CN114454161A - Manipulator rigid-flexible coupling system based on spherical motor drive - Google Patents

Manipulator rigid-flexible coupling system based on spherical motor drive Download PDF

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CN114454161A
CN114454161A CN202210013643.4A CN202210013643A CN114454161A CN 114454161 A CN114454161 A CN 114454161A CN 202210013643 A CN202210013643 A CN 202210013643A CN 114454161 A CN114454161 A CN 114454161A
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manipulator
rigid
flexible coupling
spherical motor
coupling system
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CN114454161B (en
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过希文
刘亚坤
王群京
李国丽
文彥
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Anhui 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
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/161—Hardware, e.g. neural networks, fuzzy logic, interfaces, processor
    • 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
    • B25J9/163—Programme controls characterised by the control loop learning, adaptive, model based, rule based expert control

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Abstract

The invention provides a manipulator rigid-flexible coupling system based on spherical motor driving, which takes a permanent magnet spherical motor as a driving motor of the manipulator rigid-flexible coupling system, and adopts a flexible mechanical arm to replace a rigid output shaft of the permanent magnet spherical motor to be connected with a paw to form a manipulator. The rigid-flexible coupling system mainly comprises a manipulator rigid-flexible coupling dynamics module and an inversion sliding mode control module. Establishing a partial differential dynamics model of the manipulator under a follow-up coordinate system based on a Hamilton method, and constructing the rigid-flexible coupling dynamics module by combining a rigid shaft dynamics model of the permanent magnet spherical motor; estimating unknown interference by selecting a proper nonlinear gain function, and designing the inversion sliding mode control module; the invention realizes the modeling and control of the manipulator rigid-flexible coupling system driven by the spherical motor, avoids the stress analysis of a complex system, realizes the high-precision robust control and can provide a flexible solution for the manufacturing industry and the service industry.

Description

Manipulator rigid-flexible coupling system based on spherical motor drive
Technical Field
The invention belongs to the technical field of application of special motors, and relates to modeling and control of a manipulator rigid-flexible coupling system driven by a permanent magnet spherical motor.
Background
On the one hand, with the continuous development of modern industrial technology, devices for executing multi-degree-of-freedom high-precision motion are widely applied to the fields of robot joints, intelligent medical instruments, satellite attitude control and the like. The device is usually realized by a complicated transmission mechanism formed by overlapping a plurality of single-degree-of-freedom motors. On the other hand, the permanent magnet spherical motor is a novel special motor capable of realizing multi-degree-of-freedom movement and orientation, does not need an additional mechanical transmission mechanism, and is an important scheme for solving the miniaturization and integration of a multi-dimensional movement device. Meanwhile, robots with high precision, strong adaptability and heavy load have become a development trend in the future. Therefore, the method has very important scientific significance and engineering practical value for carrying out rigid-flexible coupling system modeling and high-precision control research on the manipulator driven by the permanent magnet spherical motor.
The flexible mechanical arm provided in the prior art replaces a rigid output shaft of the permanent magnet spherical motor to be connected with the mechanical arm to form the mechanical arm, an arm rod of the mechanical arm is generally made of light materials, the response is fast, the energy consumption is low, the operation is flexible, the working efficiency is high, and the advantages of the flexible mechanical arm are gradually highlighted in the fields of medical treatment, high-precision manufacturing industry, space exploration and the like. However, due to the reduction of rigidity, the flexible connecting rod mechanical arm can generate elastic deformation during movement, the nonlinearity of a system becomes more complex, and meanwhile, the load changes along with the change of tasks, so that uncertainty exists, and great challenge is brought to the high-precision movement control design. Therefore, the high-precision motion control of the rigid-flexible coupling system of the manipulator is also a research hotspot in recent years.
Although some control theories at the present stage are relatively mature, such as: PID control, variable structure control, decoupling control and the like widely adopted in the industry, but due to the influence of factors such as uncertain interference outside the system and the like, a slip phenomenon easily occurs during movement, so that the system generates a large position error, a limit ring is generated, and the system can even be directly crashed in severe cases. In combination with the above factors, the manipulator system at the present stage has the following defects:
1. the output shaft of the traditional motor is of a rigid structure, and in the actual use process, a complex transmission mechanism needs to be combined, so that the defects of small load dead weight ratio, high energy consumption and the like exist, the application requirements of light weight, high speed, low energy consumption and the like are difficult to meet, and the working range is limited.
