Dual Fixed-Time Second-Order Sliding Mode Control Method for Input-Constrained Fruit and Vegetable Picking Manipulator

Abstract

The present invention provides a dual fixed-time second-order sliding mode control method for a fruit and vegetable picking robotic arm with input constraints, belonging to the technical field of agricultural robots. It solves the technical problems of input saturation and uncertainty existing in the fruit and vegetable picking robotic arm system. The technical solution is as follows: Step 1, define the desired motion trajectory of the end effector of the robotic arm; Step 2, establish the dynamic model of the fruit and vegetable picking robotic arm with input constraints; Step 3, design a fixed-time disturbance observer; Step 4, design a fixed-time convergence auxiliary system to compensate for input saturation; Step 5, define a sliding mode surface based on the estimated value of the fixed-time disturbance observer and the state quantity of the fixed-time convergence auxiliary system, and design a dual fixed-time second-order sliding mode control method for the fruit and vegetable picking robotic arm with input constraints; Step 6, conduct simulation and prototype experiment research. The beneficial effect of the present invention is: realizing the high-precision anti-saturation control of the fruit and vegetable picking robotic arm system with input constraints and improving the fruit and vegetable picking efficiency.

Description

Technical Field

[0001] The present invention relates to the technical field of agricultural robots, and in particular to a dual fixed-time second-order sliding mode control method for a fruit and vegetable picking manipulator with input constraints. Background Art

[0002] With the continuous progress of agricultural technology, various technological achievements have been widely applied in all aspects of agricultural production, especially in the field of intelligent agricultural robots. Fruit and vegetable picking, as a labor-intensive task, usually requires a large amount of manual input, with high labor intensity and a complex working environment. The application of fruit and vegetable picking manipulators can not only significantly improve the picking efficiency and quality but also effectively reduce the labor intensity and production costs.

[0003] As a multi-input multi-output strongly coupled non-linear system, the manipulator has uncertain factors such as modeling errors and external disturbances. During the rapid picking process, the increase in the inertial force of the manipulator will reduce the system stability and control accuracy. Although traditional control methods can achieve the rapid motion control of the manipulator, they require accurate modeling, and it is difficult to avoid external disturbances during the rapid picking action of the manipulator. Sliding mode control is well-known for its good robustness to external disturbances and uncertainties. Second-order sliding mode can suppress chattering, extend the relative order, and is simpler to design compared to any-order sliding mode control while retaining the good performance of traditional first-order sliding mode. After fruits and vegetables mature, they need to be picked in a timely manner, and the picking window period of some fruits and vegetables is short. Therefore, the control system of the picking manipulator must converge within a definite and sufficiently short time to quickly complete the picking task and reduce energy consumption. Most sliding mode control schemes adopt finite-time convergence, while the fixed-time control method can ensure that the system convergence time is bounded under any initial state, solving the problem that the trajectory tracking time extends infinitely with the increase of the initial conditions and improving the control accuracy and rapidity. In addition, in the actual control system of the manipulator, the inherent physical constraints of the actuator are likely to cause the problem of control input saturation, affecting the dynamic control performance and stability of the system and even causing damage to the mechanism. It is crucial to design a dual fixed-time second-order sliding mode control method for a fruit and vegetable picking manipulator with input constraints to achieve its high-precision anti-saturation control by combining a disturbance observer and an input saturation auxiliary system. Summary of the Invention

[0004] The purpose of the present invention is to provide a dual fixed-time second-order sliding mode control method for a fruit and vegetable picking manipulator with input constraints. Aiming at the problems of input saturation and uncertainty existing in the fruit and vegetable picking manipulator system, a dual fixed-time second-order sliding mode control method for a fruit and vegetable picking manipulator with input constraints is proposed to ensure the fixed-time convergence of the observation error and tracking error of the control system, so as to achieve the high-precision anti-saturation control of the fruit and vegetable picking manipulator system with input constraints.

