A predefined time sliding mode control method for a robot trajectory tracking system

Abstract

This invention discloses a predefined-time sliding mode control method for a robotic arm trajectory tracking system. Most existing control methods rely on the initial state of the system and rarely consider the real-time performance of the robotic arm system and the ability to complete the task within a predefined time. The method of this invention includes: establishing a dynamic model of the robotic arm system; establishing a full-drive system model of the robotic arm system; designing a predefined-time sliding mode controller; designing a predefined-time sliding mode controller for the robotic arm system based on the full-drive system method; stability analysis of the system during the approach phase; and stability analysis of the system during the sliding phase. Based on predefined-time control theory, the full-drive system method, and sliding mode control theory, this invention considers external disturbances to the robotic arm system and allows the desired performance to be achieved within a given time. It designs a predefined-time sliding mode controller based on the full-drive system method, achieving effective control of the robotic arm system's trajectory tracking.

Description

Technical Field

[0001] This invention belongs to the field of robotic arm trajectory tracking technology. Specifically, it designs a predefined time sliding mode control method based on the all-drive system approach for robotic arm systems. By designing a predefined time sliding mode controller based on the all-drive system approach for a robotic arm system with external disturbances, effective control of the robotic arm trajectory tracking system is achieved, making it applicable to robotic arm trajectory tracking control in various fields. Background Technology

[0002] With the advancement of technology and the rapid development of industrialization, robotic arms are widely used in industries such as manufacturing, space exploration, and medical surgery. Their trajectory tracking control has become a crucial research area in automation. For robotic arm trajectory tracking control systems, traditional sliding mode control methods rely on the system's initial state, and the characteristics of sliding mode control can cause system chattering, reducing the lifespan of the actual system. Therefore, designing a safe and effective control method for robotic arm trajectory tracking control systems is essential for the development of various industries.

[0003] Considering practical realities, designing a convergence law can prevent the controller from operating in harsh environments, improving its stability and lifespan, and reducing the operational difficulty for workers. On the other hand, thanks to the development of the all-drive system approach, the system model can be transformed into a first-order error system, providing arbitrary design freedom. Furthermore, real-time requirements must be considered in practical applications, increasing the performance demands on the robotic arm trajectory tracking control system. The robotic arm trajectory tracking control system is an indispensable part of modern automated industry, playing a crucial role in manufacturing and logistics. Its task is to ensure that the robotic arm can accurately perform tasks along a predetermined path; therefore, the safe and efficient control of the robotic arm trajectory is paramount.

[0004] Existing research methods for robotic arm trajectory tracking control rarely consider the combined effects of the system's initial state and external disturbances. Disturbances can negatively impact the system. Furthermore, the timing of joint position tracking during robotic arm operation is crucial. Without considering the system's stability upper bound, a high degree of determinism in system behavior cannot be provided, hindering effective adjustment of joint positions and potentially leading to erroneous execution in other components. Therefore, a novel control method is needed to safely and effectively control robotic arm trajectory tracking. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of existing sliding mode control methods and to consider the inability of the system to stabilize within a predefined time. It proposes a predefined time sliding mode control based on an all-drive system method to achieve effective control of the robotic arm.

[0006] Based on the all-drive system method, predefined time control theory and sliding mode control theory, this invention designs a predefined time sliding mode controller based on the all-drive system method, considering the need to suppress external disturbances and the system's requirement for predefined time stability, thus realizing effective control of robotic arm trajectory tracking.

