CN108693776B - Robust control method of three-degree-of-freedom Delta parallel robot - Google Patents

Robust control method of three-degree-of-freedom Delta parallel robot Download PDF

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CN108693776B
CN108693776B CN201810824730.1A CN201810824730A CN108693776B CN 108693776 B CN108693776 B CN 108693776B CN 201810824730 A CN201810824730 A CN 201810824730A CN 108693776 B CN108693776 B CN 108693776B
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parallel robot
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robot
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CN108693776A (en
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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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    • 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 robust control method of a three-degree-of-freedom Delta parallel robot, which separates out items containing uncertainty 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; 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; selecting a positive definite diagonal matrix, and designing a P.D. control link in a controller for compensating an initial position error; solving a function representing the upper bound information of an uncertain item of the system according to an item construction function related to uncertainty in a three-degree-of-freedom Delta parallel robot dynamics model, and constructing an uncertainty compensation link by using the function so as to compensate the uncertainty existing in the system; finally, a robust controller is given. The technical problem that the traditional control method is often based on an accurate dynamic model and is difficult to achieve the actual control purpose is solved.

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 error
Figure GDA0002633159860000031
A 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 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, 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 system
Figure GDA0002633159860000051
And
Figure GDA0002633159860000052
and 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 Kp＝diag[kpi]3×3，Kv＝diag[kvi]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:
Figure GDA0002633159860000053
wherein q ∈ R3In order to be the active joint angle vector,
Figure GDA0002633159860000054
in order to be the active joint angular velocity vector,
Figure GDA0002633159860000055
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,
Figure GDA0002633159860000061
being the coriolis force/centrifugal force term of the system,
Figure GDA0002633159860000062
is a diagonally symmetric matrix, G (q, σ, t) is the gravity term of the system,
Figure GDA0002633159860000063
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:
Figure GDA0002633159860000064
Figure GDA0002633159860000065
Figure GDA0002633159860000066
Figure GDA0002633159860000067
wherein the content of the first and second substances,
Figure GDA0002633159860000068
and
Figure GDA0002633159860000069
nominal term, referred to as Delta parallel robot System,. DELTA.M (q, σ, t),
Figure GDA00026331598600000610
Δ G (q, σ, t) and
Figure GDA00026331598600000611
referred to as the uncertainty term of the Delta parallel robotic system.
When Delta parallel robot is in working processWhen there is no uncertain factor in the data,
Figure GDA00026331598600000612
Figure GDA00026331598600000613
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 is such that:
M(q,σ,t)>σI (6)
assume 2:
for arbitrary q ∈ R3Always present constant γjJ is 0,1,2, and γ0>0，γ 1,20 or more, such that: II M (q, sigma, 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、
Figure GDA0002633159860000071
And
Figure GDA0002633159860000072
wherein q isd:[t0,∞)→R3Represents a desired position, and qdIs C2The process is carried out continuously,
Figure GDA0002633159860000073
period of time ofWhen the speed is to be observed,
Figure GDA0002633159860000074
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:
Figure GDA0002633159860000075
Figure GDA0002633159860000076
then:
Figure GDA0002633159860000077
and step 3:
Figure GDA0002633159860000081
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:
assume that 3:
there is a positive definite function p R3×R3×R→R+Positive definite matrix S ═ diag [ S ]i]3×3，si>0，ks＝λmin(S), i ═ 1,2,3, such that:
Figure GDA0002633159860000082
wherein:
Figure GDA0002633159860000083
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.
Figure GDA0002633159860000084
In formula (15):
Figure GDA0002633159860000085
Figure GDA0002633159860000086
in formula (17) > 0.
And 5:
considering a tracking error vector of
Figure GDA0002633159860000091
A robust trajectory tracking controller for a three-degree-of-freedom Delta parallel robot is provided:
Figure GDA0002633159860000092
wherein:
Figure GDA0002633159860000093
Figure GDA0002633159860000094
Figure GDA0002633159860000095
in formula (21):
Figure GDA0002633159860000096
Figure GDA0002633159860000097
in the formula (22), the reaction mixture is,>0, positive definite diagonal matrix Kp＝diag[kpi]3×3And k ispi>0，Kv＝diag[kvi]3×3And k isvi>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 that
Figure GDA0002633159860000101
Where τ is equal to P1+P2When t → ∞,e→0。
if the robot system has both initial position error and uncertainty, let τ be P1+P2+P3When t → ∞,eand 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 vector
Figure GDA0002633159860000102
Satisfies 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，
Figure GDA0002633159860000103
And (| hollow)e(t0)||<r is when
Figure GDA0002633159860000104
When the temperature of the water is higher than the set temperature,
Figure GDA0002633159860000105
is formed in which
Figure GDA0002633159860000106
The following is given as the demonstration process:
the Lyapunov function was constructed as:
Figure GDA0002633159860000107
according to assumption 1, there are:
Figure GDA0002633159860000108
wherein:
Figure GDA0002633159860000111
in the formula (26), the parameter kv、si、kpAndσall real numbers are greater than zero, and all sequential major-minor types are greater than zero, then psiiIs a positive definite matrix. Order to
Figure GDA0002633159860000112
(i.e. theS>0)：
Figure GDA0002633159860000113
From the equations (25) and (27), V is a positive definite matrix.
