CN115847485A - Design method of obstacle avoidance controller based on TDE (time domain reflectometry) for constraint cable driving mechanical arm - Google Patents

Design method of obstacle avoidance controller based on TDE (time domain reflectometry) for constraint cable driving mechanical arm Download PDF

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CN115847485A
CN115847485A CN202211531109.9A CN202211531109A CN115847485A CN 115847485 A CN115847485 A CN 115847485A CN 202211531109 A CN202211531109 A CN 202211531109A CN 115847485 A CN115847485 A CN 115847485A
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mechanical arm
controller
safety
state
cable
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商巍
郭永达
李立军
章正飞
李靖
刘宇帆
王喻林
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Zhejiang Qiantang Robot And Intelligent Equipment Research Co ltd
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Zhejiang Qiantang Robot And Intelligent Equipment Research Co ltd
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Abstract

The invention discloses a design method of an obstacle avoidance controller based on TDE (time delay and earth) for restraining a cable-driven mechanical arm, which takes an operator as a system operation main body under the condition of safety guarantee, and when the environment is judged to be in a dangerous or unknown state, a cable-driven mechanical arm control system triggers an adaptive state feedback controller, and the machine operation is taken as the operation main body, so that the misoperation of a human operator can be avoided. First, a set of safety constraints are set to ensure that the state or output of the cable driven robot meets the requirements of these constraints to ensure "safety" of the robot, and the remaining lumped system dynamics are estimated and compensated using time lag estimation. Secondly, errors caused by estimation are considered in the design of the controller, and the robustness of the whole shared control system is effectively improved. The self-adaptive fixed time state feedback control effectively ensures that the shared control system realizes quick, accurate and robust convergence in operation, so that the cable-driven mechanical arm meets the safety constraint and the safety performance of the cable-driven mechanical arm is guaranteed.

Description

Design method of obstacle avoidance controller based on TDE (time domain reflectometry) for constraint cable driving mechanical arm
Technical Field
The invention belongs to the technical field of mechanical arm control, and relates to a self-adaptive fixed time state feedback sharing control method based on TDE (time domain equalization) for restraining a cable-driven mechanical arm.
Background
Shared control is a control architecture that combines manual operation inputs and feedback control inputs for a non-linear system such as a cable driven robotic arm. It has the same meaning as described in the well-known anti-lock brake system. In normal situations, the human operator is responsible for managing the system, whereas in emergency situations, i.e. where the system is in a defined "dangerous" situation, the feedback controller may take the initiative for the control of the system.
There are many representative applications of shared control, such as a human-robot system, a mobile robot, and a multi-robot system. The main objective of shared control is to ensure the "safety" of the system, while the main problem of "safety" is to avoid obstacles. At present, a plurality of famous methods can solve the obstacle avoidance problem of the robot, such as a local path planning algorithm based on an artificial potential field and a control algorithm based on an obstacle Lyapunov function. However, when the robot passes through a narrow passage, the artificial potential field based local path planning algorithm tends to result in local minimization and oscillatory motion. Control algorithms based on the barrier lyapunov function cannot allow the system state to reach the boundary of the allowable space of the robot. In addition to the above-mentioned robotic field, the shared control concept is also applied in other engineering fields, such as medical operations, smart wheelchairs, assisted driving cars, airplane flight, spacecraft rendezvous, assembly industries. In these applications, a continuous scalar function is typically used to ensure a smooth transition between the human operator and the feedback controller. The control authority is then assigned to the human operator and the feedback controller using a simple shared hysteresis switching function with no switching oscillations. The performance of the controlled system in the above study was demonstrated theoretically. However, in actual engineering, a system of the cable-driven mechanical arm is nonlinear in nature, and a dynamic model often has parameter uncertainty, so that the closed-loop performance of the actual system cannot be guaranteed by adopting a non-adaptive state feedback sharing control method, and the robustness of the control system needs to be improved by means of an advanced control technology. In addition, the response speed of a system of a real cable-driven robot arm is extremely fast, and thus an advanced control method having a fast convergence performance should be studied for a shared control system.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a design method of an adaptive fixed time state feedback shared controller based on a time delay estimation method (TDE) for restraining a cable driving mechanical arm.
Aiming at the safety performance problem of a cable driving mechanical arm, a self-adaptive fixed time state feedback controller based on shared control is designed by adopting a time delay estimation method. The method utilizes the constraint to control the mechanical arm to be kept in a theoretically defined safety area, and the design of shared control ensures that the cable-driven mechanical arm can always meet the constraint condition. Meanwhile, the robustness performance, convergence speed and precision of the system are effectively improved by the combined application of self-adaption and fixed time.
