CN111152225A - Uncertain mechanical arm fixed time trajectory tracking control method with input saturation - Google Patents

Uncertain mechanical arm fixed time trajectory tracking control method with input saturation Download PDF

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CN111152225A
CN111152225A CN202010044241.1A CN202010044241A CN111152225A CN 111152225 A CN111152225 A CN 111152225A CN 202010044241 A CN202010044241 A CN 202010044241A CN 111152225 A CN111152225 A CN 111152225A
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mechanical arm
tracking
tracking error
joint
establishing
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CN111152225B (en
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孙亮
王智冬
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University of Science and Technology Beijing USTB
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/1633—Programme controls characterised by the control loop compliant, force, torque control, e.g. combined with position control

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Abstract

The invention provides an uncertain mechanical arm fixed time trajectory tracking control method with input saturation, which can overcome the influence of the input saturation effect and uncertainty of mechanical arm control and improve the robustness of a system. The method comprises the following steps: establishing a dynamic equation of an n-degree-of-freedom rotary joint rigid mechanical arm system with a viscous friction term; establishing a tracking error kinetic equation considering parameter uncertainty according to the established kinetic equation and the mechanical arm track tracking error signal; establishing a state space model of a mechanical arm track tracking error; establishing a fixed time interference observer according to the established state space model; establishing a fixed-time nonsingular terminal sliding mode surface according to the track tracking error signal of the mechanical arm; preliminarily determining control torque instructions of driving motors of all joints of the mechanical arm, and combining the control torque output range of the actuator to obtain the control torque of all joints of the mechanical arm. The invention relates to the field of mechanical arm control.

Description

Uncertain mechanical arm fixed time trajectory tracking control method with input saturation
Technical Field
The invention relates to the field of mechanical arm control, in particular to an uncertain mechanical arm fixed time trajectory tracking control method with input saturation.
Background
The mechanical arm has important significance for improving the production automation degree and improving the national manufacturing level, and is widely applied to various fields such as production, scientific research, service and the like at present, so that the mechanical arm plays an important role. It has good properties to replace some of the burdensome work for human beings, such as industrial assembly, automotive manufacturing, space exploration, modern agriculture and forestry, etc. The control system plays a key role in various tasks based on the mechanical arm, and the goals of continuously improving the precision and robustness of the control system and reducing the power consumption are the goals of the field of automatic control.
The convergence performance of the tracking error is one of important indexes for describing the performance of the control system, so that the tail end position of the mechanical arm needs to realize the rapid convergence of the tracking error so as to save control energy, reduce transient response time and improve the efficiency of executing tasks. In order to improve the convergence performance of the system, the finite time control technology is gradually favored by extensive researchers. The finite time control technology puts further requirements on the convergence speed of the system in a system stable domain, so that the convergence time is limited in an artificially controllable range, and the response performance of the system is greatly improved. Meanwhile, in order to improve the robustness of the system, a finite time control technology is often combined with a disturbance observer technology, and the uncertainty in the mechanical arm system estimated by using the disturbance observer can overcome the adverse effect on the control system caused by system modeling errors and unknown external disturbance. In order to give clear information about convergence time, a fixed time control technology is further proposed in the framework of limited time convergence, and the design of a control system is greatly facilitated. However, the system is less robust due to uncertainty in the robotic arm system and input saturation effects.
Disclosure of Invention
The invention provides an uncertain mechanical arm fixed time trajectory tracking control method with input saturation, which can overcome the influence of a mechanical arm control input saturation effect and uncertainty while meeting the convergence requirement of fixed time and improve the robustness of a system.
In order to solve the technical problem, an embodiment of the present invention provides a method for controlling trajectory tracking of a fixed time of an uncertain mechanical arm with input saturation, including:
establishing a dynamic equation of an n-degree-of-freedom rotary joint rigid mechanical arm system with a viscous friction term;
according to the established kinetic equation and the tracking error signal of the mechanical arm track, establishing a tracking error kinetic equation considering parameter uncertainty, wherein the parameter uncertainty comprises: dynamic modeling errors and unknown external disturbances;
establishing a state space model of the tracking error of the mechanical arm track according to a tracking error kinetic equation considering parameter uncertainty;
establishing a fixed time interference observer according to the established state space model;
establishing a fixed-time nonsingular terminal sliding mode surface according to the track tracking error signal of the mechanical arm;
establishing a nonlinear anti-saturation compensator according to the actuator saturation effect parameters;
according to the established fixed time disturbance observer, the fixed time nonsingular terminal sliding mode surface and the nonlinear anti-saturation compensator, preliminarily determining a control torque instruction of each joint driving motor of the mechanical arm, and combining a control torque output range of an actuator to obtain control torque of each joint of the mechanical arm.