[0005] The inventive concept of the present invention is as follows: perform inverse kinematic analysis on a fruit and vegetable picking robotic arm for picking tasks, obtain the forward kinematics and Jacobian matrix of the fruit and vegetable picking robotic arm, and define the desired motion trajectory of the end effector of the robotic arm. Next, establish a dynamic model of the input-constrained fruit and vegetable picking robotic arm considering uncertainties such as modeling errors and complex agricultural environment disturbances. Secondly, design a fixed-time disturbance observer to accurately estimate and compensate for the lumped uncertain terms such as the modeling error and external disturbance of the system within a fixed time. By introducing a fractional power term containing an auxiliary state variable, construct a new type of fixed-time convergent auxiliary system to compensate for input saturation. Combine the observer and the auxiliary system, define a sliding mode surface based on the estimated value of the fixed-time disturbance observer and the state variable of the fixed-time convergent auxiliary system, and propose a double fixed-time second-order sliding mode control method to simultaneously achieve the fixed-time convergence of the observation error and tracking error of the input-constrained fruit and vegetable picking robotic arm control system, enhance its robustness, and weaken the chattering of the sliding mode control. Finally, take the agricultural picking robotic arm as an example, conduct simulation and prototype experiments to verify that the designed control method can achieve real-time and accurate tracking of the target picking point trajectory. The present invention can effectively enhance the robustness of the robotic arm control system for fruit and vegetable picking tasks to achieve high-precision anti-saturation control of the input-constrained fruit and vegetable picking robotic arm system.

[0006] In order to achieve the above-mentioned inventive purpose, the technical solution adopted by the present invention is specifically as follows: a double fixed-time second-order sliding mode control method for an input-constrained fruit and vegetable picking robotic arm, comprising the following steps:

[0007] Step 1, perform inverse kinematic analysis on a fruit and vegetable picking robotic arm for picking tasks, obtain the forward kinematics and Jacobian matrix of the fruit and vegetable picking robotic arm, and define the desired motion trajectory of the end effector of the robotic arm;

[0008] Step 2, establish a dynamic model of the input-constrained fruit and vegetable picking robotic arm considering uncertainties such as modeling errors and complex agricultural environment disturbances;

[0009] Step 3, for the dynamic model in Step 2, design a fixed-time disturbance observer to accurately estimate and compensate for the lumped uncertain terms D such as the modeling error and external disturbance of the system within a fixed time;

[0010] Step 4, for the dynamic model in Step 2, construct a new type of fixed-time convergent auxiliary system to compensate for input saturation by introducing a fractional power term containing an auxiliary state variable ξ;

[0011] Step 5, combine the disturbance observer designed in Step 3 and the auxiliary system designed in Step 4, and define based on the estimated value of the fixed-time disturbance observer A sliding mode surface related to the state variable ξ of the fixed-time convergence auxiliary system is designed, and a double fixed-time second-order sliding mode control method for the input-limited fruit and vegetable picking manipulator is proposed to achieve real-time and accurate tracking of the target picking point trajectory;

[0012] Step 6: Taking the agricultural picking manipulator as an example, simulation and prototype experiments are carried out to verify the correctness and effectiveness of the designed control method.

[0013] Furthermore, in step 2, for the fruit and vegetable picking manipulator facing the picking task, considering uncertain factors such as modeling errors and complex agricultural environment disturbances, the dynamic model of the input-limited fruit and vegetable picking manipulator in the joint space is expressed as:

[0014]

[0015] In the formula, are the pose, velocity, and acceleration vectors of each active joint of the manipulator respectively, M(q) ∈ R n×n is the inertia matrix, is the Coriolis force and centrifugal force term, G(q) ∈ R n is the gravity term, τ ∈ R n is the driving torque of each driving motor, and d represents the disturbance term composed of system modeling errors and external disturbances. τ c =(τ c1 , τ c2 ,..., τ cn ) T represents the control input command of the manipulator dynamic control system, is the difference before and after the control input saturation constraint. The actual input τ of the control system is the saturation function form of τ c , that is

[0016]

[0017] For the convenience of control system design, M(q), G(q) are respectively abbreviated as M, C, G. Define the lumped uncertainty D = -M -1 d, then the dynamic model (1) is rewritten as

[0018]

[0019] Furthermore, in step 2, the following assumptions are made for the input-limited fruit and vegetable picking manipulator:

[0020] It is assumed that the lumped uncertainty D of the manipulator system = [D 1 , D 2 ,..., D i T is bounded and first-order differentiable, and its first-order derivative ​Bounded, there exists a constant satisfying

[0021] Furthermore, in step 3, a novel fixed-time disturbance observer is designed to accurately estimate and compensate the lumped uncertainties of the system within a fixed time

[0022]

[0023] wherein λ 1 ,λ 2 ,λ 3 >0, p>1, is the estimate of L; is the estimate of the lumped uncertainty D of the system; s 0 is the auxiliary sliding mode variable.

