CN115477259A - Cross coupling sliding mode control forklift synchronous control method - Google Patents
Cross coupling sliding mode control forklift synchronous control method Download PDFInfo
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Abstract
The invention discloses a cross-coupling sliding mode control forklift synchronous control method which comprises the steps of establishing an input and output sliding mode controller and establishing a cross-coupling controller. The cross-coupling sliding mode control forklift synchronization method designed by the invention achieves good control effect on high-precision synchronous control of the forklift and suppression of synchronous errors during synchronous steering of double trucks, and has good robustness under the condition of existence of interference of complicated operation conditions of the forklift.
Description
Technical Field
The invention belongs to the technical field of electric pallet forklifts, and particularly relates to a cross-coupling sliding-mode control forklift synchronous control method.
Background
The forklift is an engineering vehicle widely applied to ports, stations, airports, goods yards, factory workshops, warehouses, circulation centers and distribution centers, and is high-efficiency equipment essential for loading and unloading, carrying operation, pallet transportation and container transportation of pallet goods in cabins, carriages and containers. Forklifts can generally be divided into two broad categories: diesel fork truck and electric fork truck, and the factor influence such as artifical substitution of benefit, environmental protection upgrading, electronic upgrading, electric fork truck is becoming the mainstream transport vehicle of handling trade and is being used for replacing diesel fork truck. And the electric forklift has difficulty in performing precise motion control of the desired position of the forklift.
The patent name: a series-parallel electric pallet forklift and a motion method thereof (patent application number 201910530060.7) propose a series-parallel electric pallet forklift adopting a mechanical coupling mode and a motion method thereof, but the structure is hard synchronous by a mechanical structure, and the problem that the mechanical structure is distorted when the motion is asynchronous exists.
Disclosure of Invention
The invention aims to provide a fork truck synchronous control method based on cross coupling sliding mode control, and the problem of accurate motion control of a fork truck expected position is solved.
The technical scheme provided by the invention is as follows:
a fork truck synchronous control method controlled by a cross coupling sliding mode,
step one, establishing a dynamic model of the electric pallet forklift
Firstly, establishing a fixed ground coordinate system with X and Y as identifiers, and secondly, establishing a body coordinate system with the mass center P of the forklift as mass points, the X axis as the longitudinal direction of the movement of the forklift, and the Y axis as the lateral direction of the movement of the forklift;
delta denotes the angle of rotation of the front wheel of the fork-lift truck, beta f Indicating the slip angle, beta, of the front wheel r1 、β r2 Representing the slip angle of the two rear wheels, the slip angle producing a lateral force F in the tire yf 、F yr1 、F yr2 The tire controls the transverse movement of the forklift under the action of transverse lateral force;
the tire force of each forklift acts on a forklift mass center point P, beta represents a forklift mass center lateral deviation angle, psi represents a forklift mass center course angle, U and V respectively represent the transverse speed and the longitudinal speed of the forklift mass center P in a forklift body coordinate system, V represents a mass center vector motion speed, r represents a forklift mass center yaw angular velocity, and l represents the forklift mass center yaw angular velocity f Representing the distance of the centre of mass to the central axis of the front wheel, lr representing the distance of the centre of mass to the central axis of the rear wheel, d r The wheelbases of two rear wheels of the forklift are shown;
the following assumptions are made for the truck tires,
1) For a high speed truck, the lateral force of the tires is at a slip angle beta f 、β r1 、β r2 When the angle is less than 4 degrees, the angle is regarded as a lineThe sex is increased;
2) Slip angle beta of tire of forklift during movement f 、β r1 、β r2 Extremely small;
on the basis of the above assumptions, a mathematical model of the forklift is obtained as follows:
F y ＝m*α y (1)
wherein the tire lateral force is defined by F y Indicating that the fork truck mass is m and the lateral acceleration is alpha y It is shown that,
F＝I z *α r (2)
wherein the resultant force of the transverse rolling rotation of the forklift is represented by F, and the transverse rolling inertia of the forklift body is represented by I z Showing that the yaw rotation acceleration of the forklift body is alpha r It is shown that,
can be rewritten by the above formula (1)
Can be rewritten by the above formula (2)
