CN109062039A - A kind of adaptive robust control method of Three Degree Of Freedom Delta parallel robot - Google Patents

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

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CN109062039A
CN109062039A CN201810824709.1A CN201810824709A CN109062039A CN 109062039 A CN109062039 A CN 109062039A CN 201810824709 A CN201810824709 A CN 201810824709A CN 109062039 A CN109062039 A CN 109062039A
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赵睿英
惠记庄
武琳琳
李梦
张红俊
雷景媛
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Changan University
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    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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Abstract

The invention discloses a kind of adaptive robust control methods of Three Degree Of Freedom Delta parallel robot, it is separated probabilistic item is contained in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic model, respectively obtains 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 model, for compensating to nominal robot system；By choosing positive definite diagonal matrix, the P.D. controlling unit in controller is designed, for compensating to initial position error；Further according to, with uncertain related item, construction represents the function of system indeterminate upper bound information, and verifies hypothesis in Three Degree Of Freedom Delta parallel manipulator human occupant dynamic model；The adaptive law of item is established with dead zone and revealed to Selecting All Parameters, is used for the probabilistic upper bound information of On-line Estimation；According to function and adaptive law, the uncertainty in system is compensated；Finally, self-adaptive robust controller is provided.

Description

Adaptive robust control method of three-degree-of-freedom Delta parallel robot
Technical Field
The invention belongs to the field of parallel robot motion control, and particularly relates to a three-degree-of-freedom Delta parallel robot adaptive robust control method.
Background
With the application of the Delta parallel robot in high precision fields such as processing and manufacturing, microelectronics, medical rehabilitation, intelligent logistics and the like, the requirement of the Delta parallel robot on control precision and anti-interference capability is higher and higher. Delta parallel robot is a multi-link chain type parallel structure, a driven arm of the Delta parallel robot usually adopts a slender rod piece made of light materials, when the Delta parallel robot works at high speed, residual vibration can be caused by the joint clearance and the elastic deformation of the slender rod piece, and the vibration phenomenon can seriously affect the precision and the stability of the movement. Meanwhile, a great deal of uncertainty exists in the practical work of the Delta parallel robot, such as: dynamic parameters of system change, nonlinear joint friction interference, disturbance of external random loads and the like, and the uncertain factors influence the control precision and the working efficiency. Therefore, research on a Delta parallel robot dynamic control method with uncertainty becomes a research focus in the field.
For a Delta parallel robot system with uncertainty, when a robust controller is designed by a traditional robust control method, an upper bound of the uncertainty needs to be known, and a robust controller gain is constructed according to the upper bound of the uncertainty, so that the obtained robust controller can deal with the worst situation of the robot system, and the stability and the control precision of the system with accurate definition are further ensured. But Delta parallel robots do not always work in the "worst case" and therefore robust controllers designed according to the upper bound of uncertainty are somewhat conservative. The traditional self-adaptive control method can carry out system identification on uncertain parameters of a system, and designs a controller by utilizing the identified parameters. However, adaptive control cannot solve the problem of uncertainty of the system due to disturbance or unmodeled parts, and therefore, research on a control method of the Delta parallel robot system is always of interest to those skilled in the art.
Disclosure of Invention
Aiming at the defects or shortcomings of the prior art, the invention aims to provide a self-adaptive robust control method of a three-degree-of-freedom Delta parallel robot by combining robust control and self-adaptive control from a brand-new angle so as to solve the problem that the traditional robust control method is often based on accurate uncertainty upper bound information and the problem that the traditional self-adaptive control cannot solve the uncertainty of a system caused by disturbance or an unmodeled part.
In order to realize the task, the invention adopts the following technical scheme to realize the following steps:
a self-adaptive robust control method of a three-degree-of-freedom Delta parallel robot is characterized by comprising the following steps of:
step 1, separating out uncertain items in a three-degree-of-freedom Delta parallel robot dynamic model to respectively obtain a nominal item and an uncertain item of a parallel robot system;
step 2, establishing a nominal compensation link in the controller according to a nominal item in the three-degree-of-freedom Delta parallel robot dynamics model, and compensating the nominal robot system;
step 3, selecting a positive definite diagonal matrix, and designing a P.D. control link in the controller for compensating the initial position error;
step 4, constructing a function representing the upper bound information of the uncertainty item of the system according to the uncertainty-related item in the three-degree-of-freedom Delta parallel robot dynamics model, and verifying the hypothesis;
step 5, selecting parameters, and establishing a self-adaptive law with dead zones and leakage items for online estimation of upper bound information of uncertainty;
step 6, constructing an uncertain compensation link according to the function and the self-adaptive law, and compensating the uncertainty in the system;
and 7, finally, providing the adaptive robust controller.
