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

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

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 robust control method of a three-degree-of-freedom Delta parallel robot.

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.

At present, the dynamic control method for the Delta parallel robot with uncertainty mainly comprises a linear control method and a nonlinear control method. The linear control method, such as PID control, calculation torque control, etc., realizes the control of the robot after the nonlinear system model is linearized. The methods depend on an accurate dynamic model of the system and a determined working condition, the uncertainty of the system is ignored, the linear control methods have the disadvantages along with the gradual complexity of the control process, the simple linear control method is often difficult to achieve the control requirement, and the robustness is poor. Therefore, in recent years, nonlinear control methods have become important in the field. The nonlinear control method for the Delta parallel robot system comprises a variable structure control method, a linear feedback control method, an adaptive control method and the like, wherein the control methods have some problems, such as discontinuous switching characteristics exist in the process of structure switching in a sliding mode variable structure control method, and the unavoidable buffeting phenomenon is easily caused, and the linear feedback control method is not complete in compensation of the linear system due to incomplete knowledge of a dynamic model of the Delta parallel robot. While the intelligent control methods such as neural network and fuzzy control are still in the preliminary stage in the field of Delta parallel robot motion control at present, although the intelligent control theory makes remarkable progress in Delta parallel robot engineering application, problems still exist, such as selection of the number of hidden layers and the number of neurons of the neural network, high-frequency vibration phenomenon existing in the fuzzy control, difficulty in establishing a complete fuzzy control rule by single fuzzy control and the like, and further exploration needs to be carried out on the problems.

The American scholars Leitmann puts forward concepts of consistent and bounded property and finally consistent and bounded property on the basis of theories such as system optimization, game theory and the like, and puts forward a Min-Max Lyapunov control method, namely a Leitmann method, aiming at the problems of nonlinearity and uncertainty of a system. Leitmann discusses the robustness of a non-deterministic system with a state variable delay phenomenon and the stability of a linear system with the state delay phenomenon under the condition of no structure matching respectively, and combines a dispersion control theory to analyze the control of the non-linear coupling system with uncertainty, thereby providing a new idea for the research of a Delta parallel robot dynamic control method with strong coupling and nonlinearity. Therefore, the research on the three-degree-of-freedom Delta parallel robot dynamic control strategy with uncertainty has been a focus of attention of those skilled in the art.

Disclosure of Invention

Aiming at the defects or shortcomings of the prior art, the invention aims to provide a robust control method of a three-degree-of-freedom Delta parallel robot based on a Leitmann method so as to solve the problem that the traditional control method is often based on an accurate dynamic model and is difficult to achieve the actual control purpose.

In order to realize the task, the invention adopts the following technical scheme to realize the following steps:

a 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, solving a function representing the upper bound information of the uncertain items of the system according to the item construction function related to the uncertainty in the three-degree-of-freedom Delta parallel robot dynamic model, and constructing an uncertainty compensation link by using the function for compensating the uncertainty in the system;

and 5, finally, providing the robust controller.

The robust control method of the three-degree-of-freedom Delta parallel robot has the beneficial effects that in the designed 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 both initial position error and uncertainty, the uncertainty in the system can be compensated by adding an uncertainty compensation link in the controller, so that the system meets the consistent bounded performance index and the consistent final bounded performance index.

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 a simulation result of a Delta parallel robot trajectory tracking error e;

FIG. 7 shows the Delta parallel robot trajectory tracking errorA simulation result graph;

FIG. 8 is a diagram showing simulation results of Delta parallel robot operation trajectories;

the technical solution of the present invention will be 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 arm_iB_iIs 1_aLength B of the follower arm_iC_iIs 1_bThe 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, the robust control method for a three-degree-of-freedom Delta parallel robot provided in this embodiment includes the steps of:

step 1, separating out uncertain items in three-degree-of-freedom Delta parallel robot dynamic model to respectively obtain nominal items of 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 model₁。

Step 3, selecting a positive definite diagonal matrix K_p＝diag[k_pi]_3×3，K_v＝diag[k_vi]_3×3Designing P.D. control link P in controller₂。

