn because external disturbance and controlled quentity controlled variable get the full passing track weight-normality, this Guidance Law can correction of deviation gradually, finally completes omnidistance guidance.

The present embodiment angle of fall provided in fore-and-aft plane constrains in the guidance example within the scope of-150 ° ~ 0 °.First provide and expect that the angle of fall is 0 ° for strike static target point ,-30 °, the simulation scenarios of-90 ° and-150 °, trajectory tilt angle initial value is-3 °.Fig. 5 gives simulation result, can be seen by the aerial vehicle trajectory provided and trajectory tilt angle change curve, the terminal guidance method that a kind of band angle of fall based on Bezier of the present embodiment retrains can realize the precision strike to target under meeting the angle of fall on a large scale and retraining.Simultaneously in the strike moment, aircraft has higher speed, improves damage effectiveness.The angle of attack variation curve provided by Fig. 5 (d) is found out, when there is larger turning trend in flight path, and angle of attack instruction α _moccurred obvious fluctuation, this also causes aircraft speed to have significantly decaying.

Fig. 6 gives and guides effect when having larger initial deviation.Initial trajectory inclination angle is chosen as 25 ° respectively, 10 ° ,-10 ° and-25 °.Desired trajectory angle set is-60 °.As can be seen from simulation result, even if there is larger initial deviation, the zero-miss guidance under aircraft can realize being with the angle of fall to retrain under the terminal guidance method that a kind of band angle of fall based on Bezier of the present embodiment retrains is guided.Can be seen by angle of attack variation curve, occur that controlled quentity controlled variable is saturated at the mission phase initial stage, the terminal guidance method that a kind of band angle of fall based on Bezier of the present embodiment retrains still can ensure the target-bound trend of aircraft, finally completes strike task smoothly.

Embodiment 2: the present embodiment is for two-dimensional space, the angle of fall provided in fore-and-aft plane constrains in the guidance example within the scope of-180 ° ~-150 °, verifies the validity of the terminal guidance method that a kind of band angle of fall based on Bezier of the present embodiment retrains under the large angle of fall with this.

Step 1 is with embodiment 1.

Step 2, carries out kinematic trajectory planning based on Bezier.

Aircraft range x _m, height y _mconcrete form as shown in formula (9)-(10), polynomial parameters is determined by formula (12)-(13).

When aircraft needs to deal head-on blows target with larger angle (-150 ° ~-180 °), employing saves the Guidance provided and there will be larger miss distance.From the geometric properties of three rank Beziers, in a flight path initial stage very long segment distance, the change of trajectory tilt angle is less, until just there is larger motor-driven turning close to during target, now there is larger fluctuation change in curve second derivative, namely demand of transshipping increases, and easily occurs that controlled quentity controlled variable is saturated.Once occur saturated, in shorter distance, even if adopt track weight-normality to draw also cannot ensure that aircraft can to expect angle of fall precision strike target.

Adopt two section of three rank Bezier splicing greatly can improve the dirigibility of curve planning.But along with the increase of curve quantity, the calculated amount also corresponding increase that parameter regulates, when the Bezier of employing two sections splicing carries out trajectory planning design, needs the parameter regulated to be increased to 7, comprising the Bezier parameter k of first paragraph curve ₁₁, k ₁₂, the terminal (x of first paragraph curve _mid, y _mid), slope K _midand the Bezier parameter k of second segment curve ₂₁, k ₂₂.Fig. 3 gives the geometric locus in two sections of Bezier splicing situations.

In order to ensure the smooth connection of two Beziers, require that curve single order is led even second order and led continuous parameters, curve is at intermediate point (x _mid, y _mid) place meets single order and lead continuous parameters condition, namely now first paragraph is consistent in intermediate point place tangential direction with second segment Bezier.

Provide intermediate point (x below _mid, y _mid) defining method.

