CN115179300A - Flexible mechanical arm trajectory tracking control method for preset time - Google Patents

Flexible mechanical arm trajectory tracking control method for preset time Download PDF

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CN115179300A
CN115179300A CN202211039499.8A CN202211039499A CN115179300A CN 115179300 A CN115179300 A CN 115179300A CN 202211039499 A CN202211039499 A CN 202211039499A CN 115179300 A CN115179300 A CN 115179300A
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mechanical arm
motor
track
trajectory
preset time
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王洋
管艳
谢飞
王震
陈明淑
田怀谷
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Xijing University
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Xijing University
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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
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning
    • 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
    • B25J9/1635—Programme controls characterised by the control loop flexible-arm control

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Abstract

A flexible mechanical arm track tracking control method with preset time aims at the track tracking control and vibration problems of a flexible mechanical arm; firstly, a Fourier track planning function is designed for a flexible mechanical arm joint system model, belongs to open-loop control, and can eliminate the residual vibration error of the mechanical arm by adjusting coefficient parameters; on the basis, a nonsingular preset time sliding mode surface is provided by considering a dynamic loop in the motor, and finally, a preset time sliding mode controller is designed based on the provided sliding mode surface, so that the motor track tracks the motor track subjected to Fourier programming in preset time; the control method is not influenced by the initial value of the system, debugging and parameter adjustment are not needed, the control design is simple, and the performance is more stable and reliable.