In order to achieve the purpose, the invention provides the following technical scheme:
according to the design method of the TDE-based adaptive fixed time state feedback shared controller for the constraint cable-driven mechanical arm, the control method comprises the following steps:
step 1, constructing a dynamic model for describing the state of a mechanical arm of a machine;
step 2, estimating unknown parameters in the dynamic model by using a time delay estimation method;
step 3, designing a fixed time state feedback sharing controller according to the dynamic model, comprising three parts: the method comprises the steps of dividing a space region, describing the characteristics of shared control, designing an adaptive fixed time state feedback controller and designing a shared controller.
Further, the dynamic model of the mechanical arm with n degrees of freedom in step 1 is represented as:
Figure BDA0003976047810000021
Figure BDA0003976047810000022
Figure BDA0003976047810000023
wherein J and d m Is the motor inertia and damping matrix, q and theta are the joint and motor position vectors,
Figure BDA0003976047810000024
Figure BDA0003976047810000025
denotes the first and second derivatives of q and θ, respectively, M (q) is an inertia matrix, and>
Figure BDA0003976047810000026
is a Coriolis/centrifuge matrix, g (q) is the attractive force, and>
Figure BDA0003976047810000027
is the friction vector u s And τ s Control torque and joint compliance torque, d, respectively, given to the motor s To damp the matrix, k s Is a joint stiffness matrix, τ d Representing a lumped unknown uncertainty;
to facilitate the use of the TDE scheme, i.e. based on the time delay estimation method, (2) is substituted into (1), and a constant parameter is applied
Figure BDA0003976047810000031
Obtaining:
Figure BDA0003976047810000032
wherein the expression of f is:
Figure BDA0003976047810000033
further, the unknown parameter in the step 2 is f, and the estimated value is obtained by using a TED scheme
Figure BDA0003976047810000034
Figure BDA0003976047810000035
Where Δ t is the delay time, the dynamic model (4) is substituted into (6) and is available:
Figure BDA0003976047810000036
as can be seen from (6) and (7), the purpose of the TED scheme is to estimate the lumped system dynamics using only the time-lag values of the control and acceleration signals, and then to give a model-free scheme;
in engineering applications, u s (t- Δ t) may be represented by u s Is obtained by numerical differentiation
Figure BDA0003976047810000037
Figure BDA0003976047810000038
Wherein, time node t 0 ≧ 2 Δ t, at the initial stage, the time t ≦ 2 Δ t, q (t) has the actual measured value, and q (t-2 Δ t) is manually zeroed, possibly resulting in strong fluctuations, and therefore (8) is used to mitigate the strong fluctuations that may exist.
Further, the specific implementation manner of dividing the space region and describing the characteristics of the sharing control in step 3 is as follows;
the space in which the state of the robotic arm is allowed to exist is defined as an empty allowance set Θ, which is defined by a set of linear inequalities, i.e.
Figure BDA0003976047810000041
Wherein, the matrix
Figure BDA0003976047810000042
Matrix->
Figure BDA0003976047810000043
And
Figure BDA0003976047810000044
i ∈ {1,2,. Multidot.m }, m denotes the number of rows of S and T, n denotes the number of columns of the state variable p, and if m > n, the matrices S and T satisfy ≧>
Figure BDA0003976047810000045
Wherein->
Figure BDA0003976047810000046
j∈{1,2,...,l}，r 1 ,r 2 ,...,r l ∈{1,2,...,m}，l∈[n+1,m]；
According to the distance and speed of the mechanical arm when the mechanical arm reaches the boundary, the whole state space can be divided into three subspaces, namely a safety set, a hysteresis set and a danger set; setting constraint conditions, setting the mechanical arm in a safety and hysteresis set for the state meeting the constraint conditions, and setting the mechanical arm in a danger set for the mechanical arm not meeting the constraint conditions, and actively constraining the ith group:
x i ＝S i p+T i ≤0(11)
in the formula, x i Representing the position coordinates, S i Is non-exotic in that it is,
Figure BDA0003976047810000047
then, a safety set, a hysteresis set and a danger set definition are given:
Figure BDA0003976047810000048
Figure BDA0003976047810000049
or
Figure BDA00039760478100000410
Figure BDA00039760478100000411
Wherein,
Figure BDA00039760478100000412
is to x i Is derived, is taken>
Figure BDA00039760478100000413