When employing two sections of Beziers splicing tactful, intermediate point (x _mid, y _mid) coordinate and rate of curve K _mid, need to determine as tuning parameter, generally speaking, the selection of intermediate point will meet two conditions: condition 1 is the length as far as possible reducing first paragraph track, and condition 2 is for make whole-process control amount less as far as possible.Selection for centre does not have unique optimum solution, therefore in the process selected, is adjusted to principle with reduced parameter.

Below for the strike task under the constraint of the large angle of fall, provide Bezier intermediate point (x _mid, y _mid) and slope K _mida kind of building method:

1. intermediate point (x is first determined _mid, y _mid) coordinate.It chooses mode as shown in Figure 4, conveniently chooses, and make intermediate point and impact point be positioned on same ordinate, its horizontal ordinate selection range is in 40% ~ 60% of total flying height.

2. the rate of curve K at intermediate point place is determined _mid.Corresponding corner cut is chosen in 25% ~ 30% scope expecting the angle of fall.

Incorporation engineering practical experience, selects y here _mid=y ₀-50% △ y K _mid=γ ₀-tan (30% △ γ).

In formula: △ y=y ₀-y _b, △ γ=γ ₀-γ _f.

When Bezier angle at the end constraint satisfaction interval (-90 °, 0 °] time, Bezier parametric configuration method as formula (23), shown in formula (28).

When Bezier angle at the end constraint satisfaction interval [-180 ° ,-90 °) time, Bezier parametric configuration method is as shown in formula (25)-(28).

Step 3 is with embodiment 1.

Numerical simulation is carried out to the strike situation in two dimensional surface, chooses initial trajectory inclination angle and be-3 °.Simulation result as shown in Figure 7.Red-label point wherein in Fig. 7 (a) represents the splice point of two Beziers, can see the geometric properties due to Bezier, and this splice point is that air route must through point.As can be seen from simulation result, this Guidance can realize the aircraft guiding under the constraint of the large angle of fall, and reaches higher precision, and wherein trajectory tilt angle error is not more than 0.8 °, in error allowed band.At end of flight, overload instruction does not occur saturated.

Notice the terminal guidance method that a kind of band angle of fall based on Bezier of application the present embodiment retrains, at tie point place, track only meets the condition of continuity, and do not meet smoothness condition, i.e. Second Order Continuous condition, owing to comprising the second derivative x of aerial vehicle trajectory curve in controlled quentity controlled variable expression-form ", z ", saltus step can be there is in this brief acceleration instruction in junction, but general hopping amplitude is less, can obtains level and smooth preferably by adding second-order lag link, not affecting control system tracking effect.

Embodiment 3:

The present embodiment provides the guidance example in three dimensions, verifies the guidance effect of terminal guidance method in three dimensions that a kind of band angle of fall based on Bezier of the present embodiment retrains.

Step 1, ignores earth rotation, sets up aircraft particle dynamics kinematical equation as shown in formula (3)-(8).

Step 2, carries out kinematic trajectory planning based on Bezier.

The present invention adopts Bezier aerial vehicle trajectory to be drawn to the position and angle restriction that are planned to and realize aircraft end.Preferably three rank Bezier structure Bezier tracks.If given reference mark coordinate is followed successively by (x _a, y _a, z _a), (px _a, py _a, pz _a), (px _b, py _b, pz _b), (x _b, y _b, z _b), wherein for convenience of representing, by controlling polygon starting point (x _a, y _a, z _a) and (x _b, y _b, z _b) be denoted as end points, (px _a, py _a, pz _a) and (px _b, py _b, pz _b) be still denoted as reference mark.

Aircraft range x _m, height y _m, horizontal journey z _mconcrete form as shown in formula (9)-(11), polynomial parameters is determined by formula (12)-(14).

In the construction process of Bezier, in order to the selection of reduced parameter, Bezier parameter selection method is shown in embodiment 1.