Description

Flexible mechanical arm trajectory tracking control method for preset time
Technical Field
The invention relates to the technical field of control, in particular to a flexible mechanical arm track tracking control method for preset time.
Background
The flexible joint mechanical arm is widely applied to industries such as medical treatment, aerospace and machinery and social practices due to the characteristics of flexibility, rapidness, accuracy and the like, but the flexible joint mechanical arm can generate a vibration problem which is difficult to control, so that the operation precision and the production efficiency of the mechanical arm are reduced. The research shows that the trajectory planning is one of the core technologies of the robot mechanical arm for controlling vibration. Track planning is an effective open-loop control method for suppressing vibration, common track planning functions include cubic interpolation and quintic interpolation polynomials, and some complex functions and other functions exist. The trajectory planning function needs to meet certain initial conditions, and the cubic polynomial can ensure the smooth and continuous joint angles and angular velocities, but cannot meet the boundary conditions of acceleration; the fifth-order polynomial can satisfy the boundary conditions of the angle, the angular velocity, and the angular acceleration, but includes a high-order function, easily causing vibration.
The finite time sliding mode controller is closely related to the initial conditions of the system, and the system with unknown initial conditions cannot be controlled. In order to overcome the difficulty faced by the limited time, a method for stabilizing the fixed time is provided, which can control the system to a stable state without being influenced by the initial value of the system, but needs to adjust the parameters for multiple times to achieve the optimal control effect.
The patent application CN108789418a discloses a control method of a flexible mechanical arm, which combines a trajectory tracking controller and a vibration suppression controller to realize trajectory tracking and vibration suppression of the flexible mechanical arm, and firstly, a sliding mode variable structure controller is designed to track the trajectory of the mechanical arm; then, a correlation algorithm is applied to perform parameter optimization to suppress the vibration. However, the algorithm is complex with many implementation parameters.
Disclosure of Invention
In order to solve the defects in the prior art, the invention aims to provide a flexible mechanical arm trajectory tracking control method for a preset time, provides Fourier trajectory planning, belongs to open-loop control, and compared with closed-loop control, has the advantages of simple control design and more stable and reliable performance, can enable the flexible mechanical arm to track the planned trajectory as soon as possible in motion, and cannot generate a buffeting phenomenon after the flexible mechanical arm moves.
In order to achieve the purpose, the invention adopts the technical scheme that:
a flexible mechanical arm track tracking control method for preset time specifically comprises the following steps:
step 1, establishing a dynamic model of a motor rotor, a dynamic model of a joint end and an error tracking dynamic model on the basis of complex dynamic behaviors of a flexible mechanical arm and assuming that a flexible joint is a linear spring;
step 2: under the condition of not considering the internal dynamics of the motor, a Fourier track plan is designed for the mechanical arm, and buffeting is eliminated by using an open loop control method;
and 3, on the basis of the step 1 and the step 2, considering the internal dynamics of the motor, designing a nonsingular preset time sliding mode control in order to enable the motor track to track the planned Fourier track, and finally enabling the position tracking error of the mechanical arm to be converged to zero in preset time.
The specific method of the step 1 comprises the following steps:
based on the complex dynamic behavior of the flexible mechanical arm, assuming that the flexible joint is a linear spring, a dynamic model of the motor rotor and a dynamic model of the joint end are established as follows
Figure BDA0003820427500000021
Where θ represents the rotation angle of the joint, q represents the motor angle, τ m Representing the driving torque input by the motor, K representing the torsional stiffness coefficient, J l Representing the total amount of inertia at the joint end, τ f Denotes the viscous friction torque, J m Representing an amount of inertia of the rotor of the motor;
let x 1 ＝θ，
Figure BDA0003820427500000022
x 3 ＝q，
Figure BDA0003820427500000023
Expressing the form of the differential equation of the formula (1) asForm of the equation of state
Figure BDA0003820427500000024
Assume the desired trajectory of the robot arm motor is
Figure BDA0003820427500000025
The error of the motor's trajectory from the desired trajectory is expressed as
Figure BDA0003820427500000026
Then, define
Figure BDA0003820427500000027
The error dynamic equation of the flexible mechanical arm is defined as
Figure BDA0003820427500000031
The specific method of the step 2 comprises the following steps:
considering the influence of the track planning on the flexible joint only without considering the internal dynamics of the motor, wherein the mechanical arm system model of the track planning is
Figure BDA0003820427500000032
Order to
