CN105174061B - Double pendulum crane length of a game optimal trajectory planning method based on pseudo- spectrometry - Google Patents

Abstract

A global time-optimal trajectory planning method for a double pendulum crane based on the pseudospectral method. To solve the automatic control problem of nonlinear double pendulum bridge crane system, the method has good trolley positioning and two-stage load swing elimination performance. First, transform the kinematics model of the system to facilitate the subsequent analysis. Afterwards, a variety of constraints including the two-stage swing angle and the upper limit of the trolley speed and acceleration are considered, and a corresponding optimization problem is constructed. Then, the optimization problem with constraints is transformed into an easier-to-solve nonlinear programming problem by using the Gaussian pseudospectral method, and the time-optimal trolley trajectory is obtained. The invention uses the idea of the pseudo-spectral method to process and transform the complex time optimal problem, reducing the difficulty of solving; meanwhile, the method can obtain the global time optimal result, which can greatly improve the working efficiency of the crane system. Simulation and experimental results show that the invention can achieve good control effect and has good practical application value.

Description

Global Time Optimal Trajectory Planning Method for Double Pendulum Crane Based on Pseudospectral Method

technical field

The invention belongs to the technical field of automatic control of nonlinear underactuated electromechanical systems, and in particular relates to a global time optimal trajectory planning method for a double pendulum crane based on a pseudo-spectral method.

Background technique

In the industrial production process, in order to transport the load to the desired position, various types of crane systems, including bridge cranes, cantilever cranes, tower cranes, and marine cranes, are widely used. In order to simplify the mechanical structure of the crane system, the load is often not directly controlled, but indirectly dragged to the target position through the movement of the trolley. The result of this structure is that the control input dimension of the crane system is smaller than the dimension of the degrees of freedom to be controlled. A system with this characteristic is the so-called underactuated system ^[1] . Compared with the full-drive system, the under-actuated system is often more difficult to control due to the existence of under-actuated characteristics, and in actual production, the crane system is often operated by experienced workers. If there is a misoperation, it may cause the load to swing violently and cause a collision, or even a safety accident. Therefore, the research on the automatic control method of the crane system has practical significance and wide application value, and has attracted the attention of many scholars.

For the overhead crane system, the main control objectives include two aspects, namely fast and accurate trolley positioning and load swing suppression and elimination. However, these two aspects are usually contradictory, that is, too fast trolley movement often leads to large load swings. Therefore, it is very difficult to realize the control objectives of these two aspects at the same time. In order to obtain a better control effect, scholars at home and abroad have proposed many automatic control methods for crane systems. Tuan et al. proposed a control method based on partial feedback linearization ^[2-3] , which can simplify the control algorithm design of the overhead crane system. In literature [4], [5], Singhose et al. used the idea of input shaping to control the crane system, which can effectively suppress the residual swing of the load. In order to deal with uncertain external disturbances, researchers use sliding mode algorithm to control the crane system ^[6-7] , which can obtain good robustness. Hu Zhou et al. proposed a nonlinear information fusion control method ^[8] , which can deal with the problem of controller input saturation and realize high-performance control of the crane system. Literature [9], [10] proposed a control strategy based on energy and passivity, which can achieve better results. In addition, in recent years, some intelligent control methods including genetic algorithm ^[11] and fuzzy control ^[12] have also been applied in the field of crane control.

As we all know, the load swing of the crane system is caused by the acceleration and deceleration motion of the trolley, and there is a strong coupling between the movement of the trolley and the swing of the load. Based on this, a suitable trajectory can be planned for the trolley, and when the trolley moves according to the trajectory, the goal of fast and accurate positioning can be achieved. At the same time, considering the requirements of swing angle suppression and elimination, in the trajectory planning process, through in-depth analysis and rational use of the coupling relationship between the motion of the trolley and the swing of the load, a trajectory of the trolley with the ability to eliminate swing can be planned. In this way, the dual goals of fast and accurate positioning of the trolley and load swing elimination can be achieved. Based on this idea, researchers have proposed many crane trajectory planning methods ^[13-17] . For example, in [13], Uchiyama et al. proposed an open-loop control strategy for jib cranes. In order to eliminate the residual swing, they planned an S-shaped trajectory for the horizontal motion of the jib. Literature [14] proposed a trajectory planning method based on phase plane analysis, which can better suppress the load swing and eliminate the residual swing.

