CN105740503B - The optimum design method of six axis vibration-isolating platforms - Google Patents
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Abstract
A kind of method that the present invention discloses uniform Jacobian matrix of scale that general six axis parallel institution is generated based on dual quaterion, optimizes the movenent performance of six axis vibration-isolating platforms.The standard of scaling factor is defined as the ratio between the norm of two quaternary numbers by the rotation and movement that end effector is respectively indicated using two quaternary numbers, exports the uniform Jacobian matrix of scale.Using local motion isotropic as performance indicator, the vibration-isolating platform after optimization has the smallest movement coupling.
Description
Technical field
The invention belongs to the kinematics of parallel institution, optimization design research field, are based especially on the vibration isolation of parallel institution
The movenent performance optimization design of platform.
Background technique
High-precision spatial system is accurately directed to the very quiet environment of needs, space interferometer, laser communication gear etc.
High-precision spacecraft need to reach nanoscale kinetic stability.Reach this strict requirements, many Spacecraft guidance and controls and control
Method processed is used to reduce influence of the spacecraft micro-vibration to precision instrument.A kind of technology of most important one be in interference source and
Vibrating isolation system is installed between precision instrument.The six axis vibrating isolation systems based on Stewart mechanism are a kind of preferable solutions, closely
Have many researchers over year and has carried out research about this platform.Stewart mechanism is as a kind of six-degree-of-freedom parallel
Structure has the advantages such as compact-sized, high rigidity, low inertia.However, so far, vibrating the performances such as attenuation rate and vibration isolation bandwidth
Index still has greatly improved space, and one of the main reasons is that the movenent performance of vibration-isolating platform is not optimal.Poor fortune
It is more complicated with controller design that dynamic performance handles the kinetic model of vibration-isolating platform, it is difficult to realize design effect.Therefore,
In order to further increase anti-vibration performance, the optimization design to vibration-isolating platform is indispensable.
For the optimization design of six axis motion platforms, there are many kinds of these movenent performance indexs of movenent performance index at present
All rely on kinematics Jacobian matrix.Jacobian matrix reflects the mapping relations between system input speed and output speed.
All rows are all the Pu Lvke vectors for describing certain lines in space in traditional Jacobian matrix, include rotation and mobile two portions
Point.Since physical unit that is mobile and rotating is inconsistent, traditional Jacobian matrix is usually heterogeneous.It is non-homogeneous based on scale
The optimization design of Jacobi will lead to unreasonable result.Numerous researchers use a variety of methods and are dedicated to solving this asking
Topic, is broadly divided into two classes, size scaling method and separation matrix method.Size scaling method is by Jacobian matrix divided by a length
Value, obtains the uniform Jacobian matrix of scale, which is also referred to as characteristic length, natural length etc..For example, Ma etc. will be moved
The center of platform to the joint of the platform distance as scaling value.Kim etc. and LIU etc. uses three points on moving platform
Its position and posture are described, the uniform Jacobian matrix of scale is exported, belongs to implicit scaling method.The suggestions such as KHAN will scale
Ratio value is optimized as design variable.Separation matrix method is that Jacobian matrix is separated into rotation and mobile two parts,
Two parts are optimized respectively.For example, using the method define two motion sensitivity indexs in Cardou etc.: maximum
Rotation and maximum are mobile.The problem of Jacobian matrix homogenizes at present solves still without thorough, and size scaling method relates to how really
The problem of determining scale value, now still without a criterion, and partition method can not combine the rotation of six shaft platforms and mobile property
The optimization of energy.
Scaling method is that a comparison is reasonable and widely used method, lacks the standard of determining zoom factor at present.
The inconsistency of the mathematical notation for having its source in rigid body translation and rotation of scale inhomogeneities, this makes rotation and mobile amount
Guiding principle is inconsistent.For example, frequently, rigid body translation indicates that Rigid Body in Rotation With is then indicated with Eulerian angles with trivector.In order to solve this
A problem needs to parameterize the rotation of rigid body and movement using unified mathematical tool.Vibration-isolating platform is a micro-nano
Movement mechanism has six-freedom degree locomitivity, has very strong movement coupling between each freedom degree.In order to improve dynamics
The precision of modeling and linearisation enhances the effect of vibration-isolating platform active control, it is necessary to be reduced to the movement coupling of platform most
It is small.