After Bezier parameter is determined, corresponding reference mark coordinate can be obtained:

$p{z}_i=\frac{y_A-y_B-z_A\tan {\mathrm{\&chi;}}_0+z_B\tan {\mathrm{\&chi;}}_f}{\tan {\mathrm{\&chi;}}_f-\tan {\mathrm{\&chi;}}_0}---\left(39\right)$

$p{y}_i=\frac{y_A\tan {\mathrm{\&chi;}}_f-\tan {\mathrm{\&chi;}}_0[z_A-y_B-z_A\tan {\mathrm{\&chi;}}_0+z_B\tan {\mathrm{\&chi;}}_f)]}{\tan {\mathrm{\&chi;}}_f-\tan {\mathrm{\&chi;}}_0}---\left(40\right)$

$\left\{\begin{array}{c}p{x}_A=k_1(p{x}_i-x_A)+x_A\\ p{x}_B=k_2(x_B-p{x}_i)+p{x}_i\\ p{y}_A=\frac{\tan {\mathrm{\&gamma;}}_0}{\cos {\mathrm{\&chi;}}_0}(p{x}_A-x_A)+y_A\\ p{y}_B=\frac{\tan {\mathrm{\&gamma;}}_f}{\cos {\mathrm{\&chi;}}_f}(p{x}_B-x_B)+y_B\\ p{z}_A=\tan (-{\mathrm{\&chi;}}_0)(p{x}_A-x_A)+z_A\\ p{z}_B=\tan (-{\mathrm{\&chi;}}_f)(p{x}_B-x_B)+z_B\end{array}\right.---\left(41\right)$

Step 3, the aerial vehicle trajectory provided based on step 2 solves angle of attack _m, angle of heel μ _mguidance command.

Step 3.1 adopts inverse dynamics to solve angle of attack _m, angle of heel μ _mguidance command.

Because aircraft dive section flying height is generally monotone decreasing, rule of thumb, employing highly replaces t _go, building model as independent variable can Simplified analysis, more meets engineering practice simultaneously, decreases by t _goestimate the error introduced.Aircraft particle power and kinematical equation are expressed as with height y _mequation for independent variable:

$x_m^{\&prime;}=\frac{d{x}_m}{d{y}_m}=\cot {\mathrm{\&gamma;}}_m\cos {\mathrm{\&chi;}}_m---\left(42\right)$

$z_m^{\&prime;}=\frac{d{z}_m}{d{y}_m}=-\cot {\mathrm{\&gamma;}}_m\sin {\mathrm{\&chi;}}_m---\left(43\right)$

$V_m^{\&prime;}=\frac{d{V}_m}{d{y}_m}=-\frac{D_m+m_{m}g\sin {\mathrm{\&gamma;}}_m}{m_m{V}_m\sin {\mathrm{\&gamma;}}_m}---\left(44\right)$

${\mathrm{\&gamma;}}_m^{\&prime;}=\frac{d{\mathrm{\&gamma;}}_m}{d{y}_m}=\frac{a_y-g\cos {\mathrm{\&gamma;}}_m}{{V_m}^2\sin {\mathrm{\&gamma;}}_m}---\left(45\right)$

${\mathrm{\&chi;}}_m^{\&prime;}=\frac{d{\mathrm{\&chi;}}_m}{d{y}_m}=\frac{-a_z}{{V_m}^2\cos {\mathrm{\&gamma;}}_m\sin {\mathrm{\&gamma;}}_m}---\left(46\right)$

In formula: ' represent y _mdifferentiate, a _y, a _zrepresent acceleration that is longitudinal and normal direction, they have following expression-form:

$a_y=\frac{L_m\cos {\mathrm{\&mu;}}_m}{m},a_z=\frac{L_m\sin {\mathrm{\&mu;}}_m}{m_m}---\left(47\right)$

By formula (45)-(47) can its acceleration expression formula of reverse as follows:

$a_y=V_m^2\sin {\mathrm{\&gamma;}}_m{\mathrm{\&gamma;}}_m^{\&prime;}+g\cos {\mathrm{\&gamma;}}_m---\left(48\right)$

$a_z={-V}_m^2\sin {\mathrm{\&gamma;}}_m\cos {\mathrm{\&gamma;}}_m{\mathrm{\&chi;}}_m^{\&prime;}---\left(49\right)$

Namely acceleration instruction can by instant position, speed, angle and γ ' _m, χ ' _mexpression obtains, and utilizes inverse dynamics theoretical, continues differentiate then can obtain γ ' to formula (42)-(43) _m, χ ' _mexpression formula:

γ′ _m＝-sin ²γ _m(cosχ _mx″-sinχ _mz″)

(50)

χ′ _m＝-tanγ _m(sinχ _mx″ _m+cosχ _mz″ _m)

In conjunction with formula (50), formula (18) instead can inquire into obtain Aircraft Angle of Attack α _mwith angle of heel μ _minstruction.

