CN102519473B - Mixed sine maneuvering path guiding method for high-paddle fundamental frequency satellite - Google Patents

Abstract

The invention relates to a mixed sine maneuvering path guiding method for a high-paddle fundamental frequency satellite. The mixed sine maneuvering path guiding method for a high-paddle fundamental frequency satellite is characterized in that based on reasonable distribution of maneuvering time and stabilization time, according to a moment and an angular momentum capacity of an execution mechanism, an angular acceleration curve is designed, wherein the angular acceleration curve shows an angular acceleration change rule that an angular acceleration is changed into zero from a positive value, then is changed into a negative value from zero and then is changed into zero from the negative value; and one half of a whole period of a sine curve mixed step function is a positive value curve part and the other half is a negative value curve part. The mixed sine maneuvering path guiding method for a high-paddle fundamental frequency satellite retains the advantage of small excitation of a sine maneuvering path on a flexible paddle, combines the advantage of time optimality of a step function, harmonizes the contradiction between maneuvering rapidity and paddle excitation smoothness, is especially suitable for providing path program for a large-angle, rapid and high-paddle fundamental frequency satellite in imaging breadth increasing, instant observation on a emergency area or stereo imaging rapid maneuvering, and satisfies requirements of rapid and stable maneuvering performances of a high-paddle fundamental frequency satellite.

Description

A kind of sinusoidal motor-driven path guide method of mixing that is suitable for the higher satellite of windsurfing fundamental frequency

Technical field

The invention belongs to Spacecraft Attitude Control field, the path guide method while relating to a kind of flexible satellite fast reserve.

Background technology

Aspect satellite fast reserve, in order to increase imaging fabric width, when accident area is realized instant observation or carried out three-dimensional imaging, require regular side-sway automotive and pitching motor-driven.When the motor-driven angle of satellite is large, stabilization time is short, Stability index is when high, the windsurfing flexible vibration that mobile process evokes will become the principal element of restriction mobility.For the lower satellite of windsurfing fundamental frequency, sinusoidal motor-driven path can effectively reduce the excitation of large angle maneuver to flexible appendage, obtains higher stability.Although sinusoidal motor-driven path is compared with time optimal Bang-Bang path, required motor-driven arrival time will be grown, but for the lower satellite of windsurfing fundamental frequency, delaying on the time kept in reserve can be contained in the outstanding benefit obtaining aspect the vibration of reduction windsurfing in sinusoidal motor-driven path.But, for the higher situation faster of motor-driven arrival time that requires of fundamental frequency simultaneously, adopt effect that sinusoidal motor-driven path obtains aspect less in windsurfing vibration completely and sacrifice compared with motor-driven arrival time, not obvious to the contribution of fast reserve performance.Therefore need to for fundamental frequency higher require simultaneously motor-driven arrival time faster situation find new solution.

Summary of the invention

Technology of the present invention is dealt with problems and is: overcome the deficiencies in the prior art, what a kind of fast response time, good stability were provided mixes the motor-driven path guide method of satellite of stepped curve based on sine.

Technical solution of the present invention is: a kind of sinusoidal motor-driven path guide method of mixing that is suitable for the higher satellite of windsurfing fundamental frequency, and step is as follows:

(1) determine time kept in reserve t according to the motor-driven and stable time index of the attitude of satellite _max;

(2) determine maximum motor-driven angular acceleration a according to the topworks's ability configuring on satellite _max;

(3) determine maximum motor-driven limited angular velocity omega according to the topworks's angular momentum capacity configuring on satellite and sensor range _max;

(4) the path motor-driven attitude of satellite is divided into the time for [t ₁t ₂] the rate of acceleration section of increasing progressively, time be [t ₂t ₃] even accelerating sections, time be [t ₃t ₄] rate of acceleration decline fraction, time be [t ₄t ₅] at the uniform velocity coasting-flight phase, time be [t ₅t ₆] the rate of deceleration section of increasing progressively, time be [t ₆t ₇] even braking section, time be [t ₇t ₈] rate of deceleration decline fraction, wherein t ₁for the motor-driven start time of satellite,

t ₃=t ₂+ Δ T, t ₅=t ₄+ Δ t,

t ₇=t ₆+ Δ T,

t is the cycle of sine function, and the initial value of T is

Δ T is the even acceleration time, and the initial value of Δ T is

Δ t is coasting time, and the initial value of Δ t is $\mathrm{\Δt}=\frac{{2\mathrm{\π\ω}}_{\max }}{(\mathrm{\π}-2)a_{\max }}+\frac{4\left|{\mathrm{\θ}}_d\right|}{(\mathrm{\π}-2){\mathrm{\ω}}_{\max }}-\frac{6-\mathrm{\π}}{\mathrm{\π}-2}{t}_{\max };$

(5) be a according to the cycle T of sine function, maximum motor-driven angular acceleration _maxand even acceleration time Δ T, the coasting time Δ t of satellite calculate respectively the attitude angle acceleration a of satellite based on sine mixing step function in each section, and corresponding attitude angular velocity ω and attitude angle θ, wherein:

(51) the rate of acceleration section of increasing progressively, $a=a_{\max }\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1)\right),$ $\mathrm{\&omega;}=\frac{a_{\max }T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1)\right)),$ wherein θ _dfor motor-driven angle; If the Δ T < 0 obtaining according to calculation of initial value, upgrades

