CN105045285B - Geometric design method of the Spherical Ring around formation control - Google Patents
Geometric design method of the Spherical Ring around formation control Download PDFInfo
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Abstract
The present invention is a kind of geometric design method of Spherical Ring around formation control, is comprised the following steps：A) sphere under inertial coodinate system, circular orbit and movable body are dynamically represented again under orbital coordinate system；B) target sphere is expanded into the sphere cluster that different curve functional value is represented；C) movable body is calculated to the range error and its derivative of target sphere by toroidal function value, the controling power designed on globally normal vector realizes that sphere is logged in；D) error and its derivative between the latitude corresponding with expecting track of the latitude of sphere where calculating movable body, the controling power designed on globally warp direction realize surround movement；E) movable body is calculated around the axle broad sense anglec of rotation and its derivative, is designed the controling power on globally weft direction and is completed to form into columns；F) combining step c) e) calculates the controling power input of movable body.The present invention is dynamically especially suitable to the sphere, circular orbit and movable body described under orbital coordinate system.This method is simple and reliable, precision is higher, available for tasks such as collaboration detections.
Description
Technical field
The present invention relates to a kind of Spherical Ring around formation control geometric design method.
Background technology
Multiple movable bodies composition mobile sensor network of install sensor can realize inexpensive and high performance information
Collection, thus receive the concern of lot of domestic and foreign scientific research institution and famous scholar.Early in 1997, NASA began to deeply open
Explore the Mars (NASA, " Robotic Mars Exploration ", http are realized in exhibition using multiple Marsokhod://
www.nasa.gov/mission_pages/mars-pathfinder/index.html).Last decade, even more having formulated one is
The plan of exploring the Mars of row.Many underwater moving body Collect jointly marine organisms groups are successively repeatedly carried out in Princeton University
The experiment of body information, effect very notable (PrincetonUniversity, " Adaptive Sampling and
Prediction”.http://www.princeton.edu/dcsl/asap/).Recently, with China's moon exploration program progressively
Carry out, development of the country to the Detection Techniques that cooperate also increasingly is paid close attention to.In order to fully using limited movable body come real
The now data acquisition in large-scale region and the precision for ensureing collection information to greatest extent, it usually needs each motion of planning
The track of body and require that multiple movement bodies form certain formation on given track, i.e., around formation control technology.
Currently, focus primarily upon method for designing in planar tracks around formation control technology (Chen Yangyang, Tian Yuping are based on
The multirobot trailing formation control design case method of track extension, the patent No.：ZL201010552508.4；Chen Yangyang, Tian Yuping,
The trailing formation control method patent No.s of multiple movement bodies in three dimensions:ZL200910184547.0).However, in deep-sea
The shoal of fish, the collection information of micropopulation carry out information gathering around target group round and round, it is necessary to which multiple movement bodies are three-dimensional in all directions；
Space celestial body detecting then need multiple movement bodies as around the star along a certain Spherical Ring around tested star formation flight.This is just produced
New control problem is given birth to, realization logs in target sphere and expects to keep expecting while circular orbit surround movement along on sphere
Formation, i.e., so-called Spherical Ring is around formation control problem, it is clear that existing method can not solve problems.It is furthermore noted that
Existing circular formation control method needs to use different design of control law for two-way/oriented information exchange.However, real
Communication equipment in border is because by external interference, two-way communication can become oriented communication often, and end of interrupt communication is again extensive
It is two-way communication again.For oriented/two-way information exchange topology, if using different tracking formation control devices, this is undoubtedly
Many troubles can be brought to realizing.
Therefore, it is adaptable to which the Spherical Ring of oriented/two-way information interaction topology will more have around the method for designing of formation control
There is realistic meaning.But there is not yet such control method at present.
The content of the invention
Technical problem：It is an object of the invention to provide a kind of Spherical Ring around the geometric design method of formation control, this method
Simple and reliable, precision is higher, available for complex tasks such as movable body collaboration detections.
Technical scheme：The present invention is a kind of geometric design method of Spherical Ring around formation control, is particularly suitable for use in track seat
Sphere, the circular orbit on sphere and the movable body dynamic of the lower description of mark system.
