CN114454161B - Manipulator rigid-flexible coupling system based on spherical motor drive - Google Patents
Manipulator rigid-flexible coupling system based on spherical motor drive Download PDFInfo
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Abstract
The invention provides a rigid-flexible coupling system of a manipulator based on spherical motor driving, which takes a permanent-magnet spherical motor as a driving motor of the rigid-flexible coupling system of the manipulator, and adopts a flexible mechanical arm to replace a rigid output shaft of the permanent-magnet spherical motor to be connected with a claw so as to form the manipulator. The rigid-flexible coupling system mainly comprises a mechanical arm 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 Hamiltonian method, and constructing the rigid-flexible coupling dynamics module by combining a permanent magnet spherical motor rigid shaft dynamics model; estimating unknown interference by selecting a proper nonlinear gain function, and designing the inversion sliding mode control module; the invention realizes modeling and control of the rigid-flexible coupling system of the manipulator driven by the spherical motor, avoids stress analysis of a complex system, realizes high-precision robust control, and can provide flexible solutions for manufacturing industry and service industry.
Description
Technical Field
The invention belongs to the technical field of special motor application, and relates to modeling and control of a rigid-flexible coupling system of a manipulator driven by a permanent magnet spherical motor.
Background
On the one hand, with the continuous development of modern industrial technology, the device for executing the multi-degree-of-freedom high-precision motion is widely applied to the fields of robot joints, intelligent medical instruments, satellite attitude control and the like. Such devices are typically implemented using multiple single degree of freedom motors superimposed on a complex drive mechanism. On the other hand, the permanent magnet spherical motor is used as a novel special motor capable of realizing multi-degree-of-freedom motion and orientation, an additional mechanical transmission mechanism is not needed, and the novel special motor is an important scheme for solving the miniaturization and integration of a multi-dimensional motion device. Meanwhile, robots with high precision, strong adaptability and heavy load have become a future development trend. Therefore, the method has very important scientific significance and engineering practical value in carrying out rigid-flexible coupling system modeling and high-precision control research on the manipulator driven by the permanent magnet spherical motor.
The mechanical arm is formed by adopting the flexible mechanical arm to replace the rigid output shaft of the permanent magnet spherical motor in the prior art, and the arm rod of the mechanical arm is generally made of light materials, so that the mechanical arm has the advantages of quick response, low energy consumption, flexible operation, high working efficiency and the like, and gradually stands out 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 the system becomes more complex, meanwhile, the load changes along with the task change, uncertainty exists, and a great challenge is brought to the high-precision movement control design. Therefore, the rigid-flexible coupling system high-precision motion control of the manipulator is also a research hot spot in recent years.
Although some control theory at this stage is relatively mature, such as: PID control, variable structure control, decoupling control and the like which are widely adopted in the industry, but due to the influence of factors such as uncertain disturbance and the like outside the system, the phenomenon of slipping is easy to occur during movement, so that a larger position error is generated in the system, limit rings are further generated, and even the system is likely to be crashed directly in serious cases. The manipulator system at the present stage has the following defects by combining the above factors:
1. the output shaft of the traditional motor is of a rigid structure, and in the actual use process, a complex transmission mechanism is required to be combined, so that the defects of small load-to-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 load of a controller is greatly increased, and a larger starting torque is needed; meanwhile, aiming at the influence of external uncertainty interference factors, a simplified static model is basically adopted for analysis, the dynamic influence suffered in the motion of the system 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 an actual engineering environment.
Disclosure of Invention
In order to avoid the defects of the technology, the invention provides a manipulator formed by adopting a flexible mechanical arm to replace a rigid output shaft of the permanent magnet spherical motor to be connected with a claw. The rigid-flexible coupling system mainly comprises a mechanical arm 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 Hamiltonian method, and constructing the rigid-flexible coupling dynamics module by combining a rigid output 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; finally, modeling and control of the rigid-flexible coupling system of the manipulator driven by the spherical motor are realized through the combined simulation of dynamic simulation software ADAMS, finite element software ANSYS and a scientific computing tool MATLAB, stress analysis of a complex system is avoided, high-precision robust control is realized, and a flexible solution can be provided for manufacturing industry and service industry.
