CN105676636B - A kind of redundancy space manipulator Multipurpose Optimal Method based on NSGA-II algorithm - Google Patents
A kind of redundancy space manipulator Multipurpose Optimal Method based on NSGA-II algorithm Download PDFInfo
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Abstract
The redundancy space manipulator Multipurpose Optimal Method based on NSGA-II algorithm that the embodiment of the invention provides a kind of, it include: to convert the transfer task of space manipulator operating space point-to-point to joint space from cartesian space, parameterized treatment is carried out to joint variable using sinusoidal seven preserving Interpolation Using methods, the argument sequence of acquisition is encoded as the individual in NSGA-II algorithm population；Each joint moment mean value and value is minimum, maximum joint moment minimum and joint maximum angular rate minimum are as optimization aim, using NSGA-II algorithm to space manipulator running track Optimization Solution, it is concentrated from the Pareto non-domination solution obtained and chooses optimal solution, running track described in the solution can be such that each joint moment mean value and maximum joint moment and joint maximum angular rate is optimized.The technical solution provided according to embodiments of the present invention can make space manipulator multiple target according to the operation of above-mentioned planning path while be optimised when executing point-to-point transfer task in operating space under the conditions of meeting task restriction.
Description
[technical field]
The present invention relates to automated control technology more particularly to a kind of redundancy space mechanisms based on NSGA-II algorithm
Arm Multipurpose Optimal Method.
[background technique]
When the movement of redundancy space manipulator end effector, task space dimension is less than its joint space dimension.
So joint driven torque meets mission requirements when redundancy space manipulator end effector is moved by certain track
Solution be not unique.Just because of the presence of its redundancy so that we can optimize it in the case where meeting main motion task
Ilities index.
It is very important firstly, being optimized to the joint driven torque of space manipulator, is on the one hand from worked
The safety of journey considers, if the joint driven torque that requires during the motion of space manipulator is excessive or transfinites, can make
The decline of space manipulator dynamic performance, algorithm failure, and accelerate space manipulator because joint stress it is excessive caused by mechanical knot
Structure performance degradation rate even damages space manipulator itself when serious；On the other hand being desirable to space manipulator can be with smaller
Joint driven torque complete as defined in operation sequence, to further decrease energy consumption, changing the space manipulator moment faces fuel
Situation limited, supply is difficult, while space mechanism can be improved in space manipulator maximum joint moment during reducing task
The carrying load ability of arm.
The joint angular speed that should also be as concern space manipulator while considering space manipulator joint driven torque,
When space manipulator executes heavy load transfer task, if the larger shake that can cause space manipulator of joint angular speed, thus
Reduce the task execution precision of space manipulator.
So being optimized simultaneously to many index of space manipulator very necessary.
[summary of the invention]
In view of this, to propose a kind of redundancy space manipulator multiple target based on NSGA-II excellent for the embodiment of the present invention
Change method, so that space manipulator makes being reduced with value for each joint moment mean value under the premise of completion task, thus
Energy consumption needed for reducing space manipulator task execution, mitigates the mechanism wear of space manipulator, while guaranteeing space manipulator
The positioning accuracy of end improves space manipulator and is executing carrying load ability when loading transfer task.
The embodiment of the present invention proposes a kind of redundancy space manipulator Multipurpose Optimal Method based on NSGA-II, packet
It includes:
Parameterized treatment is carried out to space manipulator joint variable using sinusoidal seven preserving Interpolation Using methods, by system of polynomials
Number Sequence carries out the individual in coding acquisition NSGA-II algorithm population；
Each joint moment mean value of space manipulator and value is minimum, maximum joint moment minimum and joint maximum angular rate
The minimum optimization aim as NSGA-II algorithm；
According to the optimization aim, space manipulator running track is optimized using NSGA-II algorithm, is obtained
The non-dominant disaggregation of Pareto；It is concentrated according to certain principle from obtained Pareto non-domination solution and chooses solution, the described fortune of the solution
Row track can make space manipulator joint moment mean value and maximum joint moment and joint maximum angular rate while be optimized.
In the above method, method individual in NSGA-II Advanced group species is obtained are as follows: utilize sinusoidal seven preserving Interpolation Using sides
Method carries out interpolation traversal to each joint, and, speed continuous according to space manipulator track is continuous etc. requires, and is based on polynomial interopolation
Method obtains space manipulator joint angle expression formula, initial when space manipulator being recycled to execute the transfer task of point-to-point, termination
Certain several parameter list shows by multinomial coefficient for the joint angles of state, angular speed, angular acceleration constraint condition, ginseng obtained
Number i.e. coding obtains the individual of population used in NSGA-II algorithm.
Use the expression formula for the space manipulator joint angle that seven preserving Interpolation Using methods sinusoidal in the above method obtain are as follows:
Wherein,
c1=(θi_max-θi_min)/2
c2=(θi_max+θi_min)/2
θi(T) joint angle in i-th of joint, i=1,2 ..., n, a are indicatedi0,ai1,...ai7For seven order polynomial coefficients,
θi_maxAnd θi_minThe respectively maximum and minimum value of space manipulator i-th of joint angle under the conditions of meeting mission requirements；
Initial, the termination joint angles, angular speed, angle acceleration when the transfer task of point-to-point are executed using space manipulator
Degree constraint condition establishes joint angle constraint equation are as follows:
qint=θint
qdes=θdes
Wherein qint,Respectively the space manipulator described in space manipulator joint angle expression formula is held
Initial joint angles when the transfer task of row point-to-point, angular speed and angular acceleration；qdes,Respectively space machine
The space manipulator described in the expression formula of tool shoulder joint angle executes the termination joint angles when transfer task of point-to-point, angle
Speed and angular acceleration；θint,θdesIt is given initial and termination the joint angles of task；
By ai6And ai7It is selected as the optimal control parameter of NSGA-II algorithm, space manipulator joint angle is carried out seven times sinusoidal
Polynomial interopolation, obtained 8 multinomial coefficients pass through ai6And ai7Two unknown parameters indicate, are specifically expressed as follows:
ai1=ai2=0
Wherein tfFor planning time；Then argument sequence is expressed as a=[a16,a17,a26,…a76,a77], to therein each
Parameter carries out binary coding, and code length is n binary digit, then each individual of population in the NSGA-II algorithm obtained
It is indicated by 14 × n binary digit.