Δ T=0,

if the T+2 Δ T < T obtaining according to calculation of initial value _min, upgrade T=T _min, renewal maximum angular acceleration is

Δ T=0, Δ t=0, T _minby the shortest patient time kept in reserve;

(52) even accelerating sections, a=a _max, $\mathrm{\&omega;}=\frac{a_{\max }T}{2\mathrm{\&pi;}}+a_{\max }(t-t_2),$

$\mathrm{\&theta;}=\frac{a_{\max }{T}^2(\mathrm{\&pi;}-2)}{8{\mathrm{\&pi;}}^2}+\frac{a_{\max }T(t-t_2)}{2\mathrm{\&pi;}}+\frac{a_{\max }{(t-t_2)}^2}{2},$ Wherein the value of T, Δ T, Δ t is with the rate of acceleration section of increasing progressively;

(53) rate of acceleration decline fraction, $a=a_{\max }\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_3+\frac{T}{4})\right),$

$\mathrm{\&omega;}=a_{\max }\mathrm{\&Delta;T}+\frac{a_{\max }T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_3+\frac{T}{4})\right)),$ $\mathrm{\&theta;}\left(t_4\right)=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+\frac{a_{\max }\mathrm{\&Delta;TT}}{4}+\frac{a_{\max }\mathrm{T\&Delta;T}}{2\mathrm{\&pi;}}+\frac{a_{\max }{T}^2}{4\mathrm{\&pi;}},$

Wherein the value of T, Δ T, Δ t is with the rate of acceleration section of increasing progressively;

(54) coasting-flight phase at the uniform velocity, a=0, $\mathrm{\&omega;}=\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T},$

$\mathrm{\&theta;}=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+a_{\max }\mathrm{T\&Delta;T}(\frac{1}{4}+\frac{1}{2\mathrm{\&pi;}})+\frac{a_{\max }{T}^2}{4\mathrm{\&pi;}}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})(t-t_4),$ Wherein the value of T, Δ T, Δ t is with the rate of acceleration section of increasing progressively;

(55) rate of deceleration section of increasing progressively, $a=a_{\max }\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-\mathrm{\&Delta;T})\right),$

$\mathrm{\&omega;}=a_{\max }\mathrm{\&Delta;T}+\frac{a_{\max }T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-\mathrm{\&Delta;T})\right)),$

$\mathrm{\&theta;}=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+a_{\max }\mathrm{T\&Delta;T}(\frac{1}{4}+\frac{1}{2\mathrm{\&pi;}})+\frac{a_{\max }{T}^2}{4\mathrm{\&pi;}}+\frac{a_{\max }T}{\mathrm{\&pi;}}\mathrm{\&Delta;t}+a_{\max }\mathrm{\&Delta;T}(t-t_5+\mathrm{\&Delta;t})$

$+\frac{a_{\max }T}{2\mathrm{\&pi;}}(t-t_5)-\frac{a_{\max }{T}^2}{4{\mathrm{\&pi;}}^2}\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-\mathrm{\&Delta;T})\right),$

If the Δ t < 0 going out according to initial calculation, upgrades

Δ t=0; If the Δ T < 0 obtaining according to calculation of initial value, upgrades

Δ T=0, $\mathrm{\&Delta;t}=\frac{\mathrm{\&pi;}}{a_{\max }T}\left(\right|{\mathrm{\&theta;}}_d|-\frac{\left|a_{\max }\right|T^2}{2\mathrm{\&pi;}});$ If $\mathrm{\&Delta;t}=\frac{\mathrm{\&pi;}}{a_{\max }T}\left(\right|{\mathrm{\&theta;}}_d|-\frac{\left|a_{\max }\right|T^2}{2\mathrm{\&pi;}})<0,$ Upgrade $T=\sqrt{2\mathrm{\&pi;}\left|\frac{{\mathrm{\&theta;}}_d}{a_{\max }}\right|},$ ΔT＝0，Δt＝0；

(56) even braking section, a=-a _max, $\mathrm{\&omega;}=\frac{a_{\max }T}{2\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T}-a_{\max }(t-t_6),$

$\begin{array}{c}\mathrm{\&theta;}=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+\frac{a_{\max }\mathrm{\&Delta;TT}(\mathrm{\&pi;}+1)}{2\mathrm{\&pi;}}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})\mathrm{\&Delta;t}+\frac{a_{\max }{T}^2(3\mathrm{\&pi;}+2)}{{8\mathrm{\&pi;}}^2}\\ +(\frac{a_{\max }T}{2\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})(t-t_6)-\frac{a_{\max }{(t-t_6)}^2}{2}\end{array},$ Wherein the value of T, Δ T, Δ t is with the rate of deceleration section of increasing progressively;

(57) rate of deceleration decline fraction, $a=a_{\max }\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-2\mathrm{\&Delta;T})\right),$

$\mathrm{\&omega;}=\frac{a_{\max }\&CenterDot;T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-2\mathrm{\&Delta;T})\right)),$

$\mathrm{\&theta;}=\frac{a_{\max }\mathrm{\&Delta;TT}(\mathrm{\&pi;}+2)}{2\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T\&Delta;t}+\frac{{3a}_{\max }{T}^2}{8\mathrm{\&pi;}}+a_{\max }{\mathrm{\&Delta;T}}^2$

$+\frac{a_{\max }T}{2\mathrm{\&pi;}}(t-t_7+2\mathrm{\&Delta;t})-\frac{a_{\max }{T}^2}{{4\mathrm{\&pi;}}^2}\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-2\mathrm{\&Delta;T})\right),$

Wherein the value of T, Δ T, Δ t is with the rate of deceleration section of increasing progressively;

(6) the described path of tracking step (5) when the motor-driven control of satellite.