The adaptive robust control method of the three-degree-of-freedom Delta parallel robot has the beneficial effects that in the designed adaptive robust control method, if the initial position error and uncertainty do not exist in the robot system, the track tracking error of the parallel robot can reach the performance of consistent asymptotic stability by a single nominal compensation link in the controller. If the robot system only has initial position error, the uncertainty factor in the system is zero, and the robot system can meet the control performance index by adding a P.D. control link in a nominal compensation link in the controller. If the robot system has initial position error, uncertainty and unknown uncertainty upper bound, the uncertainty in the uncertainty compensation link and the adaptive rate compensatable system in the controller is added, so that the system meets the consistent bounded and consistent final bounded performance indexes.
Drawings
FIG. 1 is a schematic diagram of a spatial structure of a DELTA robot;
FIG. 2 is a schematic diagram of a robust controller design of a DELTA robot;
FIG. 3 is a diagram showing a simulation result of the angular displacement of a Delta parallel robot joint;
FIG. 4 is a diagram of a simulation result of the angular velocity of the joint of the Delta parallel robot;
FIG. 5 is a diagram of a simulation result of Delta parallel robot control input torque;
FIG. 6 is a diagram of adaptive parametersA simulation result graph;
FIG. 7 is a graph of the relationship between the upper bound of uncertain parameters and the maximum value of the adaptive parameter estimate;
FIG. 8 is a diagram of a simulation result of a Delta parallel robot trajectory tracking error e;
FIG. 9 shows the Delta parallel robot trajectory tracking errorA simulation result graph;
FIG. 10 shows the results of the Delta parallel robot operation trajectory simulation.
The technical solution of the present invention is further clearly and completely described below with reference to the accompanying drawings and examples.
Detailed Description
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, and it is obvious that the drawings in the following description are only some preferred embodiments of the present invention, and the present invention is not limited to these embodiments.
First, robot introduction is performed: in the embodiment, a very common parallel robot with few degrees of freedom, namely a three-degree-of-freedom Delta parallel robot, is adopted as a research object for analysis.
Fig. 1 shows a schematic structural diagram of a three-degree-of-freedom Delta parallel robot in a working plane, and a rectangular coordinate system established in a working space.
Wherein, O-A1A2A3Being a static platform, O' -C1C2C3The movable platform is an equilateral triangle. O-XYZ is a static platform system (base coordinate system), O '-x' y 'z' is a movable platform system, O, O 'is respectively positioned at the geometric centers of the static platform system and the movable platform system, and the axial upper direction of Z, z' is a positive direction. A. the1、A2、A3And the joint is positioned at the intersection point of the motor shaft and the axis of the driving arm and is called as the driving joint of the parallel robot. B is1、B2、B3At the intersection point of the master arm axis and the slave arm axis, C1、C2、C3Is positioned at the intersection point of the axis of the driven arm and the movable platform. Defining the length A of the robot's active armiBiIs 1aLength B of the follower armiCiIs 1bThe external circle radiuses of the movable platform and the static platform are R and R respectively. Theta1、θ2、θ3Opening angle of the active arm to the stationary platform, q1、q2、q3Is the active joint corner.
As shown in fig. 2, this embodiment provides an adaptive robust control method for a three-degree-of-freedom Delta parallel robot, which includes the steps of:
step 1, mixingSeparating the uncertain items in the Delta parallel robot dynamics model to respectively obtain the nominal items of the parallel robot systemAndand uncertainty terms Δ M, Δ C, Δ G, and Δ F.
Step 2, establishing a nominal compensation link P in the controller according to a nominal item in the three-degree-of-freedom Delta parallel robot dynamics model1For compensating the nominal robot system.
Step 3, selecting a positive definite diagonal matrix Kp＝diag[kpi]3×3，Kv＝diag[kvi]3×3Designing P.D. control link P in controller2For compensating for the initial position error.
