CN113985738A - Gradient neural network cooperative control of non-convex constraint omnidirectional four-wheel mobile mechanical arm repetitive motion - Google Patents
Gradient neural network cooperative control of non-convex constraint omnidirectional four-wheel mobile mechanical arm repetitive motion Download PDFInfo
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Abstract
The invention discloses a non-convex edge boundary constrained orthogonal repetitive motion scheme of an omnidirectional four-wheel mobile manipulator, which utilizes a gradient neural network to construct a kinematic model to solve the orthogonal repetitive motion scheme, and the method comprises the following steps: a. measuring data of wheels of a mobile mechanical arm and the mechanical arm in an initial state; b. given a desired trajectory of the mobile robotic arm; c. establishing a kinematic equation of the omnidirectional mobile platform based on integrity constraint, obtaining a kinematic equation under a mechanical arm base coordinate system through space coordinate transformation, and establishing an overall kinematic equation of the mobile mechanical arm by combining models of the mobile platform and the mechanical arm; d. an orthogonal quadratic programming model of non-convex joints, wheel speed limitation, track tracking and repeated motion is defined aiming at the track tracking of the mobile mechanical arm; e. a transport mechanics equation is constructed by a gradient descent method and speed compensation to solve an orthogonal repetitive motion scheme, the orthogonal repetitive motion scheme can eliminate position errors, and the omnidirectional moving mechanical arm can accurately realize repetitive motion.
Description
Technical Field
The invention relates to the field of mobile robots, in particular to an omnidirectional four-wheel mobile manipulator trajectory tracking control method based on kinematics and a repetitive motion gradient neural network.
Background
With the progress of artificial intelligence technology, the intelligent robot industry has been developed vigorously. The intelligent robot is a robot system which can comprehensively simulate people in the aspects of perception, thinking and effect and can replace human to greatly expand the body and hand in various fields. When the mobile mechanical arm works in a dynamic and unknown complex environment, the system should have complete autonomy, that is, the system has sensing capability, planning capability, action capability and coordination capability, so in the aspect of theoretical research of the mobile mechanical arm, the problems to be solved include trajectory planning, motion control, cooperative control and the like. The motion control of the mobile mechanical arm can be divided into three types, namely point stabilization, path following and trajectory tracking according to different control targets, wherein the trajectory tracking repetitive motion control of the mobile mechanical arm is a hotspot and difficulty of research in the control field in recent years.
Most of the theoretical research on the mobile mechanical arm at the present stage is based on two-wheel mobile mechanical arms or three-wheel mobile mechanical arms, most models of the robot are based on dynamic modeling, the dynamic modeling of the four-wheel mobile mechanical arm is complex, and the dynamics of a mobile platform and the dynamic model of the mechanical arm need to be analyzed. The two models are difficult to integrate in one system, so most researchers adopt two control algorithms to respectively control the two subsystems, and the cooperative control of the mobile platform and the mechanical arm is difficult to realize. The mobile platform and the mechanical arm overall kinematics model can more easily complete the cooperative control task. In addition, kinematic modeling is relatively simple compared to dynamic modeling of the system. At present, most of the establishment of the overall kinematic models of the mobile platform and the mechanical arm based on non-integrity constraint leads to the controllable degree of freedom of the mobile mechanical arm being less than the overall degree of freedom of the mobile mechanical arm. However, an omni-directional mobile robotic arm established based on integrity constraints can move in all directions, such as sideways movement occurs, and the total degree of freedom of the mobile robotic arm is equal to the controllable degree of freedom. Therefore, the invention establishes a kinematics model of the omnidirectional Maclam wheel mobile mechanical arm based on integrity constraint, integrates the omnidirectional mobile platform and the mechanical arm in a system based on a world coordinate system through space coordinate transformation, provides a track orthogonal repetitive motion method of the omnidirectional four-wheel mobile mechanical arm, and realizes accurate track tracking repetitive motion control of the omnidirectional four-wheel mobile mechanical arm by utilizing a gradient neural network.
