A Global Finite-Time Motion Control Method for Robots Based on Adaptive Composite Observer

Abstract

This invention belongs to the field of robot motion control technology and discloses a global finite-time motion control method for robots based on an adaptive composite observer. The method includes: Step 1: Designing an adaptive composite observer based on finite-time sliding mode manifolds to comprehensively compensate for uncertainties in robot kinematics and dynamics; Step 2: Then, designing a non-singular end-effector sliding mode motion controller based on an adaptive neural network and a preset performance error affine design to directly stabilize motion errors in the task space; Step 3: Finally, achieving global finite-time stability of the entire observer-control system using the Lyapunov design method, ensuring that observer errors and motion control errors converge rapidly to zero within a finite time, while satisfying the system's preset transient and steady-state response characteristics. This invention can rapidly converge robot end-effector motion errors to zero within a finite time, effectively improving robot efficiency and execution accuracy.

Description

Technical Field

[0001] This invention belongs to the field of robot motion control technology, and particularly relates to a global finite-time motion control method for robots based on an adaptive composite observer. Background Technology

[0002] Currently, most robotic applications are defined within a task space, such as Cartesian space or image space, and are always simultaneously affected by uncertain kinematics and dynamics. While joint space controllers can perform most tasks well by identifying and calibrating kinematic and dynamic parameters, they cannot cope with changes and uncertainties in the robot model and operating environment. Furthermore, when robots manipulate objects with various shapes, sizes, and mass properties, it is impossible to identify the accurate parameters of all targets in advance. Just as humans can easily manipulate any unknown object without precise kinematic and dynamic knowledge of the arm and the target object, relying on their visual perception and self-learning abilities, developing advanced task space motion controllers to address the uncertainties of robot kinematics and dynamics and directly stabilize end-effector tracking errors is crucial for achieving intelligent interaction and operation.

[0003] Currently, many advanced control technologies are used to address uncertainties in robot dynamics, including active disturbance rejection control (ADRC), observer-based robust control, adaptive neural network control, and variable structure sliding mode control. However, these methods can only guarantee that the robot's motion control error asymptotically converges to zero or a bounded interval, and cannot meet the high-precision tracking requirements of real-time tasks. Therefore, finite-time convergence control technologies have been gradually developed and applied, such as H... ∞ While control technologies exist, they all assume that the robot's kinematic parameters can be precisely obtained in advance to transform the task space trajectory from the position level to the joint space via inverse kinematics. However, existing control technologies aimed at stabilizing robot kinematic and dynamic uncertainties, such as approximate Jacobian matrix transpose feedback control and data-driven learning control, also suffer from problems such as the difficulty in calculating the precise linear regression matrix for uncertain dynamics. Furthermore, they do not simultaneously consider the steady-state and transient response characteristics during motion control, failing to guarantee the robot's work efficiency and execution accuracy in practical engineering. Summary of the Invention

[0004] The purpose of this invention is to provide a global finite-time motion control method for robots based on an adaptive composite observer, so as to solve the above-mentioned technical problems.

[0005] To address the aforementioned technical problems, the specific technical solution of the robot global finite-time motion control method based on an adaptive composite observer of the present invention is as follows:

[0006] A global finite-time motion control method for a robot based on an adaptive composite observer includes the following steps:

[0007] Step 1: Based on the popular design of finite-time sliding mode, an adaptive composite observer is designed to comprehensively compensate for the uncertainties in robot kinematics and dynamics;

[0008] Step 2: Then, based on the adaptive neural network and the preset performance error affine design, a non-singular terminal sliding mode motion controller is designed to directly stabilize the motion error in the task space;

[0009] Step 3: Finally, the global finite-time stability of the entire observation-control system is achieved through the Lyapunov design method, so that the observer error and motion control error converge to zero quickly within a finite time and satisfy the system's pre-set transient and steady-state response characteristics.

[0010] Furthermore, step 1 includes the following specific steps:

[0011] Constructing the kinematic and dynamic models of the robot:

[0012]

[0013]

[0014] In formula (1)-(2) Represents the velocity of movement within the mission space. and These represent the joint's spatial position and velocity, respectively. Represents the Jacobian matrix. Represents kinematic parameters, Represents the kinematic linear regression matrix; Represents the acceleration of joint spatial motion. These represent the inertia matrix, the Coriolis force and centrifugal force matrices, and the gravitational torque vector, respectively. Represents an unknown torque. It is the joint control torque;

[0015] Define an auxiliary regression matrix and two vectors as follows:

[0016]