Step 4, solving a function rho representing the uncertainty item upper bound information of the system according to an uncertainty-related item construction function phi in a three-degree-of-freedom Delta parallel robot dynamics model, and constructing an uncertainty compensation link P by using the function rho₃。

And 5, finally, giving a 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:

(1)

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 R^pFor uncertain parameter vectors existing in the robot system, sigma belongs to R^pIs 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 constantSuch that:

assume 2:

for arbitrary q ∈ R³Always present constant γ_jJ is 0,1,2, and γ₀>0，γ_1,20 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 γ_jAnd 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 q^d、AndwhereinRepresents a desired position, and q^dIs 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-q^d(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 K_p＝diag[k_pi]_3×3And k is_pi>0，K_v＝diag[k_vi]_3×3And k is_vi>0，i＝1,2,3。

And 4, step 4:

assume that 3:

there is a positive definite function p R³×R³×R→R₊Positive definite matrix S ═ diag [ S ]_i]_3×3，s_i>0，k_s＝λ_min(S), i ═ 1,2,3, such that:

wherein:

in assumption 3, the function φ represents all uncertainty-related terms in the three-degree-of-freedom Delta parallel robot dynamics model, whose upper bound information can be represented by the function ρ. If the system does not have any uncertainty, φ ≡ 0.

In formula (15):

in formula (17), ε > 0.

And 5:

considering a tracking error vector ofA robust trajectory tracking controller for a three-degree-of-freedom Delta parallel robot is provided:

wherein:

in formula (21):

in formula (22), ε>0, positive definite diagonal matrix K_p＝diag[k_pi]_3×3And k is_pi>0，K_v＝diag[k_vi]_3×3And k is_vi>0，i＝1,2,3。

In equation (18) of the designed robust trajectory tracking controller, if there is no initial position error or uncertainty in the robot system, let τ be P₁The track tracking error of the parallel robot can reach the performance of consistent asymptotic stability.

If only the initial position error exists in the robot system, Δ M, Δ C, Δ G, and Δ F are all zero, and the function ρ may be selected to be 0, so thatWhere τ is equal to P₁+P₂When t → ∞,

if the robot system has both initial position error and uncertainty, let τ be P₁+P₂+P₃When t → ∞,and satisfying the consistent bounded performance index and the consistent final bounded performance index.

First, stability demonstration

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 (18) can enable the trajectory tracking error vectorSatisfies the following conditions:

(1) consistent and bounded: for any given r>0, andwhen t is>t₀When there is a positive real number d (r):0<d(r)<Infinity, such thatThis is true.

(2) Consistency ends up bounded: for any givenAnd isWhen in useWhen the temperature of the water is higher than the set temperature,is formed in which

The following is given as the demonstration process:

the Lyapunov function was constructed as:

according to assumption 1, there are:

wherein:

in the formula (26), the parameter k_v、s_i、k_pAndall real numbers are greater than zero, and all sequential major-minor types are greater than zero, then psi_iIs a positive definite matrix. Order to(i.e. the)：

From the equations (25) and (27), V is a positive definite matrix.

From assumption 2:

for the first term on the right of the inequality, q: e + q^dTape-in (28):

‖q‖＝||e+q^d||≤‖e‖+max_t||q^d|| (29)

‖q‖²≤‖e‖²+2‖e‖max_t||q^d||+(max_t||q^d||)²(30)

simultaneously:

wherein,

for the second term on the right of the inequality:

thus, there is a constantSuch that:

wherein:

in formula (33)For strictly positive real numbers, allThe chosen Lyapunov function is a monotonically decreasing function. Therefore, according to equations (27) and (33), V is constructed as an effective lyapunov function.

The derivative of the lyapunov function V is:

formula (32) is substituted into formula (37):

bringing (38) into engagement formula (22) with (23) when | > epsilon:

when | ≦ epsilon,

therefore, assuming 3, equations (39) and (40) show that:

bringing formula (41) into formula (38):

wherein,with respect to the formula (38),is a bounded normal number with respect to ε, and thereforeWhen large enough, the derivative of the lyapunov function is negative, i.e.:

according to The literature (Chen Y.. On The diagnostic Performance of The Uncertaindynamic Systems [ J.)]International Journal of Control, 1986, 43 (5): 1557-1579) when the derivative of the Lyapunov function satisfies equation (43), equation (18) of the controller enables the trajectory tracking error vectorSatisfying consistent bounding, i.e. for any given r>0, andwhen t is>t₀When there isThe positive real number d (r) is as shown in equation (44) such thatThis is true.