Consider orbital coordinate system Wio={ oi,xi,yi,ziUnder on the sphere that describes N number of movable body composition circular formation control
System processed, wherein oiPositioned at the centre of sphere of the corresponding target sphere of the i-th movable body, ziAxle is put down with expecting on object ball where circular orbit
The normal vector in face is consistent.Under orbital coordinate system, the motion of movable body can regard the motion of rigid body as, meet Newton's second law,
Disturbed simultaneously by flow field known to space：
Wherein pi=[pix,piy,piz]T∈WioRepresent position coordinates of i-th of movable body in orbital coordinate system, vi=
[vix,viy,viz]T∈WioRepresent the speed of i-th of movable body, miFor its quality, ui=[uix,uiy,uiz]T∈WioIt is i-th of fortune
The controling power input of kinetoplast, fpi(pi, t)=[fpix(pi,t),fpiy(pi,t),fpiz(pi,t)]T∈WioRepresentation space flow field
Velocity vector, fvi(pi,vi)=[fvix(pi,vi),fviy(pi,vi),fviz(pi,vi)]T∈WioRepresent the celestial body being subject to during motion
Gravitation, i=1 ..., n.For example, in the equation of motion of satellite gravitation fvi(pi,vi) can be write asWherein μ is celestial body gravitational constant,For perturbation acceleration；It is diversion in the C-W equations of motion of star
fvi(pi,vi) can be write asWherein ω is the star that is diversion
The angular speed moved along circular orbit.Fig. 1 is tied to the conversion of orbital coordinate system for inertial coordinate.
Multiple movement bodies are in motion of forming into columns, and the information exchange between movable body is essential, and we are with having here
Xiang TuTo describe, whereinFor set of node,For the set of directed edge.If section
PointThere is directed edge and reach nodeThis means that i-th of movable body can obtain the information of j-th of movable body, j-th
Movable body is the adjacent node of i-th of movable body.The adjacent node collection of i-th of movable body is sharedRepresent.We say node
Node is reached in the presence of a directed walkI.e. there is one group of directed edge in orderFrom nodeTo nodeWhereinIt isThe set of middle k+1 different nodes.If all nodes of other in digraph
Node can be reached by directed walkThen nodeIt is referred to as global accessible point.If nodeWith nodeIn the presence of double
To connection side, i.e., corresponding adjacency matrix A=[aij] in aij=aji=1, other aij=aji=0, corresponding information exchange is opened up
Flutter as two-dimensional plot.If nodeTo nodeIn the presence of unidirectional connection side, i.e., corresponding adjacency matrix A=[aij] in aij=1, its
He is aij=0, corresponding information exchange topology is digraph.The Laplacian matrixes of figure can be expressed as L=[lij], wherein lii
=Σj≠iaijAnd lij=-aij.Left side (wherein owns for the topologically corresponding two-dimensional plot of information exchange between 5 movable bodies in Fig. 2
Node is global accessible point), right side is digraph (its interior joint { υ1,υ2,υ3Be global accessible point).During design, we are once
Provide multiple movement bodies information interaction relation, then later each moment movable body iAll it is constant, and corresponding
Two-way/digraph contains global accessible point.In engineer applied, we can be according to actual conditions flexibly using the side of communication
Formula, the mode of sensing and the mode that is combined of two ways realize the information exchange between multiple movement bodies.
For the target sphere of the i-th movable body described under orbital coordinate systemIt is expressed as with equation
Wherein ρiFor the radius of ball.ForOn expectation circular orbitUnder orbital coordinate systemIt is exactly just on ball
Certain one dimensional line, desired circular orbit is used hereCorresponding latitudeTo describeThe purpose of the present invention is just
It is the information according to obtained adjacent motion body, designing the controling power of each movable body makes its motion on corresponding target sphere
Certain formation is kept while expecting Circular test, between movable body.
In the present invention, such as lower section is used for the formation position relation between each movable body for being moved along target trajectory
Formula is provided：Make ziAxle is rotary shaft, li0Perpendicular to rotary shaft and to pass through the vector of i-th of movable body initial position, li0With
xioiziThe angle theta of planei0∈ [0,2 π) it is defined as the initial angle that movable body is pivoted, θi(t) for i-th of movable body around rotation
The angle of rotating shaft rotation.Broad sense anglec of rotation ξi(θi(t) it is) on rotation angle θi(t) linear function, i.e.,Wherein bi≠ 0 HeDetermined for constant and by formation.Keep expecting formation position between each movable body
Relation refers to：
ξi(θi(t))-ξj(θj(t))=0.