From assumption 2:
Figure GDA0002633159860000114
for the first term on the right of the inequality, q: e + qdTape-in (28):
‖q‖＝||e+qd||≤‖e‖+maxt||qd|| (29)
‖q‖2≤‖e‖2+2‖e‖maxt||qd||+(maxt||qd||)2 (30)
simultaneously:
Figure GDA0002633159860000115
wherein the content of the first and second substances,
Figure GDA0002633159860000116
for the second term on the right of the inequality:
Figure GDA0002633159860000121
thus, there is a constant
Figure GDA0002633159860000122
Such that:
Figure GDA0002633159860000123
wherein:
Figure GDA0002633159860000124
Figure GDA0002633159860000125
Figure GDA0002633159860000126
in formula (33)
Figure GDA0002633159860000127
For strictly positive real numbers, alleThe 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:
Figure GDA0002633159860000128
formula (32) is substituted into formula (37):
Figure GDA0002633159860000129
Figure GDA0002633159860000131
bringing formula (22) and formula (23) into formula (38), i.e. when | >:
Figure GDA0002633159860000132
when | ≦ the,
Figure GDA0002633159860000133
therefore, assuming 3, equations (39) and (40) show that:
Figure GDA0002633159860000134
bringing formula (41) into formula (38):
Figure GDA0002633159860000135
Figure GDA0002633159860000141
wherein the content of the first and second substances,λ＝min{Kv,SKp}. With respect to the formula (38),λand are all bounded normal numbers, so when | luminancee||2When large enough, the derivative of the lyapunov function is negative, i.e.:
Figure GDA0002633159860000142
according to The literature (Chen Y.. On The diagnostic Performance of Uncertian dynamic Systems [ J.)]International Journal of Control, 1986, 43 (5): 1557-1579) when the derivative of the Lyapunov function satisfies equation (43), the controller's equation (18) enables the tracking error vectoreSatisfying consistent bounding, i.e. for any given r>0, and | purplee(t0)||<r when t>t0When there is a positive real number d (r) as shown in formula (44), so that | Ye(t0)||<d (r) holds.
Figure GDA0002633159860000143
Wherein R [/4 [ ]λ]1/2. At the same time, tracking error vectoreConsistent final bounding is also satisfied. I.e. for any given r>0，
Figure GDA0002633159860000144
And is
Figure GDA0002633159860000145
When in use
Figure GDA0002633159860000146
When the temperature of the water is higher than the set temperature,
Figure GDA0002633159860000147
this is true.
Wherein:
Figure GDA0002633159860000148
Figure GDA0002633159860000151
Figure GDA0002633159860000152
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
Figure GDA0002633159860000153
To an external load
Figure GDA0002633159860000154
Wherein the content of the first and second substances,
Figure GDA0002633159860000155
Figure GDA0002633159860000156
and
Figure GDA0002633159860000157
is 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。
According to assumption 1, the uncertainty in M (-), C (-), G (-), F (-), is separated into a constructor
Figure GDA0002633159860000158
Figure GDA0002633159860000159
From equation (48), the function positive definite function ρ is solved:
Figure GDA00026331598600001510
wherein, sigmamSum-sigmaFRepresenting 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:
Figure GDA0002633159860000161
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, 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 arma＝1.193kg, mass of follower arm 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],＝0.1。
the nominal parameters were chosen as follows:
Figure GDA0002633159860000162
choosing uncertain parameters as follows:
Figure GDA0002633159860000163
Figure GDA0002633159860000164
Δm＝Δf＝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,
Figure GDA0002633159860000165
Figure GDA0002633159860000166
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 controlling the input, the tracking error is diverged from about 0.01 to 0.2m through 5.5s, and after 5s, the error is continuously increased without convergence in 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 inputs
Figure GDA0002633159860000171
When τ is equal to P, the simulation result of (1)1Error in tracking of track for control input
Figure GDA0002633159860000172
Diverges from about 0.3m/s to 0.4m/s over 5.5s, and
Figure GDA0002633159860000173
and fail to converge within a limited time. When τ is equal to P1+P2Error in tracking of track for control input
Figure GDA0002633159860000174
Decreasing from 0.3m/s to 0.1m/s over 1.5s, with increasing simulation time, error
Figure GDA0002633159860000175
The curve has a tendency to rise. When τ is equal to P1+P2+P3Error in tracking of track for control input
Figure GDA0002633159860000176
After 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 XdWhen τ is equal to P1+P2+P3To control the input, the end effector mayAnd rapidly moving from the initial deviation position to the vicinity of the target track and basically moving according to the target track.