[0024] Taking the first derivative of and substituting (3) into it, we can obtain

[0025]

[0026] Combined with the fixed-time disturbance observer, it can be further written in the following form

[0027]

[0028] Let we can obtain a second-order nonlinear system

[0029]

[0030] For system (3), if the fixed-time disturbance observer is designed as equation (4), the estimation error of the observer converges to the origin within a fixed time, and the minimum convergence time T 1 satisfies

[0031] T 1 ≤max{[2ε 1 / 2 / λ 1 +(λ 2 (p - 1)ε p-1 ) -1 [1+(w i / W i -w i h(λ 1 ) / λ 1 ) -1} (8)

[0032] wherein h(λ 1 )=λ 1 -1 +(2e / wi λ 1 ) 1 / 3 , ε = (λ 1 / λ 2 ) 1 / (p+1 / 2) 。

[0033] By selecting appropriate parameters, it converges rapidly to the origin within a fixed time T 1 . When T ≥ T 1 , the observer can accurately obtain the estimated value of the lumped uncertainty and the estimation time is independent of the initial value of the system perturbation.

[0034] Furthermore, in step 4, to solve the input saturation problem, the control input of the manipulator is constrained, and a fractional power term of the auxiliary state quantity is introduced to construct the following new fixed-time convergence auxiliary system

[0035]

[0036] where is bounded, and there exists A 0 such that A 0 ≥ ||ψ||, ξ is the auxiliary state generated by the auxiliary system, A 1 > 0, A 2 > 0, 0 < c 1 < 1, c 2 > 1.

[0037] Select the Lyapunov function V 1 as follows:

[0038]

[0039] To make the Lyapunov function V 1 stable, it can be obtained that the auxiliary state converges to zero within a fixed time T 2 , and the form of T 2 is as follows:

[0040]

[0041] Furthermore, in step 5, according to the dynamic model (3) and the auxiliary system (9), define the error e 1 = q - q d - ξ, and by taking the derivative of e 1 the following tracking error system of the active joint can be obtained

[0042]

[0043] where q d is the desired trajectory,

[0044] Define the estimated value based on the fixed-time disturbance observer The sliding surface with the state quantity ξ of the fixed-time convergence auxiliary system

[0045]

[0046] where α 1 ,α 2 ,α 3 >0, γ 1 >1, 0<γ 2 <1.

[0047] Combined with the second-order sliding mode control, design a new type of sliding surface, and its form is as follows

[0048]

[0049] where β 1 ,β 2 ,β 3 >0, θ 1 >1, 0<θ 2 <1.

[0050] Design a fixed-time convergence reaching law for σ

[0051]

[0052] where k 1 ,k 2 ,k 3 >0, μ 1 >1, 0<μ 2 <1, K 1 >0.

[0053] Furthermore, in the step 5, based on the fixed-time disturbance observer designed in step 3 and the fixed-time convergence auxiliary system designed in step 4, construct the Lyapunov function V 2 as follows:

[0054]

[0055] Make the Lyapunov function V 2 tend to be stable, and obtain the double fixed-time second-order sliding mode control law τ c in the following form

[0056]

[0057] where,

[0058]

[0059] The system will converge to σ = 0 at a fixed time, and the convergence time is T σmax . When the sliding surface σ converges to 0, combining with Equation (14), we can obtain

[0060]

[0061] At this time, the sliding surface s will converge to 0 at a fixed time, and its convergence time is T smax . If s = 0, combining with Equation (13), we can obtain

[0062]

[0063] At this time, the tracking error e of the system 1 will converge to the vicinity of the origin at a fixed time, and the convergence time is T emax .

[0064] Furthermore, in step 5, considering the tracking error system (12), the control input τ is obtained by combining the designed fixed-time disturbance observer and the fixed-time convergence auxiliary system c As shown in (17), the trajectory tracking error of the manipulator can converge to the vicinity of the origin and remain stable within a fixed time, and the convergence time T satisfies

[0065] T ≤ T 1 + T 2 + T 3 (20)

[0066] where, T 1 is as shown in Equation (8), T 2 is as shown in Equation (11), T 3 = T σmax + T smax + T emax

[0067] T σmax = k 2 -1 [(μ 1 - 1) -1 ln(1 + k 2 / k 1 )+(1 - μ 2 ) -1 ln(1 + k 2 / k 3 )],

[0068] T smax = β 2 -1 [(θ 1 - 1) -1 ln(1 + β 2 / β1 ) + (1 - θ 2 ) -1 ln(1 + β 2 / β 3 )],

[0069] T emax = α 2 -1 [(γ 1 - 1) -1 ln(1 + α 2 / α 1 )+(1 - γ 2 ) -1 ln(1 + α 2 / α 3 )].