In formula (42)-(46) ' all represent height y _mdifferentiate, and Bezier expression formula is all expressed as the function of intermediate variable τ, utilizes compound function derivation law to carry out change and arranges:

$x^{\&prime;}=\frac{\mathrm{dx}/\mathrm{d\&tau;}}{\mathrm{dy}/\mathrm{d\&tau;}}---\left(51\right)$

$z^{\&prime;}=\frac{\mathrm{dz}/\mathrm{d\&tau;}}{\mathrm{dy}/\mathrm{d\&tau;}}---\left(52\right)$

$x^{\&prime;\&prime;}=\frac{d^{2}x/d{\mathrm{\&tau;}}^2-(\mathrm{dx}/\mathrm{d\&tau;})(d^{2}y/d{\mathrm{\&tau;}}^2)/\mathrm{dy}/\mathrm{d\&tau;}}{{(\mathrm{dy}/\mathrm{d\&tau;})}^2}---\left(53\right)$

$z^{\&prime;\&prime;}=\frac{d^{2}z/d{\mathrm{\&tau;}}^2-(\mathrm{dz}/\mathrm{d\&tau;})(d^{2}y/d{\mathrm{\&tau;}}^2)/\mathrm{dy}/\mathrm{d\&tau;}}{{(\mathrm{dy}/\mathrm{d\&tau;})}^2}---\left(54\right)$

Step 3.2, carries out Real-time Feedback to aircraft the present situation, repeatedly repeats step 2 and step 3.1 and realizes the track weight-normality carried out online based on Bezier model and draw, inquire into the instruction of subsequent time acceleration, until complete omnidistance guidance.Guarantee that when controlled quentity controlled variable is saturated aircraft realizes the precision strike met under impact angle constraint, and interference and environmental uncertainty has certain robustness to external world.

First, when given starting condition, the guiding result under difference expects angle of fall constraint is given; Then by simulating, verifying under same end conswtraint, the terminal guidance method that a kind of band angle of fall based on Bezier of the present embodiment retrains has the guidance performance in larger initial deviation situation.

Fig. 8 (a)-Fig. 8 (e) gives for the simulation result under the constraint of the different angle of falls in three dimensions, and Fig. 8 (a)-Fig. 8 (e) sets forth flight path curve, trajectory tilt angle γ _mchange curve, trajectory deflection angle χ _mchange curve, angle of attack _mchange curve, angle of heel μ _mchange curve.

Fig. 8 (a)-Fig. 8 (e) gives for the simulation result under different initial angle constraint in three dimensions, and Fig. 9 (a)-Fig. 9 (e) sets forth flight path curve, trajectory tilt angle γ _mchange curve, trajectory deflection angle χ _mchange curve, angle of attack _mchange curve, angle of heel μ _mchange curve.

By Fig. 8 (a)-Fig. 8 (e), the simulation result of Fig. 9 (a)-Fig. 9 (e) can be found out, even if when having larger initial angle deviation, the terminal guidance method that a kind of band angle of fall based on Bezier of the present embodiment retrains still has good guiding performance in three-dimensional strike situation, the strike task under angle of fall constraint on a large scale can be realized when ensureing less miss distance simultaneously, trajectory tilt angle and the trajectory deflection angle Error Absolute Value of collision moment are no more than 0.8 °, in error allowed band.The angle of attack and angle of heel instruction smoothly, are easy to rudder surface control system and follow the tracks of.

