Self-adaptive trajectory tracking control method for mechanical arm

Abstract

The invention discloses a self-adaptive trajectory tracking control method for a mechanical arm, which is used for solving the control problem of a mechanical arm system with time-varying state constraint, external disturbance and actuator fault, and comprises the following steps of: proposing to combine a preset performance function and a logarithmic obstacle Lyapunov function, the time-varying full-state constraint problem of the system is solved; secondly, designing a disturbance observer with boundary characteristics to approach lumped interference, designing an actual controller in combination with a backstepping method, and meanwhile, designing a compensation variable to compensate a first-order filter error; finally, on the basis of the Cruz-Ortiz theory, it is proved that the system still meets the global fixed time stability condition under the time-varying constraint; the method is of great significance to the control problem of the mechanical arm with the time-varying state constraint and the actuator fault.

Description

Technical Field

[0001] The present application relates to a trajectory tracking control method for a robotic arm, and more specifically, to a trajectory tracking control method for a robotic arm subject to time-varying state constraints, actuator faults, and external disturbances. Background Art

[0002] Robotic arms play an important role in modern industrial production. Robotic arms are required to complete tasks in scenarios such as loading and unloading machine tools and automobile production and manufacturing. Traditional control methods such as PID control and computed torque control can solve simple control problems, but they cannot maintain good dynamic performance and disturbance rejection ability. Therefore, it is necessary to improve the traditional methods or design new methods. In the prior art, although there are many control methods for solving state constraints and actuator faults, most of them focus on solving single faults or fixed constraint conditions. In addition, the existing methods still need to be improved in terms of computational complexity and stability. Summary of the Invention

[0003] The present application proposes an adaptive trajectory tracking control method for a robotic arm using a time-varying barrier Lyapunov function, which can achieve fixed-time stability of the system and high-precision tracking of the target trajectory under complex working conditions where the robotic arm simultaneously has unknown disturbances, actuator faults, and time-varying state constraints.

[0004] To solve the above technical problems, the technical solution of the present application is as follows:

[0005] 1. Propose to combine a preset performance function and a logarithmic barrier Lyapunov function to solve the time-varying full-state constraint problem of the system.

[0006] 2. Design a disturbance observer with boundary constraint characteristics to approximate the lumped disturbance (including external disturbances, actuator faults, and unknown non-linear functions), which significantly improves the robustness and anti-disturbance ability of the system.

[0007] 3. Combine the disturbance observer and the backstepping method to recursively obtain the actual controller, and design a compensation variable to offset the influence of the error introduced by the first-order filter.

[0008] 4. Based on the Cruz-Ortiz theory, prove that the system still satisfies global fixed-time stability under time-varying constraints.

[0009] The present application provides an adaptive trajectory tracking control method for a robotic arm, which includes the following steps:

[0010] A1. Combine the actuator fault and the state constraint to establish a new robotic arm system;

[0011] A2. Use a disturbance observer with boundary characteristics to estimate the lumped disturbance existing in the new robotic arm system;

[0012] A3. Generate an adaptive controller based on the time-varying logarithmic barrier Lyapunov function and the disturbance observer, and control the new robotic arm system so that the trajectory tracking error of the robotic arm converges to the set steady-state accuracy within a fixed time.

[0013] In some embodiments, in step A1, the steps of establishing the new robotic arm system include:

[0014] Introduce a robotic arm system, whose dynamic equation is:

[0015]

[0016] The system state is defined as Define the control input u = V e , where q represents the angular displacement of the robotic arm, I represents the motor current, V e represents the input voltage, J represents the motor rotor inertia, K τ represents the torque coefficient, m represents the mass of a single rod, M0 represents the load mass, L0 represents the length of a single rod, G represents the gravity, B0 represents the viscous friction coefficient, L represents the armature inductance, K B represents the back electromotive force coefficient, and R represents the armature resistance;

[0017] Combining the external disturbance and actuator fault, simplifying the robotic arm system (33) gives:

[0018]

[0019] where g2 and g3 are regarded as known constants, f( x 2), f( x 3) are unknown nonlinear function terms, and d2 and d3 represent external disturbances;

[0020] Set the state constraint condition as:

[0021] M = {x i ∈R, |x i | < k ci (t), i = 1....3} (3)

[0022] where k ci is a nonlinear time-varying function, y represents the control output of the system, and v represents the actuator fault, and its form is as follows:

[0023] v = δu + ω (4)

[0024] where u represents the actual input of the system, 0 < δ < 1, and ω represents the unknown bias disturbance of the actuator;

[0025] Transforming and arranging the robotic arm system (34) gives the new robotic arm system, and its equations are as follows:

[0026]

[0027] Among them, the lumped disturbance D2 = f2( x 2) + d2, D3 = (δ - 1)u + f3( x 3) + d3 + ω.