[0017] In formula (3) This represents the kinematic regression matrix after first-order filtering. This represents the end position after filtering, with constant coefficients ω1 > 0 and ω2 > 0. These are estimated values ​​of kinematic parameters; known. Equivalent to parameter estimation error

[0018] Based on the tracking error e1=xx of the robot's end effector d The nominal expected velocity of the robot joint space. and acceleration They are respectively

[0019]

[0020]

[0021] In formula (5)-(6) and Let represent the nominal expected velocity and acceleration of the robot's end effector within the task space, respectively. and These represent the actual pose and the desired pose of the end effector, respectively. Based on this definition, the joint space velocity tracking error is... To design the unknown torque τ based on the kinematic estimation results d The observer, let η = M(q)e2, define an auxiliary system variable as

[0022]

[0023] In formula (7), ω3 > 0, and then an adaptive finite-time sliding mode observer for uncertain dynamics is designed.

[0024]

[0025] In formula (8) and Representing τ d And the estimated value of ξ, β d Represents τ d The supremum of the time derivative is defined by the gain coefficients γ1, γ2, and α1 all being greater than zero, where 0 < p < 1; this algorithm can mitigate uncertain dynamic observation errors. It converges to zero within a finite amount of time.

[0026] Furthermore, step 1 includes an algorithm for estimating the robot's kinematic parameters, the algorithm steps of which are as follows:

[0027]

[0028] In formula (4) Representing the positive definite gain matrix, this algorithm reduces the kinematic parameter estimation error. It converges to zero within a finite amount of time.

[0029] Furthermore, step 2 includes the following specific steps:

[0030] To meet the pre-defined tracking response characteristics -δ imin μ i (t)<e 1i (t)<S imax μ i (t), t≥0, design the equivalent affine term for motion control error e1. for

[0031]

[0032] In formula (9), Φ i (χ 1i )=(δ imax exp(χ 1i )-δ imin exp(-χ 1i )) / (exp(χ 1i )+exp(-χ 1i )) is an increasing function, and the error response characteristic function μ i (t)=(μ i0 -μ i∞ )exp(-σ i t)+μ i∞ ,

[0033] Where the constant coefficient δ imin S imax σ i μ i0 μ i∞ All are greater than zero;

[0034] Define an RBF neural network satisfy

[0035] W T Ω(z)+∈(z)=f(z)#(10)

[0036] In formula (10), f(z) represents the dynamic parameter perturbation and random error term. It is the input to the neural network. The expected weights and the number of hidden layer neurons are c. It is a Gaussian radial basis function vector. It is a bounded error vector that satisfies

[0037] Design the terminal sliding surface s(t),

[0038]

[0039] In formula (11), Λ=diag{Λ1,...,Λ n} represents a positive definite diagonal matrix, 1 < κ < 2. Based on this, the non-singular terminal sliding mode motion controller for the robot is designed as follows:

[0040]

[0041]

[0042]

[0043] And design an adaptive update law for the weights of the RBF neural network.

[0044]

[0045] In formula (12), Π2(e2)=diag{||e 21 || 1-κ ,…,||e 2n || 1-κ The coefficients κ0, κ1, κ2, κ3, and κ4 are all positive. It is an estimate of W and In formula (13) This represents the positive definite gain matrix; the algorithm enables the robot motion error e1 to converge to zero within a finite time while satisfying the pre-set tracking response performance.

[0046] Furthermore, step 3 includes the following specific steps:

[0047] The stability of the entire observation-control system is analyzed using the Lyapunov design method. First, focusing on the kinematic parameter estimation algorithm formula (4), when... Sometimes Zhengding, De Established, when When designing Lyapunov functions And derive

[0048]

[0049] In formula (14) Obtain the estimation error of robot kinematic parameters Satisfies global finite-time stability;

[0050] Then, regarding the uncertain dynamics observer formula (8) and the non-singular terminal sliding mode motion controller formulas (11) to (13), since... Design Lyapunov functions And derive

[0051]

[0052] In formula (15) Get s(t), It will converge to the bounded interval within a finite time.

[0053] Finally, design the Lyapunov function. And derive

[0054]

[0055] In formula (16) Obtain the error of the robot dynamics observer Both the tracking error e1 in the task space and the tracking error e1 will converge to zero in a finite amount of time.

[0056] The robot global finite-time motion control method based on adaptive composite observer of the present invention has the following advantages: The robot global finite-time motion control method based on adaptive composite observer of the present invention can quickly converge the robot end motion error to zero within a finite time, realize comprehensive compensation for the kinematic and dynamic uncertainties of the robot and its working environment, and at the same time meet the system transient and steady-state response characteristics preset in actual engineering, and ultimately effectively improve the robot's working efficiency and execution accuracy. Attached Figure Description

[0057] Figure 1 This is a flowchart of the robot global finite-time motion control method based on an adaptive composite observer according to the present invention.

