CN109291047B - Based on axis invariant and the inverse solution modeling method of DH parameter 1R/2R/3R - Google Patents
Based on axis invariant and the inverse solution modeling method of DH parameter 1R/2R/3R Download PDFInfo
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Abstract
The invention discloses one kind based on axis invariant and the inverse solution modeling of D-H parameter 1R/2R/3R robot pose and calculation method.Method of the invention solves the problems, such as that the inverse solution of the 1R posture based on axis invariant, 2R the and 3R posture based on axis invariant and D-H parameter demonstrate the correctness of this method against solution, and through CE3 rover engineer application based on natural system of coordinates.It is characterized by: the expression with succinct chain notation and axis invariant, has the function of pseudocode, physical meaning is accurate, ensure that the reliability of Project Realization；It based on the structural parameters of axis invariant, does not need to establish middle coordinate system, avoids measurement error caused by introducing middle coordinate system, ensure that the accuracy of the inverse solution of posture.Simultaneously as realizing the parametrization of coordinate system, polarity, structure parameter, the versatility of engineer application ensure that.
Description
Technical field
The present invention relates to a kind of inverse solution modeling of multi-axis robot posture and calculation methods, belong to robotic technology field.
Background technique
When being solved using name D-H system and D-H Parameters Computer device people system kinematics are inverse, due to there is machining and filling
With error, robot system absolute fix and accuracy of attitude determination is caused to be far below the repeatable accuracy of system；Meanwhile D-H system establish and
D-H parameter determination process is more loaded down with trivial details, and when degree of freedom in system is higher, it is low to have been manually done this process reliability.Therefore, it is necessary to solve
The determination problem of robot system D-H system and D-H parameter is certainly completed by computer.Meanwhile high-precision D-H system and D-H parameter
It is the basis that robot carries out accurate operation, and " teaching-reproduction " (Teaching and Playback) robot is to autonomous
The basis of robot development.
Summary of the invention
Technical problem to be solved by the invention is to provide one kind to be based on axis invariant and D-H parameter 1R/2R/3R robot
The inverse solution modeling of posture and calculation method, avoid measurement error caused by introducing middle coordinate system, guarantee the accuracy of the inverse solution of posture.
In order to solve the above technical problems, the invention adopts the following technical scheme:
One kind is based on axis invariant and the inverse solution modeling method of DH parameter 1R/2R/3R, characterized in that
For controlling multi-axis machine device, the multi-axis machine device includes that rod piece set and joint are gathered, the rod piece
Rod piece in set is combined through the joint of joint set, and joint set is converted into corresponding axis set, joint
The corresponding sub- axis set at the axis set in a joint in set, the axis of the axis set includes translation shaft and rotation axis two
Seed type；
The description multi-axis machine device is corresponded to using the axis set, and establishes power using the axis set
Equation is learned, to control this multi-axis machine device；
It is reference with natural system of coordinates when system is in zero-bit, measurement obtains connecting rodAnd the reference axis of rod piece l
VectorWhen moving secondary motion, axial vectorIt is invariant；Axial vectorAnd joint variableUniquely determine kinematic pair
Rotation relation；
When given joint variableAfter rotational angle, just, cosine and its half-angle just, cosine be constant；For convenience
Expression, note
It is obtained by formula (1)
Definition
Then
Given kinematic chainiln, establish the robot 3D vector posture equation based on axis invariant:
Formula (5) be aboutN tie up 2 rank multinomial equations；Expression-form power symbol in formulaIt indicatesX
Power；Upper right corner footmark ∧ orIndicate separator；It is axis invariantMultiplication cross matrix；1 is three-dimensional unit matrix；
Vector expression takes axial vector；
Establish axis invariantSecond order polynomial:
Formula (6) be aboutWithMultilinear equation, be axis invariantSecond order polynomial；For
Rotational transformation matrix；Given nature zero-bit vectorAsZero reference, thenAndRespectively indicate zero
Bit vector and radial vector；Formula (6) is asSymmetric partIndicate null axis
Tensor, skew-symmetric partIndicate radial axial tensor, respectively with axial apposition tensorIt is orthogonal, thus really
Fixed three-dimensional natural shaft space；
Formula (6) is expressed as
The robot pose equation that must be standardized by formula (7):
In formula,iQnIndicate posture, axial vector It is axis invariantMultiplication cross matrix.
If kinematic pairR indicates revolute pair,iQnIndicate posture, only three independent freedom degrees；Then when |iln|=3
When, there are the inverse solutions of 3R posture；
Given unit vectorIt is obtained by formula (8)
In formula,For unit vectorIn the projection vector of earth coordinates.
If Indicate it needs to be determined that direction, then when |iln| when=2, there are the inverse solutions of 2R posture；
Given unit vectorAndIt is obtained by formula (8)
In formula,For unit vectorIn the projection vector of earth coordinates.
Given unit vectorAndIt is obtained by formula (8)
If Indicate desired projection, then when |iln| when=1, there are the inverse solutions of 1R posture；
It is obtained by formula (5) and formula (10)
Make the unit vector of consolidationWith desired unit vectorProjectionIt is optimal, meetThe smallest solution
For
Wherein:ForWithAngle.