For the crane system, although the above methods can achieve better control effects under ideal conditions, these methods all assume that the mass of the hook can be ignored and the load can be regarded as a mass point, and the load swing of the crane system is regarded as a simple pendulum system. If the mass of the hook is too large to be ignored, or the shape of the load is so large that it cannot simply be regarded as a mass point, in this case, the swing of the crane system will show a double pendulum phenomenon, that is, the hook moves around the trolley for one level. Swing, while the load undergoes a secondary swing around the hook. At this time, the above-mentioned single-pendulum crane control method cannot achieve satisfactory control performance, and there are only control strategies for crane systems that rarely consider the double-pendulum effect. Literature [18-20] successfully extended the method of input shaping to the double pendulum crane system by analyzing the natural frequency of the double pendulum crane system. Under the premise of considering a series of constraints such as system swing angle constraints and trolley speed constraints, Sun Ning et al. proposed an optimal trajectory planning method for double pendulum cranes based on differential flat theory ^[21] . Guo Weiping proposed a passivity-based double pendulum crane control strategy by analyzing the energy of the crane system ^[22] .

Although the above control strategies can realize the control of the double-pendulum crane system, they cannot achieve the optimal control effect of the global time, that is, they cannot guarantee the maximum operating efficiency of the crane system. Therefore, it is necessary to design a suitable control method to improve the working efficiency of the crane system.

Contents of the invention

The purpose of the present invention is to solve the above-mentioned deficiencies existing in the trajectory planning method of the existing double-pendulum effect bridge crane system, and to provide a global-time optimal trajectory planning method for double-pendulum cranes based on the pseudo-spectral method.

The present invention is dedicated to analyzing the kinematic model of the bridge crane system with double pendulum effects, and proposes a global time optimal trajectory planning method for double pendulum cranes based on the pseudo-spectral method, and obtains a trolley with global time optimum The trajectory, while completing the precise positioning of the trolley, realizes the rapid suppression and elimination of the two-stage load swing, and it is applied to the actual crane platform for experiments, which can greatly improve the working efficiency of the crane system.

The global time optimal trajectory planning method for a double pendulum crane based on the pseudo-spectrum method provided by the present invention includes:

1. Analyze trajectory constraints and construct corresponding optimization problems

Analyzing the control objective of the crane system, considering various constraints including the two-stage swing angle and the upper limit of the speed and acceleration of the trolley, the following optimization problem with the transportation time as the cost function is obtained:

Among them, x(t) represents the position of the trolley, x _f represents the target position of the trolley, t in brackets represents the time, and (t) after the variable indicates that the variable is a variable about time. For the sake of simplicity, it is omitted in the formula (t) in most variables, T represents the total time to complete the delivery, min represents the minimum, and st is followed by constraints that need to be considered; Respectively represent the first and second derivatives of the trolley position x(t) with respect to time, that is, the trolley speed and acceleration; v _max and a _max respectively represent the allowable maximum speed and maximum acceleration of the trolley; θ ₁ (t) , θ ₂ (t) represent the primary and secondary pendulum angles respectively, Indicates the primary and secondary angular velocities; θ _1max , θ _2max represent the maximum allowed swing angles of the primary and secondary during transportation, ω _1max , ω _2max represent the maximum allowable angular velocities of the primary and secondary.

2. Acceleration drive model establishment and optimization problem transformation

By analyzing and utilizing the double pendulum bridge crane system, the following acceleration drive system model is obtained:

Among them, ζ represents the full state vector of the system, which is defined as follows:

where x(t), represent the position and velocity of the trolley respectively, θ ₁ (t), Indicates the primary swing angle and angular velocity, θ ₂ (t), Indicates the secondary pendulum angle and angular velocity, the superscript T in the brackets represents the transposition operation of the matrix; u(t) represents the system input of the system, and is the acceleration of the trolley; f(ζ), h(ζ) both represent functions with the full state vector ζ of the system as an independent variable, obtained from the kinematic equation of the crane system, see (7) for the specific form; is the derivative of the full state vector of the system with respect to time.

Using the above acceleration-driven system model, the original optimization problem is transformed into the following form:

Among them, ζ represents the full state vector of the system, u(t) represents the system input of the system, and is the acceleration of the trolley; the t in the brackets represents the time, and the (t) behind the variable indicates that the variable is a variable about time. For the sake of simplicity, most of the variables (t) are omitted in the formula, and T represents the total time to complete the delivery , min means the minimum, st is followed by the constraints that need to be considered; the superscript T of the vector represents the transpose operation of the matrix; x _f represents the given target position of the trolley, Indicates the speed of the trolley, v _max and a _max respectively represent the maximum speed and maximum acceleration allowed by the trolley; θ ₁ (t), θ ₂ (t) represent the primary and secondary swing angles respectively, Indicates the primary and secondary angular velocities; θ _1max , θ _2max represent the maximum allowed swing angles of the primary and secondary during transportation, ω _1max , ω _2max represent the maximum allowable angular velocities of the primary and secondary.

3. Trajectory planning based on Gaussian pseudospectral method

Using the idea of Gaussian pseudospectral method to process and solve the optimization problem in the second step, the specific steps are as follows:

3.1. First, use the Lagrangian interpolation method to select the discrete system state trajectory and input trajectory at the Legendre-Gauss (Legendre-Gauss, LG) point, and express the corresponding Approximate trajectory model.