When the movement of the end effector of redundancy space manipulator, task space dimension is tieed up less than its joint space
Number.So the operating path of redundancy space manipulator end effector will not be unique.Therefore since the presence of redundancy makes
We can optimize simultaneously multiple indexs of space manipulator.
The optimization aim of the NSGA-II are as follows:
Wherein, joint moment mean value and Z are denoted as f1(x), torque maximum of TmaxIt is denoted as f2(x), maximum joint angular speedIt is denoted as f3(x), wherein qi(T),Joint angles, the angular speed in respectively i-th joint
And angular acceleration, τiIt (T) is driving moment required for i-th of joint of current operating path down space mechanical arm, qimax,The given constraint of the joint angles, angular speed and angular acceleration in respectively i-th joint, τimaxFor space
The given constraint of the joint moment in i-th of joint of mechanical arm；
Joint moment mean value computation formula is as follows:
Wherein, i-th of joint of i=1,2 ..., n representation space mechanical arm, Z are space manipulator in task execution mistake
In journey the mean value of all joint moments and,For the mean value of i-th of joint torque in task implementation procedure of space manipulator；
The mean value of i-th of joint moment of the space manipulatorAre as follows:
Wherein, τiFor joint moment vector τ=(τ of space manipulator1,τ2,…,τn)TIn i-th of component, tfFor rule
Draw the time；
Basic equation of the joint moment vector τ by space manipulator in joint space acquires, the space
Mechanical arm is as follows in the Basic equation of joint space:
In formula, θ indicates joint angle sequence, is n dimensional vector；D(θ)∈Rn×nFor inertial matrix in its joint space；For its coriolis force and centrifugal force vectors matrix；G(θ)∈Rn×1For its gravity item；τ=(τ1,τ2,…,τn)T
For its joint moment vector.
The maximum of T of space manipulator joint driven torque during task executionmaxExpression formula are as follows:
Tmax=Max (τi)
The expression formula of space manipulator joint angular speed is obtained by the polynomial expression derivation at joint of mechanical arm angle, i.e., under
Shown in formula:
Space manipulator maximum joint angle velocity expression are as follows:
Wherein,Indicate the joint angular speed in i-th of joint；
In the above method, according to the optimization aim, space manipulator running track is carried out using NSGA-II algorithm excellent
Change and solve, concentrated according to certain principle from the Pareto non-domination solution obtained and choose solution, running track described in the solution can make
Space manipulator joint moment mean value and maximum joint moment and joint maximum angular rate are optimized simultaneously；
Carrying out multiple-objection optimization using the NSGA-II algorithm, specific step is as follows:
1) parameter needed for initializing NSGA-II algorithm, population at individual quantity NIND, mutation operator size Pm, selection operator
Size Pc, crossover operator size GGAP, maximum number of iterations Gen, corpuscular velocity upper limit Vmax, lower limit VminAnd in Problem Areas
The problem of value bound etc., and generate at random NIND it is individual；
2) by system of polynomials Number Sequence a=[a16,a17,a26,…a76,a77] encoded, as in population one by one
Body, each coefficient coding length are 20, totally 14 multinomial coefficients, i.e., each individual lengths are 280, obtain initial population Pt,
Wherein t=0；
3) to PtCalculating target function value obtains the vector (f being made of three fitness values1(x),f2(x),f3(x)),
Quick non-dominated ranking is carried out to individual according to the fitness vector, obtains non-dominant collection, is distinguished in each non-dominant concentration
Crowded angle value is calculated to each individual, steps are as follows for specific calculating:
3.1, the crowding i of each pointdIt is set to 0；
3.2, the one-component for randomly selecting fitness vector sorts according to the component, by the two of boundary individual crowdings
It is denoted as infinite, i.e. od=id=∞；
3.3, other individuals are carried out with the calculating of crowding:
Wherein, idIndicate the crowding of i point,Indicate j-th of fitness component of i+1 individual,Indicate i-1
J-th of fitness component of individual；
4) crowding according to non-dominant collection locating for individual and individual, is selected by tournament method, is selected
For the individual arrived for the intersection after carrying out, mutation operation, the specific rules of tournament method are as follows:
4.1, NIND × Pc individual is selected at random from population；
4.2, foundation is selected non-dominant set and crowding locating for individual and is selected, and preferential selection is top dog
Set in individual, then select the biggish individual of crowding from these individuals, be repeated up to and select NIND individual；
5) crossover operation is carried out according to probability GGAP to the population that step 3) obtains, carries out mutation operation according to probability P m, obtains
Progeny population Qt；
6) merge progeny population QtWith parent population PtAs new population I, t=t+1 is enabled；
7) to all individual calculating target function values in population I, fitness vector is obtained.According to fitness vector to a
Body carries out quick non-dominated ranking, obtains non-dominant collection Zi, crowding is calculated to each individual respectively in each non-dominant concentration
Value obtains new parent population P according to rule selectiont, specific rules are as follows:
7.1, non-dominant collection Z is investigatedi, wherein i=1；
If 7.2, ZiMiddle individual amount N is greater than NIND, then by ZiThe middle crowded angle value of individual choice biggish NIND are put into newly
Parent population Pt, carry out step 9)；
If 7.3, ZiMiddle individual amount N is equal to NIND, then by ZiMiddle individual is put into population Pt, carry out step 9)；
If 7.4, ZiMiddle individual amount N is less than NIND, then by ZiMiddle individual is put into population Pt, enable NIND=NIND-N, i=i
+ 1, continuation judges since 7.2；
8) judge whether t reaches maximum genetic algebra Gen, turn to step 9) if meeting；Otherwise step 3) is turned to；
9) P is exportedtIn all Pareto non-dominant disaggregation of the individual as problem, algorithm stopping.