Disclosure of Invention
The invention discloses an omnidirectional four-wheel mobile mechanical arm track tracking repetitive control method of a gradient neural network, which is characterized in that an overall kinematic equation of a four-wheel mobile mechanical arm is established under a world coordinate system based on integrity constraint, an expected motion track is designed in a reachable space range of a mobile mechanical arm, a vector type error function is defined based on a difference value between the expected motion track and an actual motion track function, and the smaller the absolute value of the error function is, the better the track tracking task is finished. The method comprises the steps of constructing multiple subtasks of trajectory tracking, repetitive motion and joint and wheel speed limitation into quadratic programming orthogonal repetitive motion schemes of equality constraint and inequality constraint, constructing a gradient neural network by using a speed compensation and gradient descent method to solve the multiple subtask quadratic programming schemes, and popularizing the joint and wheel speed constraint to a non-convex constraint range. Orthogonal projection matrix is introduced into orthogonal repeated motion scheme
Theoretically eliminating the position error and decoupling the coupling relationship of the position error and the joint drift. The technical scheme of the invention is as follows by combining the attached drawings of the specification:
the gradient neural network control method based on the omnidirectional four-wheel mobile mechanical arm repetitive motion scheme of the non-convex boundary constraint comprises the following specific steps:
s1: acquiring initial angle data of four wheels of an omnidirectional four-wheel mobile mechanical arm and initial angle data of a four-degree-of-freedom mechanical arm;
s2: designing an expected motion track of the omnidirectional four-wheel mobile mechanical arm according to the requirement of a designer;
s3: obtaining a kinematics equation of the four-degree-of-freedom mechanical arm under a base coordinate system through space coordinate transformation, establishing a motion model of the omnidirectional moving platform based on integrity constraint, and obtaining an overall kinematics equation of the omnidirectional moving mechanical arm under a world coordinate system through coordinate transformation by combining the kinematics model of the omnidirectional moving platform and the kinematics model of the mechanical arm;
s4: the inequality constraints of track tracking, repeated motion and joints are converted into a quadratic programming problem of orthogonal repeated motion, and the speed constraints and non-convex projection functions of the joints and wheels are integrated into the speed of the joints and wheels to be kept in a specified range;
s5: an omnidirectional four-wheel moving mechanical arm kinematics model is constructed by combining an omnidirectional moving mechanical arm orthogonal repetitive motion scheme and a gradient neural network model, and the introduction of an orthogonal projection matrix is proved to decouple the coupling relation of position error and joint drift, so that the position error is eliminated theoretically, and the omnidirectional moving mechanical arm can complete track tracking and repetitive tasks more accurately;
the specific process of step S1 is:
in the experiment, hardware parameters of the four-wheel mobile mechanical arm are required to be referred, the height of the mobile platform is measured through a meter ruler, and the working range of each mechanical arm is measured under the condition of power failure. Wherein the parameters of each joint are respectively as follows: the working range of the base of the shaft 1 is from +90 degrees to-90 degrees, and the maximum speed is 320 degrees/s; the working range of the large arm of the shaft 2 is 0-85 degrees, and the maximum speed is 320 degrees/s; the working range of the small arm of the shaft 3 is from minus 10 degrees to plus 95 degrees, and the maximum speed is 320 degrees/s; the shaft 4 rotates within the working range of +90 degrees to-90 degrees, and the maximum speed is 480 degrees/s;
the specific process of step S2 is:
the desired trajectory of the end effector of the four-wheel mobile robot arm is designed based on the measurement values in step S1, ensuring that it cannot exceed the reach of the joints of the mobile robot arm. Wherein the mathematical expression of the desired trajectory is as follows:
rxd＝0.4*cos(4*π*(sin(0.5*π*t/10))2+π/6)-0.4*cos(π/6)
rxd＝0.4*sin(2*π*(sin(0.5*π*t/10))2+π/6)+0.4*sin(π/6)
rxd＝0.2362
rd＝[rxd；rxd；rxd]
in the formula, rxdIs rdProjecting the curve on an x plane; r isydIs rdProjecting the curve on a y plane; r iszdIs rdThe curve is projected in the z plane.
