Adaptive Fixed-Time Motion Control Method for Mobile Manipulators Based on Delayed Observers

Abstract

The present invention belongs to the field of robot motion control, and discloses a design method for an adaptive fixed-time motion controller of a mobile manipulator based on a delay observer, including Step 1: establishing an equivalent dynamic model of the mobile manipulator, designing a delay observer to achieve model-free estimation compensation of generalized dynamic uncertainties, and improving the calculation and execution efficiency of the controller; Step 2: designing a fixed-time nonsingular terminal sliding mode controller and its gain parameter adaptation law to achieve fixed-time convergence of the motion error of the mobile manipulator, so that the stabilization time of the controller is only determined by the controller design parameters, improving the convergence speed of the control algorithm and not depending on the initial conditions of the system state; Step 3: achieving fixed-time stability of the entire observation-control closed-loop system through the Lyapunov design method, eliminating the dependence on prior knowledge such as the supremum of dynamic uncertainties and the supremum of the delay observer, and improving the response speed and accuracy of the motion control of the mobile manipulator.

Description

Technical Field

[0001] The present invention belongs to the technical field of robot motion control, and particularly relates to an adaptive fixed-time motion control method for a mobile manipulator based on a delay observer. Background Art

[0002] Integrating capabilities such as perception, operation, and movement has become a research hotspot and development direction in robotics in recent years. A mobile manipulator composed of a manipulator arm and a mobile platform simultaneously has the abilities of flexible movement and dexterous operation, and it is gradually applied to fields such as industry, agriculture, and services, with broad application prospects. However, the strong nonlinear coupling complex dynamic factors between the mobile platform and the manipulator arm directly affect the motion control performance of the mobile manipulator. How to develop an efficient and advanced autonomous motion controller remains the main challenge to demonstrate its application potential. When the mobile manipulator performs tasks in an unstructured environment, unknown loads and external disturbances will also cause its dynamics to change in real time. Therefore, it is of great engineering significance and practical value to conduct design research on its motion control method.

[0003] Currently, many advanced control methods have been used to solve the motion control problem under the influence of uncertain dynamics of robots, such as active disturbance rejection controllers, sliding mode robust controllers, and neural network controllers, etc. However, most of these control methods only consider the consistent bounded convergence region of the motion error, and rarely involve the analysis and design of the convergence speed of the motion error. However, in order to effectively solve problems such as time-varying dynamics and external disturbances faced by the mobile manipulator, the motion controller design needs to make its motion error converge quickly. Although methods such as terminal sliding mode controllers can achieve finite-time convergence of the motion controller, that is, the controller stabilization time is bounded and depends on the design parameters and the initial conditions of the system state, such methods cannot be directly applied to complex systems where the initial state is difficult to directly obtain. In addition, most of the above controllers rely on the dynamic model of the robot system or its learning approximation black box model, and the required complex computational burden will also hinder their implementation and application on the mobile manipulator. In order to improve the anti-interference ability of the controller, control methods based on disturbance observers have also been gradually applied to complex robot systems, but the observer gain values mostly depend on the supremum of the disturbance that is difficult to directly obtain, which limits their application in practical engineering. Summary of the Invention

[0004] The purpose of the present invention is to provide an adaptive fixed-time motion control method for a mobile manipulator based on a delay observer to solve the above technical problems.

[0005] To solve the above technical problems, the specific technical solution of the adaptive fixed-time motion control method for a mobile manipulator based on a delay observer of the present invention is as follows:

[0006] An adaptive fixed-time motion control method for a mobile manipulator based on a delay observer, comprising the following steps:

[0007] Step 1: Establish an equivalent dynamic model of the mobile manipulator, design a delay observer to achieve model-free estimation compensation of the generalized dynamic uncertainty, and simplify the computational complexity of the control algorithm to improve the execution efficiency of the controller;

[0008] Step 2: Design a fixed-time nonsingular terminal sliding mode controller and its gain parameter adaptation law to achieve fixed-time convergence of the motion error of the mobile manipulator, so that the stabilization time of the controller is only determined by the controller design parameters, improving the convergence speed of the control algorithm and not depending on the initial conditions of the system state;

[0009] Step 3: Achieve the fixed-time stability of the entire closed-loop observation-control system through the Lyapunov design method, eliminate the dependence on prior knowledge such as the supremum of the dynamic uncertainty and the supremum of the delay observer, and improve the response speed and accuracy of the motion control of the mobile manipulator.