The relation between the lateral force of the tire and the rotation angle linearity of the front wheel can be obtained
F yr ＝K f *β f (5-1)
F yr1 ＝K r1 *β r1 (5-2)
F yr2 ＝K r2 *β r2 (5-3)
Wherein, the lateral force of the front tire, the lateral force of the rear tire 1 and the lateral force of the rear tire 2 are respectively represented by F yf 、F yr1 、F yr2 The cornering stiffness of the front wheel, the cornering stiffness of the rear wheel 1 and the cornering stiffness of the rear wheel 2 are represented by K f 、K r1 、K r2 The slip angles of the front wheel, the rear wheel 1 and the rear wheel 2 are represented by beta f 、β r1 、β r2 It is shown that the process of the present invention,
since the slip angle is small during actual travel of the forklift, U = V × cos β ≈ V, V = V × sin β ≈ V × β, while d may be approximately assumed r Neglecting in comparison with the running length of the forklift, and making d r =0, slip angle of front wheel, slip angle of rear wheel 1, rear wheel 2 can be found
Substituting formulae (6) to (8) into the above formulae (5-1), (5-2) and (5-3) to obtain
Rear wheel 1 cornering stiffness K r1 And rear wheel 2 cornering stiffness K r2 Are identical, therefore, F yr1 ＝F yr2 In addition, let K r1 ＝K r2 ＝K r ，F yr ＝F yr1 ＝F yr2 ，
Can be rewritten by the above formula (3)
Can be rewritten by the above formula (4)
Due to beta when the forklift runs f Very small, let sin beta f ≈0，cosβ f 1, and the above formula (12) can be rewritten
By substituting expressions (9), (10) and (11) into expressions (13) and (14)
The relationship between the heading angle psi of the forklift and the yaw angular velocity r of the mass center of the forklift
From the relationship between the speed v and the acceleration alpha of the fork truck
From the relation of the coordinate system of the vehicle body and the coordinate system of the ground
Selecting a mass center side slip angle beta, a yaw angular velocity f, a yaw angle psi and a mass center motion velocity v as state variables of the system; selecting a front wheel corner delta and a vehicle body acceleration alpha as system inputs; selecting the speed X in the X direction and the speed Y in the Y direction of the forklift in a fixed coordinate system as the output of the system to establish a state space equation to obtain
Step two, establishing an input/output sliding mode controller
Output to the system in the above equations (21) and (22)
Derived to obtain
The input delta and alpha in the above formulas (23) and (24) can be collated
Defining errors
Wherein, the groundThe surface coordinate system X coordinate is represented by X, the ground coordinate system Y coordinate is represented by Y, and the desired ground coordinate system X coordinate is represented by X d The desired ground coordinate system Y coordinate is represented by Y d Expressing that deviation of the expected coordinate and the actual coordinate on the X coordinate is expressed by eX, and deviation of the expected coordinate and the actual coordinate on the Y coordinate is expressed by eY;
defining sliding mode functions
Wherein the sliding mode function 1 is represented by S 1 Expressed, the sliding mode function 2 is represented by S 2 The coefficient of convergence speed of the sliding mode is represented by C 1 The coefficient of convergence speed of the sliding mode is represented by C 2 Represents;
separating input vectors delta and alpha and designing input and output sliding mode control rate
Wherein the switching term function is represented by sgn, and the switching term coefficient 1 is represented by eta 1 Expressed by the switching term coefficient 2 of eta 2 Is represented by v 1 、v 2 Obtained by the formula (29);
step three: establishment of cross-coupled controller
Simultaneously, the expected X-axis speed and the Y-axis speed of the forklift 1 are given, the speed is different from the actual X-axis speed and the actual Y-axis speed of the forklift 1 to obtain the self motion error of the forklift 1, the self input and output sliding mode controller of the forklift 1 controls the forklift 1 to move through the self motion error of the forklift 1, the expected X-axis speed and the Y-axis speed of the forklift 2 are calculated through the same expected X-axis speed and Y-axis speed of the forklift 1 to obtain the self motion error of the forklift 2, and the self input and output sliding mode controller of the forklift 2 controls the forklift 2 to move through the self motion error of the forklift 2;
the actual speed of the X and Y axes of the forklift 1 is different from the actual speed of the X and Y axes of the forklift 2 to obtain a cross coupling motion error of the forklift 1, the cross coupling input and output sliding mode controller of the forklift 1 controls the forklift 1 to finish synchronous deviation calibration through the cross coupling motion error of the forklift 1, the same actual speed of the X and Y axes of the forklift 2 is different from the actual speed of the X and Y axes of the forklift 1 to obtain a cross coupling motion error of the forklift 2, and the cross coupling input and output sliding mode controller of the forklift 2 controls the forklift 2 to finish synchronous deviation calibration through the cross coupling motion error of the forklift 2.