2. Most of the existing control systems need a high-gain negative feedback link, so that the calculation burden of a controller is greatly increased, and a larger starting moment is needed; meanwhile, aiming at the influence of external uncertain interference factors, a simplified static model is basically adopted for analysis, the dynamic influence on the system motion cannot be completely and truly reflected, the compensation capability is limited, and further high-precision control cannot be realized, so that the method is not suitable for the actual engineering environment.
Disclosure of Invention
In order to avoid the defects of the technology, the invention provides a manipulator which is formed by connecting a flexible mechanical arm to a paw instead of a rigid output shaft of the permanent magnet spherical motor. The rigid-flexible coupling system mainly comprises a manipulator rigid-flexible coupling dynamics module and an inversion sliding mode control module. Establishing a partial differential dynamics model of the manipulator under a follow-up coordinate system based on a Hamilton method, and constructing the rigid-flexible coupling dynamics module by combining a dynamic model of a rigid output shaft of the permanent magnet spherical motor; estimating unknown interference by selecting a proper nonlinear gain function, and designing the inversion sliding mode control module; finally, modeling and control of the spherical motor-driven manipulator rigid-flexible coupling system are realized through dynamic simulation software ADAMS, finite element software ANSYS and scientific computing tool MATLAB combined simulation, stress analysis on a complex system is avoided, high-precision robust control is realized, and a flexible solution can be provided for the manufacturing industry and the service industry.
The invention adopts the following technical scheme for solving the technical problems:
a manipulator rigid-flexible coupling system based on spherical motor driving uses a permanent magnet spherical motor as a driving motor of the manipulator rigid-flexible coupling system, and adopts a flexible manipulator to replace a rigid output shaft of the permanent magnet spherical motor to be connected with a paw to form a manipulator. The manipulator rigid-flexible coupling system modeling module mainly comprises a manipulator rigid-flexible coupling dynamics module and an inversion sliding mode control module.
The manipulator rigid-flexible coupling dynamics module controls torque tau through system external disturbance d and a permanent magnet spherical motoriIs an input variable; considering that the manipulator has elastic deformation, its actual displacement offset zetai(p) is the rotation angle theta of the rigid-flexible coupling system of the manipulatoriPlus the elastic deformation phii(p), i.e.. zetai(p)＝θi+Φi(p) is an output variable; establishing a partial differential kinetic model of the manipulator rigid-flexible coupling system based on a Hamilton analysis method by combining a permanent magnet spherical motor rigid shaft kinetic model;
the inversion sliding mode control module is used for controlling the inversion sliding mode to rotate at a desired rotation angle thetaidFor inputting variable, the rotation angle theta of the rigid-flexible coupling system of the manipulatoriAnd feeding back the output variable for the control module. Error z is tracked according to position1Velocity tracking error
Figure BDA0003458862580000021
Setting the virtual control amount to z2And a sliding mode switching surface function s for setting the virtual control quantity as z2And a sliding mode switching surface function s, such that s is z2Ensuring that the system reaches an ideal sliding mode; by constructing parameter c1And c2And defining a Lyapunov verification function V by combining with a control law1、V2The output error is limited in a smaller range, and the control torque tau of the permanent magnet spherical motor calculated in real time is obtainedi。
Furthermore, the manipulator rigid-flexible coupling dynamics module is combined with a permanent magnet spherical motor rigid shaft dynamics model and is modeled based on a Hamilton analysis method.
