Anti-swing self-adaptive fuzzy sliding mode device for hoisting load of hydraulic mechanical arm and application of anti-swing self-adaptive fuzzy sliding mode device

Abstract

The invention provides a design method and application of a hydraulic mechanical arm hoisting load anti-swing self-adaptive fuzzy sliding mode controller, belongs to the technical field of engineering machinery, and aims at solving the problem that switching items of sliding mode control are discontinuous, fuzzy approximation is carried out on the switching items of the sliding mode control by combining a fuzzy control method and self-adaptive sliding mode control, and therefore the switching items of the sliding mode control are not continuous. And the problem of discontinuity is solved. The design method comprises the following steps that 1, a hydraulic mechanical arm hoisting system model is established, and a self-adaptive sliding mode anti-swing controller is designed on the basis of the hydraulic mechanical arm hoisting system model; and 2, introducing a fuzzy control method into the adaptive sliding mode anti-swing controller, and establishing an adaptive fuzzy sliding mode controller. The self-adaptive fuzzy sliding mode controller designed by the invention has an obvious swing control effect on the hoisting load in the movement process, the control speed is high, the hoisting load is stable and operates under the action of the controller, the swing amplitude of the hoisting load is kept in an acceptable range, the sudden change condition is avoided, and the control accuracy is high. And the requirements of rapid elimination of hoisting swinging and accurate operation are met.

Description

Technical Field

[0001] The invention belongs to the technical field of engineering machinery, and in particular relates to a hydraulic mechanical arm weight lifting anti-sway adaptive fuzzy sliding mode device and its application. Background Art

[0002] The lifting system of a hydraulic manipulator is nonlinear and complex, making it difficult to establish a precise mathematical model. Traditional PID control cannot achieve the desired control effect. While existing fuzzy control methods can effectively address the uncertainty of the system model, control accuracy and performance are also limited by the designer's experience. Furthermore, fuzzy control algorithms are computationally intensive, and improving control accuracy requires slowing down decision-making, which impacts the system's dynamic quality. Summary of the Invention

[0003] Aiming at the problem of discontinuity of switching items in sliding mode control, the present invention combines fuzzy control method with adaptive sliding mode control to perform fuzzy approximation on the switching items of sliding mode control to solve the discontinuity problem, and proposes an adaptive fuzzy sliding mode device for anti-sway of hydraulic manipulator lifting weight and its application.

[0004] The technical solution adopted in the present invention is:

[0005] In a first aspect, the present invention provides a method for designing an adaptive fuzzy sliding mode controller for anti-swaying of a hydraulic manipulator lifting weight, the method comprising the following steps:

[0006] Step 1: Establish a hydraulic manipulator lifting system model and design an adaptive sliding mode anti-sway controller based on the hydraulic manipulator lifting system model;

[0007] Step 2: Introduce the fuzzy control method into the adaptive sliding mode anti-sway controller and establish an adaptive fuzzy sliding mode controller.

[0008] Preferably, the dynamic expression of the hydraulic mechanical arm lifting system model established in step 1 is as follows:

[0009]

[0010] Among them, u represents the control quantity of hoisting operation, is the system state vector, representing the swing angle, angular velocity, and angular acceleration of the load respectively, and z1, z2, z3, and d are uncertain quantities.

[0011] More preferably, the design process of the adaptive sliding mode anti-sway controller is as follows:

[0012] (1) Design sliding mode function:

[0013]

[0014] Among them, λ1 and λ2 are sliding surface parameters, and e represents the swing angle error of the hanging weight, which is expressed as follows:

[0015] e=α-α d (3)

[0016] Among them, α is the current swing angle of the load, α d is the target value of the hoisting swing angle;

[0017] Combining equations (2) and (3) we get:

[0018]

[0019] Define the first Lyapunov function as:

[0020]

[0021] Derivative (5) and substitute it into (5) to obtain:

[0022]

[0023] Define the control rate as:

[0024]

[0025] Among them, u z is the switching control quantity, u s1 and u s2 is the equivalent control quantity, k s and η are adjustable parameters, sgn(*) is the sign function;

[0026] Substituting formula (6) into formula (7), we get:

[0027]

[0028] (2) Adopt the adaptive control algorithm to estimate z1, z2, z3, assuming that the parameter z i The estimated error is:

[0029]

[0030] in For z i The estimation error of For z i The estimated value of the hydraulic manipulator arm lifting system control rate is:

[0031]

[0032] Define the second Lyapunov function as:

[0033]

[0034] In the above formula, γ1, γ2, and γ3 are unknown coefficients. Derivative equation 11 and substitute equation 10 into it to obtain:

[0035]

[0036] Assume that the rate of change of z1, z2, and z3 is slow, that is, (i=1,2,3), then The adaptive rate is:

[0037]

[0038] Substituting formula (13) into formula (12) yields:

[0039]

[0040] More preferably, in step 2, the expression of the adaptive fuzzy sliding mode controller established is as follows:

[0041]

[0042] where 0≤ω i ≤1 (i=1, 2, 3) is the weight coefficient of each fuzzy rule, r * The definition of is shown in formula (21).

[0043] More preferably, the derivation process of the expression of the adaptive fuzzy sliding mode controller is as follows:

[0044] By using the fuzzy language ability, the fuzzy control method is used to approximate the switching term, and the sliding mode function s is used as the input variable of the fuzzy control. s2 As the output of fuzzy control, the fuzzy rules adopted by fuzzy control are as follows:

[0045] R1: if s is P, then u s2 =-r

[0046] R2: if s is Z, then u s2 =0

[0047] R3: if s is N, then u s2 =r

[0048] Among them, r is the parameter of the designed fuzzy system, r>0; N, Z, P are fuzzy subsets on the domain, and the fuzzy output u s2 for:

[0049]

[0050] where 0≤ω i≤1 (i=1, 2, 3) is the weight coefficient of each fuzzy rule. The fuzzy input adopts a triangular membership function, which is taken as the membership degree of the fuzzy input variable s on the corresponding fuzzy subset, that is, ω1=μP(s), ω2=μZ(s), ω3=μN(s), μ is an adjustable coefficient. For any sliding mode function, the following function is available:

[0051] ω1+ω2+ω3=1 (16)

[0052] The switching control rate is:

[0053] u s2 =-r(ω1-ω3) (17)

[0054] For any sliding mode function, the following holds:

[0055] s(ω1-ω3)=|s|·|ω1-ω3| (18)

[0056] Substituting formula (18) into formula (14) we can get

[0057]

[0058] From the above formula, we can see that when r satisfies

[0059]

[0060] At this time, the hydraulic manipulator lifting system is stable, but d is an unknown parameter, and Equation (20) cannot guarantee it. Assuming that there is r * Satisfying formula (21),

[0061]

[0062] Where ε is the smallest normal number. Let r * The valuation is The estimation error is

[0063]

[0064] The control rate at this time is:

[0065]

[0066] Define the Lyapunov function as:

[0067]

[0068] In the above formula, γ4 is the unknown coefficient. The time derivative of formula (24) can be obtained as follows:

[0069]

[0070] Take the adaptive rate as:

[0071]

[0072] At this time there

[0073]

[0074] It can be concluded that the hydraulic manipulator arm lifting system is stable.

[0075] In the second aspect, the present invention protects an adaptive fuzzy sliding mode control method for preventing the swing of a hydraulic manipulator arm weight, which uses a designed adaptive fuzzy sliding mode controller to control the swing angle of the weight during movement and keep the swing amplitude of the weight within an allowable range.

[0076] In a third aspect, the present invention protects an adaptive fuzzy sliding mode control device for preventing the weight from swinging when lifting a hydraulic manipulator arm, the control device comprising:

[0077] The control module calls the designed adaptive fuzzy sliding mode controller to realize the control of the swing angle of the hoist;

[0078] The display module is used to input the parameters of the adaptive fuzzy sliding mode controller and display the control results of the hoisting weight swing angle.

[0079] Beneficial effects:

[0080] The adaptive fuzzy sliding mode controller designed in this invention effectively and quickly controls the sway of a load during movement. Under the controller's control, the load maintains stability and operation, and the sway amplitude remains within an acceptable range without sudden changes, meeting the requirements for rapid elimination of sway and precise operation. The designed sway angle controller is insensitive to changes in system parameters, effectively reducing jitter when the sliding mode surface switches. BRIEF DESCRIPTION OF THE DRAWINGS

[0081] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0082] Figure 1 This is a model diagram of the hydraulic robotic arm lifting system.