According to specific tasks demand, suitable individual is selected from Pareto non-domination solution concentration, as space mechanism
Shoulder joint angle expression formula coefficient, the space manipulator operating path after being optimized.The path makes space manipulator complete to appoint
Make being reduced with value for each joint moment mean value under the premise of business, thus energy needed for reducing space manipulator task execution
Consumption, mitigates the mechanism wear of space manipulator, while ensure that the positioning accuracy of space manipulator end, improves space mechanism
Arm is executing carrying load ability when loading transfer task.
As can be seen from the above technical solutions, the embodiment of the present invention has the advantages that
In the technical solution of the embodiment of the present invention, using sinusoidal seven preserving Interpolation Using methods to space manipulator joint variable
Parameterized treatment is carried out, system of polynomials Number Sequence is subjected to the individual in coding acquisition NSGA-II algorithm population, by space mechanism
Each joint moment mean value of arm and value are minimum, maximum joint moment is minimum and joint maximum angular rate minimum is calculated as NSGA-II
The optimization aim of method optimizes space manipulator running track using NSGA-II algorithm, and it is non-dominant to obtain Pareto
Disaggregation, running track described in the solution which concentrates can make space manipulator joint moment mean value and most
Large joint torque and joint maximum angular rate are optimized simultaneously, therefore can make space manipulator under the premise of completion task
So that each joint moment mean value is reduced with value, so that energy consumption needed for reducing space manipulator task execution, mitigates empty
The mechanism wear of room machine arm, while guaranteeing the positioning accuracy of space manipulator end, it improves space manipulator and is executing load
Carrying load ability when transfer task.
[Detailed description of the invention]
In order to illustrate the technical solution of the embodiments of the present invention more clearly, below will be to needed in the embodiment attached
Figure is briefly described, it should be apparent that, drawings in the following description are only some embodiments of the invention, for this field
For those of ordinary skill, without any creative labor, it can also be obtained according to these attached drawings other attached
Figure.
Fig. 1 is the redundancy space manipulator multiple-objection optimization based on NSGA-II algorithm that the embodiment of the present invention is proposed
The flow diagram of method；
Fig. 2 is the multiple degrees of freedom space manipulator model schematic based on Space Operators description；
Fig. 3 is the inverse dynamics calculation flow chart based on spatial operator algebra；
Fig. 4 is that the space manipulator multiple-objection optimization based on NSGA-II algorithm proposed using the embodiment of the present invention is calculated
The flow chart of method；
Fig. 5 is seven freedom space manipulator DH coordinate system schematic diagram in the embodiment of the present invention；
Fig. 6 is the average value of joint moment mean value sum in population in iteration searching process of the embodiment of the present invention with genetic algebra
Change curve schematic diagram；
Fig. 7 is the average value of joint maximum moment in population in iteration searching process of the embodiment of the present invention with genetic algebra
Change curve schematic diagram；
Fig. 8 is the average value of maximum joint angular speed in population in iteration searching process of the embodiment of the present invention with genetic algebra
Change curve schematic diagram；
[specific embodiment]
For a better understanding of the technical solution of the present invention, being retouched in detail to the embodiment of the present invention with reference to the accompanying drawing
It states.
It will be appreciated that described embodiments are only a part of the embodiments of the present invention, instead of all the embodiments.Base
Embodiment in the present invention, it is obtained by those of ordinary skill in the art without making creative efforts it is all its
Its embodiment, shall fall within the protection scope of the present invention.
Redundancy space manipulator Multipurpose Optimal Method based on NSGA-II algorithm mainly includes three parts: first
The transfer task of space manipulator operating space point-to-point is converted from cartesian space to joint space, using sine more than seven times
Item formula interpolation method carries out parameterized treatment to space manipulator joint variable, and system of polynomials Number Sequence is carried out coding acquisition
Individual in NSGA-II algorithm population；Secondly most by each joint moment mean value of space manipulator and minimum, the maximum joint moment of value
Optimization aim small and that joint maximum angular rate minimum is as NSGA-II algorithm；Finally according to the optimization aim, utilize
NSGA-II algorithm optimizes space manipulator running track, obtains the non-dominant disaggregation of Pareto；According to certain principle
It is concentrated from obtained Pareto non-domination solution and chooses solution, running track described in the solution can make space manipulator joint moment equal
Value and maximum joint moment and joint maximum angular rate are optimized simultaneously.
The embodiment of the present invention provides a kind of redundancy space manipulator Multipurpose Optimal Method based on NSGA-II algorithm,
Referring to FIG. 1, its redundancy space manipulator multiple-objection optimization based on NSGA-II algorithm for being proposed by the embodiment of the present invention
The flow diagram of method, as shown in Figure 1, method includes the following steps:
Step 101, the transfer task of point-to-point in space manipulator operating space is converted from cartesian space to joint
Space carries out parameterized treatment to space manipulator joint variable using sinusoidal seven preserving Interpolation Using methods, by multinomial coefficient
Sequence carries out the individual in coding acquisition NSGA-II algorithm population.
Specifically, interpolation traversal is carried out to each joint first with sinusoidal seven preserving Interpolation Using methods, according to space
The requirements such as mechanical arm track is continuous, speed is continuous, it is as follows to establish space manipulator joint angle expression formula based on polynomial interpolation:
Wherein,
c1=(θi_max-θi_min)/2
c2=(θi_max+θi_min)/2
θi(T) joint angle in i-th of joint is indicated, i=1,2 ..., 7, ai0,ai1,...ai7For seven order polynomial systems
Number, θi_maxAnd θi_minThe respectively maximum and minimum value of space manipulator i-th of joint angle under the conditions of meeting mission requirements；
According to the polynomial expression of space manipulator joint angle, derivation obtains the speed in each joint of space manipulator, adds
Speed polynomial expression are as follows:
Utilize joint angles, the angular speed, angular acceleration constraint condition q that space manipulator is initial and terminatesint=θin,t
qdes=θdes,6 joint angle constraint equations are established, and in space manipulator joint angle table
Up in formula, polynomial unknowm coefficient is 8, therefore selects two parameter a thereini6And ai7Control as NSGA-II algorithm
Parameter processed, i.e. a=[ai6,ai7], i=1,2 ..., n are as undetermined parameter, and wherein n is the number of degrees of freedom of space manipulator.