[0028] In some embodiments, step A2 further includes:

[0029] Designing a disturbance observer based on a boundary function as:

[0030]

[0031] i = 2, 3, e i = x i - v i , κ 1i > 0, κ 2i > 0, 0 < γ1 < 1, 2 > γ2 > 1; is a boundary function, and its design form is as follows:

[0032]

[0033] Among them, ε i is a designed extremely small positive number;

[0034] Using the disturbance observer, the lumped disturbance term in the new robotic arm system can be estimated, such that: And the estimation error can converge to 0 within a fixed time, and the upper bound of the convergence time is computable and only affected by the set parameters.

[0035] In some embodiments, step A3 includes:

[0036] A31. Designing multiple virtual controllers according to a time-varying logarithmic barrier Lyapunov function;

[0037] A32. Designing the adaptive controller according to the virtual controller and the disturbance observer;

[0038] A33. The adaptive controller performs trajectory tracking control on the new robotic arm system.

[0039] In some embodiments, step A31 includes:

[0040] Designing a time-varying logarithmic barrier Lyapunov function as:

[0041]

[0042] wherein is the set performance function, η i0 > η i1 > 0, a i > 0, and by adjusting the parameters, holds at any time, wherein rd is the set desired trajectory; is the output result of the virtual controller α 2,3 after passing through a first-order filter;

[0043] In some embodiments, design represents the estimation error. Since both λ2 and λ3 are close to the minimum value of 0, it can be known that there exists a positive real number Γ such that Γ > |max(λ2, λ3)| holds;

[0044] Generate the first virtual controller in combination with the time-varying logarithmic barrier Lyapunov function:

[0045]

[0046] wherein

[0047]

[0048] Design the first-order filter as:

[0049]

[0050] where the virtual controller α i is the input, τ i is a set positive real number, is the filter output, and finally r1 is expressed as the compensation variable of the filter and is designed as:

[0051]

[0052] In some embodiments, generate the second virtual controller in combination with the time-varying logarithmic barrier Lyapunov function:

[0053]

[0054] wherein,

[0055]

[0056] The corresponding r2 compensation variable is designed as:

[0057]

[0058] In some embodiments, step A32 includes:

[0059] Design the adaptive controller according to the virtual controller and the disturbance observer:

[0060]

[0061] where The corresponding performance function is designed as: Beneficial effects

[0062] 1. When there are complex working conditions such as time-varying state constraints, actuator failures, and external disturbances in the system, global fixed-time stability is still achieved, and the upper bound of the convergence time only depends on the preset parameters, improving the applicability and stability of the control algorithm.

[0063] 2. The control method only has one adaptative variable updated in real time, streamlining the operation process, reducing the computational load, and being suitable for use in real-time control scenarios.

[0064] 3. By designing a preset performance function and a disturbance observer, the tracking error is strictly constrained within a preset range, significantly enhancing the dynamic response speed and anti-interference ability of the system. Brief description of the drawings

[0065] Figure 1 is the flowchart of the adaptive control method designed in this application;

[0066] Figure 2 is Figure 1 the specific flowchart described in step 1 in

[0067] Figure 3 is Figure 1 the specific flowchart described in step 2 in

[0068] Figure 4 is Figure 1 the specific flowchart described in step 3 in

[0069] Figure 5 shows the tracking effect of the angular displacement of the robotic arm and the set reference trajectory, intuitively demonstrating key performance indicators such as the convergence speed and tracking accuracy of the system;

[0070] Figure 6 are respectively the change trajectory of the angular velocity of the robotic arm and the fluctuation curve of the motor current over time, and both satisfy the condition of time-varying constraints at any moment;

