CN105171758A - Self-adaptive finite time convergence sliding-mode control method of robot - Google Patents

Abstract

The invention relates to a self-adaptive finite time convergence sliding-mode control method of a robot and belongs to the technical field of control. The method comprises steps as follows: establishing a robot kinetic equation considering uncertainty, determining a sliding mode surface, then introducing n-dimensional self-adaptive updating rate, calculating value of active power/moment required for driving the robot finally, and driving a robot system based on the value so as to enable generalized coordinate vectors of the robot to converge to a steady-state or trace command signals in finite time. The method has the characteristics and benefits as follows: firstly, the problem of chattering of the sliding-mode control is solved greatly while high-precision control is realized; secondly, possible influence of different dynamic characteristics in all directions of freedom degrees of the robot system can be eliminated; thirdly, the problem of moment saturation and chattering of the robot system at the starting stage can be solved; fourthly, kinetic compensation guaranteeing real-time performance of the control can be realized. The method is applicable to the robot systems with structure types of parallel connection, series connection, series-parallel connection and the like.

Description

A kind of self adaptation finite time convergence control sliding-mode control of robot

Technical field

The invention belongs to control technology field, particularly a kind of self adaptation finite time convergence control sliding-mode control of robot, is suitable for the robust control method containing probabilistic robot.

Background technology

General robot system, comprises the robot system of the structure types such as parallel robot, serial manipulator and series-parallel robot.The generalized coordinates vector q of the final purpose Shi Shi robot that robot system controls can reach stable state or trace command signal q _d; For realizing this object, need to obtain the generalized coordinates vector q of robot system reality and the generalized velocity vector of robot system reality value, and then calculate the value of active force needed for driven machine people or moment according to concrete control method, and then driven machine people completes the operational motion of needs.

Robot system belongs to typical complicated MIMO nonlinear systems, due to the simplification of its Dynamic Modeling process, make inevitably there is probabilistic impact, comprising: friction, external disturbance, noise, systematic parameter fluctuation, fault, Unmarried pregnancy etc.Part uncertainty for robot system in prior art carried out analysis, but considered that the analysis of fault effects is then relatively deficient.Before, the applicant has carried out detailed analysis (see the applicant's pertinent literature MengQiang for the relevant uncertainty of parallel robot system, ZhangTao, GaoXiang, etal. " Adaptiveslidingmodefault ?tolerantcontrolofuncertainStewartplatformbasedonoff ?linemulti ?bodydynamics ", IEEE/ASMETransactionsonMechatronics, 2014:19 (3): 882 ?894; QiangMeng, TaoZhang, Jing ?yanSong, " Adaptiveslidingmodecontrolfortrajectorytrackingofuncerta inStewartplatformbasedonpositionmeasurementsonly; " Robotica, accepted.).

Above-mentioned document adopts Kane method, sets up the kinetics equation of parallel robot system such as formula (1.1):

$M\left(q\right)\stackrel{\·\·}{q}+C(q,\stackrel{\·}{q})+G\left(q\right)=\mathrm{\τ}---\left(1.1\right)$

Wherein, M (q) ties up matrix for parallel robot system quality n × n, is expressed as M (q) ∈ R ^{n × n}(below by * general reference, as * ∈ R ^{n × n}representing matrix * belongs to n × n and ties up matrix); for parallel robot system Ge Shi/centripetal term coefficient vector; G (q) ∈ R ^{n × 1}for parallel robot system gravity item vector; τ ∈ R ^{n × 1}for parallel robot system active force/moment; Q ∈ R ^{n × 1}for parallel robot system generalized coordinates vector; for parallel robot system generalized velocity vector; represent parallel robot system generalized acceleration vector; N represents parallel robot system dimension, and is positive integer.

(consider that mechanical system may exist the impacts such as friction, external disturbance, noise, systematic parameter fluctuation, fault, Unmarried pregnancy, its kinetics equation can not completely accurately, therefore) formula (1.1) being revised as the probabilistic parallel robot kinetics equation of consideration is formula (1.2):

$(M_0+\mathrm{\Δ}M)\stackrel{\·\·}{q}+(C_0+\mathrm{\Δ}C)+(G_0+\mathrm{\Δ}G)={\mathrm{\δ}}_a(\mathrm{\τ}+{\mathrm{\δ}}_f)---\left(1.2\right)$

Wherein, M ₀∈ R ^{n × n}, C ₀∈ R ^{n × 1}, G ₀∈ R ^{n × 1}represent respectively in parallel robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; Δ M ∈ R ^{n × n}, Δ C ∈ R ^{n × 1}, Δ G ∈ R ^{n × 1}then represent that (Δ M, Δ C, Δ G characterize the uncertain size that parallel robot system exists, generally can computational item M for the indeterminate of parallel robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector respectively ₀, C ₀, G ₀40% within); δ _a∈ R ^{n × n}represent multiplicative fault (δ _a,isize determine according to the degree of actuator failures, effective span is [0,1], δ _a,i=1 represents i-th actuator fault-free, δ _a,i=0 represents that i-th actuator is entirely ineffective, 0 < δ _a,i< 1 represents i-th actuator partial failure); δ _f∈ R ^{n × 1}for additivity fault (δ _f,isize determine according to the degree of actuator failures, effective range must be less than maximum, force or the moment of actuator); (indeterminate Δ M, Δ C, the producing cause of Δ G has: in actual physics system, and friction, external disturbance, noise, systematic parameter fluctuation etc. all can cause parallel robot system kinetics equation to there is indeterminate; In modeling analysis process, in order to simplify calculating, generally only considering the dominant dynamic characteristic of system, therefore, the indeterminate that Unmarried pregnancy is introduced can be there is.δ _aand δ _fproducing cause be mainly the induced fault of system actuators);

Formula (1.2) arranges further as formula (1.3)

$M_0\stackrel{\&CenterDot;\&CenterDot;}{q}+C_0+G_0=\mathrm{\&tau;}+{\mathrm{\&delta;}}_{total}---\left(1.3\right)$

Wherein, δ _total∈ R ^{n × 1}for total indeterminate is such as formula (1.4):

${\mathrm{\&delta;}}_{total}=\left({\mathrm{\&delta;}}_a\right(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)-\mathrm{\&tau;})-\mathrm{\&Delta;}G-\mathrm{\&Delta;}C-\mathrm{\&Delta;}M\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(1.4\right)$

If the constraint of robot indeterminate (pertinent literature is visible: Liu Jinkun. and Sliding mode variable structure control MATLAB emulates. publishing house of Tsing-Hua University: 386 ?397.), as formula (1.5)

$\left|\right|\mathrm{\&rho;}\left|\right|<b_1+b_2\left|\right|q\left|\right|+b_3\left|\right|\stackrel{\&CenterDot;}{q}|{|}^2---\left(1.5\right)$

Wherein, ρ ∈ R ^{n × 1}for rigid machine people indeterminate; b ₁, b ₂, b ₃for positive number (b ₁, b ₂, b ₃total indeterminate ρ of existing according to system of value determine; When different when, meet the b of constraint formulations (1.5) ₁, b ₂, b ₃minimum of a value may change, therefore b in actual applications ₁, b ₂, b ₃be generally unknown, a span roughly can be estimated at most); || * || for getting norm operation; (*) ²for ask * square;

For the robot system of n dimension, the characteristic in its each free degree direction may have larger difference, therefore, formula (1.5) is comparatively conservative for the estimation of systematic uncertainty, is difficult to the control accuracy and the requirement of moment flatness that meet each free degree different qualities robot system.

Finite time convergence control (Terminal) sliding formwork controls, and due to the characteristic of its finite time convergence control, is widely used in industrial circle.At present, conventional Terminal sliding formwork control also to be generalized to robot field (pertinent literature is visible: Liu Jinkun. Sliding mode variable structure control MATLAB emulates. publishing house of Tsing-Hua University: 386 ?397.).Liu Jinkun professor for finite time convergence control (Terminal) sliding-mode control of Robot Design such as formula (2.1) ?(2.2):

Determine sliding-mode surface S ∈ R ^{n × 1}such as formula (2.1):

$S=\mathrm{\&epsiv;}+\stackrel{\&CenterDot;}{\mathrm{\&Lambda;}}{\mathrm{\&epsiv;}}^{p/q}---\left(2.1\right)$

Wherein, ε=q-q _d∈ R ^{n × 1}represent generalized coordinates error vector, q and q _dfor the generalized coordinates vector of reality and the generalized coordinates vector of expectation; represent * derivative (in such as formula: represent the derivative of ε); P and q is setting value, gets positive odd number and meets Λ=diag [λ ₁... λ _n] for Parameter adjustable joint diagonal matrix; λ ₁... λ _nfor setting value, get positive number; Diag [x, y] represents by element x, the diagonal matrix of y composition;

Calculate the value of the active force/moment needed for driven machine people such as formula (2.2):

$\begin{array}{c}\mathrm{\&tau;}=M_0{\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+C_0+G_0-\frac{q}{p}{M}_0{\mathrm{\&Lambda;}}^{-1}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{2-p/q}\right)\\ -\frac{{\&lsqb;s^T\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_0^{-1}\&rsqb;}^T}{\left|\right|s^T\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_0^{-1}|{|}^2}\left|\right|s\left|\right|\left|\right|\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_0^{-1}\left|\right|\left(b_1+b_2\left|\right|q\left|\right|+b_3\left|\right|\stackrel{\&CenterDot;}{q}|{|}^2\right)\end{array}---\left(2.2\right)$

In formula, τ ∈ R ^{n × 1}for the robot active force/moment calculated; for the generalized acceleration vector that robot system is expected;

Summary of the invention

The object of the invention is to overcome above-mentioned the deficiencies in the prior art, a kind of self adaptation finite time convergence control (Terminal) sliding-mode control of robot is provided.The method, on the basis that conventional finite time Convergence (Terminal) sliding formwork controls, introduces adaptive updates rate, while realizing high accuracy control, dramatically reduces the electric shock problem that sliding formwork controls; By determining the sliding-mode surface that n ties up and proposing the adaptive updates rate of n dimension, can be used to solve each free degree direction of robot system different dynamic properties influence that may exist; The adaptive updates rate of the n dimension proposed introduces a function increased progressively in time at the effect initial stage, with solve robot system startup stage the saturated and Flutter Problem of moment; Propose in several Practical Project, in order to ensure the Dynamics Compensation scheme of realtime control.