Figure BDA0003820427500000033
Equation (5) can be expressed as:
Figure BDA0003820427500000034
the trajectory planning of the mechanical arm meets the following boundary conditions:
Figure BDA0003820427500000035
under the influence of a trajectory planning theory and an initial value, a trajectory planning function based on Fourier series is designed as follows:
Figure BDA0003820427500000036
wherein,
Figure BDA0003820427500000037
representing the expected trajectory of the joint, ω n ＝nπ/t f ，n＝1,2,3,q f ，q 0 Is the initial and target positions of the joint, t f Is the time of the trajectory planning.
The specific method in the step 3 comprises the following steps:
in order to allow the motor to track the upper planned fourier trajectory within a predetermined time, taking into account the internal dynamics of the motor, the predetermined time sliding mode of equation (2) is designed as follows:
Figure BDA0003820427500000038
wherein T is s Is a predetermined time parameter, alpha is more than 0 and less than 1, and a non-singular function is defined as
Figure BDA0003820427500000039
The non-singular functions of the design are differentiable, and therefore,
Figure BDA00038204275000000310
the derivative of (d) is expressed as:
Figure BDA0003820427500000041
wherein
Figure BDA0003820427500000042
And delta = alpha, and the ratio of delta = alpha,
according to the sliding surface equation (10), the controller is designed to:
Figure BDA0003820427500000043
wherein 0 < alpha s ＜1。
Compared with the prior art, the invention has the following advantages:
(1) Compared with the quintic interpolation function trajectory planning, the proposed Fourier method can eliminate the residual vibration error of the flexible mechanical arm.
(2) The Fourier trajectory planning belongs to open-loop control, and compared with closed-loop control, the control design is simple, and the performance is more stable and reliable.
(3) On the basis, a dynamic loop in the motor is considered, and a novel nonsingular preset time sliding mode surface is provided, and the sliding mode surface can enable the motor track to track and carry out Fourier track planning in a preset time.
Drawings
Fig. 1 is a system model diagram of a flexible joint robot arm according to the present invention.
Fig. 2 is a schematic diagram of a motor plan of the present invention.
Fig. 3 is a schematic diagram of the joint and motor trajectory error of the present invention.
Fig. 4 is a schematic diagram of a sliding mode surface s simulation according to the present invention.
FIG. 5 is a simulation diagram of the controller u according to the present invention.
FIG. 6 shows the error e of the present invention 1 And (5) simulation schematic diagram.
FIG. 7 shows the error e of the present invention 2 And (5) simulation schematic diagram.
FIG. 8 illustrates the motor trajectory q tracking the desired trajectory in accordance with the present invention
Figure BDA0003820427500000044
And (5) simulation schematic diagram.
FIG. 9 is a flow chart of the steps performed in the present invention.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings.
A flexible mechanical arm track tracking control method for preset time specifically comprises the following steps:
step 1, establishing a dynamic model of a motor rotor, a dynamic model of a joint end and an error tracking dynamic model under the assumption that a flexible joint is a linear spring due to the complex dynamic behavior of the flexible mechanical arm:
fig. 1 shows a system model of a flexible joint robot arm, and due to the complex dynamic behavior of the flexible robot arm, assuming that the flexible joint is a linear spring, a dynamic model of a motor rotor and a dynamic model of a joint end are established as follows:
Figure BDA0003820427500000051
where θ represents the rotation angle of the joint, q represents the motor angle, and τ m Representing the driving torque input by the motor, K representing the torsional stiffness coefficient, which is generally obtained experimentally, J l Representing the total amount of inertia at the joint end, τ f Expressing viscous friction torque, J m Representing an amount of inertia of the rotor of the motor;
let x 1 ＝θ，
Figure BDA0003820427500000052
x 3 ＝q，
Figure BDA0003820427500000053
Expressing the differential equation form of the formula (1) as a form of a state equation
Figure BDA0003820427500000054
Assume that the desired trajectory of the robot arm motor is
Figure BDA0003820427500000055
The error of the trajectory of the motor from the desired trajectory can be expressed as:
Figure BDA0003820427500000056
then, define
Figure BDA0003820427500000057
The error dynamic equation of the flexible mechanical arm can be defined as:
Figure BDA0003820427500000058
step 2: under the condition of not considering the internal dynamics of a motor, a Fourier track plan is designed for the mechanical arm, and buffeting elimination is carried out by utilizing an open loop control method:
the internal dynamics of the motor are not considered, and only the influence of the trajectory planning on the flexible joint is considered. The mechanical arm system model for trajectory planning is as follows:
Figure BDA0003820427500000061
order to
Figure BDA0003820427500000062
Equation (13) can be expressed as:
Figure BDA0003820427500000063
the trajectory planning of the mechanical arm meets the following boundary conditions:
Figure BDA0003820427500000064
under the influence of a trajectory planning theory and an initial value, a trajectory planning function based on Fourier series is designed as follows:
Figure BDA0003820427500000065
wherein
Figure BDA0003820427500000066
Representing the expected trajectory of the joint, ω n ＝nπ/t f ，n＝1,2,3,q f ，q 0 Is the initial and target positions of the joint, t f Is the time of the trajectory planning.
The angular velocity and the angular acceleration of the motor track can be obtained by solving the first lead and the second lead of the formula (8)
Figure BDA0003820427500000067
Figure BDA0003820427500000068
Combining equations (7) (8) (8-1) and (8-2) can yield:
Figure BDA0003820427500000069
the parameter alpha can be designed 0 ,α 1 ,α 3 ,α 5 Comprises the following steps:
Figure BDA0003820427500000071
step 3, on the basis of the step 1 and the step 2, considering the internal dynamics of the motor, designing a nonsingular preset time sliding mode control for enabling the motor track to track the planned Fourier track, and finally enabling the position tracking error of the mechanical arm to be converged to zero in a preset time:
in order to allow the motor to track the upper planned fourier trajectory within a predetermined time, taking into account the internal dynamics of the motor, the predetermined time sliding mode surface of equation (2) can be designed as follows:
Figure BDA0003820427500000072
wherein T is s Is a predetermined time parameter, 0 < alpha < 1. The nonsingular function is defined as:
Figure BDA0003820427500000073
the non-singular functions of the design are differentiable, and therefore,
Figure BDA0003820427500000074
the derivative of (d) can be expressed as:
Figure BDA0003820427500000075
wherein
Figure BDA0003820427500000076
And δ = α.
Derivation of formula (9) can result in
Figure BDA0003820427500000077
As can be seen from error equations (4) and (9):
Figure BDA0003820427500000078
according to sliding mode surface equation (9), the predetermined time controller can be designed as follows:
Figure BDA0003820427500000081
the stability at a predetermined time proves that:
the first step is as follows: the stability of the slip form surface is proved:
one Lyapunov function is chosen as follows:
Figure BDA0003820427500000082
derivation of equation (5) can be found:
Figure BDA0003820427500000083
therefore, from the predetermined time stability theorem it is known that:
Figure BDA0003820427500000084
v＝s＝0，t≤T s (16)
the second step is that: the motor track tracking stability proves that:
the combinations (9) and (16) are known
Figure BDA0003820427500000085
When | e 1 When | ≧ δ
Figure BDA0003820427500000086
Wherein
Figure BDA0003820427500000087
When | e 1 If | is less than δ, then there is
|e 1 |≤δ，ift≥εT s +γ(T e -T s ) (19)
Wherein
Figure BDA0003820427500000088
Let epsilon < gamma, zoom in on time
|e 1 |≤δ，ift≥γT s (20)
Is combined with (20)
s＝0，t≥γT s (21)
Then, combining equations (9) and (10), it can be known that:
Figure BDA0003820427500000091
the lyapunov function is as follows:
Figure BDA0003820427500000092
the derivation of equation (23) can be:
Figure BDA0003820427500000093
when delta E (0,1/e) and alpha E (0,1), there are:
Figure BDA0003820427500000094
Figure BDA0003820427500000095
consider equations (20) and
Figure BDA0003820427500000096
it is possible to obtain:
Figure BDA0003820427500000097
then, combining the equations (23), (24), (25), (26) and (27), it is possible to obtain
Figure BDA0003820427500000098
Wherein g is e1 ＝(δ -2α /2(1-α-δlnδ))δ＞0
From equation (28), it can be derived that H is less than zero
Figure BDA0003820427500000099
Thus, at | e 1 Under the condition that | < delta, when T is more than or equal to gamma T s H (∞) =0.
End of certification
The simulation results are shown in fig. 2-8. Fig. 2 is a designed fourier motor plan, and fig. 3 is a joint end residual error situation, which is compared with a quintic interpolation trajectory plan, respectively, so that it can be seen that the designed plan can eliminate residual vibration errors; fig. 4 shows a designed non-singular predetermined time sliding mode surface, it can be seen that the sliding mode surface reaches zero within a predetermined time t =1 s; FIG. 5 shows a controller of the design; fig. 6 and 7 show the error condition of the fourier plan of the motor trajectory tracking, and it can be seen that the error reaches zero at the predetermined time t =1 s; fig. 8 is a state diagram of motor trajectory tracking, and it can be clearly seen that the motor trajectory can track the fourier trajectory plan at t =1 s.
In summary, the design of the invention aims at the track tracking control and vibration problems of the flexible mechanical arm, firstly, a Fourier track planning function is designed for a joint system model, and the residual vibration error of the mechanical arm can be eliminated by adjusting parameters. Then, considering the dynamic loop in the motor, a nonsingular predetermined time sliding mode surface is provided, and finally, a predetermined time sliding mode controller is designed based on the provided sliding mode surface, and like the existing fixed time sliding mode controller, the provided predetermined time sliding mode controller has a fast convergence speed independent of the initial conditions of the system, and unlike the fixed time sliding mode controller, the convergence time of the provided method is predefined, which means that the provided method can directly reach the expected convergence time without using a trial and error method to select control parameters. Has wide application, and can be applied in the fields of medical treatment, aerospace, industrial production and the like.

