CN102968665A - Forward kinematics solving method of parallel robot - Google Patents

Abstract

The invention discloses a forward kinematics solving method of a parallel robot. When the method is applied to solve the forward kinematics problem of the parallel robot, the characteristic of easily solving inverse kinematics of the method is utilized, the forward kinematics solving problem of the parallel robot is converted into the equivalent minimal optimization problem, and the numerical optimization method is used for solving the problem. The method is integrated with the advantage of differential evolution having excellent global exploration performance on the solution space and the characteristic of pattern search having good local development capability on the solution space; and the mutual mechanisms of the two optimization algorithms are merged so that the search performance of the algorithms is improved. Compared with the traditional method, the method provided by the invention has high optimal solution searching accuracy and high rate of convergence while solving the forward kinematics problem of a 6-SPS type parallel robot; and simultaneously, the method also can be popularized to solve the forward kinematics problems of the parallel robots of other types.

Description

The positive motion of parallel robot is learned method for solving

Technical field

The present invention proposes the 6-SPS parallel robot positive motion of a kind of Schema-based search (Pattern Search, PS)-differential evolution (Differential Evolution, DE) and learns method for solving, belongs to the Robotics field.

Background technology

With the serial machine people relatively, parallel robot has the characteristics such as high rigidity, high fine motion precision and large load-bearing capacity, therefore is used widely in some high-end technology fields such as micro-manipulating robot, parallel machine and coordinate measuring machine etc.6-SPS Kinematics of Parallel Robot problem can be divided into inverse kinematics and positive motion to be learned, and wherein the former is that the bar of finding the solution its each pull bar in parallel according to position and the attitude of moving platform on the parallel robot is long; Latter is opposite, and positive motion knowledge topic is position and the attitude of determining moving platform in the robot according to the bar length of each pull bar in parallel.The positive motion knowledge topic of 6-SPS parallel robot is to its Workspace Analysis, and the aspects such as error compensation of trajectory planning and mechanism have effect of the utmost importance.

The people is opposite with serial machine, it is easier that parallel robot is found the solution its inverse kinetics solution, have certain difficulty but obtain positive motion analytic solution by Complete Method, therefore some method of value solving such as Newton-Raphson method (the gloomy process of iteration of newton-pressgang), neural network and genetic algorithm etc. are applied in parallel robot positive motion knowledge topic in recent years.Newton-Raphson method solution procedure is relatively simple, but algorithm is found the solution minimum problems based on gradient thought, and its optimizing result depends on the just Rational choice of state of value; Neural net method makes up the long variable space of parallel robot bar to the mapping relations in moving platform pose operating variable space by learning network power, threshold parameter, but the method needs a large amount of training sample learning networks and generalization ability of network performance to be difficult to obtain fine assurance; Genetic algorithm (Genetic Algorithm, GA) be to utilize its overall parallel search characteristic to solve its positive motion knowledge topic, this algorithm is less demanding to the detail of problem solving, but Common Genetic Algorithm its gene and control parameter when dealing with problems are difficult to select, and algorithm has precocity and speed of convergence waits defective slowly, affects solving precision and the speed of positive motion knowledge topic.

Differential evolution algorithm is a kind of random paralleling Direct search algorithm that is newly proposed in nineteen ninety-five by Storn R and Price K, its initial imagination is for solving the Chebyshev polynomials problem, some are non-linear to solving in rear discovery, have the performance of powerful global optimizing ability during the complicated optimum problem of non-differentiability, the application of therefore achieving success in the engineering problems such as some numerical optimizations, power economy load distribution.There are some researches show that differential evolution algorithm is in most of numerical value Benchmark(benchmark test) test result is better than Common Genetic Algorithm, particle group optimizing method Search Results in the problem.

Differential evolution algorithm is a kind of novel intelligent optimization algorithm, invites researcher's broad interest also to obtain larger progress because it has powerful search performance.Compare with some other bionic optimization method, the differential evolution algorithm process simply is easy to realize, very strong search capability is arranged, and need the parameter of setting few, same parameter setting can be used in many different problems, so differential evolution algorithm can be used in the 6-SPS parallel robot positive motion knowledge topic solution procedure.Though conventional difference algorithm has stronger global optimization performance, but still there is the precocious defective that easily is absorbed in Local Extremum, therefore when using difference algorithm in solution 6-SPS type parallel robot positive motion knowledge topic, also need makes improvements.

Summary of the invention

The object of the invention provides a kind of positive motion of parallel robot and learns method for solving, and the method can search out optimum or approach optimum 6-SPS parallel robot positive motion in suitable time learns solution, and the method also portable is applied in other Numerical Optimization.

To achieve these goals, technical scheme of the present invention is as follows:

A kind of positive motion of parallel robot is learned method for solving, and wherein, the method specific implementation step is as follows:

S1, algorithm population P (t) and algorithm parameter initialization make current evolutionary generation t=1, and maximum evolutionary generation T is set;

S2, each individual fitness function value of assessment population P (t);

S3, the optimum individual of population is carried out pattern search as initial point, and its result is substituted the optimum individual of former colony;

S4, population at individual is carried out differential evolution operation generate the new P of colony (t+1);

S5, the individual fitness function value of the new P of colony of evaluation (t+1);

S6, evolutionary generation t=t+1 return execution in step S3 and continue iterative evolution, until evolutionary generation t＞T, end loop;

S7: output optimum individual result.

Preferably, learn in the method for solving at the positive motion of above-mentioned parallel robot, described parallel manipulator people is 6-SPS type parallel robot, comprise moving platform, lower stationary platform and be connected to described upper moving platform and lower stationary platform between 6 driving pull bars.