It is the same circular orbit that three movable bodies are evenly distributed on same sphere shown in Fig. 3, can sets hereI=1,2,3.Three movable bodies are moved concentric in the different target of radius with " one " word formation
The circular orbit (as shown in Figure 4) of Same Latitude value on sphere, here ξi(θi)=θi, i=1,2,3.
The present invention design philosophy be, by the expectation circular orbit on the target sphere described under inertial coodinate system, sphere with
And movable body is dynamically represented again under orbital coordinate system；Target sphere is expanded on curved surface by concentric compand again
Function fsi(pi) equivalent sphere cluster, the movable scope of i-th of movable body is determined by the regularity of curved surface.Design movable body edge
Sphere normal vector NiOn controling power so that be initially located at ΩiIn movable body move all the time in ΩiAnd to target sphere
Apart from dis(t) and its to the derivative of time0 is reduced to, logging in for target sphere is realized；Movable body is designed along sphere
Warp direction BiOn controling power so that movable body be located at sphere on latitude withCorresponding expectation latitudeBetween latitude errorAnd its to the derivative of time0 is reduced to, surround movement is realized.When movable body motion is on respective target sphere
Expectation circular orbit, the formation motion of multiple movement bodies just deteriorates to movable body and reaches one along the Position And Velocity of track motion
Cause.The information of adjacent motion body is obtained according to information exchange, movable body is designed along sphere weft direction TiOn controling power portion
Divide and cause broad sense anglec of rotation ξiAnd its derivative (t)Reach the formation campaign unanimously to realize multiple movement bodies.
Concretely：
The present invention is a kind of geometric design method of Spherical Ring around formation control, is comprised the following steps：
A) sphere under inertial coodinate system, circular orbit and movable body are dynamically represented again under orbital coordinate system；
B) target sphere is expanded into the sphere cluster that different curve functional value is represented；
C) movable body is calculated to the range error and its derivative of target sphere by toroidal function value, designs globally normal vector
On controling power realize that sphere is logged in；
D) error and its derivative between the latitude corresponding with expecting track of the latitude of sphere where calculating movable body, design edge
Controling power on sphere warp direction realizes surround movement；
E) movable body is calculated around the axle broad sense anglec of rotation and its derivative, is designed the controling power on globally weft direction and is completed to compile
Team；
F) combining step c)-e) control of obtained movable body globally on normal vector, warp direction and weft direction
Power part, connection row solve the controling power input of movable body.
Wherein described target sphere, the circular orbit on sphere and movable body is dynamically required for by inertial Cartesian coordinates system
It is transformed under orbital coordinate system and redescribes.
Wherein described step a) comprises the following steps：
A1) centre of sphere of the origin translation of inertial coodinate system to target sphere, its z-axis are rotated to parallel to expectation circular orbit
The normal vector of place plane, so as to obtain orbital coordinate system and coordinate transfer matrix；
A2 target sphere, circular orbit and movable body dynamic) are redescribed under orbital coordinate system.
Wherein described step b) comprises the following steps：
B1) by concentric compand sphere, target sphere is expanded into one group of equivalent sphere；
B2) according to the regularity of curved surface, it is determined that the range of movement of correspondence movable body；
B3 toroidal function) is built on range of movement, the sphere cluster expanded is taken different values by toroidal function
To represent.
Wherein described step c) comprises the following steps：
C1) position by movable body and toroidal function, calculate movable body to the range error of target sphere；
C2) by derivative of the range error to the time, the variable quantity of range error is calculated；
C3) by range error and range error variable quantity, the controling power part of design movable body globally on normal vector.
Wherein described step d) comprises the following steps：
D1) by the position of movable body, between the latitude latitude corresponding with expecting track where calculating movable body on sphere
Latitude error；
D2) by derivative of the latitude error to the time, the variable quantity of latitude error is calculated；
D3) by the variable quantity of latitude error and latitude error, the controling power portion of design movable body globally on warp direction
Point.