[0010] Further, the Step 1 includes the following specific steps:

[0011] Step 1.1: Construct the dynamic equation of the Mecanum wheel mobile manipulator;

[0012] Step 1.2: Deduce the equivalent dynamic model of the mobile manipulator;

[0013] Step 1.3: Design a delay observer for the generalized uncertain dynamics of the mobile manipulator.

[0014] Further, the dynamic equation of the Mecanum wheel mobile manipulator constructed in the Step 1.1 is as follows:

[0015]

[0016] In formula (1) represents the generalized joint space coordinates of the mobile manipulator, where q b =[x, y, φ] T and q b =[θ1, …, θ n T represent the joint space coordinates of the mobile platform and the manipulator respectively; represents the inertia matrix, represents the Coriolis force and centrifugal force vector, represents the gravity moment vector, represents the unknown torque vector caused by friction, system dynamic uncertainty and external disturbance, is the joint driving torque of the mobile manipulator, represents the input conversion matrix.​

[0017] Further, the equivalent dynamic model of the mobile manipulator is derived in Step 1.2 as follows:

[0018]

[0019] In formula (2), represents a user-defined positive definite diagonal inertia matrix, τ = E(q)τ * represents the equivalent input torque of the mobile manipulator, is the generalized uncertain dynamics of the mobile manipulator.

[0020] Further, the delay observer for the generalized uncertain dynamics of the mobile manipulator designed in Step 1.3 is as follows:

[0021]

[0022] In formula (3), is 's estimated value, L represents the delay time and is often set to the sampling period of the control system; this observer can make the uncertain dynamics estimation error converge to a bounded region, that is, there exists a constant such that the inequality holds.

[0023] Further, Step 2 includes the following specific steps:

[0024] Step 2.1: Design a fixed-time non-singular terminal sliding mode manifold;

[0025] Step 2.2: Design a fixed-time non-singular terminal sliding mode controller for the mobile manipulator;

[0026] Step 2.3: Design the controller parameter adaptation law.

[0027] Further, the fixed-time non-singular terminal sliding mode manifold designed in Step 2.1 is as follows:

[0028]

[0029] In formula (4), e = q - q d represents the motion error of the mobile manipulator, where q d represents the desired motion trajectory; and represent diagonal positive definite gain matrices, u1, v1, p1, and q1 are positive odd numbers and satisfy the inequality 1 < p1 / q1 < 2, u1 / v1 > p1 / q1,

[0030] Define an unknown constant Satisfy simultaneously

[0031]

[0032] In formula (5), is a positive definite diagonal matrix, represents the matrix the minimum eigenvalue of.

[0033] Furthermore, the fixed-time nonsingular terminal sliding mode control

[0034] controller designed in step 2.2 is as follows:

[0035]

[0036] Furthermore, the controller parameter adaptation law designed in step 2.3 is as follows:

[0037]

[0038] In formulas (6) and (7), τ s and τ r represent the sliding control law and the approaching control law respectively, and are positive definite diagonal gain matrices, u2, v2, p2, and q2 are positive odd numbers and satisfy the inequalities u2 > v2, p2 > q2; represents the estimated value of, and σ0 > 0, σ1 > 0, and σ2 > 0 are the constant coefficients of the controller.

[0039] Furthermore, step 3 includes the following specific steps:

[0040] Derive the dynamic equation of the delay observer from formulas (2) and (3) as follows

[0041]

[0042] When the controller delay time L is small enough, in formula (8), is bounded;

[0043] Substitute into formula (8) to obtain

[0044]

[0045] In formula (9), and are both bounded; Therefore, when the custom inertia matrix Satisfied When, the solution of the equivalent differential equation corresponding to formula (9) will asymptotically converge to a bounded region, that is, the inequality Holds;

[0046] Define the Lyapunov function as Take the first-order derivative with respect to time to get

[0047]

[0048] In formula (10)

[0049]

[0050] It can be seen from this that the nonsingular terminal sliding mode manifold s will converge to a bounded region within a fixed time t, that is

[0051]

[0052] In formula (11) Is a constant and satisfies

[0053] Therefore, it can be seen from formula (4) that the motion error e of the mobile manipulator will converge to a bounded region within a fixed time, that is, the controller stabilization time is only determined by its design parameters and is independent of the initial conditions of the system state. And it can be seen from the proof process that the stability of the entire observation-control system does not depend on the prior knowledge of the supremum of the dynamic uncertainty and the supremum of the delay observer.