The invention adopts the following technical scheme for solving the technical problems:
a rigid-flexible coupling system of a manipulator based on spherical motor drive uses a permanent-magnet spherical motor as a driving motor of the rigid-flexible coupling system of the manipulator, and a flexible mechanical arm is adopted to replace a rigid output shaft of the permanent-magnet spherical motor to be connected with a gripper to form the 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 mechanical arm rigid-flexible coupling dynamics module controls the torque tau by using a system external disturbance d and a permanent magnet spherical motor i Is an input variable; considering that the manipulator has elastic deformation, the actual displacement offset ζ thereof i (p) is the rotation angle theta of the rigid-flexible coupling system of the manipulator i Adding an elastic deformation quantity phi i (p), i.e. ζ i (p)＝θ i +Φ i (p) is an output variable; combining a permanent magnet spherical motor rigid shaft dynamics model, and establishing a partial differential dynamics model of the manipulator rigid-flexible coupling system based on a Hamiltonian analysis method;
the inversion sliding mode control module is controlled by a desired rotation angle theta id To input variable, the rotation angle theta of the rigid-flexible coupling system of the manipulator i Is the inverse of the control moduleAnd feeding out the output variable. Based on position tracking error z 1 Speed tracking errorSetting the virtual control quantity as z 2 And a sliding mode switching surface function s, setting a virtual control quantity as z 2 And a sliding mode switching surface function s such that s=z 2 Ensuring that the system reaches an ideal sliding mode; by constructing parameter c 1 And c 2 And combining control law to define Lyapunov verification function V 1 、V 2 The output error is limited in a smaller range, and the control torque tau of the permanent magnet spherical motor calculated in real time is obtained i 。
Furthermore, the manipulator rigid-flexible coupling dynamics module is combined with a permanent magnet spherical motor rigid shaft dynamics model, and modeling is performed based on a Hamiltonian analysis method.
Defining the flexible bending of the origin of the manipulator at any moment to be zero, namely the elastic deformation phi i (0, t) =0, and the change rate of the origin flexible bending along the vector r direction at any time is zero, thereby obtainingCombination of actual displacement offset ζ of manipulator i (p) the expression for the manipulator using the Hamiltonian principle is:
wherein t is 1 And t 2 For a given moment of two motion states,and->The variation of the work of the total kinetic energy of the manipulator, potential energy caused by elastic deformation and non-conservative force is respectively;
in combination with the length L of the flexible manipulator and the motion characteristics of the manipulator, formula (1) can be written as:
wherein the method comprises the steps of The weight of the rho flexible mechanical arm in unit length is m, the weight of the paw is J i The center rotational inertia of the permanent magnet spherical motor is represented by F, the product value of the weight m of the gripper and the vector length of the flexible mechanical arm, and EI is the bending stiffness of the mechanical arm;
due toAnd->Is a linearly independent variable, then there is a=b=c=d=0, and equation (2) can be written as:
wherein the method comprises the steps of
Combining the Lagrangian second equation and coordinate transformation, the manipulator rigid-flexible coupling dynamics model based on spherical motor driving is shown as formula (4):
in the formula (4), k is an adjustment parameter of the rigid-flexible coupling system of the manipulator and is more than 0;
further, the inversion sliding mode control module is used for controlling the inversion sliding mode according to the positionPut tracking error z 1 Speed tracking errorSetting the virtual control quantity as z 2 And a sliding mode switching surface function s; according to the adjustment parameter c 1 、c 2 In combination with the control law, a Lyapunov function V is defined 1 、V 2 The method comprises the steps of carrying out a first treatment on the surface of the The dynamic model type (4) of the rigid-flexible coupling system combined with the manipulator can be expressed as follows:
wherein x is 1 And x 2 Respectively representing the state variables of the rotation angle and the rotation angular velocity of the rigid-flexible coupling system of the manipulator, d t ＝-M -1 (x 1 ) d, and |d t D is not more than; the rotation angle error z of the rigid-flexible coupling system of the manipulator can be known by the method (4) 1 ＝x 1 -x 1d The rotational angular velocity error can be defined asDefinition of the first Lyapunov function V 1 Virtual control amount z 2 The method comprises the following steps:
z 2 ＝x 2 +c 1 z 1 -x 1d (7)
defining a sliding mode switching plane function s=z in combination with sliding mode variable structure control 2 And define a second Lyapunov function V 2 Deriving s:
in order to makeThe design controller outputs the control torque tau of the permanent magnet spherical motor i The method comprises the following steps:
wherein c 2 Is a positive constant greater than zero, eta is a constant term and eta is more than or equal to D; from (9), it can be seen thatMeets the design requirements.