The present invention's advantage is compared with prior art: with fundamental frequency compared with the fast reserve process of the complicated satellite of high flexible annex in, adopt the sine of the inventive method to mix the motor-driven path of step, on the basis of reasonable distribution time kept in reserve and stabilization time, according to the moment of topworks and angular momentum capacity, design be just afterwards first zero again for negative be zero angular acceleration curve again, the part that value is positive and negative is respectively the sinusoidal curve mixing step function curve of half period.The inventive method, compared with traditional bang-bang path and standard sine path, had both weakened the impact of flexible appendage on satellite degree of stability under large angle maneuver, had shortened again the time kept in reserve, thereby had ensured the fast and stable time after fast reserve.

Accompanying drawing explanation

Fig. 1 is the FB(flow block) of the inventive method;

Fig. 2 is the motor-driven path that adopts the inventive method to obtain;

Fig. 3 is the motor-driven path of bang-bang satellite axis of rolling attitude angular velocity curve;

Fig. 4 is the motor-driven path of standard sine satellite axis of rolling attitude angular velocity curve;

Fig. 5 is the satellite axis of rolling attitude angular velocity curve that adopts the inventive method to obtain;

Fig. 6 is satellite sailboard flexible mode coordinate displacement under the motor-driven path of bang-bang;

Fig. 7 is satellite sailboard flexible mode coordinate displacement under sinusoidal motor-driven path;

Fig. 8 is the satellite sailboard flexible mode coordinate displacement that adopts the inventive method to obtain.

Embodiment

Current surveillance satellite has For Large Angle Rapid Maneuvering and stable demand, and motor-driven angle is large, and stabilization time is short, and stability index is high, and satellite is in order to meet fast reserve ability, conventionally improves windsurfing height, and fundamental frequency is higher.Therefore, need to adopt to guarantee time kept in reserve rapidity, can guarantee again the motor-driven path of less excitation windsurfing flexible vibration.The inventive method has retained sinusoidal motor-driven path to the less advantage of flexible windsurfing excitation, combine again the time optimal feature of step function simultaneously, in slowing down the motor-driven jigger panel vibration that puts in place, optimize the time kept in reserve as far as possible, coordinate motor-driven rapidity and windsurfing and encouraged the contradiction between mild property, be particularly suitable for wide-angle, quick, the motor-driven path design of the higher satellite of windsurfing fundamental frequency.

As shown in Figure 1, be the process flow diagram of the inventive method.

In the present invention, angular acceleration adopts the sinusoidal step function form of mixing, and becomes accelerating sections even accelerating sections a=a _max, wherein a is angular acceleration, a _maxfor the motor-driven angular acceleration of maximum, be limited to topworks's ability and sensor range; T is the planning acceleration cycle; The even acceleration time is Δ T.Therefore, are acceleration time and deceleration time

consider time kept in reserve constraint, suppose that the maximum time kept in reserve is t _max, the situation that large angle maneuver and maximum angular rate are limited, supposes that limited maximum angular velocity is ω _max, the time of at the uniform velocity sliding between accelerating sections and braking section is Δ t.

Attitude angular velocity after finishing according to accelerating sections

known

determine the funtcional relationship between trigonometric function cycle T and even acceleration time Δ T according to this inequality,

the maximum time kept in reserve is t _max, known time kept in reserve T+2 Δ T+ Δ t≤t _max.Determine the funtcional relationship between trigonometric function cycle T, even acceleration time Δ T, coasting time Δ t, i.e. T+2 Δ T+ Δ t=t according to this inequality _max.

If T+2 Δ T>=T _min, can be further motor-driven angle θ as required _ddetermine coasting time Δ t, $\mathrm{\&Delta;t}=\frac{{2\mathrm{\&pi;\&omega;}}_{\max }}{(\mathrm{\&pi;}-2)a_{\max }}+\frac{4\left|{\mathrm{\&theta;}}_d\right|}{(\mathrm{\&pi;}-2){\mathrm{\&omega;}}_{\max }}-\frac{6-\mathrm{\&pi;}}{\mathrm{\&pi;}-2}{t}_{\max },$ T in formula _minfor the minimum period of setting according to the time kept in reserve, θ _dfor motor-driven angle, and then calculate $T=\frac{2\mathrm{\&pi;}(t_{\max }-\frac{{\mathrm{\&omega;}}_{\max }}{a_{\max }}-\frac{\left|{\mathrm{\&theta;}}_d\right|}{{\mathrm{\&omega;}}_{\max }})}{\mathrm{\&pi;}-2},$ $\mathrm{\&Delta;T}=\frac{{\mathrm{\&omega;}}_{\max }}{a_{\max }}-\frac{T}{\mathrm{\&pi;}};$ If the coasting time Δ t < 0 calculating is obviously inadvisable, make Δ t=0, according to time function relation