It will be directed toward problem based on the 2R of axis invariant and be converted into the direction problem of the 2R based on D-H parameter, orient inverse solution and calculate
Step are as follows:
For giving 2R swivel-chainWherein,L, k are rod piece；By first unit vectorIt is directed toward
It is expected that unit vectorFor unit vectorIn the projection vector of D-H system；Seek the natural joint coordinate of rod pieceAnd
φl:
Wherein arrangeNatural system of coordinatesCorresponding D-H system is denoted asAccording to
The number of D-H coordinate system is accustomed to, kinematic pairCorresponding axis is denoted asIndex habit i.e. in D-H system defers to father's index,
It is different that sub- index is deferred to from the parameter under natural coordinates system；Rotational angle isWhen, definition:WhereinIt is by axisTo the torsional angle of axis l ' z；
D-H parameter index is enabled to defer to sub- index, i.e.,CauseTherefore it is indicated with D-H parameter
?
It is obtained by formula (36) last line
In formula, if withIndicate attribute occupy-place, then the expression-form in formulaIndicate that member accesses symbol；
Therefore have
Have
Wherein:
Therefore have
It is obtained by formula (36) the first row
Therefore have
I.e.
Wherein:
Factor (38 and formula (42) not necessarily meet the 2nd row of formula (36), obtained by formula (38) and formula (42)And φl
It is only possible to solve；The 2nd row that substitution formula (36) will likely be solved again is really solved if still setting up.
Given 3R swivel-chainAnd expectation postureAxis invariant sequenceJoint is asked to become
Measure sequenceWherein,L, k are rod piece；It converts the 3R pose problem based on axis invariant to and is joined based on D-H
Several 3R pose problems, the inverse solution calculating formula of posture are as follows:
It is obtained by formula (38) and formula (42)By?Therefore have
In formula,Representing matrixThe 4th row element.
Advantageous effects of the invention:
Method of the invention solves the inverse solution of the 1R posture based on axis invariant, based on axis based on natural system of coordinates
2R the and 3R posture of invariant and D-H parameter demonstrates the correct of this method against solution problem, and through CE3 rover engineer application
Property.It is characterized by: the expression with succinct chain notation and axis invariant, has the function of pseudocode, physical meaning
Accurately, it ensure that the reliability of Project Realization；Based on the structural parameters of axis invariant, does not need to establish middle coordinate system, avoid
Measurement error caused by middle coordinate system is introduced, ensure that the accuracy of the inverse solution of posture.Simultaneously as realizing coordinate system, pole
Property, the parametrization of structure parameter, ensure that the versatility of engineer application.
Detailed description of the invention
Fig. 1 natural system of coordinates and axis chain；
Fig. 2 fixing axle invariant；
Fig. 3 lunar surface rover solar wing coordinate system；
The mechanical interference of Fig. 4 antenna and solar wing；
Fig. 5 decouples the inverse solution of 2 groups of postures of mechanical arm；
Fig. 6 lunar surface rover 2DOF mast.
Specific embodiment
The invention will be further described below in conjunction with the accompanying drawings.Following embodiment is only used for clearly illustrating the present invention
Technical solution, and not intended to limit the protection scope of the present invention.
In engineer application, natural system of coordinates is not only simple, conveniently, but also helps to improve engineering survey precision, enhances
The versatility of modeling.Meanwhile the kinematics of multiple axes system and the difficulty of Dynamic Modeling are primarily due to the presence of rotation, and rotate
The key of description is rotation axis.The present invention is based on natural system of coordinates, the inverse solution modeling of posture for studying 1R, 2R and 3R is asked with resolving
Topic.Main purpose is laid the foundation for subsequent multiple axes system inverse kinematics of the elaboration based on axis invariant.
Define 1 natural coordinates axis: title is coaxial with kinematic axis or measurement axis, and the unit reference axis with fixed origin is certainly
Right reference axis, also known as nature reference axis.
Define 2 naturals system of coordinates: as shown in Figure 1, if multiple axes system D is in zero-bit, all Descartes's body coordinate system directions
Unanimously, and body coordinate origin is located on the axis of kinematic axis, then the coordinate system is natural coordinates system, abbreviation natural coordinates
System.
Natural system of coordinates advantage is: (1) coordinate system easily determines；(2) joint variable when zero-bit is zero；(3) zero-bit
When posture it is consistent；(4) it is not easily introduced measurement accumulated error.
By definition 2 it is found that when system is in zero-bit, the natural system of coordinates and pedestal of all rod pieces or the direction of system, the world
Unanimously.System is in zero-bitWhen, natural system of coordinatesAround axial vectorRotational angleIt willGo to F[l]；?Under coordinate vector withIn F[l]Under coordinate vectorIt is identical, that is, have
Known by above formula,OrIndependent of adjacent coordinate systemAnd F[l]；Therefore claimOrFor axis invariant.
When not emphasizing invariance, coordinate vector (abbreviation axial vector) can be referred to as.OrCharacterization is bodyIt is total with body l
Some reference units coordinate vectors, with reference pointAnd OlIt is unrelated.BodyIt is rod piece or axis with body l.
Axis invariant and reference axis have essential distinction:
(1) reference axis is that have the reference direction of zero-bit and unit scales, can describe the position being translatable in the direction, but
Rotational angle around the direction cannot completely be described, because reference axis itself does not have radial reference direction, i.e., there is no characterizations
The zero-bit of rotation.In practical application, requiring supplementation with the radial reference of the axis.Such as: Descartes system F [l] in, turn around lx
It is dynamic, it need to be with reference to zero-bit with ly or lz.Reference axis itself is 1D, and 3 orthogonal 1D reference axis constitute Descartes's frame of 3D.