The self adaptation finite time convergence control sliding-mode control of a kind of robot that the present invention proposes, it is characterized in that, be applicable to the robot system of parallel robot, serial manipulator and series-parallel robot structure type, the method comprises the following steps:

1) the probabilistic robot dynamics's equation of consideration is set up:

Adopt traditional power Epidemiological Analysis method, set up the kinetics equation of robot system such as formula (1):

$M\left(q\right)\stackrel{\&CenterDot;\&CenterDot;}{q}+C(q,\stackrel{\&CenterDot;}{q})+G\left(q\right)=\mathrm{\&tau;}---\left(1\right)$

Wherein, M (q) ∈ R ^{n × n}for robot system mass matrix; for robot system Ge Shi/centripetal term coefficient vector; G (q) ∈ R ^{n × 1}for robot system gravity item vector; τ ∈ R ^{n × 1}for robot system active force/moment; Q ∈ R ^{n × 1}for robot system generalized coordinates vector; for robot system generalized velocity vector; represent robot system generalized acceleration vector; N represents robot system dimension, and is positive integer;

Formula (1) is introduced indeterminate, is revised as and considers that probabilistic robot dynamics's equation is formula (2):

$(M_0+\mathrm{\&Delta;}M)\stackrel{\&CenterDot;\&CenterDot;}{q}+(C_0+\mathrm{\&Delta;}C)+(G_0+\mathrm{\&Delta;}G)={\mathrm{\&delta;}}_a(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)---\left(2\right),$

Wherein, M ₀∈ R ^{n × n}, C ₀∈ R ^{n × 1}, G ₀∈ R ^{n × 1}represent respectively in robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; Δ M ∈ R ^{n × n}, Δ C ∈ R ^{n × 1}, Δ G ∈ R ^{n × 1}then represent the indeterminate of robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector respectively; δ _a∈ R ^{n × n}represent multiplicative fault; δ _f∈ R ^{n × 1}for additivity fault;

Formula (2) arranges as formula (3) further:

$M_0\stackrel{\&CenterDot;\&CenterDot;}{q}+C_0+G_0=\mathrm{\&tau;}+{\mathrm{\&delta;}}_{total}---\left(3\right),$

Wherein, δ _total∈ R ^{n × 1}for total indeterminate is such as formula (4):

${\mathrm{\&delta;}}_{total}=\left({\mathrm{\&delta;}}_a\right(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)-\mathrm{\&tau;})-\mathrm{\&Delta;}G-\mathrm{\&Delta;}C-\mathrm{\&Delta;}M\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(4\right),$

Consider the dynamics of robot system, this total indeterminate δ _totalmeet constraint formula (5):

$\left|{\mathrm{\&delta;}}_{total,i}\right|<b_{1,i}+b_{2,i}\left|q_i\right|+b_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(5\right),$

Wherein, δ _{total, i}∈ R represents n n dimensional vector n δ _totali-th element (* ∈ R represents that * is real number, such as, in formula (5): δ _{total, i}∈ R represents δ _{total, i}for real number; (*) _irepresent i-th element of (*)); b _{1, i}, b _{2, i}, b _{3, i}for positive number (b _{1, i}, b _{2, i}, b _{3, i}total indeterminate δ of existing according to system of value _{total, i}determine); | * | be the operation that takes absolute value; (*) ²for ask * square;

2) sliding-mode surface S ∈ R is determined ^{n × 1}such as formula (6):

$S=\mathrm{\&epsiv;}+\mathrm{\&Lambda;}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q}---\left(6\right),$

Wherein, ε=q-q _d∈ R ^{n × 1}represent generalized coordinates error vector, q and q _dfor the generalized coordinates vector of reality and the generalized coordinates vector of expectation; represent the derivative of *; P and q is setting value, gets positive odd number and meets Λ=diag [λ ₁... λ _n] for Parameter adjustable joint diagonal matrix; λ ₁... λ _nfor setting value, get positive number; Diag [x, y] represents by element x, the diagonal matrix of y composition;

3) the adaptive updates rate of n dimension is introduced such as formula (7) ?(11):

${\widehat{\mathrm{\&Gamma;}}}_{adaptive,i}=\frac{2}{\mathrm{\&pi;}}arctan\left(c_{i}t\right)({\hat{b}}_{1,i}+{\hat{b}}_{2,i}|q_i|+{\hat{b}}_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2)---\left(7\right),$

$W=\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_0^{-1}---\left(8\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{1,i}=d_{1,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|---\left(9\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{2,i}=d_{2,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|\left|q_i\right|---\left(10\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{3,i}=d_{3,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right||{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(11\right),$

Formula (7) is made up of two parts: be one to increase progressively in time and the function of bounded, the saturated and Flutter Problem by the moment startup stage of solving robot system; then be used for estimating the total indeterminate δ of robot system adaptively _{total, i}upper dividing value;

Wherein, for adaptive estimate; d _{1, i}∈ R, d _{2, i}∈ R, d _{3, i}∈ R is the setting value in i-th free degree direction, and gets positive number; Arctan (*) is arctan function; c _ibe the setting value in i-th free degree direction, and get positive number; π is pi; T is time variable; for compute matrix; || * || be norm operation; (*) ^-1representing matrix (*) inverse; represent the derivative of *;

4) value of the active force/moment needed for driven machine people is calculated:

According to the generalized coordinates vector q of robot system reality and the generalized velocity vector of robot system reality value, in on-line computer device people dynamics can computational item M ₀, C ₀, G ₀value; Then, according to step 1) ?3) in the probabilistic robot dynamics's equation of consideration, constraint, sliding-mode surface and adaptive updates rate, calculate the value of the active force/moment needed for driven machine people such as formula (12):

${\mathrm{\&tau;}}_{antsmc}=M_0{\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+C_0+G_0-\frac{q}{p}{M}_0{\mathrm{\&Lambda;}}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{2-p/q}-\frac{diag\left(s^{T}W\right)}{{\left(\right|\left|s^{T}W\right||+\mathrm{\&xi;})}^2}\left|\right|s\left|\right|\left|\right|W\left|\right|{\widehat{\mathrm{\&Gamma;}}}_{adaptive}---\left(12\right)$

In formula, τ _antsmc∈ R ^{n × 1}for the robot active force/moment calculated; for the generalized acceleration vector that robot system is expected; ξ ∈ R is setting value, gets less positive number; (*) ^tfor getting the transposition of matrix/vector (*);

With active force/moment (12) driven machine robot system, the generalized coordinates of robot vector q is made to realize Finite-time convergence to stable state or trace command signal q _d.

The self adaptation finite time convergence control sliding-mode control of the second robot that the present invention proposes, it is characterized in that, be applicable to the robot system of parallel robot, serial manipulator and series-parallel robot structure type, the method comprises the following steps:

1) the probabilistic robot dynamics's equation of consideration is set up:

Adopt traditional power Epidemiological Analysis method, set up the kinetics equation of robot system such as formula (1):

$M\left(q\right)\stackrel{\&CenterDot;\&CenterDot;}{q}+C(q,\stackrel{\&CenterDot;}{q})+G\left(q\right)=\mathrm{\&tau;}---\left(1\right)$

Wherein, M (q) ∈ R ^{n × n}for robot system mass matrix; for robot system Ge Shi/centripetal term coefficient vector; G (q) ∈ R ^{n × 1}for robot system gravity item vector; τ ∈ R ^{n × 1}for robot system active force/moment; Q ∈ R ^{n × 1}for robot system generalized coordinates vector; for robot system generalized velocity vector; represent robot system generalized acceleration vector; N represents robot system dimension, and is positive integer;

Formula (1) is introduced indeterminate, is revised as and considers that probabilistic robot dynamics's equation is formula (2):

$(M_0+\mathrm{\&Delta;}M)\stackrel{\&CenterDot;\&CenterDot;}{q}+(C_0+\mathrm{\&Delta;}C)+(G_0+\mathrm{\&Delta;}G)={\mathrm{\&delta;}}_a(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)---\left(2\right),$

Wherein, M ₀∈ R ^{n × n}, C ₀∈ R ^{n × 1}, G ₀∈ R ^{n × 1}represent respectively in robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; Δ M ∈ R ^{n × n}, Δ C ∈ R ^{n × 1}, Δ G ∈ R ^{n × 1}then represent the indeterminate of robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector respectively; δ _a∈ R ^{n × n}represent multiplicative fault; δ _f∈ R ^{n × 1}for additivity fault;

Formula (2) arranges as formula (3) further:

$M_0\stackrel{\&CenterDot;\&CenterDot;}{q}+C_0+G_0=\mathrm{\&tau;}+{\mathrm{\&delta;}}_{total}---\left(3\right),$

Wherein, δ _total∈ R ^{n × 1}for total indeterminate is such as formula (4):

${\mathrm{\&delta;}}_{total}=\left({\mathrm{\&delta;}}_a\right(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)-\mathrm{\&tau;})-\mathrm{\&Delta;}G-\mathrm{\&Delta;}C-\mathrm{\&Delta;}M\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(4\right),$

Consider the dynamics of robot system, this total indeterminate δ _totalmeet constraint formula (5):

$\left|{\mathrm{\&delta;}}_{total,i}\right|<b_{1,i}+b_{2,i}\left|q_i\right|+b_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(5\right),$

Wherein, δ _{total, i}∈ R represents n n dimensional vector n δ _totali-th element (* ∈ R represents that * is real number, such as, in formula (5): δ _{total, i}∈ R represents δ _{total, i}for real number; (*) _irepresent i-th element of (*)); b _{1, i}, b _{2, i}, b _{3, i}for positive number (b _{1, i}, b _{2, i}, b _{3, i}total indeterminate δ of existing according to system of value _{total, i}determine); | * | be the operation that takes absolute value; (*) ²for ask * square;

2) sliding-mode surface S ∈ R is determined ^{n × 1}such as formula (6):

$S=\mathrm{\&epsiv;}+\stackrel{\&CenterDot;}{\mathrm{\&Lambda;}}{\mathrm{\&epsiv;}}^{p/q}---\left(6\right),$

Wherein, ε=q-q _d∈ R ^{n × 1}represent generalized coordinates error vector, q and q _dfor the generalized coordinates vector of reality and the generalized coordinates vector of expectation; represent the derivative of *; P and q is setting value, gets positive odd number and meets Λ=diag [λ ₁... λ _n] for Parameter adjustable joint diagonal matrix; λ ₁... λ _nfor setting value, get positive number; Diag [x, y] represents by element x, the diagonal matrix of y composition;

3) the adaptive updates rate of n dimension is introduced such as formula (7) ?(11):

${\widehat{\mathrm{\&Gamma;}}}_{adaptive,i}=\frac{2}{\mathrm{\&pi;}}arctan\left(c_{i}t\right)({\hat{b}}_{1,i}+{\hat{b}}_{2,i}|q_i|+{\hat{b}}_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2)---\left(7\right),$

$W=\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_0^{-1}\left(t_{N-1}\right)---{\left(8\right)}^{,},$

${\stackrel{\&CenterDot;}{\hat{b}}}_{1,i}=d_{1,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|---\left(9\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{2,i}=d_{2,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|\left|q_i\right|---\left(10\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{3,i}=d_{3,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right||{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(11\right),$

Formula (7) is made up of two parts: be one to increase progressively in time and the function of bounded, the saturated and Flutter Problem by the moment startup stage of solving robot system; then be used for estimating the total indeterminate δ of robot system adaptively _{total, i}upper dividing value;

for adaptive estimate; d _{1, i}∈ R, d _{2, i}∈ R, d _{3, i}∈ R is the setting value in i-th free degree direction, and gets positive number; Arctan (*) is arctan function; c _ibe the setting value in i-th free degree direction, and get positive number; π is pi; T is time variable; for compute matrix; || * || be norm operation; (*) ^-1representing matrix (*) inverse; represent the derivative of *; M ₀(t _n-1) represent at t _n-1moment robot system mass matrix can computational item; t _n-1for system dynamics calculates the setting value in moment; t _n-1middle subscript N is positive integer and value is (as N=1, t _n-1=t ₀, represent initial time); t _ffor the moment of robot system stop motion; INT (*) represents floor operation, namely gets the maximum integer part being no more than real number (*); Δ t=t _n-t _n-1for the setting value of interval time, get Δ t>=t _c;

4) value of the active force/moment needed for driven machine people is calculated:

According to the generalized coordinates vector q of robot system reality and the generalized velocity vector of robot system reality value, adopt separated in time (interval time is Δ t=t _n-t _n-1) in calculating robot's dynamics can computational item (M ₀(t _n-1), C ₀(t _n-1), G ₀(t _n-1)) value; Then, according to step 1) ?3) in the probabilistic robot dynamics's equation of consideration, constraint, sliding-mode surface and adaptive updates rate, calculate the value of the active force/moment needed for driven machine people such as formula (12) ' shown in:

Work as t _n-1+ t _c≤ t < t _n+ t _ctime

${\mathrm{\&tau;}}_{antsmc}=M_0\left(t_{N-1}\right){\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+C_0\left(t_{N-1}\right)+G_0\left(t_{N-1}\right)-\frac{q}{p}{M}_0\left(t_{N-1}\right){\mathrm{\&Lambda;}}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{2-p/q}-\frac{diag\left(s^{T}W\right)}{{\left(\right|\left|s^{T}W\right||+\mathrm{\&xi;})}^2}\left|\right|s\left|\right|\left|\right|W\left|\right|{\widehat{\mathrm{\&Gamma;}}}_{adaptive}---{\left(12\right)}^{,}$

In formula, C ₀(t _n-1), G ₀(t _n-1) be illustrated respectively in t _n-1in moment robot system Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; t _crepresent in calculating robot mass of system matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item required time (its concrete numerical value is relevant with dynamic analysis precision and hardware platform); t _nfor system dynamics calculates the setting value in moment; τ _antsmc∈ R ^{n × 1}for the robot active force/moment calculated; for the generalized acceleration vector that robot system is expected; ξ ∈ R is setting value, gets less positive number; (*) ^tfor getting the transposition of matrix/vector (*);

With active force/moment (12) ' driven machine robot system, the generalized coordinates of robot vector q is made to realize Finite-time convergence to stable state or trace command signal q _d.

