CN108693776A - A kind of robust control method of Three Degree Of Freedom Delta parallel robots - Google Patents

A kind of robust control method of Three Degree Of Freedom Delta parallel robots Download PDF

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CN108693776A
CN108693776A CN201810824730.1A CN201810824730A CN108693776A CN 108693776 A CN108693776 A CN 108693776A CN 201810824730 A CN201810824730 A CN 201810824730A CN 108693776 A CN108693776 A CN 108693776A
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惠记庄
武琳琳
赵睿英
张红俊
李梦
雷景媛
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Changan University
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Abstract

The invention discloses a kind of robust control methods of Three Degree Of Freedom Delta parallel robots, will be separated containing probabilistic item in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models, respectively obtain the nominal item and indeterminate of parallel robot system;The nominal compensation tache in controller is established according to the nominal item in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models, for being compensated to nominal robot system;Positive definite diagonal matrix is chosen, the P.D. controlling units in controller are designed, for being compensated to initial position error;According in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models with uncertain related item constructed fuction, solve the function for representing system indeterminate upper bound information, and the construction of function uncertainty compensation tache is utilized, to uncertain present in compensation system;Finally, robust controller is provided.Solves the technical issues of traditional control method is often based upon accurate kinetic model, is extremely difficult to practical control purpose.

Description

A kind of robust control method of Three Degree Of Freedom Delta parallel robots
Technical field
The invention belongs to parallel robot motion control field more particularly to a kind of Three Degree Of Freedom Delta parallel robots Robust control method.
Background technology
As Delta parallel robots are in the high-precision pointed collar domain such as processing and manufacturing, microelectronics, medical rehabilitation, Intelligent logistics Using requirement of the Delta parallel robots to control accuracy and anti-interference ability is higher and higher.Delta parallel manipulators are artificially more Connecting rod chain type parallel-connection structure, the slender rod piece of slave arm generally use light material, in high speed operation joint space with it is thin The flexible deformation of long pole can cause residual oscillation, which will seriously affect the precision and stability of movement.Meanwhile There is a large amount of uncertainties in actual operation for Delta parallel robots, such as:It is the kinetic parameter of system change, non-thread Property the joint-friction power interference and disturbance etc. of random external load, these uncertain factors affect control accuracy and work is imitated Rate.Therefore, for the research weight with the research of probabilistic Delta parallel robots dynamic control method as the field Point.
Currently, including mainly Linear Control for the dynamic control method with probabilistic Delta parallel robots Method and nonlinear control method.Linear control method such as PID control, computed moment control etc. is by by nonlinear system mould After Linear, then realize the control to robot.These methods are dependent on the accurate kinetic model of system and the work determined Condition has ignored the uncertainty of system, with control process it is increasingly sophisticated, these linear control methods just show disadvantage End, simple linear control method are often extremely difficult to control and require, poor robustness.Therefore, nonlinear control method in recent years As the research emphasis in the field.Nonlinear control method for Delta parallel robot systems includes variable-structure control side Method, linear Feedback Control method and self-adaptation control method etc., these control methods are more or less to there are problems that, such as In synovial membrane variable structure control method, there are discontinuous switching characteristic during " structure " switching, easily causing can not eliminate Chattering phenomenon, due to not exclusively understanding the kinetic models of Delta parallel robots in linear Feedback Control method, to line The compensation of sexual system is often halfway.And the intelligent control methods such as neural network, fuzzy control are at present in Delta parallel machines Device people motion control field is still in the elementary step, although Intelligent Control Theory takes in Delta parallel robot engineer applications Significant progress was obtained, but there are still some problems, such as the implicit number of layers of neural network and the selection of neuron number, Fuzzy Control High-frequency vibration phenomenon present in system, single fuzzy control are difficult to establish complete fuzzy control rule etc., are needed to these Problem is further explored.