[0070] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0071] (1) The present invention does not need to use a disturbance observer to simultaneously estimate and compensate uncertain terms such as external disturbances and saturation errors of the system, reducing the computational burden of the observer. A fixed-time disturbance observer is designed to accurately estimate and compensate the lumped uncertain terms such as the modeling error and external disturbances of the manipulator system within a fixed time.

[0072] (2) The present invention constructs a new fixed-time convergent auxiliary system by introducing a fractional power term containing an auxiliary state variable to solve the input saturation problem existing in the manipulator system.

[0073] (3) The present invention defines a sliding mode surface based on the estimated value of the fixed-time disturbance observer and the state variable of the fixed-time convergent auxiliary system to simultaneously achieve the fixed-time convergence of the observation error and tracking error of the input-limited fruit and vegetable picking manipulator control system and weaken the chattering of the sliding mode control. The fixed-time second-order sliding mode control method strategy is extended to a high-order picking manipulator control system with multiple inputs and multiple outputs.

[0074] (4) The present invention theoretically proves the global fixed-time convergence of the closed-loop control system, and the upper bound of the convergence time is independent of the initial state of the system, and calculates the minimum upper bound of the convergence time of the system error. In addition, the effectiveness of the proposed control method is further verified through simulation and experiments on the prototype of the agricultural picking manipulator. BRIEF DESCRIPTION OF THE DRAWINGS

[0075] The drawings are used to provide a further understanding of the present invention, and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention, and do not constitute a limitation to the present invention.

[0076] Figure 1 It is a flowchart of the dual fixed-time second-order sliding mode control method for the input-limited fruit and vegetable picking manipulator in the present invention.

[0077] Figure 2 This is a schematic diagram of the hardware experimental platform for the prototype system of the agricultural picking robotic arm in the present invention.

[0078] Figure 3 This is a graph showing the estimation of the lumped uncertain disturbance term by the fixed-time disturbance observer in the present invention.

[0079] Figure 4 This is a graph showing the trajectory tracking of joint 1 of the robotic arm under a smaller initial state provided by an embodiment of the present invention.

[0080] Figure 5 This is a graph showing the trajectory tracking error of joint 1 of the robotic arm under a smaller initial state provided by an embodiment of the present invention.

[0081] Figure 6 This is a graph showing the trajectory tracking of joint 1 of the robotic arm under a larger initial state provided by an embodiment of the present invention.

[0082] Figure 7 This is a graph showing the trajectory tracking error of joint 1 of the robotic arm under a larger initial state provided by an embodiment of the present invention.

[0083] Figure 8 This is a graph showing the control input torque of joint 1 of the input-limited robotic arm provided by an embodiment of the present invention.

[0084] Figure 9 This is a graph showing the experimental results of the trajectory tracking error of joint 1 of the prototype of the agricultural picking robotic arm provided by an embodiment of the present invention. Detailed implementation manners

[0085] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Of course, the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0086] Embodiment 1

[0087] Refer to Figures 1 to 9 , taking the agricultural picking robotic arm as an example, a simulation study is carried out on the traditional control method. In order to more clearly display the simulation results, the position of joint 1 of the fruit and vegetable picking robotic arm is selected in this embodiment to show the tracking performance of the traditional controller. Taking the Figure 2 shown prototype of the agricultural picking robotic arm as the research object, the following four traditional control methods are used for the design and performance test of the control system considering input saturation. Set τ d =cos2t - sint, and the model error is ΔM = [0.5, 0.5, 0.5, 0.5, 0.5, 0.5] TWhen considering input limitations, the driving force range of the robotic arm joint 1 is limited to [-5, 5] (N·m).

[0088] To verify the superiority of the method of the present invention compared with the traditional control method in high-precision trajectory tracking, the traditional finite-time second-order sliding mode control method FT-SOSMC with an observer is denoted as Controller 1. This method is based on the finite-time sliding mode control theory and designs a second-order sliding mode controller with an observer. Figure 4 and Figure 5 respectively show the tracking curve and the trajectory tracking error curve of this traditional method when the initial angle of joint 1 of the system is 0.25 rad. Figure 6 and Figure 7 are respectively the trajectory tracking curve and the trajectory tracking error curve of this traditional method when the initial angle of joint 1 of the system is 1.5 rad. It can be analyzed from the figure that this traditional method can better achieve the tracking effect of the reference trajectory, and the trajectory tracking error can converge within a finite time.