In sum, the terminal guidance method that a kind of band angle of fall based on Bezier disclosed by the invention retrains, when complex environment and controlled quentity controlled variable saturated can precision strike under two dimensional surface and three dimensions realize retraining the fixed ground target band angle of fall, there is very high engineer applied and be worth.

Scope is not only confined to three embodiments that the present invention provides, embodiment for explaining the present invention, all changes with the present invention under same principle and design condition or revise all within protection domain disclosed by the invention.

Claims (5)

1., based on the terminal guidance method that the band angle of fall of Bezier retrains, it is characterized in that: comprise the steps,

Step 1, ignores earth rotation, sets up aircraft particle dynamics kinematical equation,

${\stackrel{\&CenterDot;}{x}}_m=V_m\cos {\mathrm{\&gamma;}}_m\cos {\mathrm{\&chi;}}_m---\left(1\right)$

${\stackrel{\&CenterDot;}{y}}_m=V_m\sin {\mathrm{\&gamma;}}_m---\left(2\right)$

${\stackrel{\&CenterDot;}{z}}_m=-V_m\cos {\mathrm{\&gamma;}}_m\sin {\mathrm{\&chi;}}_m---\left(3\right)$

${\stackrel{\&CenterDot;}{V}}_m=-\frac{D_m}{m_m}-g\cos {\mathrm{\&gamma;}}_m---\left(4\right)$

${\stackrel{\&CenterDot;}{\mathrm{\&gamma;}}}_m=\frac{L_m\cos {\mathrm{\&mu;}}_m}{m_m{V}_m}-\frac{g\cos {\mathrm{\&gamma;}}_m}{V_m}---\left(5\right)$

${\stackrel{\&CenterDot;}{\mathrm{\&chi;}}}_m=-\frac{L_m\sin {\mathrm{\&mu;}}_m}{m_m{V}_m\cos {\mathrm{\&gamma;}}_m}---\left(6\right)$

Wherein, x _m, y _m, z _mfor the position coordinates of aircraft under inertial system; V _m, γ _m, χ _mbe respectively speed, trajectory tilt angle, trajectory deflection angle; G is acceleration of gravity; μ _mfor angle of heel; L _m, D _mbe respectively lift and resistance, wherein, $L_m=\hat{q}{S}_{\mathrm{ref}}{C}_{\mathrm{Lm}},D_m=\hat{q}{\hat{S}}_{\mathrm{ref}}{C}_{\mathrm{Dm}},q=0.5\mathrm{\&rho;}{V}_m^2;$ S _reffor the area of reference of aircraft; ρ is atmospheric density; C _lm, C _dmbe respectively lift coefficient and resistance coefficient, lift coefficient C _lm, resistance coefficient C _dmabout angle of attack _mwith the function of Mach number Ma;

Step 2, carries out kinematic trajectory planning based on Bezier;

Current positional information and velocity reversal and terminal location and angle restriction determination aerial vehicle trajectory, structure Bezier track, described track is required to be the Bezier on more than three rank, preferably three rank Beziers structure Bezier tracks;

If given reference mark coordinate is followed successively by (x _a, y _a, z _a), (px _a, py _a, pz _a), (px _b, py _b, pz _b), (x _b, y _b, z _b), three rank bezier curve equations of aircraft coordinate can be obtained:

For convenience of representing, by controlling polygon starting point (x _a, y _a, z _a) and (x _b, y _b, z _b) be denoted as end points, (px _a, py _a, pz _a) and (px _b, py _b, pz _b) be still denoted as reference mark;

x _m＝a _xτ ³+b _xτ ²+c _xτ+d _x(7)

y _m＝a _yτ ³+b _yτ ²+c _yτ+d _y(8)

z _m＝a _zτ ³+b _zτ ²+c _zτ+d _z(9)

Wherein: τ ∈ [0,1] is intermediate variable, and the multinomial coefficient in formula (7)-(9) is determined by following formula (10)-(12):

d _x＝x _A

c _x＝3(px _A-x _A)

(10)

b _x＝3(px _B-px _A)-c _x

a _x＝x _B-x _A-c _x-b _x

d _y＝y _A

c _y＝3(py _A-y _A)