Therefore the technical solution for needing one kind new is to solve the above problems.
Summary of the invention
The purpose of the present invention is in view of the deficienciess of the prior art, providing a kind of side of six axis vibration-isolating platform optimization designs
Method and result.
To solve the above problems, following technical solution can be used in six axis vibration-isolating platform optimization designs of the invention:
A kind of uniform Jacobian matrix generation method of scale of the multiaxis parallel institution based on dual quaterion, described is more
Axis parallel institution includes six telescopic rods in parallel of upper mounting plate, lower platform and the upper and lower platform of connection, which is characterized in that the party
Method includes the following steps:
(1), the rotation of rigid body is indicated with quaternary number:
For a quaternary number q, if | | q | |=1, which is referred to as unit quaternary number, and unit quaternion set is closed
In quaternary number multiplication be a groupIn order to illustrate unit quaternion and proper orthogonal groupRelationship, define a list
Position quaternary numberWherein, n is unit vector, and θ is the amplitude of rigid body rotary shaft, rigid body rotation
Shaft is determined by unit vector n and its amplitude θ；For anyProductAnd mapping x → ε x ε*:It is of equal value with rotation R (θ, n)；Wherein, x is gyration vector, and R is the unique Rotation matrix of Rigid Body in Rotation With；
(2), the movement of rigid body is indicated with quaternary number:
In order to establish mobile the contacting with unit quaternion of rigid body, by the special Euclidean group of dual quaterion expression
Thought, according to the displacement vector t of rigid body and indicate unit turn quaternary number ε construct a quaternary numberThe two quaternarys
Number ε, λ will meet two constraint equations of unit dual quaterion:
In formulaIndicate quaternary number real part；
Quaternary number λ may be constructed such that by the displacement vector t and quaternary number ε of rigid body:
λ=c1T ε, or λ=c2εt
C in formula1And c2It is any non-zero constant；
(3), the standard of scaling factor:
Due to quaternary numberIt is normed algebra, can be measured with the mould of quaternary number, therefore, by scaling factor
Standard is defined as the ratio between two quaternary number norms
D=| | ε | |/| | λ | |；
(4), according to the determination of the above-mentioned size scaling factor, which is imported to the ruler of multiaxis parallel institution
Uniform Jacobian matrix is spent, the result of the uniform Jacobian matrix of multiaxis parallel institution scale is obtained
X=(ε λ)TThe rotation and movement for indicating end effector, are the pose coordinate of end effector, end effector
Generalized velocity useIt indicating, vector L indicates to drive diarthrodial coordinate,It is driving speed.
Compared with the existing technology, the physical quantity system of Rigid Body in Rotation With and rigid body movement is carried out present invention employs quaternary number
One, for the size scaling problem generated in unification, equally size scaling directly can be carried out using the ratio between norm of quaternary number
The determination of the factor, due to either rotating and moving physical quantity or the size scaling factor is to use quaternary number as parameter list
Show, rotation and the mobile parametrization carried out using unified mathematical tool hence for rigid body can be to six axis vibration-isolating platforms
Design optimizes.
Further, according to the above method, the export of the uniform Jacobi of scale of six axis vibration-isolating platforms, movenent performance are realized
Optimization process and result:
(1), the result of the uniform Jacobian matrix of scale of six axis vibration-isolating platforms
The scaling factor of six axis vibration-isolating platforms is d=| | ε | |/| | λ | |=1/ | | t | |, it is the origin of two platforms
The inverse of distance；The concrete form of the uniform Jacobian matrix of scale of six axis vibration-isolating platforms are as follows:
In formula, ai(i=1,2 ... 6) and bi(i=1,2 ... 6) be respectively vibration-isolating platform upper and lower flexural pivot position vector,
fi=λ+ε ai-bi(i=1,2 ... 6), are customized quaternary numbers by ε.