Step 4, constructing a function representing the upper bound information of the uncertain item of the system according to the item construction function phi related to uncertainty in the three-degree-of-freedom Delta parallel robot dynamics modelAnd verifies hypothesis 3.
Step 5, selecting parameters kappa and k1、k2、k3E and ξ, establishing an adaptive law with dead zones and leakage terms for online estimation of upper bound information of uncertainty.
Step 6, according to the functionAnd adaptation law, construct P3For compensating for uncertainties in the system.
And 7, finally, giving an adaptive robust controller tau as P1+P2+P3。
The following is a detailed implementation of each step:
step 1:
the three-degree-of-freedom Delta parallel robot dynamics model with uncertainty is expressed as:
wherein q ∈ R3In order to be the active joint angle vector,in order to be the active joint angular velocity vector,is the active joint angular acceleration vector. Sigma belongs to RpFor uncertain parameter vectors existing in the robot system, sigma belongs to RpIs a tight set of uncertain parameters, representing a bound for uncertainty. M (q, sigma, t) is the inertia matrix of the robot system,being the coriolis force/centrifugal force term of the system, is a diagonally symmetric matrix, G (q, σ, t) is the gravity term of the system,and tau (t) is the input torque of the system. M (-), C (-), G (-), and F (-), are continuous or measurable with respect to time t Leeberg.
For the design of the subsequent controller, M (-), C (-), G (-), and F (-) in equation (3.3) are decomposed into:
wherein,andnominal term, referred to as Delta parallel robot System,. DELTA.M (q, σ, t),Δ G (q, σ, t) andreferred to as the uncertainty term of the Delta parallel robotic system.
When the Delta parallel robot has no uncertain factors in the working process,
to simplify the derivation process, the arguments in the partial formula below are omitted in the case where no ambiguity arises.
Wherein the inertia matrix satisfies:
assume that 1:
the inertial matrix M (q, σ, t) is a positive definite matrix, i.e., for any q ∈ R3Existence of a constantσ>0, such that:
M(q,σ,t)>σI (6)
assume 2:
for arbitrary q ∈ R3Always present constant γjJ is 0,1,2, and γ0>0，γ1,20 or more, such that:
‖M(q,σ,t)‖<γ0+γ1‖q‖+γ2‖q‖2(7)
for a serial-parallel robot connected by a revolute pair and a sliding pair, the inertia matrix M (q, sigma, t) is only related to the mass inertia parameters, and the positions of the sliding joint and the revolute joint. Thus, there is always a set of constants γjAnd enabling the Euclidean norm of the mass inertia matrix of the serial-parallel robot to satisfy the formula (7).
Step 2:
setting the expected track of the Delta parallel robot with three degrees of freedom as qd、Andwherein q isd:[t0,∞)→R3Represents a desired position, and qdIs C2The process is carried out continuously,in order to be able to take the desired speed,is the desired acceleration.
The trajectory tracking error of the system is defined as:
e：＝q-qd(8)
thus, the velocity tracking error and acceleration tracking error of the system can be expressed as:
then:
and step 3:
wherein, positive definite diagonal matrix Kp＝diag[kpi]3×3And k ispi>0，Kv＝diag[kvi]3×3And k isvi>0，i＝1,2,3。
And 4, step 4: (to construct a function satisfying assumption 3)
Assume that 3:
(1) there is a known positive definite function Γ (·) (0, ∞)k×R3×R3×R→R+And an unknown vector α ∈ (0, ∞)kSo that:
wherein,
in formula (14), the positive definite matrix S ═ diag [ S ]i]3×3，si>0，ks＝λmin(S)，i＝1,2,3。
(2) For allFunction(s)Satisfies the following conditions: (i) c1(ii) concave function with respect to α, i.e. for any of α1，α2：
(3) Function(s)Is a no drop function with respect to α.
And 5:
the self-adaptation law with dead zones is designed as follows:
equation (16) is an adaptation rate with dead band design and leakage terms,in order to adapt the parameters to the application, is composed ofThe ith element of the vector, i ═ 1,2, …, k1，k2，k3∈Rk×kAnd k is1，k2，k3>0，κ∈R,κ>0,∈∈R,∈>0。
When in useNot into the range of size e,being non-negative, leakyDesigned in an exponential form such thatExponentially decays towards a value of 0, ifConstant establishment of t>t0，i＝1,2,…,k。
Dead zone portion (Into a range of size e) may simplify the control algorithm.