[0089] To verify that the present invention can better weaken the sliding mode control chattering and effectively solve the input saturation problem, the traditional second-order sliding mode control method without introducing a fixed-time convergence auxiliary system is denoted as Controller 2, the fixed-time second-order sliding mode control method Fx-SOSMC without an observer is denoted as Controller 3, and the finite-time first-order sliding mode control method FTSMC without an observer is denoted as Controller 4. Figure 8 shows the torque curve of the control input of joint 1 in the case of input saturation for each traditional control method. It can be analyzed that Controller 2 significantly violates the constraint condition of the torque being [-5, 5] (N·m). In addition, due to the use of the traditional first-order sliding mode control method in the method of Controller 4, obvious sliding mode control chattering appears. The method of Controller 3 adopts the second-order sliding mode control method, which effectively weakens the sliding mode control chattering. However, since it does not introduce a fixed-time disturbance observer to further reduce the burden of the sliding mode control to overcome the uncertain factors of the robotic arm system, the control accuracy and the ability to suppress the sliding mode control of this method need to be improved in the case of input limitations.

[0090] The simulation results show that the above traditional control methods all have certain limitations in dealing with the trajectory tracking problem of the robotic arm in a complex agricultural environment. Especially when dealing with input saturation and external disturbances, the control effect and system robustness of the traditional methods still need to be improved. These comparison results provide an important basis for verifying the effectiveness of the method of the present invention in subsequent embodiments.

[0091] Embodiment 2

[0092] See Figures 1 to 9, this embodiment provides its technical solution as a double fixed-time second-order sliding mode control method for an input-constrained fruit and vegetable picking manipulator. Kinematic inverse solution analysis is performed on the fruit and vegetable picking manipulator for the picking task to obtain the forward kinematics and Jacobian matrix of the fruit and vegetable picking manipulator, and the desired motion trajectory of the end effector of the manipulator is defined. Next, a dynamic model of the input-constrained fruit and vegetable picking manipulator considering uncertainties such as modeling errors and complex agricultural environment disturbances is established. Secondly, a fixed-time disturbance observer is designed to accurately estimate and compensate the lumped uncertain terms such as the modeling error and external disturbance of the system within a fixed time. By introducing a fractional power term containing an auxiliary state variable, a new fixed-time convergent auxiliary system is constructed to compensate for input saturation. Combining the disturbance observer and the auxiliary system, a sliding mode surface based on the estimated value of the fixed-time disturbance observer and the state variable of the fixed-time convergent auxiliary system is defined, and a double fixed-time second-order sliding mode control method for the input-constrained fruit and vegetable picking manipulator is designed to simultaneously achieve the fixed-time convergence of the observation error and tracking error of the input-constrained fruit and vegetable picking manipulator control system, enhance its robustness, and weaken the chattering of the sliding mode control. Finally, taking the agricultural picking manipulator as an example, simulation and prototype experiments are carried out to verify that the designed control method can achieve real-time and accurate tracking of the target picking point trajectory. The present invention can effectively enhance the robustness of the manipulator control system for fruit and vegetable picking tasks to achieve high-precision anti-saturation control of the input-constrained fruit and vegetable picking manipulator system.

[0093] The method flow of this embodiment is as Figure 1 shown, and the specific method is as follows:

[0094] 1. For the fruit and vegetable picking manipulator for the picking task, considering uncertain factors such as modeling errors and complex agricultural environment disturbances, the dynamic model of the input-constrained fruit and vegetable picking manipulator in the joint space is expressed as:

[0095]

[0096] In the formula, are the pose, velocity, and acceleration vectors of each active joint of the manipulator respectively, M(q) ∈ R n×n is the inertia matrix, is the Coriolis force and centrifugal force term, G(q) ∈ R n is the gravity term, τ ∈ R n is the driving torque of each drive motor, and d represents the disturbance term composed of the system modeling error and external disturbance. τ c =(τ c1 , τ c2 ,..., τ cn ) T represents the control input command of the manipulator dynamic control system, is the difference before and after the control input saturation constraint.

[0097] In this embodiment, considering the inherent physical constraints of the robotic arm actuator, the actual input τ of each joint control system is τ c in the form of a saturation function, that is

[0098]

[0099] Define the lumped uncertainty D = -M -1 d, then the dynamic model (1) can be rewritten as

[0100]

[0101] In this embodiment, assume that the lumped uncertainty D of the robotic arm system is D = [D 1 , D 2 ,..., D i T is bounded and first-order differentiable, and its first derivative is bounded. There exists a constant that satisfies

[0102] 2. Design a new fixed-time disturbance observer to accurately estimate and compensate the lumped uncertainty of the system within a fixed time

[0103]

[0104] where λ 1 , λ 2 , λ 3 > 0, p > 1, is the estimate of L; is the estimate of the lumped uncertainty D of the system; s 0 is the auxiliary sliding mode variable.