The self adaptation finite time convergence control sliding-mode control of the third robot that the present invention proposes, it is characterized in that, be applicable to the robot system of parallel robot, serial manipulator and series-parallel robot structure type, the method comprises the following steps:

1) the probabilistic robot dynamics's equation of consideration is set up:

Adopt traditional power Epidemiological Analysis method, set up the kinetics equation of robot system such as formula (1):

$M\left(q\right)\stackrel{\&CenterDot;\&CenterDot;}{q}+C(q,\stackrel{\&CenterDot;}{q})+G\left(q\right)=\mathrm{\&tau;}---\left(1\right)$

Wherein, M (q) ∈ R ^{n × n}for robot system mass matrix; for robot system Ge Shi/centripetal term coefficient vector; G (q) ∈ R ^{n × 1}for robot system gravity item vector; τ ∈ R ^{n × 1}for robot system active force/moment; Q ∈ R ^{n × 1}for robot system generalized coordinates vector; for robot system generalized velocity vector; represent robot system generalized acceleration vector; N represents robot system dimension, and is positive integer;

Formula (1) is introduced indeterminate, is revised as and considers that probabilistic robot dynamics's equation is formula (2):

$(M_0+\mathrm{\&Delta;}M)\stackrel{\&CenterDot;\&CenterDot;}{q}+(C_0+\mathrm{\&Delta;}C)+(G_0+\mathrm{\&Delta;}G)={\mathrm{\&delta;}}_a(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)---\left(2\right),$

Wherein, M ₀∈ R ^{n × n}, C ₀∈ R ^{n × 1}, G ₀∈ R ^{n × 1}represent respectively in robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; Δ M ∈ R ^{n × n}, Δ C ∈ R ^{n × 1}, Δ G ∈ R ^{n × 1}then represent the indeterminate of robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector respectively; δ _a∈ R ^{n × n}represent multiplicative fault; δ _f∈ R ^{n × 1}for additivity fault;

Formula (2) arranges as formula (3) further:

$M_0\stackrel{\&CenterDot;\&CenterDot;}{q}+C_0+G_0=\mathrm{\&tau;}+{\mathrm{\&delta;}}_{total}---\left(3\right),$

Wherein, δ _total∈ R ^{n × 1}for total indeterminate is such as formula (4):

${\mathrm{\&delta;}}_{total}=\left({\mathrm{\&delta;}}_a\right(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)-\mathrm{\&tau;})-\mathrm{\&Delta;}G-\mathrm{\&Delta;}C-\mathrm{\&Delta;}M\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(4\right),$

Consider the dynamics of robot system, this total indeterminate δ _totalmeet constraint formula (5):

$\left|{\mathrm{\&delta;}}_{total,i}\right|<b_{1,i}+b_{2,i}\left|q_i\right|+b_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(5\right),$

Wherein, δ _{total, i}∈ R represents n n dimensional vector n δ _totali-th element (* ∈ R represents that * is real number, such as, in formula (5): δ _{total, i}∈ R represents δ _{total, i}for real number; (*) _irepresent i-th element of (*)); b _{1, i}, b _{2, i}, b _{3, i}for positive number (b _{1, i}, b _{2, i}, b _{3, i}total indeterminate δ of existing according to system of value _{total, i}determine); | * | be the operation that takes absolute value; (*) ²for ask * square;

2) sliding-mode surface S ∈ R is determined ^{n × 1}such as formula (6):

$S=\mathrm{\&epsiv;}+\mathrm{\&Lambda;}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q}---\left(6\right),$

Wherein, ε=q-q _d∈ R ^{n × 1}represent generalized coordinates error vector, q and q _dfor the generalized coordinates vector of reality and the generalized coordinates vector of expectation; represent the derivative of *; P and q is setting value, gets positive odd number and meets Λ=diag [λ ₁... λ _n] for Parameter adjustable joint diagonal matrix; λ ₁... λ _nfor setting value, get positive number; Diag [x, y] represents by element x, the diagonal matrix of y composition;

3) the adaptive updates rate of n dimension is introduced such as formula (7) ?(11):

${\widehat{\mathrm{\&Gamma;}}}_{adaptive,i}=\frac{2}{\mathrm{\&pi;}}arctan\left(c_{i}t\right)({\hat{b}}_{1,i}+{\hat{b}}_{2,i}|q_i|+{\hat{b}}_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2)---\left(7\right),$

$W=\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_{0,d}^{-1}---{\left(8\right)}^{,,},$

${\stackrel{\&CenterDot;}{\hat{b}}}_{1,i}=d_{1,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|---\left(9\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{2,i}=d_{2,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|\left|q_i\right|---\left(10\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{3,i}=d_{3,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right||{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(11\right),$

Formula (7) is made up of two parts: be one to increase progressively in time and the function of bounded, the saturated and Flutter Problem by the moment startup stage of solving robot system; then be used for estimating the total indeterminate δ of robot system adaptively _{total, i}upper dividing value;

Wherein, for adaptive estimate; d _{1, i}∈ R, d _{2, i}∈ R, d _{3, i}∈ R is the setting value in i-th free degree direction, and gets positive number; Arctan (*) is arctan function; c _ibe the setting value in i-th free degree direction, and get positive number; π is pi; T is time variable; for compute matrix; || * || be norm operation; (*) ^-1representing matrix (*) inverse; represent the derivative of *; M _{0, d}what represent the robot system mass matrix in the corresponding moment of off-line calculation can computational item;

4) value of the active force/moment needed for driven machine people is calculated:

According to the generalized coordinates vector q that robot system is expected _dand the generalized velocity vector that robot system is expected value, first off-line in calculating robot's dynamics can computational item (M _{0, d}, C _{0, d}, G _{0, d}) value, and result to be stored, when actual motion, directly calls (the M in corresponding moment _{0, d}, C _{0, d}, G _{0, d}) value; Then, according to step 1) ?3) in the probabilistic robot dynamics's equation of consideration, constraint, sliding-mode surface and adaptive updates rate, calculate the value of the active force/moment needed for driven machine people such as formula (12) " shown in:

${\mathrm{\&tau;}}_{antsmc}=M_{0,d}{\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+C_{0,d}+G_{0,d}-\frac{q}{p}{M}_{0,d}{\mathrm{\&Lambda;}}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{2-p/q}-\frac{diag\left(s^{T}W\right)}{{\left(\right|\left|s^{T}W\right||+\mathrm{\&xi;})}^2}\left|\right|s\left|\right|\left|\right|W\left|\right|{\widehat{\mathrm{\&Gamma;}}}_{adaptive}---{\left(12\right)}^{,,},$

In formula, C _{0, d}, G _{0, d}represent respectively in the robot system Ge Shi in the corresponding moment of off-line calculation/centripetal term coefficient vector, gravity item vector can computational item; τ _antsmc∈ R ^{n × 1}for the robot active force/moment calculated; for the generalized acceleration vector that robot system is expected; ξ ∈ R is setting value, gets less positive number; (*) ^tfor getting the transposition of matrix/vector (*);

With active force/moment (12) " driven machine robot system, make the generalized coordinates of robot vector q realize Finite-time convergence to stable state or trace command signal q _d.

Compared with prior art, feature of the present invention and beneficial effect are:

One, after the present invention considers many uncertainties, particularly fault effects that robot exists, robot is uncertain may there is larger variation along with the time.On the basis that conventional finite time Convergence (Terminal) sliding formwork controls, introduce adaptive updates rate, while realizing high accuracy control, dramatically reduce the electric shock problem that sliding formwork controls;

Its two: by determining the sliding-mode surface that n ties up and proposing the adaptive updates rate of n dimension, can be used to solve each free degree direction of robot system different dynamic properties influence that may exist;

Its three, n dimension proposed adaptive updates rate introduce one at the effect initial stage and increase progressively in time and the function of bounded, with solve robot system startup stage the saturated and Flutter Problem of moment;

Its four, propose in several Practical Project, in order to ensure the Dynamics Compensation scheme of realtime control.

Relevant achievement in research is extended to general robot field by the present invention, is applicable to the robot system of the structure types such as parallel robot, serial manipulator and series-parallel robot.

Accompanying drawing explanation

Fig. 1 is the realization flow block diagram of the method for the invention;

Fig. 2 is the routine 2 side chain mechanical arm sketch of this method embodiment.

Fig. 3 is the tracking error curve of joint 1 under the self adaptation finite time convergence control sliding-mode control " C3 " that traditional finite time convergence control sliding-mode control " C1 ", traditional finite time convergence control sliding-mode control " C2 " and the present invention propose acts on of the present embodiment;

Fig. 4 is the tracking error curve of joint 2 under the self adaptation finite time convergence control sliding-mode control " C3 " that traditional finite time convergence control sliding-mode control " C1 ", traditional finite time convergence control sliding-mode control " C2 " and the present invention propose acts on of the present embodiment;

Fig. 5 is traditional finite time convergence control sliding-mode control " C1 " at the control inputs at joint 1 and joint 2 place;

Fig. 6 is traditional finite time convergence control sliding-mode control " C2 " at the control inputs at joint 1 and joint 2 place;

Fig. 7 is the self adaptation finite time convergence control sliding-mode control " C3 " that proposes of the present invention at the control inputs at joint 1 and joint 2 place.

Detailed description of the invention

Below in conjunction with drawings and Examples, the present invention is further described.