American scholar Leitmann proposed on the basis of system optimization, game theory scheduling theory uniform bound, it is consistent most The concept of whole bounded, for the non-linear and uncertain problem of system, it is proposed that Min-Max Lyapunov control methods, again Claim Leitmann methods.Leitmann is discussed respectively under the conditions of no structure matching, existence variable time delay phenomenon it is non- The Linear System Stability of the robustness and existence time delay phenomenon of deterministic system, in conjunction with decentralised control theory to containing not Deterministic System with Nonlinear Coupling control is analyzed, for close coupling and nonlinear Delta parallel robots dynamic Control method research provides new approaches.Therefore, to probabilistic Three Degree Of Freedom Delta parallel robot dynamic controls Strategy is studied always those skilled in the art hot spot of interest.
Invention content
It is in view of the above-mentioned drawbacks of the prior art or insufficient, it is an object of the present invention to be carried based on Leitmann methods For a kind of robust control method of Three Degree Of Freedom Delta parallel robots, it is often based upon accurately with solving traditional control method Kinetic model, the problem of being extremely difficult to practical control purpose.
In order to realize that above-mentioned task, the present invention are achieved using following technical solution:
A kind of robust control method of Three Degree Of Freedom Delta parallel robots, which is characterized in that according to the following steps into Row:
Step 1, it will be separated containing probabilistic item in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models, Respectively obtain the nominal item and indeterminate of parallel robot system;
Step 2, the mark in controller is established according to the nominal item in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models Claim compensation tache, for being compensated to nominal robot system;
Step 3, positive definite diagonal matrix is chosen, the P.D. controlling units in controller are designed, for initial position error It compensates;
Step 4, according in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models letter is constructed with uncertain related item Number solves and represents the function of system indeterminate upper bound information, and utilizes the construction of function uncertainty compensation tache, for pair Uncertainty in system compensates;
Step 5, finally, robust controller is provided.
The robust control method of the Three Degree Of Freedom Delta parallel robots of the present invention, the advantageous effect brought is, set In the robust control method of meter, if there is no initial position errors with uncertainty in robot system, in controller individually Nominal compensation tache the track following error of parallel robot can be made to reach the performance of uniformly asymptotic stability.If system of robot There is only when initial position error in system, the uncertain factor in system is zero, and nominal compensation tache adds P.D. in controller Controlling unit can make robot system meet Control performance standard.If not only there is initial position error in robot system but also deposited In uncertainty, in addition the uncertain compensation tache in controller can compensate for the uncertainty in system, system is made to meet Uniform bound and Uniform Ultimate Boundedness energy index.
Description of the drawings
Fig. 1 is the space structure simplified schematic diagram of DELTA robots;
Fig. 2 is the robust Controller Design simplified schematic diagram of DELTA robots;
Fig. 3 is Delta parallel robot joint angle Displacement simulation result figures;
Fig. 4 is Delta parallel robot joint angle velocity simulation result figures;
Fig. 5 is that Delta parallel robots control input torque simulation result diagram;
Fig. 6 is Delta parallel robot track following error e simulation result diagrams;
Fig. 7 is Delta parallel robot track following errorsSimulation result diagram;
Fig. 8 is Delta parallel robot running orbit simulation result diagrams;
Technical scheme of the present invention work is further clearly and completely described below in conjunction with drawings and examples.
Specific implementation mode
In order to more clearly explain the embodiment of the invention or the technical proposal in the existing technology, to embodiment or will show below There is attached drawing needed in technology description to be briefly described, it should be apparent that, the accompanying drawings in the following description is only this Some preferred embodiments of invention, the present invention is not limited to these Examples.
Robot introduction is carried out first:The present embodiment uses a kind of very common Limited-DOF Parallel Robot --- and three Degree of freedom Delta parallel robots are analyzed as research object.
Shown in FIG. 1 is structure schematic diagram of the Three Degree Of Freedom Delta parallel robots in working face, and in work Make the rectangular coordinate system established in space.
Wherein, O-A1A2A3For silent flatform, O '-C1C2C3For moving platform, silent flatform and moving platform are equilateral triangle.O- XYZ is silent flatform system (basis coordinates system), and O '-x ' y ' z ' are moving platform system, and O, O ' are located at quiet, moving platform system geometric center, Z, z ' axis upward direction is set as positive direction.A1,A2,A3Positioned at the intersection point of motor shaft and master arm axis, referred to as parallel robot Active joint.B1,B2,B3Positioned at the intersection point of master arm axis and slave arm axis, C1,C2,C3Positioned at slave arm axis and move The intersection point of platform.