Preferably, learn in the method for solving at the positive motion of above-mentioned parallel robot, the concrete grammar of described step S2 comprises:

(1) sets up the parallel robot inverse kinematics model, ask for each individual corresponding pull bar long vector l that respectively drives _iAnd bar is long | l _i| (i=1,2 ..., 6);

(2) fitness function of definition algorithm: long according to the driving pull bar bar that parallel robot is given

Corresponding respectively to drive the pull bar bar long with asked for population at individual by step (1) inverse kinematics model | l _i| (i=1,2 ..., 6), defining the long error function of each bar is the fitness function of algorithm;

(3) fitness function according to step (2) calculates each individual fitness value f, and fitness value is less to show that individual corresponding positive motion solution precision is higher:

$\min f\left(x\right)=\underset{i=1}{\overset{6}{\mathrm{\&Sigma;}}}|{\left|l_i\right|}^2-{\left|l_i^{\mathrm{ref}}\right|}^2|=\underset{i=1}{\overset{6}{\mathrm{\&Sigma;}}}\left|\begin{array}{c}{\mathrm{<figure-callout\; id="2"\; label="X\; P"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">X</figure-callout>}}_P^{}+{\mathrm{<figure-callout\; id="2"\; label="Y\; P"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">Y</figure-callout>}}_P^{}+{\mathrm{<figure-callout\; id="2"\; label="Z\; P"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">Z</figure-callout>}}_P^{}+{\mathrm{<figure-callout\; id="2"\; label="b\; ix"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{ix}}^{}+{\mathrm{<figure-callout\; id="2"\; label="b\; iy"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{iy}}^{}+{\mathrm{<figure-callout\; id="2"\; label="B\; iX"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">B</figure-callout>}}_{\mathrm{iX}}^{}+{\mathrm{<figure-callout\; id="2"\; label="B\; iY"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">B</figure-callout>}}_{\mathrm{iY}}^{}\\ +2(d_{11}{b}_{\mathrm{ix}}+d_{12}{b}_{\mathrm{iy}})(X_P-B_{\mathrm{iX}})+2(d_{21}{b}_{\mathrm{ix}}+d_{22}{b}_{\mathrm{iy}})(Y_P-B_{\mathrm{iy}})\\ +2(d_{31}{b}_{\mathrm{ix}}+d_{32}{b}_{\mathrm{iy}})Z_P-2(X_P{B}_{\mathrm{iX}}+Y_P{B}_{\mathrm{iy}})-{\left|l_i^{\mathrm{ref}}\right|}^2\end{array}\right|$

In the formula, x represents population at individual, i.e. parameter x to be optimized=[x ₁, x ₂..., x ₆] ^T=(X _P, Y _P, Z _P, α, beta, gamma) ^TP=[X _PY _PZ _P] ^TBe upper moving platform moving coordinate system initial point o position vector in lower stationary platform fixed coordinate system O-XYZ, b _IxAnd b _IyBe respectively x, the y axial coordinate of each hinge point of moving platform in moving coordinate system o-xyz, B _IxAnd B _IyBe respectively each hinge point of lower stationary platform X, Y-axis coordinate in fixed coordinate system O-XYZ, d _Ij(i=1,2,3; J=1,2) be element among the lower attitude matrix T, wherein α, β and γ are respectively the moving platform coordinate system with respect to the independent corner of lower stationary platform coordinate system, c _α=cos α, s _α=sin α is the concise and to the point literary style of α attitude angle, c _β=cos β, s _β=sin β is the concise and to the point literary style of β attitude angle, c _γ=cos γ, s _γ=sin _γBe the concise and to the point literary style of γ attitude angle

$T=\left[\begin{array}{ccc}{d}_{11}& d_{12}& d_{13}\\ d_{21}& d_{22}& d_{23}\\ d_{31}& d_{32}& d_{33}\end{array}\right]=\left[\begin{array}{ccc}{c}_{\mathrm{\&gamma;}}{c}_{\mathrm{\&beta;}}& c_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{s}_{\mathrm{\&alpha;}}-s_{\mathrm{\&gamma;}}{c}_{\mathrm{\&alpha;}}& c_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{c}_{\mathrm{\&alpha;}}+s_{\mathrm{\&gamma;}}{s}_{\mathrm{\&alpha;}}\\ s_{\mathrm{\&gamma;}}{c}_{\mathrm{\&beta;}}& s_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{s}_{\mathrm{\&alpha;}}-c_{\mathrm{\&gamma;}}{c}_{\mathrm{\&alpha;}}& s_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{c}_{\mathrm{\&alpha;}}-c_{\mathrm{\&gamma;}}{s}_{\mathrm{\&alpha;}}\\ {-s}_{\mathrm{\&beta;}}& c_{\mathrm{\&beta;}}{s}_{\mathrm{\&alpha;}}& c_{\mathrm{\&beta;}}{c}_{\mathrm{\&alpha;}}\end{array}\right]$

Preferably, learn in the method for solving at the positive motion of above-mentioned parallel robot, among the described step S3, the optimum individual of population carried out pattern search according to lower step:

(1) the given optimum individual position x of colony ⁽¹⁾∈ R ⁿ, its axial coordinate direction is e _j(j=1,2 ..., n), make y ⁽¹⁾=x ⁽¹⁾, k=1;

(2) each component j is axially searched for successively, if f is (y ^(j)+ δ e _j) be better than f (y ^(j)), y ^(j+1)=y ^(j)+ δ e _jIf f is (y ^(j)-δ e _j) be better than f (y ^(j)), y ^(j+1)=y ^(j)-δ e _jOtherwise y ^(j+1)=y ^(j)

(3) if f is (y ⁽ⁿ⁺¹⁾) be better than f (x ^(k)), put x ^(k+1)=y ⁽ⁿ⁺¹⁾, make y ⁽¹⁾=x ^(k+1)+ α (x ^(k+1)-x ^(k)), k=k+1, j=1 turns step (2), otherwise carries out step (4);

(4) if δ≤ε, iteration stopping then, invocation point x ^(k)Otherwise δ=β δ, y ⁽¹⁾=x ^(k), x ^(k+1)=x ^(k), k=k+1, j=1 turns step (2).