[0083] Figure 2 It is a schematic diagram of the S membership function.

[0084] Figure 3are the simulation results; (a) change of the swing angle of the hanging weight; (b) estimated value of ks; (c) estimated value of Z1; (d) estimated value of Z2; (e) estimated value of Z3.

[0085] Figure 4 It is the change of the hoisting weight swing angle.

[0086] Figure 5 This is a diagram showing the change in the hoisting weight swing angle; (a) the hoisting weight swing angle when the hoisting weight is 200 kg; (b) the hoisting weight swing angle when the hoisting weight is 300 kg. DETAILED DESCRIPTION

[0087] Various exemplary embodiments of the present invention are now described in detail. This detailed description should not be considered as a limitation of the present invention, but should be understood as a more detailed description of certain aspects, features, and embodiments of the present invention. It should be understood that the terms used in the present invention are only for describing specific embodiments and are not intended to limit the present invention.

[0088] In addition, for numerical ranges in the present invention, it is understood that each intervening value between the upper and lower limits of the range is also specifically disclosed. Each smaller range between any stated value or intervening value in a stated range and any other stated value or intervening value in the stated range is also included in the present invention. The upper and lower limits of these smaller ranges may independently be included or excluded in the range.

[0089] Unless otherwise indicated, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the art. Although only preferred methods and materials are described herein, any methods and materials similar or equivalent to those described herein may also be used in the practice or testing of the present invention. All documents mentioned in this specification are incorporated by reference to disclose and describe the methods and / or materials associated with the documents. In the event of any conflict with any incorporated document, the contents of this specification shall prevail.

[0090] It will be apparent to those skilled in the art that various modifications and variations can be made to the specific embodiments described herein without departing from the scope or spirit of the invention. Other embodiments derived from the description of the invention will be apparent to those skilled in the art. The description and examples are intended to be exemplary only.

[0091] 1. Design of adaptive sliding mode anti-sway controller

[0092] The simplified model of the hydraulic manipulator crane system with fixed rope length is as follows: Figure 1As shown, the length of the rope is l, the swing angle of the load is α, θ is the amplitude angle of the manipulator at any time, the coordinates of the center of gravity of the load P are P(xP, yP), and the mass is M. Based on the dynamic equation of the hydraulic manipulator lifting system, the mathematical model of the differential model is converted into the general form of the state equation:

[0093]

[0094] Among them, u represents the control quantity of hoisting operation, is the system state vector, representing the swing angle, angular velocity, and angular acceleration of the load respectively, and z1, z2, z3, and d are uncertain quantities.

[0095] The swing angle error of the hanging weight is defined as:

[0096] e=α-α d (2)

[0097] The sliding mode function is designed as:

[0098]

[0099] Among them, α is the current swing angle of the load, α d is the target value of the hoisting swing angle.

[0100] In formula 3, λ1 and λ2 are sliding surface parameters; from formula 2 and formula 3, we can get

[0101]

[0102] Define the first Lyapunov function as:

[0103]

[0104] Derivative of Equation 5 and substitute into Equation 4 to obtain

[0105]

[0106] Define the control rate as:

[0107]

[0108] Among them, u z is the switching control quantity, u s1 and u s2 is the equivalent control quantity, k s and η are adjustable parameters, and sgn(*) is the sign function.

[0109] Substituting equation 6 into equation 7, we get

[0110]

[0111] At this time, the system is stable, but considering that z1, z2, and z3 are unknown parameters in the system, the control rate in Equation 7 cannot be achieved. Therefore, an adaptive control algorithm is used to estimate the parameters. Assume that the parameter z i The estimated error is

[0112]

[0113] in For z i The estimation error of For z i The estimated value of . At this time, the control rate of the system is

[0114]

[0115] Define the second Lyapunov function as:

[0116]

[0117] In the above formula, γ1, γ2, and γ3 are unknown coefficients, and all are positive. Taking the derivative of formula 11 and substituting 10 into it, we can get:

[0118]

[0119] Assume that the rate of change of z1, z2, and z3 is slow, that is, (i=1,2,3), then Therefore, the adaptive rate is

[0120]