The optimization process of (2) six axis vibration-isolating platforms
Solve design variable ra、rb, α and β so that Condition Number of Jacobian Matrix:
Reach minimum value
Design variable need to meet constraint condition:
H=1, ra≥0,rb≥0,0≤α≤2π/3,0≤β≤2π/3；
Platform radius is ra, pedestal radius is rb, when between upper mounting plate, pedestal without rotation, height is that h, α and β distinguish between the two
For the central angle of the adjacent flexural pivot position of upper and lower platform；
Calculate JdAll singular values:
In formula
Z=(rarbsin[(α-β)/2])2,
Due to the value of exchange parameter α and β, or exchange raAnd rbValue, σi(Jd) (i=1,2 ... 6) remain unchanged；And
And as α=β, σ2(Jd)=0, mechanism singularity；Therefore increase by two constraint conditions on the basis of former optimization problem to reduce
Improve optimization efficiency in the space of search:
The optimum results of (3) six axis vibration-isolating platforms
Optimal design parameters are
ra=0.84, rbπ/3=0.84, α=0, β=2；
From optimum results as can be seen that the optimal Stewart mechanism of kinematics isotropism is a 3-3SPS configuration,
There is the position of three pairs of flexural pivots to be overlapped on upper and lower platform.When vibration-isolating platform equal proportion scales, isotropic property retention is constant.
Detailed description of the invention
Fig. 1 is the geometric description figure of six axis vibration-isolating platforms in the present invention.
Fig. 2 is optimization algorithm convergence process figure in the present invention.
Fig. 3 is the preferred configuration figure of vibration-isolating platform in the present invention.
Specific embodiment
In the following with reference to the drawings and specific embodiments, the present invention is furture elucidated, it should be understood that these embodiments are merely to illustrate
It the present invention rather than limits the scope of the invention, after the present invention has been read, those skilled in the art are to of the invention each
The modification of kind equivalent form falls within the application range as defined in the appended claims.
1 dual quaterion and the uniform Jacobian matrix of scale
In mathematics and mechanics, the set of dual quaterion is a Clifford algebra, can be with spinor, 4 × 4 homogeneous
Transformation matrix is mutually converted, and can be used to indicate that the displacement of Rigid Body In Space.Dual quaterion is the orderly of two quaternary numbers
Right, the quaternary number of real part indicates rotation, and the quaternary number of antithesis part contains movement.But directly by unit antithesis four
When pose coordinate of first number as vibration-isolating platform end effector, export Jacobian matrix be still it is heterogeneous, this isWhat the geometric properties of geometry determined.Then, we combine dual quaterion and scaling method, export general in parallel
The uniform Jacobian matrix of the scale of mechanism.
Rigid Body in Rotation With can be by unique Rotation matrixIt indicates.R is the linear operation determined by rotary shaft, rotation
Shaft is determined by unit vector n and its amplitude θ.When specifying these elements, rotation can be expressed as R (θ, n).
In order to describe vector x around the rotation of n, x is decomposed into two parts by us:
X=(xn) n+ (n × x) × n (1)
Due to vector n, n × x and (n × x) × n be it is mutually orthogonal, then the rotation of vector x are as follows:
R (x)=(xn) n+R [(n × x) × n]=(xn) n+ (n × x) sin θ+[(n × x) × n] cos θ (2)
Above-mentioned equation can also be write as well-known Euler-Rodrigues form
R (x)=x+ (n × x) sin θ+[n × (n × x)] (1-cos θ) (3)
Consider setBy a vector q and a scalar q when its element0Composition to (q q0), it indicates are as follows: q
=(q q0)=(q1 q2 q3 q0) or q=q1i+q2j+q3k+q0.Wherein, i2=j2=k2=-1, and ij=-ji=k, jk=-
Kj=i, ki=-ik=j.So, for any q=(q q0) and p=(p p0), quaternion product
(q, p) → qp=(q0p+p0q+q×p q0p0-q·p) (4)
It is q, the bilinear form of p.The operation be it is combinative, since comprising cross product, without commutative, this makes
It is an Associative algcbra.Set with this structureIt is named asIts element is known as quaternary number.One quaternary number q
=(q, q0) component part q and q0Regard the imaginary part of q as respectivelyAnd real partIt is different from plural number
, the imaginary part of q is thenOn vector.Importantly, oneOn vector be also quaternary that a real part is zero
Number, meets all algorithms of quaternary number.