Preferably, learn in the method for solving at the positive motion of above-mentioned parallel robot, among the described step S4, population at individual is carried out the differential evolution operation generate the new P of colony (t+1), namely

$\begin{array}{cc}{v}_i^t=x_{r1}^t+F(x_{r2}^t-x_{r3}^t)& (i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,m)\end{array}$

In the formula, r1, r2, r3 ∈ 1,2 ..., m} is different and also different from target sequence number i, and F ∈ [0,2] is zoom factor; Carry out according to the following equation interlace operation

$u_{\mathrm{ij}}^t=\left\{\begin{array}{cc}{v}_{\mathrm{ij}}^t& \mathrm{rand}(\&CenterDot;)\&le;\mathrm{CR}|j=\mathrm{randn}\\ x_{\mathrm{ij}}^t& \mathrm{otherwise}\end{array}\right.$

In the formula, rand () is uniform random number between [0,1], and CR ∈ [0,1] is crossover probability, randn be 1,2 ..., choose random quantity among the n}; At last to individuality

With Be at war with, select the conduct individuality of lower generation of the more excellent individuality of fitness

Namely

Compared with prior art, the present invention proposes a kind of 6-SPS type parallel robot positive motion knowledge topic method for solving of Schema-based search-differential evolution.The DE/rand/1 differential evolution algorithm is strong to solution space overall situation exploring ability, but local development ability a little less than, the characteristic of the slow even stagnation behavior that may occur evolving of speed of convergence; Pattern search is as a kind of deterministic local search approach, and its Local Search performance is strong, but the pattern search initial point selects to affect its search solution result, and different Search Results are caused in different initial points position, easily are absorbed in the local minimum characteristic; This method combines the DE/rand/1 differential evolution algorithm and has stronger overall situation exploration performance, and the pattern search method has preferably local development ability, and mechanism merges mutually, the initial position that in per generation evolution, operates as pattern search with the optimum individual when former generation colony, further local exploitation searches the more excellent individuality of performance to solution space, and replace optimum individual in the former colony, improve the algorithm optimization performance.The method has excellent performance in solving 6-SPS type parallel robot positive motion knowledge topic, also can be applicable to other complicated Numerical Optimization simultaneously.

Description of drawings

In order to be illustrated more clearly in the embodiment of the invention or technical scheme of the prior art, the below will do to introduce simply to the accompanying drawing of required use in embodiment or the description of the Prior Art, apparently, accompanying drawing in the following describes only is some embodiments of the present invention, for those of ordinary skills, under the prerequisite of not paying creative work, can also obtain according to these accompanying drawings other accompanying drawing.

Figure 1 shows that 6-SPS type parallel robot mechanism coordinate diagram;

Figure 2 shows that the flow diagram of Schema-based search-differential evolution blending algorithm in the embodiment of the invention;

Figure 3 shows that three kinds of algorithms of different find the solution 6-SPS type parallel robot positive motion and learn to separate 10 average fitness functional values of independent operating evolution curve comparison diagram;

Figure 4 shows that Schema-based search-differential evolution algorithm typical case fitness function evolution curve map.

Embodiment

Facts have proved that every kind of single searching method all has self-defect when solving complicated optimum problem because of himself natural mode, if with two kinds of different searching algorithms mutually mechanism merge and consist of hybrid algorithm, its search performance is better than the search performance of single algorithm usually.Pattern search is a kind of effective, simple and direct determinacy searching method, it is moved by exploration and moves two parts with model utility and form, move with motion of defect modes by continuous detection close to extreme point gradually, so pattern search fast convergence rate, local search ability is strong.Based on this, the present invention is dissolved into the pattern search operation in the differential evolution algorithm, proposes a kind of numerical optimization of Schema-based search-differential evolution blending algorithm, and is successfully applied in the 6-SPS type parallel robot positive motion knowledge topic solution procedure.

Join shown in Figure 2ly, the positive motion that the embodiment of the invention discloses a kind of parallel robot is learned method for solving, and the method specific implementation step is as follows:

S1, algorithm population P (t) and algorithm parameter initialization make current evolutionary generation t=1, and maximum evolutionary generation T is set;

S2, each individual fitness function value of assessment population P (t);

S3, the optimum individual of population is carried out pattern search as initial point, and its result is substituted the optimum individual of former colony;

S4, population at individual is carried out differential evolution operation generate the new P of colony (t+1);

S5, the individual fitness function value of the new P of colony of evaluation (t+1);

S6, evolutionary generation t=t+1 return execution in step S3 and continue iterative evolution, until evolutionary generation t＞T, end loop;

S7: output optimum individual result.

The method utilizes 6-SPS type parallel robot easily to obtain the characteristics of inverse kinetics solution, its positive motion is learned the minimization problem that Solve problems is converted into an equivalence, by the long error of the calculate driving pull bar bar that returns, by pattern search-differential evolution blending algorithm 6 unknown pose parameters of upper moving platform might value find out in the candidate solution set of combination one group optimum or near optimum solution, make the 6-SPS type parallel robot of definition drive the long error function value minimum of pull bar bar.Pattern search-differential evolution blending algorithm take full advantage of differential evolution algorithm to solution space overall situation exploring ability strong and pattern search fast convergence rate, the excellent advantage of Local Search performance, mutually merge the Optimal performance that search mechanisms has separately improved algorithm.