[0007] The specific steps of this invention are as follows:

[0008] Step 1: Establish a dynamic model of the robotic arm system;

[0009] First, based on the principles of dynamics, a dynamic model with external disturbances was established:

[0010]

[0011] Where q = [q1, q2, ..., q n ] T This indicates the joint position of the robotic arm. This indicates the joint angular velocity of the robotic arm. Let M(q) represent the joint angular acceleration of the robotic arm, with the superscript T indicating the transpose of the matrix and the subscript n indicating the nth joint of the robotic arm. n×n The inertia matrix of the robotic arm is represented. Let G(q) ∈ R represent the centrifugal force and Coriolis force of the robotic arm. n×1 Let τ ∈ R represent the gravitational term of the robotic arm. n×1 τ represents the control torque of the robotic arm. d ∈R n×1 This indicates that the external disturbance is bounded and satisfies Where a0, a1, and a2 are unknown positive constants, and ||·|| denotes the L2 norm. The symbol R represents the Euclidean space. n×n Let R represent an n×n dimensional matrix. n×1 This represents an n-dimensional column vector.

[0012] Step 2: Design a predefined time sliding surface function;

[0013] Define position error

[0014] γ=q d -q

[0015] Design predefined time sliding mode variables

[0016]

[0017] The final predefined time sliding surface function is

[0018]

[0019] Where, s∈R n×1 a = diag{a1, a2, ... a n} represents the constant diagonal matrix to be designed; T f V1 is a predefined time constant during the sliding phase, β∈(0,1) is a constant parameter, π is pi, and V1 is a Lyapunov function.

[0020] Step 3: Establish a first-order all-drive system model;

[0021] Define speed error

[0022]

[0023] According to the all-wheel drive system approach, the first-order all-wheel drive system takes the following form:

[0024]

[0025] in, Then, we can get

[0026]

[0027] Where v is part of the controller to be designed, and A is a constant matrix to be designed.

[0028] Step 4: Design the reaching law

[0029] The boundary layer method is used to mitigate chattering caused by sliding mode control characteristics, resulting in the following piecewise function.

[0030]

[0031] Where sgn is the sign function and φ is the boundary layer thickness.

[0032] Step 5: Design of a predefined time sliding mode controller for the robotic arm system based on the all-drive system approach;

[0033] The following design (v) ensures that the robotic arm system can reach the sliding surface within a predefined time during the approach phase.

[0034]

[0035] Where T s It is a predefined time constant during the sliding phase, where ρ∈(0,1) is a constant parameter. It is s a The derivative of V2 with respect to time t is a Lyapunov function.

[0036] Substituting v and the designed predefined time sliding mode controller into the controller derived from the first-order all-drive system method, we obtain the following controller.

[0037]

[0038] Step 6: Stability analysis of the system during the approach phase

[0039] Based on Lyapunov stability theory, the following Lyapunov functions are selected.

[0040]

[0041] Where s∈R n×1 During the approach phase of the system, the derivative of V1 with respect to time is:

[0042]

[0043] V1 is positive definite. To make the system asymptotically stable and the system state approach the sliding surface in a predefined time, it is only necessary to make V1 positive definite.

[0044]

[0045] Since -H(s)s≤0, let Then one can obtain It can then be proven that, under the approach law of the designed all-drive system-based method, the system is stable in the arrival phase at a predefined time, and the predefined time is T. s .

[0046] Step 7: Stability analysis of the system during the sliding phase

[0047] Based on Lyapunov stability theory, the following Lyapunov functions are selected.

[0048]

[0049] Where γ is the defined position error, and when the system is on the sliding surface, s = 0, at which point the derivative of V2 with respect to time is...

[0050]

[0051] V2 is positive definite. To achieve asymptotic stability of the sliding surface and convergence of the system state to the equilibrium point within a predefined time, it is only necessary to make V2 positive definite.

[0052]

[0053] Due to matrix -aγ T γ≤0, thus obtaining At this point, it can be proven that under the designed sliding surface, the system is asymptotically stable and satisfies the designed predefined time performance index, that is, the system is stable at the predefined time T. f The inner edge of the sliding surface converges to the equilibrium point.