Defining the flexible bending of the mechanical arm at any time origin to be zero, namely the elastic deformation phii(0, t) is 0, and the rate of change of the flexible bending from the origin at any time along the direction of the vector r is zero, so that
Figure BDA0003458862580000031
Zeta combined with actual displacement offset of manipulatori(p) obtaining an expression of the manipulator by using Hamilton principle as follows:
Figure BDA0003458862580000032
wherein t is1And t2Given the time instants of the two motion states,
Figure BDA0003458862580000033
and
Figure BDA0003458862580000034
the total kinetic energy of the manipulator, potential energy caused by elastic deformation and the variation of work done by non-conservative force are respectively;
in combination with the length L of the flexible manipulator and the motion characteristics of the manipulator, equation (1) can be written as:
Figure BDA0003458862580000035
wherein
Figure BDA0003458862580000036
Figure BDA0003458862580000037
Rho flexible mechanical arm unit length mass, m is paw self weight, JiThe central moment of inertia of the permanent magnet spherical motor is shown, F is the product value of the weight m of the paw and the vector length of the flexible mechanical arm, and EI is the bending rigidity of the manipulator;
due to the fact that
Figure BDA0003458862580000038
And
Figure BDA0003458862580000039
are linearly independent variables, then a ═ B ═ C ═ D ═ 0, and formula (2) can be written as:
Figure BDA00034588625800000310
wherein
Figure BDA00034588625800000311
Combining a Lagrange second equation and coordinate transformation, the rigid-flexible coupling dynamic model of the manipulator based on the spherical motor drive is shown as a formula (4):
Figure BDA00034588625800000312
in the formula (4), k is an adjustment parameter of the manipulator rigid-flexible coupling system and is greater than 0;
further, the inversion sliding mode control module tracks error z according to position1Velocity tracking error
Figure BDA00034588625800000313
Setting the virtual control amount to z2And a sliding mode switching surface function s; according to the adjusting parameter c1、c2Defining Lyapunov function V by combining with control law1、V2(ii) a The combined manipulator rigid-flexible coupling system dynamics model formula (4) can be expressed as:
Figure BDA0003458862580000041
wherein x1And x2Respectively representing the state variables of the rotation angle and the rotation angular velocity of the rigid-flexible coupling system of the manipulator, dt＝-M-1(x1) d, and | dtLess than or equal to D; the rotation angle error z of the manipulator rigid-flexible coupling system can be known from the formula (4)1＝x1-x1dTherefore, the error of the rotational angular velocity can be defined as
Figure BDA0003458862580000042
Defining a first Lyapunov function V1Virtual control quantity z2Comprises the following steps:
Figure BDA0003458862580000043
z2＝x2+c1z1-x1d (7)
combining sliding mode variable structure control to define sliding mode switching surface function s as z2And a second Lyapunov function V is defined2And obtaining the following by derivation of s:
Figure BDA0003458862580000044
Figure BDA0003458862580000045
Figure BDA0003458862580000046
to make it possible to
Figure BDA0003458862580000047
Designing controller to output control torque tau of permanent magnet spherical motoriComprises the following steps:
Figure BDA0003458862580000048
wherein c is2Is a normal number larger than zero, eta is a constant term and eta is larger than or equal to D; from the formula (9)
Figure BDA0003458862580000049
Meets the design requirements.
Compared with the prior art, the invention has the beneficial effects that:
1. the invention provides a rigid output shaft structure using a flexible mechanical arm to replace a permanent magnet spherical motor, which can overcome the defects of high rigidity, low flexibility and the like of the traditional mechanical arm. The spherical motor has high rigidity, good controllability and good mechanical arm toughness. The system integrates the advantages of the system and has wider development space.
2. The method utilizes a Hamilton analysis method to establish a partial differential dynamic model of the manipulator rigid-flexible coupling system, can more accurately reflect the dynamic characteristics of the flexible member, avoids complex stress analysis on the system, and simultaneously can accurately describe the distribution parameter characteristics of the manipulator, thereby facilitating the design of the controller.
3. According to the invention, the stability of the inversion sliding mode control strategy is verified by adopting a Lyapunov stability principle, the stability of the manipulator rigid-flexible coupling system in the motion process can be improved, and meanwhile, more ideal tracking output capability is obtained.
Drawings
Fig. 1 is a block diagram of a control structure of a rigid-flexible coupling system of a manipulator based on a spherical motor drive.
FIG. 2 is a flow chart and effect graph of modeling by dynamic software ADAMS combined with finite element software ANSYS according to the present invention.
Fig. 3 is a track and speed tracking curve of the manipulator rigid-flexible coupling system during the tilting motion.
Fig. 4 is a track tracing curve of the three coordinate axis direction expected track, the rigid output axis track and the manipulator rigid-flexible coupling system track when the manipulator is in no-load.
Fig. 5 shows the tracking error of the robot of the present invention when it is empty.
Fig. 6 is a track tracing curve of the desired track of three coordinate axis directions, the rigid output axis track and the rigid-flexible coupling system track of the manipulator when the manipulator is loaded.