Wherein,at the same time, tracking error vectorConsistent final bounding is also satisfied. I.e. for any given r>0，And isWhen in useWhen the temperature of the water is higher than the set temperature,this is true.

Wherein:

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.m_O′、ΔF₁、ΔF₂And Δ F₃As an uncertainty term over time. The uncertain parameter vector is defined as: σ ═ Δ m_O＇,ΔF₁,ΔF₂,ΔF₃]^T。

According to assumption 1, the uncertainty in M (-), C (-), G (-), F (-), is separated into a constructor

From equation (48), the function positive definite function ρ is solved:

wherein, sigma_mSum-sigma_FRepresenting the maximum degree of influence of the uncertain parameter on the robot system.

Setting a target track needing to be tracked by a Delta parallel robot working platform as follows:

the three-degree-of-freedom Delta parallel robot has the following structural parameters:

length l of the active arm_aThe radius R of the circumscribed circle of the static platform is 180mm, the radius R of the circumscribed circle of the movable platform is 100mm, and the quality parameters of the robot are as follows: mass m of active arm_a1.193kg, driven arm mass m_b1.178kg, moving platform mass m_O，＝4.3225kg。

The control parameters of the controller are selected as follows:

K_v＝diag[1,1,1]，K_p＝diag[1,1,1]，S＝diag[8,8,8],ε＝0.1。

the nominal parameters were chosen as follows:

choosing uncertain parameters as follows: Δ_m＝Δ_f＝0.5。∑_m＝0.5，∑_F0.5. Setting the initial values of simulation as follows: q. q.s₀＝[0.5434 0.5434 0.9639]^T, The simulation results are shown in fig. 3-8.

Fig. 3 and 4 show simulation results of the angular displacement and the angular velocity of the active joint of the Delta parallel robot, and fig. 5 shows simulation results of input moments at three active joint angles.

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. 6-8 for control input versus control effect.

FIG. 6 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 diverges from around 0.01 to 0.2m over 5.5s, and after 5s, the error continues to increase without convergence within the simulation time. When τ is equal to P₁+P₂In order to control the input, the tracking error decreases from about 0.01m to about 0.05m after 3 seconds, and as time increases, the error remains about 0.05m and fails to converge to 0 m. When τ is equal to P₁+P₂+P₃In order to control the input, the system enters and is maintained in the range around 0m after 0.4s from around 0.01 m.

FIG. 7 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 0.3m/s to 0.4m/s over 5.5s, andand fail to converge within a limited time. When τ is equal to P₁+P₂Error in tracking of track for control inputDecreasing from 0.3m/s to 0.1m/s over 1.5s, with increasing simulation time, errorThe curve has a tendency to rise. When τ is equal to P₁+P₂+P₃Error in tracking of track for control inputAfter 0.4s, the flow rate decreases from 0.3m/s to a value close to 0 m/s.

FIG. 8 shows the simulation result of the trajectory of the end effector of the Delta parallel robot under three control inputs, when τ is equal to P₁And τ ═ P₁+P₂In order to control input, the trajectory of the end effector cannot be controlled to track the target trajectory X^dWhen τ is equal to P₁+P₂+P₃To control the input, the end effector may be moved rapidly from the initial offset position to near the target trajectory and substantially follow the target trajectory.

Simulation results show that: in the case of an initial position error and uncertainty of the system, the proposed trajectory tracking controller can control the trajectory of the end effector to track the target trajectory while making the trajectory tracking error meet the consistent and eventually bounded.

Claims (1)

1. A 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, solving a function representing the upper bound information of the uncertain items of the system according to the item construction function related to the uncertainty in the three-degree-of-freedom Delta parallel robot dynamic model, and constructing an uncertainty compensation link by using the function for compensating the uncertainty in the system;

and 5, finally, providing the robust controller.