Step 6:
in formula (17):
wherein, positive definite diagonal matrix Kp＝diag[kpi]3×3And k ispi>0，Kv＝diag[kvi]3×3And k isvi>0，i＝1,2,3，kp＝λmin(Kp),kp＝λmin(Kv)，ksp＝kskp,ε>0，ξ>0。
And 7:
considering a tracking error vector ofAn adaptive robust trajectory tracking controller for a three-degree-of-freedom Delta parallel robot is provided:
in the equation (20), the controller is divided into three parts, and if there is an initial position error or uncertainty in the robot system, let τ be P1+P2+P3The tracking error vector can be adjusted to t → ∞ timeSatisfying consistent bounding and consistent final bounding.
When only initial unknown errors exist in the system, Δ M ≡ 0, Δ C ≡ 0, Δ G ≡ 0 and Δ F ≡ 0 functions may be chosen thatSo that P is30, where τ is P1+P2When t → ∞,e→0。
if no initial position error and uncertainty exists in the system,let τ be P1When t is>t0When the temperature of the water is higher than the set temperature,this is always true.
First, stability demonstration
1. Stability proof conclusions are given first:
if the three-degree-of-freedom Delta parallel robot dynamics model (1) meets the assumptions 1-3, the controller design (20) can enable the trajectory tracking error vectorSatisfies the following conditions:
(1) consistent and bounded: for any given r>0, and | purplee(t0)<r when t>t0When there is a positive real number d (r) 0<d(r)<Infinity, making | purplee(t)||<d (r) holds.
(2) Consistency ends up bounded: for any given r>0，And (| hollow)e(t0)||<r is whenWhen the temperature of the water is higher than the set temperature,is formed in which
2. The demonstration process is as follows:
the Lyapunov function was constructed as:
the derivative of the lyapunov function V is:
the first term in analytical formula (22):
according to formula (11):
in formula (23):
bringing formula (19) into formula (23):
according to assumption 3, there are:
bringing formulae (23) to (27) into formula (22) includes:
the adaptive rate (16) is introduced into the channel (28) by:
(1) when in useWhen the temperature of the water is higher than the set temperature,
if the number of the first and second antennas is greater than the predetermined number,comprises the following steps:
the third term in the pair of equations (29) is:
the fourth term in the pair of equations (29) is:
due to the fact thatAccording to the formulae (30) to (31), there are:
wherein,
if the number of the first and second antennas is greater than the predetermined number,comprises the following steps:
wherein,
(2) when in useIn time, there are:
if the number of the first and second antennas is greater than the predetermined number,comprises the following steps:
because of the fact thatTherefore, it is not only easy to useThen:
wherein ψ ∈.
If the number of the first and second antennas is greater than the predetermined number,comprises the following steps:
the derivative of the Lyapunov function according to equations (32), (33), (36) and (37)Comprises the following steps:
where ρ is1＝ρ，ρ2Psi or rho2＝0，ρ3θ. For equation (38), when | satisfies:
negative values, i.e.:
according to the literature (Chen Y., Zhang X., Adaptive Robust Adaptive conductivity Control for Mechanical Systems [ J]Journal of the Franklin institute,2010, 347 (1): 69-86) when the derivative of the Lyapunov function is presentWhen the formula (40) is satisfied, the tracking error vectorAnd adaptive parametersWith consistent boundingSex;
wherein:
γ1＝
min{λmin(M),λmin((κk1)-1)},γ2＝min{λmax(M),λmax((κk1)-1)}。
at the same time, the trajectory tracking error vectorAnd adaptive parametersConsistent final bounding is also satisfied;
second, dynamic model simulation
In MATLAB software, a dynamic model of the three-degree-of-freedom Delta parallel robot and a designed controller are simulated by using an ode15i function.
The uncertain factors suffered by the parallel robot are assumed as the quality parameters of the moving platform To an external loadWherein,andis a nominal term,. DELTA.mo＇、ΔF1、ΔF2And Δ F3As an uncertainty term over time.