and the motor-driven angle that puts in place $\mathrm{\&theta;}\left(t_8\right)=\frac{a_{\max }\mathrm{\&Delta;TT}}{2}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})\mathrm{\&Delta;t}+\frac{a_{\max }{T}^2}{2\mathrm{\&pi;}}+\frac{a_{\max }\mathrm{T\&Delta;T}}{\mathrm{\&pi;}}+a_{\max }{\mathrm{\&Delta;T}}^2,$ Obtain with Δ t=0 homographic solution

if T+2 Δ T < is T _min, show maximum motor-driven angular acceleration a _maxshould be less compared with the angular acceleration under topworks's ability and sensor limit of range, therefore upgrade a _max, make T=T _min, Δ T=0, Δ t=0, calculates

The derivation in motor-driven path is as follows:

(1) the rate of acceleration section of increasing progressively: [t ₁t ₂], t ₁for the motor-driven start time,

by angular acceleration a at time period [t ₁t] upper to time integral, can obtain as calculated

equally to angular velocity omega at time period [t ₁t] upper to time integral, can obtain at time t ₁and t ₂point has respectively, a (t ₁)=0, a (t ₂)=a _max, ω (t ₁)=0, $\mathrm{\&omega;}\left(t_2\right)=\frac{a_{\max }T}{2\mathrm{\&pi;}},$ θ(t ₁)＝0， $\mathrm{\&theta;}\left(t_2\right)=\frac{a_{\max }{T}^2(\mathrm{\&pi;}-2)}{{8\mathrm{\&pi;}}^2}.$

(2) even accelerating sections: [t ₂t ₃], t ₃=t ₂+ Δ T; A=a _max; Warp is at time period [t ₂t] upper angular acceleration a carries out respectively integration and quadratic integral, can obtain attitude angular velocity and attitude angle is respectively $\mathrm{\&omega;}=\frac{a_{\max }T}{2\mathrm{\&pi;}}+a_{\max }(t-t_2);$ $\mathrm{\&theta;}=\frac{a_{\max }{T}^2(\mathrm{\&pi;}-2)}{{8\mathrm{\&pi;}}^2}+\frac{a_{\max }T(t-t_2)}{2\mathrm{\&pi;}}+\frac{a_{\max }{(t-t_2)}^2}{2},$ At time t ₃point, $\mathrm{\&omega;}\left(t_3\right)=\frac{a_{\max }T}{2\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T},$ $\mathrm{\&theta;}\left(t_3\right)=\frac{a_{\max }\&CenterDot;T^2(\mathrm{\&pi;}-2)}{{8\mathrm{\&pi;}}^2}+\frac{a_{\max }\mathrm{T\&Delta;T}}{2\mathrm{\&pi;}}+\frac{a_{\max }\&CenterDot;{\mathrm{\&Delta;T}}^2}{2}.$

(3) rate of acceleration decline fraction: [t ₄t ₄], $t_4=t_3+\frac{T}{4};$ $a=a_{\max }\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_3+\frac{T}{4})\right);$ By angular acceleration a at time period [t ₃t] upper to time integral, can obtain as calculated $\mathrm{\&omega;}=a_{\max }\mathrm{\&Delta;T}+\frac{a_{\max }T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_3+\frac{T}{4})\right));$ To angular velocity omega at time period [t ₃t] upper to time integral, $\mathrm{\&theta;}=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+a_{\max }\mathrm{\&Delta;T}(t-t_3)+\frac{a_{\max }T}{2\mathrm{\&pi;}}(t-t_3+\frac{T}{4}+\mathrm{\&Delta;T})-\frac{a_{\max }{T}^2}{{4\mathrm{\&pi;}}^2}\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_3+\frac{T}{4})\right),$ At time t ₄point, a (t ₄)=0, $\mathrm{\&omega;}\left(t_4\right)=\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T},$ $\mathrm{\&theta;}\left(t_4\right)=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+\frac{a_{\max }\mathrm{\&Delta;TT}}{4}+\frac{a_{\max }\mathrm{T\&Delta;T}}{2\mathrm{\&pi;}}+\frac{a_{\max }{T}^2}{4\mathrm{\&pi;}}.$

(4) coasting-flight phase at the uniform velocity: [t ₄t ₅], t ₅=t ₄+ Δ t; A=0; As calculated,

$\mathrm{\&theta;}=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+a_{\max }\mathrm{\&Delta;TT}(\frac{1}{4}+\frac{1}{2\mathrm{\&pi;}})+\frac{a_{\max }{T}^2}{4\mathrm{\&pi;}}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})(t-t_4),$ At time t ₅point $\mathrm{\&omega;}\left(t_5\right)=\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T},$ $\mathrm{\&theta;}\left(t_5\right)=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+\frac{a_{\max }\mathrm{T\&Delta;T}}{4}+\frac{a_{\max }{T}^2}{4\mathrm{\&pi;}}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})\mathrm{\&Delta;t}.$