The self adaptation finite time convergence control sliding-mode control of a kind of robot that the present invention proposes, it is characterized in that, be applicable to the robot system of parallel robot, serial manipulator and series-parallel robot structure type, the method comprises the following steps:

1) the probabilistic robot dynamics's equation of consideration is set up:

Adopt traditional power Epidemiological Analysis method (traditional dynamic analysis method adopt following any one: Kane method, Niu Dun ?Euler method, Niu Dun ?Lagrangian method, the principle of virtual work etc.), set up the kinetics equation of robot system such as formula (1):

$M\left(q\right)\stackrel{\&CenterDot;\&CenterDot;}{q}+C(q,\stackrel{\&CenterDot;}{q})+G\left(q\right)=\mathrm{\&tau;}---\left(1\right)$

Wherein, M (q) ∈ R ^{n × n}for robot system mass matrix; for robot system Ge Shi/centripetal term coefficient vector; G (q) ∈ R ^{n × 1}for robot system gravity item vector; τ ∈ R ^{n × 1}for robot system active force/moment; Q ∈ R ^{n × 1}for robot system generalized coordinates vector; for robot system generalized velocity vector; represent robot system generalized acceleration vector; N represents robot system dimension, and is positive integer;

(consider that mechanical system may exist the impacts such as friction, external disturbance, noise, systematic parameter fluctuation, fault, Unmarried pregnancy, its kinetics equation can not completely accurately, therefore,) formula (1) is introduced indeterminate, be revised as and consider that probabilistic robot dynamics's equation is formula (2):

$(M_0+\mathrm{\&Delta;}M)\stackrel{\&CenterDot;\&CenterDot;}{q}+(C_0+\mathrm{\&Delta;}C)+(G_0+\mathrm{\&Delta;}G)={\mathrm{\&delta;}}_a(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)---\left(2\right)$

Wherein, M ₀∈ R ^{n × n}, C ₀∈ R ^{n × 1}, G ₀∈ R ^{n × 1}represent respectively in robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; Δ M ∈ R ^{n × n}, Δ C ∈ R ^{n × 1}, Δ G ∈ R ^{n × 1}then represent the indeterminate (the uncertain size that Δ M, Δ C, Δ G characterization system exists) of robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector respectively; δ _a∈ R ^{n × n}represent multiplicative fault (δ _a,isize determine according to the degree of actuator failures); δ _f∈ R ^{n × 1}for additivity fault (δ _f,isize determine according to the degree of actuator failures);

(indeterminate Δ M, Δ C, the producing cause of Δ G has: in actual physics system, and friction, external disturbance, noise, systematic parameter fluctuation etc. all can cause robot system kinetics equation to there is indeterminate; In modeling analysis process, in order to simplify calculating, generally only considering the dominant dynamic characteristic of system, therefore, the indeterminate that Unmarried pregnancy is introduced can be there is.δ _aand δ _fproducing cause be mainly the induced fault of system actuators);

Formula (2) arranges as formula (3) further:

$M_0\stackrel{\&CenterDot;\&CenterDot;}{q}+C_0+G_0=\mathrm{\&tau;}+{\mathrm{\&delta;}}_{total}---\left(3\right)$

Wherein, δ _total∈ R ^{n × 1}for total indeterminate is such as formula (4):

${\mathrm{\&delta;}}_{total}=\left({\mathrm{\&delta;}}_a\right(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)-\mathrm{\&tau;})-\mathrm{\&Delta;}G-\mathrm{\&Delta;}C-\mathrm{\&Delta;}M\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(4\right)$

Consider the dynamics of robot system, this total indeterminate δ _totalmeet constraint formula (5)

$\left|{\mathrm{\&delta;}}_{total,i}\right|<b_{1,i}+b_{2,i}\left|q_i\right|+b_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(5\right)$

Wherein, δ _{total, i}∈ R represents n n dimensional vector n δ _totali-th element (* ∈ R represents that * is real number, such as, in formula (5): δ _{total, i}∈ R represents δ _{total, i}for real number; (*) _irepresent i-th element of (*)); b _{1, i}, b _{2, i}, b _{3, i}for positive number (b _{1, i}, b _{2, i}, b _{3, i}total indeterminate δ of existing according to system of value _{total, i}determine); | * | be the operation that takes absolute value; (*) ²for ask * square;

(formula (5), for the description of robot system indeterminate, has taken into full account and the different dynamic properties influence that each free degree direction of robot system may exist has greatly reduced the conservative to robot system uncertainty estimation.Certainly, for the system that each free degree directional characteristic difference is little or identical, this constraint formula is applicable equally);

2) sliding-mode surface S ∈ R is determined ^{n × 1}such as formula (6):

$S=\mathrm{\&epsiv;}+\mathrm{\&Lambda;}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q}---\left(6\right)$

Wherein, ε=q-q _d∈ R ^{n × 1}represent generalized coordinates error vector, q and q _dfor the generalized coordinates vector of reality and the generalized coordinates vector of expectation; represent * derivative (in such as formula: represent the derivative of ε); P and q is setting value, gets positive odd number and meets regulate according to actual conditions with q, robot stabilized for value principle realizing); Λ=diag [λ ₁... λ _n] for Parameter adjustable joint diagonal matrix; λ ₁... λ _nfor setting value, get positive number (λ ₁... λ _nneed regulate according to actual conditions, robot stabilized for value principle realizing); Diag [x, y] represents by element x, the diagonal matrix of y composition;

3) the adaptive updates rate of n dimension is introduced such as formula (7) ?(11):

(the present invention realizes high accuracy to control and the electric shock problem alleviating sliding formwork control, on the basis that conventional finite time Convergence (Terminal) sliding formwork controls, introduces adaptive updates rate; For solving each free degree direction of robot system different dynamic properties influence that may exist, adaptive updates rate and sliding-mode surface are designed to n ties up; For solve robot system startup stage the saturated and Flutter Problem of moment, the adaptive updates rate of described n dimension startup stage also introduce one and increase progressively in time and the function of bounded; )

${\widehat{\mathrm{\&Gamma;}}}_{adaptive,i}=\frac{2}{\mathrm{\&pi;}}arctan\left(c_{i}t\right)({\hat{b}}_{1,i}+{\hat{b}}_{2,i}|q_i|+{\hat{b}}_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2)---\left(7\right)$

$W=\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_0^{-1}---\left(8\right)$

${\stackrel{\&CenterDot;}{\hat{b}}}_{1,i}=d_{1,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|---\left(9\right)$

${\stackrel{\&CenterDot;}{\hat{b}}}_{2,i}=d_{2,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|\left|q_i\right|---\left(10\right)$

${\stackrel{\&CenterDot;}{\hat{b}}}_{3,i}=d_{3,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right||{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(11\right)$

Formula (7) is made up of two parts: be one to increase progressively in time and the function of bounded, the saturated and Flutter Problem by the moment startup stage of solving robot system; then be used for estimating the total indeterminate δ of robot system adaptively _{total, i}upper dividing value;

Wherein, for adaptive estimate; d _{1, i}∈ R, d _{2, i}∈ R, d _{3, i}∈ R is the setting value in i-th free degree direction, and gets positive number (d _{1, i}, d _{2, i}, d _{3, i}need regulate according to actual conditions, to ensure that the stable and control effects satisfaction of robot system is for principle); Arctan (*) is arctan function; c _ibe the setting value in i-th free degree direction, and get positive number (c _ineed regulate according to actual conditions, with the moment startup stage of alleviating robot system, saturated and Flutter Problem is for principle); π is pi; T is time variable; for compute matrix; || * || be norm operation; (*) ^-1representing matrix (*) inverse (in such as formula: representing matrix M ₀inverse); represent * derivative (in such as formula: represent derivative);

4) value of the active force/moment needed for driven machine people is calculated:

According to the generalized coordinates vector q of robot system reality and the generalized velocity vector of robot system reality value, in on-line computer device people dynamics can computational item M ₀, C ₀, G ₀value; Then, according to step 1) ?3) in the probabilistic robot dynamics's equation of consideration, constraint, sliding-mode surface and adaptive updates rate, calculate the value of the active force/moment needed for driven machine people such as formula (12):

${\mathrm{\&tau;}}_{antsmc}=M_0{\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+C_0+G_0-\frac{q}{p}{M}_0{\mathrm{\&Lambda;}}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{2-p/q}-\frac{diag\left(s^{T}W\right)}{{\left(\right|\left|s^{T}W\right||+\mathrm{\&xi;})}^2}\left|\right|s\left|\right|\left|\right|W\left|\right|{\widehat{\mathrm{\&Gamma;}}}_{adaptive}---\left(12\right)$

In formula, τ _antsmc∈ R ^{n × 1}for the robot active force/moment calculated; for the generalized acceleration vector that robot system is expected; ξ ∈ R is setting value, gets less positive number (ξ value belongs to prior art, need regulate according to actual conditions, to alleviate the flutter of active force/moment for value principle on the basis ensureing control effects); (*) ^tfor getting the transposition of matrix/vector (*) (in such as formula: s ^ttransposition for amount of orientation s);

With active force/moment (12) driven machine robot system, the generalized coordinates of robot vector q is made to realize Finite-time convergence to stable state or trace command signal q _d.

In described formula (8) and (12), in robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item M ₀, C ₀, G ₀have multiple way of realization, what the most important thing is consideration robot dynamics calculating is the situation that real-time and movement locus change.The method adopted at least comprises several as follows:

The present invention realize above-mentioned steps 3) ?4) second method be:

The method is applicable to system dynamics and cannot calculates in real time and movement locus changes little situation (t _ccomparatively large, and at Δ t>=t interval time _cin inner machine robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item M ₀, C ₀, G ₀change little), be specially:

According to the generalized coordinates vector q of robot system reality and the generalized velocity vector of robot system reality value, adopt separated in time (interval time is Δ t=t _n-t _n-1) in calculating robot's dynamics can computational item (M ₀(t _n-1), C ₀(t _n-1), G ₀(t _n-1)) value; Then, according to step 1) ?3) in the probabilistic robot dynamics's equation of consideration, constraint, sliding-mode surface and adaptive updates rate, calculate the value of the active force/moment needed for driven machine people such as formula (8) ' to (12) ' shown in, (except (8) ' and (12) ' except, all same first method of other formula):

Work as t _n-1+ t _c≤ t < t _n+ t _ctime

$W=\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_0^{-1}\left(t_{N-1}\right)---{\left(8\right)}^{,}$

${\mathrm{\&tau;}}_{antsmc}=M_0\left(t_{N-1}\right){\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+C_0\left(t_{N-1}\right)+G_0\left(t_{N-1}\right)-\frac{q}{p}{M}_0\left(t_{N-1}\right){\mathrm{\&Lambda;}}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{2-p/q}-\frac{diag\left(s^{T}W\right)}{{\left(\right|\left|s^{T}W\right||+\mathrm{\&xi;})}^2}\left|\right|s\left|\right|\left|\right|W\left|\right|{\widehat{\mathrm{\&Gamma;}}}_{adaptive}---{\left(12\right)}^{,}$

In formula, for compute matrix; || * || be norm operation; (*) ^-1representing matrix (*) inverse; represent the derivative of *; t _n-1and t _nfor system dynamics calculates the setting value in moment; t _n-1and t _nmiddle subscript N is positive integer and value is (as N=1, t _n-1=t ₀, represent initial time); t _ffor the moment of robot system stop motion; INT (*) represents floor operation, namely gets the maximum integer part being no more than real number (*); Δ t=t _n-t _n-1for the setting value of interval time (gets Δ t>=t _c, and interval time Δ t inner machine robot system mass matrix, Ge Shi/centripetal term coefficient vector, in gravity item vector can computational item M ₀, C ₀, G ₀change little); M ₀(t _n-1), C ₀(t _n-1), G ₀(t _n-1) be illustrated respectively in t _n-1in moment robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; t _crepresent in calculating robot mass of system matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item required time, its concrete numerical value is relevant with dynamic analysis precision and hardware platform; T represents current time;

If system dynamics cannot calculate and the situation (t that changes greatly of movement locus in real time _ccomparatively large, and at Δ t>=t interval time _cin inner machine robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item M ₀, C ₀, G ₀change greatly), the present invention realize above-mentioned steps 3) ?4) the third method be:

According to the generalized coordinates vector q that robot system is expected _dand the generalized velocity vector that robot system is expected value, first off-line in calculating robot's dynamics can computational item (M _{0, d}, C _{0, d}, G _{0, d}) value, and result to be stored, when actual motion, directly calls (the M in corresponding moment _{0, d}, C _{0, d}, G _{0, d}) value; Then, according to step 1) ?3) in the probabilistic robot dynamics's equation of consideration, constraint, sliding-mode surface and adaptive updates rate, calculate the value of the active force/moment needed for driven machine people such as formula (8) " to (12) " shown in ((except (8) " to (12) " except, all same first method of other formula):

$W=\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_{0,d}^{-1}---{\left(8\right)}^{,,}$

${\mathrm{\&tau;}}_{antsmc}=M_{0,d}\stackrel{\&CenterDot;\&CenterDot;}{q}+C_{0,d}+G_{0,d}-\frac{q}{p}{M}_{0,d}{\mathrm{\&Lambda;}}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{2-p/q}-\frac{diag\left(s^{T}W\right)}{{\left(\right|\left|s^{T}W\right||+\mathrm{\&xi;})}^2}\left|\right|s\left|\right|\left|\right|W\left|\right|{\widehat{\mathrm{\&Gamma;}}}_{adaptive}---{\left(12\right)}^{,,}$

In formula, M _{0, d}, C _{0, d}, G _{0, d}represent respectively in the robot system mass matrix in the corresponding moment of off-line calculation, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item.