Compared with the prior art, the invention has the beneficial effects that:
1. the invention provides a rigid output shaft structure with a flexible mechanical arm instead of 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 and good controllability, and the manipulator has good toughness. The system integrates the advantages of the system and has wider development space.
2. According to the invention, the partial differential dynamics model of the manipulator rigid-flexible coupling system is established by using the Hamiltonian analysis method, so that the dynamics characteristic of the flexible component can be reflected more accurately, the complex stress analysis on the system is avoided, and meanwhile, the model can accurately describe the distribution parameter characteristics of the manipulator, so that the design of a controller is facilitated.
3. According to the inversion sliding mode control strategy adopted by the invention, the stability of the inversion sliding mode control strategy is verified by adopting the Lyapunov stability principle, so that the stability of the mechanical arm in the rigid-flexible coupling system movement process can be improved, and meanwhile, the ideal tracking output capability is obtained.
Drawings
Fig. 1 is a control structure block diagram of a rigid-flexible coupling system of a manipulator based on spherical motor driving.
FIG. 2 is a flow chart and effect graph of the invention modeled by the dynamics software ADAMS in combination with the finite element software ANSYS.
Fig. 3 is a graph of trajectory and velocity tracking during tilting motion of the manipulator rigid-flexible coupling system of the present invention.
Fig. 4 is a graph showing the expected track, the track of a rigid output shaft and the track of a rigid-flexible coupling system of the manipulator in three coordinate axis directions when the manipulator is in idle load.
Fig. 5 is a graph of the track following error of the robot arm in accordance with the present invention when empty.
Fig. 6 is a graph of the expected track, the track of the rigid output shaft and the track of the rigid-flexible coupling system of the manipulator in three coordinate axis directions when the manipulator is loaded.
Fig. 7 is a graph of the track following error when the manipulator of the present invention is loaded.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, the technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
The permanent magnet spherical motor comprises a stator part and a rotor part. 4 layers of axially magnetized cylindrical NdFeB (NdFeB) permanent magnets are symmetrically embedded on the rotor parallel to the equatorial plane. 10 cylindrical permanent magnets are arranged on each layer, the included angle between layers is 30 degrees, the included angle between columns is 36 degrees, and the polarities of adjacent permanent magnets are crossed; 2 layers of coils are symmetrically embedded on the stator parallel to the equatorial plane, each layer of coils has 12 layers, the included angle between layers is 44 degrees, and the included angle between columns is 30 degrees; the lower half part of the stator is embedded with 13 support rods in 2 layers. The rotor shaft can realize three-degree-of-freedom high-precision motion, namely pitch, yaw and spin motion. Limited by its structure, a tilting motion of 67 degrees and a spinning motion of 360 degrees can be achieved at maximum. The three-dimensional model of the mechanism is shown in figure 2, and consists of 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 move in three degrees of freedom in space, and the effect similar to a robot joint can be achieved by implementing the method.