(5) rate of deceleration section of increasing progressively: [t ₅t ₆], $t_6=t_5+\frac{T}{4};$ $a=a_{\max }\&CenterDot;\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-\mathrm{\&Delta;T})\right);$ By angular acceleration a at time period [t ₅t] upper to time integral, can obtain as calculated $\mathrm{\&omega;}=a_{\max }\&CenterDot;\mathrm{\&Delta;T}+\frac{a_{\max }T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-\mathrm{\&Delta;T})\right));$ By angular velocity omega at time period [t ₅t] upper integral, $\begin{array}{c}\mathrm{\&theta;}=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+a_{\max }\mathrm{T\&Delta;T}(\frac{1}{4}+\frac{1}{2\mathrm{\&pi;}})+\frac{a_{\max }{T}^2}{4\mathrm{\&pi;}}+\frac{a_{\max }T}{\mathrm{\&pi;}}\mathrm{\&Delta;t}+a_{\max }\mathrm{\&Delta;T}(t-t_5+\mathrm{\&Delta;t})\\ +\frac{a_{\max }T}{2\mathrm{\&pi;}}(t-t_5)-\frac{a_{\max }{T}^2}{4{\mathrm{\&pi;}}^2}\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-\mathrm{\&Delta;T})\right)\end{array};$ At time point t ₆upper, a (t ₆the a of)=- _max, $\mathrm{\&omega;}\left(t_6\right)=\frac{a_{\max }T}{2\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T},$

$\mathrm{\&theta;}\left(t_6\right)=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+\frac{a_{\max }\mathrm{\&Delta;TT}}{2}+\frac{a_{\max }\mathrm{\&Delta;TT}}{2\mathrm{\&pi;}}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})\mathrm{\&Delta;t}+\frac{{3a}_{\max }{T}^2}{8\mathrm{\&pi;}}+\frac{a_{\max }{T}^2}{4{\mathrm{\&pi;}}^2}.$

(6) even braking section: [t ₆t ₇], t ₇=t ₆+ Δ T; A=-a _max; By angular acceleration a at time period [t ₆t] above can obtain time integral

by angular velocity omega at time period [t ₆t] upper integral, can obtain as calculated

$\mathrm{\&theta;}=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+\frac{a_{\max }\mathrm{\&Delta;TT}(\mathrm{\&pi;}+1)}{2\mathrm{\&pi;}}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})\mathrm{\&Delta;t}+\frac{a_{\max }{T}^2(3\mathrm{\&pi;}+2)}{{8\mathrm{\&pi;}}^2}$ ；

$+(\frac{a_{\max }T}{2\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})(t-t_6)-\frac{a_{\max }{(t-t_6)}^2}{2}$

At time point t ₇upper, can be calculated,

$\mathrm{\&theta;}\left(t_7\right)=\frac{a_{\max }\mathrm{\&Delta;TT}}{2}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})\mathrm{\&Delta;t}+\frac{{3a}_{\max }{T}^2}{8\mathrm{\&pi;}}+\frac{a_{\max }\mathrm{T\&Delta;T}}{\mathrm{\&pi;}}+a_{\max }{\mathrm{\&Delta;T}}^2+\frac{a_{\max }{T}^2}{{4\mathrm{\&pi;}}^2}.$

(7) rate of deceleration decline fraction: [t ₇t ₈], $t_8=t_7+\frac{T}{4};$ $a=a_{\max }\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-2\mathrm{\&Delta;T})\right);$ By angular acceleration a at time period [t ₇t] above to time integral, can obtain as calculated, $\mathrm{\&omega;}=\frac{a_{\max }\&CenterDot;T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-2\mathrm{\&Delta;T})\right));$ By angular velocity omega at time period [t ₇t] upper integral, can obtain as calculated

$\mathrm{\&theta;}=\frac{a_{\max }\mathrm{\&Delta;TT}(\mathrm{\&pi;}+2)}{2\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T\&Delta;t}+\frac{{3a}_{\max }{T}^2}{8\mathrm{\&pi;}}+a_{\max }{\mathrm{\&Delta;T}}^2$

$+\frac{a_{\max }T}{2\mathrm{\&pi;}}(t-t_7+2\mathrm{\&Delta;t})-\frac{a_{\max }{T}^2}{{4\mathrm{\&pi;}}^2}\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-2\mathrm{\&Delta;T})\right);$

Wherein ω (t ₈)=0, $\mathrm{\&theta;}\left(t_8\right)=\frac{a_{\max }\mathrm{\&Delta;TT}}{2}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})\mathrm{\&Delta;t}+\frac{a_{\max }{T}^2}{2\mathrm{\&pi;}}+\frac{a_{\max }\mathrm{T\&Delta;T}}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;}{T}^2.$

The acceleration calculation in above-mentioned each stage is only considered the situation of forward angular movement, and angle as motor-driven in reality need to be got negative for bearing acceleration.