(2) axis invariant is the mikey reference axis of 3D, its own is exactly a frame.Its own has radial reference
Axis refers to zero-bit.Solid axes and the radial reference axis of its own can determine Descartes's frame.Solid axes can be with
Reflect kinematic axis and measure three of axis and refers to attribute substantially.
The axial vector of no chain index is denoted as by existing documentAnd referred to as Euler's axis (Euler Axis), it is corresponding to close
Section angle is known as Eulerian angles (Euler Angle).Why the application no longer continues to use Euler's axis, and referred to as axis invariant, be because
Have for axis invariant with properties:
[1] rotation transformation battle array is givenBecause it is real matrix, mould is unit, therefore it has a factual investigation λ1And
Two complex eigenvalue λ being conjugated each other2=eiφAnd λ3=e-iφ；Wherein: i is pure imaginary number.Therefore, | λ1|·||λ2||·||λ3||
=1, obtain λ1=1.Axial vectorIt is factual investigation λ1=1 corresponding characteristic vector, is invariant；
[2] it is 3D reference axis, not only there is axial reference direction, but also there is radial direction to refer to zero-bit, will saves and give in 3.3.1
To illustrate.
[3] under natural system of coordinates:That is axis invariantIt is very special vector, it leads the time
Number also has invariance, and has very good mathematical operations performance；
For axis invariant, absolute derivative is exactly its Relative Derivations.Because axis invariant is the nature with invariance
Reference axis, therefore its absolute derivative perseverance is zero vector.Therefore, axis invariant has the invariance to time diffusion.Have:
[4] in natural coordinates system, pass through axial vectorAnd joint variableRotational coordinates battle array can be described directlyIt is not necessary to establish respective system for the rod piece in addition to root.Meanwhile needing the root coordinate system that defines for ginseng with unique
It examines, the measurement accuracy of system structure parameter can be improved；
[5] axial vector is appliedSuperior operational, by establish include topological structure, coordinate system, polarity, structure parameter and power
Learn the unified multiple axes system kinematics and kinetic model of the risk management of parameter.
Because of base vector elIt is and F[l]Any vector of consolidation, base vectorBe withAny vector of consolidation, againIt is F[l]AndShared unit vector, thereforeIt is F[l]AndShared base vector.Therefore, axis invariantIt is F[l]AndAltogether
Some refers to base.Axis invariant is the natural coordinates base of parametrization, is the primitive of multiple axes system.The translation of fixing axle invariant with
Translation and the rotation for rotating the coordinate system consolidated with it are of equal value.
It is reference with natural system of coordinates when system is in zero-bit, measurement obtains coordinate vectorIn kinematic pair
When movement, axial vectorIt is invariant；Axial vectorAnd joint variableUniquely determine kinematic pairRotation relation.
Therefore, using natural coordinates system, when system is in zero-bit, only a public referential need to be determined, without
Respective body coordinate system must be determined for each rod piece in system, because they are uniquely determined by axis invariant and natural coordinates.When
Carry out network analysis when, in addition to pedestal system, with rod piece consolidation other naturals system of coordinates only occur in it is conceptive, and with it is actual
It measures unrelated.Natural coordinates system is multiple axes system (MAS) theory analysis and engineering effect:
(1) the structural parameters measurement of system needs to measure with unified referential；Otherwise, not only engineering survey process is tired
It is trivial, and introduce different system and can introduce bigger measurement error.
(2) natural coordinates system is applied, in addition to root rod piece, the natural coordinates system of other rod pieces is by structure parameter and joint
Variable determines naturally, facilitates the kinematics and dynamics analysis of MAS system.
(3) in engineering, it can realize using optical measuring apparatus such as laser trackers to the accurate of fixing axle invariant
Measurement.
(4) due to the special case that kinematic pair R and P, screw pair H, Contact Pair O are cylindrical pair C, can simplify using cylindrical pair
MAS kinematics and kinetics analysis.
Define 3 invariants: the amount measured independent of one group of coordinate system is referred to as invariant.
Define 4 rotational coordinates vectors: around coordinate vectorTurn to Angle PositionCoordinate vectorFor
Define 5 translation coordinate vectors: along coordinate vectorIt is translatable to line positionCoordinate vectorFor
Define 6 natural coordinates: using natural coordinates axial vector as reference direction, the Angle Position of relative system zero-bit or line position
It sets, is denoted as ql, referred to as natural coordinates；The amount mapped one by one with natural coordinates is referred to as joint variable；Wherein:
Define 7 mechanical zeros: for kinematic pairT is carved at the beginning0When, the zero-bit of joint absolute encoderIt is different
It is set to zero, which is known as mechanical zero；
Therefore jointControl amountFor
Define 8 proper motion vectors: will be by natural coordinates axial vectorAnd natural coordinates qlDetermining vectorReferred to as certainly
Right motion vector.Wherein:
Proper motion vector realizes the Unified Expression of axis translation and rotation.It will be determined by natural coordinates axial vector and joint
Vector, such asReferred to as free movement vector, also known as free spiral rotation.Obviously, axial vectorBe it is specific from
By spiral.
Define 9 joint spaces: with joint natural coordinates qlThe space of expression is known as joint space.
Define 10 configuration spaces: the cartesian space of expression position and posture (abbreviation pose) is referred to as configuration space, is double
Vector space or the space 6D.
It defines 11 natural joint spaces: being reference with natural system of coordinates, pass through joint variableIt indicates, in system zero-bit
Must haveJoint space, referred to as natural joint space.