Wherein described step e) comprises the following steps：
E1 rotary shaft and initial rotation angle) are provided；
E2) position by movable body and speed, calculate movable body and are pivoted angle；
E3) require to determine the functional relation between the broad sense anglec of rotation and the anglec of rotation according to forming into columns, calculate the broad sense anglec of rotation and its
To the derivative of time；
E4) the information of the adjacent motion body obtained by information exchange, the control of design movable body globally on weft direction
Power part.
Wherein described step f) comprises the following steps：
F1) combining step c)-e) calculate movable body controling power input；
F2) the controling power input of movable body is sent in slave computer by host computer, motion is completed by servo-drive system
Control.
Beneficial effect：The characteristics of this method has simple and reliable, precision higher and is easy to practice, available for many fortune
The complex tasks such as kinetoplast collaboration detection.
Brief description of the drawings
Fig. 1 is dynamic for sphere, circular orbit and the movable body described under coordinate transform and orbital coordinate system；
Fig. 2 is the topologically corresponding two-dimensional plot of information exchange and digraph；
Fig. 3 is the signal for the same circular orbit that three movable bodies are evenly distributed on same sphere；
Fig. 4 is that three movable bodies move the circular orbit on the different target concentric spherical of radius with " one " word formation；
Fig. 5 is the equivalent sphere cluster that concentric compand target sphere is obtained；
Fig. 6 is geometry designs flow chart of the Spherical Ring around motion of forming into columns.
In figure above：WI：Inertial coodinate system；oI：The origin of inertial coodinate system；xI：The x-axis of inertial coodinate system；yI：It is used
The y-axis of property coordinate system；zI：The z-axis of inertial coodinate system；piI：I-th of movable body is in WIPosition coordinates；viI：I-th of movable body
In WISpeed；Qit：Coordinate translation matrix；Wit：Inertial coodinate system translates QitObtained coordinate system；oit：WitOrigin；xit：
WitX-axis；yit：WitY-axis；zit：WitZ-axis；Qir：Coordinate spin matrix；Wio：WitRotate QirObtained orbital coordinate system；
oi：WioOrigin；xi：WioX-axis；yi：WioY-axis；zi：WioZ-axis；pi：I-th of movable body is in WioPosition coordinates；vi：
I-th of movable body is in WioSpeed；1st movable body；2nd movable body；3rd movable body；4th fortune
Kinetoplast；5th movable body；p1：1st movable body is in WioPosition coordinates；p2：2nd movable body is in WioPosition sit
Mark；p3：3rd movable body is in WioPosition coordinates；v1：1st movable body is in WioSpeed；v2：2nd movable body is in Wio's
Speed；v3：3rd movable body is in WioSpeed；The corresponding target sphere of 1st movable body；2nd movable body pair
The target sphere answered；The corresponding target sphere of 3rd movable body；The corresponding target sphere of i-th of movable body；Ni：
Sphere normal vector；WithThe length of compand；Along NiCompress in directionThe sphere that length is obtained；Along NiCompress in directionThe sphere that length is obtained；fis(pi)：Toroidal function.
Embodiment
Fig. 6 is the design flow diagram of the present invention, is made up of module P1, P2, P3, P4, P5 and P6, each module is described below：
1) module P1
Because the premise of design control law of the present invention is target sphere, the expectation circular orbit on sphere and movable body dynamic
Need what is described under orbital coordinate system.And in practice either target sphere and expect circular orbit, or movable body dynamic
Often provided under inertial coordinate.Module P1 is used to realize inertial coordinate to the coordinate transform of orbit coordinate.