Space manipulator joint angle is substituted into space manipulator constraint equation for remaining six multinomial in its speed, acceleration expression formula
Coefficient ai6And ai7Two parameters indicate, are specifically expressed as follows:
ai1=ai2=0
Wherein, θintWith θdesIt is the joint angles that the given space manipulator of task is initial and terminates；tfFor planning time；
So far the expression formula of space manipulator joint angle is by ai6And ai7Two parameters determine, by argument sequence a=[a16,a17,a26,…
a76,a77] encoded, as the individual in algorithm population.
Step 102, each joint moment mean value of space manipulator and value is minimum, maximum joint moment minimum and joint are most
Big optimization aim of the angular speed minimum as NSGA-II algorithm.
Specifically, each joint moment mean value of space manipulator and value is minimum, maximum joint moment minimum and joint are most
Big optimization aim of the angular speed minimum as NSGA-II algorithm, optimizes redundancy space manipulator joint moment.
The optimization aim of the NSGA-II are as follows:
Wherein, joint moment mean value and Z are denoted as f1(x), torque maximum of TmaxIt is denoted as f2(x), maximum joint angular speedIt is denoted as f3(x), wherein qi(T),Joint angles, the angular speed in respectively i-th joint
And angular acceleration, τiIt (T) is driving moment required for i-th of joint of current operating path down space mechanical arm, qimax,The given constraint of the joint angles, angular speed and angular acceleration in respectively i-th joint, τimaxFor space
The given constraint of the joint moment in i-th of joint of mechanical arm；
Joint moment mean value computation formula is as follows:
Wherein, i-th of joint of the representation space mechanical arm of i=1,2 ..., 7, Z are space manipulator in task execution mistake
In journey the mean value of all joint moments and,For the mean value of i-th of joint torque in task implementation procedure of space manipulator；
The mean value of i-th of joint moment of the space manipulatorAre as follows:
Wherein, τiFor joint moment vector τ=(τ of space manipulator1,τ2,…,τn)TIn i-th of component, tfFor rule
Draw the time；
Basic equation of the joint moment vector τ by space manipulator in joint space acquires, the space
Mechanical arm is as follows in the Basic equation of joint space:
In formula, θ indicates joint angle sequence, is n dimensional vector；D(θ)∈Rn×nFor inertial matrix in its joint space；For its coriolis force and centrifugal force vectors matrix；G(θ)∈Rn×1For its gravity item；τ=(τ1,τ2,…,τn)T
For its joint moment vector.
The maximum of T of space manipulator joint driven torque during task executionmaxExpression formula are as follows:
Tmax=Max (τi)
The expression formula of space manipulator joint angular speed is obtained by the polynomial expression derivation at joint of mechanical arm angle, i.e., under
Shown in formula:
Space manipulator maximum joint angle velocity expression are as follows:
Wherein,Indicate the joint angular speed in i-th of joint；
Fig. 2 is the multiple degrees of freedom space manipulator model schematic based on Space Operators description, and the present embodiment uses dynamic
Mechanics fundamental equation is namely based on the model foundation.Space Manipulator System is each as based on Fig. 2 obtains the present embodiment
Symbol is expressed as follows: ΣIFor inertial coodinate system, all recursive operations of space manipulator are carried out relative to inertial coodinate system
's；ΣkFor kth bar coordinate system, it is defined on joint；JkFor joint k, JnFor space manipulator pedestal, J0For space manipulator end
End and extraneous connecting place；CkFor kth bar centroid position；akFor joint JkTo k bar mass center CkVector；bkFor k bar mass center CkTo pass
Save Jk+1Vector；pkFor vector of the joint k under inertial system；M (k) is kth bar quality；PcIt (k) is point JkTo CkVector.
Fig. 3 is the inverse dynamics calculation flow chart based on spatial operator algebra, show that space space manipulator is closing by Fig. 3
Save the Basic equation in space, the specific steps are as follows:
The speed, acceleration and power and torque that method carrys out representation space mechanical arm are described first with spinor:
Wherein, ωk,vk,Nk,FkRespectively represent the angular speed in k-th of joint, speed, angular acceleration, acceleration,
Torque and power.
The inertia mass matrix of definition space kth bar are as follows:
Wherein IkInertial tensor matrix for connecting rod k relative to kth joint coordinate system, mkFor the quality of connecting rod k,
For the antisymmetric matrix of the centroid vector of connecting rod k, E is three-dimensional unit matrix.Define joint k state transition matrix be H (k)=
[hT(k) 00 0], wherein h (k) is the rotation axial vector in joint, is three dimensional vectors, when k-th of freedom degree is prismatic pair
When, H (k)=[0 00 hT(k)], when there is connected pedestal, since pedestal is that imaginary hinge links with inertial system, there is H (n+1)
=diag [1,1,1,1,1,1].
Joint velocity, acceleration stepping type are as follows, wherein k=n, n-1 ..., 1
Joint power, torque stepping type are as follows, wherein k=1,2 ..., n
F (k)=φ (k, k-1) f (k-1)+M (k) α (k)+b (k)
T (k)=H (k) f (k)
Wherein, a (k), b (k) respectively indicate the coriolis force and centrifugal force of space manipulator, and φ (k+1, k) is power and torque
Recursion Operator, φT(k+1, k) is that velocity and acceleration Recursion Operator such as following formula indicates:
It is vector of the adjacent segment k+1 to joint k, wherein definingH (k) is the state of joint k
Transition matrix.
A (k), b (k) respectively indicate the coriolis force and centrifugal force of space manipulator, when joint is rotary hinge:
When joint is mobile hinge, a (k) is expressed as follows:
The system speed operator for defining multiple degrees of freedom space manipulator is V=[V (1) ... V (n-1), V (n)]T, and with
Acceleration operator, coriolis force operator, centrifugal force operator, power operator, the torque operator point of same form representation space mechanical arm
Not Wei α, a, b, f, T, then it is available:
F=φ (M α+b)
T=Hf
Wherein M is space manipulator mass matrix operator, and H is state, as projected matrix operator, and φ is space transfer operator.
