CN109062039B - Adaptive robust control method of three-degree-of-freedom Delta parallel robot - Google Patents

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

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CN109062039B
CN109062039B CN201810824709.1A CN201810824709A CN109062039B CN 109062039 B CN109062039 B CN 109062039B CN 201810824709 A CN201810824709 A CN 201810824709A CN 109062039 B CN109062039 B CN 109062039B
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uncertainty
parallel robot
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nominal
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CN109062039A (en
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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 self-adaptive robust control method of a three-degree-of-freedom Delta parallel robot, which separates out an item containing uncertainty in a three-degree-of-freedom Delta parallel robot dynamic model to respectively obtain a nominal item and an uncertainty 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; designing a P.D. control link in a controller by selecting a positive definite diagonal matrix, wherein the P.D. control link is used for compensating the initial position error; then, according to the items related to uncertainty in the three-degree-of-freedom Delta parallel robot dynamics model, a function representing the upper bound information of the uncertainty items of the system is constructed, and the hypothesis is verified; selecting parameters, establishing a self-adaptive law with dead zones and leakage items, and using the self-adaptive law for estimating upper bound information of uncertainty on line; according to the function and the self-adaptive law, the uncertainty in the system is compensated; finally, an adaptive robust controller is given.

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 parameters
Figure BDA0001742191020000031
A 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 error
Figure BDA0001742191020000032
A 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 moving platform system, O, O ' are respectively positioned at a static position,The geometric center of the movable platform is set as the positive direction in the axial direction of Z, z'. 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, separating out uncertain items in three-degree-of-freedom Delta parallel robot dynamic model to respectively obtain nominal items of parallel robot system
Figure BDA0001742191020000041
And
Figure BDA0001742191020000042
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 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 an uncertainty representing system according to a term construction function phi related to uncertainty in a three-degree-of-freedom Delta parallel robot dynamics modelFunction of item upper bound information
Figure BDA0001742191020000051
And verifies hypothesis 3.
Step 5, selecting parameters kappa and k1、k2、k3And e and xi, establishing an adaptive law with dead zones and leakage items, and using the adaptive law for estimating the upper bound information of uncertainty on line.
Step 6, according to the function
Figure BDA0001742191020000052
And 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:
Figure BDA0001742191020000053
wherein q ∈ R3In order to be the active joint angle vector,
Figure BDA0001742191020000054
in order to be the active joint angular velocity vector,
Figure BDA0001742191020000055
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 BDA0001742191020000056
being the coriolis force/centrifugal force term of the system,
Figure BDA0001742191020000057
Figure BDA0001742191020000061
is a diagonally symmetric matrix, G (q, σ, t) is the gravity term of the system,
Figure BDA0001742191020000062
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 BDA0001742191020000063
Figure BDA0001742191020000064
Figure BDA0001742191020000065
Figure BDA0001742191020000066
wherein the content of the first and second substances,
Figure BDA0001742191020000067
and
Figure BDA0001742191020000068
nominal term, referred to as Delta parallel robot System,. DELTA.M (q, σ, t),
Figure BDA0001742191020000069
Δ G (q, σ, t) and
Figure BDA00017421910200000610
referred to as the uncertainty term of the Delta parallel robotic system.
When the Delta parallel robot has no uncertain factors in the working process,
Figure BDA00017421910200000611
Figure BDA00017421910200000612
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，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 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 BDA0001742191020000071
And
Figure BDA0001742191020000072
wherein q isd:[t0,∞)→R3Represents a desired position, and qdIs C2The process is carried out continuously,
Figure BDA0001742191020000073
in order to be able to take the desired speed,
Figure BDA0001742191020000074
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 BDA0001742191020000075
Figure BDA0001742191020000076
then:
Figure BDA0001742191020000077
and step 3:
Figure BDA0001742191020000081
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
Figure BDA0001742191020000082
)
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:
Figure BDA0001742191020000083
wherein the content of the first and second substances,
Figure BDA0001742191020000084
in formula (14), the positive definite matrix S ═ diag [ S ]i]3×3，si>0，ks＝λmin(S)，i＝1,2,3。
(2) For all
Figure BDA0001742191020000085
Function(s)
Figure BDA0001742191020000086
Satisfies the following conditions: (i) c1(ii) a (ii) Concave function with respect to alpha, i.e. for arbitrary alpha1，α2：
Figure BDA0001742191020000087
(3) Function(s)
Figure BDA0001742191020000088
Is a non-decreasing function with respect to alpha.
And 5:
the self-adaptation law with dead zones is designed as follows:
Figure BDA0001742191020000091
equation (16) is an adaptation rate with dead band design and leakage terms,
Figure BDA0001742191020000092
in order to adapt the parameters to the application,
Figure BDA0001742191020000093
Figure BDA0001742191020000094
is composed of
Figure BDA0001742191020000095
The 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 use
Figure BDA0001742191020000096
Not into the range of size e,
Figure BDA0001742191020000097
being non-negative, leaky
Figure BDA0001742191020000098
Designed in an exponential form such that
Figure BDA0001742191020000099
Exponentially decays towards a value of 0, if
Figure BDA00017421910200000910
Constant establishment of t>t0，i＝1,2,…,k。
Dead zone portion (
Figure BDA00017421910200000911
Into a range of size e) may simplify the control algorithm.