As shown in Fig. 2, given chain linkOrigin OlBy position vectorThe axial vector of constraintFor fixed axial vector, note
ForWherein:
Axial vectorIt is the natural reference axis of joint natural coordinates.CauseIt is axis invariant, therefore claimsIt is constant for fixing axle
Amount, it characterizes kinematic pairStructural relation, that is, natural coordinates axis has been determined.Fixing axle invariantIt is chain linkStructure
The natural description of parameter.
Define 12 natural coordinates shaft spaces: using fixing axle invariant as nature reference axis, with corresponding natural coordinates table
The space shown is known as natural coordinates shaft space, referred to as natural shaft space.It is the 3d space with 1 freedom degree.
As shown in Fig. 2,AndNot because of rod piece ΩlMovement and change, be constant structural reference amount.It has determined
Axis l is relative to axisFive structural parameters；With joint variable qlTogether, rod piece Ω is completely expressedlThe position 6D shape.It is givenWhen, the natural system of coordinates of rod piece consolidation can be by structural parametersAnd joint variableUniquely
It determines.Claim axis invariantFixing axle invariantJoint variableAndFor natural invariant.Obviously, not by fixing axle
VariableAnd joint variableThe joint nature invariant of compositionWith by coordinate systemTo F[l]Determining space bit
ShapeWith mapping relations one by one, i.e.,
Given multiple axes system D={ T, A, B, K, F, NT }, in system zero-bit, as long as establishing pedestal system or inertial system, with
And the reference point O on each axisl, other member coordinates also determine naturally.Substantially, it is only necessary to determine pedestal system or inertial system.
A given structure diagram with closed chain connected by kinematic pair, can select any of circuit kinematic pair,
The stator and mover that form the kinematic pair is separated；To obtain a loop-free tree, referred to as Span
Tree.T indicates the span tree with direction, to describe the topological relation of tree chain movement.
I is structural parameters；A is axis sequence, and F is rod piece referential sequence, and B is rod piece body sequence, and K is kinematic pair type sequence
Column, NT are the sequence, that is, non-tree for constraining axis.To take axis sequenceMember.Revolute pair R, prismatic pair P, screw pair H, Contact Pair
O is the special case of cylindrical pair C.
The basic topology symbol and operation for describing kinematic chain are the bases for constituting kinematic chain topology notation, and definition is such as
Under:
[1] kinematic chain by partial ordering set (] mark.
【2】A [l]For the member for taking axis sequence A；Because there is axis name l unique number to correspond toA [l]Serial number, thereforeA [l]It calculates
Complexity is O (1).
【3】For the father's axis for taking axis l；AxisComputation complexity be O (1).Computation complexity O () indicates calculating process
Number of operations, the number for being often referred to floating multiplication and adding.With floating multiplication with plus number expression computation complexity it is very loaded down with trivial details, therefore often
Using the primary operational number in algorithm cyclic process；Such as: the number of the operations such as joint position, speed, acceleration.
【4】To take axis sequenceMember；Computation complexity is O (1).
【5】llkTo take the kinematic chain by axis l to axis k, output is expressed asAndRadix note
For |llk|。llkImplementation procedure: it executesIfThen executeOtherwise, terminate.llkComputation complexity be O (|llk|)。
【6】lL is the son for taking axis l.The operation indicatesIn find the address k of member l；To obtain the sub- A of axis l[k]。
CauseWithout partial order structure, thereforelThe computation complexity of l is
【7】lL, which indicates to obtain, closes subtree by what axis l and its subtree were constituted, l L is the subtree without l；Recurrence executeslL is calculated
Complexity is
[8] branch, the increase of subtree and non-tree arc and delete operation are also necessary component part；To pass through dynamic
Span tree and Dynamic Graph describe primary topology.In branchllkIn, ifThen remember
I.e.Indicate the son that member m is taken in branch.
Define following formula or expression-form:
Axis and rod piece have one-to-one correspondence property；The attribute amount of between centersAnd the attribute amount between rod pieceWith partial order.
Agreement:Indicate attribute occupy-place；If attribute p or P be about position,It is interpreted as coordinate system's
Origin is to F[l]Origin；If attribute p or P be about direction,It is interpreted as coordinate systemTo F[l]。
AndIt should be interpreted as the function about time t respectivelyAndAndAndIt is t0Moment
Constant or constant array.But romanAndIt should be regarded as constant or constant array.
Arrange in the application: in kinematic chain symbolic operation system, attribute variable or constant with partial order, nominally
Index comprising indicating partial order；Comprising the upper left corner and lower right corner index or include the upper right corner and lower right corner index；They
Direction always by upper left corner index to lower right corner index, or by upper right corner index to lower right corner index, be narration in the application
Simplicity omits the description in direction sometimes, even if omitting, those skilled in the art are by character expression it will also be appreciated that this Shen
Please in use each parameter, for certain attribute accord with, their direction is always by the upper left corner index of partial order index to the lower right corner
Index, or by upper right corner index to lower right corner index.Such as:It can sketch (to indicate the vector that is translatable by k to l)；Indicate (by
K is to l's) line position；Indicate (by k to l's) translation vector；Wherein: r indicates that " translation " attribute symbol, remaining attribute symbol correspond to
Are as follows: attribute, which accords with φ, indicates " rotation "；Attribute, which accords with Q, indicates " rotational transformation matrix "；Attribute, which accords with l, indicates " kinematic chain "；Attribute accords with u table
Show " unit vector "；Attribute, which accords with w, indicates " angular speed "；Footmark is that i indicates inertial coodinate system or earth coordinates；Other footmarks can
Think other letters, or number.