Further, the established kinetic equation is expressed as:
Figure BDA0002368804830000021
wherein x is,
Figure BDA0002368804830000022
Respectively representing the angle, the angular velocity and the angular acceleration of the mechanical arm joint; m (x),
Figure BDA0002368804830000023
D. g (x) respectively representing an actual positive definite inertia matrix, a centrifugal force and Coriolis force matrix, a joint viscous friction coefficient matrix and a gravity vector of the mechanical arm; m0(x)、C0(x)、D0、g0(x) Respectively representing estimated values of a positive definite inertia matrix, a centrifugal force and Coriolis force matrix, a joint viscous friction coefficient matrix and a gravity vector of the mechanical arm; mΔ(x)、CΔ(x)、DΔ(x)、gΔ(x) Respectively representing positive definite inertia matrix, centrifugal force and Coriolis force matrix, joint viscous friction coefficient matrix and gravity vector of mechanical armEstimating an error; u is the control moment of each joint of the mechanical arm; d is the external disturbance moment applied to the mechanical arm.
Further, the control torque u satisfies the input limited constraint of:
Figure BDA0002368804830000031
Figure BDA0002368804830000032
wherein u isiRepresents the actual control moment, u, of the ith robot arm jointi0The theoretical control moment of the ith mechanical arm joint obtained by the track tracking control method is shown, n is the number of the mechanical arm joints,
Figure BDA0002368804830000033
and
Figure BDA0002368804830000034
respectively, the minimum and maximum control torques that the actuator is capable of outputting.
Further, the establishing of the tracking error kinetic equation considering the uncertainty of the parameters according to the established kinetic equation and the tracking error signal of the mechanical arm trajectory comprises:
according to the expected tracking angle x of each joint of the mechanical armdDesired tracking angular velocity
Figure BDA0002368804830000035
And the measured angle x and angular velocity of each joint of the mechanical arm
Figure BDA0002368804830000036
Calculating the tracking error e of the mechanical arm track position as x-xdAnd velocity tracking error
Figure BDA0002368804830000037
According to the established kinetic equation and the tracking error signal of the mechanical arm trackNumber e and
Figure BDA0002368804830000038
and establishing a tracking error kinetic equation considering parameter uncertainty.
Further, the established tracking error kinetic equation is expressed as:
Figure BDA0002368804830000039
wherein,
Figure BDA00023688048300000310
respectively representing an angular velocity tracking error and an angular acceleration tracking error; x is the number ofd、
Figure BDA00023688048300000311
Respectively representing a desired tracking angle, a desired tracking angular velocity and a desired tracking angular acceleration;
Figure BDA00023688048300000312
representing a kinematically known term related to a desired tracking angle, a desired tracking angular velocity, and a desired tracking angular acceleration;
Figure BDA00023688048300000313
representing the entire set of unknown terms in the tracking error dynamics equation.
Further, the state space model is established as:
Figure BDA00023688048300000314
wherein x is1Indicating position tracking error, x1＝e；
Figure BDA00023688048300000315
And x2Are indicative of a speed tracking error and,
Figure BDA00023688048300000316
η＝-M0 -1(C0+D0)x2-M0 -1h0representing a known term in a kinetic equation; delta-M0 -1ω represents the aggregate disturbance and δ contains both parametric uncertainty and external moment disturbances.
Further, the fixed-time disturbance observer is established as:
Figure BDA0002368804830000041
wherein,
Figure BDA0002368804830000042
and
Figure BDA0002368804830000043
respectively represent x2And an estimate of δ;
Figure BDA0002368804830000044
and
Figure BDA0002368804830000045
are respectively as
Figure BDA0002368804830000046
And
Figure BDA0002368804830000047
the rate of change of (c);
Figure BDA0002368804830000048
representing an angular velocity estimation error; m is1、n1、m2、n2And gamma represent the gain of the fixed time disturbance observer; p is a radical of1、p2、q1And q is2All represent fixed time disturbance observer fractional power parameters; the function sig (-) is of the form: sigz(y)＝|y|zSign (y) a sign function denoted y.
Further, the established fixed-time nonsingular terminal sliding mode surface s is expressed as:
Figure BDA0002368804830000049
wherein, α1,α2For fixed time nonsingular terminal sliding mode face gain coefficient, β ═ β1,β2,…βn]TRepresenting variable-structure sliding-mode terms used to cope with singularity problems, which constitute the parameter βiExpressed as:
Figure BDA00023688048300000410
wherein, i is 1, …, n, x1iRepresenting the velocity tracking error of the ith mechanical arm joint; judgment of conditions
Figure BDA00023688048300000411
γ1、γ2And κ both represent constant coefficients; coefficient of constant
Figure BDA00023688048300000412
Figure BDA00023688048300000413
Further, the nonlinear anti-saturation compensator is established as:
Figure BDA00023688048300000414
wherein ξ is the state of the nonlinear anti-saturation compensator, k is a positive constant coefficient, u isΔRepresenting the actuator saturation effect parameter, uΔ＝u-u0；u0And the control torque command of each joint driving motor of the mechanical arm is represented.