As shown in fig. 1, the inversion 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 communication can be carried out between the six-axis gyroscope position sensor and a PC controller through Bluetooth, output position and speed signals can be measured in real time, and a graphical interaction interface of the PC controller can change control parameters in real time;
the control problem presented by the invention is that the rotation angle theta of the rigid-flexible coupling system of the manipulator i Tracking the desired rotation angle θ id Rotational angular velocityTracking the desired rotational angular velocity +.>Deriving an error signal z from the desired signal and the rotation signal 1 、/>In the specific embodiment, firstly, a mechanical arm rigid-flexible coupling system dynamics model is established through dynamics characteristics and material characteristics of a permanent magnet spherical motor and a mechanical arm:
(1) Describing the relative motion between a rotor and a stator on the premise of not considering the influence of friction force for the permanent magnet spherical motor, and respectively defining a fixed stator coordinate system O-X 0 Y 0 Z 0 And a follow-up rotor coordinate system O-X i Y i Z i . Each movement of the rotor can be broken down into three coordinate rotations relative to the stator coordinate system. First, O-X 0 Y 0 Z 0 Around X 0 The axis rotates by an angle alpha to obtain O-X 1 Y 1 Z 1 A coordinate system; next, around Y 1 The axis rotates by beta angle to obtain O-X 2 Y 2 Z 2 A coordinate system; finally, around Z 2 The axis rotates by gamma angle to obtain a rotor coordinate system O-X 3 Y 3 Z 3 . The angular displacement vector of the rotor in three coordinate rotations is theta i ＝[α β γ] T ；
The rigid body system with the rotor as the rotation center unchanged and the centroid coincident with the centroid is deduced based on the dynamic equation combining the Lagrangian second equation and the Karl Dandelion rotation as follows:
e in formula (1) i For total kinetic energy of rotor τ i ＝[τ x τ y τ z ] T The control torque of the permanent magnet spherical motor expressed in the stator coordinate system can be expressed as the rotation angular speed of the rotorThe rotational acceleration can be expressed as +.>Since the spherical motor structure is regarded as a rigid body and highly symmetrical, the rotor has a moment of inertia component J with respect to the stator coordinate system x ＝J y ≈J z . Let P be the rotation matrix of the spherical motor rotor, in the stator coordinate system O-X 0 Y 0 Z 0 Lower rotor angular velocity omega i ＝[ω x ω y ω z ] T Can be expressed as:
according to the above formula (2) and the law of conservation of energy, the rotor-to-stator coordinate system O-X can be obtained 0 Y 0 Z 0 The total kinetic energy of (2) is:
the kinetic equation of the permanent magnet spherical motor obtained according to the formulas (1) to (3) is:
where d is the control interference term,is a permanent magnet spherical motor inertia array, < >>The non-linear term for the corresponding permanent magnet spherical motor includes the coriolis force and centrifugal force terms.
When the rigid output shaft at the tail end of the spherical motor is replaced by a manipulator, the manipulator can be regarded as an Euler-Bernoulli beam when the control of the variable load is considered, and a manipulator follow-up coordinate system is established as o-x i y i z i The 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 when elastic deformation is not considered i G is gravity acceleration; one end of the manipulator with the length L is fixedly connected to the driving motor, the other end of the manipulator is a free end and is provided with a load which can be regarded as mass points, the load is regarded as a paw and a clamping object thereof in the design, and ρ is the linear density (namely the mass per unit length) of the flexible arm.
The torque tau is controlled by the permanent-magnet spherical motor in the rotary motion i Driving, wherein the mechanical arm is clamped on the output shaft of the permanent magnet spherical motor to move along with the output shaft; let p point be any point of the manipulator, r be o-x in the following coordinate system of the point p before the manipulator is deformed i y i z i Position vector of phi i (r, t) is the elastic deformation at p during movement due to arm flexibility. To derive the dynamic equation of the manipulator, the Hamiltonian method is used for continuousThe manipulator displacement system of (1) establishes a partial differential dynamics model, and the manipulator is defined in a follow-up coordinate system o-x i y i z i The flexible bending of the origin at any moment is zero, and phi is obtained i (0, t) =0; and the change rate of the origin flexible bending edge vector r at any moment is known to be zero, thus obtainingThe boundary condition may be expressed as:
Φ i (0)＝Φ x (0)＝Φ y (0)＝Φ z (0)＝0 (5)
can be approximated to the manipulator in the following coordinate system o-x i y i z i Any point p (X, y, z) on vector r is in inertial coordinate system O-X 0 Y 0 Z 0 The following is expressed:
ζ i (p)＝Φ i (p)+θ i (6)
wherein ζ i (p) is an offset considering the elastic deformation of the manipulator, and is obtained by the formulas (5), (6):
ζ i (0)＝0 (7)
from (11), it can be seen thatAccording to the Hamiltonian principle:
wherein t is 1 And t 2 For a given moment of two motion states,and->The kinetic energy, potential energy and non-conservative force do work respectively. The rotational kinetic energy of the permanent magnet spherical motor is known as E i The kinetic energy of the flexible mechanical arm is +.>The kinetic energy of the paw is->The total kinetic energy E of the manipulator k The method comprises the following steps:
wherein m is the weight of the paw; in order to facilitate deriving elastic potential energy, the deformation of the manipulator is subjected to linear elastic assumption, and based on the elastic mechanics theory, the potential energy E of a system caused by the elastic deformation of the manipulator p The method comprises the following steps:
the EI is the bending rigidity of the manipulator, and the manipulator cannot utilize rigid body dynamics to study because the manipulator generates 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 rigid-flexible coupling system of the manipulator is as follows:
W＝τ i θ i +Fζ i (L) (15)
wherein F is the product value of the weight m of the paw and the vector length of the flexible mechanical arm; expanding the first term of formula (12) to obtain
Wherein:
then
Expanding the second term of formula (12) according toThe method can obtain:
finally, the third term in the formula (12) is developed to obtain:
the partial differential equation of the manipulator system is obtained through mathematical derivation, so that complex analysis on the system can be avoided, and the partial differential equation can be obtained according to the analysis:
zeta is to i (0)＝0、And->The following formula can be carried:
wherein the method comprises the steps of According to the hamilton principle, equation (12) can be modified as:
due toThe independent variables, i.e. the linearity of the terms in the above equation, are independent, then a=b=c=d=0, resulting in a manipulator partial differential kinetic model:
wherein the method comprises the steps ofAccording to the partial differential model type (25) of the manipulator, in order to realize angular velocity response and inhibit deformation of the flexible manipulator, the mechanical manipulator rigid-flexible coupling system dynamics model can be deduced by combining the permanent magnet spherical motor rotor dynamics model as follows:
in the formula (26), k is an adjustment parameter of the rigid-flexible coupling system of the manipulator and k is more than 0; the observation shows that the rigid body rotation of the permanent magnet spherical motor and the elastic deformation of the manipulator are in kinematic coupling, and the coupling dynamics model accords with a momentum conservation equation containing an elastic theory.
And decomposing the complex nonlinear system into subsystems with the system order not exceeding by utilizing an inversion sliding mode control strategy, and then respectively designing a Lyapunov verification function and an intermediate virtual control quantity for each subsystem, and always 'backing' to the whole system until the design of the whole control law is completed. In the actual running 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 stability, high precision and the like, has the advantages of quick dynamic response and strong anti-interference capability, and can be used for online real-time estimation of the control torque tau of the rigid-flexible coupling system of the manipulator i And improving the controllability and robustness of the system.
The inversion sliding mode controller is characterized in that the inversion sliding mode controller is based on a position tracking error z 1 Speed tracking errorSetting the virtual control quantity as z 2 And a sliding mode switching surface function s; according to the adjustment parameter c 1 、c 2 Lyapunov validation function V is defined in conjunction with control law 1 、V 2 The inversion design method and the sliding mode control are organically combined to jointly inhibit the influence of external disturbance and parameter perturbation, so that the whole closed-loop system meets expected dynamic and static performance indexes, and the output error is limited in a small range. The model can be expressed in terms of the following state variables in combination with the manipulator rigid-flexible coupling system dynamics model (26)
Wherein x is 1 And x 2 Respectively represent rigid-flexible coupling of mechanical armRotational angle and rotational angular velocity state variables of the system, d t ＝-M -1 (x 1 ) d, and |d t D is not more than; from (26), the rotation angle error z of the rigid-flexible coupling system of the manipulator can be known 1 ＝x 1 -x 1d The rotational angular velocity error can be defined asDefining a first Lyapunov function as:
taking x 2 ＝-c 1 z 1 +x 1d +z 2 Wherein c 1 >0，z 2 Is the virtual control quantity z 2 ＝x 2 +c 1 z 1 -x 1d ；
Then
If z 2 =0, thenTherefore, the second step of inversion design is to combine the inversion control method with sliding mode control to overcome the +.>M -1 (x 1 )τ i The application range of the inversion control method can be enlarged, and the disturbance of the model is robust. Defining sliding surfaces as s=z in combination with sliding mode variable structure control 2 Defining a second Lyapunov function as:
defining sliding surfaces as s=z in combination with sliding mode variable structure control 2 Obtaining:
in order to makeMeanwhile, the system modeling error can be reduced, the accuracy of control driving is improved, and the controller is designed to output the control torque tau of the permanent magnet spherical motor i The method comprises the following steps:
wherein c 2 Is a positive constant greater than zero, eta is a constant term and eta is more than or equal to D;
Then
i.e.Form V of exponential convergence can be obtained by the same 2 (t)＝V 2 (0)e -ηt The method comprises the steps of carrying out a first treatment on the surface of the And due toTo meet design requirements->And z is at t → ≡infinity 1 、z 2 All exponentially converging, i.e. z 1 0 and z 2 -0; and due to z 2 ＝x 2 +c 1 z 1 -x 1d Meets the design requirement.