The cross-coupling sliding mode control forklift synchronous control method achieves good control effect on high-precision synchronous control of the forklift and suppression of synchronous errors during double-vehicle synchronous steering, and has good robustness under the condition that interference of complicated operation conditions of the forklift exists.
Drawings
FIG. 1 is a schematic structural view of an electric pallet fork lift system of the present invention;
FIG. 2 is a diagram of a dynamic stress analysis of a forklift truck according to the present invention;
FIG. 3 is a block diagram of a cross-coupled input/output sliding mode control method according to the present invention;
FIG. 4 is an ideal path diagram for the synchronous operation of the forklift truck according to the present invention;
FIG. 5 is a diagram of the simulation results of the no-load synchronous motion of the present invention;
fig. 6 is a diagram of the simulation result of the 2T load synchronous motion of the present invention.
Detailed Description
In order to make the technical means, the creation characteristics, the achievement purposes and the effects of the invention easy to understand, the invention is further described with the specific embodiments.
A cross-coupling sliding mode control forklift synchronous control method comprises the steps of establishing an input and output sliding mode controller and establishing a cross-coupling controller.
The cross-coupling sliding mode control method of the electric pallet forklift comprises the following specific processes:
the method comprises the following steps: establishment of dynamic model of electric pallet forklift
The actual condition of the forklift operation scene is complex, the variable friction coefficient between the tires and the ground when the forklift runs on different roads influences the acceleration and deceleration movement of the forklift, and the backlash shake of the steering transmission structure influences the steering movement of the forklift system. The sliding mode controller has high response speed and robustness to interference caused by external corner shaking and friction force change, so that the input and output sliding mode controller is adopted to cope with the complicated operation working conditions of the forklift.
Referring to fig. 1, the electric pallet forklift system structure includes a forklift head 9, a pallet fork 8, a forklift front wheel 4, a forklift rear wheel 7, a steering servo motor 3, a planetary reducer 2, a planetary gear 1, a sun gear 6, a straight servo motor 5 and other components. The steering servo motor 3 is decelerated by the planetary reducer 2 and then rotates on the sun gear 6 through the planetary gear 1 to realize the transverse rotation of the front wheel 4 of the forklift; the main shaft of the straight servo motor 5 is connected with the middle shaft of the front wheel 4 of the forklift, and is a power device for realizing the forward and backward movement of the forklift.