Define the length A of robot master armiBiFor la, the length B of slave armiCiFor lb, the circumscribed circle half of dynamic and static platform Diameter is respectively r, R.θ1,θ2,θ3It is master arm to the subtended angle of silent flatform, q1,q2,q3For main movable joint corner.
As shown in Fig. 2, a kind of robust control method for Three Degree Of Freedom Delta parallel robots that the present embodiment provides, it should Method includes the steps that being:
Step 1, it will be separated containing probabilistic item in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models, Respectively obtain the nominal item of parallel robot systemWithWith indeterminate Δ M, Δ C, Δ G and Δ F.
Step 2, the mark in controller is established according to the nominal item in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models Claim compensation tache P1。
Step 3, positive definite diagonal matrix K is chosenp=diag[kpi]3×3, Kv=diag[kvi]3×3, design in controller P.D. controlling unit P2。
Step 4, according in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models letter is constructed with uncertain related item Number φ solves the function ρ for representing system indeterminate upper bound information, and constructs uncertain compensation tache P using function ρ3。
Step 5, finally, robust controller τ=P is provided1+P2+P3。
It is the detailed implementation content of each step below:
Step 1:
It is expressed as with probabilistic Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models:
(1)
Wherein, q ∈ R3It is vectorial for main movable joint angle,For main movable joint angular velocity vector,For actively Joint angle vector acceleration.σ∈Σ∈RpFor the vector of uncertain parameter present in robot system, Σ ∈ RpFor uncertain ginseng Several compacts, and represents probabilistic boundary.M (q, σ, t) is robot system inertial matrix,For system Coriolis force/centrifugal force item,For skew symmetric matrix, G (q, σ, t) is the gravity item of system,For the external disturbance suffered by system, τ (t) is system input torque.M (), C (), G () and F () are equal Continuously or about time t Lebesgue measurable.
For the design of subsequent controllers, M (), C (), G () and the F () in formula (3.3) are decomposed into:
Wherein,WithReferred to as Delta parallel robot systems Nominal item, Δ M (q, σ, t),Δ G (q, σ, t) andReferred to as Delta parallel robots The indeterminate of system.
When uncertain factor is not present during the work time in Delta parallel robots,
In order to simplify derivation, the case where not producing ambiguity, the independent variable in following sections formula can omit.
Wherein, inertial matrix meets:
Assuming that 1:
Inertial matrix M (q, σ, t) is positive definite matrix, i.e., to arbitrary q ∈ R3, there are a constantsSo that:
Assuming that 2:
To arbitrary q ∈ R3, there is always constant γj, j=0,1,2, and γ0>0, γ1,2>=0 so that:
‖M(q,σ,t)‖<γ0+γ1‖q‖+γ2‖q‖2 (7)
For the series and parallel robot connected with sliding pair by revolute pair, inertial matrix M (q, σ, t) is only used with quality Property parameter, arthrodia are related to the position of cradle head.Therefore, there is always one group of constant γj, enable series and parallel machine hostage The European norm of amount inertial matrix meets formula (7).
Step 2:
If the desired trajectory of Three Degree Of Freedom Delta parallel robots is qd,WithWherein Indicate desired locations, and qdFor C2Continuously,For desired speed,It is expected acceleration.
The track following error of definition system is:
e:=q-qd (8)
Therefore, the speed tracing error of system can be expressed as with acceleration tracking error:
Then:
Step 3:
Wherein, positive definite diagonal matrix Kp=diag&#91;kpi]3×3And kpi>0, Kv=diag&#91;kvi]3×3And kvi>0, i=1,2,3.
Step 4:
Assuming that 3:
There are a positive definite integral form ρ:R3×R3×R→R+, positive definite matrix S=diag&#91;si]3×3, si>0, ks=λmin(S), i =1,2,3 so that:
Wherein:
Assuming that in 3, function phi represent in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models it is all with it is uncertain Related item, upper bound information available functions ρ are indicated.If uncertain factor is not present in system, φ ≡ 0.