That as shown in Fig. 1 left figures is W under inertial coodinate systemI={ oI,xI,yI,zIThe target sphere of description, the phase on sphere
Hope circular orbit and movable body dynamic.By inertial coodinate system by coordinate translation by the origin o of coordinate systemIMove to target sphere
The centre of sphere (as shown in Fig. 1 middle graphs).The coordinate system W after translation is rotated againit={ oi,xit,yit,zitZitAxle so that the axle
WithThe normal vector of plane is consistent where upper expectation circular orbit, and then obtains orbital coordinate system Wio={ oi,xi,yi,ziAnd sit
Mark transformation matrix (as shown in Fig. 1 right figures).By transformation matrix of coordinates, target sphere, sphere are redescribed under orbital coordinate system
On expectation circular orbit and movable body dynamic.Module P1 specifically follows these steps to realize：
The first step：By the origin o of inertial coodinate systemIThe centre of sphere of target sphere is moved to, coordinate translation matrix Q is obtainedit.Again
The z of coordinate system after rotation translationitAxle cause the axle withThe normal vector of plane is consistent where upper expectation circular orbit, and then
To orbital coordinate system and coordinate spin matrix Qir, calculate the transformation matrix of coordinates Q that inertial coordinate is tied to orbital coordinate systemi
Qi=QirQit；
Second step：Under orbital coordinate system, by target sphereIt is expressed as on longitude μi∈ [- π, π] and latitudeSmooth function
Wherein On expectation circular orbitWithLatitude valueTo describe, the position p of i-th of movable bodyiWith speed viFor
pi=QipiI,
vi=QiviI,
Wherein piIAnd viIPosition and speed coordinate of i-th of movable body under inertial coodinate system are represented respectively.
2) module P2
As shown in figure 5, the sphere described under orbital coordinate system Near, willOn every bit along cross should
The direction of the vertical sphere of point translates (compression and expansion) different real number λi, we can obtain different spheres (such as：With
).Module P2 specifically follows these steps to realize：
The first step,Near, willOn every bit along cross the vertical sphere of point direction (the method direction of sphere)
NiThe different real number λ of translationiSphere after being expandedI.e.
Second step, by extending after sphere need to meet regularity conditions, we select i-th movable scope ΩiFor space
In all satisfaction-ρi＜ λiThe set of ＜ ε ＜ ∞ points.
3rd step, due to set omegaiIn every bit belong to ΩiIn one extension sphere, we can be in ΩiOn
Build toroidal function
And then, ΩiΩ can be expressed asi={ pi∈Woi|-ρi＜ fis(pi) ＜ ε.It is on toroidal function fisOne
Bar equivalence sphere, i.e.,Then fis(pi)=λi；Whenfis(pi)=0.
3) module P3
Module P3 is for designing movable body along the controling power part on sphere normal vector so that movable body to object ball
The range error in face is reduced to the requirement for meeting design, while ensureing movable body all the time in movable range of motion, specifically
Design procedure is as follows：
The first step：By toroidal function fisWith the current location p of movable bodyi, movable body is calculated to the distance mistake of target sphere
Poor dis(t)
Second step：By orbital fisCalculate sphere normal vector Ni
By NiPosition and speed with movable body, calculate range error dis(t) to the derivative of time
3rd step：Design controling power of the movable body globally on normal vector
Wherein
Control parameter k1＞ 0,Function ψi(dis) it is to be initially located at Ω for guaranteeiI-th of movable body all the time in ΩiIn
Move and finally move in target sphere.
4) module P4
Module P4 is for designing movable body along the controling power part on sphere warp direction so that ball where movable body
The latitude error between latitude latitude corresponding with expecting circular orbit on face is reduced to the requirement for meeting design, specific design step
It is rapid as follows：
The first step：By the current location p of movable bodyi, calculate the latitude that movable body is located on sphere
By the corresponding dimension of expectation trackCalculate dimension error
Second step：By dimension errorWith movable body dynamic, dimension error is calculatedTo the derivative of time
Wherein BiVectorial for sphere warp direction, expression is as follows:
3rd step：Movable body is designed along the controling power on sphere warp direction
Wherein
Control parameter k2,k3＞ 0.
5) module P5
Module P5 obtains the broad sense anglec of rotation ξ of adjacent motion body according to information exchangeiAnd its derivativeTo design i-th
Movable body is along the controling power on sphere weft direction to realize formation, and specific design procedure is as follows：
The first step：Provide ziAxle is rotary shaft, and the anglec of rotation of movable body is defined as around ziAxle rotated counterclockwise by angle.Motion
The Initial Value definition of the anglec of rotation of body is θi0
Second step：Calculate the rotation angle θ of movable bodyi(t)
Wherein, TiIt is sphere weft direction vector, expression formula is as follows：
Second step：Rotation angle θi(t) with expectation formation, broad sense anglec of rotation ξ is definedi(t) the parameter b ini≠ 0 HeCalculate
Broad sense anglec of rotation ξi(t)
Then, derivative of the broad sense arc length to the time is calculated
4th step：According to the broad sense anglec of rotation and its derivative of the adjacent motion body obtained by information exchange, movable body is designed
Controling power along on sphere weft direction
Wherein
For desired broad sense angular velocity of rotation；a0∈ { 0,1 }, works as a0Represented when=1 in formation task to the anglec of rotation
Speed is required, otherwise a0=0；
Control parameter k4,k6＞ 0；If the information commutative Topology containing global accessible point is two-way, control parameter k5Choosing
Select the constant more than 0；If the information commutative Topology containing global accessible point is unidirectional, control parameter k5Need to meet
Wherein μi, i=1,2 ..., n are-L characteristic values.