And b 2 ＞b 1 ＞0；
The s-closed loop of the mechanical arm system under the control of the shared controller and the h-closed loop of the mechanical arm system under the manual operation are respectively composed of (4) and
Figure BDA00039760478100000414
describe, in addition, with Ω h And Ω s Respectively representing the limit sets omega-limit of the h-closed loop system and the s-closed loop system; Θ is a set of allowable configurations, u, given and compact according to the kinetic model (4) s (u h ,u f )∈R n Is an external input, where u h Is given a manually operated input, u f Is an input to the feedback controller; then, the design of the share control is to find a feedback controller, a safe subset and a share function, so that the mechanical arm maintains the following properties:
a) The structure of the robot arm remains in Θ at all times and a safety subset is defined for the robot arm
Figure BDA0003976047810000051
Wherein
Figure BDA0003976047810000052
Is forward invariant;
b)u s the target of manual operation cannot be changed;
c) If the state of the robot arm remains in the safety subset, u s ＝u h 。
Further, the specific implementation manner of designing the adaptive fixed time state feedback controller is as follows;
position coordinate x i The method is defined by the following steps (11),
Figure BDA0003976047810000053
representing a state feedback controller, the dynamical model (4) can therefore be written as: />
Figure BDA0003976047810000054
To eliminate the pair x i Is constrained, defines variables
Figure BDA0003976047810000055
Comprises the following steps:
Figure BDA0003976047810000056
wherein,
Figure BDA0003976047810000057
is relative to->
Figure BDA0003976047810000058
Is defined as:
Figure BDA0003976047810000059
in the formula,
Figure BDA00039760478100000510
Figure BDA00039760478100000511
is an intermediate variable, ε is a sufficiently small positive number;
in trajectory tracking, the feedback controller shares the desired state information of manual operation, so that for the feedback controller, in the case of free operation, the desired position q is d (t) is known and can be represented by u h Calculating;
it should be noted that it is preferable that,
Figure BDA00039760478100000512
is a smooth function, is>
Figure BDA00039760478100000513
Is less than 0,j e {1,2>
Figure BDA00039760478100000514
And &>
Figure BDA00039760478100000515
Is present as
Figure BDA00039760478100000516
Wherein,
Figure BDA00039760478100000517
in that
Figure BDA0003976047810000061
In space, assume >>
Figure BDA0003976047810000062
Is omega h In group i constraint, based on the number of x groups in the group i>
Figure BDA0003976047810000063
In the secure subset R s A mapping of (5) is indicated as->
Figure BDA0003976047810000064
And is defined as:
Figure BDA0003976047810000065
thus, for the ith set of constraints, Ω will be h In the secure subset R s The mapping in (1) is defined as:
Figure BDA0003976047810000066
an error of a first derivative of a position vector defining a joint is
Figure BDA0003976047810000067
By a variable z i And &>
Figure BDA0003976047810000068
Obtaining a system error model:
Figure BDA0003976047810000069
wherein
Figure BDA00039760478100000610
/>
Figure BDA00039760478100000611
Wherein diag refers to a diagonal matrix, and the virtual control input is designed according to a frame designed by a classical backstepping method
Figure BDA00039760478100000612
Comprises the following steps:
Figure BDA00039760478100000613
wherein alpha is 1 ＞0，α 2 ＞0，β＝[β 1 ,β 2 ,...,β n ] T Is defined as:
Figure BDA00039760478100000614
in the formula,
Figure BDA00039760478100000615
is->
Figure BDA00039760478100000616
On line j of (a), and +>
Figure BDA00039760478100000617
Figure BDA00039760478100000618
γ 1 ＞1，0＜γ 2 ＜1，ε z Is a normal number with a smaller value; in addition, the time of the virtual control input (20) is differentiated:
Figure BDA00039760478100000619
wherein
Figure BDA00039760478100000620
Comprises the following steps:
Figure BDA0003976047810000071
definition of
Figure BDA0003976047810000072
The error model (19) can be expressed as:
Figure BDA0003976047810000073
further consider errors caused by TED schemes
Figure BDA0003976047810000074
Figure BDA0003976047810000075
Is the upper limit of the error, and then based on the model (22), the adaptive fixed-time feedback control is designed to:
Figure BDA0003976047810000076
Figure BDA0003976047810000077
Figure BDA0003976047810000078
wherein k is 1 (0) > 0 and k 2 (0)＞0；k 0 ，κ 1 ，κ 2 ，ξ 1 ，ξ 2 ，σ 1 ，σ 2 ，u 1 Are all normal numbers.