The below is clearly and completely described the technical scheme in the embodiment of the invention.

1.6-SPS type parallel robot positive motion knowledge topic is found the solution mathematical description

6-SPS type parallel manipulator artificial one typical Stewart platform structure.Lower platform is the symmetrical hexagon of inequilateral on this parallel institution, and wherein lower platform is stationary platform, and upper mounting plate can carry out the 6DOF motion in work space, and two platforms adopt 6 to have same structure and be connected by gimbal suspension with the pull bar of stroke up and down.Fig. 1 is the corresponding mechanism of 6-SPS parallel robot coordinate diagram, b among the figure _i(i=1,2 ..., 6) and be each hinge point of upper moving platform, B _i(i=1,2 ..., 6) and be each hinge point of lower stationary platform, b ₁Be the initial point of upper moving platform coordinate system o-xyz, Be upper moving platform coordinate system x axle positive dirction, B ₁Be the initial point of lower stationary platform coordinate system O-XYZ,

Be lower stationary platform coordinate system X-axis positive dirction.But arbitrary vectorial R' through type (1) transforms to vectorial R under the lower fixed coordinate system O-XYZ in upper moving coordinate system o-xyz thus

R=TR'+P (1)

P=[X in the formula _PY _PZ _P] ^TBe the position vector of upper mounting plate moving coordinate system initial point o in fixed coordinate system O-XYZ, T is the attitude matrix of upper mounting plate, wherein

$T=\left[\begin{array}{ccc}{d}_{11}& d_{12}& d_{13}\\ d_{21}& d_{22}& d_{23}\\ d_{31}& d_{32}& d_{33}\end{array}\right]=\left[\begin{array}{ccc}{c}_{\mathrm{\&gamma;}}{c}_{\mathrm{\&beta;}}& c_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{s}_{\mathrm{\&alpha;}}-s_{\mathrm{\&gamma;}}{c}_{\mathrm{\&alpha;}}& c_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{c}_{\mathrm{\&alpha;}}+s_{\mathrm{\&gamma;}}{s}_{\mathrm{\&alpha;}}\\ s_{\mathrm{\&gamma;}}{c}_{\mathrm{\&beta;}}& s_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{s}_{\mathrm{\&alpha;}}-c_{\mathrm{\&gamma;}}{c}_{\mathrm{\&alpha;}}& s_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{c}_{\mathrm{\&alpha;}}-c_{\mathrm{\&gamma;}}{s}_{\mathrm{\&alpha;}}\\ {-s}_{\mathrm{\&beta;}}& c_{\mathrm{\&beta;}}{s}_{\mathrm{\&alpha;}}& c_{\mathrm{\&beta;}}{c}_{\mathrm{\&alpha;}}\end{array}\right]---\left(2\right)$

Following formula α, β and γ are respectively the moving platform coordinate system with respect to the independent corner of lower stationary platform coordinate system, c _γ=cos γ, s _γ=sin γ is the concise and to the point literary style of γ attitude angle, and α, β angle are similar.

Given upper moving platform attitude matrix T and the position vector P of upper moving platform coordinate system o-xyz initial point o in lower stationary platform coordinate system O-XYZ, and the upper moving platform of definition and each hinge point of lower stationary platform with respect to they separately the position coordinate value of coordinate system be respectively b _iAnd B _i(i=1,2 ..., 6), can in lower stationary platform coordinate system, each drive pull bar long vector l according to formula (1) _iAnd bar is long | l _i| be respectively

$l_i=\stackrel{\&RightArrow;}{B_i{b}_i}=\left[\begin{array}{c}{d}_{11}{b}_{\mathrm{ix}}+d_{12}{b}_{\mathrm{iy}}+X_P-B_{\mathrm{iX}}\\ d_{21}{b}_{\mathrm{ix}}+d_{22}{b}_{\mathrm{iy}}+Y_P-B_{\mathrm{iY}}\\ d_{31}{b}_{\mathrm{ix}}+d_{32}{b}_{\mathrm{iy}}+Z_P\end{array}\right]=\left[\begin{array}{c}{l}_{\mathrm{iX}}\\ l_{\mathrm{iY}}\\ l_{\mathrm{iZ}}\end{array}\right]---\left(3\right)$

$\begin{array}{cc}\left|l_i\right|=\sqrt{{\mathrm{<figure-callout\; id="2"\; label="l\; iX"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">l</figure-callout>}}_{\mathrm{iX}}^{}+{\mathrm{<figure-callout\; id="2"\; label="l\; iY"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">l</figure-callout>}}_{\mathrm{iY}}^{}+l_{\mathrm{iZ}}^2}& (i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,6)\end{array}---\left(4\right)$

In the formula, b _IxAnd b _IyBe respectively x, the y axial coordinate of each hinge point of upper mounting plate in moving coordinate system o-xyz, B _IxAnd B _IyBe respectively each hinge point of lower platform X, Y-axis coordinate in fixed coordinate system O-XYZ.

6-SPS type parallel robot positive motion knowledge topic Numerical Methods Solve is exactly will be in the candidate solution set that the unknown pose parameter of upper moving platform might make up, find optimum or make the robot of definition drive the long error function minimum of pull bar bar near optimum solution, can make up fitness function and be

${(X_P,Y_P,Z_P,\mathrm{\&alpha;},\mathrm{\&beta;},\mathrm{\&gamma;})}_{\mathrm{opt}}=\arg \min \left\{\underset{i=1}{\overset{6}{\mathrm{\&Sigma;}}}\right|{\left|l_i\right|}^2-{\left|l_i^{\mathrm{ref}}\right|}^2\left|\right\}---\left(5\right)$

In the formula,

For the driving pull bar bar that the 6-SPS parallel robot is given long.