FIG. 7 illustrates the trajectory tracking error of the robot of the present invention when loaded.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
The permanent magnet spherical motor comprises a stator and a rotor. And 4 layers of axially magnetized cylindrical neodymium iron boron (NdFeB) permanent magnets are symmetrically embedded in the rotor parallel to the equatorial plane. Each layer of 10 cylindrical permanent magnets is provided, the included angle between layers is 30 degrees, the included angle between columns is 36 degrees, and the polarities of the adjacent permanent magnets are arranged in a crossed manner; 2 layers of coils are symmetrically embedded in the stator parallel to the equatorial plane, each layer is 12, the included angle between layers is 44 degrees, and the included angle between columns is 30 degrees; the lower half of the stator is embedded into 2 layers of 13 support rods. The rotor shaft can realize three-degree-of-freedom high-precision motion, namely pitching, yawing and spinning motion. Due to the structural limitation, 67 degrees of tilting motion and 360 degrees of spinning motion can be realized at most. The three-dimensional model of the mechanism is shown in figure 2 and comprises a permanent magnet spherical motor rotor, a flexible mechanical arm and a paw. The permanent magnet spherical motor rotor is connected with the paw through the flexible mechanical arm connecting rod, so that the paw can do three-degree-of-freedom motion in space.
As shown in fig. 1, the inverse sliding mode control in this embodiment includes a virtual control law module, a Lyapunov stability verification module, and a sliding mode variable structure control module. The experimental device mainly comprises a mechanical system and a measuring system: the mechanical system comprises a rigid-flexible coupling system and a driving circuit board; the measuring system comprises a six-axis gyroscope position sensor, serial port communication can be carried out between the six-axis gyroscope position sensor and a PC controller through Bluetooth, position and speed signals are measured and output in real time, and a graphical interactive interface of the PC controller can change control parameters in real time;
the invention provides a control problem of a rotation angle theta of a rigid-flexible coupling system of a manipulatoriTracking desired angle of rotation thetaidAngular velocity of rotation
Figure BDA0003458862580000061
Tracking desired rotational angular velocity
Figure BDA0003458862580000062
Deriving the error signal z from the desired signal and the rotation signal1、
Figure BDA0003458862580000063
In a specific embodiment, firstly, a mechanical arm rigid-flexible coupling system dynamic model is established through the dynamic characteristics and material characteristics of a permanent magnet spherical motor and a mechanical arm:
(1) describing the relative motion between the rotor and the stator aiming at the permanent magnet spherical motor under the premise of not considering the influence of friction force, and respectively defining a fixed stator coordinate system O-X0Y0Z0And a following rotor coordinate system O-XiYiZi. Each movement of the rotor can be decomposed into three coordinate rotations relative to the stator coordinate system. First, O-X0Y0Z0Around X0The shaft rotates by an angle alpha to obtain O-X1Y1Z1A coordinate system; secondly, wind Y1The shaft rotates by a beta angle to obtain O-X2Y2Z2A coordinate system; finally, winding around Z2The shaft rotates by an angle of gamma to obtain a rotor coordinate system O-X3Y3Z3. The angular displacement vector of the rotor in three coordinate rotations is thetai＝[α β γ]T；
Regarding the rotor as a rigid system with the rotation center unchanged and the centroid coinciding with the centroid, the following is derived based on a dynamic equation combining the lagrangian second equation and the kardan angular rotation:
Figure BDA0003458862580000064
e in the formula (1)iFor total kinetic energy of the rotor, τi＝[τx τy τz]TThe control torque of the permanent magnet spherical motor under the stator coordinate system is expressed, and the rotation angular speed of the rotor can be expressed as
Figure BDA0003458862580000065
The angular acceleration of rotation can be expressed as
Figure BDA0003458862580000066
Because the spherical motor structure is regarded as a rigid body and is highly symmetrical, the rotor is relative to the stator coordinate systemComponent of moment of inertia Jx＝Jy≈Jz. Let P be a spherical motor rotor rotation matrix and in a stator coordinate system O-X0Y0Z0Lower rotor angular velocity ωi＝[ωx ωy ωz]TCan be expressed as:
Figure BDA0003458862580000067
according to the formula (2) and the law of conservation of energy, a coordinate system O-X of the rotor relative to the stator can be obtained0Y0Z0The total kinetic energy of (c) is:
Figure BDA0003458862580000068
the kinetic equation of the permanent magnet spherical motor obtained according to the formulas (1) to (3) is as follows:
Figure BDA0003458862580000071
wherein d is a control interference term and d is a control interference term,
Figure BDA0003458862580000072
is an inertia array of a permanent magnet spherical motor,
Figure BDA0003458862580000073
the non-linear terms of the corresponding permanent magnet spherical motor comprise Copeng force and centrifugal force terms.