Embodiment 1

Take the path design of the motor-driven 70 ° of processes of certain typical complex satellite axis of rolling as example, suppose satellite moment of inertia 2962kg.m ², topworks is control-moment gyro, and the maximum moment that can provide is 20Nm, and angular momentum capacity is Nms, and sensor is gyro, and maximum angle measurement speed is 2.5 °/s, and requiring motor-driven and stabilization time is 80s.The maximum time kept in reserve of obtaining satellite according to step (1) is 40s, and the motor-driven angular acceleration of maximum that obtains satellite according to step (2) is 0.3869 °/s ², obtaining maximum angular rate according to step (3) is 2.5 °/s.Consider and in Project Realization, need to leave certain surplus for gyro to measure and topworks, therefore choosing maximum motor-driven angular acceleration when design path is a _max=0.3 °/s ², maximum angular rate ω _max=2.4 °/s.According to step (4), calculate the sine trigonometric function cycle $T=\frac{2\mathrm{\&pi;}(t_{\max }-\frac{{\mathrm{\&omega;}}_{\max }}{a_{\max }}-\frac{{\mathrm{\&theta;}}_d}{{\mathrm{\&omega;}}_{\max }})}{\mathrm{\&pi;}-2}=15.5943s.$ Choose minimum period T _min=8s, the even acceleration time according to step (5), calculate coasting time $\mathrm{\&Delta;t}=\frac{{2\mathrm{\&pi;\&omega;}}_{\max }}{(\mathrm{\&pi;}-2)a_{\max }}+\frac{4\left|{\mathrm{\&theta;}}_d\right|}{(\mathrm{\&pi;}-2){\mathrm{\&omega;}}_{\max }}-\frac{6-\mathrm{\&pi;}}{\mathrm{\&pi;}-2}{t}_{\max }=18.3333s.$ Suppose t ₁=0s, calculates motor-driven path as follows:

The rate of acceleration section of increasing progressively, time [0 3.8986] s, a=0.3sin (0.1283 π t) °/s ²,

ω＝0.7446(1-cos(0.1283πt))，

θ＝0.7446(t-2.4819sin(0.1283πt))；

Even accelerating sections, time [3.89866.9348] s, a=0.3 °/s ²,

ω＝0.7446+0.3(t-3.8986)°/s，

θ＝1.0548+7.446(t-3.8986)+0.15(t-3.8986) ²°；

Rate of acceleration decline fraction, time [6.934810.8334] s, a=0.3sin (0.1283 (t-3.0362)) °/s ²,

ω＝0.9109+0.7446(1-cos(0.1283(t-6.9348)))°/s，

θ＝3.6434+1.6555(t-6.9348)-1.8480sin(0.4029(t-6.9348)))°；

Coasting-flight phase, time [10.833429.1667] s, a=0 °/s ²,

ω＝2.4001°/s，

θ＝13+2.4(t-10.8334)°；

The rate of deceleration section of increasing progressively, time [29.166733.0653] s, a=0.3sin (0.1283 π (t-21.3695)) °/s ²,

ω＝1.6555-0.7446cos(0.1283π(t-21.3695))°/s，

θ＝56.9999+1.6554(t-29.1667)-1.848sin(0.1283π(t-21.3695))；

Even braking section, time [33.065336.1015] s, a=-0.3 °/s ²,

ω＝1.6555-0.3(t-33.0653)°/s，

θ＝65.3019+1.6554(t-33.0653)-0.15(t-33.0653) ²；

Rate of deceleration decline fraction, time [36.101540.0001] s, a=0.3sin (0.1283 π (t-24.4057)) °/s ²,

ω＝0.7446-0.7446cos(0.1283π(t-24.4057))°/s，

θ＝67.0974+0.7446(t-36.1015)-1.848sin(0.1283π(t-24.4057))。

The direction of attitude angle acceleration a is identical with the actual motor-driven direction of satellite.

Figure 2 shows that according to the reserve road diametal curve of the attitude angle acceleration of above result of calculation design, attitude angular velocity, attitude angle.

Embodiment 2

Take 70 ° of processes of certain typical complex satellite axis of rolling 80s fast reserve as example, suppose satellite moment of inertia 2962kg.m ², flexible windsurfing fundamental frequency 0.5Hz, control cycle is 0.1s, by calculating, chooses sine trigonometric function cycle T=20s, maximum angular acceleration a _max=0.3 °/s ², maximum angular rate ω _max2.4 °/s.Find by mathematical simulation, as shown in Figure 3, under the motor-driven path of bang-bang, in mobile process, the maximum tracking error of attitude angular velocity is 1.2 °/s; As shown in Figure 4, under sinusoidal motor-driven path, in mobile process, the maximum tracking error of attitude angular velocity is 0.7 °/s; As shown in Figure 5, mix under the motor-driven path of stepped curve based on sine, in mobile process, the maximum tracking error of attitude angular velocity is 0.9 °/s; Significantly reduce compared with the motor-driven path of bang-bang.As shown in Figure 6, under the motor-driven path of bang-bang, the modal coordinate displacement of flexible windsurfing is 0.14 left and right to the maximum; As shown in Figure 7, the modal coordinate displacement based on sinusoidal motor-driven path downwarp windsurfing is 0.08 left and right to the maximum.As shown in Figure 8, the modal coordinate displacement of the motor-driven path downwarp windsurfing based on sine mixing stepped curve is 0.08 left and right to the maximum.According to computational analysis, under sinusoidal motor-driven path, the motor-driven required time that puts in place is 41.7332, and as shown in Figure 2, under this method, the motor-driven required time that puts in place is 40.0001.Visible, this method has larger improvement compared with the motor-driven path of standard bang-bang aspect the vibration that reduces flexible windsurfing in mobile process, reducing there is larger improvement aspect the time kept in reserve compared with the motor-driven path of standard sine, and stability is more or less the same, thereby improved the quick performance of satellite.