Space manipulator Calculating Torque during Rotary formula can be finally derived by are as follows:
Wherein:
MG=H φ M φTHT
C=H φ (M φTa+b)
MGThe general mass matrix of representation space mechanical arm, the non-linear force matrix of C representation space mechanical arm.Space mechanism
For arm in the Basic equation of joint space, D (θ) is M hereinG,It is equal in this calculation formula
C, since gravity is minimum in space, therefore gravity item G (θ) is negligible.
Step 103, according to the optimization aim, space manipulator running track is optimized using NSGA-II algorithm
It solves, obtains the non-dominant disaggregation of Pareto；It is concentrated according to certain principle from obtained Pareto non-domination solution and chooses solution, the solution institute
The running track of description can make space manipulator joint moment mean value and maximum joint moment and joint maximum angular rate while obtain
To optimization.
Specifically, Fig. 4 is the space manipulator multiple target based on NSGA-II algorithm proposed using the embodiment of the present invention
The flow chart of optimization algorithm；As shown in figure 4, the specific step that NSGA-II algorithm optimizes multiple indexs of space manipulator
It is rapid as follows:
Carrying out multiple-objection optimization using the NSGA-II algorithm, specific step is as follows:
1) parameter needed for initializing NSGA-II algorithm, population at individual quantity NIND, mutation operator size Pm, selection operator
Size Pc, crossover operator size GGAP, maximum number of iterations Gen, corpuscular velocity upper limit Vmax, lower limit VminAnd in Problem Areas
The problem of value bound etc., and generate at random NIND it is individual；
2) by system of polynomials Number Sequence a=[a16,a17,a26,…a76,a77] encoded, as in population one by one
Body, each coefficient coding length are 20, totally 14 multinomial coefficients, i.e., each individual lengths are 280, obtain initial population Pt,
Wherein t=0；
3) to Pt calculating target function value, the vector (f being made of three fitness values is obtained1(x),f2(x),f3(x)),
Quick non-dominated ranking is carried out to individual according to the fitness vector, obtains non-dominant collection, is distinguished in each non-dominant concentration
Crowded angle value is calculated to each individual, steps are as follows for specific calculating:
3.1, the crowding i of each pointdIt is set to 0；
3.2, the one-component for randomly selecting fitness vector sorts according to the component, by the two of boundary individual crowdings
It is denoted as infinite, i.e. od=ld=∞；
3.3, other individuals are carried out with the calculating of crowding:
Wherein, idIndicate the crowding of i point,Indicate j-th of fitness component of i+1 individual,Indicate i-1
J-th of fitness component of individual；
4) crowding according to non-dominant collection locating for individual and individual, is selected by tournament method, is selected
For the individual arrived for the intersection after carrying out, mutation operation, the specific rules of tournament method are as follows:
4.1, NIND × Pc individual is selected at random from population；
4.2, foundation is selected non-dominant set and crowding locating for individual and is selected, and preferential selection is top dog
Set in individual, then select the biggish individual of crowding from these individuals, be repeated up to and select NIND individual；
5) crossover operation is carried out according to probability GGAP to the population that step 3) obtains, carries out mutation operation according to probability P m, obtains
Progeny population Qt；
6) merge progeny population QtWith parent population PtAs new population I, t=t+1 is enabled；
7) to all individual calculating target function values in population I, fitness vector is obtained.According to fitness vector to a
Body carries out quick non-dominated ranking, obtains non-dominant collection Zi, crowding is calculated to each individual respectively in each non-dominant concentration
Value obtains new parent population P according to rule selectiont, specific rules are as follows:
7.1, non-dominant collection Z is investigatedi, wherein i=1；
If 7.2, ZiMiddle individual amount N is greater than NIND, then by ZiThe middle crowded angle value of individual choice biggish NIND are put into newly
Parent population Pt, carry out step 9)；
If 7.3, ZiMiddle individual amount N is equal to NIND, then by ZiMiddle individual is put into population Pt, carry out step 9)；
If 7.4, ZiMiddle individual amount N is less than NIND, then by ZiMiddle individual is put into population Pt, enable NIND=NIND-N, i=i
+ 1, continuation judges since 7.2；
8) judge whether t reaches maximum genetic algebra Gen, turn to step 9) if meeting；Otherwise step 3) is turned to；
9) P is exportedtIn all Pareto non-dominant disaggregation of the individual as problem, algorithm stopping.
By the Pareto non-domination solution concentrate individual as space manipulator joint angle expression formula parameter to get to
Space manipulator operating path after optimization.The path optimizes many index of space manipulator.
For having carried out numerical value with the space manipulator of a seven freedom for specific research object and having imitated in specific embodiment
True research, the space manipulator are made of seven rotary joints and the connection of two root long straight-arm bars, and the structure of space manipulator has
Symmetry, wherein seven freedom space manipulator DH coordinate system schematic diagram is as shown in Figure 5 in the embodiment of the present invention.Space in Fig. 5
The length of each rod piece of mechanical arm is d1=1.2m；d2=0.53m；d3=0.53m；d4=0.52m；d5=0.53m；d6=0.53m；
d7=1.2m；a3=5.8m；a4=5.8m；Table one is space manipulator D-H parameter list, as follows:
Table one
| Connecting rod i | θi(°) | di(m) | ai-1(m) | αi-1(°) |
| 1 | θ1(0) | d1 | 0 | 90 |
| 2 | θ2(90) | d2 | 0 | -90 |
| 3 | θ3(0) | 0 | a3 | 0 |
| 4 | θ4(0) | d3+d4+d5 | a4 | 0 |
| 5 | θ5(0) | 0 | 0 | 90 |
| 6 | θ6(-90) | d6 | 0 | -90 |
| 7 | θ7(0) | d7 | 0 | 0 |
By the DH parameter of the available space manipulator of table one.In addition, first joint coordinate system position of mechanical arm is opposite
Coordinate in base coordinate system center is rca0={ 0.2,0,2 }, attitude misalignment are { 0,0,0 }.