Quaternary number q*=(- q, q0) it is known as q=(q, q0) conjugation.Map q → q*It is vector spaceAutomorphism,
Due to (qp)*=p*q*, then it is the antiatomorphism of Algebraic Structure.Due to
Be two positive numbers and, then define a quaternary number norm be scalarIt is reasonable.Obviously,
When q=0, | | q | |=0.Moreover, to any
This means that normPresence makeFor a normed algebra.
For a quaternary number q, if | | q | |=1, which is referred to as unit quaternary number, unit quaternion collection
Closing about quaternary number multiplication is a groupIn order to illustrate unit quaternion withRelationship, define a unit four
First numberFor anyProductAnd mapping x → ε x ε*:It is of equal value with rotation R (θ, n).According to following calculating:
εxε*=((xn) n+ (n × x) sin θ+[(n × x) × n] cos θ 0) (7)
It can be seen that equation (7) and equation (2) are consistent.
In order to establish mobile the contacting with unit quaternion of rigid body, indicated by dual quaterionThought, root
According to displacement vector t and indicate that unit turn quaternary number ε constructs a quaternary numberThe two quaternary numbers will meet unit pair
Two constraint equations of even quaternary number:
λ may be constructed such that by vector t and quaternary number ε:
λ=c1T ε, or λ=c2εt (9)
C in formula (9)1And c2It is any non-zero constant.
Due to quaternary numberIt is normed algebra, can be measured with the mould of quaternary number, therefore, by scaling factor
It is reasonable that standard, which is defined as the ratio between two quaternary number norms,
D=| | ε | |/| | λ | | (10)
In theory of mechanisms, the operation between vector space Euclid norm and quaternary number norm is often used.If
Quaternary number is regarded asMiddle element, then quaternary number norm hasOn Euclid norm meaning.Define quaternary number
Inner product:Operation it is as follows
(q, p) → qp=q0p0+q1p1+q2p2+q3p3 (11)
Since quaternary number conjugation, norm and inner product are linear operators, obtain extracting Jacobian matrix from kinematical equation
Transform:
For general six axis parallel institution, with X=(ε λ)TThe rotation and movement for indicating end effector are that end executes
The generalized velocity of the pose coordinate of device, end effector is usedIt indicating, vector L indicates to drive diarthrodial coordinate,
It is driving speed.The generalized Jacobian J of parallel institution isWithBetween Linear Mapping matrix:
Initially set up the closed loop pose equation of parallel institution:
F (L, ε, t)=0 (14)
The time-derivative of accounting equation (14), convolution (9), it is available aboutWithEquation:
For parallel institution, matrixGenerally it is reversible, it is available
According to the definition (10) of scaling factor and generalized Jacobian (16), the uniform Jacobean matrix of scale is exported
Battle array:
The uniform Jacobian matrix of scale of 2 six axis vibration-isolating platforms
Six axis vibration-isolating platforms are designed based on Stewart mechanism, as shown in Figure 1.Control force will be installed on platform
Square gyro or sensitive load, pedestal will be mounted on satelloid.It is connected by flexural pivot with six legs between platform and pedestal.Every
One actuator is installed respectively in leg.Upper and lower flexural pivot is respectively mounted in a plane, by circle distribution.Platform radius is ra, base
Seat radius is rb, when between upper mounting plate, pedestal without rotation, height is h between the two.
Establish two rectangular coordinate systems: global coordinate system Oxyz and local coordinate system O ' x ' y ' z '.Oxyz is fixedly arranged at pedestal
The center point, O ' x ' y ' z ' are fixedly arranged at platform the center point.Position of the platform flexural pivot center in Oxyz is denoted as vector ai(i=1,2 ...