The content not being described in detail in instructions of the present invention belongs to those skilled in the art's known technology.

Claims (1)

1. the sinusoidal motor-driven path guide method of mixing that is suitable for the higher satellite of windsurfing fundamental frequency, is characterized in that step is as follows:

(1) determine time kept in reserve t according to the motor-driven and stable time index of the attitude of satellite _max;

(2) determine maximum motor-driven angular acceleration a according to the topworks's ability configuring on satellite _max;

(3) determine maximum motor-driven limited angular velocity omega according to the topworks's angular momentum capacity configuring on satellite and sensor range _max;

(4) the path motor-driven attitude of satellite is divided into the time for [t ₁t ₂] the rate of acceleration section of increasing progressively, time be [t ₂t ₃] even accelerating sections, time be [t ₃t ₄] rate of acceleration decline fraction, time be [t ₄t ₅] at the uniform velocity coasting-flight phase, time be [t ₅t ₆] the rate of deceleration section of increasing progressively, time be [t ₆t ₇] even braking section, time be [t ₇t ₈] rate of deceleration decline fraction, wherein t ₁for the motor-driven start time of satellite,

t ₃=t ₂+ Δ T,

t ₅=t ₄+ Δ t,

t ₇=t ₆+ Δ T,

t is the cycle of sine function, and the initial value of T is $\frac{2\mathrm{\&pi;}(t_{\max }-\frac{{\mathrm{\&omega;}}_{\max }}{a_{\max }}-\frac{\left|{\mathrm{\&theta;}}_d\right|}{{\mathrm{\&omega;}}_{\max }})}{\mathrm{\&pi;}-2},$ Δ T is the even acceleration time, and the initial value of Δ T is $\frac{{\mathrm{\&omega;}}_{\max }}{a_{\max }}-\frac{T}{\mathrm{\&pi;}},$ Δ t is coasting time, and the initial value of Δ t is $\mathrm{\&Delta;t}=\frac{{2\mathrm{\&pi;\&omega;}}_{\max }}{(\mathrm{\&pi;}-2)a_{\max }}+\frac{4\left|{\mathrm{\&theta;}}_d\right|}{(\mathrm{\&pi;}-2){\mathrm{\&omega;}}_{\max }}-\frac{6-\mathrm{\&pi;}}{\mathrm{\&pi;}-2}{t}_{\max },$ θ _dfor motor-driven angle;

(5) be a according to the cycle T of sine function, maximum motor-driven angular acceleration _maxand even acceleration time Δ T, the coasting time Δ t of satellite calculate respectively the attitude angle acceleration a of satellite based on sine mixing step function in each section, and corresponding attitude angular velocity ω and attitude angle θ, wherein:

(51) the rate of acceleration section of increasing progressively, $a=a_{\max }\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1)\right),\mathrm{\&omega;}=\frac{a_{\max }T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1)\right)),$ $\mathrm{\&theta;}=\frac{a_{\max }\&CenterDot;T}{2\mathrm{\&pi;}}((t-t_1)-\frac{T}{2\mathrm{\&pi;}}\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1)\right));$ If the Δ T < 0 obtaining according to calculation of initial value, upgrades $T=\left|\frac{{\mathrm{\&pi;\&omega;}}_{\max }}{a_{\max }}\right|,$ ΔT＝0， $\mathrm{\&Delta;t}=\frac{\mathrm{\&pi;}}{a_{\max }T}\left(\right|{\mathrm{\&theta;}}_d|-\frac{\left|a_{\max }\right|T^2}{2\mathrm{\&pi;}});$ If the T+2 Δ T < T obtaining according to calculation of initial value _min, upgrade T=T _min, renewal maximum angular acceleration is

Δ T=0, Δ t=0, T _minby the shortest patient time kept in reserve;

(52) even accelerating sections, a=a _max, $\mathrm{\&omega;}=\frac{a_{\max }T}{2\mathrm{\&pi;}}+a_{\max }(t-t_2),$ $\mathrm{\&theta;}=\frac{a_{\max }{T}^2(\mathrm{\&pi;}-2)}{{8\mathrm{\&pi;}}^2}+\frac{a_{\max }T(t-t_2)}{2\mathrm{\&pi;}}+\frac{a_{\max }{(t-t_2)}^2}{2},$ Wherein the value of T, Δ T, Δ t is with the rate of acceleration section of increasing progressively;

(53) rate of acceleration decline fraction, $a=a_{\max }\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_3+\frac{T}{4})\right),$ $\mathrm{\&omega;}=a_{\max }\mathrm{\&Delta;T}+\frac{a_{\max }T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_3+\frac{T}{4})\right)),\mathrm{\&theta;}=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+\frac{a_{\max }\mathrm{\&Delta;TT}}{4}+\frac{a_{\max }\mathrm{T\&Delta;T}}{2\mathrm{\&pi;}}+\frac{a_{\max }{T}^2}{4\mathrm{\&pi;}},$ Wherein the value of T, Δ T, Δ t is with the rate of acceleration section of increasing progressively;