When the rigid output shaft at the tail end of the spherical motor is replaced by the manipulator, the manipulator can be regarded as an Euler-Bernoulli beam when the control of the variable load of the manipulator is considered, and a manipulator following coordinate system o-x is establishediyiziThe connection point of the manipulator and the spherical motor is the origin of a follow-up coordinate system, and the rotation angle of the manipulator is theta without considering the elastic deformationiG is the acceleration of gravity; one end of the L-shaped manipulator is fixedly connected with the driving motor, and the other end of the L-shaped manipulator is a free end and is provided with a visible partFor the load of the mass point, the load is considered in this design as the gripper and its gripping object, and ρ is the linear density (i.e., mass per unit length) of the flexible arm.
The rotary motion is controlled by a permanent magnet spherical motoriDriving, wherein the manipulator is clamped on an output shaft of the permanent magnet spherical motor to move along with the permanent magnet spherical motor; let p be any point of the manipulator, r be o-x of the point p before the manipulator deforms in the follow-up coordinate systemiyiziPosition vector of phii(r, t) is the elastic deformation at p due to the flexibility of the arm during motion. In order to derive a dynamic equation of the manipulator, a partial differential dynamic model is established for a continuous manipulator displacement system by adopting a Hamilton method, and the manipulator is defined in a follow-up coordinate system o-xiyiziThe flexible bending of the origin at any time is zero to obtain phii(0, t) ═ 0; and knowing that the change rate of the flexible bending of the origin along the vector r at any moment is zero to obtain
Figure BDA0003458862580000074
The boundary conditions can be expressed as:
Φi(0)＝Φx(0)＝Φy(0)＝Φz(0)＝0 (5)
can approximate the manipulator on a follow-up coordinate system o-xiyiziAny point p (X, y, z) on the vector r is in the inertial frame O-X0Y0Z0The following is expressed as:
ζi(p)＝Φi(p)+θi (6)
wherein ζi(p) the amount of deflection in consideration of the elastic deformation of the hand is obtained from equations (5) and (6):
ζi(0)＝0 (7)
Figure BDA0003458862580000075
Figure BDA0003458862580000076
Figure BDA0003458862580000077
Figure BDA0003458862580000078
from the formula (11)
Figure BDA0003458862580000079
According to Hamilton principle, the method comprises the following steps:
Figure BDA0003458862580000081
wherein t is1And t2Given the time instants of the two motion states,
Figure BDA0003458862580000082
and
Figure BDA0003458862580000083
which are the variations of kinetic energy, potential energy and non-conservative force. The rotation kinetic energy of the known permanent magnet spherical motor is EiThe kinetic energy of the flexible mechanical arm is
Figure BDA0003458862580000084
The kinetic energy of the paw is
Figure BDA0003458862580000085
The total kinetic energy E of the manipulatorkComprises the following steps:
Figure BDA0003458862580000086
wherein m is the weight of the paw; in order to facilitate the derivation of elastic potential energy, the deformation of the manipulator is assumed to be linear elastic, and based on the theory of elastic mechanics, the potential energy E of the system caused by the elastic deformation of the manipulatorpComprises the following steps:
Figure BDA0003458862580000087
the EI is the bending rigidity of the manipulator, and the manipulator can not utilize rigid body dynamics research because the manipulator can generate larger bending deformation and stronger residual vibration in the motion process; in addition, due to the existence of the permanent magnet spherical motor and the paw, the non-conservative force acting W of the manipulator rigid-flexible coupling system is as follows:
W＝τiθi+Fζi(L) (15)
f is the product value of the weight m of the paw and the vector length of the flexible mechanical arm; the first term of the formula (12) is developed
Figure BDA0003458862580000088
Wherein:
Figure BDA0003458862580000089
Figure BDA00034588625800000810
then
Figure BDA00034588625800000811
Expanding the second term of equation (12) according to
Figure BDA00034588625800000812
The following can be obtained:
Figure BDA00034588625800000813
finally, the third term in the formula (12) is expanded to obtain:
Figure BDA0003458862580000091
the partial differential equation of the manipulator system is obtained through mathematical derivation, so that the complex analysis of the system can be avoided, and the following can be obtained according to the analysis:
Figure BDA0003458862580000092
will ζi(0)＝0、
Figure BDA0003458862580000093
And
Figure BDA0003458862580000094
bringing into the above formula can obtain:
Figure BDA0003458862580000095
wherein
Figure BDA0003458862580000096
Figure BDA0003458862580000097
According to the Hamiltonian principle, equation (12) can be modified as follows:
Figure BDA0003458862580000098
due to the fact that
Figure BDA0003458862580000099
The independent variables, i.e., the terms in the above formula are linearly independent, then a ═ B ═ C ═ D ═ 0, results in a manipulator partial differential kinetic model:
Figure BDA00034588625800000910
wherein
Figure BDA00034588625800000911
According to the partial differential model formula (25) of the manipulator, in order to realize angular velocity response and inhibit the deformation of the flexible manipulator, a dynamic model of a rigid-flexible coupling system of the manipulator can be deduced by combining a rotor dynamic model of a permanent magnet spherical motor as follows:
Figure BDA00034588625800000912
in the formula (26), k is an adjustment parameter of the manipulator rigid-flexible coupling system and is greater than 0; the observation shows that the rigid body rotation of the permanent magnet spherical motor and the elastic deformation of the mechanical arm are kinematically coupled, and the coupling kinetic model conforms to the momentum conservation equation containing the elastic theory.