[0121] Substituting the adaptive control rate of Equation 13 into Equation 12, we get:

[0122]

[0123] 2. Fuzzy controller design

[0124] When the control rate of the adaptive sliding mode control switches, the function within the control rate is discontinuous, which will cause the sliding mode control system to vibrate. In order to eliminate the vibration phenomenon, the fuzzy control theory is combined with the adaptive sliding mode control method, and the ability of fuzzy language is utilized to approximate the switching term using a fuzzy control system to weaken the system vibration. The sliding mode function s is used as the fuzzy input variable, u s2 As the output of fuzzy control. Using the translation-width principle: To reduce system jitter, when the system state is about to reach the sliding surface, the control effect is reduced; when the system state deviates from the sliding surface, the control effect on the system is enhanced to reduce the error. In summary, the fuzzy rules used by the fuzzy system are as follows:

[0125] R1: if s is P, then u s2 =-r

[0126] R2: if s is Z, then u s2 =0

[0127] R3: if s is N, then u s2 =r

[0128] Where r is the parameter of the designed fuzzy system, r>0; N (negative), Z (zero), P (positive) are fuzzy subsets on the domain. The membership function of S is as follows Figure 2 As shown, product reasoning and weighted average defuzzifier are used, and the fuzzy output u s2 for:

[0129]

[0130] where 0≤ω i ≤1 (i=1, 2, 3) is the weight coefficient of each fuzzy rule. Since the fuzzy input uses a triangular membership function, μ is an adjustable coefficient, which is generally taken as the membership degree of the fuzzy input variable s on the corresponding fuzzy subset.

[0131] That is, ω1=μ P (s),ω2=μ Z (s),ω3=μ N (s), for any sliding mode function, there are the following functions:

[0132] ω1+ω2+ω3=1 (16)

[0133] The switching control rate is

[0134] u s2 =-r(ω1-ω3) (17)

[0135] Depend on Figure 2 It can be seen that for any sliding mode function s:

[0136] (1) When s≥s0, the fuzzy rule is R1, at which ω1=1, ω2=ω3=0, and s(ω1-ω3)>0;

[0137] (2) When 0<s<s0, the fuzzy rules are R1 and R2, at which time 0<ω1<1, 0<ω2<1, ω3=0, s(ω1-ω3)>0;

[0138] (3) When s = 0, the fuzzy rule is R2, at which point ω1 = 0, ω2 = 1, ω3 = 0, and s(ω1 - ω3) = 0;

[0139] (4) When -s0<s<0, the fuzzy rules are R2 and R3, at this time ω1=0, 0<ω2<1, 0<ω3<1, s(ω1-ω3)>0;

[0140] (5) When s<-s0, the fuzzy rule is R3, at this time ω1=ω2=0, ω3=1, s(ω1-ω3)>0.

[0141] Therefore, for any sliding mode function, the following formula holds true

[0142] s(ω1-ω3)=|s|·|ω1-ω3| (18)

[0143] Substituting Equation 18 into Equation 14, we can obtain

[0144]

[0145] From the above formula, we can see that when r satisfies

[0146]

[0147] At this point the system is stable, but d is an unknown parameter and Equation 20 cannot guarantee this. Assume that there is r * Satisfying Equation 21,

[0148]

[0149] Where ε is the smallest normal number. Let r * The valuation is The estimation error is

[0150]

[0151] The control rate at this time is:

[0152]

[0153] Define the Lyapunov function as:

[0154]

[0155] In the above formula, γ4 is the unknown coefficient. The time derivative of formula 24 can be obtained as follows:

[0156]

[0157] Take the adaptive rate as:

[0158]

[0159] At this time there

[0160]

[0161] where 0≤ω i ≤1 (i=1, 2, 3) is the weight coefficient of each fuzzy rule, r * The definition of is shown in formula (21), and the system is stable at this time.

[0162] 3. Simulation Analysis

[0163] The initial values of the system parameters and their boundary settings are shown in Table 1, and the parameter settings of the adaptive fuzzy sliding mode controller are shown in Table 2.

