CN108908341A - Redundant robot's repetitive motion planning method based on secondary radical final state attracting performance index - Google Patents
Redundant robot's repetitive motion planning method based on secondary radical final state attracting performance index Download PDFInfo
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
- B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
- B25J9/00—Programme-controlled manipulators
- B25J9/16—Programme controls
- B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
- B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning
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Abstract
A kind of redundant robot's repetitive motion planning method based on secondary radical final state attracting performance index gives the target terminal track r of end effector of robot in cartesian spaced(t), it and provides the expectation in each joint and backs angle, θd(0)；Repeating motion for robot, design it is a kind of make current joint vector and it is expected joint vector deviation have secondary radical final state attracting performance performance indicator, by converting time-varying matrix equation Solve problems for the double optimization problem of redundant robot's trajectory planning, using finite value final state neural network as solver, the present invention realizes the repeating motion planning tasks of redundant robot's finite time convergence control in the case where initial position deviates.The present invention has the characteristics that finite time convergence control, computational accuracy are high, is easily achieved.
Description
Technical field
The present invention relates to industrial robot motion planning technologies, specifically, provide a kind of finite time convergence energy index,
Initial position deviates redundant robot's repeating motion controlling party under desired trajectory situation, based on finite value final state neural network
Method.
Background technique
Industrial robot can be moved any object by desired spatial pose, be wanted to complete a specific operation
It asks.The joint number of self-movement, the freedom of motion of referred to as this mechanism are capable of in its mechanism.Six-freedom degree is that have to complete
The minimum degree of freedom number of space orientation ability, the freedom degree that redundant robot possesses exceed the freedom degree of required by task.Using superfluous
The flexibility and fault-tolerance of redundant robot can be enhanced in complementary characteristic, avoids the barrier in operating environment, avoids in robot
Portion's singular configuration adapts to complicated working environment and changeable job requirements.
Redundant robot has extra freedom degree, and same end effector position can correspond to the group of a variety of joint vectors
It closes.The basic problem of robot real time kinematics control is redundancy parsing, and conventional way is the parsing scheme based on pseudoinverse.It examines
Consider the relationship of each joint angles of n degree of freedom robot and end effector displacement
R (t)=f (θ (t)) (1)
Wherein, r (t) is the terminal track of end effector in cartesian coordinate system, and θ (t) indicates joint angles, f (θ)
It is robotic end effector motion profile function.Relationship between end cartesian space and the differential of joint space is
Wherein,WithIt is respective time-derivative respectively,It is robot Jacobian matrix.By calculating J
The pseudoinverse of (θ) obtains the least-norm solution of joint velocity variable
Here, J+=JT(JJT)-1It is the pseudoinverse of Jacobian matrix.
Minimum speed norm performance indicator with equality constraint as motion planning objective function (D.E.Whitney,
Resolved motion rate control of manipulators and human prostheses (executor and people
The movement rate of work artificial limb controls), IEEE Trans.Man-Machine Syst., 1969,10 (2):47-53) it is
In formula, A is positive definite weighting matrix, solves the planning problem demand solution following equation group
Its solution is
As can be seen that formula (3) are special case of the formula (6) as A=I, I is unit matrix.
Redundant analysis scheme based on double optimization (Quadratic Optimization, QP) also attracts attention,
F.T.Cheng, T.-H.Cheng were in proposition joint zero deflection performance indicator (F.T.Cheng, T.-H.Chen, and in 1994
Y.-Y.Sun, Resolving manipulator redundancy under inequality constraints (inequality
Redundant robot's method for planning track under constraint condition), IEEE Trans.Robotics Automat., 1994,10 (1)：
65-71)：
Wherein C, H are adjusting matrix.
Can the performance of redundant robot's trajectory planning is directly related to robot realize given end task.Work as end
The motion profile of actuator is closure, and after robot completes end task, each joint angle variable is in space
In track not necessarily close.There may be undesirable joint position shapes for this non-repeatability problem, so that redundant robot is last
The case where repetition operation that end seal closes track is occurred beyond expection, result even in the generation of accident and dangerous situation.Using most
It cannot be guaranteed the repeatability of movement for extensive pseudo- inverse control method.In order to complete repeating motion, the side of autokinesis is generallyd use
Method is made up, and autokinesis is adjusted and often inefficient (is detailed in Klein C A and Huang C, Review of
Pseudo inverse control for use with kinematically redundant manipulators (is based on
Redundant robot's motion planning of pseudo- inverse control method), IEEE Trans.Syst.Man.Cybern.1983,13 (2):245-
250；Tchon K,Janiak M.Repeatable approximation of the Jacobian pseudo-inverse
(repeatable the approaching of Jacobi puppet inverse matrix), Systems and Control Letters, 2009,58 (12):849-856).