The specification of symbols of the application and agreement are according to the partial order of kinematic chain, chain link be kinematic chain basic unit this two
What a principle determined, reflect the substantive characteristics of kinematic chain.Chain index expression is connection relationship, the reference of upper right index characterization
System.It is succinct using this symbolic formulation, accurate, convenient for exchange and wirtiting.Meanwhile they are the notations of structuring,
The element and relationship for forming each attribute amount are contained, is convenient for computer disposal, lays the foundation for computer auto-building modle.Index
Meaning needs the background i.e. context accorded with by attribute to be understood；Such as: if attribute symbol is translation type, the upper left corner refers to
Mark origin and the direction of indicates coordinate system；If attribute symbol is rotary type, the direction of upper left corner index expression coordinate system.
(1)lSPoint S in rod piece l；And the point S in S representation space.
(2)The origin O of rod piece kkTo the origin O of rod piece llTranslation vector；
In natural system of coordinates F[k]Under coordinate vector, i.e., by the coordinate vector of k to l；
(3)Origin OkTo point lSTranslation vector；
In F[k]Under coordinate vector；
(4)Origin OkTo the translation vector of point S；
In F[k]Under coordinate vector；
(5)Connecting rodAnd the kinematic pair of rod piece l；
Kinematic pairAxial vector；
AndExist respectivelyAnd F[l]Under coordinate vector；It is axis invariant, is a structural constant；
For gyration vector, gyration vector/angle vectorIt is free vector, i.e., the vector can free shift；
(6)Along axisLine position (translation position),
Around axisAngle Position, i.e. joint angle, joint variable are scalar；
(7) when lower left corner index is 0, mechanical zero is indicated；Such as:
Translation shaftMechanical zero,
Rotation axisMechanical zero；
(8) 0- three-dimensional null matrix；1- three-dimensional unit matrix；
(9) arrange: " " indicate continuation character；Indicate attribute occupy-place；Then
Power symbolIt indicatesX power；Upper right corner footmark ∧ orIndicate separator；Such as:OrForX power.
It indicatesTransposition, indicate to set transposition, not to member execute transposition；Such as:
For projection symbol, indicate vector or second-order tensor to the projection vector or projection sequence of reference base, i.e. coordinate vector
Or coordinate array, projection are dot-product operation " "；Such as: position vectorIn coordinate system F[k]In projection vector be denoted as
For multiplication cross symbol；Such as:It is axis invariantMultiplication cross matrix；Give any vectorMultiplication cross matrix beMultiplication cross matrix is second-order tensor.
The priority that multiplication cross accords with operation is higher than projection symbolPriority.Projection symbolPriority be higher than member access symbolOrMember accesses symbolPriority is accorded with higher than power
(10) projection vector of the unit vector in earth coordinatesUnit zero-bit vector
(11)By origin when zero-bitTo origin OlTranslation vector, and rememberIndicate position construction parameter.
(12)iQl, the rotation transformation battle array of opposite absolute space；
(13) using natural coordinates axial vector as reference direction, the Angle Position or line position of relative system zero-bit are denoted as ql, claim
For natural coordinates；Joint variableNatural joint coordinate is φl；
(14) orderly set r=[1,4,3,2] is given for oneT, remember r[x]Expression takes the xth row element of set r.Often
Note [x], [y], [z] and [w] expression takes the column element of the 1st, 2,3 and 4.
(15)iljIndicate the kinematic chain by i to j；llkTo take the kinematic chain by axis l to axis k；
Given kinematic chainIf n indicates Descartes's rectangular system, claimFor cartesian axis
Chain；If n indicates nature reference axis, claimFor natural axis chain.
(16) Rodrigues quaternary number expression-form:
Euler's quaternary number expression-form:
Quaternary number (also referred to as axis quaternary number) expression-form of invariant
1, based on the 1R posture of axis invariant against solution method
Projection is measurement of the rotating vector under linear space.Given chain linkR is revolute pair；Control is closed
Save variableMake the unit vector of consolidationWith desired unit vectorProjectionIt is optimal；Wherein:ForWith
Angle.The problem is referred to as to project inverse solution problem.
Work as given angleAfterwards, just, cosine and its half-angle just, cosine be constant；For convenience of expression, note
Agreement:Note D-H parameter index defers to father's index, under natural coordinates system
Parameter to defer to sub- index different.Rotational angle isWhen, definition:
It is obtained by formula (1)
Definition
Therefore have
Given kinematic chainiln, establish the 3D vector posture equation based on axis invariant:
Formula (5) be aboutN tie up 2 rank multinomial equations.Vector expression takes axial vector.
Establish axis invariantSecond order polynomial:
Formula (6) be aboutWithMultilinear equation, be axis invariantSecond order polynomial.It is given
Natural zero-bit vectorAsZero reference, thenAndRespectively indicate zero-bit vector and radial vector.
Formula (6) is asSymmetric partIndicate zero-bit axial tensor, skew-symmetric partIndicate radial axial tensor, respectively with axial apposition tensorIt is orthogonal, so that it is determined that three-dimensional nature axis is empty
Between；Formula (6) contains only a sine and cos operation, 6 long-pending operations and 6 and operation, computation complexity are low；Meanwhile passing through axis
InvariantAnd joint variableRealize coordinate system and polar parametrization.