[0071] Figure 7 is the trajectory curve graph of the tracking error of each state variable, verifying that the method of this application has significant advantages in convergence speed and control accuracy;

[0072] Figure 8 Shows the locus curve of the input voltage, whose amplitude is affected by the partial failure of the actuator but still remains bounded. Detailed implementation

[0073] The present application will be described in detail below with reference to the accompanying drawings and in conjunction with embodiments. Each example is provided by way of explanation of the present application rather than limiting the present application. In fact, those skilled in the art will appreciate that modifications and variations can be made to the present application without departing from the scope or spirit thereof. For example, features shown or described as part of one embodiment can be used in another embodiment to yield yet another embodiment. Accordingly, it is intended that the present application cover such modifications and variations that fall within the scope of the appended claims and their equivalents.

[0074] In the description of the present application, the orientation or positional relationships indicated by the terms "longitudinal", "transverse", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", etc. are based on the orientation or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present application rather than requiring the present application to be constructed and operated in a specific orientation. Therefore, it should not be construed as a limitation of the present application. The terms "connected", "connected to", and "disposed" used in the present application should be understood in a broad sense. For example, it can be a fixed connection or a detachable connection; it can be directly connected or indirectly connected through an intermediate component; it can be a wired connection, a radio connection, or a wireless communication signal connection. For those of ordinary skill in the art, the specific meanings of the above terms can be understood according to specific circumstances.

[0075] One or more examples of the present application are shown in the accompanying drawings. The detailed description uses numerical and alphabetical labels to refer to features in the drawings. Similar or like labels in the drawings and the description have been used to refer to similar or like parts of the present application. As used herein, terms such as "first", "second", and "third" can be used interchangeably to distinguish one component from another and are not intended to indicate the position or importance of individual components.

[0076] The implementation process of the present application will be further described in detail below in conjunction with the Figure 1 design flow chart. The robotic arm adaptive trajectory tracking control method of the present application includes the following steps:

[0077] Step 1: Combine the preset performance function as a logarithmic barrier Lyapunov function.

[0078] Step 2: Design a disturbance observer with boundary characteristics, and then combine with the backstepping method to obtain the actual controller, and design a compensation variable to offset the error introduced by the first-order filter.

[0079] Step 3: Prove that the system satisfies global fixed-time stability under time-varying constraints.

[0080] According to Figure 2 it can be known that the specific steps of Step 1 are as follows:

[0081] Step 1.1: Describe the robotic arm system, actuator faults, and state constraints

[0082] For a class of single-link flexible robotic arm systems with external disturbances, state constraints, and actuator faults, the system state is defined as Define the control input u = V e , where q represents the angular displacement of the robotic arm, I represents the motor current, V e represents the input voltage, and the dynamic equation of the robotic arm system is as follows:

[0083]

[0084] where J represents the inertia of the motor rotor, K τ represents the torque coefficient, m represents the mass of the single rod, M0 represents the load mass, L0 represents the length of the single rod, G represents the gravity, B0 represents the viscous friction coefficient, L represents the armature inductance, K B represents the back electromotive force coefficient, and R represents the armature resistance.

[0085] Finally, assume that there are external disturbances and actuator faults in the robotic arm system. Simplifying equation (1) gives:

[0086]

[0087] where g2 and g3 are regarded as known constants, f( x 2), f( x 3) are unknown nonlinear function terms, d2 and d3 represent external disturbances, and the state constraint conditions are set as:

[0088] M = {x i ∈ R, |x i | < k ci (t), i = 1....3} (17)

[0089] where k ci is a nonlinear time-varying function, y represents the control output of the system, v represents the actuator fault, and its fault form is as follows:

[0090] v = δu + ω (18)

[0091] where \(u\) represents the actual input of the system. When \(\delta = 0\), the actuator fails completely; when \(\delta = 1\), the actuator is intact; when \(0\lt\delta\lt1\), the actuator fails partially. This application studies the case when the actuator fails partially, and \(\delta\) is assumed to be an unknown and slowly time-varying real number. \(\omega\) represents the additive unknown disturbance of the actuator.