Step 6:
Figure BDA00017421910200000912
in formula (17):
Figure BDA00017421910200000913
Figure BDA00017421910200000914
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 of
Figure BDA0001742191020000101
An adaptive robust trajectory tracking controller for a three-degree-of-freedom Delta parallel robot is provided:
Figure BDA0001742191020000102
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 → ∞ time
Figure BDA0001742191020000103
Satisfying 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 that
Figure BDA0001742191020000104
So that P is30, where τ is P1+P2When t → ∞,e→0。
if no initial position error and uncertainty exists in the system,
Figure BDA0001742191020000105
let τ be P1When t is>t0When the temperature of the water is higher than the set temperature,
Figure BDA0001742191020000106
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 vector
Figure BDA0001742191020000107
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 BDA0001742191020000111
And (| hollow)e(t0)||<r is when
Figure BDA0001742191020000112
When the temperature of the water is higher than the set temperature,
Figure BDA0001742191020000113
is formed in which
Figure BDA0001742191020000114
2. The demonstration process is as follows:
the Lyapunov function was constructed as:
Figure BDA0001742191020000115
the derivative of the lyapunov function V is:
Figure BDA0001742191020000116
the first term in analytical formula (22):
Figure BDA0001742191020000117
according to formula (11):
Figure BDA0001742191020000118
in formula (23):
Figure BDA0001742191020000119
Figure BDA00017421910200001110
Figure BDA0001742191020000121
bringing formula (19) into formula (23):
Figure BDA0001742191020000122
according to assumption 3, there are:
Figure BDA0001742191020000123
bringing formulae (23) to (27) into formula (22) includes:
Figure BDA0001742191020000124
the adaptive rate (16) is introduced into the channel (28) by:
(1) when in use
Figure BDA0001742191020000125
When the temperature of the water is higher than the set temperature,
Figure BDA0001742191020000126
Figure BDA0001742191020000131
if the number of the first and second antennas is greater than the predetermined number,
Figure BDA0001742191020000132
comprises the following steps:
Figure BDA0001742191020000133
Figure BDA0001742191020000141
the third term in the pair of equations (29) is:
Figure BDA0001742191020000142
the fourth term in the pair of equations (29) is:
Figure BDA0001742191020000143
due to the fact that
Figure BDA0001742191020000144
According to the formulae (30) to (31), there are:
Figure BDA0001742191020000145
Figure BDA0001742191020000151
wherein the content of the first and second substances,
Figure BDA0001742191020000152
Figure BDA0001742191020000153
if the number of the first and second antennas is greater than the predetermined number,
Figure BDA0001742191020000154
comprises the following steps:
Figure BDA0001742191020000155
Figure BDA0001742191020000161
wherein the content of the first and second substances,
Figure BDA0001742191020000162
Figure BDA0001742191020000163
(2) when in use
Figure BDA0001742191020000164
In time, there are:
Figure BDA0001742191020000165
if the number of the first and second antennas is greater than the predetermined number,
Figure BDA0001742191020000171
comprises the following steps:
Figure BDA0001742191020000172
because of the fact that
Figure BDA0001742191020000173
Therefore, it is not only easy to use
Figure BDA0001742191020000174
Then:
Figure BDA0001742191020000175
Figure BDA0001742191020000181
wherein ψ ∈.
If the number of the first and second antennas is greater than the predetermined number,
Figure BDA0001742191020000182
comprises the following steps:
Figure BDA0001742191020000183
Figure BDA0001742191020000191
the derivative of the Lyapunov function according to equations (32), (33), (36) and (37)
Figure BDA0001742191020000192
Comprises the following steps:
Figure BDA0001742191020000193
wherein the content of the first and second substances,ρ1＝ρ，ρ2psi or rho2＝0，ρ3θ. For equation (38), when | satisfies:
Figure BDA0001742191020000194
Figure BDA0001742191020000195
negative values, i.e.:
Figure BDA0001742191020000196
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 present
Figure BDA0001742191020000197
When the formula (40) is satisfied, the tracking error vector
Figure BDA0001742191020000198
And adaptive parameters
Figure BDA0001742191020000199
Have consistent and bounded;
Figure BDA0001742191020000201
wherein:
Figure BDA0001742191020000202
γ1＝
min{λmin(M),λmin((κk1)-1)},γ2＝min{λmax(M),λmax((κk1)-1)}。
at the same time, the trajectory tracking error vector
Figure BDA0001742191020000203
And adaptive parameters
Figure BDA0001742191020000204
Consistent final bounding is also satisfied;
Figure BDA0001742191020000205
Figure BDA0001742191020000206
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 BDA0001742191020000207
Figure BDA0001742191020000208
To an external load
Figure BDA0001742191020000209
Wherein the content of the first and second substances,
Figure BDA00017421910200002010
and
Figure BDA00017421910200002011
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. Setting a target track needing to be tracked by a Delta parallel robot working platform as follows:
Figure BDA0001742191020000211
according to hypothesis 3, function
Figure BDA0001742191020000212