[0164] Table 1 Parameters of adaptive fuzzy sliding mode controller

[0165]

[0166] Table 2 Initial values and boundary values of system parameters

[0167]

[0168] The length of the steel wire rope of the hydraulic manipulator lifting system is set to 1m, the mass of the lifting weight is 100kg, the extension displacement of the variable amplitude hydraulic cylinder is controlled to be 0.13m, and the extension of the telescopic hydraulic cylinder is controlled to be 0.3m. The simulation results are as follows Figure 3 As shown:

[0169] Figure (a) shows the swing angle curve during the lifting process. It can be seen from the figure that under the action of the adaptive fuzzy sliding mode controller, the maximum swing angle of the hydraulic manipulator during the lifting process is about 0.45°. The system quickly adjusts and controls the swing angle of the lifting weight to stabilize. After stabilization, the swing angle of the lifting weight is less than 0.1 degrees. Figures (b), (c), (d), and (e) are k s The estimated values of z1, z2, and z3 are within the set limits. The above simulation results show that the designed controller has a strong control effect on the swing angle of the hanging weight, and there is no sudden change in the swing angle of the hanging weight during the entire movement process, and the stability is good.

[0170] The wire rope length of the hydraulic mechanical arm lifting system is set to 2m, the mass of the lifting weight is 100kg, the extension displacement of the variable amplitude hydraulic cylinder is controlled to be 0.13m, and the extension of the telescopic hydraulic cylinder is controlled to be 0.3m. The simulation experiment is carried out, and the results are as follows. Figure 4 shown

[0171] Depend on Figure 4 It can be seen that the swing angle of the load will increase by simply changing the length of the wire rope, and its movement trend is consistent with the swing angle trend of the load with a wire rope length of 1m; the swing angle of the load reaches its maximum at 0.3s, at which time the controller takes control and the system's swing angle gradually decreases and stabilizes at 0.15° at 0.6s.

[0172] The wire rope length of the hydraulic mechanical arm lifting system is set to 1.5m, the weight of the lifting is 200kg and 300kg respectively, the extension displacement of the variable amplitude hydraulic cylinder is controlled to be 0.13m, the extension of the telescopic hydraulic cylinder is controlled to be 0.3m, and the simulation time is set to 10s for the simulation experiment. The results are as follows Figure 5 shown.

[0173] Depend on Figure 5 As shown in (a) and (b), the swing amplitude change trends of the load under different masses are consistent, and the change law of the load swing angle under different load masses is consistent with the law obtained in Chapter 2, which verifies the correctness of the simulation experiment; the swing angle of the load in the figure has the maximum swing angle of 0.45° and 0.55° respectively after 0.5s of simulation, and the swing angle of the load is controlled at 4.5s. The swing elimination time is short, and the swing elimination effect is better than PID control and fuzzy PID control.

[0174] In summary, the adaptive fuzzy sliding mode controller effectively and quickly controls the weight's swing during motion. Under the controller's control, the weight maintains stability and operation, and the weight's swing amplitude remains within an acceptable range without sudden changes, meeting the requirements for rapid elimination of weight swings and precise operation. The designed weight swing angle controller is insensitive to changes in system parameters and effectively reduces jitter when the sliding mode surface switches.

[0175] The above-described embodiments are only preferred specific implementation methods of the present invention, and the protection scope of the present invention is not limited thereto. Any simple changes or equivalent replacements of the technical solutions that can be obviously obtained by any technician familiar with the field within the technical scope disclosed in the present invention fall within the protection scope of the present invention.

Claims

1. A design method for an adaptive fuzzy sliding mode controller for anti-sway of a hydraulic manipulator lifting weight, characterized in that: The design approach includes the following steps: Step 1: Establish a hydraulic manipulator lifting system model and design an adaptive sliding mode anti-sway controller based on the hydraulic manipulator lifting system model; Step 2: Introduce the fuzzy control method into the adaptive sliding mode anti-sway controller and establish an adaptive fuzzy sliding mode controller.

2. The design method of a hydraulic manipulator weight anti-sway adaptive fuzzy sliding mode controller according to claim 1 is characterized in that: The dynamic expression of the hydraulic manipulator lifting system model established in step 1 is as follows: Among them, u represents the control quantity of hoisting operation, is the system state vector, representing the swing angle, angular velocity, and angular acceleration of the load respectively, and z1, z2, z3, and d are uncertain quantities.