Formula (6) is represented by
The posture equation that must be standardized by formula (7)
Given kinematic chainiln, consider formula (8), if iQnIndicate posture, only three independent freedom degrees；Then when |iln| when=3, there are the inverse solutions of 3R posture.Given unit vectorIt is obtained by formula (8)
If Indicate it needs to be determined that direction, then when |iln| when=2, there are the inverse solutions of 2R posture.To order
Bit vectorAndIt is obtained by formula (8)
If Indicate desired projection, then when |iln| when=1, there are the inverse solutions of 1R posture.
It is obtained by formula (5) and formula (10)
I.e.
Wherein:
IfSolution formula (11)
IfIt is expression of first degree that formula (11), which is degenerated:
It is obtained by formula (14)
Note formula (13) radical sign part
Because of τlBe aboutContinuous function.CauseTherefore AboutMonotone decreasing.WhenWhen, it is obtained by formula (12) and formula (16)
It is obtained by formula (17)
At this point, meetingThe smallest solution is
It to the inverse solution modeling of CE3 solar wing posture and is resolved using the above method below:
CE3 rover solar wing system p, O as shown in Figure 3pPositioned at revolute paircRpAxis centre, xpCross revolute paircRpAxis
And it is directed toward before rover to ypIt is directed toward on the left of rover, zpIt is determined by right hand rule, i.e. direction+Y photoarray normal direction.It patrols
Visual organ system is denoted as c.
Wherein:- solar wing angle of rotation,Sf、Sr- it is the preceding points outside of solar wing respectively and rear outer
Side point；- rover system origin OcTo solar wing system origin OpPosition vector coordinate under rover system；- patrol
Visual organ to the sun unit vector in the case where navigation is n coordinate.
It is obtained by formula (5)
Any point S coordinate under its system is denoted as on solar wingThen there is homogeneous coordinate transformation
Note rover Relative Navigation system rotation transformation battle array benQc, then havenQp=nQc·cQp, therefore have
cuS=cQn·nuS, (22)
puS=pQn·nuS。 (23)
Remember that device day vector is with respect to solar wing elevation angleIt is determined by formula (23)
Note is denoted as by solar wing normal direction to sun unit vector angleThen have
The solar wing control of CE3 rover includes both of which:
1. solar wing adjusts control
Solar wing adjusts control and refers to: givenMinimum thresholdControlIt is enough should to guarantee that solar wing generates
Power, guarantee that solar wing will not be overheated due to solar radiation again, i.e.,τ can be solved by formula (13) or (15)p。
Obviously,
2. solar wing optimum control
Solar wing optimum control refers to: controlGuarantee the maximum generated energy of solar wing.τ can be solved by formula (19)p, show
So,Below by the correctness of special case verification expression (13), formula (15) and formula (19).If
Formula (26) are substituted into formula (18) to obtain
Formula (26) are substituted into formula (19) to obtain
WhenWhen, it is obtained by formula (27)It is obtained by formula (28)
WhenWhen, it is obtained by formula (27)It is obtained by formula (28)
WhenWhen, it is obtained by formula (27)It is obtained by formula (28)
Obviously, the above results are consistent with intuitive physical meaning, it was demonstrated that the 1R based on axis invariant projects inverse solution method
Correctness.
By the inverse solution of above-mentioned solar wing it is found that there are two groups of optimal solutions.Since solar wing rotational angle is by structural constraint, the sun
Wing temperature restraint, solar wing and number pass antennas or omnidirectional antenna there may be mechanical interference, need to solar wing operation interval into
Row limits.Solar wing is controlled in the operation interval of permission, guarantees the maximization of generated energy.
As shown in figure 4, easily blocking the transmission of electromagnetic wave because solar wing is closer away from number biography antenna and omnidirectional antenna, causing to count
It passes communication or directional communication is interrupted or power attenuation, referred to as the mechanical interference of solar wing and antenna.Avoiding mechanical interference is to patrol
The basic constraint condition of visual organ mission planning, mast control, solar wing control.
The method for judging rover number biography antenna and solar wing or omnidirectional antenna and solar wing mechanical interference is as follows: note omnidirectional
Transmitting antenna and receiving antenna vertex are respectively SlAnd Sr, numeration passes antenna beam (Wave beam) axis and surface of emission intersection point is S.
At rover system c, S is establishedlRay equation, S to tracking telemetry and command stationrRay equation, S to data receiving station to tracking telemetry and command station
Ray equation solves intersection point by directional communication or digital transmission communication ray equation and solar wing plane equation.If intersection point exist and
In sun aerofoil, then it is considered as mechanical interference.The starting point for remembering ray is A, and ray unit vector iscnt, parameter t, correspondence
Point be denoted ascrt, ray parameter equation is at rover system c
crt=crA+cntT, (29)
I.e.
Inside angle point is B before remembering solar wing, and solar wing normal direction iscnp, ray is denoted as with any intersection point of solar wing planecrt。
Solar wing plane equation is
(crt-crB)T·cnp=0, (31)
I.e.
Formula (31) are substituted into formula (29) to obtain
In formula (33)When, illustrate that ray is orthogonal with solar wing normal direction, it is clear that there is no interference, i.e.,
CauseFormula (33) substitution formula (29) can be obtained into ray and solar wing plane point of intersectioncrt。
IfThen detect ray and solar wing
Interference.Certainly, it when Project Realization, needs to carry out more ray detections, and considers to interfere threshold degree.
The control of CE3 rover solar wing is rover task grouping, basic group of rover remote operating control system
At part.