A dynamic force analysis diagram of a forklift is shown in figure 2,
firstly, establishing a fixed ground coordinate system with X and Y as identifiers, and secondly, establishing a body coordinate system with the mass center P of the forklift as mass points, the X axis as the longitudinal direction of the movement of the forklift, and the Y axis as the lateral direction of the movement of the forklift;
delta denotes the angle of rotation of the front wheel of the fork-lift truck, beta f Indicating the slip angle, beta, of the front wheel r1 、β 2 Representing the slip angle of the two rear wheels, the slip angle generating a lateral force F in the tire yf 、F yr1 、F yr2 The tire controls the transverse movement of the forklift under the action of transverse lateral force;
the tire force of each forklift acts on a forklift mass center point P, beta represents a forklift mass center slip angle, psi represents a forklift mass center heading angle, U and V respectively represent the transverse speed and the longitudinal speed of the forklift mass center P in a vehicle body coordinate system, V represents the vector movement speed of the forklift mass center P, r represents the yaw angular speed of the forklift mass center, and l represents the lateral speed and the longitudinal speed of the forklift mass center P in the vehicle body coordinate system f Representing the distance of the centre of mass from the central axis of the front wheel, l r Representing the distance of the centre of mass to the central axis of the rear wheel, d r To representTwo rear wheel wheelbases of the forklift;
the following assumptions are made for the truck tires,
1) For a forklift moving at high speed, the lateral force of the tyre is at the slip angle beta f 、β r1 、β r2 A linear increase is considered when less than 4 °;
2) Slip angle beta of tire of forklift during movement f 、β r1 、β r2 Extremely small;
on the basis of the above assumptions, a mathematical model of the forklift is obtained as follows:
F y ＝m*α y (1)
wherein the tire lateral force is defined by F y Indicating that the fork truck mass is m and the lateral acceleration is alpha y It is shown that the process of the present invention,
F＝I z *α r (2)
wherein the resultant yaw rotation force of the forklift is represented by F, and the yaw rotation inertia of the forklift body is represented by I z Indicating that the yaw rotation acceleration of the forklift body is alpha r It is shown that,
can be rewritten by the above formula (1)
Can be rewritten by the above formula (2)
The relation between the lateral force of the tire and the rotation angle linearity of the front wheel can be obtained
F yr ＝K f *β f (5-1)
F yr1 ＝K r1 *β r1 (5-2)
F yr2 ＝K r2 *β r2 (5-3)
Wherein, the lateral force of the front tire, the lateral force of the rear tire 1 and the lateral force of the rear tire 2 are respectively represented by F yf 、F yr1 、F yr2 The front wheel cornering stiffness, the rear wheel 1 cornering stiffness and the rear wheel 2 cornering stiffness are represented by K f 、K r1 、K r2 The slip angles of the front wheel, the rear wheel 1 and the rear wheel 2 are represented by beta f 、β r1 、β r2 It is shown that the process of the present invention,
due to the small slip angle during the actual travel of the fork truck, U = V × cos β ≈ V, V = V × sin β ≈ V × β, while d may be approximated, with r Neglecting in comparison with the running length of the forklift, and making d r =0 get beta r1 ≈β r2 The slip angles of the front wheel and the rear wheels 1 and 2 can be obtained
Substituting formulae (6) to (8) into the above formulae (5-1), (5-2) and (5-3)
Rear wheel 1 cornering stiffness K r1 And rear wheel 2 cornering stiffness K r2 Same, therefore, F yr1 ＝F yr2 In addition, let K r1 ＝K r2 ＝K r ，F yr ＝F yr1 ＝F yr2 ，
Can be rewritten by the above formula (3)
Can be rewritten by the above-mentioned formula (4)
Due to beta when the forklift runs f Very small, let sin beta f ≈0，cosβ f 1, and the above formula (12) can be rewritten
By substituting expressions (9), (10) and (11) into expressions (13) and (14)
The relationship between the heading angle psi of the forklift and the yaw angular velocity r of the mass center of the forklift
From the relationship between the speed v and the acceleration alpha of the forklift
From the relationship between the coordinate system of the vehicle body and the coordinate system of the ground
Selecting a mass center side slip angle beta, a yaw angle speed r, a yaw angle psi and a mass center motion speed v as state variables of the system; selecting a front wheel corner delta and a vehicle body acceleration alpha as system inputs; selecting the X-direction speed X and the Y-direction speed Y of the forklift in a fixed coordinate system as the output of the system to establish a state space equation
Step two, establishing an input/output sliding mode controller
Output to the system in the above equations (21) and (22)
Derived to obtain
To the above ( 23 ) The input delta and alpha in the formula (24) can be obtained by sorting
Defining errors
Wherein the X coordinate of the ground coordinate system is represented by X, the Y coordinate of the ground coordinate system is represented by Y, and the X coordinate of the expected ground coordinate system is represented by X d The desired ground coordinate system Y coordinate is represented by Y d Expressing that deviation of the expected coordinate and the actual coordinate on the X coordinate is expressed by eX, and deviation of the expected coordinate and the actual coordinate on the Y coordinate is expressed by eY;
defining sliding mode functions
Wherein the sliding mode function 1 is represented by S 1 Expressed, the sliding mode function 2 is represented by S 2 The coefficient of convergence speed of the sliding mode is represented by C 1 The coefficient of convergence speed of the sliding mode is represented by C 2 And (4) showing.