Example:
firstly, a three-dimensional model of a permanent magnet spherical motor rotor body is built in SOLIWORKS software, a flexible mechanical arm is built for part of component flexible treatment by utilizing dynamic simulation software ADAMS/AutoFlex module, then a mechanical arm rigid-flexible coupling system dynamic model is built by adding quality attributes, constraints and claws, and the model is compared with ANSYS simulation results to verify the correctness of the model; then, taking the influence of factors such as modeling, measurement uncertainty, load change and the like into consideration, 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, through a paw component load experiment, the superiority of the invention is verified.
During the experiment, the rigid-flexible coupling system of the given manipulator expects a rotation angle theta id Desired rotational angular velocityThe method is characterized in that no-load and loaded conditions of the manipulator are set, and the position track information of the rigid output shaft and the rigid-flexible coupling system of the permanent magnet spherical motor is compared through a comparison experiment, so that the effectiveness of the method is verified. Selecting nominal parameters of the controller comprisesUnit deg/s, initial position θ id ＝[0 0 0] T The method comprises the steps of carrying out a first treatment on the surface of the Parameter c 1 ＝c 2 ＝[20 10 15] T Arm length l=63 mm, linear density ρ=0.2 kg/m, manipulator bending stiffness ei=3N/m 2 Control disturbance d= [0.3sin (pi t) 0.2sin (2 pi t) 0.25cos (pi t)]In addition, the no-load experiment takes m=0.35 kg, the loaded experiment takes m=0.5 kg, and the gravity acceleration g takes 9.8m/s 2 。
Fig. 2 (a) and (b) are a modeling flowchart and an effect diagram according to the present invention. Fig. 3 is a graph of the rotational angle and angular velocity of the manipulator rigid-flexible coupling system of the present invention during tilting motion. The expected rotation angle is set to be 0.5rad, the track tracking curve can basically reach the target set value, the jitter is small, and the accuracy of the mechanical arm rigid-flexible coupling system dynamics model is further verified.
The three curves in fig. 4 are respectively the expected track of the three coordinate axis directions when the manipulator is in idle load, the track of the rigid output shaft and the track tracking curve of the rigid-flexible coupling system of the manipulator. By comparison, the rigid output shaft of the permanent magnet spherical motor is free from the influence of flexible deformation, and the deflection angle of the rigid output shaft is smooth; the initial deflection angle of the manipulator rigid-flexible coupling system provided by the invention is changed in a shaking way up and down along a given track curve. Fig. 5 shows tracking errors of the rigid output shaft, the rigid-flexible coupling system of the manipulator and the expected track, 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, the desired trajectory was tracked, thereby confirming the effectiveness of the inverted sliding mode controller of the present invention.
The three curves in fig. 6 are respectively the expected track in the three coordinate axis directions when the manipulator is loaded, the track of the rigid output shaft and the track tracking curve of the rigid-flexible coupling system of the manipulator. As can be seen from the figure, as the load increases, the ductility of the flexible mechanical arm increases, and the range of angular misalignment increases. Meanwhile, the jitter frequency of the corresponding track tracking error curve in fig. 7 is also observed to be faster than that of the corresponding track tracking error curve in no-load state, and the phenomenon is caused by larger flexible deformation of the flexible mechanical arm in the working process. Finally, the expected track is tracked, so that the inversion sliding mode controller has stronger robustness for different load variables.
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.