Section 3.2. Next, by deriving the approximated trajectory model, the derivative of the system state is represented by the derivative of the Lagrangian polynomial.

3.3. Subsequently, the original system kinematics model is converted into a series of polynomial equations by using the discrete trajectory model and its derivatives; using Gaussian integrals, the boundary conditions in the optimization problem in the second step are also expressed in the form of polynomial equations.

Section 3.4. Finally, the time-optimal trajectory planning problem is transformed into a nonlinear programming problem with algebraic constraints, and the global optimal time and optimal trajectory can be obtained by solving it.

4. Trajectory tracking

Measure the position and speed signal x(t) of the trolley through the code disc or laser sensor, Using the optimal time reference trajectory of the trolley to be tracked and the corresponding velocity trajectory obtained in step 3.4, select a proportional-derivative (PD) controller as follows:

Among them, F(t) represents the driving force acting on the trolley, x _r (t), represent the reference displacement trajectory and velocity trajectory respectively, k _p and k _d are the positive control gains that need to be adjusted. Using the controller, the corresponding real-time control signal can be calculated to drive the crane to move and complete the control target.

Theoretical basis and derivation process of the inventive method

1. Analyze trajectory constraints and construct corresponding optimization problems

The kinematic model of an overhead crane with double pendulum effect is as follows:

Among them, m ₁ and m ₂ represent the mass of the hook and the load respectively, M represents the mass of the trolley; x(t) represents the displacement of the trolley, Indicates the second derivative of x(t) with respect to time, that is, the acceleration of the trolley; t represents time, and (t) after the variable indicates that the variable is a variable with respect to time. For the sake of simplicity, most of the variables (t) are omitted in the formula ); θ ₁ (t), θ ₂ (t) represent the primary and secondary swing angles (swing angle of the hook and the swing angle of the load around the hook), is the corresponding angular velocity, is the angular acceleration; l ₁ is the length of the sling, l ₂ is the equivalent rope length, that is, the distance between the center of mass of the load and the center of mass of the hook; g is the acceleration of gravity.

For formulas (1) and (2), divide both sides by (m ₁ +m ₂ )l ₁ and m ₂ l ₂ respectively, and simplify to get:

Equations (3) and (4) describe the coupling relationship between the displacement x(t) of the trolley and the two-stage swing angles θ ₁ (t), θ ₂ (t) of the system, that is, the influence of the movement of the trolley on the swing of the load . Through in-depth analysis of the coupling relationship, planning a trolley trajectory with the ability to eliminate swing is the basis of the present invention.

In order to complete the time-optimal trajectory planning, the present invention will systematically consider the trajectory constraints of the following aspects ^[21] , considering the objectives, physical constraints and safety of the crane system during actual work:

1) In order to achieve fast and accurate positioning of the trolley, the trolley starts to move from the initial position x ₀ , and reaches the target position x _f after a time T, and the speed and acceleration of the trolley at the start time and the end time are both 0, that is

Among them, T represents the time required for the transportation process; for the initial position, x ₀ =0 is selected here without loss of generality.

2) Considering that the output of the actual motor is bounded, the speed and acceleration of the trolley should be kept within an appropriate range during the transportation process, that is,

Among them, v _max and a _max represent the allowable maximum speed and acceleration of the trolley respectively, Represents the absolute value of the speed and acceleration of the trolley.

3) In order to ensure that the load can be directly processed in the next step at the end of the transportation, there should be no residual swing after the trolley reaches the target position and the angular velocity should be 0, that is,

4) In order to avoid the collision caused by the violent swing of the load, during the transportation process, the swing angle and angular velocity of the two-stage swing should be kept within the allowable range, that is,

Among them, θ _1max and θ _1max represent the maximum angles allowed by the primary and secondary swing angles respectively; ω _1max and ω _2max represent the maximum angular velocities allowed by the primary and secondary swing angles.

In summary, the following optimization problem is constructed:

Among them, min means the minimum, and s.t. is followed by the constraints that need to be considered. Next, the optimization problem will be solved by the pseudospectral method, and a time-optimal trajectory will be planned for the trolley.