Further, the preliminarily determined control torque command of each joint driving motor of the mechanical arm is expressed as:
Figure BDA00023688048300000415
wherein,
Figure BDA00023688048300000416
is the intermediate variable(s) of the variable,
Figure BDA00023688048300000417
a representation definition;
Figure BDA00023688048300000418
represents the rate of change of variable structure sliding mode term β;
Figure BDA00023688048300000419
is composed of
Figure BDA00023688048300000420
In the form of a short-hand writing of (1),
Figure BDA00023688048300000421
the technical scheme of the invention has the following beneficial effects:
in the scheme, the fixed time trajectory tracking controller consists of a fixed time disturbance observer, a fixed time nonsingular terminal sliding mode surface and a nonlinear anti-saturation compensator, aiming at the uncertainty of the mechanical arm system caused by the estimation and tracking error of physical parameters, the fixed time disturbance observer is adopted in the controller to compensate unknown total disturbance terms, and the observation error of the total disturbance terms can be ensured to be converged to zero within fixed time, so that the mechanical arm system has good robustness; a nonlinear anti-saturation compensator is adopted in a controller aiming at the input saturation problem to deal with the saturation effect of mechanical arm control input. Therefore, under the condition that parameter uncertainty and unknown external interference exist, the trajectory tracking error of the mechanical arm with saturation control input constraint can meet the requirement of consistent and final bounded fixed time convergence.
Drawings
Fig. 1 is a schematic flow chart of a fixed time trajectory tracking control method for an uncertain manipulator with input saturation according to an embodiment of the present invention;
FIG. 2 is a schematic diagram illustrating a method for tracking and controlling a fixed time trajectory of an uncertain manipulator with input saturation according to an embodiment of the present invention;
fig. 3 is a schematic structural diagram of a two-degree-of-freedom rigid mechanical arm according to an embodiment of the present invention;
FIG. 4 is a schematic diagram illustrating a tracking response curve of positions of joints of a robot arm according to an embodiment of the present invention;
FIG. 5 is a schematic diagram illustrating velocity tracking response curves of joints of a robot arm according to an embodiment of the present invention;
FIG. 6 is a schematic diagram illustrating a variation curve of control input torque of each joint of a robot arm according to an embodiment of the present invention;
FIG. 7 is a schematic diagram of a disturbance total set observation error variation curve according to an embodiment of the present invention;
FIG. 8 is a schematic diagram of an acceleration observation error variation curve according to an embodiment of the present invention;
FIG. 9 is a schematic diagram of a variation curve of a position tracking error according to an embodiment of the present invention;
fig. 10 is a schematic diagram of a tracking response curve of positions of joints of a mechanical arm based on a finite time observer according to an embodiment of the present invention;
fig. 11 is a schematic diagram of a velocity tracking response curve of each joint of a mechanical arm based on a finite time observer according to an embodiment of the present invention;
FIG. 12 is a schematic diagram of an observation error variation curve of a disturbance total set based on a finite time observer according to an embodiment of the present invention;
fig. 13 is a schematic diagram of a tracking response curve of the position of each joint of the mechanical arm based on a linear extended observer according to an embodiment of the present invention;
fig. 14 is a schematic diagram of a velocity tracking response curve of each joint of a mechanical arm based on a linear extended observer according to an embodiment of the present invention;
fig. 15 is a schematic view of an observation error variation curve of a disturbance total set based on a finite time observer according to an embodiment of the present invention.
Detailed Description
In order to make the technical problems, technical solutions and advantages of the present invention more apparent, the following detailed description is given with reference to the accompanying drawings and specific embodiments.
As shown in fig. 1, a method for tracking and controlling a fixed-time trajectory of an uncertain mechanical arm with input saturation according to an embodiment of the present invention includes:
s1, establishing a dynamic equation of the n-degree-of-freedom rotary joint rigid mechanical arm system with the viscous friction term;
s2, establishing a tracking error kinetic equation considering parameter uncertainty according to the established kinetic equation and the mechanical arm track tracking error signal, wherein the parameter uncertainty comprises: dynamic modeling errors and unknown external disturbances;
s3, establishing a state space model of the mechanical arm track tracking error according to a tracking error kinetic equation considering parameter uncertainty;
s4, establishing a fixed time disturbance observer according to the established state space model;
s5, establishing a fixed-time nonsingular terminal sliding mode surface according to the mechanical arm track tracking error signal;
s6, establishing a nonlinear anti-saturation compensator according to the actuator saturation effect parameters;
and S7, preliminarily determining control moment instructions of driving motors of each joint of the mechanical arm according to the established fixed-time disturbance observer, the established fixed-time nonsingular terminal sliding mode surface and the established nonlinear anti-saturation compensator, and combining the control moment output range of the actuator to obtain the control moment of each joint of the mechanical arm.