(54) coasting-flight phase at the uniform velocity, a=0, $\mathrm{\&omega;}=\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T},$ $\mathrm{\&theta;}=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}+a_{\max }\mathrm{T\&Delta;T}(\frac{1}{4}+\frac{1}{2\mathrm{\&pi;}})+\frac{a_{\max }{T}^2}{4\mathrm{\&pi;}}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})(t-t_4),$ Wherein the value of T, Δ T, Δ t is with the rate of acceleration section of increasing progressively;

(55) rate of deceleration section of increasing progressively, $a=a_{\max }\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-\mathrm{\&Delta;T})\right),$ $\mathrm{\&omega;}=a_{\max }\mathrm{\&Delta;T}+\frac{a_{\max }T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-\mathrm{\&Delta;T})\right)),$ $\begin{array}{c}\mathrm{\&theta;}=\frac{a_{\max }{\mathrm{\&Delta;T}}^2}{2}{a}_{\max }\mathrm{T\&Delta;T}(\frac{1}{4}+\frac{1}{2\mathrm{\&pi;}})+\frac{a_{\max }{T}^2}{4\mathrm{\&pi;}}+\frac{a_{\max }T}{\mathrm{\&pi;}}\mathrm{\&Delta;t}+a_{\max }\mathrm{\&Delta;T}(t-t_5+\mathrm{\&Delta;t})\\ +\frac{a_{\max }T}{2\mathrm{\&pi;}}(t-t_5)-\frac{a_{\max }{T}^2}{{4\mathrm{\&pi;}}^2}\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-\mathrm{\&Delta;T})\right)\end{array},$

If the Δ t < 0 going out according to initial calculation, upgrades $T=\frac{2\mathrm{\&pi;}}{2+\mathrm{\&pi;}}(\frac{\left|{\mathrm{\&theta;}}_d\right|}{{\mathrm{\&omega;}}_{\max }}-\frac{{\mathrm{\&omega;}}_{\max }}{a_{\max }}),\mathrm{\&Delta;T}=\frac{{\mathrm{\&omega;}}_{\max }}{a_{\max }}-\frac{T}{\mathrm{\&pi;}},$ Δ t=0; If the Δ T < 0 obtaining according to calculation of initial value, upgrades

Δ T=0, $\mathrm{\&Delta;t}=\frac{\mathrm{\&pi;}}{a_{\max }T}\left(\right|{\mathrm{\&theta;}}_d|-\frac{\left|a_{\max }\right|T^2}{2\mathrm{\&pi;}});$ If $\mathrm{\&Delta;t}=\frac{\mathrm{\&pi;}}{a_{\max }T}\left(\right|{\mathrm{\&theta;}}_d|-\frac{\left|a_{\max }\right|T^2}{2\mathrm{\&pi;}})<0,$ Upgrade $T=\sqrt{2\mathrm{\&pi;}\left|\frac{{\mathrm{\&theta;}}_d}{a_{\max }}\right|,}$ ΔT＝0，Δt＝0；

(56) even braking section, a=-a _max, $\mathrm{\&omega;}=\frac{a_{\max }T}{2\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T}-a_{\max }(t-t_6),$ $\begin{array}{c}\mathrm{\&theta;}=\frac{a_{\max }\mathrm{\&Delta;}{T}^2}{2}+\frac{a_{\max }\mathrm{\&Delta;TT}(\mathrm{\&pi;}+1)}{2\mathrm{\&pi;}}+(\frac{a_{\max }T}{\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})\mathrm{\&Delta;t}+\frac{a_{\max }{T}^2(3\mathrm{\&pi;}+2)}{{8\mathrm{\&pi;}}^2}\\ +(\frac{a_{\max }T}{2\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T})(t-t_6)-\frac{a_{\max }{(t-t_6)}^2}{2}\end{array},$ Wherein the value of T, Δ T, Δ t is with the rate of deceleration section of increasing progressively;

(57) rate of deceleration decline fraction, $a=a_{\max }\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-2\mathrm{\&Delta;T})\right),$ $\mathrm{\&omega;}=\frac{a_{\max }\&CenterDot;T}{2\mathrm{\&pi;}}(1-\cos \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-2\mathrm{\&Delta;T})\right)),$ $\begin{array}{c}\mathrm{\&theta;}=\frac{a_{\max }\mathrm{\&Delta;TT}(\mathrm{\&pi;}+2)}{2\mathrm{\&pi;}}+a_{\max }\mathrm{\&Delta;T\&Delta;t}+\frac{{3a}_{\max }{T}^2}{8\mathrm{\&pi;}}+a_{\max }{\mathrm{\&Delta;T}}^2\\ +\frac{a_{\max }T}{2\mathrm{\&pi;}}(t-t_7+2\mathrm{\&Delta;t})-\frac{a_{\max }{T}^2}{{4\mathrm{\&pi;}}^2}\sin \left(\frac{2\mathrm{\&pi;}}{T}(t-t_1-\mathrm{\&Delta;t}-2\mathrm{\&Delta;T})\right)\end{array},$

Wherein the value of T, Δ T, Δ t is with the rate of deceleration section of increasing progressively;

(6) the described path of tracking step (5) when the motor-driven control of satellite.

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