Separating input vectors delta and alpha and designing input and output sliding mode control rate
Wherein the switching term function is represented by sgn, and the switching term coefficient 1 is represented by eta 1 Expressed by a switching term coefficient 2 of η 2 Is represented by v 1 、v 2 Obtained by the formula (29)
Further, stability analysis is performed on the forklift system
Taking Lyapunov function
Derived from the above equation (30)
Substituting the above expression (31) into the expressions (25) and (28) in the fourth step, and commanding the step
It is apparent that the following formula
The Lyapunov theorem shows that the system Lyapunov is stable. Indicating that within a limited time, S 1 →0，S 2 → 0. The limited time accessibility of sliding variables has been demonstrated.
Further, depending on the sliding conditions, once the system states are confined to the pre-specified S =0 sliding mode region, they can slide along the sliding surface S towards the origin. Here, the sliding manifold is constructed in the ground coordinate system error space, so the ground coordinate system error will converge progressively to zero along S. It can be seen from the formula (33) that e is → ∞ times t → ∞ X → 0 and e Y → 0. The self position error and the cross coupling synchronous error of the forklift synchronous system are gradually converged to zero.
Step three: establishment of cross-coupled controller
The cross coupling controller is widely applied to position synchronization of multiple motors and has the advantages of high synchronization precision, strong robustness and the like. The invention applies a cross-coupling sliding mode controller to eliminate the motion deviation of the two vehicles.
A block diagram of the cross-coupled input/output sliding mode control method of the present invention is shown in fig. 3.
Referring to fig. 3, the expected X-axis speed and the desired Y-axis speed of the forklift 1 are given at the same time, the speed is subtracted from the actual X-axis speed and the actual Y-axis speed of the forklift 1 to obtain the self motion error of the forklift 1, and the self input and output sliding mode controller of the forklift 1 controls the forklift 1 to move through the self motion error of the forklift 1. The same expecting X, Y axle speed by fork truck 1, Y axle speed calculation obtains fork truck 2's expecting X, Y axle speed, and this speed makes the difference with fork truck 2 self X, Y axle actual speed and obtains fork truck 2 self motion error, and fork truck 2 self input/output sliding mode controller passes through fork truck 2 self motion error control fork truck 2 and moves.
Furthermore, a black dotted line frame represents a cross coupling controller part, the actual speed of the X axis and the actual speed of the Y axis of the forklift 1 are different from the actual speed of the X axis and the actual speed of the Y axis of the forklift 2 to obtain a cross coupling motion error of the forklift 1, and the cross coupling input and output sliding mode controller of the forklift 1 controls the forklift 1 to complete synchronous deviation calibration through the cross coupling motion error of the forklift 1. The same actual speeds of the X and Y axes of the forklift 2 and the actual speeds of the X and Y axes of the forklift 1 are differed to obtain a cross-coupling motion error of the forklift 2, and the cross-coupling input and output sliding mode controller of the forklift 2 controls the forklift 2 to complete synchronous deviation calibration through the cross-coupling motion error of the forklift 2.
The accuracy of the invention is verified by simulation experiments.
Experiment 1: no-load double-forklift synchronous motion simulation experiment
The cross-coupling sliding-mode control forklift synchronous control method established according to the method carries out simulation experiments on a forklift dynamic model, and the effectiveness of the method is verified.
Simulating a factory transportation environment, starting from a point a, loading and unloading at points b, c and d, and finally moving to a point e to terminate movement by a forklift synchronous system, wherein the two forklifts keep synchronous during movement, an ideal path diagram of the forklift synchronous operation is shown in fig. 4, and a point P of the mass center of the two forklifts is used for subsequently evaluating the index of the movement synchronization performance.