Embodiment

The present embodiment method with the mechanical arm of routine 2 side chain (typical serial machine robot system) for control object, 2 branched structures as shown in Figure 2, wherein, r ₁, r ₂, J ₁, J ₂, m ₁, m ₂, q ₁, q ₂represent side chain 1 length, side chain 2 length, side chain 1 inertia, side chain 2 inertia, side chain 1 quality, side chain 2 quality, side chain 1 corner, side chain 2 corner respectively.

Its main structure parameters r given ₁, r ₂, J ₁, J ₂, m ₁and m ₂be respectively 1m, 0.8m, 5kgm, 5kgm, 0.5kg and 1.5kg.Assuming that the movement locus q expected _{d, rob}=[q _{d, rob, 1}, q _{d, rob, 2}] ^tand initial attitude q _{0, rob}be respectively

q _d,rob,1＝a ₁sin(ω ₁t)+a ₂cos(ω ₂t)+a ₃sin(ω ₃t)+a ₄cos(ω ₄t)+a ₅sin(ω ₅t)+a ₆cos(ω ₆t)(3.1)

q _d,rob,2＝b ₁sin(ω ₁t)+b ₂cos(ω ₂t)+b ₃sin(ω ₃t)+b ₄cos(ω ₄t)+b ₅sin(ω ₅t)+b ₆cos(ω ₆t)(3.2)

q _0,rob＝[1.5,-1.5] ^T(3.3)

In formula, (a ₁~ a ₆), (b ₁~ b ₆), and (ω ₁~ ω ₆) be set as respectively (1,1,0.01,1,0.01,0.001), (?1 ， ?1 ， ?0.01 ， ?1 ， ?0.01 ， ?0.001), and (0.1,0.2,0.4,0.8,1.6,3.2); Sin (*) represents the sine value asking (*); Cos (*) represents the cosine value asking (*);

The present embodiment, to the self adaptation finite time convergence control sliding-mode control of this mechanical arm, comprises the following steps: 1) set up and consider probabilistic robot dynamics's equation:

Adopt Niu Dun ?Euler method, set up the kinetics equation of this mechanical arm such as formula (3.4):

$M_{rob}\left(q_{rob}\right){\stackrel{\&CenterDot;\&CenterDot;}{q}}_{rob}+C_{rob}(q_{rob},{\stackrel{\&CenterDot;}{q}}_{rob})+G_{rob}\left(q_{rob}\right)={\mathrm{\&tau;}}_{rob}---\left(3.4\right)$

Wherein,

q _rob＝[q ₁,q ₂] ^T

M ₁₁＝(m ₁+m ₂)r ₁ ²+m ₂r ₂ ²+2m ₂r ₁r ₂cos(q ₂)+J ₁

M ₁₂＝M ₂₁＝m ₂r ₂ ²+m ₂r ₁r ₂cos(q ₂)

M ₂₂＝m ₂r ₂ ²+J ₂

$C_1=-m_2{r}_1{r}_{2}sin\left(q_2\right)({\stackrel{\&CenterDot;}{q}}_1^2+2{\stackrel{\&CenterDot;}{q}}_1{\stackrel{\&CenterDot;}{q}}_2)$

(3.5)

$C_2=m_2{r}_1{r}_2\sin \left(q_2\right){\stackrel{\&CenterDot;}{q}}_2^2$

G _1＝((m ₁+m ₂)r ₁cos(q ₂)+m ₂r ₂cos(q ₁+q ₂))g

G ₂＝m ₂r ₂cos(q ₁+q ₂)g

In formula, M _ijrepresenting matrix M _rob(q _rob) (i, j) individual element; C _iand G _ithen represent respectively and G _rob(q _rob) i-th element.M _rob(q _rob) ∈ R ^{2 × 2}for this mechanical arm mass matrix; for this mechanical arm Ge Shi/centripetal term coefficient vector; G _rob(q _rob) ∈ R ^{2 × 1}for this mechanical arm gravity item vector; τ _rob∈ R ^{2 × 1}for this mechanical arm active force/moment; q _rob∈ R ^{2 × 1}for the generalized coordinates vector of this mechanical arm; for the generalized velocity vector of this mechanical arm; represent the generalized acceleration vector of this mechanical arm; This mechanical arm dimension is 2; G is acceleration of gravity;

Formula (3.4) is introduced indeterminate, is revised as and considers that probabilistic Manipulator Dynamics equation is formula (3.6):

$(M_{0,rob}+{\mathrm{\&Delta;M}}_{rob}){\stackrel{\&CenterDot;\&CenterDot;}{q}}_{rob}+(C_{0,rob}+{\mathrm{\&Delta;C}}_{rob})+(G_{0,rob}+{\mathrm{\&Delta;G}}_{rob})={\mathrm{\&delta;}}_{a,rob}({\mathrm{\&tau;}}_{rob}+{\mathrm{\&delta;}}_{f,rob})---\left(3.6\right)$

Wherein, M _{0, rob}∈ R ^{2 × 2}, C _{0, rob}∈ R ^{2 × 1}, G _{0, rob}∈ R ^{2 × 1}represent respectively in mechanical arm mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item (can estimate in the present embodiment/can the Manipulator Dynamics parameter of calculated value be: ${\hat{m}}_1=0.4kg,{\hat{m}}_2=1.2kg,{\hat{J}}_1=4kgm,{\hat{J}}_2=4kgm,$ In formula, for actual m ₁, m ₂, J ₁, J ₂estimate/can calculated value; M _{0, rob}, C _{0, rob}, G _{0, rob}can by r ₁, r ₂, and actual motion track q _robbring formula (3.5) into calculate); Δ M _rob∈ R ^{2 × 2}, Δ C _rob∈ R ^{2 × 1}, Δ G _rob∈ R ^{2 × 1}then represent the indeterminate of mechanical arm mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector respectively; (M in the present embodiment _rob(q _rob), g _rob(q _rob) can by main structure parameters r ₁, r ₂, J ₁, J ₂, m ₁, m ₂and actual motion track q _robbring formula (3.5) into calculate; Δ M _rob, Δ C _rob, Δ G _robcan respectively by Δ M _rob=M _rob(q _rob)-M _{0, rob}, Δ G _rob=G _rob(q _rob)-G _{0, rob}calculate); δ _{a, rob}∈ R ^{2 × 2}represent multiplicative fault (δ _{a, rob, i}size determine according to the degree of actuator failures, effective span is [0,1], and the present embodiment value is δ _{a, rob, i}=1); δ _{f, rob}∈ R ^{2 × 1}for additivity fault (δ _{f, rob, i}size determine according to the degree of actuator failures, the present embodiment value is δ _{f, rob, i}=0);

Formula (3.6) arranges further as formula (3.7)

$M_{0,rob}{\stackrel{\&CenterDot;\&CenterDot;}{q}}_{rob}+C_{0,rob}+G_{0,rob}={\mathrm{\&tau;}}_{rob}+{\mathrm{\&delta;}}_{total,rob}---\left(3.7\right)$

Wherein, δ _{total, rob}∈ R ^{2 × 1}for total indeterminate is such as formula (3.8):

${\mathrm{\&delta;}}_{total,rob}=\left({\mathrm{\&delta;}}_{a,rob}\right({\mathrm{\&tau;}}_{rob}+{\mathrm{\&delta;}}_{f,rob})-{\mathrm{\&tau;}}_{rob})-{\mathrm{\&Delta;G}}_{rob}-{\mathrm{\&Delta;C}}_{rob}-{\mathrm{\&Delta;M}}_{rob}{\stackrel{\&CenterDot;\&CenterDot;}{q}}_{rob}---\left(3.8\right)$

Consider the dynamics of mechanical arm, this total indeterminate δ _{total, rob}meet constraint formula (3.9)

$\left|{\mathrm{\&delta;}}_{total,rob,i}\right|<b_{1,rob,i}+b_{2,rob,i}\left|q_{rob,i}\right|+b_{3,rob,i}|{\stackrel{\&CenterDot;}{q}}_{rob,i}{|}^2---\left(3.9\right)$

Wherein, δ _{total, rob, i}∈ R represents 2 n dimensional vector n δ _{total, rob}i-th element; b _{1, rob, i}, b _{2, rob, i}, b _{3, rob, i}span can be estimated as 1≤b _{1, rob, i}≤ 10,8≤b _{2, rob, i}≤ 80,3≤b _{3, rob, i}≤ 30 (when different when, meet the b of constraint formulations (3.9) _{1, rob, i}, b _{2, rob, i}, b _{3, rob, i}minimum of a value may change, therefore b in actual applications _{1, rob, i}, b _{2, rob, i}, b _{3, rob, i}be generally unknown, a span roughly can be estimated at most); | * | be the operation that takes absolute value; (*) ²for ask * square;

2) sliding-mode surface S is determined _rob∈ R ^{n × 1}such as formula (3.10):

$S_{rob}={\mathrm{\&epsiv;}}_{rob}+{\mathrm{\&Lambda;}}_{rob}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_{rob}^{p/q}---\left(3.10\right)$

Wherein, ε _rob=q _rob-q _{d, rob}∈ R ^{2 × 1}represent the generalized coordinates error vector of mechanical arm, q _roband q _{d, rob}for the generalized coordinates vector of mechanical arm reality and the generalized coordinates vector of expectation; P and q is respectively p=9 in the present embodiment value; Q=5; Λ _rob=diag [λ _{1, rob}, λ _{2, rob}] for Parameter adjustable joint diagonal matrix; λ _{1, rob}, λ _{2, rob}150 are in the present embodiment value; Diag [λ _{1, rob}, λ _{2, rob}] represent by element λ _{1, rob}, λ _{2, rob}the diagonal matrix of composition;

3) the adaptive updates rate of n dimension is introduced such as formula (3.11) ?(3.15):

${\widehat{\mathrm{\&Gamma;}}}_{adaptive,rob,i}=\frac{2}{\mathrm{\&pi;}}arctan\left(c_{rob,i}t\right)({\hat{b}}_{1,rob,i}+{\hat{b}}_{2,rob,i}|q_{rob,i}|+{\hat{b}}_{3,rob,i}|{\stackrel{\&CenterDot;}{q}}_{rob,i}{|}^2)---\left(3.11\right)$