The specific process of step S3 is:
s301: in order to describe the relative position and direction relationship of the links of the mobile robot arm, a coordinate system needs to be established on each link according to the joint structure of the robot arm. Establishing a kinematic model of the mobile mechanical arm by using a D-H method, and performing homogeneous transformation on a connecting rod coordinate system { i } relative to { i-1}i-1TiSo-called connecting rod conversion, in which the angle of rotation alpha of the shaft is designedi-1Length of connecting rod ai-1Offset distance d of connecting rodiThe joint variable thetaiAnd thus can be decomposed into sub-transformations of the coordinate system { i }, each of which depends on only one link parameter, then there are:
i-1Ti＝Rot(x,αi-1)Trans(x,ai-1)Rot(z,θi)Trans(z,di)
obtaining a mechanical arm kinematics equation based on a base coordinate system through coordinate transformation:
wherein d is1Is the length of the connecting rod 1; d2Is the length of the connecting rod 2; d3Is the length of the connecting rod 3; d4Is the length of the connecting rod 4; c. C1＝cosφ1，s1＝sinφ1，c23＝cos(φ2+φ3)，s23＝sin(φ2+φ3)；
S302: the mobile platform selects Mecanum wheels as driving wheels, and a four-wheel full-drive mode is adopted in the aspect of power. Resolving the motion of a Mecanum wheel chassis of a mobile platform into three independent variables for description; firstly, calculating the speed of the axle center position of each wheel; calculating the speed of the roller of which the wheel is in contact with the ground according to the result of the first step; and calculating the real rotating speed of the wheel according to the result of the second step. Finally, obtaining a positive solution of the four-wheel omnidirectional kinematics model:
wherein a is the distance from the middle point of the front and rear wheels to the front and rear wheels, b is the distance from the middle point of the front and rear wheels to the middle point of the four wheels,
indicating the direction of movement of the X axis, is equivalent to
I.e., the left-right direction, is defined as positive to the right,
indicating the direction of Y-axis motion, is equivalent to
I.e., the front-rear direction, defines forward as positive, and ω represents the angular velocity of rotation of the yaw axis, which is equivalent to
Defining counterclockwise as positive, these quantities are the velocities of the geometric centers (the diagonals of the rectangle) of the four wheels.
Representing the speed of each wheel.
S303: by using the transformation matrix of the base coordinate system to the world coordinate system, the overall kinematic equation of the mobile robot arm with respect to the world coordinate system can be obtained:
differentiating the time t by the formula to obtain a kinematic equation of the mobile manipulator arm under the world coordinate system as follows:
wherein, the Jacobian matrix
The final formulation is the following simplified kinematic equation:
wherein the content of the first and second substances,
and the angle vector of the mobile mechanical arm is represented, and comprises the rotation angle of the wheel of the mobile platform and the rotation angle of each joint of the mechanical arm.
The specific process of step S4 is:
the method comprises the following steps of constructing a multi-subtask equation and inequality constraint of track tracking, repetitive motion and joint and wheel speed constraint into an orthogonal repetitive motion scheme, constructing an omnidirectional moving mechanical arm repetitive motion kinematics model according to the orthogonal repetitive motion scheme of the omnidirectional four-wheel moving mechanical arm and a gradient neural network control method, and designing the orthogonal projection repetitive motion scheme of the gradient neural network with non-convex joint constraint as follows:
in the formula (I), the compound is shown in the specification,
representing a pseudo-inverse operation of a matrix, I being an identity matrix; c ═ mu3(pc-pc(0))；μ2(sin(α)-sinα(0))；μ1(φ-φ(0))]； Q＝[D,0；0,I]∈R(n+m)×(m+2)，D＝[B；Acosα]∈Rm×2；pcAnd alpha is the connecting point (X) of the omnidirectional moving platform and the manipulator respectivelyd,Yd0) and the heading angle of the mobile platform; p is a radical ofc(0) And sin α (0) are each pcAnd an initial value of α; mu.s1,μ2,μ3Respectively are feedback coefficients; n and m are dimensional coefficients of the matrix; q. q.sT＝QTc and P ═ QTQ is a coefficient matrix for ensuring that the omnidirectional mobile mechanical arm realizes repeated motion; superscript T represents the transpose operation of the matrix; Ω represents a set of constraints that limit joint and wheel speeds to a specified range;
the kinematics model of the omni-directional mobile mechanical arm based on the orthogonal repetitive motion scheme corresponding to the gradient neural network is designed as follows:
in the formula, eta is a recursion coefficient;
to complete a recursive functional variable;
is a non-convex projection function; delta is a convergence coefficient;
realizing a repetitive motion coefficient matrix for the omnidirectional mobile mechanical arm;
by orthogonal projection matrix
Lead of (2)In theory, the position error converges to zero, and the quantitative coupling relation between the position error and the joint drift is decoupled. Under the control of the model, the omnidirectional four-wheel mobile mechanical arm can more accurately complete track tracking and repeated motion tasks.