Is selected and a function
Figure BDA0001742191020000213
Related, selection function
Figure BDA0001742191020000214
Comprises the following steps:
Figure BDA0001742191020000215
wherein α is 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:
Figure BDA0001742191020000221
choosing uncertain parameters as follows:
Figure BDA0001742191020000222
Figure BDA0001742191020000223
Figure BDA0001742191020000224
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,
Figure BDA0001742191020000225
Figure BDA0001742191020000226
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 parameters
Figure BDA0001742191020000227
And (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 controlling the input, the tracking error diverges from about 0.01m through 0.8sTo 0.2m, there was no convergence within 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 inputs
Figure BDA0001742191020000228
When τ is equal to P, the simulation result of (1)1Error in tracking of track for control input
Figure BDA0001742191020000229
Diverges from about 0.31m/s to 1m/s over 0.8s, increases after 0.8s, and
Figure BDA00017421910200002210
and fail to converge within a limited time. When τ is equal to P1+P2Error in tracking of track for control input
Figure BDA00017421910200002211
Always oscillating around 0.1m/s, and with the increase of simulation time, the error
Figure BDA00017421910200002212
There is an increasing trend. When τ is equal to P1+P2+P3Error in tracking of track for control input
Figure BDA0001742191020000231
After 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;
(1) there is a known positive definite function Γ (·): (0, ∞)k×R3×R3×R→R+And an unknown vector α ∈ (0, ∞)kSo that:
Figure FDA0002761843880000011
wherein the content of the first and second substances,
Figure FDA0002761843880000012
in the formula (14), the positive definite matrix S ═diag[si]3×3，si＞0，ks＝λmin(S)，i＝1，2，3；
(2) For all
Figure FDA0002761843880000013
Function(s)
Figure FDA0002761843880000014
Satisfies the following conditions: (i) c1(ii) a (ii) Concave function with respect to alpha, i.e. for arbitrary alpha1，α2：
Figure FDA0002761843880000015
(3) Function(s)
Figure FDA0002761843880000016
Is a non-decreasing function with respect to α;
wherein q ∈ R3In order to be the active joint angle vector,
Figure FDA0002761843880000021
in order to be the active joint angular velocity vector,
Figure FDA0002761843880000022
is the active joint angular acceleration vector; sigma e is sigma e RpFor uncertain parameter vectors existing in the robot system, sigma belongs to RpIs a tight set of uncertain parameters representing the uncertainty bound; m (q, sigma, t) is the inertia matrix of the robot system,
Figure FDA0002761843880000023
being the coriolis force/centrifugal force term of the system,
Figure FDA0002761843880000024
is a diagonally symmetric matrix, G (q, σ, t) is the gravity term of the system,
Figure FDA0002761843880000025
epsilon (t) is the input torque of the system, and epsilon (t) is the external interference borne by the system; m (-), C (-), G (-), and F (-), all either continuously 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 FDA0002761843880000026
Figure FDA0002761843880000027
Figure FDA0002761843880000028
Figure FDA0002761843880000029
wherein the content of the first and second substances,
Figure FDA00027618438800000210
and
Figure FDA00027618438800000211
nominal term, referred to as Delta parallel robot System,. DELTA.M (q, σ, t),
Figure FDA00027618438800000212
Δ G (q, σ, t) and
Figure FDA00027618438800000213
an uncertainty term referred to as the Delta parallel robot system;
period for setting three-degree-of-freedom Delta parallel robotThe observation track is qd、
Figure FDA00027618438800000214
And
Figure FDA00027618438800000215
wherein q isd：[t0，∞)→R3Represents a desired position and qdIs C2The process is carried out continuously,
Figure FDA00027618438800000216
in order to be able to take the desired speed,
Figure FDA00027618438800000217
is a 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 FDA0002761843880000031
Figure FDA0002761843880000032
then:
Figure FDA0002761843880000033
Figure FDA0002761843880000034
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；
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;
designing an adaptation rate with dead zone design and leakage terms:
Figure FDA0002761843880000035
wherein the content of the first and second substances,
Figure FDA0002761843880000036
in order to adapt the parameters to the application,
Figure FDA0002761843880000037
Figure FDA0002761843880000038
is composed of
Figure FDA0002761843880000039
The ith element of the vector, i ═ 1,2, …, k1，k2，k3∈Rk×kAnd k is1，k2，k3＞0，κ∈R，κ＞0，∈∈R，∈＞0；
Step 6, constructing an uncertainty compensation link according to the function and the self-adaptive law, and compensating the uncertainty in the system;
Figure FDA00027618438800000310
in formula (17):
Figure FDA00027618438800000311
Figure FDA0002761843880000041
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；
Step 7, finally, providing an adaptive robust controller;
considering a tracking error vector of
Figure FDA0002761843880000042
A self-adaptive robust trajectory tracking controller for a three-degree-of-freedom Delta parallel robot is provided:
Figure FDA0002761843880000043
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