$W_{rob}={\mathrm{\&Lambda;}}_{rob}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_{rob}^{p/q-1}\right)M_{0,rob}^{-1}---\left(3.12\right)$

${\stackrel{\&CenterDot;}{\hat{b}}}_{1,rob,i}=d_{1,rob,i}\&CenterDot;\frac{p}{q}\left|\right|W_{rob}\left|\right|\left|s_{rob,i}\right|---\left(3.13\right)$

${\stackrel{\&CenterDot;}{\hat{b}}}_{2,rob,i}=d_{2,rob,i}\&CenterDot;\frac{p}{q}\left|\right|W_{rob}\left|\right|\left|s_{rob,i}\right|\left|q_{rob,i}\right|---\left(3.14\right)$

${\stackrel{\&CenterDot;}{\hat{b}}}_{3,rob,i}=d_{3,rob,i}\&CenterDot;\frac{p}{q}\left|\right|W_{rob}\left|\right|\left|s_{rob,i}\right||{\stackrel{\&CenterDot;}{q}}_{rob,i}{|}^2---\left(3.15\right)$

Formula (3.11) is made up of two parts: be one to increase progressively in time and the function of bounded, the saturated and Flutter Problem by the moment startup stage of solving mechanical arm; then be used for estimating the total indeterminate δ of mechanical arm adaptively _{total, rob, i}upper dividing value;

Wherein, for adaptive estimate, the present embodiment value is (in formula, i value is 1 and 2; be respectively with initial value); d _{1, rob, i}∈ R, d _{2, rob, i}∈ R, d _{3, rob, i}∈ R is the setting value in i-th free degree direction, and the present embodiment value is d _{1, rob, i}=5d _{2, rob, i}=0.1d _{3, rob, i}=1 (in formula, i value is 1 and 2); Arctan (*) is arctan function; c _{rob, i}be the setting value in i-th free degree direction, the present embodiment value is c _{rob, i}=5 (in formula, i value is 1 and 2); π is pi; T is time variable; for compute matrix; || * || be norm operation; (*) ^-1representing matrix (*) inverse (in such as formula: representing matrix M _{0, rob}inverse); represent * derivative (in such as formula: represent derivative);

4) value of the active force/moment needed for driven machine people is calculated:

According to the generalized coordinates vector q of mechanical arm reality _roband the generalized velocity vector of mechanical arm reality value, in on-line computer mechanical arm dynamics can computational item M _{0, rob}, C _{0, rob}, G _{0, rob}value; Then, according to step 1) ?3) in the probabilistic robot dynamics's equation of consideration, constraint, sliding-mode surface and adaptive updates rate, calculate the value of the active force/moment needed for driving machine mechanical arm such as formula (3.16):

${\mathrm{\&tau;}}_{antsmc,rob}=M_{0,rob}{\stackrel{\&CenterDot;\&CenterDot;}{q}}_{d,rob}+C_{0,rob}+G_{0,rob}-\frac{q}{p}{M}_{0,rob}{\mathrm{\&Lambda;}}_{rob}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_{rob}^{2-p/q}-\frac{diag\left(s_{rob}^T{W}_{rob}\right)}{{\left(\right|\left|s_{rob}^T{W}_{rob}\right||+{\mathrm{\&xi;}}_{rob})}^2}\left|\right|s_{rob}\left|\right|\left|\right|W_{rob}\left|\right|{\widehat{\mathrm{\&Gamma;}}}_{adaptive,rob}---\left(3.16\right)$

In formula, τ _{antsmc, rob}∈ R ^{2 × 1}for the mechanical arm active force/moment calculated; for the generalized acceleration vector that mechanical arm is expected; ξ _rob∈ R is setting value, gets less positive number, and the present embodiment value is ξ _rob=0.001; (*) ^tfor getting the transposition of matrix/vector (*);

By active force/moment (3.16) driving machine mechanical arm, make the generalized coordinates vector q of mechanical arm _robrealize Finite-time convergence to stable state or trace command signal q _{d, rob}.

The effect analysis that the present embodiment method realizes is as follows:

In order to the superiority of method proposed by the invention is described, design three kinds of its results of control program as figure 3 ?shown in 7: traditional finite time convergence control sliding-mode control " C1 " (b _{1, c1, rob}, b _{2, c1, rob}, b _{3, c1, rob}select little value); Traditional finite time convergence control sliding-mode control " C2 " (b _{1, c2, rob}, b _{2, c2, rob}, b _{3, c2, rob}the large value of choosing); The self adaptation finite time convergence control sliding-mode control " C3 " that the present invention proposes is the present embodiment method.Wherein, traditional finite time convergence control sliding-mode control " C1 " and " C2 " as formula (3.17) and (3.18) (reference literature: Liu Jinkun. Sliding mode variable structure control MATLAB emulates. publishing house of Tsing-Hua University: formula (2.2) in 386 ?397.-corresponding literary compositions)

${\mathrm{\&tau;}}_{c1,rob}=M_{0,rob}{\stackrel{\&CenterDot;\&CenterDot;}{q}}_{d,rob}+C_{0,rob}+G_{0,rob}-\frac{q}{p}{M}_{0,rob}{\mathrm{\&Lambda;}}_{rob}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_{rob}^{2-p/q}-\frac{{\&lsqb;s_{rob}^T{W}_{rob}\&rsqb;}^T}{{\left(\right|\left|s_{rob}^T{W}_{rob}\right||+{\mathrm{\&xi;}}_{rob})}^2}\left|\right|s_{rob}\left|\right|\left|\right|W_{rob}\left|\right|{\mathrm{\&Gamma;}}_{c1,rob}$

(3.17)

${\mathrm{\&tau;}}_{c2,rob}=M_{0,rob}{\stackrel{\&CenterDot;\&CenterDot;}{q}}_{d,rob}+C_{0,rob}+G_{0,rob}-\frac{q}{p}{M}_{0,rob}{\mathrm{\&Lambda;}}_{rob}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}_{rob}^{2-p/q}-\frac{{\&lsqb;s_{rob}^T{W}_{rob}\&rsqb;}^T}{{\left(\right|\left|s_{rob}^T{W}_{rob}\right||+{\mathrm{\&xi;}}_{rob})}^2}\left|\right|s_{rob}\left|\right|\left|\right|W_{rob}\left|\right|{\mathrm{\&Gamma;}}_{c2,rob}$

(3.18)

Wherein,

${\mathrm{\&Gamma;}}_{c1,rob}=b_{1,c1,rob}+b_{2,c1,rob}\left|\right|q_{rob}\left|\right|+b_{3,c1,rob}\left|\right|{\stackrel{\&CenterDot;}{q}}_{rob}|{|}^2---\left(3.19\right)$

${\mathrm{\&Gamma;}}_{c2,rob}=b_{1,c2,rob}+b_{2,c2,rob}\left|\right|q_{rob}\left|\right|+b_{3,c2,rob}\left|\right|{\stackrel{\&CenterDot;}{q}}_{rob}|{|}^2---\left(3.20\right)$

In formula, b _{1, c1, rob}∈ R, b _{2, c1, rob}∈ R, b _{3, c1, rob}∈ R is the design parameter (b of traditional finite time convergence control sliding-mode control " C1 " _{1, c1, rob}, b _{2, c1, rob}, b _{3, c1, rob}for positive number, during value, consider total indeterminate δ that system exists _{total, rob, i}and get less value, the present embodiment value is b _{1, c1, rob}=1, b _{2, c1, rob}=8, b _{3, c1, rob}=3); Γ _{c1, rob}according to the q of mechanical arm reality _robwith calculate according to formula (3.19); b _{1, c2, rob}∈ R, b _{2, c2, rob}∈ R, b _{3, c2, rob}∈ R is the design parameter (b of traditional finite time convergence control sliding-mode control " C2 " _{1, c2, rob}, b _{2, c2, rob}, b _{3, c2, rob}for positive number, during value, consider total indeterminate δ that system exists _{total, rob, i}and get larger value, the present embodiment value is b _{1, c2, rob}=10, b _{2, c2, rob}=80, b _{3, c2, rob}=30); Γ _{c2, rob}according to the q of mechanical arm reality _robwith calculate according to formula (3.20);

In addition, in order to contrast the quality of above-mentioned three control methods liberally, other setup parameter of above-mentioned two kinds of methods all adopts the numerical value identical with the present embodiment

$A_{rob}=\left[\begin{array}{cc}150& 0\\ 0& 150\end{array}\right],q=5,p=9,{\mathrm{\&xi;}}_{rob}=0.001,{\mathrm{\&tau;}}_{max}=40---\left(3.21\right)$

In formula, τ _maxrepresent the maximum moment of mechanical arm.

Under the effect of three kinds of control methods, mechanical arm all can realize the tracking in finite time (2s).But the uncertain upper bound due to setting is less than actual systematic uncertainty, the tracking characteristics of control method " C1 " 3s ?cannot ensure between 6s.Simple consideration track following characteristic, the tracking effect of control method " C2 " and " C3 " is all very satisfactory.In addition, from the smoothness properties of moment/power, the output of control method " C3 " is the most level and smooth, and the output of control method " C1 " is comparatively level and smooth, and the output of control method " C2 " then exists serious concussion and saturation problem.

In sum, relative to traditional finite time convergence control (Terminal) sliding-mode control (" C1 " and " C2 "), self adaptation finite time convergence control (Terminal) sliding-mode control (" C3 ") of a kind of robot proposed by the invention, can ensure to there is the flatness that high tracking accuracy and power/moment export obviously advantage simultaneously.Further illustrate as follows in conjunction with each figure:

In Fig. 3, " C1_eq1 (" solid line " represents) " is the tracking error curve of joint 1 under the effect of traditional finite time convergence control sliding-mode control " C1 "; " C2_eq1 (" dotted line " represents) " is the tracking error curve of joint 1 under the effect of traditional finite time convergence control sliding-mode control " C2 "; " C3_eq1 (" point " represents) " is the tracking error curve of joint 1 under the self adaptation finite time convergence control sliding-mode control " C3 " that the present invention proposes acts on.

Remarks: under the effect of three kinds of control methods, mechanical arm all can realize the tracking in finite time (2s).But the uncertain upper bound due to setting is less than actual systematic uncertainty, the tracking characteristics of control method " C1 " 3s ?cannot ensure between 6s.Simple consideration track following characteristic, the tracking effect of control method " C2 " and " C3 " is all very satisfactory.

In Fig. 4, " C1_eq2 (" solid line " represents) " is the tracking error curve of joint 2 under the effect of traditional finite time convergence control sliding-mode control " C1 "; " C2_eq2 (" dotted line " represents) " is the tracking error curve of joint 2 under the effect of traditional finite time convergence control sliding-mode control " C2 "; " C3_eq2 (" point " represents) " is the tracking error curve of joint 2 under the self adaptation finite time convergence control sliding-mode control " C3 " that the present invention proposes acts on.

Remarks: under the effect of three kinds of control methods, mechanical arm all can realize the tracking in finite time (2s).But the uncertain upper bound due to setting is less than actual systematic uncertainty, the tracking characteristics of control method " C1 " 3s ?cannot ensure between 6s.Simple consideration track following characteristic, the tracking effect of control method " C2 " and " C3 " is all very satisfactory.

In Fig. 5, " C1_u1 (" solid line " represents) " is traditional finite time convergence control sliding-mode control " C1 " control inputs at joint 1 place; " C1_u2 (" dotted line " represents) " is traditional finite time convergence control sliding-mode control " C1 " control inputs at joint 2 place.

Remarks: from the smoothness properties of moment/power, the output of control method " C1 " is comparatively level and smooth.