[0105] For the system (3), if the fixed-time disturbance observer is designed as equation (4), then the estimation error of the observer converges to the origin within a fixed time, and the minimum convergence time T 1 satisfies

[0106] T 1 ≤ max{[2ε 1 / 2 / λ 1 +(λ 2 (p - 1)ε p-1 ) -1 [1+(w i / W i -w i h(λ 1 ) / λ 1 ) -1} (25)

[0107] where​ h(λ 1 ) = λ 1 -1 +(2e / w i λ 1 ) 1 / 3 , ε = (λ 1 / λ 2 ) 1 / (p+1 / 2) .

[0108] In this embodiment, the observer parameters λ 1 = 30, λ 2 = 40, λ 3 = 20, p = 1.5.

[0109] 3. To solve the input saturation problem, the control input of the robotic arm is constrained, and a fractional power term of the auxiliary state quantity is introduced to construct the following new fixed-time convergence auxiliary system

[0110]

[0111] where is bounded, and there exists A 0 such that A 0 ≥ ||ψ||, ξ is the auxiliary state generated by the auxiliary system, A 1 > 0, A 2 > 0, 0 < c 1 < 1, c 2 > 1.

[0112] If the fixed-time convergence auxiliary system is designed as Equation (6), the auxiliary state converges to zero within the fixed time T 2

[0113]

[0114] In this embodiment, the parameters of the auxiliary system are selected as A 1 = 0.5, A 2 = 1.5, c 1 = 0.55, c 1 = 1.28.

[0115] 4. Define a sliding mode surface based on the estimated value of the fixed-time disturbance observer and the state quantity of the fixed-time convergence auxiliary system, and propose a double fixed-time second-order sliding mode control method to simultaneously achieve the fixed-time convergence of the observation error and tracking error of the input-limited fruit and vegetable picking robotic arm control system, enhance its robustness, and weaken the chattering of the sliding mode control.

[0116] According to the dynamic model (3) and the auxiliary system (6), define the error e 1 = q - q d - ξ, for e 1 ​The derivative yields the following tracking error system of the active joint

[0117]

[0118] where q d is the desired trajectory,

[0119] In this embodiment, the sliding surface is defined based on the estimated value of the fixed-time disturbance observer and the state variable ξ of the fixed-time convergence auxiliary system

[0120]

[0121] where α 1 ,α 2 ,α 3 >0, γ 1 >1, 0<γ 2 <1.

[0122] Combined with second-order sliding mode control, a new sliding surface is designed, and its form is as follows

[0123]

[0124] where β 1 ,β 2 ,β 3 >0, θ 1 >1, 0<θ 2 <1.

[0125] A fixed-time convergence reaching law is designed for σ

[0126]

[0127] where k 1 ,k 2 ,k 3 >0, μ 1 >1, 0<μ 2 <1, K 1 >0;

[0128] According to the equivalent control design method, the expression of the control law τ c can be obtained

[0129]

[0130] where,

[0131]

[0132] In this embodiment, the selected controller parameters are α 1 = 3, α2 = 3, α 3 = 3, γ 1 = 1.5, γ 2 = 0.85, β 1 = 4.2, β 2 = 4.5, β 3 = 4.2, θ 1 = 1.5, θ 2 = 0.45, k 1 = 10, k 2 = 5, k 3 = 10, μ 1 = 1.5, μ 2 = 0.45, K 1 = 1. The selected first-order sliding surface is

[0133] Considering the tracking error system (8), combining the designed fixed-time disturbance observer and the fixed-time convergence auxiliary system to obtain the control input τ c As shown in (12), the trajectory tracking error of the manipulator can converge to the vicinity of the origin and remain stable within a fixed time, and the convergence time T satisfies

[0134] T ≤ T 1 + T 2 + T 3 (33)

[0135] where, T 1 As shown in equation (5), T 2 As shown in equation (7), T 3 = T σmax + T smax + T emax

[0136] T σmax = k 2 -1 [(μ 1 - 1) -1 ln(1 + k 2 / k 1 ) + (1 - μ 2 ) -1 ln(1 + k 2 / k 3 )],

[0137] T smax = β 2 -1 [(θ 1 - 1) -1 ln(1 + β 2 / β 1 ) + (1 - θ 2 )-1 ln(1 + β 2 / β 3 )],

[0138] T emax = α 2 -1 [(γ 1 - 1) -1 ln(1 + α 2 / α 1 ) + (1 - γ 2 ) -1 ln(1 + α 2 / α 3 )].