[0058] Figure 2 This is a schematic diagram illustrating the robot end-effector trajectory tracking effect of the present invention;

[0059] Figure 3 This is a schematic diagram illustrating the convergence effect of the robot's end effector motion error according to the present invention. Detailed Implementation

[0060] To better understand the purpose, structure, and function of this invention, the following detailed description of a robot global finite-time motion control method based on an adaptive composite observer, with reference to the accompanying drawings, is provided.

[0061] This invention comprehensively observes and compensates for uncertainties in robot kinematics and dynamics, and then designs a non-singular end-effector sliding mode motion controller based on a preset performance error affine and adaptive neural network. This enables the global finite-time rapid convergence of end-effector motion errors and satisfies preset system transient and steady-state response constraints, which can effectively improve the robot's working efficiency and execution accuracy.

[0062] like Figure 1 As shown, the specific implementation of the present invention includes the following steps:

[0063] S1. An adaptive composite observer based on the popular design of finite-time sliding mode is used to comprehensively compensate for uncertainties in robot kinematics and dynamics;

[0064] Constructing the kinematic and dynamic models of the robot:

[0065]

[0066]

[0067] In formula (1)-(2) Represents the velocity of movement within the mission space. and These represent the joint's spatial position and velocity, respectively. Represents the Jacobian matrix. Represents kinematic parameters, Represents the kinematic linear regression matrix; Represents the acceleration of joint spatial motion. These represent the inertia matrix, the Coriolis force and centrifugal force matrices, and the gravitational torque vector, respectively. Represents an unknown torque. It is the joint control torque.

[0068] Define an auxiliary regression matrix and two vectors as follows:

[0069]

[0070] In formula (3) This represents the kinematic regression matrix after first-order filtering. This represents the end position after filtering, with constant coefficients ω1 > 0 and ω2 > 0. These are estimated values ​​of kinematic parameters. (Known) Equivalent to parameter estimation error

[0071] Based on this, the algorithm for estimating the robot's kinematic parameters is designed as follows:

[0072]

[0073] In formula (4) Representing the positive definite gain matrix, this algorithm can reduce the kinematic parameter estimation error. It converges to zero within a finite amount of time.

[0074] Based on the tracking error e1=xx of the robot's end effector d The nominal expected velocity of the robot joint space. and acceleration They are respectively

[0075]

[0076]

[0077] In formula (5)-(6) and Let represent the nominal expected velocity and acceleration of the robot's end effector within the task space, respectively. and These represent the actual pose and the desired pose of the end effector, respectively. Based on this definition, the joint space velocity tracking error is... To design the unknown torque τ based on the kinematic estimation results d The observer, let η = M(q)e2, define an auxiliary system variable as

[0078]

[0079] In formula (7), ω3 > 0. Then, an adaptive finite-time sliding mode observer for uncertain dynamics is designed.

[0080]

[0081] In formula (8) and Representing τ d And the estimated value of ξ, β d Represents τ d The supremum of the time derivative is defined by the gain coefficients γ1, γ2, and α1 all being greater than zero, where 0 < p < 1; this algorithm can mitigate uncertain dynamic observation errors. It converges to zero within a finite amount of time.

[0082] S2. Then, based on the adaptive neural network and the preset performance error affine design, a non-singular terminal sliding mode motion controller is designed to directly stabilize the motion error in the task space;

[0083] To meet the pre-defined tracking response characteristics -δ imin μ i (t)<e 1i (t)<δ imax μ i (t), t≥0, design the equivalent affine term for motion control error e1. for

[0084]

[0085] In formula (9), Φ i (χ 1i )=(δ imax exp(χ 1i)-δ imin exp(-χ 1i )) / (exp(χ 1i )+exp(-χ 1i )) is an increasing function, and the error response characteristic function μ i (t)=(μ i0 -μ i∞ )exp(-σ i t)+μ i∞ The constant coefficient δ imin δ imax σ i μ i0 μ i∞ All are greater than zero.

[0086] Define an RBF neural network satisfy

[0087] W T Ω(z)+∈(z)=f(z)#(10)

[0088] In formula (10), f(z) represents the dynamic parameter perturbation and random error term. It is the input to the neural network. The expected weights and the number of hidden layer neurons are c. It is a Gaussian radial basis function vector. It is a bounded error vector that satisfies

[0089] To quickly stabilize robot end effector motion errors within a finite time, an end effector sliding surface s(t) is designed.