In formula (15):
In formula (17), ε >0.
Step 5:
Consider that tracking error vector isIt now provides a kind of for Three Degree Of Freedom Delta parallel robots Robust trajectory tracking control device:
Wherein:
In formula (21):
In formula (22), ε >0, positive definite diagonal matrix Kp=diag&#91;kpi]3×3And kpi>0, Kv=diag&#91;kvi]3×3And kvi> 0, i=1,2,3.
In the formula (18) of designed robust trajectory tracking control device, if there is no initial positions to miss in robot system When difference and uncertainty, τ=P is enabled1The track following error of parallel robot can be made to reach the performance of uniformly asymptotic stability.
If there is only when initial position error in robot system, Δ M, Δ C, Δ G and Δ F are zero, can choose letter Number ρ=0 so thatτ=P at this time1+P2, when can make t → ∞,
If not only there is initial position error in robot system but also had uncertain, τ=P is enabled1+P2+P3, can make t → When ∞,Meet uniform bound and Uniform Ultimate Boundedness energy index.
One, stability proves
First providing stability proves conclusion:
If Three Degree Of Freedom Delta parallel manipulators human occupant dynamic model (1), which meets, assumes 1-3, controller design (18) can So that track following error vectorMeet:
(1) Uniform boundedness:For any given r>0, andWork as t>t0When, there are a positive real numbers d(r):0<d(r)<∞ so thatIt sets up.
(2) Uniform Ultimate Boundedness:For any givenAndWhenWhen,It sets up, wherein
It is as follows that proof procedure is given below:
Constructing liapunov function is:
According to assuming 1, have:
Wherein:
In formula (26), parameter kv,si,kpWithIt is the real number more than zero, the Principal Minor Sequence is all higher than zero, then ψiFor positive definite Matrix.It enables(i.e.):
The V known to formula (25) and formula (27) is positive definite matrix.
By assuming 2:
For first item on the right of inequality, by q:=e+qdBring formula (28) into:
‖ q ‖=s &#124;&#124;e+qd||≤‖e‖+maxt||qd|| (29)
‖q‖2≤‖e‖2+2‖e‖maxt||qd||+(maxt||qd||)2 (30)
Simultaneously:
Wherein,
For Section 2 on the right of inequality:
Therefore there are constantsSo that:
Wherein:
In formula (33)For stringent positive real number, then to allSelected liapunov function is passed for dullness Subtraction function.Therefore, according to formula (27) and formula (33), the V of construction is an effective liapunov function.
The derivative of liapunov function V is:
Formula (32) is substituted into formula (37):
Bring formula (22) and formula (23) into formula (38), i.e., as ‖ μ ‖ >When ε:
As ‖ μ ‖≤ε,
Therefore, according to assuming 3, formula (39) is understood with formula (40):
Bring formula (41) into formula (38):
Wherein,For formula (38),It is the normal number of bounded with ε, therefore works asFoot When enough big, the derivative of liapunov function is negative value, i.e.,:
According to document (Chen Y..On The Deterministic Performance of Uncertain Dynamical Systems&#91;J&#93;.International Journal of Control, 1986,43 (5):1557-1579), When the derivative of liapunov function meets formula (43), the formula (18) of controller can make track following error vectorMeet Uniform bound, i.e., for any given r>0, andWork as t>t0When, there are positive real number d (r) such as formula (44) institutes Show so thatIt sets up.
Wherein,Meanwhile tracking error vectorAlso meet uniform ultimate bounded.I.e. for appointing Meaning gives r>0,AndWhenWhen,It sets up.
Wherein:
Two, kinetic model emulates
In MATLAB softwares, using ode15i function pair Three Degree Of Freedom Delta parallel robots kinetic model with The controller of design is emulated.
Assuming that the uncertain factor that parallel robot is subject to is the mass parameter of moving platform With external loadingWherein, WithFor nominal item, Δ mO′,ΔF1,ΔF2With Δ F3For the indeterminate changed over time.Uncertain parameter vector is defined as: σ=s &#91;ΔmO ',ΔF1,ΔF2,ΔF3]T。
According to assuming 1, the indeterminate in M (), C (), G (), F () is separated into constructed fuction
By formula (48), solved function positive definite integral form ρ:
Wherein, ΣmAnd ΣFRepresent biggest impact degree of the uncertain parameter to robot system.