2. Acceleration drive model establishment and optimization problem transformation

In order to facilitate the subsequent trajectory planning, the system model of the double pendulum crane and the optimization problem (5) are transformed first. For this purpose, the system full state vector ζ(t) is defined as follows:

where x(t), represent the position and velocity of the trolley respectively, θ ₁ (t), Indicates the primary swing angle and angular velocity, θ ₂ (t), Indicates the secondary swing angle and angular velocity, and the superscript T in the brackets represents the transposition operation of the matrix. According to formulas (3) and (4), the acceleration of the trolley is taken as the input of the system. At this point, the kinematics model is transformed into the following form:

in, Indicates the derivative of ζ(t) with respect to time; u(t) is the acceleration of the trolley f(ζ), h(ζ) represent auxiliary functions about ζ(t), the specific form is as follows:

Among them, for the convenience of description, the following auxiliary variables A, B, C, D are defined:

In the above formula, the following simplified form is used:

S ₁ = sinθ ₁ , S ₂ = sinθ ₂ , C ₁ = cosθ ₁ , C ₂ = cosθ ₂ ,

S _1-2 ＝sin(θ ₁ -θ ₂ ),C _1-2 ＝cos(θ ₁ -θ ₂ ).

Using the obtained acceleration drive system model (6), the original optimization problem (5) can be transformed into:

Among them, ζ represents the full state vector of the system, u(t) represents the system input of the system, and is the acceleration of the trolley; the t in the brackets represents the time, and the (t) behind the variable indicates that the variable is a variable about time. For the sake of simplicity, most of the variables (t) are omitted in the formula, and T represents the total time to complete the delivery , min means the minimum, st is followed by the constraints that need to be considered; the superscript T of the vector represents the transpose operation of the matrix; x _f represents the given target position of the trolley, Indicates the speed of the trolley, v _max and a _max respectively represent the maximum speed and maximum acceleration allowed by the trolley; θ ₁ (t), θ ₂ (t) represent the primary and secondary swing angles respectively, Indicates the primary and secondary angular velocities; θ _1max , θ _2max represent the maximum allowed swing angles of the primary and secondary during transportation, ω _1max , ω _2max represent the maximum allowable angular velocities of the primary and secondary.

Next, the optimization problem will be solved by the pseudospectral method, and a time-optimal trajectory will be planned for the trolley.

3. Trajectory planning based on Gaussian pseudospectral method

In order to meet the requirements of the Gaussian pseudospectral method, it is first necessary to use coordinate transformation to transform the time interval corresponding to the trajectory from t∈[0,T] to the interval τ∈[-1,1], that is

Here τ denotes an auxiliary variable like time. Then select K Legendre-Gauss (Legendre-Gauss, LG) points {τ ₁ ,τ ₂ ,...,τ _K }∈(-1,1) to form a point sequence. Here, τ ₁ , τ ₂ ,...,τ _K represent the selected LG points, and the subscript indicates that the serial numbers of the points are 1, 2,..., K; K is the number of selected LG points. The selection of the LG point is obtained by solving the zero point of the K-order Legendre polynomial. At the same time, add τ ₀ ＝-1 to the first position of the point column, and the discrete expression of the system state quantity and input quantity to be planned is as follows:

ζ(τ ₀ ),ζ(τ ₁ ),ζ(τ ₂ ),...,ζ(τ _K ),

u(τ ₀ ),u(τ ₁ ),u(τ ₂ ),...,u(τ _K ),

Using the K+1 nodes, construct K+1 Lagrangian interpolation polynomials, the specific form is as follows:

in, Represents the Lagrangian difference polynomial with serial number i, i∈{0,1,...,K}; here, τ∈[-1,1]; Indicates the multiplication symbol, that is, starting from the item with sequence number j=0, multiplying to the item of j=K, and skipping the item of j=i in the process. Using formula (11) and the values of the system state quantity and input quantity at the LG point, the state quantity trajectory and input quantity trajectory of the system are approximately expressed in the following way:

Among them, ζ(τ _i ), u(τ _i ) represent the system state quantity and input quantity at τ=τ _i respectively; Indicates the accumulation symbol, that is, the item with sequence number i=0 is accumulated to the item with i=K. Calculate the derivative of formula (12), and use the specific form of the interpolation function in formula (11), calculate and simplify, and obtain the derivative of the state quantity trajectory as follows:

in, Indicates the state trajectory derivative value at τ=τ _k ; Representing τ=τ _k , the Lagrangian polynomial The derivative value of , the specific form is as follows:

Using equations (13), (14) and the trajectory value at the LG point, the differential equation constraints in the optimization problem (9) Carry out discretization and approximation processing, the specific results are as follows:

where k∈{0,1,2,...,K}. Equation (15) has an algebraically constrained form. Next, the boundary condition constraints in the optimization problem also need to be converted into the form of algebraic constraints, where the boundary constraints at time 0 are directly rewritten as follows:

ζ(0)=[0 0 0 0 0 0] ^T .

To express the boundary constraints at the end of the transport process, τ _K+1 =1 is defined. It can be seen from formula (10) that τ _K+1 =1 corresponds to the delivery end time t=T. Using Gaussian integrals, this boundary constraint is expressed as:

Among them, ζ(τ ₀ ) is the above-mentioned initial state vector of the system; w _k represents the kth Legendreweight (Legendreweight), and the specific value is obtained when solving the LG point.