And decomposing the complex nonlinear system into subsystems with the order not exceeding the system order by utilizing an inversion sliding mode control strategy, then respectively designing a Lyapunov verification function and an intermediate virtual control quantity for each subsystem, and backing to the whole system until the design of the whole control law is completed. In the actual operation process, aiming at the problems that the state variable of the system is difficult to accurately measure, the system cannot smoothly and accurately output and the like, the control algorithm provided by the invention has the advantages of good stability, high speed and precision, fast dynamic response, strong anti-interference capability and the like, and can carry out online real-time estimation on the control torque tau of the manipulator rigid-flexible coupling systemiAnd to improve the controllability and robustness of the system.
The inverse sliding mode controller is characterized by tracking error z according to position1Velocity tracking error
Figure BDA0003458862580000101
Setting the virtual control amount to z2And a sliding mode switching surface function s; according to the adjusting parameter c1、c2Defining Lyapunov verification function V by combining control law1、V2An inversion design methodThe influence of external disturbance and parameter perturbation is suppressed together with sliding mode control, so that the whole closed-loop system meets expected dynamic and static performance indexes, and the output error is limited in a very small range. The dynamic model (26) of the rigid-flexible coupling system of the manipulator can be represented as the following state variable forms
Figure BDA0003458862580000102
Wherein x1And x2Respectively representing the state variables of the rotation angle and the rotation angular velocity of the rigid-flexible coupling system of the manipulator, dt＝-M-1(x1) d, and | dtLess than or equal to D; the rotation angle error z of the rigid-flexible coupling system of the manipulator can be known from the formula (26)1＝x1-x1dTherefore, the error of the rotational angular velocity can be defined as
Figure BDA0003458862580000103
The first Lyapunov function is defined as:
Figure BDA0003458862580000104
get x2＝-c1z1+x1d+z2Wherein c is1>0，z2For a virtual control quantity z2＝x2+c1z1-x1d；
Then
Figure BDA0003458862580000105
If z is2When the value is equal to 0, then
Figure BDA0003458862580000106
So the second step of inverse design is to combine the inverse control method and sliding mode control to overcome the controlled object
Figure BDA0003458862580000107
M-1(x1)τiThe accurate modeling information disturbance can not only enlarge the application range of the inversion control method, but also make the disturbance to the model have robustness. Defining the slip plane as s-z in combination with sliding mode variable structure control2Defining a second Lyapunov function as:
Figure BDA0003458862580000108
defining the slip plane as s-z in combination with sliding mode variable structure control2Obtaining:
Figure BDA0003458862580000111
Figure BDA0003458862580000112
to make it possible to
Figure BDA0003458862580000113
Meanwhile, the system modeling error can be reduced, the control driving accuracy is improved, and the controller is designed to output the control torque tau of the permanent magnet spherical motoriComprises the following steps:
Figure BDA0003458862580000114
wherein, c2Is a normal number larger than zero, eta is a constant term and eta is larger than or equal to D;
Then
Figure BDA0003458862580000115
namely, it is
Figure BDA0003458862580000116
In the same way, form V of exponential convergence can be obtained2(t)＝V2(0)e-ηt(ii) a And due to
Figure BDA0003458862580000117
To meet the design requirements
Figure BDA0003458862580000118
Then and z when t → ∞ time1、z2All converge exponentially, i.e. z1→ 0 and z2→ 0; and due to z2＝x2+c1z1-x1dAnd meets the design requirements.