The behaviour control of solar wing shown by 3D scene, can intuitively reflect " day the moon " and earth station, rover
Posture, solar wing motion state.Not only enable users to accurately to hold rover it is in-orbit when scene state, but also help to mention
The reliability of high software.In emulation testing, can be used to analyze search coverage, lunar surface landforms, detection time section, solar wing
And left solar wing power generation performance etc. makes an inspection tour the adaptability of detection mission with lunar surface, can optimize the design of rover power-supply system.
2, the inverse solution of 2R and 3R posture based on axis invariant and D-H parameter
For any one rod piece, D-H parameter only has 3 structural parameters and 1 joint variable, is conducive to simplify posture equation
The member that disappears.Since D-H parameter is usually name, it is difficult to obtain accurate engineering parameter, need through fixing axle invariant
Precise measurement, and corresponding accurately D-H system and D-H parameter are obtained by calculating.Therefore, 2R based on axis invariant be directed toward with
3R pose problem can be converted into the 2R based on D-H parameter and be directed toward and 3R pose problem.
Given 2R swivel-chainBy first unit vectorIt is directed toward expectation unit vectorIt asksAnd
φl, this is to orient inverse solution problem.
Wherein arrangeNatural system of coordinatesCorresponding D-H system is denoted asAccording to D-
The number of H coordinate system is accustomed to, kinematic pairCorresponding axis is denoted asIndex habit i.e. in D-H system defers to father's index, with
It is different that parameter under natural coordinates system defers to sub- index；Rotational angle isWhen, definition:
WhereinIt is by axisTo the torsional angle of axis l ' z；
If D-H parameter index is enabled to defer to sub- index,CauseTherefore it is indicated with D-H parameter
?
It is obtained by formula (36) last line
Therefore have
Have
Wherein:
Therefore have
It is obtained by formula (36) the first row
Therefore have
I.e.
Wherein:
Factor (38) and formula (42) not necessarily meet the 2nd row of formula (36), are obtained by formula (38) and formula (42)And φl
It is only possible to solve；Therefore need to solve the 2nd row of substitution formula (36)；If still setting up, true solution just can be obtained.
Given 3R swivel-chainAnd expectation postureAxis invariant sequenceJoint is asked to become
Measure sequenceThis is 3R posture against solution problem.
It is obtained by formula (38) and formula (42)By?Therefore have
So far, solve the problems, such as that the posture based on cartesian axis chain lacks versatility against solution method.By formula (42) and formula
(44) usually there are two groups of solutions known to, as shown in Figure 5.
Embodiment
The inverse solution modeling of CE3 data set structure posture is carried out using the above method below and is resolved:
As shown in fig. 6, the data set structure swivel-chain of CE3 rover isclm=(c, d, m], axis invariant sequence be [cnd,dnm].Ground data receiving station unit vector iscuS.Seek its angle sequence [φd,φm]。
It is by the structural parameters that accurate measurement obtains axis expression in terms of invariants
Relationship based on natural system of coordinates Yu D-H system, F={ F[l]| l ∈ A },Wherein: F[l]
For natural system of coordinates, F[l′]For D-H system；And have
Determine intermediate pointAnd D-H system origin Ol′。
It enablesAnd zl′Pass through axis invariant respectivelyWithAnd
It is defined asTo nlCommon vertical line.It is axisUnit coordinate vector.For indicatingZero
Position direction.
It enablesBy defining for axis torsional angle
It enablesIt is defined by articulation angle
Wherein: alAnd clRespectively axisTo the wheelbase and offset distance of axis l, alFor axisTo the torsional angle of axis l,For axisZero
Position.
In conclusion passing through fixing axle invariantWithExpress D-H parameter with can be convenient
AndZero-bit can be expressed simultaneously
Formula (45) are substituted into formula (48) and formula (49) obtains mast D-H parameter, as shown in table table 1-1.
Table 1-1 mast D-H parameter
Parameter in table is substituted into formula (39) respectively and formula (43) obtains
Formula (52) are substituted into formula (40) and obtain two groups of solutions
Formula (53) are substituted into formula (40) to obtain
The 2nd performing check because needing to substitute into formula (36) just can be solved really, therefore φlAt most there are two groups of solutions.
Consideration formula (50) and formula (51) pass through special case verification expression (54) and the correctness of formula (55):
When numerical value is calculated due to there is digital truncated error, no solution may cause；At this time, it may be necessary to willIn addition one
Fractional increments, then recalculate, to guarantee the existence of solution.
CE3 data set structure control module is shown through emulation, after adjustment rover yaw, is carried out number and is passed day line traffic control, antenna
Wave beam axially refers to earthwards always.Rover longitude and latitude is [- 28.6,40.06] °, and antenna beam direction is directed toward east always
Southern position.When rover longitude and latitude is [28.6,40.06] °, antenna beam direction is directed toward southwestern orientation always.Different
Longitude and latitude, it is correct that number passes antenna control result.
The above is only a preferred embodiment of the present invention, it is noted that for the ordinary skill people of the art
For member, without departing from the technical principles of the invention, several improvement and deformations can also be made, these improvement and deformations
Also it should be regarded as protection scope of the present invention.