[0090]

[0091] In formula (11), Λ=diag{Λ1,...,Λ n} represents a positive definite diagonal matrix, 1 < κ < 2. Based on this, the non-singular terminal sliding mode motion controller for the robot is designed as follows:

[0092]

[0093]

[0094]

[0095] And design an adaptive update law for the weights of the RBF neural network.

[0096]

[0097] In formula (12), Π2(e2)=diag{||e 21 ||1-κ ,…,||e 2n || 1-κ The coefficients κ0, κ1, κ2, κ3, and κ4 are all positive. It is an estimate of W and In formula (13) This represents the positive definite gain matrix; the algorithm enables the robot motion error e1 to converge to zero within a finite time while satisfying the pre-set tracking response performance.

[0098] S3. Finally, the global finite-time stability of the entire observation-control system is achieved through the Lyapunov design method, so that the observer error and motion control error converge to zero rapidly within a finite time and satisfy the system's pre-set transient and steady-state response characteristics.

[0099] The stability of the entire observation-control system is analyzed using the Lyapunov design method. First, focusing on the kinematic parameter estimation algorithm formula (4), when... Sometimes If the condition is correct, then we can obtain... Established. When When designing Lyapunov functions And derive

[0100]

[0101] In formula (14) Obtain the estimation error of robot kinematic parameters It satisfies global finite-time stability.

[0102] Then, regarding the uncertain dynamics observer formula (8) and the non-singular terminal sliding mode motion controller formulas (11) to (13), since... Design Lyapunov functions And derive

[0103]

[0104] In formula (15) Get s(t), It will converge to the bounded interval within a finite time.

[0105] Finally, further design of Lyapunov functions. And derive

[0106]

[0107] In formula (16) Obtain the error of the robot dynamics observer Both the tracking error e1 in the task space and the tracking error e1 will converge to zero in a finite amount of time.

[0108] As can be seen from the proof process described in formulas (14) to (16), the robot motion controller based on the adaptive composite observer designed in this invention satisfies the global finite-time stability of the observation-control closed-loop system, which means that the robot end motion error can be quickly converged to zero in a short time.

[0109] Example:

[0110] like Figure 1 As shown, the robot specifically implemented in this invention is a six-degree-of-freedom robotic arm with kinematic parameters d1 = 0.140m, a2 = 0.375m, a3 = 0.345m, d4 = 0.122m, d5 = 0.122m, d6 = 0.083m. The masses of the robot links are m1 = 4.785kg, m2 = 5.485kg, m1 = 3.101kg, m1 = 2.568kg, m1 = 2.568kg, m1 = 0.414kg, and the moment of inertia of the robot links is I1 = 1.368kg·m. 2 I2 = 3.819 kg·m 2 I3 = 1.844 kg·m 2 I4 = 0.145 kg·m 2 I5 = 0.030 kg·m 2 I6 = 0.013 kg·m 2 .

[0111] In the implementation of this invention, the desired motion trajectory of the robot's end effector is designed as follows: Where r = 0.1 + 0.05cos(3ωt)m, ω = 0.1π. The initial position of the robot's end effector is set to x(0) = [0.5, 0.03, 0.25]. T m, the initial velocity of the robot's end effector is set to The initial values ​​of the robot's kinematic parameters are set to d1 = 0.140m, a2 = a3 = 0.200m, d4 = 0.070m, d5 = 0.050m, and d6 = 0.030m. Simultaneously, active perturbations are applied to each joint of the robot.

[0112] The control law of a robot global finite-time motion controller based on an adaptive composite observer is shown in equations (4), (8), (11)-(13), where the parameters of the adaptive composite observer are: ω1 = 0.01, ω2 = 0.025, p=0.5, Γ=diag{0.025, 0.1, 0.1, 0.02, 0.02, 0.02}, ω3=0.8, βd =120, α1=0.25; The parameters of the global finite-time motion controller are: δ imin =δ imax =1, μ i0 =0.1, μ i∞ =0.001, σ i =0.5, A=diag{1.5, 1.5, 1}, κ=1.5, κ0=30, κ1=25, κ2=0.4, κ3=0.5, κ4=0.02, c = 21. Based on this, the implementation effect of robot motion control can be obtained. Figure 2 and Figure 3 This demonstrates that the uncertainties in robot kinematics and dynamics can be accurately estimated and compensated within a finite time, thereby enabling the end effector motion error to converge rapidly within 2 seconds and satisfy the pre-set transient and steady-state response characteristics.