The uncertain parameter vector is defined as: σ ═ Δ mO＇,ΔF1,ΔF2,ΔF3]T. Setting a target track needing to be tracked by a Delta parallel robot working platform as follows:
according to hypothesis 3, functionIs selected and a functionRelated, selection functionComprises the following steps:
wherein, α ═ max { α1,α2,α3}。
The three-degree-of-freedom Delta parallel robot has the following structural parameters:
length l of the active armaThe radius R of the circumscribed circle of the static platform is 180mm, and the radius R of the circumscribed circle of the movable platform is 100 mm;
the quality parameters of the robot are as follows:
mass m of active arma1.193kg, driven arm mass mb1.178kg, moving platform mass mO′＝4.3225kg。
The control parameters of the controller are selected as follows:
Kv＝diag[1,1,1]，Kp＝diag[1,1,1]，S＝diag[8,8,8],ε＝0.1，κ＝0.05，k1＝10,k2＝0.3,k3＝0.5,ξ＝0.001。
the nominal parameters were chosen as follows:
choosing uncertain parameters as follows:
Δm＝0.7，Δf＝0.6。
setting the initial value positions of simulation as follows: q. q.s0＝[0.5434 0.5434 0.9639]T, The simulation results are shown in fig. 3-10.
Fig. 3 and 4 are simulation results of the angular displacement and the angular velocity of the active joint of the three-degree-of-freedom Delta parallel robot. FIG. 5 is a simulation of input moments at three active joint angles. FIG. 6 is a diagram of adaptive parametersAnd (5) simulation results. Fig. 7 reflects the influence of the value of the uncertain parameter on the maximum value of the adaptive parameter estimation.
When the three-degree-of-freedom Delta parallel robot system is influenced by initial position error and uncertainty, respectively setting tau as P1、τ＝P1+P2、τ＝P1+P2+P3The simulation results are shown in fig. 8-9 for control input versus control effect.
FIG. 8 shows the simulation result of the system tracking error e under three control inputs, when τ is equal to P1For the control input, the tracking error is diffused from 0.8s around 0.01m to 0.2m, and is not converged in the simulation time. When τ is equal to P1+P2To control the input, the tracking error oscillates around 0.1 m. When τ is equal to P1+P2+P3In order to control the input, the system enters and remains within the range around 0m after 0.2s from around 0.1 m.
FIG. 9 illustrates system trajectory tracking error under three control inputsWhen τ is equal to P, the simulation result of (1)1Error in tracking of track for control inputDiverges from about 0.31m/s to 1m/s over 0.8s, increases after 0.8s, andand fail to converge within a limited time.When τ is equal to P1+P2Error in tracking of track for control inputAlways oscillating around 0.1m/s, and with the increase of simulation time, the errorThere is an increasing trend. When τ is equal to P1+P2+P3Error in tracking of track for control inputAfter 0.5s from 0.31m/s, the flow rate decreases to a value close to 0 m/s.
In fig. 10, when τ is P1+P2+P3To control the input, the tracking target track X of the end effector track with high quality can be controlleddWhen τ is equal to P1And τ ═ P1+P2In order to control input, the track tracking purpose cannot be achieved.
Simulation results show that: the self-adaptive robust controller can effectively resist the influence caused by initial position error and uncertainty and control the track of the end effector to track the target track with high quality. Meanwhile, under the condition that uncertainty upper bound information is unknown, the adaptive robust controller can estimate the system uncertainty upper bound information on line, the conservatism of the robust control method is improved, and the track tracking error meets the requirement of consistent and bounded performance and the requirement of consistent and bounded performance.

Claims (1)

1. A self-adaptive robust control method of a three-degree-of-freedom Delta parallel robot is characterized by comprising the following steps of:
step 1, separating out uncertain items in a three-degree-of-freedom Delta parallel robot dynamic model to respectively obtain a nominal item and an uncertain item of a parallel robot system;
step 2, establishing a nominal compensation link in the controller according to a nominal item in the three-degree-of-freedom Delta parallel robot dynamics model, and compensating the nominal robot system;
step 3, selecting a positive definite diagonal matrix, and designing a P.D. control link in the controller for compensating the initial position error;
step 4, constructing a function representing the upper bound information of the uncertainty item of the system according to the uncertainty-related item in the three-degree-of-freedom Delta parallel robot dynamics model, and verifying the hypothesis;
step 5, selecting parameters, and establishing a self-adaptive law with dead zones and leakage items for online estimation of upper bound information of uncertainty;
step 6, constructing an uncertainty compensation link according to the function and the self-adaptive law, and compensating the uncertainty in the system;
and 7, finally, providing the adaptive robust controller.
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