In order to execute repeating motion task, repeating motion index is introduced as Optimality Criteria, forms repeating motion planning
(Repetitive motion planning, RMP) scheme (Zhang Y, Wang J, Xia Y.A dual neural
network for redundancy resolution of kinematically redundant manipulators
Subject to joint limits and joint velocity limits (is limited based on joint angles and angular speed
Redundant robot's method for planning track), IEEE Trans Neural Netw., 2003,14 (3):658-667).It is common to repeat
Motion index is following asymptotic convergence performance index AOC (Asympototically-Convengent Optimality
Criterion)：
Wherein β is constant coefficient, and θ (0) is the joint angles of initial time.Here quadratic programming (QP) uses recurrent neural
Network (RNN) calculation method solves.Common Neural Networks Solution device has asymptotic convergence performance, as long as it is enough to calculate the time
It is long, effective solution can be obtained.
Redundancy parsing problem is described based on quadratic programming, document has been delivered and often recurrent neural network has been used to solve.Phase
Than in the recurrent neural network with asymptotic convergence dynamic characteristic, final state, which restrains dynamic characteristic, has finite time convergence, tool
There are fast convergence rate, the high advantage of computational accuracy.It is worth noting that redundant robot's trajectory planning problem is to be attributed to time-varying
Computational problem, and the effective ways for solving time-varying problem are calculated using the recurrent neural network with finite time convergence energy
Method.In addition, the neural network for having delivered the finite time convergence control in document mostly uses linear incentive function, or there is infinitary value
Excitation function, in actual implementation, due to finite energy, when infinitary value excitation function neural fusion, there is also essential difficult.
Summary of the invention
The present invention provide a kind of finite time convergence control, computational accuracy are high, be easily achieved optimizing index is attracted based on final state
Redundant robot's motion planning method.Repeating motion performance optimizing index of the invention takes final state to attract optimizing index, passes through
Time-varying matrix equation Solve problems are converted by the double optimization problem of redundant robot's trajectory planning, with finite value final state nerve
Network is as solver, in the case where initial position offset, realizes the repeating motion rule of redundant robot's finite time convergence control
The task of drawing.
In order to realize the purpose, the present invention provides the following technical solution：
A kind of redundant robot's repetitive motion planning method based on secondary radical final state attracting performance index, including it is following
Step：
(1) the target terminal track r of end effector of robot is given in cartesian spaced(t), and each pass is provided
The expectation of section backs joint angles θd(0)；
(2) for the robot of repeating motion, initial joint angles are θ (0)=θ0, initial joint angles θ0It can not
It is same as it is expected back joint angles, i.e. θ0≠θd(0)；
(3) redundant robot's repeating motion planning is described as quadratic programming problem, to current joint vector and the phase of sening as an envoy to
It hopes joint vector deviation that there is the performance indicator of final state attracting performance, forms repeating motion programme：
Wherein,
0,0 < δ < 1 of κ > 0, ε > is constant coefficient；Sgn () indicates sign function, θ (t)-θd(0) joint angles are indicated
And it is expected that the joint angles deviation that backs, since robot initial position is not on desired trajectory, end effector speed with
The left side of the relational expression of joint velocity is needed plus a feedback deviation amount, i.e. rd- f (θ), the departure indicate physical end rail
There are errors between mark and expectation terminal track；Due to the convergence relation of arteface, this error can constantly reduce, until
It is zero, parameter κ is used to adjust the rate that end effector moves to desired trajectory, and J (θ) is provided according to robot DH parameter
Jacobian matrix, f (θ) are robotic end effector motion profile functions；
(4) dynamic characteristic equation of finite value final state neural network described below is constructed
Wherein, ε > 0,0 < δ < 1, formula (10) right side functions take finite value, and when error E is intended to infinite, functional value becomes
To in ± ε, therefore, the neural network as described in formula (10) is referred to as finite value neural network, when formula (10) expression one is limited
Between collective system, error convergence is in zero time
Wherein,
(5) Lagrangian is defined
Wherein, λ is Lagrange multiplier vector.AboutLocal derviation is sought respectively with λ, and enabling it is zero, is arranged
W (t) Y (t)=v (13)
Wherein,
I is unit matrix；
(6) it is the quadratic programming problem in solution procedure (3), defines error by formula (13)
E (t)=W (t) Y (t)-v
Finite value final state neural network equation is obtained according to formula (10)
Wherein E obtains each joint angles of robot by neural computing process for error variance.
Matrix operation expression therein calculates each element, such as takes absolute value | | it indicates to each of matrix
Element takes absolute value, and matrix division expression is divided by by element.