6), position of the pedestal flexural pivot central point in Oxyz is denoted as vector bi(i=1,2 ... 6).The Rotation matrix R of platform opposite base
It is parameterized with unit quaternion ε, it is meant that Rai=ε aiε*.O ' x ' y ' z ' indicates relative to the position of Oxyz with vector t, every
The length l of legi(i=1,2 ... 6) indicate, direction unit vector ei(i=1,2 ... 6) is indicated.
First according to the closed-loop vector figure in Fig. 1, the closed loop location equation of vibration-isolating platform is established
liei=t+ ε aiε*-bi(i=1,2 ... 6) (18)
ε a in equation (18)iε*It is vector aiCoordinate representation in coordinate system Oxyz.According to formula (9) construct another four
Equation (18) right side is multiplied ε, obtained by first number λ=t ε
lieiε=λ+ε ai-biε (i=1,2 ... 6) (19)
In order to eliminate the vector e in equation (19)i, it is noted thatAndSoTherefore, calculating side
The inner product of journey (19) and itself, obtains
The relationship of vibration-isolating platform input speed and platform speed in order to obtain defines four by equation (20) to time derivation
First number fi=λ+ε ai-bi(i=1,2 ... 6), and abbreviation can obtain by ε
It is isolated from equation (21) belowWithAccording to transform (12), can obtainWithFurther obtain the explicit shape about vibration-isolating platform input speed and end effector generalized velocity
Formula
There are identical form in equation (22) and (15).At this point, generalized Jacobian is one 6 × 8 matrix
Scaling factor is d=| | ε | |/| | λ | |=1/ | | t | |, the initial point distance of precisely two coordinate systems is fallen
Number.Finally, the uniform Jacobian matrix of scale of six axis vibration-isolating platform of derived space
The uniform Jacobian matrix J of scaledIt is still 6 × 8 matrixes, every a line is no longer the Pu Lv of certain lines in space
Gram vector, but the leg long vector of vibration-isolating platform and the combination of dual quaterion.It can not find out it in three-dimensional space at present
In physics or mechanics meaning, however due to JdThe dimension of middle element is 1, therefore six singular values of the matrix are dimensionless
, it will not change with the variation of physical unit, the optimization process and result of vibration-isolating platform will be provided based on this below.
3 optimization problems
Every leg of vibration-isolating platform it is independent move can all make platform position and posture generate variation, each freedom of motion it
Between have very strong coupling.The strong coupling precision that will reduce kinetics equation linearisation, increases the difficulty of Large System Control.In order to
The movement coupling for reducing vibration-isolating platform, selects performance optimizing index of the kinematics isotropism as vibration-isolating platform.It examines simultaneously
Consider the space that vibration-isolating platform only has several microns, only need to optimize the geometric parameter that the platform is in initialization position, this
When optimizing index be known as local motion isotropism.
In order to reduce the number of optimization design variable, on platform and pedestal the position at flexural pivot center according to Cyclic Symmetry side
Formula is arranged on two circles.When vibration-isolating platform is in initialization position, t=(0 0 h)T.Therefore, the geometry knot of vibration-isolating platform
Structure can be by parameter ra、rb, α, β and h completely determine, as shown in Figure 1.Due to JdIt is nondimensional, vibration-isolating platform progress equal proportion
When scaling, JdRemain constant.Then, similar with scaling method, the scaling factor of design variable is set as flat
The height h of platform, is equivalent to vibration-isolating platform equal proportion zooming to h=1.Do not remove only the dimension of optimization design variable in this way,
And the optimization problem of quintuple space is made to be reduced to four-dimensional problem, reduce the time that optimization calculates.