According to the uncertain mechanical arm fixed time trajectory tracking control method with input saturation, a fixed time trajectory tracking controller is composed of a fixed time disturbance observer, a fixed time nonsingular terminal sliding mode surface and a nonlinear anti-saturation compensator, aiming at the uncertainty of a mechanical arm system caused by the estimation and tracking errors of physical parameters, the fixed time disturbance observer is adopted in the controller to compensate unknown total disturbance items, and the observation errors of the total disturbance items can be guaranteed to be converged to zero in fixed time, so that the mechanical arm system has good robustness; a nonlinear anti-saturation compensator is adopted in a controller aiming at the input saturation problem to deal with the saturation effect of mechanical arm control input. Therefore, under the condition that parameter uncertainty and unknown external interference exist, the trajectory tracking error of the mechanical arm with saturation control input constraint can meet the requirement of consistent and final bounded fixed time convergence.
In this embodiment, consistent and ultimately bounded fixed time convergence means that both tracking and observation error values can converge to a small neighborhood of the origin in a fixed time. .
The method for tracking and controlling the fixed time trajectory of the uncertain mechanical arm with input saturation provided by the embodiment of the invention has the advantages of faster convergence performance, higher steady-state precision and better anti-saturation control performance, and can be easily expanded to other second-order nonlinear mechanical systems, thereby greatly improving the universality of the method.
In order to better understand the method for tracking and controlling the fixed-time trajectory of the uncertain manipulator with input saturation provided by the embodiment of the present invention, as shown in fig. 1 and fig. 2, the method specifically includes the following steps:
s1, establishing a dynamic equation (also called as a dynamic model) of the n-degree-of-freedom rotary joint rigid mechanical arm system with the viscous friction term; wherein the established kinetic equation is expressed as:
Figure BDA0002368804830000071
wherein x is,
Figure BDA0002368804830000072
Respectively representing the angle, the angular velocity and the angular acceleration of the mechanical arm joint; m (x) ε Rn×n、
Figure BDA0002368804830000073
D∈Rn×n、g(x)∈Rn×1Representing actual positive definite inertia matrix, centrifugal force and coriolis force of an n-degree-of-freedom robotic armThe matrix, the joint viscous friction coefficient matrix and the gravity vector; rn×n、Rn×1Respectively representing n multiplied by n and n multiplied by 1 dimensional real number sets; m0(x)、C0(x)、D0、g0(x) Respectively representing estimated values of a positive definite inertia matrix, a centrifugal force and Coriolis force matrix, a joint viscous friction coefficient matrix and a gravity vector of the mechanical arm; mΔ(x)、CΔ(x)、DΔ(x)、gΔ(x) Respectively representing the estimation errors of a positive definite inertia matrix, a centrifugal force and Coriolis force matrix, a joint viscous friction coefficient matrix and a gravity vector of the mechanical arm; u is an element of RnControlling the moment, R, for each joint of the armnRepresenting an n-dimensional real number set; d is equal to RnIs an external disturbing moment exerted on the robot arm.
In this embodiment, in consideration of the physical characteristics of the robot joint driving motor, each joint control torque u should satisfy the following input limit constraints:
Figure BDA0002368804830000074
Figure BDA0002368804830000081
wherein u isiRepresents the actual control moment, u, of the ith robot arm jointi0The theoretical control moment of the ith mechanical arm joint obtained by the track tracking control method is shown, n is the number of the mechanical arm joints,
Figure BDA0002368804830000082
and
Figure BDA0002368804830000083
respectively, the minimum and maximum control torques that the actuator is capable of outputting.
S2, establishing a tracking error kinetic equation considering uncertainty of parameters according to the established kinetic equation and the tracking error signal of the mechanical arm trajectory, which may specifically include the following steps:
s21, according to each joint of the mechanical armDesired tracking angle xdDesired tracking angular velocity
Figure BDA0002368804830000084
And the measured angle x and angular velocity of each joint of the mechanical arm
Figure BDA0002368804830000085
Calculating the tracking error e of the mechanical arm track position as x-xdAnd velocity tracking error
Figure BDA0002368804830000086
S22, according to the established kinetic equation and the tracking error signal e and
Figure BDA0002368804830000087
establishing a tracking error kinetic equation considering parameter uncertainty, wherein the established tracking error kinetic equation is expressed as:
Figure BDA0002368804830000088
wherein,
Figure BDA0002368804830000089
respectively representing an angular velocity tracking error and an angular acceleration tracking error; x is the number ofd、
Figure BDA00023688048300000810
Respectively representing a desired tracking angle, a desired tracking angular velocity and a desired tracking angular acceleration;
Figure BDA00023688048300000811
representing a kinematically known term related to a desired tracking angle, a desired tracking angular velocity, and a desired tracking angular acceleration;
Figure BDA00023688048300000812
representing the entire set of unknown terms in the tracking error dynamics equation.