(11)

b _y＝3(py _B-py _A)-c _y

a _y＝y _B-y _A-c _y-b _y

d _z＝z _A

c _z＝3(pz _A-z _A)

(12)

b _z＝3(pz _B-pz _A)-c _z

a _z＝z _B-z _A-c _z-b _z

Step 3, the aerial vehicle trajectory provided based on step 2 solves angle of attack _m, angle of heel μ _mguidance command, and Real-time Feedback is carried out to aircraft the present situation, realize the track weight-normality carried out online based on Bezier model and draw, inquire into the instruction of subsequent time acceleration, until complete omnidistance guidance.

2. the terminal guidance method that retrains of a kind of band angle of fall based on Bezier as claimed in claim 1, is characterized in that: step 3 implementation method comprises step 3.1,3.2,

Step 3.1 adopts inverse dynamics to solve angle of attack _m, angle of heel μ _mguidance command;

After by Bezier determination aircraft movements track, utilize inverse dynamics theory to solve and guidance command, its normal acceleration a _ywith longitudinal acceleration a _zbe defined as: there is following expression-form:

$a_y=V_m^2\sin {\mathrm{\&gamma;}}_m{\mathrm{\&gamma;}}_m^{\&prime;}+g\cos {\mathrm{\&gamma;}}_m---\left(13\right)$

$a_z=-V_m^2\sin {\mathrm{\&gamma;}}_m\cos {\mathrm{\&gamma;}}_m{\mathrm{\&chi;}}_m^{\&prime;}---\left(14\right)$

Wherein, ' represent y differentiate, γ ' _m, χ ' _mthere is following expression-form:

γ′ _m＝-sin ²γ _m(cosχ _mx″-sinχ _mz″)

(15)

χ′ _m＝-tanγ _m(sinχ _mx″ _m+cosχ _mz″ _m)

In conjunction with above-mentioned inverse dynamics analysis, aircraft acceleration instruction is not only relevant to aircraft current flight state, simultaneously also relevant to the shape of flight path, namely with range x _mwith horizontal journey z _mabout height y _mfirst order derivative is relevant to second derivative, and therefore above-mentioned guidance problems is converted into trajectory planning problem in fact;

Try to achieve aircraft acceleration a _ywith a _zafter instruction, aircraft angle of heel μ _mdetermined by following formula (16):

${\mathrm{\&mu;}}_m=\arctan \frac{a_z}{a_y}---\left(16\right)$

Defined by formula (16) and aircraft acceleration and can inquire into aircraft lift L _mwith lift coefficient C _lm, and then anti-release Aircraft Angle of Attack α _minstruction;

Step 3.2, Real-time Feedback is carried out to aircraft the present situation, repeatedly repeat step 2 and step 3.1 to realize the track weight-normality carried out online based on Bezier model and draw, inquire into the instruction of subsequent time acceleration, until complete omnidistance guidance, guarantee that when controlled quentity controlled variable is saturated aircraft realizes the precision strike met under impact angle constraint, and interference and environmental uncertainty has certain robustness to external world.

3. the terminal guidance method that retrains of a kind of band angle of fall based on Bezier as claimed in claim 1 or 2, is characterized in that: described Bezier has following characteristic,

Character 1: starting point, the terminal of the starting point of Bezier, terminal and corresponding characteristic polygon overlap;

Character 2: Bezier starting point is consistent with the trend on characteristic polygon Article 1 limit and the last item limit with the tangential direction of destination county;

Character 3: the point on Bezier all drops on by its reference mark P _iamong the convex closure formed.