In summary, all constraints in the optimization problem can be expressed in the form of algebraic constraints. Based on this, the original optimization problem is transformed into a nonlinear programming problem with algebraic constraints, as follows:

min T

s.t.

ζ(0)=[0 0 0 0 0 0] ^T ,

ζ(τ)-χ≤0,-ζ(τ)-χ≤0,

u(τ)-a _max ≤0,-u(τ)-a _max ≤0

Among them, the definition of vector χ is as follows:

χ＝[∞ v _max θ _1max ω _1max θ _2max ω _2max ] ^T

Among them, ∞ represents infinity. For the above-mentioned constrained nonlinear programming problem, the sequential quadratic programming (SQP) method is chosen here to solve it, and the following time-optimal state vector sequence is obtained:

ζ(τ ₀ ),ζ(τ ₁ ),ζ(τ ₂ ),...,ζ(τ _K ),ζ(τ _K+1 ),

The above formula is the time-discrete optimal state vector sequence. Take the first two items of each vector (car displacement and trolley speed) and perform interpolation to obtain the corresponding global time optimal trolley displacement and velocity trajectory.

4. Trajectory tracking

Measure the position and speed signal x(t) of the trolley through the code disc or laser sensor, Using the optimal time reference trajectory of the trolley to be tracked and the corresponding velocity trajectory obtained in step 3, select a proportional-derivative (PD) controller as follows:

Among them, F(t) represents the driving force acting on the trolley, x _r (t), represent the reference displacement trajectory and velocity trajectory respectively, k _p and k _d are the positive control gains that need to be adjusted. Using the controller, the corresponding real-time control signal can be calculated to drive the crane to move and complete the control target.

Advantages and beneficial effects of the present invention

Aiming at the bridge crane with double pendulum effect, the invention proposes a global time optimal trajectory planning method for the double pendulum crane based on the pseudo spectrum method. Specifically, firstly, the kinematics model of the crane system is transformed into an acceleration-driven model, and based on this model, various constraints are considered to construct an optimization problem with constraints; then, the obtained optimization problem is analyzed by using the Gaussian pseudospectral method Transform it into a nonlinear programming problem that is more convenient to solve. On this basis, the time-optimal trolley trajectory can be obtained. The trajectory planning method proposed by the present invention can handle practical physical constraints such as swing angle constraints, angular velocity constraints, trolley speed constraints, and acceleration constraints very conveniently, in addition to considering the goal of swing elimination. Different from the existing method, the method proposed by the invention can obtain the global time optimal solution, which greatly improves the working efficiency of the crane system. Finally, the effectiveness of the present invention is verified through simulation and experiment.

Description of drawings:

Fig. 1 represents trajectory planning simulation result 1 (carriage displacement and speed curve) among the present invention;

Fig. 2 represents trajectory planning simulation result 2 (two-stage swing angle and angular velocity curve) among the present invention;

Fig. 3 represents trajectory planning experiment result among the present invention;

Figure 4 shows the experimental results of the linear quadratic controller;

Figure 5 shows the experimental results of polynomial trajectory planning.

detailed description:

Example 1:

Analyzing the control objective of the crane system, considering various constraints including the two-stage swing angle and the upper limit of the speed and acceleration of the trolley, the following optimization problem with the transportation time as the cost function is obtained:

Here, the target position of the trolley is selected as x _f =0.6m, and the trajectory constraints are as follows:

θ _1max ＝θ _2max ＝2deg, v _max ＝0.3m/s, ω _1max ＝ω _2max ＝5deg/s, a _max ＝0.15m/s ²

2. Acceleration drive model establishment and optimization problem transformation

Analyze and utilize the double pendulum overhead crane system to establish the following acceleration drive system model:

Since the specific expression of the model is too complicated, it is not repeated here, only the corresponding physical parameters of the crane system are given, as follows:

M=6.5kg, m ₁ =2.003kg, m ₂ =0.559kg, g=9.8m/s ² , l ₁ =0.53m, l ₂ =0.4m.

Then the optimization problem is transformed into the following form:

3. Trajectory planning based on Gaussian pseudospectral method

In order to realize the trajectory planning method based on the pseudospectral method proposed by the present invention, the GPOPS software toolbox ^[23] and the SNOPT toolbox ^[24] are used to solve the optimization problem in the second step offline and obtain the corresponding time optimal trajectory. Wherein, the Legendre-Gauss point parameter K=750 is selected. For the specific results, please refer to the description part of the simulation experiment.

4. Description of simulation experiment effect

Section 4.1, Simulation Results

In order to verify the feasibility of the trajectory planning algorithm proposed by the present invention, numerical simulation is first carried out in the MATLAB/Simulink environment. The simulation process is divided into two steps. The first step is to plan a time-optimal reference trajectory for the trolley according to this method; the second step is to assume that the trolley runs according to the reference trajectory to obtain the trajectory of the trolley and its swing angle.