Further, a specific implementation manner of designing the shared controller is as follows;
with reference to the ith set of n constraints defined in (11), the state space can be divided by (12) into three subsets, in order to eliminate ambiguity of the different sets of constraints by
Figure BDA0003976047810000079
Push the subset back
Figure BDA00039760478100000710
Coordinates;
thus, a structure consistent with the overall feasible state space is constructed
Figure BDA00039760478100000711
i∈{1,2,...,N c That is for any fixed->
Figure BDA00039760478100000712
In other words, the union of the safety set, the hysteresis set, and the hazard set in the ith group constraint, i.e., S i q+T i Less than 0; a safety set of different constraint groups is then defined, with the hysteresis set and the hazard set ≧>
Figure BDA00039760478100000713
R h ＝R-R d -R s And &>
Figure BDA00039760478100000714
Based on three subsets, in
Figure BDA00039760478100000715
Defining a shared control input on coordinates>
Figure BDA00039760478100000716
Comprises the following steps:
Figure BDA00039760478100000717
in the formula, feedback sharing function
Figure BDA00039760478100000718
Is defined as:
Figure BDA0003976047810000081
wherein
Figure BDA0003976047810000082
The invention has the beneficial effects that:
through sharing control, the safety performance of the mechanical arm is improved, and unnecessary danger caused by misoperation of an operator is prevented. A novel TDE-based adaptive fixed time state feedback sharing control method is provided, and the method has high control precision and robustness. And the design of self-adaptive parameters is adopted, the error generated by delay estimation is compensated, and the robustness of the system is greatly improved. Fixed time control is applied, the convergence speed of the controller is ensured, and the requirement of sharing control on quick convergence of the system is met.
Detailed Description
The scheme of the present invention is explained in further detail below.
A design method of a TDE-based adaptive fixed time state feedback shared controller for a constraint cable driven mechanical arm comprises the following steps:
step 1, constructing a dynamic model for describing the state of the mechanical arm of the machine.
The kinetic model of a mechanical arm with n degrees of freedom is represented as:
Figure BDA0003976047810000083
Figure BDA0003976047810000084
Figure BDA0003976047810000085
wherein J and d m Is the motor inertia and damping matrix, q and theta are the joint and motor position vectors,
Figure BDA0003976047810000086
Figure BDA0003976047810000087
denotes the first and second derivatives of q and θ, respectively, with M (q) being an inertia matrix, and->
Figure BDA0003976047810000088
Is a Coriolis/centrifuge matrix, g (q) is the gravitational force, and>
Figure BDA0003976047810000089
is the friction vector u s And τ s Control torque and joint compliance torque, d, respectively, given to the motor s To damp the matrix, k s Is a joint stiffness matrix. Tau is d Representing the lumped unknown uncertainty. />
To facilitate the use of the TDE scheme, substitute (2) into (1), apply constant parameters
Figure BDA0003976047810000091
Obtaining:
Figure BDA0003976047810000092
wherein the expression of f is:
Figure BDA0003976047810000093
the three main components of f, including residual link dynamics, motor dynamics, and collective uncertainty, are subsequently estimated using TED, considering that it is difficult to obtain using conventional methods.
The proposed control scheme does not use the dynamics models of the systems (1) - (3), but only uses the dynamics model (4) to illustrate the method of designing the controller.
And 2, step: the above-mentioned unknown parameter f is estimated using a TED scheme.
As mentioned above, f is particularly complex and difficult to obtain, inIn this section, we will use the TED scheme to find its estimate
Figure BDA0003976047810000094
Figure BDA0003976047810000095
Where Δ t is the delay time, the dynamic model (4) is substituted (6) to obtain:
Figure BDA0003976047810000096
as can be seen from (6) and (7), the main purpose of the TED scheme is to estimate the lumped system dynamics using only the time-lag values of the control and acceleration signals, and then to give a model-free scheme.
In engineering applications, u s (t- Δ t) may be represented by u s Direct time lag of (D) is obtained. Obtained by numerical differentiation
Figure BDA0003976047810000097
Figure BDA0003976047810000098
Wherein, time node t 0 Is more than or equal to 2 delta t. In the initial phase t ≦ 2 Δ t, q (t) has the actual measured value, and q (t-2 Δ t) is manually set to zero, which may lead to strong fluctuations, and thus (8) is used to mitigate the strong fluctuations that may be present. Also of interest is (8) and its initial version, namely:
Figure BDA0003976047810000101
t＞0 (9)
it is widely applied to many robust control schemes based on TDE. The numerical differentiation (9) is significant if no measures are takenThe noise effect is amplified, thereby degrading control performance. However, it has been shown in theory that it is possible to reduce the gain
Figure BDA00039760478100001010
Or an additional low pass filter may be used to solve this problem.
It can be known from (8) that the current value of the dynamic model (4) is estimated by using a time-lag system state by using a TDE scheme, so that the estimation error of the method becomes larger when a large disturbance occurs, but the estimation error can be effectively reduced by the proposed method.