6) module P6
Module P6 integration modules P3, P4 and P5 result calculate the controling power input of movable body, complete the motion of movable body
Control, is specifically realized according to following steps：
The first step：The movable body that integration module P3 is designed controling power part globally on normal vector, module P4 designs
Go out along the controling power part on sphere warp direction and module P5 and design along the controling power portion on sphere weft direction
Point, connection row solve the controling power input u of final each movable bodyi
Second step：The controling power input of movable body is sent to slave computer by host computer in the form of a command, by movable body
Servo-drive system completes the motion control to movable body, and returns to module P3.
Claims (1)
1. a kind of Spherical Ring is around the geometric design method of formation control, it is characterised in that the phase on target sphere therein, sphere
Hope that circular orbit and movable body are dynamically described under orbital coordinate system, this method comprises the following steps：
A) sphere under inertial coodinate system, circular orbit and movable body are dynamically represented again under orbital coordinate system；
B) target sphere is expanded into the sphere cluster that different curve functional value is represented；
C) movable body is calculated to the range error and its derivative of target sphere by toroidal function value, design is globally on normal vector
Controling power realizes that sphere is logged in；
D) error and its derivative between the latitude corresponding with expecting track of the latitude of sphere where calculating movable body, design is globally
Controling power on warp direction realizes surround movement；
E) movable body is calculated around the axle broad sense anglec of rotation and its derivative, is designed the controling power on globally weft direction and is completed to form into columns；
F) control of the movable body that combining step c)-step e) is obtained globally on normal vector, warp direction and weft direction
Power part, connection row solve the controling power input of movable body；
Wherein described step a) comprises the following steps：
A1) rotated to by the centre of sphere of the origin translation of inertial coodinate system to target sphere, by z-axis parallel to where expectation circular orbit
The normal vector of plane, so as to obtain orbital coordinate system and coordinate transfer matrix；
A2 target sphere, circular orbit and movable body dynamic) are redescribed under orbital coordinate system；
Wherein described step b) comprises the following steps：
B1) by concentric compand sphere, target sphere is expanded into one group of equivalent sphere；
B2) according to the regularity of curved surface, it is determined that the range of movement of correspondence movable body；
B3 toroidal function) is built on range of movement, allows the sphere cluster expanded the value different by toroidal function takes come table
Show；
Wherein described step c) comprises the following steps：
C1) position by movable body and toroidal function, calculate movable body to the range error of target sphere；
C2) by derivative of the range error to the time, the variable quantity of range error is calculated；
C3) by range error and range error variable quantity, the controling power part of design movable body globally on normal vector；
Wherein described step d) comprises the following steps：
D1) by the position of movable body, the latitude between latitude latitude corresponding with expecting track where calculating movable body on sphere
Error；
D2) by derivative of the latitude error to the time, the variable quantity of latitude error is calculated；
D3) by the variable quantity of latitude error and latitude error, the controling power part of design movable body globally on warp direction；
Wherein described step e) comprises the following steps：
E1 rotary shaft and initial rotation angle) are provided；
E2) position by movable body and speed, calculate movable body and are pivoted angle；
E3) require to determine the functional relation between the broad sense anglec of rotation and the anglec of rotation according to forming into columns, calculate the broad sense anglec of rotation and its pair when
Between derivative；
E4) the information of the adjacent motion body obtained by information exchange, the controling power portion of design movable body globally on weft direction
Point；
Wherein described step f) comprises the following steps：
F1) combining step c)-step e) calculates the controling power input of movable body；
F2) the controling power input of movable body is sent in slave computer by host computer, motion control is completed by servo-drive system.
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