Claims (1)
1. A rigid-flexible coupling system of a manipulator based on spherical motor drive is characterized in that: the permanent magnet spherical motor is used as a driving motor of a rigid-flexible coupling system of the manipulator, and a flexible mechanical arm is used for replacing a rigid output shaft of the permanent magnet spherical motor to be connected with a claw to form the 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 mechanical arm rigid-flexible coupling dynamics module controls the torque tau by using a system external disturbance d and a permanent magnet spherical motor i Is an input variable; considering that the manipulator has elastic deformation, the actual displacement offset ζ thereof i (p) is the rotation angle theta of the rigid-flexible coupling system of the manipulator i Adding an elastic deformation quantity phi i (p), i.e. ζ i (p)＝θ i +Φ i (p) is an output variable; combining a permanent magnet spherical motor rigid shaft dynamics model, and establishing a partial differential dynamics model of the manipulator rigid-flexible coupling system based on a Hamiltonian analysis method;
the inversion sliding mode control module rotates at a desired angle theta id To input variable, the rotation angle theta of the rigid-flexible coupling system of the manipulator i The output variable is fed back to the control module; based on position tracking error z 1 Speed tracking errorSetting the virtual control quantity as z 2 And a sliding mode switching surface function s such that s=z 2 Ensuring that the system reaches an ideal sliding mode; by constructing parameter c 1 And c 2 And combining control law to define Lyapunov verification function V 1 、V 2 The output error is limited in a smaller range, and the control torque tau of the permanent magnet spherical motor calculated in real time is obtained i ；
The manipulator rigid-flexible coupling dynamics module is combined with a permanent magnet spherical motor rigid shaft dynamics model, and modeling is carried out based on a Hamiltonian analysis method;
it defines the flexible bending of the origin of the manipulator at any moment to be zero, namely the elastic deformation phi i (0, t) =0, and the change rate of the origin flexible bending along the vector r direction at any time is zero, thereby obtainingCombining actual displacement offset of manipulatorQuantity zeta i (p) the expression for the manipulator using the Hamiltonian principle is:
wherein t is 1 And t 2 For a given moment of two motion states,and->The variation of the work of the total kinetic energy of the manipulator, potential energy caused by elastic deformation and non-conservative force is respectively;
in combination with the length L of the flexible manipulator and the motion characteristics of the manipulator, formula (1) can be written as:
wherein the method comprises the steps of The weight of the rho flexible mechanical arm in unit length is m, the weight of the paw is J i The center rotational inertia of the permanent magnet spherical motor is represented by F, the product value of the weight m of the gripper and the vector length of the flexible mechanical arm, and EI is the bending stiffness of the mechanical arm;
due toAnd->Is a linearly independent variable, then there is a=b=c=d=0,formula (2) is written:
wherein the method comprises the steps of
Combining the Lagrangian second equation and coordinate transformation, the manipulator rigid-flexible coupling dynamics model based on spherical motor driving is shown as formula (4):
in the formula (4), k is an adjustment parameter of the rigid-flexible coupling system of the manipulator and is more than 0;
the inversion sliding mode control module tracks the error z according to the position 1 Speed tracking errorSetting the virtual control quantity as z 2 And a sliding mode switching surface function s; according to the adjustment parameter c 1 、c 2 In combination with the control law, a Lyapunov function V is defined 1 、V 2 The method comprises the steps of carrying out a first treatment on the surface of the The dynamics model (4) of the rigid-flexible coupling system combined with the manipulator can be expressed as follows:
wherein x is 1 And x 2 Respectively representing the state variables of the rotation angle and the rotation angular velocity of the rigid-flexible coupling system of the manipulator, d t ＝-M -1 (x 1 ) d, and |d t D is not more than; the rotation angle error z of the rigid-flexible coupling system of the manipulator can be known by the method (4) 1 ＝x 1 -x 1d The rotational angular velocity error is defined asDefinition of the first Lyapunov function V 1 Virtual control amount z 2 The method comprises the following steps:
z 2 ＝x 2 +c 1 z 1 -x 1d (7)
defining a sliding mode switching plane function s=z in combination with sliding mode variable structure control 2 And define a second Lyapunov function V 2 Deriving s:
in order to makeThe design controller outputs the control torque tau of the permanent magnet spherical motor i The method comprises the following steps:
wherein c 2 Is a positive constant greater than zero, eta is a constant term and eta is more than or equal to D; from (9), it can be seen thatMeets the design requirements.
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