Table two is space manipulator mass property parameter list, is joined by the inertia mass that Space Manipulator System can be obtained in table two
Several and center-of-mass coordinate vector parameter, as follows:
Table two
Space manipulator transfer task setting of point-to-point in joint space is as follows: setting space manipulator is being transported
One group of initial joint angles during row are θint=[- 20 °, 0, -10 °, -100 °, 120 °, 180 °, 70 °], one group desired
Termination joint angles are θdes=[0,15 °, -30 °, -110 °, 140 °, 165 °, 90 °], planning time tf=20s.
Using method of the invention, by taking space manipulator is in fixed pedestal mode as an example, using multinomial coefficient as parameter,
Accounting equation in binding kinetics model, with each joint moment mean value of space manipulator in task and minimum, the maximum joint power of value
The optimization aim that square is minimum and joint maximum angular rate minimum is as NSGA-II algorithm, using NSGA-II algorithm to given sky
Room machine shoulder joint space tracking optimizes calculating, and has carried out emulation experiment by Matlab software.Given space mechanism
Relevant parameter in arm load parameter and NSGA-II algorithm are as follows: quality mload=5000kg；Inertial tensor: Iload=[16,0,
0],[0,137,0],[0,0,146]}；, maximum number of iterations Gen=100, population scale NIND=30, mutation operator size Pm
=0.03, selection operator size Pc=0.8, crossover operator size GGAP=0.85,.Constraint condition during task execution
Are as follows: the range of joint angles q is [- 180 °, 180 °], joint angular speedJoint angular accelerationJoint moment τ≤3000Nm.
Fig. 6 is the average value of joint moment mean value sum in population in iteration searching process of the embodiment of the present invention with genetic algebra
Change curve schematic diagram, as shown in fig. 6, axis of abscissas is population recruitment the number of iterations, axis of ordinates is that corresponding population is current
The corresponding space manipulator joint moment mean value of individual and divided by individual amount average value obtained in the population of the number of iterations,
It can be seen that in population iteration searching process by the figure, the average value of joint moment mean value sum is under fast speed in population
Drop, and 2.81 × 10^4 when iterating to 70 times from the outset converges to 5.54 × 10^3, and convergence process is more flat
Surely.
Fig. 7 is the average value of space mechanism shoulder joint maximum moment in iteration searching process of the embodiment of the present invention with hereditary generation
Several change curve schematic diagram, as shown in fig. 7, obtaining space machine in iteration searching process after population iteration updates 100 times
The average value of tool shoulder joint maximum moment is with the change curve of genetic algebra, and wherein axis of abscissas is population recruitment the number of iterations,
Axis of ordinates is average value of the maximum joint moment to individual amount of corresponding population current iteration number, be can be seen that by the figure
In population iteration searching process, the convergence of the average value of maximum joint moment rapidly, and when iterating to 50 times from the beginning of
When 1.55 × 10^4 converge to 2.78 × 10^3, convergence process is more steady, can illustrate to make using this method by the figure
The maximum joint moment of the space manipulator has reached relatively good effect of optimization.
Fig. 8 is the average value of space manipulator maximum joint angular speed in iteration searching process of the embodiment of the present invention with heredity
The change curve schematic diagram of algebra, as shown in figure 8, axis of abscissas is population recruitment the number of iterations, axis of ordinates is corresponding population
The maximum joint angular speed of current iteration number can be seen that in population iteration searching process the mean value of individual amount by the figure
In, the convergence of the average value of maximum joint moment rapidly, and 0.074 the converging to from the outset when iterating to 60 times
0.048, fall is about 35%.
As can be seen that carrying out multiple-objection optimization to space manipulator using NSGA-II algorithm can obtain preferably optimizing knot
Fruit.The individual that the Pareto non-domination solution finally obtained is concentrated, as the coefficient of joint angle velocity expression, is obtained by decoding
Complete space manipulator joint angle velocity expression, as shown in the table by the optimization target values for being calculated final:
Table three
As can be seen from the table, the multiple target values obtained by Optimization Solution can not be optimal simultaneously, in reality
In engineering, suitable individual can targetedly be chosen for different problems, by decoding as angular speed expression formula
Coefficient.
In conclusion the embodiment of the present invention has the advantages that
In the technical solution of the embodiment of the present invention, using sinusoidal seven preserving Interpolation Using methods to space manipulator joint variable
Parameterized treatment is carried out, system of polynomials Number Sequence is subjected to the individual in coding acquisition NSGA-II algorithm population, by space mechanism
Each joint moment mean value of arm and value are minimum, maximum joint moment is minimum and joint maximum angular rate minimum is calculated as NSGA-II
The optimization aim of method optimizes space manipulator running track using NSGA-II algorithm and asks according to the optimization aim
Solution, obtains the non-dominant disaggregation of Pareto, and running track described in the solution which concentrates can make space mechanism
Shoulder joint torque mean value and maximum joint moment and joint maximum angular rate are optimized simultaneously.It mentions according to embodiments of the present invention
The technical solution of confession, when executing point-to-point transfer task in operating space under the conditions of meeting task restriction, according to above-mentioned planning
Path operation can make space manipulator make being reduced with value for each joint moment mean value under the premise of completion task, from
And energy consumption needed for reducing space manipulator task execution, mitigate the mechanism wear of space manipulator, while guaranteeing space mechanism
The positioning accuracy of arm end improves space manipulator and is executing carrying load ability when loading transfer task.
The foregoing is merely illustrative of the preferred embodiments of the present invention, is not intended to limit the invention, all in essence of the invention
Within mind and principle, any modification, equivalent substitution, improvement and etc. done be should be included within the scope of the present invention.