And step 3: and (4) designing a fixed time state feedback sharing controller according to the dynamic model given in the step (4).
In this step, the division into three parts, namely the division into space regions and the description of the characteristics of the shared control, the design of the adaptive fixed time state feedback controller and the design of the shared controller, are mainly performed.
1) The spatial regions are divided and the characteristics of the sharing control are explained.
The space (set) in which the state of the robot arm is allowed to exist is defined as an empty allowance set Θ, which is defined by a set of linear inequalities, i.e.
Figure BDA0003976047810000102
Wherein, the matrix
Figure BDA0003976047810000103
Matrix->
Figure BDA0003976047810000104
Figure BDA0003976047810000105
And
Figure BDA0003976047810000106
i ∈ {1,2, ·, m }, m denotes the number of rows of S and T, n denotes the number of columns of the state variable p, and if m > n, the matrices S and T satisfy &>
Figure BDA0003976047810000107
Wherein->
Figure BDA0003976047810000108
j∈{1,2,...,l}，r 1 ,r 2 ,...,r l ∈{1,2,...,m}，l∈[n+1,m]。
According to the distance and the speed of the mechanical arm when the mechanical arm reaches the boundary, the whole state space can be divided into three subspaces, namely a safety set R s Lagged set R h With the danger set R d . For the ith set of active constraints:
x i ＝S i q+T i ≤0 (11)
the mechanical arm is in a safe and lagging set when the constraint is met, and the mechanical arm is in a dangerous set (collision can occur) when the constraint is not met; in the formula, S i Is non-exotic in that it is,
Figure BDA0003976047810000109
then, a safety set, a hysteresis set and a danger set are given:
Figure BDA0003976047810000111
Figure BDA0003976047810000112
or
Figure BDA0003976047810000113
Figure BDA0003976047810000114
Wherein,
Figure BDA0003976047810000115
is to x i Is derived, is taken>
Figure BDA0003976047810000116
And b 2 ＞b 1 ＞0。
The s-closed loop (the mechanical arm system under the control of the shared controller) and the h-closed loop (the mechanical arm system under the manual operation) are respectively composed of (4)
Figure BDA0003976047810000117
Describe, in addition, with Ω h And Ω s Respectively representing the limit set omega-limit of the h-closed loop and the s-closed loop system. Θ is a set of allowable configurations, u, given and compact according to the dynamic model (4) s (u h ,u f )∈R n Is an external input, where u h Is given a manually operated input, u f Is an input to the feedback controller.
Then, the design of the share control is to find a feedback controller, a safety subset and a share function that keeps the following properties for the mechanical arm:
a) The structure of the robot arm remains in Θ at all times and defines a safe subset for the robot arm
Figure BDA0003976047810000118
In which space
Figure BDA0003976047810000119
Is forward invariant;
b)u s the target of manual operation cannot be changed;
c) If the state of the robot arm remains in the safety subset, u s ＝u h 。
2) An adaptive fixed time state feedback controller is designed.
Coordinate x i The method is defined by the following (11),
Figure BDA00039760478100001110
represents a state feedback controller, so that the dynamic model (4) can be written to ≧>
Figure BDA00039760478100001111
To eliminate the pair x i Is constrained, defines variables
Figure BDA00039760478100001112
Is composed of
Figure BDA0003976047810000121
Wherein,
Figure BDA0003976047810000122
is relative to->
Figure BDA0003976047810000123
Is defined as
Figure BDA0003976047810000124
In the formula,
Figure BDA0003976047810000125
Figure BDA0003976047810000126
is an intermediate variable, ε is a sufficiently small positive number;
in trajectory tracking, the feedback controller shares the desired state information of manual operation, so that for the feedback controller, in the case of free operation, the desired position q is d (t) is known and can be represented by u h And (6) calculating.