[0054] The features and beneficial effects of this invention are:

[0055] This invention addresses the shortcomings of existing robotic arm trajectory tracking control methods in controlling the robotic arm within a predefined time and the problem of chattering caused by neglecting sliding mode control characteristics. It presents a predefined time sliding mode control method based on a fully driven system approach. This invention considers external disturbances and transforms the robotic arm model into a first-order fully driven system using the fully driven system approach, providing designable degrees of freedom. Furthermore, a predefined time sliding mode controller is designed, enabling the robotic arm to converge within a predefined time even when the initial system state is unknown. Using this method, effective control of the robotic arm can be achieved even in the presence of external disturbances and with an unknown initial system state. Detailed Implementation

[0056] Step 1: Establish the dynamic model of the robotic arm system

[0057] First, based on the principles of dynamics, a system model with external disturbances was established:

[0058]

[0059] Where q = [q1, q2, ..., q n ] T This indicates the joint position of the robotic arm. This indicates the joint angular velocity of the robotic arm. Let M(q) represent the joint angular acceleration of the robotic arm, with the superscript T indicating the transpose of the matrix and the subscript n indicating the n joints of the robotic arm. n×n The inertia matrix of the robotic arm is represented. Let G(q) ∈ R represent the centrifugal force and Coriolis force of the robotic arm. n×1 Let τ ∈ R represent the gravitational term of the robotic arm. n×1 τ represents the control torque of the robotic arm. d ∈R n×1 This indicates that the external disturbance is bounded and satisfies Where a0, a1, and a2 are unknown positive constants, and ||·|| denotes the L2 norm. The symbol R represents the Euclidean space. n×n Let R represent an n×n dimensional matrix. n×1 This represents an n-dimensional column vector.

[0060] Step 2: Design of predefined time sliding surface functions

[0061] Define position error

[0062] γ=q d -q

[0063] Design predefined time sliding mode variables

[0064]

[0065] The final predefined time sliding surface function is

[0066]

[0067] Where, s∈R n×1 a = diag{a1, a2, ... a n} represents the constant diagonal matrix to be designed; T f V1 is a predefined time constant during the sliding phase, β∈(0,1) is a constant parameter, π is pi, and V1 is a Lyapunov function.

[0068] Step 3: Establishment of the first-order all-wheel drive system model

[0069] Define speed error

[0070]

[0071] According to the all-wheel drive system approach, the first-order all-wheel drive system takes the following form:

[0072]

[0073] in, Then, we can get

[0074]

[0075] Where v is part of the controller to be designed, and A is a constant matrix to be designed.

[0076] Step 4: Design of the reaching law

[0077] The boundary layer method is used to mitigate chattering caused by sliding mode control characteristics, resulting in the following piecewise function.

[0078]

[0079] Where sgn is the sign function and φ is the boundary layer thickness.

[0080] Step 5: Design of a predefined time sliding mode controller for the robotic arm system based on the all-drive system approach.

[0081] We design the following v to ensure that the robotic arm system can reach the sliding surface within a predefined time during the approach phase.

[0082]

[0083] Where T sIt is a predefined time constant during the sliding phase, where ρ∈(0,1) is a constant parameter. It is s a The derivative of V2 with respect to time t is a Lyapunov function.

[0084] Substituting v and the designed predefined time sliding mode controller into the controller derived from the first-order all-drive system method, we obtain the following controller.

[0085]

[0086] Step 6: Stability analysis of the system during the approach phase

[0087] Based on Lyapunov stability theory, the following Lyapunov functions are selected.

[0088]

[0089] Where s∈R n×1 During the approach phase of the system, the derivative of V1 with respect to time is:

[0090]

[0091] V1 is positive definite. To make the system asymptotically stable and the system state approach the sliding surface in a predefined time, it is only necessary to make V1 positive definite.

[0092] Since -H(s)s≤0, let In turn, one can obtain It can then be proven that, under the approach law of the designed all-drive system-based method, the system is stable in the arrival phase at a predefined time, and the predefined time is T. s .

[0093] Step 7: Stability analysis of the system during the sliding phase

[0094] Based on Lyapunov stability theory, the following Lyapunov functions are selected.