The simulation results are shown in Figure 1 and Figure 2 . In Figure 1, the dotted line represents the target position of the trolley, the dotted line represents the speed constraint of the trolley, and the solid line represents the simulation result. In Figure 2, the dotted line represents the angular constraint, the dotted line represents the angular velocity constraint, and the solid line represents the simulation result. It can be seen from Figure 1 that when the trolley runs along the reference trajectory, the trolley quickly and accurately converges to the target position x _f =0.6m; at the same time, during the whole process, the speed of the trolley satisfies the set constraints . It can be seen from Figure 2 that during the transportation of the trolley, the two-stage swing angles are less than the given constraint value 2deg; at the same time, the angular velocity is also within the restricted range; and at the end of the transportation, the two-stage swing angles do not exist The goal of residual swing, i.e. fast swing elimination, is also achievable.

Section 4.2. Experimental results

Measure the position and speed signal x(t) of the trolley through the code disc or laser sensor, Using the optimal time reference trajectory of the trolley to be tracked and the corresponding velocity trajectory obtained in step 3, select a proportional-derivative (PD) controller as follows:

Among them, F(t) represents the driving force acting on the trolley, x _r (t), represent the reference displacement trajectory and velocity trajectory respectively, k _p and k _d are the positive control gains that need to be adjusted. Using the controller, the corresponding real-time control signal can be calculated to drive the crane to move and complete the control target.

In the experiment, the selected tracking controller control gain is:

k _p =750, k _d =150

The experimental results are shown in Figure 3. Among them, the dotted line represents the constraint of the swing angle, and the solid line represents the actual trajectory of the trolley and the trajectory of the swing angle. It can be seen from the figure that under the action of the PD controller, the trolley can track the reference trajectory well and achieve the control goal of fast and accurate trolley positioning. The two-stage swing angles are maintained within a given range during the entire delivery process, and there is almost no residual swing at the end of delivery. The experimental results have verified that the present invention can achieve better effects.

In order to further demonstrate the effectiveness of the present invention, as a comparison, the optimal trajectory planning method in [21] and the experimental results of the linear quadratic regulator (LQR) method are given here. Among them, the constraint selection of the trajectory planning method in literature [21] is consistent with the constraints of the method proposed in the present invention; and for the LQR method, the controller expression is as follows:

Among them, F(t) represents the driving force acting on the trolley; x(t), Indicates the position and speed of the trolley measured in real time; x _f is the given target position, set x _f =0.6m; θ ₁ (t), Indicates the first-order pendulum angle and its angular velocity; θ ₂ (t), Indicates the secondary swing angle and its angular velocity. k ₁ , k ₂ , k ₃ , k ₄ , k ₅ , and k ₆ are the corresponding control gains. See below for specific values. At the same time, the cost function of this method is selected as follows:

where X is defined as follows

e(t) represents the positioning error of the trolley, e(t)=x(t)-x _f ; the selection of matrix Q and R is as follows:

Q=diag{200,1,200,1,200,1}, R=0.05

The calculated controller gain is as follows:

k ₁ =63.2456, k ₂ =50.7765, k ₃ =-129.3086, k ₄ =-6.9634, k ₅ =19.9137, k ₆ =-6.7856.

The LQR method and the method in the literature [21], the experimental results are shown in Figure 4 and Figure 5. Among them, Figure 5 shows the experimental results using the method in literature [21], the solid line represents the experimental results, and the dotted line represents the swing angle constraint.

It can be seen from Figure 3 that when the trolley tracks the optimal trajectory planned, it only takes 4.095s to complete the transportation process, and the two-stage swings are kept within the given constraint 2deg during the whole process, and there is basically no problem when the transportation is completed. Residual swing. For the method proposed in [21], it takes 5.445s to complete the delivery process; for the LQR method, it takes 7.425s to complete the delivery process. At the same time, the maximum swing angle of the first-stage swing and the maximum swing angle of the second-stage swing caused by the LQR method reach 6.5 deg, which are much larger than the swing angle of the method proposed in the present invention. In summary, the method designed in the present invention can realize the precise positioning of the trolley and the rapid elimination of the two-stage swing of the system, and obtain good control performance.