[0139] 5. Taking the agricultural picking manipulator as an example, simulation and prototype experiments are carried out to verify that the designed control method can achieve real-time and accurate tracking of the target picking point trajectory.

[0140] To verify the effectiveness of the double fixed-time second-order sliding mode control method for the input-constrained fruit and vegetable picking manipulator proposed in the present invention, taking the prototype of the agricultural picking manipulator shown in Figure 2 as the controlled object, the method of the present invention is simulated and compared with the four traditional methods in Example 1 considering input saturation. The model error ΔM = [0.5, 0.5, 0.5, 0.5, 0.5, 0.5] is uniformly set T , τ d = cos2t - sint. When considering input constraints, the driving force range of joint 1 is limited to [-5, 5] (N·m).

[0141] Figure 3 shows the estimation curve of the lumped uncertainty by the fixed-time disturbance observer designed in the present invention. It can be analyzed that, under different initial disturbance values, the invented observer can achieve accurate estimation of the lumped uncertainty within 0.17 s, and the convergence time is independent of the initial state of the disturbance, effectively avoiding the problem of overestimating the system convergence time due to the overly long observation time T 1 of the observer.

[0142] Figure 4 and Figure 5 are respectively the tracking curve and the trajectory tracking error curve of the system when the initial angle of joint 1 is 0.25 rad, Figure 6 and Figure 7 are respectively the trajectory tracking curve and the trajectory tracking error curve of the system when the initial angle of joint 1 is 1.5 rad. It can be analyzed that the method proposed in the present invention can achieve high-precision control, and the convergence time of the trajectory tracking error does not change significantly with the change of the initial state of the system, which is much smaller than the controller 1 method that can only achieve finite-time convergence.

[0143] Figure 8 It is the torque curve of the control input of joint 1 under the condition of input saturation. Through analysis, it can be seen that the method proposed in the present invention can limit the driving force of the joint within [-5, 5] (N·m) due to the introduction of a fixed-time convergence auxiliary system. However, Controller 2 clearly violates the constraint conditions. In addition, compared with the method of Controller 4, the switching term of this method acts on the derivative of the system control input, making the input signal of the system continuous, thereby better suppressing the chattering of the sliding mode control. And compared with Controller 3, a fixed-time disturbance observer is introduced to further weaken the chattering. Using the prototype of the agricultural picking manipulator, motion control experiments were carried out on the traditional Controller 1 and the method proposed in the present invention. Figure 9 It is the experimental result curve of the trajectory tracking error of joint 1 of the prototype of the agricultural picking manipulator. The experimental results show that the proposed control method has better tracking accuracy than the traditional finite-time second-order sliding mode control method when dealing with manipulators with uncertainties and input saturation.

[0144] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A dual fixed-time second-order sliding mode control method for an input-constrained fruit and vegetable picking robot arm, characterized in that: The steps include: Step 1: Perform inverse kinematic analysis on the fruit and vegetable picking robot arm for picking tasks, obtain the kinematic forward solution and Jacobian matrix of the fruit and vegetable picking robot arm, and define the expected motion trajectory of the end effector of the robot arm; Step 2, establish a dynamic model of the input-constrained fruit and vegetable picking robot arm considering modeling errors and the uncertainty of interference in complex agricultural environments; In step 2, for the fruit and vegetable picking robot arm for picking tasks, considering the modeling error and the uncertainty of the complex agricultural environment interference, the dynamic model of the input-constrained fruit and vegetable picking robot arm in the joint space is expressed as: In the formula, are the position, velocity and acceleration vector of each active joint of the robot arm, M(q)∈R n×n is the inertia matrix, is the Coriolis force and centrifugal force, G(q)∈R n is the gravity term, τ∈R n is the driving torque of each driving motor, d is the disturbance term composed of system modeling error and external interference, τ c =(τ c1 ,τ c2 ,...,τ cn ) T Represents the control input instructions of the robot arm dynamics control system, is the difference before and after the control input saturation constraint. The actual input τ of the control system is τ c The saturation function form is M(q), G(q) is abbreviated as M, C, G respectively, and the lumped uncertainty term D = -M -1 d, then the kinetic model (1) can be rewritten as Assume that the total uncertainty of the robot system D = [D1, D2, ..., D i ] T is bounded and first-order differentiable, and its first-order derivative is is bounded, there exists a constant satisfying Step 3: Design a fixed-time disturbance observer for the dynamic model in step 2 to accurately estimate and compensate for the system modeling error and the lumped uncertainty term D of external disturbance within a fixed time; Step 4, for the dynamic model in step 2, a fixed-time convergence auxiliary system is constructed by introducing a fractional power term containing an auxiliary state quantity ξ to compensate for input saturation; Step 5: Combine the disturbance observer designed in step 3 and the auxiliary system designed in step 4 to define the fixed-time disturbance observer estimate The sliding mode surface of the state quantity ξ of the auxiliary system is converged with the fixed time, and a double fixed-time second-order sliding mode control method for the input-constrained fruit and vegetable picking robot arm is designed to achieve real-time and accurate tracking of the trajectory of the target picking point; Step 6: Conduct simulation and prototype experiments on the agricultural harvesting robot arm to verify the correctness and effectiveness of the designed control method.