S3, establishing a state space model of the mechanical arm track tracking error according to a tracking error kinetic equation considering parameter uncertainty; wherein, the established state space model is expressed as:
Figure BDA00023688048300000813
wherein x is1Indicating position tracking error, x1＝e；
Figure BDA00023688048300000814
And x2Are indicative of a speed tracking error and,
Figure BDA00023688048300000815
η＝-M0 -1(C0+D0)x2-M0 -1h0representing a known term in a kinetic equation; δ represents polymerization disturbance, δ ═ M0 -1Omega and delta comprise two parts of uncertain parameters and external moment interference and meet the assumption condition that | | | delta | | < delta | |, and1，
Figure BDA00023688048300000816
δ1>0，δ2>0。
s4, establishing a fixed time disturbance observer according to the established state space model; wherein the established fixed-time disturbance observer is represented as:
Figure BDA0002368804830000091
wherein,
Figure BDA0002368804830000092
and
Figure BDA0002368804830000093
respectively represent x2And an estimate of δ;
Figure BDA0002368804830000094
and
Figure BDA0002368804830000095
are respectively as
Figure BDA0002368804830000096
And
Figure BDA0002368804830000097
the rate of change of (c);
Figure BDA0002368804830000098
representing an angular velocity estimation error; m is1、n1、m2、n2And gamma represent the gain of the fixed time disturbance observer; p is a radical of1、p2、q1And q is2All represent fixed time disturbance observer fractional power parameters; the function sig (-) is of the form: sigz(y)＝|y|zSign (y), sign (y) the sign function expressed as y is:
Figure BDA0002368804830000099
in this embodiment, observer parameter 0 is selected<p1<1，q1>1，0<p2<1，q2>1，
Figure BDA00023688048300000910
And parameter m1，m2，n1，n2The values are selected such that the fixed time disturbance observer is herwitz.
In this embodiment, when the fixed-time disturbance observer is selected and used and the assumed condition | | | δ | ≦ δ |, is satisfied1、
Figure BDA00023688048300000911
Time, velocity tracking error x2And the aggregate disturbance term δ will be
Figure BDA00023688048300000912
And
Figure BDA00023688048300000913
estimating and satisfying angular velocity estimation error
Figure BDA00023688048300000914
And aggregate disturbance estimation error
Figure BDA00023688048300000915
Will converge to zero within a fixed time.
In this embodiment, the fixed-time disturbance observer designs the variation rate of the acceleration error and the disturbance total set estimated value based on a state space model of the mechanical arm trajectory tracking error
Figure BDA00023688048300000916
And
Figure BDA00023688048300000917
the control method is made to have global robustness to unknown disturbances.
S5, tracking error signals e and
Figure BDA00023688048300000918
establishing a fixed-time nonsingular terminal sliding mode surface; the established fixed-time nonsingular terminal sliding mode surface s is expressed as:
Figure BDA00023688048300000919
wherein, α1,α2A fixed-time nonsingular terminal sliding mode surface gain coefficient, namely a sliding mode surface gain coefficient α1、α2Satisfies the condition α1>0，α2>0；β＝[β1,β2,…βn]TRepresenting variable-structure sliding-mode terms used to cope with singularity problems, which constitute the parameter βiIs defined as:
Figure BDA00023688048300000920
wherein, i is 1, …, n, x1iRepresenting the velocity tracking error of the ith mechanical arm joint; judgment of conditions
Figure BDA00023688048300000921
Is defined as
Figure BDA0002368804830000101
γ1、γ2And κ both represent constant coefficients; and constant coefficient
Figure BDA0002368804830000102
Figure BDA0002368804830000103
γ1>1，0<γ2<1 and κ is a sufficiently small normal number.
In this embodiment, on the fixed-time nonsingular terminal sliding mode surface, the proposed variable-structure sliding mode term can avoid the singularity problem existing in the sliding mode control, so that the convergence time of the system state is unrelated to its initial state.
S6, establishing a nonlinear anti-saturation compensator according to the actuator saturation effect parameters; wherein the established nonlinear anti-saturation compensator is expressed as:
Figure BDA0002368804830000104
wherein ξ is the state of the nonlinear anti-saturation compensator, k is a positive constant coefficient, u isΔRepresenting the actuator saturation effect parameter, uΔ＝u-u0，uΔThe method is used for describing the saturation degree of the output torque of the actuator under the condition that limited constraint exists; u. of0And the control torque command of each joint driving motor of the mechanical arm is represented.
In the embodiment, the nonlinear anti-saturation compensator provided under the theoretical framework of closed-loop fixed time stability well solves the problem of saturation effect of the manipulator controller, avoids overlarge control input, further reduces the abrasion of an actual actuating mechanism, and prolongs the service life of the actual actuating mechanism.