The specific process of step S6 is:
the situation of the change of the vehicle speed of the wheels of the moving platform and the rotation angle of each joint in the process of tracking the expected track of the omnidirectional four-wheel moving mechanical arm are solved through the kinematic equation, and the parameters obtained through calculation can be applied to each motor to adjust each variable to perform track tracking repetitive motion.
Compared with the prior art, the invention has the advantages that:
the invention establishes an overall kinematic equation of the omnidirectional four-wheel mobile mechanical arm based on integrity constraint, designs track tracking, repeated motion, joint and wheel speed limitation into a multi-subtask orthogonal quadratic programming model, introduces an orthogonal projection matrix into the quadratic programming model, theoretically eliminates position errors, and decouples the coupling relation between the position errors and joint drift. Characterized in that, firstly: the traditional control of the mobile mechanical arm needs to establish a kinematic model of the system and respectively control the mobile platform and each joint mechanical arm, however, in the invention, the kinematic modeling is respectively carried out on the mobile platform and the four-degree-of-freedom mechanical arm to avoid complex kinematic modeling, and the two are integrated into one system through space coordinate transformation to realize the cooperative control of the mobile mechanical arm. Secondly, the method comprises the following steps: the omnidirectional mobile mechanical arm is established based on the integrity constraint, and different from the establishment of the mobile mechanical arm based on the non-integrity constraint, the omnidirectional mobile mechanical arm established based on the integrity constraint can move towards all directions, if the omnidirectional mobile mechanical arm moves laterally, and the total degree of freedom of the mobile mechanical arm is equal to the controllable degree of freedom. And the controllable degree of freedom of the omnidirectional mobile mechanical arm established based on the non-integrity constraint is smaller than the total degree of freedom of the mobile mechanical arm. Typically, the degree of freedom of the moving robot arm is 3, including the lateral axis, the longitudinal axis and the orientation. Thirdly, the method comprises the following steps: the invention relates the joint and wheel speed constraint of the orthogonal quadratic programming scheme and the non-convex mapping function, and can be more practicalThe situation is consistent. And an orthogonal projection matrix is introduced into the orthogonal quadratic programming scheme
The relation between the position error and the joint drift is decoupled, the position error is theoretically eliminated, and the scheme that the track tracking repeated motion of the omni-directional moving mechanical arm is accurately completed is realized. And then solving the problem of multi-subtask quadratic programming by using a speed compensation and gradient descent method, solving the problem of cooperative control of the tracking and repeated motion of the mechanical arm under a non-convex mapping function, and verifying the effectiveness of the algorithm through a simulation experiment.
Drawings
FIG. 1 is a point connection profile and course angle for a non-convex projection function orthogonal repetitive motion scheme of the present invention;
FIG. 2 is an image of angular changes of each mechanical arm of an omnidirectional four-wheel mechanical arm controlled by a gradient neural network model constructed based on a speed compensation and gradient descent method according to a non-convex projection function orthogonal repetitive motion scheme;
fig. 3 is a rotation angle change image of each wheel of the omnidirectional four-wheel mobile mechanical arm end effector controlled by the gradient neural network model constructed based on the speed compensation and gradient descent method according to the non-convex projection function orthogonal repetitive motion scheme of the invention to track an expected track;
fig. 4 is an angular velocity change image of each arm of which the four-wheel omnidirectional moving mechanical arm end effector is controlled by a gradient neural network model constructed based on a velocity compensation and gradient descent method according to the non-convex projection function orthogonal repetitive motion scheme of the invention to track an expected track;
fig. 5 is an image of the change of the rotational angular velocity of each wheel of the four-wheel omnidirectional moving mechanical arm end effector controlled by the gradient neural network model constructed based on the velocity compensation and gradient descent method according to the non-convex projection function orthogonal repetitive motion scheme of the present invention to track the desired trajectory;
fig. 6 is an error image of the gradient neural network model based on velocity compensation and gradient descent method to control the end effector of the four-wheel omnidirectional mobile manipulator to track an expected trajectory according to the non-convex projection function orthogonal repetitive motion scheme of the present invention;
FIG. 7 is a joint drift (joint error) image of an end effector of a four-wheel omnidirectional moving mechanical arm controlled by a gradient neural network model constructed based on a velocity compensation and gradient descent method according to a non-convex projection function orthogonal repetitive motion scheme;
fig. 8 is an image of a gradient neural network model based on velocity compensation and gradient descent method to control the trajectory of an end effector of a four-wheel omnidirectional mobile manipulator to track an expected trajectory according to a non-convex projection function orthogonal repetitive motion scheme of the present invention;
fig. 9 is a top view image of the track of the end effector of the four-wheel omnidirectional moving mechanical arm controlled by the gradient neural network model constructed based on the velocity compensation and gradient descent method according to the non-convex projection function orthogonal repetitive motion scheme of the present invention.