Technical concept of the invention is：The optimizing index of redundant robot's trajectory planning is designed as a kind of finite value final state
The optimizing index of attraction
Wherein,
0,0 < δ < 1 of κ > 0, ε >；Sgn () indicates sign function, θ (t)-θd(0) joint angles are indicated and it is expected back
Hold together joint angles deviation, | | representative takes absolute value to each element in matrix.Due to the relationship of arteface, φ (t) is most
Whole finite time convergence control is in zero.For the repeating motion planning problem under solving optimization index, it is whole to construct finite value described below
The dynamic characteristic equation of state neural network
Wherein, E is error variance.0,0 < δ < 1 of ε >.Formula (17) right side functions take finite value, when error E is intended to nothing
When poor, functional value is intended to ± ε.Therefore, the neural network as described in formula (17) is referred to as finite value neural network.
Beneficial effects of the present invention are mainly manifested in a kind of based on final state attraction optimizing index TOC (Teminal
Optimality Criterion), realize that the finite time convergence control of redundant robot repeats under initial position drift condition
Motion planning task.Compared to existing repetitive motion planning method, this method error convergence is fast, and time in each joint of robot
Hold together angle precision height.Compared to the recurrent neural network with asymptotic convergence characteristic, finite value final state neural network it is limited when
Between convergence property be suitable for time-varying problem solving (formula (13) be time-varying matrix equation)；The network that the Neural Networks Solution device uses
Dynamical equation (10) is finitely valued function, is easily achieved, and at low cost in engineer application, meets actual demands of engineering.
Detailed description of the invention
Fig. 1 is the flow chart provided by the invention for repeating programme.
Fig. 2 is φ () function when taking different δ values.
Fig. 3 is the redundant robot PUMA560 that the present invention repeats programme.(select from Y Zhang, Z
Zhang.Repetitive motion planning and control of redundant robot manipulators
(the repeating motion planning and control of redundant robot) .Berlin:Springer,2013:23-24)
Fig. 4 is the motion profile of redundant robot PUMA560 end effector.
Fig. 5 is the motion profile in each joint redundant robot PUMA560.
Fig. 6 is each joint angle of redundant robot PUMA560.
Fig. 7 is each joint velocity of redundant robot PUMA560.
Fig. 8 is each position error of redundant robot PUMA560 end effector.
Fig. 9 is the solution error of finite value final state neural network and asymptotic neural network.
Specific embodiment
Specific implementation step of the invention is described further with reference to the accompanying drawing.
Referring to Fig.1~Fig. 9, a kind of redundant robot's repeating motion planning based on secondary radical final state attracting performance index
Method, Fig. 1 are redundant robot's repeating motion programme flow chart, are made of following three steps：1, redundancy machine is determined
People's end effector desired trajectory and expectation back each joint angles；2, the redundancy machine that there is final state to attract optimizing index is established
People's repeating motion quadratic programming scheme；3, with finite value final state Neural Networks Solution quadratic programming problem, each joint angle rail is obtained
Mark.It is specific as follows：
1) desired trajectory is determined
The expectation of setting redundant robot PUMA560 backs joint angles
The central coordinate of circle (x=0.2m, y=0, z=0) for determining Circular test, is set as 0.2m, disc and xy for round radius
The angle of plane is π/6rad, and end effector completes Circular test time T=10s, it is contemplated that the initial position of robot does not exist
On desired motion profile, six joint angles initial values of robot are set as θ (0)=[0, -0.5, -0.1,1.5, -0.6,0
]Trad；
2) the quadratic programming scheme of redundant robot's repeating motion is established
Redundant robot's repeating motion trajectory planning is described as following quadratic programming problem, the optimization that wherein final state attracts
Index is
Sgn () indicates that sign function, κ take 2, ε to take 2；φ=θ (t)-θd(0) indicate that joint angles and expectation back pass
Save angular deviation；Since robot initial position is not on desired trajectory, the pass of end effector speed and joint velocity
It is that the left side of formula is needed plus a feedback deviation amount, i.e. rd- f (θ), the departure indicate between desired locations and physical location
There are errors；Due to the convergence relation of arteface, this error can constantly reduce, until being zero；Parameter κ is used to adjust end
Hold actuator motions to the rate of desired trajectory, J (θ) is the Jacobian matrix provided according to robot DH parameter, and f (θ) is machine
Device people's end effector motion profile function；
3) finite value final state Neural Networks Solution quadratic programming problem is constructed
Finite value final state neural network model is constructed, dynamic characteristic equation is
Wherein, E is error variance, and ε takes 2, δ to take 0.5；Formula (10) indicate a finite time convergence control system, error convergence in
Zero time is
Wherein
Define following Lagrangians：
It is rightAboutLocal derviation is sought respectively with λ, enabling it is zero, collated
W (t) Y (t)=v (13)
Wherein,
I is unit matrix；
Error variance E (t)=W (t) Y (t)-v is defined by formula (10), according to finite value final state neural network dynamic equation,
Provide finite value final state neural network equation
Wherein E obtains each joint angles of robot by neural computing process for error variance.