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;
the three-degree-of-freedom Delta parallel robot dynamic model is expressed by the formula (1);
Figure FDA0002633159850000011
wherein q ∈ R3In order to be the active joint angle vector,
Figure FDA0002633159850000012
in order to be the active joint angular velocity vector,
Figure FDA0002633159850000013
for active joint angular acceleration vector, sigma belongs to sigma e RpFor uncertain parameter vectors existing in the robot system, sigma belongs to RpIs a compact set of uncertain parameters, representing the uncertainty bound, M (q, σ, t) is the robot system inertia matrix,
Figure FDA0002633159850000014
the Coriolis/centrifugal force term of the system, G (q, σ, t) the gravitational force term of the system,
Figure FDA0002633159850000015
tau (t) is the system input torque, M (-), C (-), G (-), and F (-), which are continuous or measurable with respect to time t Leeberg, for external disturbances to the system;
for the design of the subsequent controller, the M (-), C (-), G (-), and F (-), are decomposed into:
Figure FDA0002633159850000016
Figure FDA0002633159850000017
Figure FDA0002633159850000018
Figure FDA0002633159850000019
wherein the content of the first and second substances,
Figure FDA00026331598500000110
and
Figure FDA00026331598500000111
nominal term, referred to as Delta parallel robot System,. DELTA.M (q, σ, t),
Figure FDA00026331598500000112
Δ G (q, σ, t) and
Figure FDA00026331598500000113
an uncertainty term referred to as the Delta parallel robot system;
when the Delta parallel robot has no uncertain factors in the working process,
Figure FDA0002633159850000021
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;
the expected track of the three-degree-of-freedom Delta parallel robot is qd、
Figure FDA0002633159850000022
And
Figure FDA0002633159850000023
wherein q isd：[t0，∞)→R3Represents a desired position, and qdIs C2The process is carried out continuously,
Figure FDA0002633159850000024
in order to be able to take the desired speed,
Figure FDA0002633159850000025
is a desired acceleration;
the trajectory tracking error of the system is defined as equation (8):
e：＝q-qd (8)
therefore, the velocity tracking error and the acceleration tracking error of the system can be expressed as equation (9) and equation (10):
Figure FDA0002633159850000026
Figure FDA0002633159850000027
then:
Figure FDA0002633159850000028
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;
there is a positive definite function ρ: r3×R3×R→R+Positive definite matrix S ═ diag [ S ]i]3×3，si＞0，ks＝λmin(S), i ═ 1,2,3, such that:
Figure FDA0002633159850000031
wherein:
Figure FDA0002633159850000032
phi represents all items related to uncertainty in the three-degree-of-freedom Delta parallel robot dynamic model, the upper bound information of the items can be represented by a function rho, and if the system has no uncertain factors, phi is equal to 0;
step 5, finally, the robust controller is given as a formula (18);
Figure FDA0002633159850000033
wherein:
Figure FDA0002633159850000034
Figure FDA0002633159850000035
Figure FDA0002633159850000036
in formula (21):
Figure FDA0002633159850000037
Figure FDA0002633159850000038
in equation (22), >0, positive definite diagonal matrix Kp＝diag[kpi]3×3And k ispi＞0，Kv＝diag[kvi]3×3And k isvi＞0，i＝1，2，3；
In equation (18), 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 is selected to be 0, so that
Figure FDA0002633159850000039
Where τ is equal to P1+P2T → ∞ time, e → 0;
if the robot system has both initial position error and uncertainty, let τ be P1+P2+P3Let t → ∞ then e satisfy the consistently bounded and consistently ultimately bounded performance indicators.
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