In Fig. 6, " C2_u1 (" solid line " represents) " is traditional finite time convergence control sliding-mode control " C2 " control inputs at joint 1 place; " C2_u2 (" dotted line " represents) " is traditional finite time convergence control sliding-mode control " C2 " control inputs at joint 2 place.

Remarks: from the smoothness properties of moment/power, then there is serious concussion and saturation problem in the output of control method " C2 ".

In Fig. 7, self adaptation finite time convergence control sliding-mode control " C3 " control inputs at joint 1 place that " C3_u1 (" solid line " represents) " proposes for the present invention; Self adaptation finite time convergence control sliding-mode control " C3 " control inputs at joint 2 place that " C3_u2 (" dotted line " represents) " proposes for the present invention.

Remarks: from the smoothness properties of moment/power, the output of control method " C3 " is the most level and smooth.

Claims (4)

1. a self adaptation finite time convergence control sliding-mode control for robot, is characterized in that, be applicable to the robot system of parallel robot, serial manipulator and series-parallel robot structure type, the method comprises the following steps:

1) the probabilistic robot dynamics's equation of consideration is set up:

Adopt traditional power Epidemiological Analysis method, set up the kinetics equation of robot system such as formula (1):

$M\left(q\right)\stackrel{\&CenterDot;\&CenterDot;}{q}+C(q,\stackrel{\&CenterDot;}{q})+G\left(q\right)=\mathrm{\&tau;}---\left(1\right)$

Wherein, M (q) ∈ R ^{n × n}for robot system mass matrix; for robot system Ge Shi/centripetal term coefficient vector; G (q) ∈ R ^{n × 1}for robot system gravity item vector; τ ∈ R ^{n × 1}for robot system active force/moment; Q ∈ R ^{n × 1}for robot system generalized coordinates vector; for robot system generalized velocity vector; represent robot system generalized acceleration vector; N represents robot system dimension, and is positive integer;

Formula (1) is introduced indeterminate, is revised as and considers that probabilistic robot dynamics's equation is formula (2):

$(M_0+\mathrm{\&Delta;}M)\stackrel{\&CenterDot;\&CenterDot;}{q}+(C_0+\mathrm{\&Delta;}C)+(G_0+\mathrm{\&Delta;}G)={\mathrm{\&delta;}}_a(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)---\left(2\right),$

Wherein, M ₀∈ R ^{n × n}, C ₀∈ R ^{n × 1}, G ₀∈ R ^{n × 1}represent respectively in robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; Δ M ∈ R ^{n × n}, Δ C ∈ R ^{n × 1}, Δ G ∈ R ^{n × 1}then represent the indeterminate of robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector respectively; δ _a∈ R ^{n × n}represent multiplicative fault; δ _f∈ R ^{n × 1}for additivity fault;

Formula (2) arranges as formula (3) further:

$M_0\stackrel{\&CenterDot;\&CenterDot;}{q}+C_0+G_0=\mathrm{\&tau;}+{\mathrm{\&delta;}}_{total}---\left(3\right),$

Wherein, δ _total∈ R ^{n × 1}for total indeterminate is such as formula (4):

${\mathrm{\&delta;}}_{total}=\left({\mathrm{\&delta;}}_a\right(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)-\mathrm{\&tau;})-\mathrm{\&Delta;}G-\mathrm{\&Delta;}C-\mathrm{\&Delta;}M\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(4\right),$

Consider the dynamics of robot system, this total indeterminate δ _totalmeet constraint formula (5):

$\left|{\mathrm{\&delta;}}_{total,i}\right|<b_{1,i}+b_{2,i}\left|q_i\right|+b_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(5\right),$

Wherein, δ _{total, i}∈ R represents n n dimensional vector n δ _totali-th element (* ∈ R represents that * is real number, such as, in formula (5): δ _{total, i}∈ R represents δ _{total, i}for real number; (*) _irepresent i-th element of (*)); b _{1, i}, b _{2, i}, b _{3, i}for positive number (b _{1, i}, b _{2, i}, b _{3, i}total indeterminate δ of existing according to system of value _{total, i}determine); | * | be the operation that takes absolute value; (*) ²for ask * square;

2) sliding-mode surface S ∈ R is determined ^{n × 1}such as formula (6):

$S=\mathrm{\&epsiv;}+\mathrm{\&Lambda;}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q}---\left(6\right),$

Wherein, ε=q-q _d∈ R ^{n × 1}represent generalized coordinates error vector, q and q _dfor the generalized coordinates vector of reality and the generalized coordinates vector of expectation; represent the derivative of *; P and q is setting value, gets positive odd number and meets Λ=diag [λ ₁... λ _n] for Parameter adjustable joint diagonal matrix; λ ₁... λ _nfor setting value, get positive number; Diag [x, y] represents by element x, the diagonal matrix of y composition;

3) the adaptive updates rate of n dimension is introduced such as formula (7) ?(11):

${\widehat{\mathrm{\&Gamma;}}}_{adaptive,i}=\frac{2}{\mathrm{\&pi;}}arctan\left(c_{i}t\right)({\hat{b}}_{1,i}+{\hat{b}}_{2,i}|q_i|+{\hat{b}}_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2)---\left(7\right),$

$W=\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_0^{-1}---\left(8\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{1,i}=d_{1,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|---\left(9\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{2,i}=d_{2,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|\left|q_i\right|---\left(10\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{3,i}=d_{3,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right||{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(11\right),$

Formula (7) is made up of two parts: be one to increase progressively in time and the function of bounded, the saturated and Flutter Problem by the moment startup stage of solving robot system; then be used for estimating the total indeterminate δ of robot system adaptively _{total, i}upper dividing value;

Wherein, for adaptive estimate; d _{1, i}∈ R, d _{2, i}∈ R, d _{3, i}∈ R is the setting value in i-th free degree direction, and gets positive number; Arctan (*) is arctan function; c _ibe the setting value in i-th free degree direction, and get positive number; π is pi; T is time variable; for compute matrix; || * || be norm operation; (*) ^-1representing matrix (*) inverse; represent the derivative of *;

4) value of the active force/moment needed for driven machine people is calculated:

According to the generalized coordinates vector q of robot system reality and the generalized velocity vector of robot system reality value, in on-line computer device people dynamics can computational item M ₀, C ₀, G ₀value; Then, according to step 1) ?3) in the probabilistic robot dynamics's equation of consideration, constraint, sliding-mode surface and adaptive updates rate, calculate the value of the active force/moment needed for driven machine people such as formula (12):

${\mathrm{\&tau;}}_{antsmc}=M_0{\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+C_0+G_0-\frac{q}{p}{M}_0{\mathrm{\&Lambda;}}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{2-p/q}-\frac{diag\left(s^{T}W\right)}{{\left(\right|\left|s^{T}W\right||+\mathrm{\&xi;})}^2}\left|\right|s\left|\right|\left|\right|W\left|\right|{\widehat{\mathrm{\&Gamma;}}}_{adaptive}---\left(12\right)$

In formula, τ _antsmc∈ R ^{n × 1}for the robot active force/moment calculated; for the generalized acceleration vector that robot system is expected; ξ ∈ R is setting value, gets less positive number; (*) ^tfor getting the transposition of matrix/vector (*);

With active force/moment (12) driven machine robot system, the generalized coordinates of robot vector q is made to realize Finite-time convergence to stable state or trace command signal q _d.

2. a self adaptation finite time convergence control sliding-mode control for robot, is characterized in that, be applicable to the robot system of parallel robot, serial manipulator and series-parallel robot structure type, the method comprises the following steps:

1) the probabilistic robot dynamics's equation of consideration is set up:

Adopt traditional power Epidemiological Analysis method, set up the kinetics equation of robot system such as formula (1):

$M\left(q\right)\stackrel{\&CenterDot;\&CenterDot;}{q}+C(q,\stackrel{\&CenterDot;}{q})+G\left(q\right)=\mathrm{\&tau;}---\left(1\right)$

Wherein, M (q) ∈ R ^{n × n}for robot system mass matrix; for robot system Ge Shi/centripetal term coefficient vector; G (q) ∈ R ^{n × 1}for robot system gravity item vector; τ ∈ R ^{n × 1}for robot system active force/moment; Q ∈ R ^{n × 1}for robot system generalized coordinates vector; for robot system generalized velocity vector; represent robot system generalized acceleration vector; N represents robot system dimension, and is positive integer;

Formula (1) is introduced indeterminate, is revised as and considers that probabilistic robot dynamics's equation is formula (2):

$(M_0+\mathrm{\&Delta;}M)\stackrel{\&CenterDot;\&CenterDot;}{q}+(C_0+\mathrm{\&Delta;}C)+(G_0+\mathrm{\&Delta;}G)={\mathrm{\&delta;}}_a(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)---\left(2\right),$

Wherein, M ₀∈ R ^{n × n}, C ₀∈ R ^{n × 1}, G ₀∈ R ^{n × 1}represent respectively in robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; Δ M ∈ R ^{n × n}, Δ C ∈ R ^{n × 1}, Δ G ∈ R ^{n × 1}then represent the indeterminate of robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector respectively; δ _a∈ R ^{n × n}represent multiplicative fault; δ _f∈ R ^{n × 1}for additivity fault;

Formula (2) arranges as formula (3) further:

$M_0\stackrel{\&CenterDot;\&CenterDot;}{q}+C_0+G_0=\mathrm{\&tau;}+{\mathrm{\&delta;}}_{total}---\left(3\right),$

Wherein, δ _total∈ R ^{n × 1}for total indeterminate is such as formula (4):

${\mathrm{\&delta;}}_{total}=\left({\mathrm{\&delta;}}_a\right(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)-\mathrm{\&tau;})-\mathrm{\&Delta;}G-\mathrm{\&Delta;}C-\mathrm{\&Delta;}M\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(4\right),$

Consider the dynamics of robot system, this total indeterminate δ _totalmeet constraint formula (5):

$\left|{\mathrm{\&delta;}}_{total,i}\right|<b_{1,i}+b_{2,i}\left|q_i\right|+b_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(5\right),$

Wherein, δ _{total, i}∈ R represents n n dimensional vector n δ _totali-th element (* ∈ R represents that * is real number, such as, in formula (5): δ _{total, i}∈ R represents δ _{total, i}for real number; (*) _irepresent i-th element of (*)); b _{1, i}, b _{2, i}, b _{3, i}for positive number (b _{1, i}, b _{2, i}, b _{3, i}total indeterminate δ of existing according to system of value _{total, i}determine); | * | be the operation that takes absolute value; (*) ²for ask * square;

2) sliding-mode surface S ∈ R is determined ^{n × 1}such as formula (6):

$S=\mathrm{\&epsiv;}+\mathrm{\&Lambda;}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q}---\left(6\right),$

Wherein, ε=q-q _d∈ R ^{n × 1}represent generalized coordinates error vector, q and q _dfor the generalized coordinates vector of reality and the generalized coordinates vector of expectation; represent the derivative of *; P and q is setting value, gets positive odd number and meets Λ=diag [λ ₁... λ _n] for Parameter adjustable joint diagonal matrix; λ ₁... λ _nfor setting value, get positive number; Diag [x, y] represents by element x, the diagonal matrix of y composition;

3) the adaptive updates rate of n dimension is introduced such as formula (7) ?(11):

${\widehat{\mathrm{\&Gamma;}}}_{adaptive,i}=\frac{2}{\mathrm{\&pi;}}arctan\left(c_{i}t\right)({\hat{b}}_{1,i}+{\hat{b}}_{2,i}|q_i|+{\hat{b}}_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2)---\left(7\right),$