Claims (4)

1. A flexible mechanical arm track tracking control method with preset time is characterized in that: the method specifically comprises the following steps:
step 1, establishing a dynamic model of a motor rotor, a dynamic model of a joint end and an error tracking dynamic model on the basis of complex dynamic behaviors of a flexible mechanical arm and assuming that a flexible joint is a linear spring;
step 2: under the condition of not considering the internal dynamics of the motor, a Fourier track plan is designed for the mechanical arm, and buffeting is eliminated by using an open loop control method;
and 3, considering the internal dynamics of the motor on the basis of the steps 1 and 2, designing a nonsingular preset time sliding mode control for enabling the motor track to track the planned Fourier track, and finally enabling the position tracking error of the mechanical arm to be converged to zero in preset time.
2. The method for controlling the trajectory tracking of the flexible mechanical arm at the preset time according to claim 1, wherein the method comprises the following steps: the specific method of the step 1 comprises the following steps:
based on the complex dynamic behavior of the flexible mechanical arm, assuming that the flexible joint is a linear spring, a dynamic model of the motor rotor and a dynamic model of the joint end are established as follows:
Figure FDA0003820427490000011
where θ represents the rotation angle of the joint, q represents the motor angle, τ m Representing the driving torque input by the motor, K representing the torsional stiffness coefficient, J l Representing the total amount of inertia at the joint end, τ f Indicating viscous frictionMoment, J m Representing an amount of inertia of the rotor of the motor;
let x be 1 ＝θ，
Figure FDA0003820427490000012
x 3 ＝q，
Figure FDA0003820427490000013
Expressing the differential equation form of the formula (1) as a form of a state equation
Figure FDA0003820427490000014
Assume the desired trajectory of the robot arm motor is
Figure FDA0003820427490000015
The error of the trajectory of the motor from the desired trajectory is expressed as:
Figure FDA0003820427490000021
then, define
Figure FDA0003820427490000022
The error dynamic equation of the flexible mechanical arm is defined as:
Figure FDA0003820427490000023
3. the method for controlling the trajectory tracking of the flexible mechanical arm in the preset time according to claim 1, wherein the method comprises the following steps: the specific method of the step 2 comprises the following steps:
the internal dynamic of the motor is not considered, only the influence of the track planning on the flexible joint is considered, and the mechanical arm system model of the track planning is
Figure FDA0003820427490000024
Order to
Figure FDA0003820427490000025
Equation (5) can be expressed as
Figure FDA0003820427490000026
The trajectory planning of the mechanical arm meets the following boundary conditions:
Figure FDA0003820427490000027
under the influence of a trajectory planning theory and an initial value, a trajectory planning function based on Fourier series is designed as follows:
Figure FDA0003820427490000028
wherein,
Figure FDA0003820427490000029
representing the expected trajectory of the joint, ω n ＝nπ/t f ，n＝1,2,3,q f ，q 0 Is the initial and target positions of the joint, t f Is the time of the trajectory planning.
4. The method for controlling the trajectory tracking of the flexible mechanical arm in the preset time according to claim 1, wherein the method comprises the following steps: the specific method in the step 3 comprises the following steps:
in order to allow the motor to track the upper planned fourier trajectory within a predetermined time, taking into account the internal dynamics of the motor, the predetermined time sliding mode of equation (2) is designed as follows:
Figure FDA00038204274900000210
wherein T is s Is a predetermined time parameter, alpha is more than 0 and less than 1, and the non-singular function is defined as:
Figure FDA0003820427490000031
the non-singular functions of the design are differentiable, and therefore,
Figure FDA0003820427490000032
the derivative of (d) is expressed as:
Figure FDA0003820427490000033
wherein
Figure FDA0003820427490000034
And delta = alpha, and the ratio of delta = alpha,
according to the sliding surface equation (10), the controller is designed to:
Figure FDA0003820427490000035
wherein 0 < alpha s ＜1。
CN202211039499.8A 2022-08-29 2022-08-29 Flexible mechanical arm trajectory tracking control method for preset time Pending CN115179300A (en)

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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116931436A (en) * 2023-09-11 2023-10-24 中国科学院西安光学精密机械研究所 Design method of flexible mechanism self-adaptive tracking control and vibration suppression controller

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116931436A (en) * 2023-09-11 2023-10-24 中国科学院西安光学精密机械研究所 Design method of flexible mechanism self-adaptive tracking control and vibration suppression controller
CN116931436B (en) * 2023-09-11 2024-01-30 中国科学院西安光学精密机械研究所 Design method of flexible mechanism self-adaptive tracking control and vibration suppression controller

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