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Claims (1)

1. A double-pendulum crane global time optimal trajectory planning method based on a pseudo-spectrum method is characterized by comprising the following steps:

1, analyzing track constraint and constructing corresponding optimization problem

Analyzing the control target of the crane system, and considering various constraints including two-stage swing angles and upper limit values of the speed and the acceleration of the trolley, obtaining the following optimization problem taking the conveying time as a cost function:

$\begin{array}{c}\min T\\ s.t.x\left(0\right)=\stackrel{\·}{x}\left(0\right)=\stackrel{\·\·}{x}\left(0\right)=0,\\ x\left(T\right)=x_f,\stackrel{\·}{x}\left(T\right)=\stackrel{\·\·}{x}\left(T\right)=0,\\ \left|\stackrel{\·}{x}\left(t\right)\right|\≤v_{\max },\left|\stackrel{\·\·}{x}\left(t\right)\right|\≤a_{\max },\\ {\mathrm{\θ}}_1\left(0\right)={\stackrel{\·}{\mathrm{\θ}}}_1\left(0\right)=0,{\mathrm{\θ}}_1\left(T\right)={\stackrel{\·}{\mathrm{\θ}}}_1\left(T\right)=0,\\ {\mathrm{\θ}}_2\left(0\right)={\stackrel{\·}{\mathrm{\θ}}}_2\left(0\right)=0,{\mathrm{\θ}}_2\left(T\right)={\stackrel{\·}{\mathrm{\θ}}}_2\left(T\right)=0,\\ \left|{\mathrm{\θ}}_1\left(t\right)\right|\≤{\mathrm{\θ}}_{1\max },\left|{\mathrm{\θ}}_2\left(t\right)\right|\≤{\mathrm{\θ}}_{2\max },\\ \left|{\stackrel{\·}{\mathrm{\θ}}}_1\left(t\right)\right|\≤{\mathrm{\ω}}_{1\max },\left|{\stackrel{\·}{\mathrm{\θ}}}_2\left(t\right)\right|\≤{\mathrm{\ω}}_{2\max },\end{array}---\left(5\right)$

wherein x (t) represents the position of the trolley, x_fIndicating the target position of the trolley, T in brackets indicates time, the variable is followed by (T) indicating that the variable is a variable related to time, for simplicity, most of the variables are omitted in the formula, and T indicates the total time for completing the deliveryMin represents minimum, and s.t. is followed by constraint conditions to be considered;first and second derivatives with respect to time, i.e. trolley speed and acceleration, respectively, representing trolley position x (t); v. of_max,a_maxRespectively representing the maximum speed and the maximum acceleration of the trolley; theta₁(t),θ₂(t) respectively representing a primary swing angle and a secondary swing angle,representing primary and secondary angular velocities; theta_1max,θ_2maxRepresenting the allowable primary and secondary maximum swing angles, ω, during transport_1max,ω_2maxRepresenting the allowable primary and secondary maximum angular velocities;

2, establishing an acceleration driving model and converting an optimization problem

Analyzing and utilizing a double-pendulum bridge crane system to obtain the following acceleration driving system model:

$\stackrel{\·}{\mathrm{\ζ}}=f\left(\mathrm{\ζ}\right)+h\left(\mathrm{\ζ}\right)u,---\left(6\right)$

where ζ represents an all-state vector of the system, and is specifically defined as follows:

$\mathrm{\ζ}={\left[\begin{array}{cccccc}x& \stackrel{\·}{x}& {\mathrm{\θ}}_1& {\stackrel{\·}{\mathrm{\θ}}}_1& {\mathrm{\θ}}_2& {\stackrel{\·}{\mathrm{\θ}}}_2\end{array}\right]}^T$

wherein, x (t),respectively representing position and speed of the trolley, theta₁(t),Representing first-order swing angle and angular velocity, theta₂(t),Representing a secondary swing angle and an angular speed, and the superscript T of the bracket represents the transposition operation of the matrix; u (t) represents the system input of the system, and is the trolley acceleration; f (zeta), h (zeta) are all the functions of system state vector zeta as independent variable and are used by crane systemObtaining a system kinematic equation, wherein the specific form is shown in (7);a derivative with respect to time of an all-state vector of the system;

$f\left(\mathrm{\ζ}\right)={\left[\begin{array}{cccccc}\stackrel{\·}{x}& 0& {\stackrel{\·}{\mathrm{\θ}}}_1& A& {\stackrel{\·}{\mathrm{\θ}}}_2& B\end{array}\right]}^T,h\left(\mathrm{\ζ}\right)={\left[\begin{array}{cccccc}0& 1& 0& C& 0& D\end{array}\right]}^T---\left(7\right)$

for convenience of description, the following auxiliary variables a, B, C, D are defined:

$\begin{array}{c}A=-\frac{m_2{C}_{1-2}}{l_1(m_1+m_2)-m_2{l}_1{C}_{1-2}^2}\[l_1{S}_{1-2}{\stackrel{\·}{\mathrm{\θ}}}_1^2+\frac{m_2{l}_2}{m_1+m_2}{S}_{1-2}{C}_{1-2}{\stackrel{\·}{\mathrm{\θ}}}_2^2-g(S_2-S_1{C}_{1-2})\]\\ -\frac{1}{l_1}{\mathrm{gS}}_1-\frac{m_2{l}_2}{l_1(m_1+m_2)}{S}_{1-2}{\stackrel{\·}{\mathrm{\θ}}}_2^2\\ B=\frac{m_1+m_2}{l_2(m_1+m_2)-m_2{l}_2{C}_{1-2}^2}\[\frac{m_2{l}_2}{m_1+m_2}{S}_{1-2}{C}_{1-2}{\stackrel{\·}{\mathrm{\θ}}}_2^2+l_1{S}_{1-2}{\stackrel{\·}{\mathrm{\θ}}}_1^2-g(S_2-S_1{C}_{1-2})\],\\ C=\frac{m_2{C}_{1-2}}{l_1(m_1+m_2)-m_2{l}_1{C}_{1-2}^2}\[C_2-C_1{C}_{1-2})-\frac{1}{l_1}{C}_1,\\ D=-\frac{m_1+m_2}{l_2(m_1+m_2)-m_2{l}_2{C}_{1-2}^2}(C_2-C_1{C}_{1-2}),\end{array}---\left(8\right)$

(8) in the formula, the following simplified form is used:

$\begin{array}{c}{S}_1={\mathrm{sin\θ}}_1,S_2={\mathrm{sin\θ}}_2,C_1={\mathrm{cos\θ}}_1,C_2={\mathrm{cos\θ}}_2,\\ S_{1-2}=sin({\mathrm{\θ}}_1-{\mathrm{\θ}}_2),C_{1-2}=\cos ({\mathrm{\θ}}_1-{\mathrm{\θ}}_2).\end{array};$

by utilizing the acceleration driving system model, the original optimization problem is converted into the following form:

$\begin{array}{c}\min T\\ s.t.\stackrel{\·}{\mathrm{\ζ}}=f\left(\mathrm{\ζ}\right)+h\left(\mathrm{\ζ}\right)u,\\ \mathrm{\ζ}\left(0\right)={\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\end{array}\right]}^T,\\ \mathrm{\ζ}\left(T\right)={\left[\begin{array}{cccccc}{x}_f& 0& 0& 0& 0& 0\end{array}\right]}^T,\\ \left|\stackrel{\·}{x}\left(t\right)\right|\≤v_{\max },\left|u\left(t\right)\right|\≤a_{\max },\\ \left|{\mathrm{\θ}}_1\left(t\right)\right|\≤{\mathrm{\θ}}_{1\max },\left|{\mathrm{\θ}}_2\left(t\right)\right|\≤{\mathrm{\θ}}_{2\max },\\ \left|{\stackrel{\·}{\mathrm{\θ}}}_1\left(t\right)\right|\≤{\mathrm{\ω}}_{1\max },\left|{\stackrel{\·}{\mathrm{\θ}}}_2\left(t\right)\right|\≤{\mathrm{\ω}}_{2\max }\end{array}---\left(9\right)$

where ζ represents the full state vector of the system, u (t) represents the system input of the system, and is the trolley acceleration; the superscript T of the vector represents the transposition operation of the matrix;

3, trajectory planning based on Gaussian pseudo-spectral method

And (3) processing and solving the optimization problem in the step 2 by using the idea of a Gaussian pseudo-spectrum method, wherein the method comprises the following specific steps:

3.1, selecting a discrete system state track and an input track at a Legendre-Gaussian (LG) point by using a Lagrange interpolation method, and representing a corresponding approximate track model by using a discrete track and a Lagrange interpolation polynomial;

3.2, then, carrying out derivation on the approximated track model, and expressing the derivative of the system state by a Lagrange polynomial derivative;

3.3, then, converting the original system kinematic model into a series of polynomial equations by using the discrete track model and the derivative thereof; using Gaussian integration, the boundary conditions in the optimization problem in step 2 are also expressed in the form of polynomial equation;

3.4, finally, converting the time optimal trajectory planning problem into a nonlinear planning problem with algebraic constraints, and obtaining the global optimal time and optimal trajectory by solving;

4 th, track following

Measuring position and speed signals x (t) of the trolley through a code disc or a laser sensor,and (3) selecting a proportional-derivative (PD) controller by using the optimal reference track of the trolley time to be tracked and the corresponding speed track obtained in the step (3.4) as follows:

$F\left(t\right)=-k_p\left(x\right(t)-x_r(t\left)\right)-k_d\left(\stackrel{\·}{x}\right(t)-{\stackrel{\·}{x}}_r(t\left)\right)---\left(16\right)$

wherein F (t) represents a driving force acting on the carriage, x_r(t),Respectively representing a reference displacement trajectory and a velocity trajectory, k_p,k_dIs in need of adjustmentA positive control gain of unity; by utilizing the controller, a corresponding real-time control signal can be obtained through calculation, the crane is driven to move, and a control target is completed.