$W=\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_0^{-1}\left(t_{N-1}\right)---{\left(8\right)}^{,},$

${\stackrel{\&CenterDot;}{\hat{b}}}_{1,i}=d_{1,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|---\left(9\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{2,i}=d_{2,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|\left|q_i\right|---\left(10\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{3,i}=d_{3,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right||{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(11\right),$

Formula (7) is made up of two parts: be one to increase progressively in time and the function of bounded, the saturated and Flutter Problem by the moment startup stage of solving robot system; then be used for estimating the total indeterminate δ of robot system adaptively _{total, i}upper dividing value;

for adaptive estimate; d _{1, i}∈ R, d _{2, i}∈ R, d _{3, i}∈ R is the setting value in i-th free degree direction, and gets positive number; Arctan (*) is arctan function; c _ibe the setting value in i-th free degree direction, and get positive number; π is pi; T is time variable; for compute matrix; || * || be norm operation; (*) ^-1representing matrix (*) inverse; represent the derivative of *; M ₀(t _n-1) represent at t _n-1moment robot system mass matrix can computational item; t _n-1for system dynamics calculates the setting value in moment; t _n-1middle subscript N is positive integer and value is (as N=1, t _n-1=t ₀, represent initial time); t _ffor the moment of robot system stop motion; INT (*) represents floor operation, namely gets the maximum integer part being no more than real number (*); Δ t=t _n-t _n-1for the setting value of interval time, get Δ t>=t _c;

4) value of the active force/moment needed for driven machine people is calculated:

According to the generalized coordinates vector q of robot system reality and the generalized velocity vector of robot system reality value, adopt separated in time (interval time is Δ t=t _n-t _n-1) in calculating robot's dynamics can computational item (M ₀(t _n-1), C ₀(t _n-1), G ₀(t _n-1)) value; Then, according to step 1) ?3) in the probabilistic robot dynamics's equation of consideration, constraint, sliding-mode surface and adaptive updates rate, calculate the value of the active force/moment needed for driven machine people such as formula (12) ' shown in:

Work as t _n-1+ t _c≤ t < t _n+ t _ctime

${\mathrm{\&tau;}}_{antsmc}=M_0\left(t_{N-1}\right){\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+C_0\left(t_{N-1}\right)+G_0\left(t_{N-1}\right)-\frac{q}{p}{M}_0\left(t_{N-1}\right){\mathrm{\&Lambda;}}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{2-p/q}-\frac{diag\left(s^{T}W\right)}{{\left(\right|\left|s^{T}W\right||+\mathrm{\&xi;})}^2}\left|\right|s\left|\right|\left|\right|W\left|\right|{\widehat{\mathrm{\&Gamma;}}}_{adaptive}---{\left(12\right)}^{,}$

In formula, C ₀(t _n-1), G ₀(t _n-1) be illustrated respectively in t _n-1in moment robot system Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; t _crepresent in calculating robot mass of system matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item required time; t _nfor system dynamics calculates the setting value in moment; τ _antsmc∈ R ^{n × 1}for the robot active force/moment calculated; for the generalized acceleration vector that robot system is expected; ξ ∈ R is setting value, gets less positive number; (*) ^tfor getting the transposition of matrix/vector (*);

With active force/moment (12) ' driven machine robot system, the generalized coordinates of robot vector q is made to realize Finite-time convergence to stable state or trace command signal q _d.

3. a self adaptation finite time convergence control sliding-mode control for robot, is characterized in that, be applicable to the robot system of parallel robot, serial manipulator and series-parallel robot structure type, the method comprises the following steps:

1) the probabilistic robot dynamics's equation of consideration is set up:

Adopt traditional power Epidemiological Analysis method, set up the kinetics equation of robot system such as formula (1):

$M\left(q\right)\stackrel{\&CenterDot;\&CenterDot;}{q}+C(q,\stackrel{\&CenterDot;}{q})+G\left(q\right)=\mathrm{\&tau;}---\left(1\right)$

Wherein, M (q) ∈ R ^{n × n}for robot system mass matrix; for robot system Ge Shi/centripetal term coefficient vector; G (q) ∈ R ^{n × 1}for robot system gravity item vector; τ ∈ R ^{n × 1}for robot system active force/moment; Q ∈ R ^{n × 1}for robot system generalized coordinates vector; for robot system generalized velocity vector; represent robot system generalized acceleration vector; N represents robot system dimension, and is positive integer;

Formula (1) is introduced indeterminate, is revised as and considers that probabilistic robot dynamics's equation is formula (2):

$(M_0+\mathrm{\&Delta;}M)\stackrel{\&CenterDot;\&CenterDot;}{q}+(C_0+\mathrm{\&Delta;}C)+(G_0+\mathrm{\&Delta;}G)={\mathrm{\&delta;}}_a(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)---\left(2\right),$

Wherein, M ₀∈ R ^{n × n}, C ₀∈ R ^{n × 1}, G ₀∈ R ^{n × 1}represent respectively in robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector can computational item; Δ M ∈ R ^{n × n}, Δ C ∈ R ^{n × 1}, Δ G ∈ R ^{n × 1}then represent the indeterminate of robot system mass matrix, Ge Shi/centripetal term coefficient vector, gravity item vector respectively; δ _a∈ R ^{n × n}represent multiplicative fault; δ _f∈ R ^{n × 1}for additivity fault;

Formula (2) arranges as formula (3) further:

$M_0\stackrel{\&CenterDot;\&CenterDot;}{q}+C_0+G_0=\mathrm{\&tau;}+{\mathrm{\&delta;}}_{total}---\left(3\right),$

Wherein, δ _total∈ R ^{n × 1}for total indeterminate is such as formula (4):

${\mathrm{\&delta;}}_{total}=\left({\mathrm{\&delta;}}_a\right(\mathrm{\&tau;}+{\mathrm{\&delta;}}_f)-\mathrm{\&tau;})-\mathrm{\&Delta;}G-\mathrm{\&Delta;}C-\mathrm{\&Delta;}M\stackrel{\&CenterDot;\&CenterDot;}{q}---\left(4\right),$

Consider the dynamics of robot system, this total indeterminate δ _totalmeet constraint formula (5):

$\left|{\mathrm{\&delta;}}_{total,i}\right|<b_{1,i}+b_{2,i}\left|q_i\right|+b_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(5\right),$

Wherein, δ _{total, i}∈ R represents n n dimensional vector n δ _totali-th element (* ∈ R represents that * is real number, such as, in formula (5): δ _{total, i}∈ R represents δ _{total, i}for real number; (*) _irepresent i-th element of (*)); b _{1, i}, b _{2, i}, b _{3, i}for positive number (b _{1, i}, b _{2, i}, b _{3, i}total indeterminate δ of existing according to system of value _{total, i}determine); | * | be the operation that takes absolute value; (*) ²for ask * square;

2) sliding-mode surface S ∈ R is determined ^{n × 1}such as formula (6):

$S=\mathrm{\&epsiv;}+\mathrm{\&Lambda;}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q}---\left(6\right),$

Wherein, ε=q-q _d∈ R ^{n × 1}represent generalized coordinates error vector, q and q _dfor the generalized coordinates vector of reality and the generalized coordinates vector of expectation; represent the derivative of *; P and q is setting value, gets positive odd number and meets Λ=diag [λ ₁... λ _n] for Parameter adjustable joint diagonal matrix; λ ₁... λ _nfor setting value, get positive number; Diag [x, y] represents by element x, the diagonal matrix of y composition;

3) the adaptive updates rate of n dimension is introduced such as formula (7) ?(11):

${\widehat{\mathrm{\&Gamma;}}}_{adaptive,i}=\frac{2}{\mathrm{\&pi;}}arctan\left(c_{i}t\right)({\hat{b}}_{1,i}+{\hat{b}}_{2,i}|q_i|+{\hat{b}}_{3,i}|{\stackrel{\&CenterDot;}{q}}_i{|}^2)---\left(7\right),$

$W=\mathrm{\&Lambda;}diag\left({\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{p/q-1}\right)M_{0,d}^{-1}---{\left(8\right)}^{,,},$

${\stackrel{\&CenterDot;}{\hat{b}}}_{1,i}=d_{1,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|---\left(9\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{2,i}=d_{2,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right|\left|q_i\right|---\left(10\right),$

${\stackrel{\&CenterDot;}{\hat{b}}}_{3,i}=d_{3,i}\&CenterDot;\frac{p}{q}\left|\right|W\left|\right|\left|s_i\right||{\stackrel{\&CenterDot;}{q}}_i{|}^2---\left(11\right),$

Formula (7) is made up of two parts: be one to increase progressively in time and the function of bounded, the saturated and Flutter Problem by the moment startup stage of solving robot system; then be used for estimating the total indeterminate δ of robot system adaptively _{total, i}upper dividing value;

Wherein, ${\widehat{\mathrm{\&Gamma;}}}_{adaptive,i}\&Element;R,{\hat{b}}_{1,i}\&Element;R,{\hat{b}}_{2,i}\&Element;R,{\hat{b}}_{3,i}\&Element;R$ For adaptive estimate; d _{1, i}∈ R, d _{2, i}∈ R, d _{3, i}∈ R is the setting value in i-th free degree direction, and gets positive number; Arctan (*) is arctan function; c _ibe the setting value in i-th free degree direction, and get positive number; π is pi; T is time variable; for compute matrix; || * || be norm operation; (*) ^-1representing matrix (*) inverse; represent the derivative of *; M _{0, d}what represent the robot system mass matrix in the corresponding moment of off-line calculation can computational item;

4) value of the active force/moment needed for driven machine people is calculated:

According to the generalized coordinates vector q that robot system is expected _dand the generalized velocity vector that robot system is expected value, first off-line in calculating robot's dynamics can computational item (M _{0, d}, C _{0, d}, G _{0, d}) value, and result to be stored, when actual motion, directly calls (the M in corresponding moment _{0, d}, C _{0, d}, G _{0, d}) value; Then, according to step 1) ?3) in the probabilistic robot dynamics's equation of consideration, constraint, sliding-mode surface and adaptive updates rate, calculate the value of the active force/moment needed for driven machine people such as formula (12) " shown in:

${\mathrm{\&tau;}}_{antsmc}=M_{0,d}{\stackrel{\&CenterDot;\&CenterDot;}{q}}_d+C_{0,d}+G_{0,d}-\frac{q}{p}{M}_{0,d}{\mathrm{\&Lambda;}}^{-1}{\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^{2-p/q}-\frac{diag\left(s^{T}W\right)}{{\left(\right|\left|s^{T}W\right||+\mathrm{\&xi;})}^2}\left|\right|s\left|\right|\left|\right|W\left|\right|{\widehat{\mathrm{\&Gamma;}}}_{adaptive}---{\left(12\right)}^{,,},$

In formula, C _{0, d}, G _{0, d}represent respectively in the robot system Ge Shi in the corresponding moment of off-line calculation/centripetal term coefficient vector, gravity item vector can computational item; τ _antsmc∈ R ^{n × 1}for the robot active force/moment calculated; for the generalized acceleration vector that robot system is expected; ξ ∈ R is setting value, gets less positive number; (*) ^tfor getting the transposition of matrix/vector (*);

With active force/moment (12) " driven machine robot system, make the generalized coordinates of robot vector q realize Finite-time convergence to stable state or trace command signal q _d.

4. the self adaptation finite time convergence control sliding-mode control of a kind of robot according to claim 1,2 or 3, it is characterized in that: in described step (1) formula (1), traditional dynamic analysis method adopt following any one: Kane method, Niu Dun ?Euler method, Niu Dun ?Lagrangian method, the principle of virtual work.