And (3) carrying out forklift no-load synchronous control simulation on a forklift dynamic model aiming at the forklift synchronous control method of the cross-coupling sliding mode control provided by the formula (28) and the formula shown in figure 3.
Wherein the parameters of the self-calibration input/output sliding mode controllers of the two forklifts are set to be C 1 ＝20；C 2 ＝20；η 1 ＝1.8；η 2 =0.85, the parameters of the cross-coupled input-output sliding mode controllers of two forklifts are set to C 1 ＝10；C 2 ＝10；η 1 ＝3；η 2 ＝3.5。
The parameters of the forklift system are selected as follows: the mass m =280kg of the forklift; distance L from center of mass to rear wheel f =0.5m; distance L from center of mass to front wheel r =1m; front wheel cornering stiffness K f 4000N × m/rad; rear wheel cornering stiffness K r =1500n × m/rad; horizontal swinging moment of inertia I of whole vehicle around mass center z =69.488kg/m 2; front wheel longitudinal stiffness K x =2000N × m/rad; front wheel unit lateral stiffness K y =53333n × m/rad; the gravity acceleration g =9.8N/kg; reduction ratio i of planetary reducer 1 =0.025; reduction ratio I of steering gear to sun gear 2 =0.169; backlash deviation e of steering gear and sun gear gear bacldach ＝0.0074rad。
Setting the initial coordinate of the forklift 1 in a ground coordinate system as X =0; y =0, initial body angle 0 °, fork 2 initial coordinate X =0; y = -1, the initial vehicle body angle is 0 degrees, the target speed of the forklift is set to be 3.5m/s, the forklift rotates anticlockwise for 90 degrees under a ground coordinate system in 25s-30s, 55s-60s and 85s-90s of motion simulation respectively, and steering motion of the forklift at points b, c and d is achieved.
For the experiment 1: and (3) an unloaded synchronous motion simulation result chart, as shown in fig. 5.
Referring to fig. 5, it can be seen from a simulation diagram that, under the forklift synchronization method of cross-coupling sliding mode control, the synchronous error of the movement of the centroid point P is jittered between-3 mm and 3mm, the jitter is small when the forklift synchronously steers, the error amplitude is adjusted to 0 within a certain time, and the goal of high-precision double-forklift no-load synchronization is achieved.
Experiment 2:2T load double-forklift synchronous motion simulation experiment
The cross-coupling sliding-mode control forklift synchronous control method established according to the method carries out simulation experiments on a forklift dynamic model, and the effectiveness of the method is verified.
And (3) carrying out forklift 2T load synchronous control simulation on a forklift dynamic model aiming at the forklift synchronous control method of the cross-coupling sliding mode control provided by the formula (28) and the cross-coupling sliding mode control provided by the figure 3.
Wherein the parameters of the self-calibration input/output sliding mode controllers of the two forklifts are set to be C 1 ＝40；C 2 ＝40；η 1 ＝0.4η 2 =0.3, the parameters of the cross-coupled input-output sliding mode controllers of the two forklifts are set to C 1 ＝1；C 2 ＝1；η 1 ＝8；η 2 ＝8。
The parameters of the forklift system are selected as follows: the mass m =1280kg of the forklift; distance L from center of mass to rear wheel f =1m; distance L from center of mass to front wheel r =0.5m; front wheel cornering stiffness K f = 4000Nx m/rad; rear wheel cornering stiffness K r =1500N x m/rad; horizontal swinging inertia I around mass center of whole vehicle z =69.488kg/m 2; front wheel longitudinal stiffness K x = 2000Nxm/rad; front wheel unit lateral stiffness K y =53333n × m/rad; acceleration of gravity g =9.8N/kg; reduction ratio I of planetary reducer 1 =0.025; reduction ratio I of steering gear to sun gear 2 =0.169; backlash deviation e of steering gear and sun gear gear bacldach ＝0.0074。
As shown in fig. 4, the ideal trajectory is set such that the initial coordinate of the forklift 1 in the ground coordinate system is X =0; y =0, initial body angle 0 °, fork 2 initial coordinate X =0; y = -1, the initial vehicle body angle is 0 degree, the target speed of the forklift is set to be 3.5m/s, the forklift rotates 90 degrees in the counterclockwise direction under the ground coordinate system in 25s-30s, 55s-60s and 85s-90s of motion simulation, and steering motion of the forklift at points b, c and d is achieved.