Claims (4)
1. a kind of redundancy space manipulator Multipurpose Optimal Method based on NSGA-II, which is characterized in that the method step
Include:
(1) the transfer task of space manipulator operating space point-to-point is converted from cartesian space to joint space, using just
Seven preserving Interpolation Using method of string carries out parameterized treatment to space manipulator joint variable, and system of polynomials Number Sequence is encoded
Obtain the individual in NSGA-II algorithm population；
(2) each joint moment mean value of space manipulator and value is minimum, maximum joint moment minimum and joint maximum angular rate
The minimum optimization aim as NSGA-II algorithm；
(3) according to the optimization aim, space manipulator running track is optimized using NSGA-II algorithm, is obtained
The non-dominant disaggregation of Pareto；It is concentrated according to certain principle from obtained Pareto non-domination solution and chooses solution, the principle specifically: when
When executing heavy load transfer task, joint output torque can then increase accordingly space manipulator, to guarantee space manipulator
End carrying load ability then needs preferentially to select to make the lesser solution of maximum joint moment；And when space manipulator is executing zero load
It is for reduction can be consumed, then excellent when choosing optimal solution by joint moment mean value and as main target of optimization when transfer task
First choosing makes joint moment mean value and lesser solution；Running track described in the solution can make space manipulator joint moment mean value
Optimized simultaneously with maximum joint moment and joint maximum angular rate.
2. the method according to claim 1, wherein using sinusoidal seven preserving Interpolation Using methods to space manipulator
Joint variable carries out parameterized treatment, and system of polynomials Number Sequence is carried out the individual in coding acquisition NSGA-II algorithm population, packet
It includes: interpolation traversal being carried out to each joint using sinusoidal seven preserving Interpolation Using methods, speed continuous according to space manipulator track
Degree is continuous to be required, and is obtained space manipulator joint angle expression formula based on seven preserving Interpolation Using methods of sine, is recycled space machine
Initial, the joint angles of final state, angular speed, angular acceleration constraint condition will just when tool arm executes the transfer task of point-to-point
Seven order polynomial coefficient of string is shown with certain several parameter list, and the argument sequence of acquisition is carried out coding as NSGA-II algorithm population
In individual；
Expression formula based on the space manipulator joint angle that sinusoidal seven preserving Interpolation Using methods obtain are as follows:
Wherein,
c1=(θi_max-θi_min)/2
c2=(θi_max+θi_min)/2
θi(T) joint angle in i-th of joint, i=1,2 ..., 7, a are indicatedi0,ai1,...ai7For seven order polynomial coefficients, θi_max
And θi_minThe respectively maximum and minimum value of space manipulator i-th of joint angle under the conditions of meeting mission requirements；
Using space manipulator execute point-to-point transfer task when it is initial, termination joint angles, angular speed, angular acceleration about
Beam condition establishes joint angle constraint equation are as follows:
qint=θint
qdes=θdes
Wherein qint,The respectively space manipulator execution point described in space manipulator joint angle expression formula
To point transfer task when initial joint angles, angular speed and angular acceleration；qdes,Respectively space machine
The space manipulator described in the expression formula of tool shoulder joint angle executes the termination joint angles when transfer task of point-to-point, angle
Speed and angular acceleration；θint,θdesIt is given initial and termination the joint angles of task；
By ai6And ai7It is selected as parameter, sinusoidal seven preserving Interpolation Usings are carried out to space manipulator joint angle, 8 in each expression formula
A multinomial coefficient passes through ai6And ai7Two unknown parameters indicate, are specifically expressed as follows:
ai1=ai2=0
Wherein tfFor planning time；The argument sequence is expressed as a=[a16,a17,a26,…a76,a77], to each ginseng therein
Number carries out binary codings, the requirement according to space manipulator particular task to trajectory planning precision, code length be n two into
Position processed；Each of population individual is indicated by 14 × n binary digit in the NSGA-II algorithm then obtained.
3. the method according to claim 1, wherein each joint moment mean value of space manipulator and value is minimum,
The optimization aim that maximum joint moment is minimum and joint maximum angular rate minimum is as NSGA-II algorithm, is described as follows:
The numerical value of each joint moment of space manipulator reflects its energy consumption in task implementation procedure indirectly, carries out to maximum joint moment
Optimization can then improve space manipulator and execute carrying load ability when loading transfer task, carry out to joint maximum angular rate excellent
Change the end positioning accuracy that then ensure that space manipulator, minimum, the most high point by each joint moment mean value of space manipulator and value
Save the optimization aim that torque is minimum and maximum angular rate minimum is as NSGA-II algorithm；
The optimization aim of the NSGA-II are as follows:
Wherein, joint moment mean value and Z are denoted as f1(x), torque maximum of TmaxIt is denoted as f2(x), maximum joint angular speed
It is denoted as f3(x), wherein qi(T),Joint angles, angular speed and the angle in respectively i-th joint accelerate
Degree, τiIt (T) is driving moment required for i-th of joint of current operating path down space mechanical arm, qimax,
The given constraint of the joint angles, angular speed and angular acceleration in respectively i-th joint, τimaxIt is i-th of space manipulator
The given constraint of the joint moment in joint；
Joint moment mean value and calculation formula are as follows:
Wherein, i-th of joint of the representation space mechanical arm of i=1,2 ..., 7, Z are space manipulator in task implementation procedure
The mean value of all joint moments and,For the mean value of i-th of joint torque in task implementation procedure of space manipulator；
The mean value of i-th of joint moment of the space manipulatorAre as follows:
Wherein, τiFor joint moment vector τ=(τ of space manipulator1,τ2,…,τn)TIn i-th of component, tfWhen to plan
Between；
Basic equation of the joint moment vector τ by space manipulator in joint space acquires, the space mechanism
Arm is as follows in the Basic equation of joint space:
In formula, θ indicates joint angle sequence, is n dimensional vector；D(θ)∈Rn×nFor inertial matrix in its joint space；For its coriolis force and centrifugal force vectors matrix；G(θ)∈Rn×1For its gravity item；τ=(τ1,τ2,…,τn)T
For its joint moment vector；
The maximum of T of space manipulator joint driven torque during task executionmaxExpression formula are as follows:
Tmax=Max (τi)
The expression formula of space manipulator joint angular speed is obtained by the polynomial expression derivation of its joint angle, i.e., shown in following formula:
Space mechanism shoulder joint maximum angular rate expression formula are as follows:
Wherein,Indicate the joint angular speed in i-th of joint.