[0092] Performing transformation and arrangement on (2) gives:

[0093]

[0094] where the lumped disturbance \(D_2 = f_2(x_2)+d_2\), \(D_3 = (\delta - 1)u + f_3(x_3)+d_3+\omega\).

[0095] The coordinate transformation of the system is:

[0096]

[0097] where \(r_d\) is the set desired trajectory. is the output result of the virtual controller \(\alpha\) 2,3 after passing through a first-order filter.

[0098] Step 1.2: Set the preset performance function as the barrier Lyapunov function boundary

[0099] To satisfy the time-varying full-state constraint condition, a logarithmic barrier Lyapunov function is introduced, and its form is as follows:

[0100]

[0101] where is the set performance function, \(\eta\) i0 \(\gt\eta\) i1 \(\gt0\), \(a\) i \(\gt0\), \(t\) represents time. By adjusting the parameters, holds at any time, and \(V\) i is continuously differentiable in the set \(\Omega=\{z\) i \(\lt k\) bi \}\). Equation (7) confines the system error \(z\) i within the time-varying performance function \(k\) bi , and \(\eta\) i0 , \(\eta\) i1 respectively determine the initial upper bound and the upper bound after stabilization of \(z\) i .

[0102] According to Figure 3 it can be known that the specific steps of Step 2 are as follows:

[0103] Step 2.1: Design a boundary function to construct a disturbance observer

[0104] According to the system model, the disturbance observer based on the boundary function is designed as follows:

[0105]

[0106] i = 2, 3, e i = x i - v i , κ 1i > 0, κ 2i > 0, 0 < γ1 < 1, 2 > γ2 > 1, is the boundary function, and its design form is as follows:

[0107]

[0108] where ε i is the designed extremely small positive number.

[0109] Construct the auxiliary variable Assume that at a certain moment |e i | > σ i , and select the Lyapunov function as:

[0110]

[0111] Take the derivative of it with respect to time to get:

[0112]

[0113] Also because It can be seen from this that holds, so it can be further obtained that:

[0114]

[0115] where, According to the fixed-time stability theorem, it can be known that the disturbance observer can converge to the extremely small neighborhood of [-σ i , σ i within a fixed time, i = 2, 3. And the upper bound T k of the convergence time can be calculated as:

[0116]

[0117] It can be seen from this that if the observer is used to observe the lumped disturbance component of the system. It includes external disturbances, actuator faults, and unknown nonlinear functions. In the case of unknown upper bounds of the lumped disturbance, the observer error e i can converge to the extremely small neighborhood of 0, [-ε i , ε i , i = 2, 3, within a fixed time. The upper bound of the convergence time can be calculated and is only affected by the set parameters.

[0118] When the observer error e i converges to the range of [-ε i , ε i , the following can be obtained:

[0119]

[0120] Substitute into the manipulator model and simplify to get:

[0121]

[0122] where represents the error between the lumped disturbance and the boundary function. And since both λ2 and λ3 are close to the minimum value of 0, it can be known that there exists a positive real number Γ such that Γ > |max(λ2, λ3)| holds.

[0123] Step 2.2: Design the virtual controller and compensation variables step by step through the disturbance observer and the backstepping method

[0124] To design the virtual controller for the first step, first pass through the first-order filter:

[0125]

[0126] where the virtual controller α i is the input, and τ i is a set positive real number. is the output of the filter.

[0127] According to Equation (6), take the derivative of the error z1 with respect to time and substitute it into the model to get:

[0128]

[0129] To cancel the error brought by the first-order filter introduce the compensation variable r1 for z1, and its derivative form is as follows:

[0130]

[0131] where c1 is a set positive real number.

[0132] Define Take the derivative of to get:

[0133]

[0134] Select the Lyapunov function and replace z1 in V1 with and take the derivative to get:

[0135]

[0136] in Notice is the compensation for z1. Through the compensation mechanism, the error z1 is driven closer to the origin neighborhood, so that the control accuracy of the system is further improved. Replace z1 with The filtering error is offset, the initial value of r1 is 0 and the system will tend to be within a very small neighborhood of the origin after it stabilizes.