[0095]

[0096] Where γ is the defined position error, and when the system is on the sliding surface, s = 0, at which point the derivative of V2 with respect to time is...

[0097]

[0098] V2 is positive definite. To achieve asymptotic stability of the sliding surface and convergence of the system state to the equilibrium point within a predefined time, it is only necessary to make V2 positive definite.

[0099]

[0100] Due to matrix -aγ T γ≤0, and thus we obtain At this point, it can be proven that under the designed sliding surface, the system is asymptotically stable and satisfies the designed predefined time performance index, that is, the system is stable at the predefined time T. f The inner edge of the sliding surface converges to the equilibrium point.

Claims

1. A predefined time sliding mode control method for a robotic arm trajectory tracking system, characterized in that, The method specifically includes the following steps: Step 1: Establish the dynamic model of the robotic arm system Step 2: Design a predefined time sliding surface function Define position error Design predefined time sliding mode variables in, It is a predefined time constant during the sliding phase. It is a constant parameter. Pi It is a Lyapunov function; The final time-defined sliding surface function is in, ,and It is positive definite; Step 3: Establish a first-order all-drive system model Define speed error ； For the desired trajectory, It's about time. A twice continuously differentiable function; According to the all-wheel drive system approach, the first-order all-wheel drive system takes the following form: in, Then, we get in It is part of the controller to be designed. It is a constant matrix to be designed; These are the position, angular velocity, and angular acceleration of the robotic arm; The inertia matrix of the robotic arm is represented. This represents the centrifugal force and Coriolis force of the robotic arm. For the mechanical arm's gravity term, To control the torque, For external disturbances, satisfy ,in , and It is an unknown positive constant. Represents the L2 norm; Step 4: Design the reaching law The boundary layer method is used to mitigate chattering caused by sliding mode control characteristics, resulting in the following piecewise function. in, For symbolic functions, Boundary layer thickness; Step 5: Design of a predefined time sliding mode controller for the robotic arm system based on the all-drive system approach The design is as follows: To ensure that the robotic arm system can reach the sliding surface within a predefined time during the approach phase. in It is a predefined time constant during the sliding phase. It is a constant parameter. yes Regarding time The derivative, It is a Lyapunov function; Will The predefined time sliding mode controller is then incorporated into the controller derived from the first-order all-drive system method, resulting in the following controller. 。 2. The predefined time sliding mode control method for a robotic arm trajectory tracking system according to claim 1, characterized in that: The aforementioned establishment of the dynamic model of the robotic arm system is as follows: 。 3. The predefined time sliding mode control method for a robotic arm trajectory tracking system according to claim 1, characterized in that: It also includes stability analysis of the system during the approach phase, specifically: Based on Lyapunov stability theory, the following Lyapunov functions are selected. in When the system is in the approach phase, The derivative with respect to time is It is positive definite. To make the system asymptotically stable and for the system state to approach the sliding surface in a predefined time, it is only necessary to make... ; because Let again Therefore, we can obtain At this point, it can be proven that under the approach law of the designed all-drive system-based method, the system is stable in the arrival phase at a predefined time, and the predefined time is... .

4. The predefined time sliding mode control method for a robotic arm trajectory tracking system according to claim 1, characterized in that: It also includes stability analysis of the system during the sliding phase; specifically: Based on Lyapunov stability theory, the following Lyapunov functions are selected. in, This is the defined positional error when the system is on the sliding surface. ,at this time The derivative with respect to time is It is positive definite. To achieve asymptotic stability of the sliding surface and convergence of the system state to the equilibrium point within a predefined time, it is only necessary to make... ; Due to the matrix And thus obtain At this point, it can be proven that under the designed sliding surface, the system is asymptotically stable and meets the designed predefined time performance index, that is, the system is stable within the predefined time. The inner edge of the sliding surface converges to the equilibrium point.

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