2. differential evolution algorithm (DE) ultimate principle

Differential evolution has three mainly to be operating as variation, to intersect and to select.In each iteration, algorithm utilizes differential evolution variation and interlace operation to current colony, produces an interim population; Recycling is carried out man-to-man selection based on the selection operation of Greedy idea to this two colony, thereby realizes the renewal of colony.

Differential evolution algorithm produces first m initial individual x at random in the search volume _i(i=1,2 ..., m), the individual variable number is n.To the arbitrary individual vector x of population _iPress the DE/rand/1 mode and produce the variation vector v _i

$\begin{array}{cc}{v}_i^t=x_{r1}^t+F(x_{r2}^t-x_{r3}^t)& (i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,m)\end{array}---\left(6\right)$

In the formula, t is current algebraically, r1, and r2, r3 ∈ 1,2 ..., m} is different and also different from target sequence number i, and F ∈ [0,2] is zoom factor, is used for the control difference vector

Impact, the less disturbance to individuality of difference vector is also less.

The interlace operation purpose of differential evolution algorithm is by the variation vector v _iWith individual vector x _iCombine and improve the vectorial diversity of variation, algorithm carries out interlace operation according to the following equation

$u_{\mathrm{ij}}^t=\left\{\begin{array}{cc}{v}_{\mathrm{ij}}^t& \mathrm{rand}(\&CenterDot;)\&le;\mathrm{CR}|j=\mathrm{randn}\\ x_{\mathrm{ij}}^t& \mathrm{otherwise}\end{array}\right.---\left(7\right)$

Rand () is uniform random number between [0,1] in the formula, and CR ∈ [0,1] is crossover probability, randn be 1,2 ..., choose random quantity among the n};

The selection operation of differential evolution algorithm is a kind of preference pattern based on " greediness ", namely to the vectorial u of new individuality _iWith former vector x _iBe at war with, select the more excellent individuality of fitness to enter in the colony of future generation.

3. pattern search algorithm (PS)

Pattern search method is a kind of Deterministic searching method that was proposed in 1961 by Hooke and Jeeves, and it is moved by exploration and moves two parts with model utility and form; Wherein exploration mobile then with a fixed step size along the descent direction of axial exploration with probe function, it is to directly search to seek better point along beneficial direction that model utility moves, move and motion of defect modes by continuous detection, iteration point will be close to minimal point gradually.The basic step of the method is as follows:

(1) the given optimum individual x of colony ⁽¹⁾∈ R ⁿ, n axial coordinate direction e ₁, e ₂..., e _n, initial step length δ, speedup factor α 〉=1, economy β ∈ [0,1], permissible error ε＞0 makes y ⁽¹⁾=x ⁽¹⁾, k=1, j=1;

(2) each component j is axially searched for successively, if f is (y ^(j)+ δ e _j) be better than f (y ^(j)), y ^(j+1)=y ^(j)+ δ e _jIf f is (y ^(j)-δ e _j) be better than f (y ^(j)), y ^(j+1)=y ^(j)-δ e _jOtherwise y ^(j+1)=y ^(j)

(3) if f is (y ⁽ⁿ⁺¹⁾) be better than f (x ^(k)), put x ^(k+1)=y ⁽ⁿ⁺¹⁾, make y ⁽¹⁾=x ^(k+1)+ α (x ^(k+1)-x ^(k)), k=k+1, j=1 turns step (2), otherwise carries out step (4);

(4) if δ≤ε, iteration stopping then, invocation point x ^(k)Otherwise δ=β δ, y ⁽¹⁾=x ^(k), x ^(k+1)=x ^(k), k=k+1, j=1 turns step (2).

4. pattern search-differential evolution blending algorithm is found the solution the performing step of 6-SPS parallel robot positive motion knowledge topic

In sum, invent a kind of 6-SPS type parallel robot positive motion knowledge topic method for solving of Schema-based search-differential evolution, the method concrete steps are as follows:

Step 1: parameter initialization.Make current evolutionary generation t=1, maximum evolutionary generation T is set, algorithm Population Size m, individual variable is counted n; Pattern search parameter initial step length δ, speedup factor α 〉=1, economy β ∈ [0,1], step-length allowable error ε＞0; Differential evolution zoom factor parameter F, crossover probability parameters C R; Relative its position coordinate value b of coordinate system separately of moving platform and each hinge point of lower stationary platform on the 6-SPS type parallel robot _iAnd B _i, given driving pull bar length

Deng;

Step 2: algorithm population P (t) initialization is also assessed population at individual: produce at random m initial individual x in the search volume _i(i=1,2 ..., m), calculate each individual fitness function value f according to finding the solution fitness function constructed in the 6-SPS parallel robot positive motion knowledge topic mathematical description;

$\min f\left(x\right)=\underset{i=1}{\overset{6}{\mathrm{\&Sigma;}}}|{\left|l_i\right|}^2-{\left|l_i^{\mathrm{ref}}\right|}^2|=\underset{i=1}{\overset{6}{\mathrm{\&Sigma;}}}\left|\begin{array}{c}{\mathrm{<figure-callout\; id="2"\; label="X\; P"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">X</figure-callout>}}_P^{}+{\mathrm{<figure-callout\; id="2"\; label="Y\; P"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">Y</figure-callout>}}_P^{}+{\mathrm{<figure-callout\; id="2"\; label="Z\; P"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">Z</figure-callout>}}_P^{}+{\mathrm{<figure-callout\; id="2"\; label="b\; ix"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{ix}}^{}+{\mathrm{<figure-callout\; id="2"\; label="b\; iy"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">b</figure-callout>}}_{\mathrm{iy}}^{}+{\mathrm{<figure-callout\; id="2"\; label="B\; iX"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">B</figure-callout>}}_{\mathrm{iX}}^{}+{\mathrm{<figure-callout\; id="2"\; label="B\; iY"\; filenames="HDA00002531172500011.png"\; state="\{\{state\}\}">B</figure-callout>}}_{\mathrm{iY}}^{}\\ +2(d_{11}{b}_{\mathrm{ix}}+d_{12}{b}_{\mathrm{iy}})(X_P-B_{\mathrm{iX}})+2(d_{21}{b}_{\mathrm{ix}}+d_{22}{b}_{\mathrm{iy}})(Y_P-B_{\mathrm{iy}})\\ +2(d_{31}{b}_{\mathrm{ix}}+d_{32}{b}_{\mathrm{iy}})Z_P-2(X_P{B}_{\mathrm{iX}}+Y_P{B}_{\mathrm{iy}})-{\left|l_i^{\mathrm{ref}}\right|}^2\end{array}\right|---\left(8\right)$