Example calculation:
firstly, establishing a three-dimensional model of a permanent magnet spherical motor rotor body in SOLIDWORKS software, establishing a flexible mechanical arm by utilizing an ADAMS/AutoFlex module of dynamics simulation software to perform flexible processing on part of components, then establishing a dynamics model of the manipulator rigid-flexible coupling system by adding mass attributes, constraints and paws, and comparing the dynamics model with an ANSYS simulation result to verify the correctness of the model; then, considering the influence of factors such as modeling, measurement uncertainty and load change, and controlling the manipulator rigid-flexible coupling system by adopting an inversion sliding mode control method, so that the robustness of the controller is improved; finally, the superiority of the invention is verified through a gripper component load test.
In the experimental process, the expected rotation angle theta of the manipulator rigid-flexible coupling system is givenidDesired rotational angular velocity
Figure BDA0003458862580000119
The method is characterized in that the two conditions of no load and load of the manipulator are set, and the position track information of the rigid output shaft of the permanent magnet spherical motor and the rigid-flexible coupling system is compared through a contrast experiment to verify the effectiveness of the method. Selecting nominal controller parameters including
Figure BDA00034588625800001110
Unit deg/s, initial position θid＝[0 0 0]T(ii) a Parameter c1＝c2＝[20 10 15]TThe length L of the mechanical arm is 63mm, the linear density rho is 0.2kg/m, and the bending rigidity EI of the mechanical arm is 3N/m2Control interference d ═[0.3sin(πt) 0.2sin(2πt) 0.25cos(πt)]In addition, m is 0.35kg in no-load test, 0.5kg in load test, and 9.8m/s in gravity acceleration g2。
FIGS. 2(a) and (b) are a modeling flowchart and an effect diagram according to the present invention. Fig. 3 is a graph showing the rotation angle and angular velocity of the rigid-flexible coupling system of the manipulator according to the present invention during tilting. The expected rotation angle is set to be 0.5rad, the fact that a trajectory tracking curve can basically reach a target set value can be found, the jitter is small, and the correctness of the manipulator rigid-flexible coupling system dynamic model is further verified.
In fig. 4, three curves are respectively an expected trajectory in three coordinate axis directions when the manipulator is in no-load, a rigid output axis trajectory, and a trajectory tracking curve of the manipulator rigid-flexible coupling system of the present invention. Through comparison, the deflection angle of the rigid output shaft of the permanent magnet spherical motor is smoother because the rigid output shaft is not influenced by flexible deformation; the initial deflection angle of the manipulator rigid-flexible coupling system provided by the invention is changed along a given trajectory curve in an up-and-down shaking manner. Fig. 5 shows the error between the rigid output shaft and the rigid-flexible coupling system of the manipulator and the expected trajectory tracking, and it can be found that the error of the initial part is larger, and the error range of the rigid-flexible coupling system of the manipulator is larger than the error range of the rigid output shaft due to the existence of flexible deformation. Finally, an upper expected track is tracked, so that the effectiveness of the inversion sliding mode controller is proved.
In fig. 6, three curves are respectively an expected trajectory in three coordinate axis directions when the manipulator is loaded, a rigid output axis trajectory, and a trajectory tracking curve of the manipulator rigid-flexible coupling system of the present invention. As can be seen, as the load increases, the ductility of the flexible mechanical arm will increase, and the offset angle jitter range will also increase. Meanwhile, the jitter frequency of the corresponding trajectory tracking error curve in fig. 7 is observed to be faster than that of the no-load curve, which is caused by the larger flexible deformation of the flexible mechanical arm in the working process. Finally, an upper expected track is tracked, so that the inversion sliding mode controller provided by the invention is proved to have stronger robustness for different load variables.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (3)

1. The utility model provides a manipulator rigid-flexible coupling system based on spherical motor drive which characterized in that: the permanent magnet spherical motor is used as a driving motor of a manipulator rigid-flexible coupling system, and a flexible mechanical arm is used for replacing a rigid output shaft of the permanent magnet spherical motor to be connected with a paw to form a manipulator; the modeling module of the manipulator rigid-flexible coupling system comprises a manipulator rigid-flexible coupling dynamics module and an inversion sliding mode control module;