[0054] The adaptive fixed-time motion control method for a mobile manipulator based on a delay observer of the present invention has the following advantages: The adaptive fixed-time motion control method for a mobile manipulator of the present invention can achieve the fixed-time convergence of its motion error, that is, the stabilization time of the controller is not affected by the system initial state that is difficult to obtain in practice and does not depend on the prior knowledge of the system uncertainty; At the same time, a delay observer is designed to estimate and compensate the generalized uncertainty of the system dynamics, greatly simplifying the computational complexity of the control algorithm to optimize the execution efficiency of the controller, and can effectively improve the response speed and accuracy of the motion control of the mobile manipulator. Brief Description of the Drawings

[0055] Figure 1 Is the flow chart of the adaptive fixed-time motion control method for a mobile manipulator based on a delay observer of the present invention;

[0056] Figure 2 Is the schematic diagram of the desired trajectory of the generalized joint space of the mobile manipulator of the present invention;

[0057] Figure 3It is a schematic diagram of the fixed-time convergence effect of the motion control error of the mobile manipulator of the present invention under different initial conditions. Detailed implementation manners

[0058] To better understand the purpose, structure and function of the present invention, the adaptive fixed-time motion control method of the mobile manipulator based on the delay observer of the present invention will be further described in detail below with reference to the accompanying drawings.

[0059] The present invention first designs a delay observer to compensate for the generalized uncertain dynamics of the mobile manipulator, and then designs a nonsingular terminal sliding mode controller and its gain parameter adaptive law based on this to achieve fixed-time convergence of the motion error, effectively improving the response speed and accuracy of the motion control of the mobile manipulator.

[0060] As Figure 1 shown, the specific implementation scheme of the present invention includes the following steps:

[0061] Step 1: Establish an equivalent dynamic model of the mobile manipulator, design a delay observer to achieve model-free estimation compensation of the generalized dynamic uncertainty, and simplify the calculation amount of the control algorithm to improve the execution efficiency of the controller;

[0062] Step 1.1: Construct the dynamic equation of the mobile manipulator with Mecanum wheels as follows

[0063]

[0064] In formula (1) represents the generalized joint space coordinates of the mobile manipulator, where q b =[x, y, φ] T and q b =[θ1,..., θ n T respectively represent the joint space coordinates of the mobile platform and the manipulator; represents the inertia matrix, represents the vector of Coriolis and centrifugal torques, represents the vector of gravity torques, represents the unknown torque vector caused by friction, system dynamic uncertainty and external disturbances, is the joint driving torque of the mobile manipulator, represents the input transformation matrix.

[0065] Step 1.2: Based on Step 1.1, derive the equivalent dynamic model of the mobile manipulator as follows

[0066]

[0067] In formula (2) ​represents the positive definite diagonal inertia matrix defined by the user, and τ = E(q)τ * represents the equivalent input torque of the mobile manipulator, is the generalized uncertain dynamics of the mobile manipulator.

[0068] Step 1.3: Design a delay observer for the generalized uncertain dynamics of the mobile manipulator as follows

[0069]

[0070] In formula (3), is the estimated value of, L represents the delay time and is often set to the sampling period of the control system; this observer can make the uncertain dynamics estimation error converge to a bounded region, that is, there exists a constant such that the inequality holds, and it does not involve complex calculations of the dynamic matrix of the dynamics, and the algorithm execution efficiency is high.

[0071] S2: Design a fixed-time nonsingular terminal sliding mode controller and its gain parameter adaptation law to achieve fixed-time convergence of the motion error of the mobile manipulator, so that the controller stabilization time is only determined by the controller design parameters, improving the convergence speed of the control algorithm and not depending on the initial conditions of the system state;

[0072] Step 2.1: To make the motion error of the mobile manipulator converge within a fixed time, design a fixed-time nonsingular terminal sliding mode manifold as follows

[0073]

[0074] In formula (4), e = q - q d represents the motion error of the mobile manipulator, where q d represents the desired motion trajectory; and represent diagonal positive definite gain matrices, and u1, v1, p1, and q1 are positive odd numbers and satisfy the inequality 1 < p1 / q1 < 2, u1 / v1 > p1 / q1.

[0075] Define an unknown constant while satisfying

[0076]

[0077] In formula (5), is a positive definite diagonal matrix, represents the matrix the minimum eigenvalue of.