4. according to the method described in claim 3, it is characterized in that, according to the optimization aim, using NSGA-II algorithm to sky
Room machine arm running track optimizes, and obtains the non-dominant disaggregation of Pareto；It is concentrated from the Pareto non-domination solution and presses one
Determine principle and choose solution, running track described in the solution can make space manipulator joint moment mean value and maximum joint moment and
Joint maximum angular rate is optimized simultaneously；Carrying out multiple-objection optimization using the NSGA-II algorithm, specific step is as follows:
1) parameter needed for initializing NSGA-II algorithm, population at individual quantity NIND, mutation operator size Pm, selection operator size
Pc, crossover operator size GGAP, maximum number of iterations Gen, corpuscular velocity upper limit Vmax, lower limit VminAnd asking in Problem Areas
The bound of topic value, and NIND individual is generated at random；
2) by system of polynomials Number Sequence a=[a16,a17,a26,…a76,a77] encoded, as the individual in population, often
A coefficient coding length is 20, totally 14 multinomial coefficients, i.e., each individual lengths are 280, obtains initial population Pt, wherein t
=0；
3) to Pt calculating target function value, the vector (f being made of three fitness values is obtained1(x),f2(x),f3(x)), foundation
The fitness vector carries out quick non-dominated ranking to individual, obtains non-dominant collection, in each non-dominant concentration respectively to every
Individual calculates crowded angle value, and steps are as follows for specific calculating:
3.1, the crowding i of each pointdIt is set to 0；
3.2, the one-component for randomly selecting fitness vector sorts according to the component, and the two of boundary individual crowdings are denoted as
It is infinite, i.e. od=ld=∞；
3.3, other individuals are carried out with the calculating of crowding:
Wherein, idIndicate the crowding of i pointIndicate j-th of fitness component of i+1 individual,Indicate i-1 individual
J-th of fitness component；
4) crowding according to non-dominant collection locating for individual and individual, is selected by tournament method, is selected
For individual for the intersection after carrying out, mutation operation, the specific rules of tournament method are as follows:
4.1, NIND × Pc individual is selected at random from population；
4.2, foundation is selected non-dominant set and crowding locating for individual and is selected, and preferentially selects the collection being top dog
Individual in conjunction, then the biggish individual of crowding is selected from these individuals, it is repeated up to and selects NIND individual；
5) crossover operation is carried out according to probability GGAP to the population that step 3) obtains, carries out mutation operation according to probability P m, obtains filial generation
Population Qt；
6) merge progeny population QtWith parent population PtAs new population I, t=t+1 is enabled；
7) to all individual calculating target function values in population I, obtain fitness vector, according to fitness vector to individual into
The quick non-dominated ranking of row, obtains non-dominant collection Zi, crowded angle value is calculated to each individual respectively in each non-dominant concentration, according to
Rule selection obtains new parent population Pt, specific rules are as follows:
7.1, non-dominant collection Z is investigatedi, wherein i=1；
If 7.2, ZiMiddle individual amount N is greater than NIND, then by ZiThe middle crowded angle value of individual choice biggish NIND are put into new parent
Population Pt, carry out step 9)；
If 7.3, ZiMiddle individual amount N is equal to NIND, then by ZiMiddle individual is put into population Pt, carry out step 9)；
If 7.4, ZiMiddle individual amount N is less than NIND, then by ZiMiddle individual is put into population Pt, NIND=NIND-N, i=i+1 are enabled,
Continuation judges since 7.2；
8) judge whether t reaches maximum genetic algebra Gen, turn to step 9) if meeting；Otherwise step 3) is turned to；
9) P is exportedtIn all Pareto non-dominant disaggregation of the individual as problem, algorithm stopping；
It concentrates to choose according to certain principle from the Pareto non-domination solution obtained and solve, running track described in the solution can make space
Joint of mechanical arm torque mean value and maximum joint moment and joint maximum angular rate are optimized simultaneously.
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| CN106527151B (en) * | 2017-01-09 | 2019-07-19 | 北京邮电大学 | A kind of method for searching path of bringing onto load Six-freedom-degree space mechanical arm |
| CN107457780B (en) * | 2017-06-13 | 2020-03-17 | 广州视源电子科技股份有限公司 | Method and device for controlling mechanical arm movement, storage medium and terminal equipment |
| CN108920793A (en) * | 2018-06-21 | 2018-11-30 | 北京工业大学 | A kind of robotic joint space track Multipurpose Optimal Method based on quick non-dominated ranking algorithm |
| CN111195915B (en) * | 2020-01-20 | 2022-07-15 | 深圳市鑫君特智能医疗器械有限公司 | Mechanical arm absolute positioning precision calibration method and system based on genetic algorithm |
| CN111687834B (en) * | 2020-04-30 | 2023-06-02 | 广西科技大学 | System and method for controlling reverse priority impedance of redundant mechanical arm of mobile mechanical arm |
| CN111687835B (en) * | 2020-04-30 | 2023-06-02 | 广西科技大学 | System and method for controlling reverse priority impedance of redundant mechanical arm of underwater mechanical arm |
| CN111687833B (en) * | 2020-04-30 | 2023-06-02 | 广西科技大学 | System and method for controlling impedance of inverse priority of manipulator |
| CN111687832B (en) * | 2020-04-30 | 2023-06-02 | 广西科技大学 | System and method for controlling inverse priority impedance of redundant mechanical arm of space manipulator |
| CN111958598B (en) * | 2020-08-17 | 2021-08-31 | 北方工业大学 | Multi-target fusion redundant drive parallel robot internal force optimization method and system |
| CN112757306B (en) * | 2021-01-25 | 2022-03-01 | 北京交通大学 | Inverse solution multi-solution selection and time optimal trajectory planning algorithm for mechanical arm |
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