[0137] The designed virtual controller α2 is:

[0138]

[0139] in Substituting the virtual controller into the Lyapunov function V1 yields:

[0140]

[0141] Design the virtual controller for step 2, define Construct the compensation variable r in step 2 i for:

[0142]

[0143] Select the Lyapunov function V i for:

[0144]

[0145] in The designed virtual controller α3 is:

[0146]

[0147] in Substitute it into the Lyapunov function V i And the derivative is:

[0148]

[0149] in

[0150]

[0151] Step 2.3: Obtain an adaptive controller by back-stepping the virtual controller

[0152] definition Since there is no filter error in z3, no compensation processing is required for it, and therefore no compensation variable is required.

[0153] Select the Lyapunov function V3 as

[0154]

[0155] where Design the adaptive controller u as:

[0156]

[0157] where Substitute the adaptive controller into the designed Lyapunov function and take the derivative and simplify to obtain:

[0158]

[0159] where

[0160]

[0161] According to Figure 4 It can be seen that the specific steps of step 3 are as follows:

[0162] Step 3.1: Substitute the adaptive controller u into the time-varying barrier Lyapunov function V designed in step 1 and take the derivative

[0163] Select the Lyapunov function shown below By selecting reasonable parameters such that always holds.

[0164] By referring to the theory of scholar Cruz-Ortiz, for any constant ρ with |ρ| < 1, the following inequality holds:

[0165] -[ρ 2 / (1 - ρ 2 )] ≤ -log(1 / [1 - ρ 2 )] (44)

[0166] Let and substitute it into the derivative of the Lyapunov function V to obtain:

[0167]

[0168] Step 3.2: According to the theory of scholar Cruz-Ortiz, the system is stable in a fixed time under state constraints

[0169] It can be seen that V is stable in a fixed time, and the upper bound T of the total convergence time of the system can be calculated as:

[0170]

[0171] Example: The simulation experiment of the robotic arm model (1) was carried out using the MATLAB simulation software, and the effectiveness of the proposed control method was verified. The experiment is as follows:

[0172] Set \(g_2 = 2\), \(g_3 = 10\), At the same time, set the state constraint conditions as:

[0173] \(\vert x\) i \(\vert\lt k\) ci , \(i = 1\cdots n\), \(\delta=0.8 - 0.1\cos(t)\), \(f( x 3)=0.18x^2 + 0.001x^3\), \(k c1 =0.7+(0.2e -t +0.1)\), \(k c2 =2.0e -t +0.8\), \(k c3 =1.5 + 5e -t . Set the reference signal trajectory as: \(r_d = 0.5\sin(t)+0.5\sin(0.5t)\). The corresponding control parameters are \(\eta 10 =\eta 20 =1\), \(\eta 11 =0.5\), \(\eta 21 =0.2\), the initial state of the system is \([0,0,0.05]\), \(f( x 3)+d_3 = 2x^2 - x^3\kappa 1i =0.5\), \(\varepsilon i =0.02\), \(\eta 30 =3\), \(\eta 31 =1\), \(\gamma_1 = 0.7\), \(\gamma_2 = 1.7w 21 =w 22 =w 31 =w 32 =10\), \(\tau_1=\tau_2 = 0.02\), \(p 1i =0.7\), \(p 2i =1.5\), \(a i =1\), \(c_2 = 30\), \(c_1 = 10\).

[0174] This example combines the robotic arm model with time-varying state constraints, actuator faults, and external disturbances for evolutionary simulation and experimental verification. It can be seen from the simulation image 5 that the output \(y\) quickly and accurately tracks the reference signal \(r_d\). Combining with Figure 6 it can be known that the system satisfies the full-state time-varying constraint conditions of \(\vert x i \vert\lt k ci , \(i = 1\cdots3\). Figure 7 These are the corresponding errors of each system state. According to the image, all errors do not exceed the specified performance function, that is: \(z i \lt k bi .Figure 8 is the actual controller input voltage V of the system e . Experiments show that this method makes the tracking error z1 converge to 10 within t = 2 s -3 order of magnitude, and the overshoot is less than 2%. It shows that the method significantly improves the robustness of the system against actuator faults and external disturbances.

[0175] The above are only some embodiments of the present application and are not intended to limit the present application. For those skilled in the art, various changes and modifications can be made to the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included within the protection scope of the present application.