[0078] Step 2.2: Design a fixed-time nonsingular terminal sliding mode controller for the mobile manipulator as follows Step 2.3: Design the adaptive law of the controller parameters as follows

[0079]

[0080] In formulas (6) and (7), τ s and τ r represent the sliding control law and the approaching control law respectively, and are positive definite diagonal gain matrices, and u2, v2, p2, and q2 are positive odd numbers and satisfy the inequalities u2 > v2, p2 > q2; represents the estimated value of, and σ0 > 0, σ1 > 0, and σ2 > 0 are the constant coefficients of the controller. This algorithm enables the motion error of the mobile manipulator to converge to a bounded region within a fixed time, that is, the controller stabilization time is only determined by its design parameters and is independent of the initial conditions of the system state.

[0081] S3: Achieve the fixed-time stability of the entire closed-loop system of observation and control through the Lyapunov design method, eliminate the dependence on prior knowledge such as the supremum of dynamic uncertainties and the supremum of delay observers, and improve the response speed and accuracy of the motion control of the mobile manipulator.

[0082] Derive the dynamic equation of the delay observer from formulas (2) and (3) as follows

[0083]

[0084] When the controller delay time L is small enough, in formula (8) is bounded.

[0085] Substitute into formula (8), and we can get

[0086]

[0087] In formula (9), and are both bounded; therefore, when the custom inertia matrix satisfies When, the solution of the equivalent differential equation corresponding to formula (9) will asymptotically converge to a bounded region, that is, the inequality holds.

[0088] To further achieve the fixed-time convergence of the designed controller, the Lyapunov function is defined as Taking the first-order derivative with respect to time, we can get

[0089]

[0090] It can be seen that the non-singular terminal sliding mode manifold s will converge to a bounded region within a fixed time t, that is

[0091]

[0092] In formula (11) is a constant and satisfies

[0093] Therefore, it can be seen from formula (4) that the motion error e of the mobile manipulator will converge within a fixed time, and it can be seen from the proof process that the stability of the entire observation-control system does not depend on a priori knowledge such as the supremum of dynamic uncertainties and the supremum of delay observers.

[0094] Embodiment

[0095] The flow of the mobile manipulator adaptive fixed-time motion control method based on a delay observer of the present invention is as Figure 1 shown. The specific object to be implemented is a mobile manipulator composed of a Mecanum wheeled mobile platform and a six-degree-of-freedom manipulator. Its kinematic parameters are d b = 0.140m, d1 = 0.140m, a2 = 0.375m, a3 = 0.345m, d4 = 0.122m, d5 = 0.122m, d6 = 0.083m, the link mass is m b = 76.156kg, m1 = 4.785kg, m2 = 5.485kg, m3 = 3.101kg, m4 = 2.568kg, m5 = 2.568kg, m6 = 0.414kg, the link moment of inertia is I b = 9.619kg·m 2 、I1 = 5.438kg·m 2 、I2 = 6.251kg·m 2 、I3 = 4.488kg·m 2 、I4 = 1.075kg·m 2 、I5 = 0.130kg·m 2 、I6 = 0.025kg·m 2 .

[0096] During the implementation of the present invention, the desired motion trajectory at the end of the mobile manipulator is designed as

[0097] where ω = 0.1π. Then the desired trajectory in the generalized joint space of the mobile manipulator can be obtained by the formula where is the forward kinematics of the mobile manipulator, is the pseudo-inverse of the Jacobian matrix of the mobile manipulator, and the gain parameter Λ = 1.5; the obtained desired trajectory is as shown in Figure 2 shown.

[0098] The control laws of an adaptive fixed-time motion controller for a mobile manipulator based on a delay observer are shown in formulas (3), (4), (6), and (7), where the delay time of the delay observer is set to L = 0.001, and the custom inertia matrix is The parameter values of the adaptive motion controller are: u1 / v1 = 7 / 3, p1 / q1 = 17 / 9, u2 / v2 = 9 / 5, q2 / p2 = 5 / 3, σ0 = 0.1, σ1 = 0.6, σ2 = 0.8, To verify that the settling time of the designed controller is independent of the initial conditions of the system state, four sets of initial positions in the joint space of the mobile manipulator are taken as: q1(0) = 0, q2(0) = [-π / 6, -π / 4, π / 6, π / 4, -T / 12, π / 12, -π / 9, π / 9, -π / 16, T / 16] T 、q3(0) = [π / 4, π / 6, -π / 4, -π / 6, π / 20, -π / 20, π / 15, -π / 15, π / 10, -π / 10] T 、q4(0) = [π / 3, π / 3, -π / 3, -π / 3, π / 10, -π / 10, π / 8, -π / 8, π / 6, -π / 6] T . Based on this, the implementation effect of the motion control of the mobile manipulator can be obtained, Figure 3 indicating that the motion errors of each joint of the mobile manipulator can converge quickly, and at the same time, the convergence time is not affected by the changes in the initial positions of each joint, indicating that the designed controller can effectively improve the response speed and accuracy of the motion control of the mobile manipulator.