If the target trajectory that Delta parallel robot workbenches needs track is:
The structural parameters of Three Degree Of Freedom Delta parallel robots are as follows:
The length l of master arma=200mm, the circumradius R=180mm of silent flatform, the circumradius r of moving platform The mass parameter of=100mm, robot are as follows:Master arm quality ma=1.193kg, slave arm quality mb=1.178kg is moved flat Platform quality mO,=4.3225kg.
The control parameter for choosing controller is as follows:
Kv=diag&#91;1,1,1&#93;, Kp=diag&#91;1,1,1&#93;, S=diag&#91;8,8,8&#93;, ε=0.1.
It is as follows to choose nominal parameters:
It is as follows to choose uncertain parameter: Δm=Δf=0.5.∑m=0.5, ∑F =0.5.Set emulation initial value as:q0=&#91;0.5434 0.5434 0.9639&#93;T, Simulation result is as shown in figures 3-8.
Wherein, Fig. 3 and Fig. 4 is the simulation result of the angular displacement of active joint and joint angular speed of Delta parallel robots, Fig. 5 is the input torque simulation result on three active joint angles.
When Three Degree Of Freedom Delta parallel robot systems are influenced by initial position error with uncertainty, τ is enabled respectively =P1, τ=P1+P2, τ=P1+P2+P3Input comparison control effect, simulation result are as Figure 6-Figure 8 in order to control.
The simulation result that Fig. 6 is the system trajectory tracking error e under control input at three kinds, as τ=P1When inputting in order to control, Track following error diffuses to 0.2m near 0.01 by 5.5s, and error persistently increases after 5s, does not have in simulation time There is convergence.As τ=P1+P2When inputting in order to control, track following error is decreased to 0.05m near 0.01m after 3s, and with The increase of time, error maintains near 0.05m always, cannot converge to 0m.As τ=P1+P2+P3When inputting in order to control, it is System enters after 0.4s near 0.01m and is maintained in 0m environs.
Fig. 7 is that system trajectory tracking error under input is controlled at three kindsSimulation result, as τ=P1When inputting in order to control, Track following errorIt is dissipated into 0.4m/s by 5.5s near 0.3m/s, andIt cannot restrain within the limited time.Work as τ =P1+P2When inputting in order to control, track following errorIt is decreased to 0.1m/s from 0.3m/s by 1.5s, with simulation time Increase, errorCurve has the tendency that rising.As τ=P1+P2+P3When inputting in order to control, track following errorIt is passed through from 0.3m/s It is reduced near 0m/s after crossing 0.4s.
Fig. 8 be the track emulations of the lower Delta parallel robot end effectors operation of three kinds of control inputs as a result, when τ= P1With τ=P1+P2When inputting in order to control, end effector track following target trajectory X cannot be controlledd, but work as τ=P1+P2+P3 When inputting in order to control, end effector can be moved to rapidly from initial deviation position near target trajectory, and substantially according to Target trajectory is run.
Simulation result shows:System have initial position error and it is uncertain when, the Trajectory Tracking Control that is proposed Device can control end effector track following target trajectory, at the same make track following error meet uniform bound with it is consistent final Bounded.

Claims (1)

1. a kind of robust control method of Three Degree Of Freedom Delta parallel robots, which is characterized in that follow the steps below:
Step 1, it will be separated containing probabilistic item in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models, respectively Obtain the nominal item and indeterminate of parallel robot system;
Step 2, the nominal benefit in controller is established according to the nominal item in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models Link is repaid, for being compensated to nominal robot system;
Step 3, positive definite diagonal matrix is chosen, the P.D. controlling units in controller are designed, for being carried out to initial position error Compensation;
Step 4, according in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic models with uncertain related item constructed fuction, The function for representing system indeterminate upper bound information is solved, and utilizes the construction of function uncertainty compensation tache, for being Uncertainty in system compensates;
Step 5, finally, robust controller is provided.
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