3. The design method of a hydraulic manipulator weight anti-sway adaptive fuzzy sliding mode controller according to claim 2 is characterized in that: The design process of the adaptive sliding mode anti-sway controller is as follows: (1) Design sliding mode function: Among them, λ1 and λ2 are sliding surface parameters, and e represents the swing angle error of the hanging weight, which is expressed as follows: e＝α-α d (3) Among them, α is the current swing angle of the load, α d is the target value of the hoisting swing angle. Combining equations (2) and (3) we get: Define the first Lyapunov function as: Derivative (5) and substitute it into (5) to obtain: Define the control rate as: Among them, u z is the switching control quantity, u s1 and u s2 is the equivalent control quantity, k s and η are adjustable parameters, and sgn(*) is the sign function. Substituting formula (6) into formula (7), we get: (2) Adopt the adaptive control algorithm to estimate z1, z2, z3, assuming that the parameter z i The estimated error is: in For z i The estimation error of For z i The estimated value of the hydraulic manipulator arm lifting system control rate is: Define the second Lyapunov function as: In the above formula, γ1, γ2, and γ3 are unknown coefficients. Derivative (11) and substitute (10) into the formula to obtain: Assume that the rate of change of z1, z2, and z3 is slow, that is, Then there is The adaptive rate is: Substituting Equation 13 into Equation 12, we get:

4. The design method of a hydraulic manipulator weight anti-sway adaptive fuzzy sliding mode controller according to claim 3 is characterized in that: In step 2, the expression of the adaptive fuzzy sliding mode controller is established as follows: where 0≤ω i ≤1 (i=1, 2, 3) is the weight coefficient of each fuzzy rule, r * The definition of is shown in formula (21).

5. The design method of a hydraulic manipulator weight anti-sway adaptive fuzzy sliding mode controller according to claim 4 is characterized in that: The derivation process of the expression of the adaptive fuzzy sliding mode controller is as follows: By using the fuzzy language ability, the fuzzy control method is used to approximate the switching term, and the sliding mode function s is used as the input variable of the fuzzy control. s2 As the output of fuzzy control, the fuzzy rules adopted by fuzzy control are as follows: R1:ifs is P,then u s2 =-r R2:ifs is Z,then u s2 =0 R3:ifs is N,then u s2 =r Among them, r is the parameter of the designed fuzzy system, r>0; N, Z, P are fuzzy subsets on the domain, and the fuzzy output u s2 for: where 0≤ω i ≤1 (i=1, 2, 3) is the weight coefficient of each fuzzy rule. The fuzzy input adopts a triangular membership function, which is taken as the membership degree of the fuzzy input variable s on the corresponding fuzzy subset, that is, ω1=μP(s), ω2=μZ(s), ω3=μN(s), μ is an adjustable coefficient. For any sliding mode function, the following function is available: ω1+ω2+ω3=1 (16) The switching control rate is: you s2 =-r(ω1-ω3) (17) For any sliding mode function, the following holds: s(ω1-ω3)=|s|·|ω1-ω3| (18) Substituting formula (18) into formula (14) we can get From the above formula, we can see that when r satisfies At this time, the hydraulic manipulator lifting system is stable, but d is an unknown parameter, and Equation (20) cannot guarantee it. Assuming that there is r * Satisfying formula (21), Among them, ε is the smallest normal number, let r * The valuation is The estimation error is The control rate at this time is: Define the Lyapunov function as: In the above formula, γ4 is the unknown coefficient. The time derivative of formula (24) can be obtained as follows: Take the adaptive rate as At this time there It can be concluded that the hydraulic manipulator arm lifting system is gradually stable.

6. A method for adaptive fuzzy sliding mode control of hydraulic manipulator weight anti-sway, characterized in that: The adaptive fuzzy sliding mode controller designed in claim 5 is used to control the swing angle of the hanging weight during the movement, so as to keep the swing amplitude of the hanging weight within the allowable range.

7. A hydraulic manipulator arm anti-sway adaptive fuzzy sliding mode control device, characterized in that: The control device includes: The control module calls the adaptive fuzzy sliding mode controller designed in claim 5 to realize the control of the swing angle of the hoist; The display module is used to input the parameters of the adaptive fuzzy sliding mode controller and display the control results of the hoisting weight swing angle.

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