Claims (6)
1. one kind is based on axis invariant and the inverse solution modeling method of DH parameter 1R/2R/3R, characterized in that
For controlling multi-axis machine device, the multi-axis machine device includes that rod piece set and joint are gathered, the rod piece set
In rod piece combined through the joint of joint set, joint set is converted into corresponding axis set, joint set
In the corresponding sub- axis set at the axis set in a joint, the axis of the axis set includes translation shaft and two type of rotation axis
Type；
The description multi-axis machine device is corresponded to using the axis set, and establishes dynamics side using the axis set
Journey, to control this multi-axis machine device；
It is reference with natural system of coordinates when system is in zero-bit, measurement obtains connecting rodAnd the coordinate vector of rod piece lWhen moving secondary motion, axial vectorIt is invariant；Axial vectorAnd joint variableUniquely determine the rotation of kinematic pair
Relationship；
When given joint variableAfter rotational angle, just, cosine and its half-angle just, cosine be constant；For convenience of expression,
Note
It is obtained by formula (1)
Definition
Then
Given kinematic chainiln, establish the robot 3D vector posture equation based on axis invariant:
Formula (5) be aboutN tie up 2 rank multinomial equations；Expression-form power symbol in formulaIt indicatesX times
Power；Upper right corner footmark ∧ orIndicate separator；It is axis invariantMultiplication cross matrix；1 is three-dimensional unit matrix；Vector
Expression takes axial vector；
Establish axis invariantSecond order polynomial:
Formula (6) be aboutWithMultilinear equation, be axis invariantSecond order polynomial；Become for rotation
Change matrix；Given nature zero-bit vectorAsZero reference, thenAndRespectively indicate zero-bit vector
And radial vector；Formula (6) is asSymmetric partIndicate zero-bit axial tensor,
Skew-symmetric partIndicate radial axial tensor, respectively with axial apposition tensorIt is orthogonal, so that it is determined that three
Tie up nature shaft space；
Formula (6) is expressed as
The robot pose equation that must be standardized by formula (7):
In formula,iQnIndicate posture, axial vector It is axis invariantMultiplication cross matrix.
2. according to claim 1 be based on axis invariant and the inverse solution modeling method of DH parameter 1R/2R/3R, characterized in that
If kinematic pairR indicates revolute pair,iQnIndicate posture, only three independent freedom degrees；Then when |iln| when=3,
There are the inverse solutions of 3R posture；
Given unit vectorIt is obtained by formula (8)
In formula,For unit vectorIn the projection vector of earth coordinates.
3. according to claim 1 be based on axis invariant and the inverse solution modeling method of DH parameter 1R/2R/3R, characterized in that
If Indicate it needs to be determined that direction, then when |iln| when=2, there are the inverse solutions of 2R posture；
Given unit vectorAndIt is obtained by formula (8)
In formula,For unit vectorIn the projection vector of earth coordinates.
4. according to claim 1 be based on axis invariant and the inverse solution modeling method of DH parameter 1R/2R/3R, characterized in that
Given unit vectorAndIt is obtained by formula (8)
If Indicate desired projection, then when |iln| when=1, there are the inverse solutions of 1R posture；
It is obtained by formula (5) and formula (10)
Make the unit vector of consolidationluSWith desired unit vectorProjectionIt is optimal, meetThe smallest solution is
Wherein:ForluSWithAngle.
5. according to claim 3 be based on axis invariant and the inverse solution modeling method of DH parameter 1R/2R/3R, characterized in that
It will be directed toward problem based on the 2R of axis invariant and be converted into the direction problem of the 2R based on D-H parameter, orient inverse solution and calculate step
Are as follows:
For giving 2R swivel-chainWherein,L, k are rod piece；By first unit vectorIt is directed toward expectation
Unit vectorFor unit vectorIn the projection vector of D-H system；Seek the natural joint coordinate of rod pieceAnd φl:
Wherein arrangeNatural system of coordinatesCorresponding D-H system is denoted asIt is sat according to D-H
The number of mark system is accustomed to, kinematic pairCorresponding axis is denoted asIndex habit i.e. in D-H system defers to father's index, with oneself
It is different that parameter under right coordinate system defers to sub- index；Rotational angle isWhen, definition:Its
InIt is by axisTo the torsional angle of axis l ' z；
D-H parameter index is enabled to defer to sub- index, i.e.,CauseTherefore it is indicated with D-H parameter
It is obtained by formula (36) last line
In formula, if withIndicate attribute occupy-place, then the expression-form in formulaIndicate that member accesses symbol；
Therefore have
Have
Wherein:
Therefore have
It is obtained by formula (36) the first row
Therefore have
I.e.
Wherein:
Factor (38) and formula (42) not necessarily meet the 2nd row of formula (36), are obtained by formula (38) and formula (42)And φlOnly
It may solution；The 2nd row that substitution formula (36) will likely be solved again is really solved if still setting up.
6. according to claim 5 be based on axis invariant and the inverse solution modeling method of DH parameter 1R/2R/3R, characterized in that
Given 3R swivel-chainAnd expectation postureAxis invariant sequenceSeek joint variable sequence
ColumnWherein,L, k are rod piece；It converts the 3R pose problem based on axis invariant to based on D-H parameter
3R pose problem, the inverse solution calculating formula of posture are as follows:
It is obtained by formula (38) and formula (42)By?Therefore have
In formula,Representing matrixThe 4th row element.
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Effective date of registration: 20200617 Address after: Room 8011, innovation and entrepreneurship building, national thousand talents plan, No.7, Yingcui Road, Jiangning Development Zone, Nanjing, Jiangsu Province Patentee after: Nanjing ansheman Robotics Research Institute Co., Ltd Address before: 211100 No. 2289 Tianyuan East Road, Jiangning District, Nanjing, Jiangsu (Jiangning Gao Xinyuan) Patentee before: Ju Hehua |