Therefore, the optimization design problem of vibration-isolating platform can state in this way:
Solve design variable ra、rb, α and β so that Condition Number of Jacobian Matrix
Wherein, σminAnd σmaxIt is the uniform Jacobian matrix J of scaledMinimum and maximum at initialization position is unusual
Value.As κ (Jd) close to 1 when, vibration-isolating platform the directive movement of institute more evenly, it is more independent.As κ (Jd) approach infinity when, table
Bright vibration-isolating platform becomes more uncontrollable close to unusual pose.Isotropism index can not only make the movement coupling of the platform
It reduces, while the unusual pose for avoiding it.
Design variable need to meet constraint condition
H=1, ra≥0,rb≥0,0≤α≤2π/3,0≤β≤2π/3 (26)
First constraint h=1 can guarantee platform and pedestal processing discrete state, i.e. li(i=1,2 ... 6), vibration isolation by > 0
Platform is not in that joint is unusual.Constrain ra>=0 and rb>=0 ensures non-negative structure size.Constrain 0≤α≤π/3 and 0≤β≤
π/3 are avoided that six legs of vibration-isolating platform interfere.
4 optimization process and result
This section will provide result by local motion isotropism index to vibration-isolating platform optimization design.First according to
To the setting of optimization problem in 4 sections, by the uniform Jacobian matrix J of scaledIn element indicated with design variable, that is, determine leg it is long
li(i=1,2 ... 6) and quaternary number fiThe expression formula of (i=1,2 ... 6).The position of flexural pivot point is in respective coordinate on platform and pedestal
Position a in systemi(i=1,2 ... 6) and bi(i=1,2 ... 6) is by design variable ra、rb, α and β determine:
When vibration-isolating platform is in initialization position, t=(0 0 1)T, rotate at this time and mobile quaternary number be respectively
According to formula (20), and consider aiλ=0 and biλ=0, we obtain the long l of legiWith quaternary number fi
Six long l of legiIt is mutually equal, Wo MenyongIt indicates.By formula (29)
It substitutes into (24), is easy to get Jacobian matrix J of uniform sizedForm.Although JdProject is more in matrix, but is one
A sparse matrix, and JdSingular value σ (Jd) be characteristic value square root, can according to determine conditional number change
Law.
In above formula
Z=(rarbsin[(α-β)/2])2.Further, we calculate JdAll singular values
Singular value in formula (31) is not arranged according to sequence from big to small.If can be seen that exchange parameter α and β
Value, or exchange raAnd rbValue, σi(Jd) (i=1,2 ... 6) remain unchanged.Moreover, as α=β, σ2(Jd)=0, mechanism are odd
It is different.Therefore, we can increase by two constraint conditions on the basis of former optimization problem to reduce the space of search, improve optimization
Efficiency:
Although we have obtained the sign format of all singular values, the size of each singular value is still difficult to judge.It is right
The optimization of conditional number can only be carried out by numerical method.We using MATLAB provide GAs Toolbox to the problem into
Row numerical optimization, after 90 cycle calculations, conditional number κ (Jd) 2.06 are converged to, process is as shown in the figure.Optimal design parameters
For
ra=0.84, rbπ/3=0.84, α=0, β=2 (33)
From optimum results as can be seen that the optimal Stewart mechanism of kinematics isotropism is a 3-3SPS configuration,
There is the position of three pairs of flexural pivots to be overlapped on upper and lower platform, as shown in Figure 3.When vibration-isolating platform equal proportion scales, isotropic performance is protected
It holds constant.