2. The dual fixed-time second-order sliding mode control method for input-constrained fruit and vegetable picking mechanical arm according to claim 1 is characterized in that: In step 3, a fixed-time disturbance observer is designed to accurately estimate and compensate the system lumped uncertainty within a fixed time: In the formula, λ1, λ2, λ3>0, p>1, is the estimate of L; is the estimate of the system lumped uncertainty D; s0 is the auxiliary sliding mode variable; For the system, if the fixed-time disturbance observer is designed as formula (4), the estimated error of the observer converges to the origin at a fixed time, and the minimum convergence time T1 satisfies: T1≤max{[2ε 1 / 2 / λ1+(λ2(p-1)ε p-1 ) -1 ][1+(w i / W i -w i h(λ1) / λ1) -1 ]} (5) Among them, λ1h -1 (λ1)>W i ; h(λ1)=λ1 -1 +(2e / w i λ1) 1 / 3 ,(i=1,...,6),ε=(λ1 / λ2) 1 / (p+1 / 2) 。 3. The dual fixed-time second-order sliding mode control method for input-constrained fruit and vegetable picking mechanical arm according to claim 1 is characterized in that: In step 4, in order to solve the input saturation problem and constrain the control input of the manipulator, the fractional power term of the auxiliary state quantity is introduced to construct the following fixed time convergence auxiliary system: in, is bounded, and there exists A0 such that A0 ≥ ||ψ||, ξ is the auxiliary state generated by the auxiliary system, A1>0, A2>0, 0<c1<1, c2>1; If the fixed time convergence auxiliary system is designed as formula (6), the auxiliary state converges to zero within the fixed time T2 4. The dual fixed-time second-order sliding mode control method for input-constrained fruit and vegetable picking mechanical arm according to claim 2 or 3, characterized in that: In step 5, according to the dynamic model and the auxiliary system, the error e1=qq is defined d -ξ, and the derivative of e1 gives the following tracking error system of the active joint: Among them, q d is the expected trajectory, Define the estimate based on the fixed-time disturbance observer The sliding surface of the state quantity ξ of the auxiliary system with fixed time convergence Among them, α1, α2, α3>0, γ1>1, 0<γ2<1; Combined with the second-order sliding mode control, a sliding mode surface is designed, which is as follows Among them, β1, β2, β3>0, θ1>1, 0<θ2<1; Design a fixed-time convergence law for σ Among them, k1, k2, k3>0, μ1>1, 0<μ2<1, K1>0; According to the equivalent control design method, the control law τ can be obtained: c The expression in, Considering the tracking error system (8), the control input τ is obtained by combining the designed fixed-time disturbance observer and fixed-time convergence auxiliary system c , then the trajectory tracking error of the robot converges to the vicinity of the origin within a fixed time and remains stable, and the convergence time T satisfies T≤T1+T2+T3 (13) Where, T1 is as shown in formula (5), T2 is as shown in formula (7), T3 = T σmax +T smax +T emax T σmax =k2 -1 [(μ1-1) -1 ln(1+k2 / k1)+(1-μ2) -1 ln(1+k2 / k3)], T smax =β2 -1 [(θ1-1) -1 ln(1+β2 / β1)+(1-θ2) -1 ln(1+β2 / β3)], T emax =α2 -1 [(c1-1) -1 ln(1+α2 / α1)+(1-γ2) -1 ln(1+α2 / α3)].

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