In the formula, x represents population at individual, i.e. parameter x to be optimized=[x ₁, x ₂..., x ₆] ^T=(X _P, Y _P, Z _P, α, beta, gamma) ^TP=[X _PY _PZ _P] ^TBe the position vector of upper mounting plate moving coordinate system initial point o in fixed coordinate system O-XYZ, b _IxAnd b _IyBe respectively x, the y axial coordinate of each hinge point of upper mounting plate in moving coordinate system o-xyz, B _IxAnd B _IyBe respectively each hinge point of lower platform X, Y-axis coordinate in fixed coordinate system O-XYZ, d _Ij(i=1,2,3; J=1,2) be element among the attitude matrix T, embody referring to shown in the formula (2);

Step 3: the optimum individual to population carries out the operation of pattern Local Search as initial point, and substitutes the optimum individual of former colony with the result after the search;

Step 4: population is carried out the operation of DE/rand/1 differential evolution generate the lower P of Dai Xin colony (t+1), and calculate each individual fitness function value f of new colony according to fitness function;

Step 5: evolutionary generation t=t+1, return execution in step three and continue iterative evolution, until evolutionary generation t＞T, end loop;

Step 6: output optimum individual result, note (X _P, Y _P, Z _P, α, beta, gamma) _Opt, this result is 6-SPS type parallel robot positive motion and learns solution.

Find the solution the performance of 6-SPS type parallel robot positive motion knowledge topic below by instantiation checking Schema-based search-differential evolution blending algorithm proposed by the invention.Experimental situation is 2.60GHz, 1G internal memory, Matlab 7.11.0.584 version.Obviously, described embodiment only is the present invention's part embodiment, rather than whole embodiment.Based on the embodiment among the present invention, those of ordinary skills belong to the scope of protection of the invention not making the every other embodiment that obtains under the creative work prerequisite.

To 6-SPS type parallel robot mechanism coordinate diagram as shown in Figure 1, establish each hinge point of upper and lower platform with respect to they separately the position coordinate parameters of coordinate system be respectively shown in the table 1.

Each hinge point of table 1.6-SPS parallel robot is coordinate figure (rice) for corresponding coordinate

	b 1	b 2	b 3	b 4	b 5	b 6
							b ix	0	0.10	0.50	0.50	0.20	0.10

b iy	0	0	0.20	0.30	0.50	0.10
								B 1	B 2	B 3	B 4	B 5	B 6
B ix	0	0.70	0.70	0.40	0.30	0
							B iy	0	0	0.10	0.40	0.40	0.10

If 6 driving pull bars of parallel robot length is respectively | l ₁|=1.0m, | l ₂|=1.0m, | l ₃|=1.2m, | l ₄|=1.2m, | l ₅|=1.0m and | l ₆|=1.0m.

The below adopts respectively the differential evolution algorithm of DE/rand/1 and two kinds of Different Variation strategies of DE/best/1 and the present invention is based on pattern search-differential evolution blending algorithm and finds the solution 6-SPS type parallel robot upper mounting plate pose positive motion solution, and mutual comparison search solution performance.Each method parameter arranges as follows in the experiment: dimension is upper mounting plate pose parameter n=6,6 pose parameter (X _P, Y _P, Z _P, α, beta, gamma) and the variable hunting zone is respectively X _P∈ [0.6,0.6], Y _P∈ [0.6,0.6], Z _P∈ [0.9,0.9] rice, α ∈ [40* π/180,40* π/180], β ∈ [75* π/180,75* π/180], γ ∈ [115* π/180,115* π/180] radian; The common parameter zoom factor coefficient F=0.80 of each algorithm DE operation, crossover probability CR=0.90; Pattern search operating parameter δ=0.2 in the pattern search that other the present invention proposes-differential evolution blending algorithm, α=1.0, β=0.5, ε=0.001; During justice, also value is identical for the common parameter of other of three kinds of algorithms of different, be that population at individual is counted P=30, total evolutionary generation T=1500, with the fitness function of formula (8) as upper each algorithm, independent operating is 10 times respectively, and table 2 is respectively 10 corresponding optimizing results of each method independence optimizing.