S7, preliminarily determining a control torque instruction u of each joint driving motor of the mechanical arm according to the established fixed time disturbance observer, the established fixed time nonsingular terminal sliding mode surface and the established nonlinear anti-saturation compensator0And combining the control torque output range of the actuator to obtain the control torque u of each joint of the mechanical arm.
In this embodiment, the preliminarily determined control torque command of each joint driving motor of the mechanical arm is represented as:
Figure BDA0002368804830000105
wherein,
Figure BDA0002368804830000106
Figure BDA0002368804830000107
Figure BDA0002368804830000108
a representation definition;
Figure BDA0002368804830000109
representing the rate of change of the variable structure sliding mode term β.
Figure BDA00023688048300001010
Is composed of
Figure BDA00023688048300001011
In the form of a short-hand writing of (1),
Figure BDA00023688048300001012
in this embodiment, when there is input saturation, the actual joint drive motor control torque u of the robot arm should satisfy the input limitation constraint in S1, and the finally obtained robot arm joint control torque u ═ u1,u2,…,un]TComprises the following steps:
Figure BDA0002368804830000111
then, the effectiveness and the superiority of the method for tracking and controlling the fixed time trajectory of the uncertain mechanical arm with input saturation provided by the embodiment are verified by adopting a simulation experiment, and a finite time disturbance observer and a linear expansion observer are adopted in the simulation to carry out a comparative simulation experiment. The simulation platform is carried out based on Matlab 2017b under a win10x64 bit operating system. The simulation object is a two-degree-of-freedom mechanical arm as shown in fig. 3. The adopted finite-time disturbance observation and linear expansion observer are designed into the following forms:
Figure BDA0002368804830000112
Figure BDA0002368804830000113
in the simulation case, the kinetic equation of the two-degree-of-freedom rigid mechanical arm can be established as:
Figure BDA0002368804830000114
wherein,
Figure BDA0002368804830000115
Figure BDA0002368804830000116
Figure BDA0002368804830000117
in the case of neglecting the joint friction force, it can be considered that
Figure BDA0002368804830000118
Wherein x1＝[x11x12]T，x2＝[x21x22]T，
Figure BDA0002368804830000119
p3＝m2l1lc2，p4＝m1lc2+m2l1，p5＝m2lc2，
Figure BDA00023688048300001110
The physical parameters of the two-degree-of-freedom rigid mechanical arm are shown in table 1:
TABLE 1 physical parameters of two-DOF rigid manipulator
Symbol Definition of Parameter value Symbol Definition of Parameter value
m1 Mass of connecting rod 1 2.00(kg) l1 Length of connecting rod 1 0.35(m)
m2 Mass of connecting rod 2 0.85(kg) l2 Length of connecting rod 2 0.31(m)
lc1 Distance from joint 1 to connecting rod 1 centroid 0.175(m) g Constant of gravity 9.8(m/s2)
lc2 Distance from joint 2 to connecting rod 2 centroid 0.155(m)
The external disturbance torque in the dynamic equation is set as:
d(t)＝[sin(t)+0.25sin(t) 0.5cos(t)+0.25sin(t)]T
based on the above system parameters, the initial condition of the mechanical arm is x11(0)＝x12(0)＝1(rad)，x21(0)＝x22(0) 0 (rad/s). The expected track is designed as
Figure BDA0002368804830000121
The initial state of the observer is defined as
Figure BDA0002368804830000122
The controller parameters involved in the simulation are shown in Table 2
TABLE 2 controller parameters
Figure BDA0002368804830000123
Then, the effectiveness and the superiority of the method for tracking and controlling the fixed time trajectory of the uncertain mechanical arm with input saturation provided by the embodiment are verified by adopting a simulation experiment, and a finite time disturbance observer and a linear expansion observer are adopted in the simulation to carry out a comparative simulation experiment. The simulation platform is carried out based on Matlab 2017b under a win10x64 bit operating system. The simulation object is a two-degree-of-freedom mechanical arm as shown in figure 3
Simulation results are shown in fig. 4 to fig. 15, wherein, based on the tracking response curves of the position and the velocity of the joint of the rigid mechanical arm with two degrees of freedom under the proposed uncertain mechanical arm fixed time trajectory tracking control method with input saturation, as shown in fig. 4 and fig. 9, under the control method proposed by the embodiment of the invention, the position and the velocity of the joint of the mechanical arm can track an expected trajectory in a fixed time; the output of the controller can be maintained in the constraint range of the actuator under the action of the nonlinear anti-saturation compensator, the problem that the output of the manipulator exceeds the effective input range of the actuator is solved, and all observation errors can be converged to zero in fixed time according to the observation performance of the fixed-time interference observer. Further, in comparison with fig. 10 to 15, it can be seen that the observation error in the control group takes longer time to converge to zero, which will generate larger control input in the process of convergence of the observation error to zero according to the controller form proposed in the present invention, thereby affecting the tracking performance of the system. The simulation example shows that the provided uncertain mechanical arm fixed time trajectory tracking control method with input saturation has good control performance and achieves an expected control target.