For the experiment 2: a 2T load synchronous motion simulation result chart, as shown in fig. 6.
Referring to fig. 6, it can be seen from the simulation diagram that, under the forklift synchronization method of cross-coupling sliding mode control, the motion synchronization error of the mass center point P is jittered between-6 mm and 4mm, the jitter is small when the forklift synchronously steers, the error amplitude is adjusted to 0 within a certain time, and the purpose of high-precision 2T load synchronization of the double forklifts is achieved.
In conclusion, the cross-coupling sliding mode control forklift synchronization method designed by the invention achieves a good control effect on high-precision synchronization control of the forklift and suppression of synchronization errors during synchronous steering of two trucks, and has good robustness under the condition that the forklift is interfered by complex operation conditions.
Claims (1)
1. A cross-coupling sliding mode control forklift synchronous control method is characterized in that,
step one, establishing a dynamic model of the electric pallet forklift
Firstly, establishing a fixed ground coordinate system with X and Y as identifiers, and secondly, establishing a body coordinate system with the mass center P of the forklift as mass points, the X axis as the longitudinal direction of the movement of the forklift, and the Y axis as the lateral direction of the movement of the forklift;
delta denotes the angle of rotation of the front wheel of the fork-lift truck, beta f Indicating the slip angle, beta, of the front wheel r1 、β r2 Representing the slip angle of the two rear wheels, the slip angle generating a lateral force F in the tire yf 、F yr1 、F yr2 The transverse movement of the forklift is controlled by the tire under the action of transverse lateral force;
the tire force of each forklift acts on a forklift mass center point P, beta represents a forklift mass center slip angle, psi represents a forklift mass center heading angle, U and V respectively represent the transverse speed and the longitudinal speed of the forklift mass center P in a vehicle body coordinate system, V represents the vector movement speed of the forklift mass center P, r represents the yaw angular speed of the forklift mass center, and l represents the lateral speed and the longitudinal speed of the forklift mass center P in the vehicle body coordinate system f Representing the distance of the centre of mass to the central axis of the front wheel, l r Representing the distance of the centre of mass to the central axis of the rear wheel, d r The wheelbases of two rear wheels of the forklift are shown;
the following assumptions are made for the truck tires,
1) For a forklift moving at high speed, the lateral force of the tyre is at the slip angle beta f 、β r1 、β r2 A linear increase is considered when less than 4 °;
2) Slip angle beta of tires of forklift during movement f 、β r1 、β r2 Extremely small;
on the basis of the above assumptions, the mathematical model of the forklift is obtained as follows:
F y ＝m*α y (1)
wherein the tire lateral force is defined by F y Indicating that the fork truck mass is m and the lateral acceleration is alpha y It is shown that,
F＝I z *α r (2)
wherein the resultant force of the transverse rolling rotation of the forklift is represented by F, and the transverse rolling inertia of the forklift body is represented by I z Showing that the yaw rotation acceleration of the forklift body is alpha r It is shown that,
can be rewritten by the above formula (1)
Can be rewritten by the above formula (2)
The relation between the lateral force of the tire and the rotation angle linearity of the front wheel can be obtained
F yr ＝K f *β f (5-1)
F yr1 ＝K r1 *β r1 (5-2)
F yr2 ＝/K r2 *β r2 (5-3)
Wherein, the lateral force of the front tire, the lateral force of the rear tire 1 and the lateral force of the rear tire 2 are respectively represented by F yf 、F yr1 、F yr2 The cornering stiffness of the front wheel, the cornering stiffness of the rear wheel 1 and the cornering stiffness of the rear wheel 2 are represented by K f 、K r1 、K r2 The front wheel slip angle, the rear wheel 1 slip angle and the rear wheel 2 slip angle are represented by beta f 、β r1 、β r2 It is shown that,
because the slip angle is very small in the actual running process of the forklift, the slip angle can be approximate to that of the forkliftU = V × cos β ≈ V, V = V × sin β ≈ V × β, while d r Neglecting in comparison with the running length of the forklift, and making d r =0, slip angle of front wheel, slip angle of rear wheel 1, rear wheel 2 can be obtained