4. the terminal guidance method that retrains of a kind of band angle of fall based on Bezier as claimed in claim 3, it is characterized in that: in order to adapt to the guidance requirement within the scope of-180 ° ~-150 °, to the terminal guidance method adopting the band angle of fall of Bezier splicing to retrain, need comprise the steps:

Step 1 is with claim 1;

Step 2, carries out kinematic trajectory planning based on Bezier;

Aircraft range x _m, height y _mconcrete form as shown in formula (9)-(10), polynomial parameters is determined by formula (12)-(13);

Adopt two section of three rank Bezier splicing can improve the dirigibility of curve planning; But along with the increase of curve quantity, the calculated amount also corresponding increase that parameter regulates, when the Bezier of employing two sections splicing carries out trajectory planning, needs the parameter regulated to be increased to 7, comprising the Bezier parameter k of first paragraph curve ₁₁, k ₁₂, the terminal (x of first paragraph curve _mid, y _mid), slope K _midand the Bezier parameter k of second segment curve ₂₁, k ₂₂;

In order to ensure the smooth connection of two Beziers, require that curve single order is led even second order and led continuous parameters, curve is at intermediate point (x _mid, y _mid) place meets single order and lead continuous parameters condition, namely now first paragraph is consistent in intermediate point place tangential direction with second segment Bezier;

Provide intermediate point (x _mid, y _mid) defining method; Intermediate point (x _mid, y _mid) coordinate and rate of curve K _mid, need to determine as tuning parameter, the selection of intermediate point will meet two conditions: condition 1 is the length as far as possible reducing first paragraph track, and condition 2 is for make whole-process control amount less as far as possible;

Step 3 is with claim 1.

5. the terminal guidance method that retrains of a kind of band angle of fall based on Bezier as claimed in claim 4, is characterized in that:

For the strike task under the constraint of the large angle of fall, provide Bezier intermediate point (x _mid, y _mid) and slope K _mida kind of building method,

1. intermediate point (x is first determined _mid, y _mid) coordinate; Conveniently choose, make intermediate point and impact point be positioned on same ordinate, its horizontal ordinate selection range is in 40% ~ 60% of total flying height;

2. the rate of curve K at intermediate point place is determined _mid; Corresponding corner cut is chosen in 25% ~ 30% scope expecting the angle of fall;

Incorporation engineering practical experience, selects y _mid=y ₀-50% △ y K _mid=γ ₀-tan (30% △ γ);

In formula: △ y=y ₀-y _b, △ γ=γ ₀-γ _f;

When Bezier angle at the end constraint satisfaction interval (-90 °, 0 °] time, Bezier parametric configuration method as formula (17), shown in formula (22);

When Bezier angle at the end constraint satisfaction interval [-180 ° ,-90 °) time, Bezier parametric configuration method is as shown in formula (19)-(22);

$k_1=\frac{p{x}_A-x_A}{x_B-x_A},k_2=\frac{p{x}_B-x_A}{x_B-x_A}---\left(17\right)$

0≤k ₁≤k ₂≤1 (18)

$k_1=\frac{p{x}_A-x_A}{p{x}_i-x_A},k_2=\frac{p{x}_B-x_A}{p{x}_B-x_A}---\left(19\right)$

0≤k ₁,k ₂≤1

$p{x}_i=\frac{y_A-y_B-x_A\tan {\mathrm{\&gamma;}}_0+x_B\tan {\mathrm{\&gamma;}}_f}{\tan {\mathrm{\&gamma;}}_f-\tan {\mathrm{\&gamma;}}_0}---\left(20\right)$

$p{y}_i=\frac{y_A\tan {\mathrm{\&gamma;}}_f-\tan {\mathrm{\&gamma;}}_0[x_A-y_B-x_A\tan {\mathrm{\&gamma;}}_0+x_B\tan {\mathrm{\&gamma;}}_f)]}{\tan {\mathrm{\&gamma;}}_f-\tan {\mathrm{\&gamma;}}_0}---\left(21\right)$

$\left\{\begin{array}{c}p{x}_A=k_1(p{x}_i-x_A)+x_A\\ p{x}_B=k_2(x_B-p{x}_i)+p{x}_i\\ p{y}_A=\tan {\mathrm{\&gamma;}}_0(p{x}_A-x_A)+y_A\\ p{y}_B=\tan {\mathrm{\&gamma;}}_f(p{x}_B-x_B)+y_B\end{array}\right.---\left(2\right).$