In addition, the following should be noted:
in FIG. 4, FIG. 5, FIG. 11, FIG. 14
Figure BDA0002368804830000134
Respectively represents the actual speed of the two-degree-of-freedom mechanical arm joint 1 and the joint 2,
Figure BDA0002368804830000133
respectively representing the expected speeds of the two-degree-of-freedom mechanical arm joint 1 and the joint 2; τ in FIG. 61、τ2Respectively representing the actual control moment of the two-degree-of-freedom mechanical arm joint 1 and the actual control moment of the two-degree-of-freedom mechanical arm joint 2; in FIG. 7, FIG. 12, FIG. 15
Figure BDA0002368804830000132
Respectively representing disturbance aggregate observation errors of a two-degree-of-freedom mechanical arm joint 1 and a joint 2; in FIG. 8
Figure BDA0002368804830000131
Respectively representing acceleration observation errors of a two-degree-of-freedom mechanical arm joint 1 and a mechanical arm joint 2; e in FIG. 91、e2Respectively representing the position tracking errors of a two-degree-of-freedom mechanical arm joint 1 and a joint 2; q in FIGS. 10 and 131、q2Respectively representing the actual positions/angles, q, of the two-degree-of-freedom mechanical arm joints 1 and 2d1、qd2The desired positions/angles of the two-degree-of-freedom robot joints 1, 2 are indicated, respectively.
While the foregoing is directed to the preferred embodiment of the present invention, it will be understood by those skilled in the art that various changes and modifications may be made without departing from the spirit and scope of the invention as defined in the appended claims.

Claims (10)

1. A fixed time trajectory tracking control method for an uncertain mechanical arm with input saturation is characterized by comprising the following steps:
establishing a dynamic equation of an n-degree-of-freedom rotary joint rigid mechanical arm system with a viscous friction term;
according to the established kinetic equation and the tracking error signal of the mechanical arm track, establishing a tracking error kinetic equation considering parameter uncertainty, wherein the parameter uncertainty comprises: dynamic modeling errors and unknown external disturbances;
establishing a state space model of the tracking error of the mechanical arm track according to a tracking error kinetic equation considering parameter uncertainty;
establishing a fixed time interference observer according to the established state space model;
establishing a fixed-time nonsingular terminal sliding mode surface according to the track tracking error signal of the mechanical arm;
establishing a nonlinear anti-saturation compensator according to the actuator saturation effect parameters;
according to the established fixed time disturbance observer, the fixed time nonsingular terminal sliding mode surface and the nonlinear anti-saturation compensator, preliminarily determining a control torque instruction of each joint driving motor of the mechanical arm, and combining a control torque output range of an actuator to obtain control torque of each joint of the mechanical arm.
2. The method for controlling the fixed-time trajectory tracking of the uncertain manipulator with input saturation according to claim 1, wherein the established kinetic equation is expressed as:
Figure FDA0002368804820000011
wherein x is,
Figure FDA0002368804820000012
Respectively representing the angle, the angular velocity and the angular acceleration of the mechanical arm joint; m (x),
Figure FDA0002368804820000013
D. g (x) respectively representing an actual positive definite inertia matrix, a centrifugal force and Coriolis force matrix, a joint viscous friction coefficient matrix and a gravity vector of the mechanical arm; m0(x)、C0(x)、D0、g0(x) Respectively representing positive definite inertia matrix, centrifugal force and Coriolis force matrix and joint viscous friction coefficient matrix of the mechanical armAn estimated value of the gravity vector; mΔ(x)、CΔ(x)、DΔ(x)、gΔ(x) Respectively representing the estimation errors of a positive definite inertia matrix, a centrifugal force and Coriolis force matrix, a joint viscous friction coefficient matrix and a gravity vector of the mechanical arm; u is the control moment of each joint of the mechanical arm; d is the external disturbance moment applied to the mechanical arm.
3. The method for controlling the fixed-time trajectory tracking of the uncertain manipulator with input saturation according to claim 2, wherein the input limit constraint satisfied by the control moment u is as follows:
Figure FDA00023688048200000211
Figure FDA0002368804820000021
wherein u isiRepresents the actual control moment, u, of the ith robot arm jointi0The theoretical control moment of the ith mechanical arm joint obtained by the track tracking control method is shown, n is the number of the mechanical arm joints,
Figure FDA00023688048200000212
and
Figure FDA00023688048200000213
respectively, the minimum and maximum control torques that the actuator is capable of outputting.