Claims (1)
1. a kind of optimum design method of six axis vibration-isolating platforms, the six axis vibration-isolating platforms include upper mounting plate, lower platform and company
It connects, six telescopic rods in parallel of lower platform, which is characterized in that this method is the multiaxis parallel institution based on dual quaterion
The uniform Jacobian matrix generation method of scale, include the following steps:
(1), the rotation of rigid body is indicated with quaternary number:
For a quaternary number q, if | | q | |=1, which is referred to as unit quaternary number, and unit quaternion set is about four
First number multiplication is a groupIn order to illustrate unit quaternion and proper orthogonal groupRelationship, define a unit four
First numberWherein, n be unit vector, θ be rigid body rotary shaft amplitude, rigid body rotary shaft by
Unit vector n and its amplitude θ is determined；For anyProductAnd mapping x → ε x ε*:With
It is of equal value for rotating R (θ, n)；Wherein, x is gyration vector, and R is the unique Rotation matrix of Rigid Body in Rotation With；
(2), the movement of rigid body is indicated with quaternary number:
In order to establish mobile the contacting with unit quaternion of rigid body, by the special Euclidean group of dual quaterion expressionThink of
Think, according to the displacement vector t of rigid body and indicates that unit turn quaternary number ε constructs a quaternary numberThe two quaternary numbers ε,
λ will meet two constraint equations of unit dual quaterion:
Re () indicates quaternary number real part in formula；
Quaternary number λ may be constructed such that by the displacement vector t and quaternary number ε of rigid body:
λ=c1T ε, or λ=c2εt
C in formula1And c2It is any non-zero constant；
(3), the standard of scaling factor:
Due to quaternary numberIt is normed algebra, can be measured with the mould of quaternary number, therefore, by the standard of scaling factor
It is defined as the ratio between two quaternary number norms
D=| | ε | |/| | λ | |；
(4), according to the determination of above-mentioned scaling factor, the scale that scaling factor d is imported multiaxis parallel institution is equal
Even Jacobian matrix obtains the result of the uniform Jacobian matrix of multiaxis parallel institution scale
X=(ε λ)TThe rotation and movement for indicating end effector, are the pose coordinates of end effector, end effector it is wide
Adopted speed is usedIt indicating, vector L indicates to drive diarthrodial coordinate,It is driving speed；
Realize the export of the uniform Jacobi of scale of six axis vibration-isolating platforms, movenent performance optimization process and result:
(1), the result of the uniform Jacobian matrix of scale of six axis vibration-isolating platforms
The scaling factor of six axis vibration-isolating platforms is d=| | ε | |/| | λ | |=1/ | | t | |, it is the initial point distance of two platforms
Inverse；The concrete form of the uniform Jacobian matrix of scale of six axis vibration-isolating platforms are as follows:
In formula, ai(i=1,2 ... 6) and bi(i=1,2 ... 6) be respectively vibration-isolating platform upper and lower flexural pivot position vector, fi=λ
+εai-bi(i=1,2 ... 6), are customized quaternary numbers by ε；
The optimization process of (2) six axis vibration-isolating platforms
Solve design variable ra、rb, α and β so that Condition Number of Jacobian Matrix:
Reach minimum value
Design variable need to meet constraint condition:
H=1, ra≥0,rb≥0,0≤α≤2π/3,0≤β≤2π/3；
Platform radius is ra, pedestal radius is rb, when between upper mounting plate, pedestal without rotation, height is that h, α and β are respectively between the two
The central angle of the upper and lower adjacent flexural pivot position of platform；
Calculate JdAll singular values:
In formula
Due to the value of exchange parameter α and β, or exchange raAnd rbValue, σi(Jd) (i=1,2 ... 6) remain unchanged；Moreover, working as α
When=β, σ2(Jd)=0, mechanism singularity；Therefore increase by two constraint conditions on the basis of former optimization problem to reduce search
Improve optimization efficiency in space:
The optimum results of (3) six axis vibration-isolating platforms
Optimal design parameters are
ra=0.84, rbπ/3=0.84, α=0, β=2；When vibration-isolating platform equal proportion scales, isotropic property retention is constant.
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| CN110259879B (en) * | 2019-06-12 | 2020-06-09 | 北京理工大学 | Powerless feedback vibration isolation control method and system for electric Stewart structure |
| CN111651923B (en) * | 2020-06-03 | 2023-03-28 | 北京航宇振控科技有限责任公司 | Multi-degree-of-freedom passive vibration isolation system optimization design method |
| CN111707986B (en) * | 2020-06-18 | 2022-05-27 | 重庆邮电大学 | Three-dimensional parameter estimation method based on sparse area array |
| CN112068419B (en) * | 2020-07-28 | 2021-09-03 | 北京航空航天大学 | Flexible satellite pointing tracking control method containing six-degree-of-freedom vibration isolation platform |
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