Fig. 2 is the φ () when taking different δ values.It can be seen from the figure that the functional derivative tends to be infinite as δ increases
Greatly, so that neural network final state restrains.
Redundant robot PUMA560 shown in Fig. 3 is for realizing repetition programme of the invention.The robot is by 1
Pedestal, 3 connecting rods, 6 joints are constituted, and length of connecting rod is L=[0.4318,0.0203,0.4318,0.25625]Tm。
Fig. 4 show the motion profile of end effector of robot in space.Target Circular test and machine are provided in figure
People's end effector motion profile, 1 is desired trajectory, and 2 be actual path.As can be seen that the initial position of end effector is not
On desired trajectory.As shown in figure 8, the final value location error precision of end effector is in XYZ as the time increases (T=10s)
Three sides of axis reach up to 10-7M, actual path and desired trajectory coincide.
Fig. 5 show the articulate motion profile of institute of robot.It can be seen from the figure that the track in each joint is being run
It is closure after a cycle, realizes repeating motion control.
Attract validity of the optimizing index in repeating motion planning to verify final state, robot PUMA560 is held end
Row device completes joint angle transient state track obtained in Circular test process and joint angle speed transients track difference is as shown in Figure 6,7.
As can be seen that each joint angle of redundant robot finally converges on desired joint Angle Position.The characteristic equation of asymptotic neural network
For
As T=10s, before and after robot motion joint angle and its it is expected the maximum deviation between joint angle be -1.1 ×
10-6rad；ρ=2,4,8 are taken, respectively with the final value max value of error difference for each joint angle that asymptotic Neural Networks Solution obtains
It is 3.7 × 10-4rad、1.6×10-4Rad and 4.4 × 10-5Rad, see Table 1 for details.
Table 1
For the constringency performance for comparing asymptotic convergence network Yu final state network, definition calculates error JE(t)=| | W (t) Y (t)-
v(t)||2.Fig. 9 provides the error rail for solving quadratic programming problem with finite value final state neural network and recurrent neural network respectively
Mark.It can be seen from the figure that the J when ρ takes 8, under asymptotic networkEConvergence rate just takes convergence speed when 2 with ε under final state network
Degree is suitable.J as t=0.8s, when final state network ε=2EIt is 3 × 10-5, asymptotic network takes J when ρ=2,4,8ERespectively
0.014,0.07 and 0.16.
Claims (1)
1. a kind of redundant robot's repetitive motion planning method based on secondary radical final state attracting performance index, feature exist
In the described method comprises the following steps：
(1) the target terminal track r of end effector of robot is given in cartesian spaced(t), and each joint is provided
It is expected that the joint angles θ that backsd(0)；
(2) for the robot of repeating motion, initial joint angles are θ (0)=θ0, initial joint angles θ0Different from expectation
Back joint angles, i.e. θ0≠θd(0)；
(3) redundant robot's repeating motion planning is described as quadratic programming problem, to current joint vector and the expectation pass of sening as an envoy to
The performance indicator that vector deviation has final state attracting performance is saved, repeating motion programme is formed：
Wherein,
0,0 < δ < 1 of κ > 0, ε >；Sgn () indicates sign function, θ (t)-θd(0) indicate that joint angles and expectation back joint
Angular deviation；Since robot initial position is not on desired trajectory, the relationship of end effector speed and joint velocity
The left side of formula is needed plus a feedback deviation amount, i.e. rd- f (θ), the departure indicate physical end track and desired terminal rail
There are errors between mark；Due to the convergence relation of arteface, this error can constantly reduce, until being zero；Parameter κ is used to
Adjust the rate that end effector moves to desired trajectory；J (θ) is the Jacobian matrix provided according to robot DH parameter, f
(θ) is robotic end effector motion profile function；
(4) dynamic characteristic equation of finite value final state neural network is constructed
Wherein, E is error variance；0,0 < δ < 1 of ε >, formula (2) indicate a finite time convergence control system, and error convergence is in zero
Time is
Wherein
(5) Lagrangian is defined
Wherein, λ is Lagrange multiplier vector；AboutLocal derviation is sought respectively with λ, and enabling it is zero, collated
W (t) Y (t)=v (13)
Wherein,
I is unit matrix；
(6) it is the quadratic programming problem in solution procedure (3), defines error by formula (5)
E (t)=W (t) Y (t)-v
Finite value final state neural network equation is obtained according to formula (10)
Wherein E is error variance；By neural computing process, each joint angles of robot are obtained.
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