It should be noted that it is preferable that,
Figure BDA0003976047810000127
is a smooth function, is>
Figure BDA0003976047810000128
Are less than 0,j e {1,2. Accordingly, is present>
Figure BDA0003976047810000129
And &>
Figure BDA00039760478100001210
Is present as
Figure BDA00039760478100001211
Wherein,
Figure BDA00039760478100001212
in that
Figure BDA00039760478100001213
In space, assume >>
Figure BDA00039760478100001214
Is omega h Point (2) of (c). In group i constraint, ->
Figure BDA00039760478100001215
In the secure subset R s A mapping of (5) is indicated as->
Figure BDA00039760478100001216
And is defined as
Figure BDA00039760478100001217
Thus, for the ith set of constraints, Ω h In the secure subset R s The mapping in (1) is defined as
Figure BDA00039760478100001218
Primary derivation of a position vector defining a jointHas an error of
Figure BDA00039760478100001219
By a variable z i And &>
Figure BDA00039760478100001220
Available systematic error model
Figure BDA00039760478100001221
Wherein
Figure BDA0003976047810000131
/>
Figure BDA0003976047810000132
Wherein diag refers to a diagonal matrix, a frame is designed according to a classical backstepping method, and a virtual control input is designed
Figure BDA0003976047810000133
Comprises the following steps:
Figure BDA0003976047810000134
wherein alpha is 1 ＞0，α 2 ＞0，β＝[β 1 ,β 2 ,...,β n ] T Is defined as
Figure BDA0003976047810000135
In the formula,
Figure BDA0003976047810000136
is->
Figure BDA0003976047810000137
On line j of (a), and +>
Figure BDA0003976047810000138
Figure BDA0003976047810000139
γ 1 ＞1，0＜γ 2 ＜1，ε z Is a positive constant with a smaller value. In addition, the time derivative of the virtual control input (20) is obtained
Figure BDA00039760478100001310
Wherein
Figure BDA00039760478100001311
Is composed of
Figure BDA00039760478100001312
Definition of
Figure BDA00039760478100001313
The error model (19) can be expressed as
Figure BDA00039760478100001314
Further consider errors caused by TED schemes
Figure BDA00039760478100001315
Figure BDA00039760478100001316
Is the upper limit of the error, and then based on the model (22), an adaptive fixed-time feedback control is designed as:
Figure BDA00039760478100001317
Figure BDA00039760478100001318
Figure BDA00039760478100001319
wherein k is 1 (0) > 0 and k 2 (0)＞0；k 0 ，κ 1 ，κ 2 ，ξ 1 ，ζ 2 ，σ 1 ，σ 2 ，u 1 Are all normal numbers.
3) Design sharing controller
With reference to the i-th set of n constraints defined in (11), the state space can be divided into three subsets by (12). In order to eliminate ambiguity of different group constraints by
Figure BDA0003976047810000141
Push the subset back
Figure BDA0003976047810000142
And (4) coordinates.
Thus, a structure consistent with the overall feasible state space is constructed
Figure BDA0003976047810000143
i∈{1,2,...,N c That is for any fixed->
Figure BDA0003976047810000144
Also in the ith group of constraints, is the union of the safety, lag and hazard sets, i.e., S i q+T i Is less than 0. A safety set of different constraint groups is then defined, with the hysteresis set and the hazard set being >>
Figure BDA0003976047810000145
R h ＝R-R d -R s And &>
Figure BDA0003976047810000146
Based on these subsets, are
Figure BDA0003976047810000147
Defining a shared control input on coordinates>
Figure BDA0003976047810000148
Is composed of
Figure BDA0003976047810000149
In the formula, feedback sharing function
Figure BDA00039760478100001410
Is defined as
Figure BDA00039760478100001411
Wherein
Figure BDA00039760478100001412
The specific embodiments described herein are merely illustrative of the spirit of the invention. Various modifications or additions may be made to the described embodiments or alternatives may be employed by those skilled in the art without departing from the spirit or ambit of the invention as defined in the appended claims.

Claims (6)

1. A design method of an obstacle avoidance controller based on TDE (time domain reflectometry) for constraining a cable to drive a mechanical arm is characterized by comprising the following steps of:
step 1, constructing a dynamic model for describing the state of a mechanical arm of a machine;
step 2, estimating unknown parameters in the dynamic model by using a time delay estimation method;
step 3, designing a fixed time state feedback sharing controller according to the dynamic model, comprising three parts: the method comprises the steps of dividing a space region, describing the characteristics of shared control, designing an adaptive fixed time state feedback controller and designing a shared controller.