Substituting formulae (6) to (8) into formulae (5-1), (5-2) and (5-3) above
Rear wheel 1 cornering stiffness K r1 And rear wheel 2 cornering stiffness K r2 Same, therefore, F yr1 ＝F yr2 In addition, let K r1 ＝K r2 ＝K r ，F yr ＝F yr1 ＝F yr2 ，
Can be rewritten by the above formula (3)
Can be rewritten by the above formula (4)
Due to beta when the forklift runs f Very small, let sin beta f ≈0，cosβ f 1, and the above formula (12) can be rewritten
By substituting expressions (9), (10) and (11) into expressions (13) and (14) above
The relationship between the heading angle psi of the forklift and the yaw angular velocity r of the mass center of the forklift
From the relationship between the speed v and the acceleration alpha of the fork truck
From the relationship between the coordinate system of the vehicle body and the coordinate system of the ground
Selecting a mass center side slip angle beta, a yaw angle speed r, a yaw angle psi and a mass center motion speed v as state variables of the system; selecting a front wheel corner delta and a vehicle body acceleration alpha as system inputs; selecting the X-direction speed X and the Y-direction speed Y of the forklift in a fixed coordinate system as the output of the system to establish a state space equation
Step two, establishing an input/output sliding mode controller
Output to the system in the above equations (21) and (22)
Derived by derivation
The input delta and alpha in the above formulas (23) and (24) can be collated
Defining errors
Wherein the X coordinate of the ground coordinate system is represented by X, the Y coordinate of the ground coordinate system is represented by Y, and the X coordinate of the expected ground coordinate system is represented by X d The desired ground coordinate system Y coordinate is represented by Y d The deviation of the desired coordinate from the actual coordinate on the X coordinate is represented by e X The deviation of the desired coordinate from the actual coordinate on the Y coordinate is represented by e Y Represents;
defining sliding mode functions
Wherein the sliding mode function 1 is represented by S 1 Expressed, the sliding mode function 2 is represented by S 2 The coefficient of convergence speed of the sliding mode is represented by C 1 The coefficient of convergence speed of the sliding mode is represented by C 2 Representing;
separating input vectors delta and alpha and designing input and output sliding mode control rate
Wherein the switching term function is represented by sgn, and the switching term coefficient 1 is represented by eta 1 Expressed by the switching term coefficient 2 of eta 2 Is represented by v 1 、v 2 Obtained by the formula (29);
step three: establishment of cross-coupled controller
Simultaneously, the expected X-axis speed and the Y-axis speed of the forklift 1 are given, the speed is different from the actual X-axis speed and the actual Y-axis speed of the forklift 1 to obtain the self motion error of the forklift 1, the forklift 1 is controlled to move by the self input and output sliding mode controller of the forklift 1 through the self motion error of the forklift 1, the expected X-axis speed and the Y-axis speed of the forklift 2 are calculated by the same expected X-axis speed and Y-axis speed of the forklift 1 to obtain the self motion error of the forklift 2, and the forklift 2 is controlled to move by the self input and output sliding mode controller of the forklift 2 through the self motion error of the forklift 2;
the actual speed of the X and Y axes of the forklift 1 is different from the actual speed of the X and Y axes of the forklift 2 to obtain a cross coupling motion error of the forklift 1, the cross coupling input and output sliding mode controller of the forklift 1 controls the forklift 1 to finish synchronous deviation calibration through the cross coupling motion error of the forklift 1, the same actual speed of the X and Y axes of the forklift 2 is different from the actual speed of the X and Y axes of the forklift 1 to obtain a cross coupling motion error of the forklift 2, and the cross coupling input and output sliding mode controller of the forklift 2 controls the forklift 2 to finish synchronous deviation calibration through the cross coupling motion error of the forklift 2.
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