4. The method for controlling the fixed-time trajectory tracking of the uncertain manipulator having input saturation according to claim 3, wherein the establishing of the tracking error kinetic equation considering the uncertainty of the parameters according to the established kinetic equation and the manipulator trajectory tracking error signal comprises:
according to the expected tracking angle x of each joint of the mechanical armdDesired tracking angular velocity
Figure FDA0002368804820000022
And the measured angle x and angular velocity of each joint of the mechanical arm
Figure FDA0002368804820000023
Calculating the tracking error e of the mechanical arm track position as x-xdAnd velocity tracking error
Figure FDA0002368804820000024
According to the established kinetic equation and the tracking error signal e and
Figure FDA0002368804820000025
and establishing a tracking error kinetic equation considering parameter uncertainty.
5. The method for controlling the fixed-time trajectory tracking of the uncertain manipulator with input saturation according to claim 4, wherein the established tracking error kinetic equation is expressed as:
Figure FDA0002368804820000026
wherein,
Figure FDA0002368804820000027
respectively representing an angular velocity tracking error and an angular acceleration tracking error; x is the number ofd、
Figure FDA0002368804820000028
Respectively representing a desired tracking angle, a desired tracking angular velocity and a desired tracking angular acceleration;
Figure FDA0002368804820000029
representing a kinematically known term related to a desired tracking angle, a desired tracking angular velocity, and a desired tracking angular acceleration;
Figure FDA00023688048200000210
representing the entire set of unknown terms in the tracking error dynamics equation.
6. The method for controlling the fixed-time trajectory tracking of the uncertain manipulator with input saturation according to claim 5, wherein the established state space model is expressed as:
Figure FDA0002368804820000031
wherein x is1Indicating position tracking error, x1＝e；
Figure FDA0002368804820000032
And x2Are indicative of a speed tracking error and,
Figure FDA0002368804820000033
η＝-M0 -1(C0+D0)x2-M0 -1h0representing a known term in a kinetic equation; delta-M0 -1ω represents the aggregate disturbance and δ contains both parametric uncertainty and external moment disturbances.
7. The method for controlling the trajectory tracking of the fixed time of the uncertain manipulator with input saturation according to claim 6, wherein the established fixed time disturbance observer is represented as:
Figure FDA0002368804820000034
wherein,
Figure FDA0002368804820000035
and
Figure FDA0002368804820000036
respectively represent x2And an estimate of δ;
Figure FDA0002368804820000037
and
Figure FDA0002368804820000038
are respectively as
Figure FDA0002368804820000039
And
Figure FDA00023688048200000310
the rate of change of (c);
Figure FDA00023688048200000311
representing an angular velocity estimation error; m is1、n1、m2、n2And gamma represent the gain of the fixed time disturbance observer; p is a radical of1、p2、q1And q is2All represent fixed time disturbance observer fractional power parameters; the function sig (-) is of the form: sigz(y)＝|y|zSign (y) a sign function denoted y.
8. The method for controlling the fixed-time trajectory tracking of the uncertain manipulator with input saturation according to claim 7, wherein the established fixed-time nonsingular terminal sliding mode surface s is represented as:
Figure FDA00023688048200000312
wherein, α1,α2For fixed time nonsingular terminal sliding mode face gain coefficient, β ═ β1,β2,…βn]TRepresenting variable-structure sliding-mode terms used to cope with singularity problems, which constitute the parameter βiExpressed as:
Figure FDA00023688048200000313
wherein, i is 1, …, n, x1iRepresenting the velocity tracking error of the ith mechanical arm joint; judgment of conditions
Figure FDA00023688048200000314
γ1、γ2And κ both represent constant coefficients; coefficient of constant
Figure FDA00023688048200000316
Figure FDA00023688048200000317
9. The method of controlling fixed-time trajectory tracking of an uncertain manipulator with input saturation according to claim 8, wherein the established nonlinear anti-saturation compensator is represented as:
Figure FDA00023688048200000315
wherein ξ is the state of the nonlinear anti-saturation compensator, k is a positive constant coefficient, u isΔRepresenting the actuator saturation effect parameter, uΔ＝u-u0；u0And the control torque command of each joint driving motor of the mechanical arm is represented.
10. The method for controlling the trajectory tracking of the fixed time of the uncertain manipulator with the input saturation as recited in claim 9, wherein the preliminarily determined control torque command of the driving motor of each joint of the manipulator is represented as:
Figure FDA0002368804820000041
wherein,
Figure FDA0002368804820000042
is the intermediate variable(s) of the variable,
Figure FDA0002368804820000043
a representation definition;
Figure FDA0002368804820000044
represents the rate of change of variable structure sliding mode term β;
Figure FDA0002368804820000045
is composed of
Figure FDA0002368804820000046
In the form of a short-hand writing of (1),
Figure FDA0002368804820000047
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