2. The TDE-based obstacle avoidance controller design method for constraining a cable-driven mechanical arm according to claim 1, characterized in that: the kinetic model of the mechanical arm with n degrees of freedom in step 1 is represented as:
Figure FDA0003976047800000011
Figure FDA0003976047800000012
Figure FDA0003976047800000013
in the formula, J and d m Is the motor inertia and damping matrix, q and theta are the joint and motor position vectors,
Figure FDA0003976047800000014
Figure FDA0003976047800000015
denotes the first and second derivatives of q and θ, respectively, M (q) is an inertia matrix, and>
Figure FDA0003976047800000016
is a Coriolis/centrifuge matrix, g (q) is the attractive force, and>
Figure FDA0003976047800000017
is the friction vector u s And τ s Control torque and joint flexibility torque respectively given to motor，d s As a damping matrix, k s Is a joint stiffness matrix, τ d Representing a lumped unknown uncertainty;
to facilitate the use of the TDE scheme, i.e. based on the time delay estimation method, (2) is substituted into (1), and a constant parameter is applied
Figure FDA0003976047800000018
Obtaining:
Figure FDA0003976047800000019
wherein the expression of f is:
Figure FDA00039760478000000110
3. the design method of the TDE-based obstacle avoidance controller for the constraint cable-driven mechanical arm, as claimed in claim 2, is characterized in that: step 2, the unknown parameter is f, and the estimated value is obtained by using a TED scheme
Figure FDA0003976047800000021
Figure FDA0003976047800000022
Where Δ t is the delay time, the dynamic model (4) is substituted into (6) and is available:
Figure FDA0003976047800000023
as can be seen from (6) and (7), the purpose of the TED scheme is to estimate the lumped system dynamics using only the time-lag values of the control and acceleration signals, and then to give a model-free scheme;
in the application of the engineering, the method can be used,u s (t- Δ t) may be represented by u s Is obtained by numerical differentiation
Figure FDA0003976047800000024
Figure FDA0003976047800000025
Wherein, the time node t 0 ≧ 2 Δ t, at the initial stage, time t ≦ 2 Δ t, q (t) has the actual measurement value, and q (t-2 Δ t) is manually zeroed, possibly resulting in strong fluctuations, and therefore (8) is used to mitigate the strong fluctuations that may exist.
4. The design method of the TDE-based obstacle avoidance controller for the constraint cable-driven mechanical arm, as claimed in claim 3, is characterized in that: the specific implementation manner of dividing the space region and explaining the characteristics of the sharing control in the step 3 is as follows;
the space in which the state of the robotic arm is allowed to exist is defined as an empty allowance set Θ, which is defined by a set of linear inequalities, i.e.
Figure FDA0003976047800000026
Wherein, the matrix
Figure FDA0003976047800000027
Matrix->
Figure FDA0003976047800000028
And &>
Figure FDA0003976047800000029
i ∈ {1,2., m }, m denotes the number of rows of S and T, n denotes the number of columns of the state variable p, and if m > n, the matrices S and T satisfy
Figure FDA00039760478000000210
Wherein->
Figure FDA00039760478000000211
j∈{1,2,...,l}，r 1 ,r 2 ,...,r l ∈{1,2,...,m}，l∈[n+1,m]；
According to the distance and speed of the mechanical arm when the mechanical arm reaches the boundary, the whole state space can be divided into three subspaces, namely a safety set, a hysteresis set and a danger set; setting constraint conditions, setting the mechanical arm in a safety and hysteresis set for the state meeting the constraint conditions, and setting the mechanical arm in a danger set for the mechanical arm not meeting the constraint conditions, and actively constraining the ith group:
x i ＝S i p+T i ≤0 (11)
in the formula, x i Representing the position coordinates, S i Is non-exotic in that it is,
Figure FDA0003976047800000039
then, a safety set, a hysteresis set and a danger set definition are given:
Figure FDA0003976047800000031
or
Figure FDA0003976047800000032
Figure FDA0003976047800000033
Wherein,
Figure FDA0003976047800000034
is to x i Is derived, is taken>
Figure FDA0003976047800000035
And b 2 ＞b 1 ＞0；
The s-closed loop of the mechanical arm system under the control of the shared controller and the h-closed loop of the mechanical arm system under the manual operation are respectively composed of (4) and
Figure FDA0003976047800000036
describe, in addition, with Ω h And Ω s Respectively representing the limit sets omega-limit of the h-closed loop system and the s-closed loop system; Θ is a set of allowable configurations, u, given and compact according to the kinetic model (4) s (u h ,u f )∈R n Is an external input, where u h Is given a manually operated input, u f Is an input to the feedback controller; then, the design of the share control is to find a feedback controller, a safe subset and a share function, so that the mechanical arm maintains the following properties:
a) The structure of the robot arm remains in Θ at all times and a safety subset is defined for the robot arm
Figure FDA0003976047800000037
Wherein