[0113] It is understood that the present invention has been described through some embodiments, and those skilled in the art will recognize that various changes or equivalent substitutions can be made to these features and embodiments without departing from the spirit and scope of the invention. Furthermore, under the teachings of the present invention, these features and embodiments can be modified to adapt to specific situations and materials without departing from the spirit and scope of the invention. Therefore, the present invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are within the protection scope of the present invention.

Claims

1. A global finite-time motion control method for a robot based on an adaptive composite observer, characterized in that, Includes the following steps: Step 1: Based on the popular design of finite-time sliding mode, an adaptive composite observer is designed to comprehensively compensate for the uncertainties in robot kinematics and dynamics; Step 2: Then, based on the adaptive neural network and the preset performance error affine design, a non-singular terminal sliding mode motion controller is designed to directly stabilize the motion error in the task space; To meet the pre-defined tracking response characteristics Design motion control error Equivalent affine terms for In formula (9) It is an increasing function, the error response characteristic function. Among them, constant coefficients , , , , All are greater than zero; Define an RBF neural network satisfy In formula (10) Represents the dynamic parameter perturbation and random error term. It is the input to the neural network. The expected weights and the number of hidden layer neurons are , It is a Gaussian radial basis function vector. It is a bounded error vector that satisfies ; Design terminal sliding surface , In formula (11) Represents a positive definite diagonal matrix. Based on this, the non-singular terminal sliding mode motion controller for the robot is designed as follows: And design an adaptive update law for the weights of the RBF neural network. In formula (12) ,coefficient , , , , All are positive. yes The estimated value and ;In formula (13) This represents the positive definite gain matrix; the algorithm enables the robot's motion error to be minimized. It converges to zero within a finite time while satisfying the pre-defined tracking response performance; Step 3: Finally, the global finite-time stability of the entire observation-control system is achieved through the Lyapunov design method, so that the observer error and motion control error converge to zero quickly within a finite time and satisfy the system's pre-set transient and steady-state response characteristics.

2. The robot global finite-time motion control method according to claim 1, characterized in that, Step 1 includes the following specific steps: Constructing the kinematic and dynamic models of the robot: In formula (1)-(2) Represents the velocity of movement within the mission space. and These represent the joint's spatial position and velocity, respectively. Represents the Jacobian matrix. Represents kinematic parameters, Represents the kinematic linear regression matrix; Represents the acceleration of joint spatial motion. , , These represent the inertia matrix, the Coriolis force and centrifugal force matrices, and the gravitational torque vector, respectively. Represents an unknown torque. It is the joint control torque; Define an auxiliary regression matrix and two vectors , ,as follows: In formula (3) This represents the kinematic regression matrix after first-order filtering. Represents the end position after filtering, constant coefficient , , These are estimated values ​​of kinematic parameters; known. Equivalent to parameter estimation error ; Based on tracking error of robot end effector The nominal expected velocity of the robot joint space. and acceleration They are respectively In formula (5)-(6) and Let represent the nominal expected velocity and acceleration of the robot's end effector within the task space, respectively. and Representing the actual pose and desired pose of the end effector, respectively, the joint space velocity tracking error is defined as follows: In order to design unknown torques based on kinematic estimation results The observer, Define an auxiliary system variable as In formula (7) Then, an adaptive finite-time sliding mode observer for uncertain dynamics is designed. In formula (8) and Represent and The estimated value, , , represent The upper bound of the time derivative, the gain coefficient , and All are greater than zero. ; This algorithm can minimize uncertain dynamic observation errors. It converges to zero within a finite amount of time.

3. The robot global finite-time motion control method according to claim 2, characterized in that, Step 1 includes an algorithm for estimating the robot's kinematic parameters, the steps of which are as follows: In formula (4) Representing the positive definite gain matrix, this algorithm reduces the kinematic parameter estimation error. It converges to zero within a finite amount of time.

4. The robot global finite-time motion control method according to claim 3, characterized in that, Step 3 includes the following specific steps: Stability analysis of the entire observation-control system is performed using the Lyapunov design method. Firstly, focusing on the kinematic parameter estimation algorithm formula (4), when… Sometimes Zhengding, De Established, when When designing Lyapunov functions and derive In formula (14) The estimation error of the robot's kinematic parameters is obtained. Satisfies global finite-time stability; Then, regarding the uncertain dynamics observer formula (8) and the non-singular terminal sliding mode motion controller formulas (11) to (13), since , Design Lyapunov functions and derive In formula (15) , , ;get , , It will converge to the bounded interval within a finite time. Finally, design the Lyapunov function. and derive In formula (16) The error of the robot dynamics observer is obtained. and task space tracking error All of them will converge to zero within a finite time.

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