Claims (3)
1. The gradient neural network cooperative control method generated by repeated motion of the omnidirectional four-wheel mobile mechanical arm based on non-convex boundary constraint is characterized by comprising the following steps:
s1: designing an expected trajectory equation of the omnidirectional four-wheel mobile mechanical arm according to requirements;
s2: giving an initial rotation angle of each wheel of the omnidirectional four-wheel mobile mechanical arm and an initial angle of the four-degree-of-freedom mechanical arm, and measuring the length and the width of the mobile platform;
s3: constructing an overall kinematics model of the mechanical arm of the omnidirectional four-wheel mobile platform;
s4: the method comprises the following steps of converting equality constraint and inequality constraint of track tracking, repeated motion and joint constraint into a quadratic programming model with an orthogonal repeated motion function, and integrating joint and wheel speed constraint and a non-convex projection function into joint and wheel speed which is kept in a constrained range;
s5: the kinematics model of the omnidirectional four-wheel mobile mechanical arm is constructed by combining the orthogonal repetitive motion scheme of the omnidirectional four-wheel mobile mechanical arm and the gradient neural network model, the problems of position error and joint drift are solved by utilizing an orthogonal projection matrix, the position error is eliminated, and the track tracking and repetitive tasks of the omnidirectional four-wheel mobile mechanical arm are accurately completed.
2. The method for controlling repetitive motion of an omnidirectional four-wheel mobile manipulator of a gradient neural network as claimed in claim 1, wherein the specific process of step S5 is as follows:
based on the kinematics characteristics of the omnidirectional four-wheel mobile mechanical arm, a speed-level horizontal overall kinematics model of the mobile mechanical arm is constructed, and the specific mathematical expression is as follows:
in the formula, J is a coefficient matrix of the overall kinematic model;
differentiating an actual track deduced for the omnidirectional mobile manipulator kinematics model with respect to time t;
the speed of moving four wheels and four mechanical arms of the mechanical arm in all directions.
3. An orthogonal projection matrix is introduced at S4
Converting a plurality of subtasks of the speed level overall kinematics model, joint constraint, trajectory tracking and repetitive motion of the omnidirectional four-wheel mobile mechanical arm into an orthogonal repetitive motion quadratic programming scheme:
minimize
subject to
in the formula (I), the compound is shown in the specification,
upper label
A pseudo-inverse operation of the representative matrix; i is an identity matrix; p, qTThe coefficient matrix is used for ensuring that the omnidirectional four-wheel mobile mechanical arm realizes repeated motion; superscript T represents the transpose operation of the matrix; Ω represents a set of constraints that limit joint and wheel speeds to a specified range;
the gradient neural network orthogonal projection repetitive motion scheme with non-convex joint constraint is designed as follows:
in the formula, eta is a recursion coefficient;
to complete a recursive functional variable; gamma rayΩ(. is a projection function; delta is a convergence coefficient;
realizing a repeated motion coefficient matrix for the omnidirectional four-wheel mobile mechanical arm;
by orthogonal projection matrix
The orthogonal repetitive motion scheme of the omnidirectional four-wheel mobile mechanical arm is obtained by introduction, proves that the position error is converged to zero, the position error and the joint drift are decoupled, and the feedback of the joint drift is increasedThe coefficient, the joint drift and the position error are reduced simultaneously, and the omnidirectional four-wheel mobile mechanical arm can accurately complete track tracking and repeated motion tasks under the control of the model.
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