Figure FDA0003976047800000038
Is forward invariant;
b)u s the target of manual operation cannot be changed;
c) If the state of the robot arm remains in the safety subset, u s ＝u h 。
5. The TDE-based obstacle avoidance controller design method for constraining a cable-driven mechanical arm according to claim 4, characterized in that: the specific implementation mode of designing the self-adaptive fixed time state feedback controller is as follows;
position coordinate x i The method is defined by the following (11),
Figure FDA0003976047800000041
representing a state feedback controller, the dynamical model (4) can therefore be written as:
Figure FDA0003976047800000042
to eliminate the pair x i Is constrained, defines variables
Figure FDA0003976047800000043
Comprises the following steps:
Figure FDA0003976047800000044
wherein,
Figure FDA0003976047800000045
is relative to->
Figure FDA0003976047800000046
Is defined as:
Figure FDA0003976047800000047
in the formula,
Figure FDA0003976047800000048
Figure FDA0003976047800000049
is an intermediate variable, ε is a sufficiently small positive number;
in trajectory tracking, the feedback controller shares the desired state information of manual operation, so that for the feedback controller, in the case of free operation, the desired position q is d (t) is known and can be represented by u h Calculating;
it should be noted that it is preferable that,
Figure FDA00039760478000000410
is a smoothing function>
Figure FDA00039760478000000411
Is less than 0,j e {1,2,. Ang., n }, and therefore, is greater than or equal to>
Figure FDA00039760478000000412
And &>
Figure FDA00039760478000000413
Is present as
Figure FDA00039760478000000414
Wherein,
Figure FDA00039760478000000415
in that
Figure FDA00039760478000000416
In space, on hypothesis>
Figure FDA00039760478000000417
Is omega h In group i constraint, is greater than>
Figure FDA00039760478000000418
In the secure subset R s A mapping of (5) is indicated as->
Figure FDA00039760478000000419
And is defined as:
Figure FDA00039760478000000420
thus, for the ith set of constraints, Ω h In the secure subset R s The mapping in (1) is defined as:
Figure FDA00039760478000000421
an error of a first derivative of a position vector defining a joint is
Figure FDA0003976047800000051
By a variable z i And &>
Figure FDA0003976047800000052
Obtaining a system error model:
Figure FDA0003976047800000053
wherein
Figure FDA0003976047800000054
Figure FDA0003976047800000055
Wherein diag refers to a diagonal matrix, and the virtual control input is designed according to a frame designed by a classical backstepping method
Figure FDA0003976047800000056
Comprises the following steps:
Figure FDA0003976047800000057
wherein alpha is 1 ＞0，α 2 ＞0，β＝[β 1 ,β 2 ,...,β n ] T Is defined as:
Figure FDA0003976047800000058
in the formula,
Figure FDA0003976047800000059
is->
Figure FDA00039760478000000510
Is on the jth row of (1), and->
Figure FDA00039760478000000511
Figure FDA00039760478000000512
ε z Is a normal number with a smaller value; furthermore, a time derivative of the virtual control input (20) is determined:
Figure FDA00039760478000000513
wherein
Figure FDA00039760478000000514
Comprises the following steps:
Figure FDA00039760478000000515
definition of
Figure FDA00039760478000000516
The error model (19) can be expressed as:
Figure FDA00039760478000000517
further consider errors caused by TED schemes
Figure FDA00039760478000000518
Figure FDA00039760478000000519
Is the upper limit of the error, and then based on the model (22), the adaptive fixed-time feedback control is designed to:
Figure FDA0003976047800000061
Figure FDA0003976047800000062
Figure FDA0003976047800000063
wherein k is 1 (0) > 0 and k 2 (0)＞0；k 0 ，κ 1 ，κ 2 ，ξ 1 ，ξ 2 ，σ 1 ，σ 2 ，u 1 Are all normal numbers.
6. The design method of the TDE-based obstacle avoidance controller for the constraint cable-driven mechanical arm, as recited in claim 5, wherein: the specific implementation of designing the shared controller is as follows;
with reference to the i-th set of n constraints defined in (11), the state space can be divided into three subsets by (12), in order to eliminate ambiguity of different sets of constraints
Figure FDA0003976047800000064
"push" back a subset>
Figure FDA0003976047800000065
Coordinates; />
Thus, a structure consistent with the overall feasible state space is constructed
Figure FDA0003976047800000066
This arrangement is relevant for any fixed->
Figure FDA0003976047800000067
Also in the ith group of constraints, is the union of the safety, lag and hazard sets, i.e., S i q+T i Less than 0; a safety set of different constraint groups is then defined, with the hysteresis set and the hazard set ≧>
Figure FDA0003976047800000068
R h ＝R-R d -R s And
Figure FDA0003976047800000069
based on three subsets, in
Figure FDA00039760478000000610
Defining shared control input ≥ in coordinates>
Figure FDA00039760478000000611
Comprises the following steps:
Figure FDA00039760478000000612
in the formula, feedback sharing function
Figure FDA00039760478000000613
Is defined as:
Figure FDA00039760478000000614
wherein
Figure FDA00039760478000000615
/>
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