10 optimizing results of table 2.6-SPS parallel robot positive motion knowledge topic independent operating relatively

	Optimum	On average	The poorest	Standard deviation
					DE/rand/1	5.9952E-15	1.6352E-02	1.4095E-01	4.4030E-02
DE/best/1	0	6.7718E-03	1.1286E-02	5.8282E-03
					The inventive method	6.6613E-16	1.3212E-14	8.3045E-14	2.4827E-14

The optimizing result of 10 operations can be found out from table 2, compare with the DE/rand/1 algorithm, the inventive method find the solution the optimum of 6-SPS parallel robot positive motion knowledge topic independent operating 10 times, on average, the standard deviation of the most bad optimization solution precision and optimization solution all is better than the analog value of these two kinds of methods, illustrate that Schema-based search-differential evolution blending algorithm not only has degree of precision finding the solution on the parallel robot positive motion knowledge topic, also has relative stability.With the DE/best/1 algorithm relatively, the inventive method is on average, also be better than the DE/best/1 algorithm on the standard deviation of the most bad optimization solution precision and optimization solution; Though the inventive method optimum solution result is not as good as the DE/best/1 algorithm, but in 10 independent operating results, the DE/best/1 algorithm wherein occupies the optimizing solution (establishing take convergence precision ε=1.0E-5 as condition) that can not converge on expectation for 6 times, obtaining expectation optimization solution success ratio only is 40%, so comprehensive evaluation comparison DE/best/1 algorithm obtains the performance of 6-SPS type parallel robot positive motion solution not as good as the present invention is based on pattern search-differential evolution blending algorithm.Fig. 3 is that three kinds of distinct methods are found the solution 10 average evolution curve comparison diagrams of 6-SPS parallel robot positive motion knowledge topic independent operating, and comparison curves is also verified and be the present invention is based on pattern search-differential evolution blending algorithm to finding the solution the validity of parallel robot positive motion knowledge topic among the figure.

The present invention is based on pattern search-differential evolution blending algorithm and ask for 6-SPS parallel robot positive motion and learn solution procedure for further describing, Fig. 4 is fusion mode search-differential evolution algorithm fitness value function evolutionary process figure when finding the solution positive motion knowledge topic and getting optimum solution f=6.6613E-16, and the pose parameter of corresponding 6-SPS parallel robot upper mounting plate is respectively

(X _P,Y _P,Z _P)

=(0.4402913,0.5242920,0.7288768) rice

(α,β，γ)

=(0.525408 ,-1.272296 ,-1.952770) radian

As can be seen from Figure, the fitness value function evolution curve of this blending algorithm is close to transverse axis fast, can obtain the higher 6-SPS parallel robot positive motion of solving precision and learn solution.

This fusion method provides a kind of effective method approach for solving multidimensional function overall situation numerical value optimizing problem, and other relates in the multidimensional function overall situation numerical optimization field also can be widely used in robot, Aeronautics and Astronautics etc.

In sum, method of the present invention is utilized its characteristic of easily asking for inverse kinetics solution when solving parallel robot positive motion knowledge topic, the parallel robot positive motion is learned the minimum optimization problem that Solve problems is converted into equivalence, and adopts numerical optimization to find the solution.This method combines differential evolution solution space is had the advantage that the strong overall situation is explored performance, reaches pattern search solution space is had more excellent local development ability characteristic, and the mutual mechanism of two kinds of optimized algorithms is merged the search performance that has improved algorithm.Compare with traditional method, the method that the present invention proposes has higher optimizing solution precision and speed of convergence faster when solving 6-SPS type parallel robot positive motion knowledge topic, during the method parallel robot positive motion knowledge of also can be generalized to other types is inscribed and found the solution simultaneously.

Be to be understood that, although this instructions is described according to embodiment, but be not that each embodiment only comprises an independently technical scheme, this narrating mode of instructions only is for clarity sake, those skilled in the art should make instructions as a whole, technical scheme in each embodiment also can through appropriate combination, form other embodiments that it will be appreciated by those skilled in the art that.

Above listed a series of detailed description only is specifying for feasibility embodiment of the present invention; they are not to limit protection scope of the present invention, allly do not break away from equivalent embodiment or the change that skill spirit of the present invention does and all should be included within protection scope of the present invention.

Claims (5)

1. the positive motion of a parallel robot is learned method for solving, and it is characterized in that: the method specific implementation step is as follows:

S1, algorithm population P (t) and algorithm parameter initialization make current evolutionary generation t=1, and maximum evolutionary generation T is set;

S2, each individual fitness function value of assessment population P (t);

S3, the optimum individual of population is carried out pattern search as initial point, and its result is substituted the optimum individual of former colony;

S4, population at individual is carried out differential evolution operation generate the new P of colony (t+1);

S5, the individual fitness function value of the new P of colony of evaluation (t+1);

S6, evolutionary generation t=t+1 return execution in step S3 and continue iterative evolution, until evolutionary generation t＞T, end loop;

S7, output optimum individual result.

2. the positive motion of parallel robot according to claim 1 is learned method for solving, it is characterized in that: described parallel manipulator people is for 6-SPS type parallel robot, comprise moving platform, lower stationary platform and connect described upper moving platform and lower stationary platform between 6 driving pull bars.

3. the positive motion of parallel robot according to claim 2 is learned method for solving, it is characterized in that: specifically comprise among the described step S2:

(1) sets up the parallel robot inverse kinematics model, ask for each individual corresponding pull bar long vector l that respectively drives _iAnd bar is long | l _i| (i=1,2 ..., 6);

(2) fitness function of definition algorithm: long according to the driving pull bar bar that parallel robot is given

Corresponding respectively to drive the pull bar bar long with asked for population at individual by step (1) inverse kinematics model | l _i| (i=1,2 ..., 6), defining the long error function of each bar is the fitness function of algorithm;

(3) fitness function according to step (2) calculates each individual fitness value f of population, and fitness value is less to show that individual corresponding positive motion solution precision is higher:

$\min f\left(x\right)=\underset{i=1}{\overset{6}{\mathrm{\&Sigma;}}}|{\left|l_i\right|}^2-{\left|l_i^{\mathrm{ref}}\right|}^2|=\underset{i=1}{\overset{6}{\mathrm{\&Sigma;}}}\left|\begin{array}{c}{X}_P^2+Y_P^2+Z_P^2+b_{\mathrm{ix}}^2+b_{\mathrm{iy}}^2+B_{\mathrm{iX}}^2+B_{\mathrm{iY}}^2\\ +2(d_{11}{b}_{\mathrm{ix}}+d_{12}{b}_{\mathrm{iy}})(X_P-B_{\mathrm{iX}})+2(d_{21}{b}_{\mathrm{ix}}+d_{22}{b}_{\mathrm{iy}})(Y_P-B_{\mathrm{iy}})\\ +2(d_{31}{b}_{\mathrm{ix}}+d_{32}{b}_{\mathrm{iy}})Z_P-2(X_P{B}_{\mathrm{iX}}+Y_P{B}_{\mathrm{iy}})-{\left|l_i^{\mathrm{ref}}\right|}^2\end{array}\right|$

In the formula, x represents population at individual, i.e. parameter x to be optimized=[x ₁, x ₂..., x ₆] ^T=(X _P, Y _P, Z _P, α, beta, gamma) ^TP=[X _PY _PZ _P] ^TBe upper moving platform moving coordinate system initial point o position vector in lower stationary platform fixed coordinate system O-XYZ, b _IxAnd b _IyBe respectively x, the y axial coordinate of each hinge point of moving platform in moving coordinate system o-xyz, B _IxAnd B _IyBe respectively each hinge point of lower stationary platform X, Y-axis coordinate in fixed coordinate system O-XYZ, d _Ij(i=1,2,3; J=1,2) be element among the lower attitude matrix T, wherein α, β and γ are respectively the moving platform coordinate system with respect to the independent corner of lower stationary platform coordinate system, c _α=cos α, s _α=sin α is the concise and to the point literary style of α attitude angle, c _β=cos β, s _β=sin β is the concise and to the point literary style of β attitude angle, c _γ=cos γ, s _γ=sin _γBe the concise and to the point literary style of γ attitude angle.

$T=\left[\begin{array}{ccc}{d}_{11}& d_{12}& d_{13}\\ d_{21}& d_{22}& d_{23}\\ d_{31}& d_{32}& d_{33}\end{array}\right]=\left[\begin{array}{ccc}{c}_{\mathrm{\&gamma;}}{c}_{\mathrm{\&beta;}}& c_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{s}_{\mathrm{\&alpha;}}-s_{\mathrm{\&gamma;}}{c}_{\mathrm{\&alpha;}}& c_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{c}_{\mathrm{\&alpha;}}+s_{\mathrm{\&gamma;}}{s}_{\mathrm{\&alpha;}}\\ s_{\mathrm{\&gamma;}}{c}_{\mathrm{\&beta;}}& s_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{s}_{\mathrm{\&alpha;}}-c_{\mathrm{\&gamma;}}{c}_{\mathrm{\&alpha;}}& s_{\mathrm{\&gamma;}}{s}_{\mathrm{\&beta;}}{c}_{\mathrm{\&alpha;}}-c_{\mathrm{\&gamma;}}{s}_{\mathrm{\&alpha;}}\\ {-s}_{\mathrm{\&beta;}}& c_{\mathrm{\&beta;}}{s}_{\mathrm{\&alpha;}}& c_{\mathrm{\&beta;}}{c}_{\mathrm{\&alpha;}}\end{array}\right]$

4. the positive motion of parallel robot according to claim 1 is learned method for solving, it is characterized in that: among the described step S3, the optimum individual of population is carried out pattern search according to lower step:

(1) the given optimum individual position x of colony ⁽¹⁾∈ R ⁿ, its axial coordinate direction is e _j(j=1,2 ..., n), make y ⁽¹⁾=x ⁽¹⁾, k=1;

(2) each component j is axially searched for successively, if f is (y ^(j)+ δ e _j) be better than f (y ^(j)), y ^(j+1)=y ^(j)+ δ e _jIf f is (y ^(j)-δ e _j) be better than f (y ^(j)), y ^(j+1)=y ^(j)-δ e _jOtherwise y ^(j+1)=y ^(j)

(3) if f is (y ⁽ⁿ⁺¹⁾) be better than f (x ^(k)), put x ^(k+1)=y ⁽ⁿ⁺¹⁾, make y ⁽¹⁾=y ^(k+1)+ α (x ^(k+1)-x ^(k)), k=k+1, j=1 turns step (2), otherwise carries out step (4);

(4) if δ≤ε, iteration stopping then, invocation point x ^(k)Otherwise δ=β δ, y ⁽¹⁾=x ^(k), x ^(k+1)=x ^(k), k=k+1, j=1 turns step (2).

5. the positive motion of parallel robot according to claim 1 is learned method for solving, it is characterized in that: among the described step S4, population at individual is carried out the differential evolution operation generate the new P of colony (t+1), namely

$\begin{array}{cc}{v}_i^t=x_{r1}^t+F(x_{r2}^t-x_{r3}^t)& (i=\mathrm{1,2},\&CenterDot;\&CenterDot;\&CenterDot;,m)\end{array}$

In the formula, r1, r2, r3 ∈ 1,2 ..., m} is different and also different from target sequence number i, and F ∈ [0,2] is zoom factor; Carry out according to the following equation interlace operation

$u_{\mathrm{ij}}^t=\left\{\begin{array}{cc}{v}_{\mathrm{ij}}^t& \mathrm{rand}(\&CenterDot;)\&le;\mathrm{CR}|j=\mathrm{randn}\\ x_{\mathrm{ij}}^t& \mathrm{otherwise}\end{array}\right.$

In the formula, rand () is uniform random number between [0,1], and CR ∈ [0,1] is crossover probability, randn be 1,2 ..., choose random quantity among the n}; At last to individuality

With

Be at war with, select the conduct individuality of lower generation of the more excellent individuality of fitness

Namely

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