the manipulator rigid-flexible coupling dynamics module controls the torque tau through system external disturbance d and a permanent magnet spherical motoriIs an input variable; considering that the manipulator has elastic deformation, its actual displacement offset zetai(p) is the rotation angle theta of the rigid-flexible coupling system of the manipulatoriPlus the elastic deformation phii(p), i.e.. zetai(p)＝θi+Φi(p) is an output variable; establishing a partial differential kinetic model of the manipulator rigid-flexible coupling system based on a Hamilton analysis method by combining a permanent magnet spherical motor rigid shaft kinetic model;
the inversion sliding mode control module rotates at a desired angle thetaidFor inputting variable, the rotation angle theta of the rigid-flexible coupling system of the manipulatoriAnd feeding back the output variable for the control module. Error z is tracked according to position1Velocity tracking error
Figure FDA0003458862570000016
Setting the virtual control amount to z2And a sliding mode switching surface function s, such that s is z2Ensuring that the system reaches an ideal sliding mode; by constructing parameter c1And c2And defining a Lyapunov verification function V by combining with a control law1、V2The output error is limited in a smaller range, and the control torque tau of the permanent magnet spherical motor calculated in real time is obtainedi。
2. The rigid-flexible coupling system based on the spherical motor-driven manipulator of claim 1, characterized in that: the manipulator rigid-flexible coupling dynamics module is combined with a permanent magnet spherical motor rigid shaft dynamics model and is modeled based on a Hamilton analysis method;
it defines the flexible bending of the mechanical arm at any time origin as zero, namely the elastic deformation phii(0, t) is 0, and the rate of change of the flexible bending from the origin at any time along the direction of the vector r is zero, so that
Figure FDA0003458862570000011
Zeta combined with actual displacement offset of manipulatori(p) obtaining an expression of the manipulator by using Hamilton principle as follows:
Figure FDA0003458862570000012
wherein t is1And t2Given the time instants of the two motion states,
Figure FDA0003458862570000013
and
Figure FDA0003458862570000014
the total kinetic energy of the manipulator, potential energy caused by elastic deformation and the variation of work done by non-conservative force are respectively;
in combination with the length L of the flexible manipulator and the motion characteristics of the manipulator, equation (1) can be written as:
Figure FDA0003458862570000015
wherein
Figure FDA0003458862570000021
Figure FDA0003458862570000022
Rho flexible mechanical arm unit length mass, m is paw self weight, JiThe central moment of inertia of the permanent magnet spherical motor is shown, F is the product value of the weight m of the paw and the vector length of the flexible mechanical arm, and EI is the bending rigidity of the manipulator;
due to the fact that
Figure FDA0003458862570000023
And
Figure FDA0003458862570000024
are linearly independent variables, then a ═ B ═ C ═ D ═ 0, and formula (2) can be written as:
Figure FDA0003458862570000025
wherein
Figure FDA0003458862570000026
Combining a Lagrange second equation and coordinate transformation, the rigid-flexible coupling dynamic model of the manipulator based on the spherical motor drive is shown as a formula (4):
Figure FDA0003458862570000027
in the formula (4), k is an adjustment parameter of the rigid-flexible coupling system of the manipulator and is greater than 0.
3. The rigid-flexible coupling system based on the spherical motor-driven manipulator of claim 1, characterized in that: the inversion sliding mode control module tracks error z according to position1Speed tracking error
Figure FDA0003458862570000028
Setting the virtual control quantity toz2And a sliding mode switching surface function s; according to the adjusting parameter c1、c2Defining a Lyapunov function V by combining with a control law1、V2(ii) a The combined manipulator rigid-flexible coupling system dynamic model (4) can be expressed as follows:
Figure FDA0003458862570000029
wherein x1And x2Respectively representing the state variables of the rotation angle and the rotation angular velocity of the rigid-flexible coupling system of the manipulator, dt＝-M-1(x1) d, and | dtLess than or equal to D; the rotation angle error z of the manipulator rigid-flexible coupling system can be known from the formula (4)1＝x1-x1dTherefore, the error of the rotational angular velocity can be defined as
Figure FDA00034588625700000210
Defining a first Lyapunov function V1Virtual control quantity z2Comprises the following steps:
Figure FDA00034588625700000211
z2＝x2+c1z1-x1d (7)
combining sliding mode variable structure control to define sliding mode switching surface function s as z2And a second Lyapunov function V is defined2And obtaining the following by derivation of s:
Figure FDA0003458862570000031
Figure FDA0003458862570000032
Figure FDA0003458862570000033
to make it possible to
Figure FDA0003458862570000034
Designing controller to output control torque tau of permanent magnet spherical motoriComprises the following steps:
Figure FDA0003458862570000035
wherein c is2Is a normal number greater than zero, eta is a constant term and eta is greater than or equal to D; from the formula (9)
Figure FDA0003458862570000036
Meets the design requirements.
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