Claims

1. A method for adaptive trajectory tracking control of a robotic arm, characterized in that: It includes the following steps: A1. Combine actuator faults and state constraints to establish a new robotic arm system; A2. Use a disturbance observer with boundary characteristics to estimate the lumped disturbance existing in the new robotic arm system; A3. Generate an adaptive controller according to the time-varying logarithmic barrier Lyapunov function and the disturbance observer; and control the new robotic arm system so that the trajectory tracking error of the robotic arm converges to the set steady-state accuracy within a fixed time.

2. The control method according to claim 1, characterized in that: In the step A1, the steps of establishing a new robotic arm system include: Introduce a robotic arm system, and its dynamic equation is: The system state is defined as Define the control input u = V e , where q represents the angular displacement of the robotic arm, I represents the motor current, V e represents the input voltage, J represents the inertia of the motor rotor, K τ represents the torque coefficient, m represents the mass of a single rod, M0 represents the load mass, L0 represents the length of a single rod, G represents the gravity, B0 represents the viscous friction coefficient, L represents the armature inductance, K B represents the back electromotive force coefficient, and R represents the armature resistance; Combined with external disturbances and actuator faults, the robotic arm system (33) can be simplified to obtain: Among them, g2 and g3 are regarded as known constants, f(x2) and f(x3) are unknown nonlinear function terms, and d2 and d3 represent external disturbances; Set the state constraint condition as: M = {x i ∈R, |x i | < k ci (t), i = 1....3} (49) where k ci is a non-linear time-varying function, y represents the control output of the system, and v represents the actuator fault, which is in the following form: v = δu + ω (50) Among them, u represents the actual input of the system, 0 < δ < 1, and ω represents the unknown bias disturbance of the actuator; Transform and organize the robotic arm system (34) to obtain the new robotic arm system, and its equation is as follows: Among them, the lumped disturbance D2 = f2(x2) + d2, D3 = (δ - 1)u + f3(x3) + d3 + ω.

3. The control method according to claim 2, characterized in that: The step A2 further includes: Design a disturbance observer based on a boundary function as: 0 < γ1 < 1, 2 > γ2 > 1; is a boundary function, and its design form is as follows: where ε i is a very small positive number designed; The lumped disturbance term in the new robotic arm system can be estimated by using the disturbance observer, such that: and the estimation error can converge to 0 within a fixed time, and the upper bound of the convergence time is computable and only affected by the set parameters.

4. The control method according to claim 3, characterized in that: The step A3 includes: A31. Design multiple virtual controllers according to the time-varying logarithmic barrier Lyapunov function; A32. Design the adaptive controller according to the virtual controller and the disturbance observer; A33. The adaptive controller performs trajectory tracking control on the new robotic arm system.

5. The control method according to claim 4, characterized in that: The step A31 includes: Design a time-varying logarithmic barrier Lyapunov function as: where is the set performance function, η i0 > η i1 > 0, a i > 0, and by adjusting the parameters, holds at any time. where rd is the set desired trajectory; is the output result of the virtual controller α 2,3 after passing through a first-order filter.

6. The control method according to claim 5, characterized in that: Design Denote the estimation error. Since both λ2 and λ3 are close to the minimum value of 0, it can be seen that there exists a positive real number Γ such that Γ > |max(λ2, λ3)| holds; Combine the time-varying logarithmic barrier Lyapunov function to generate the first virtual controller: wherein p 21 > 1, w 11 > 0, w 12 > 0, η 10 > η 11 > 0, a1 > 0, design a first-order filter as follows: where the virtual controller α i is the input, τ i is a set positive real number, is the filter output, and finally r1 is expressed as the compensation variable of the filter and is designed as:

7. The control method according to claim 6, characterized in that: Combine the time-varying logarithmic barrier Lyapunov function to generate the second virtual controller: Among them, η 20 > η 21 > 0, a2 > 0, and the corresponding r2 compensation variable is designed as:

8. The control method according to claim 7, characterized in that: The step A32 includes: Design the adaptive controller according to the virtual controller and the disturbance observer: Among them The corresponding performance function is designed as follows: η 30 > η 31 > 0, a3 > 0.

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