[0099] It will be understood that the present invention is described by way of some embodiments, and those skilled in the art will know that various changes or equivalent substitutions can be made to these features and embodiments without departing from the spirit and scope of the present invention. Additionally, under the teaching of the present invention, these features and embodiments can be modified to adapt to specific circumstances and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of the present application belong to the scope protected by the present invention.

Claims

1. An adaptive fixed-time motion control method for a mobile manipulator based on a delay observer, characterized in that, It includes the following steps: Step 1: Establish an equivalent dynamic model of the mobile manipulator, design a delay observer to achieve model-free estimation and compensation of generalized dynamic uncertainties, and simplify the computational complexity of the control algorithm to improve the execution efficiency of the controller; Step 2: Design a fixed-time nonsingular terminal sliding mode controller and its gain parameter adaptive law to achieve fixed-time convergence of the motion error of the mobile manipulator, so that the controller stabilization time is only determined by the controller design parameters, improve the convergence speed of the control algorithm and is independent of the initial conditions of the system state; Step 2.1: Design a fixed-time nonsingular terminal sliding mode manifold; The fixed-time nonsingular terminal sliding mode manifold designed in Step 2.1 is as follows: In Equation (4), e = q - q d represents the motion error of the mobile manipulator, where q d represents the desired motion trajectory; and represent the diagonal positive definite gain matrix, where u1, v1, p1, and q1 are positive odd numbers and satisfy the inequalities 1 < p1 / q1 < 2 and y1 / v1 > p1 / q1, Define an unknown constant Satisfy simultaneously In formula (5) is a positive definite diagonal matrix, represents the matrix 's minimum eigenvalue; Step 2.2: Design a fixed-time nonsingular terminal sliding mode controller for the mobile manipulator; The fixed-time nonsingular terminal sliding mode controller for the mobile manipulator designed in Step 2.2 is as follows: Step 2.3: Design a controller parameter adaptive law; The controller parameter adaptive law designed in Step 2.3 is as follows: In Formulas (6) and (7), τ s and τ r represent the sliding control law and the approaching control law respectively, and are positive definite diagonal gain matrices, u2, v2, p2, and q2 are positive odd numbers and satisfy the inequalities u2 > v2, p2 > q2; represents the estimated value of, σ0 > 0, σ1 > 0, and σ2 > 0 are constant coefficients of the controller; Step 3: Achieve fixed-time stability of the entire observation-control closed-loop system through the Lyapunov design method, eliminate the dependence on prior knowledge such as the supremum of dynamic uncertainties and the supremum of the delay observer, and improve the response speed and accuracy of the motion control of the mobile manipulator.

2. The adaptive fixed-time motion control method for a mobile manipulator based on a delay observer according to claim 1, wherein The specific steps included in Step 1 are as follows: Step 1.1: Construct the dynamic equation of the Mecanum wheel mobile manipulator; Step 1.2: Deduce the equivalent dynamic model of the mobile manipulator; Step 1.3: Design a delay observer for the generalized uncertain dynamics of the mobile manipulator.

3. The adaptive fixed-time motion control method for a mobile manipulator based on a delay observer according to claim 2, wherein The dynamic equation of the Mecanum wheel mobile manipulator constructed in Step 1.1 is as follows: In formula (1) represents the generalized joint space coordinates of the mobile manipulator, where q b = [x, y, φ] T and q b = [θ1, …, θ n T represent the joint space coordinates of the mobile platform and the manipulator respectively; represents the inertia matrix, represents the Coriolis and centrifugal torque vectors, represents the gravity torque vector, represents the unknown torque vector caused by friction, system dynamics uncertainty and external disturbances, is the joint driving torque of the mobile manipulator, represents the input transformation matrix.​ 4. The adaptive fixed-time motion control method for a mobile manipulator based on a delay observer according to claim 3, characterized in that The equivalent dynamic model of the mobile manipulator deduced in Step 1.2 is as follows: In formula (2) represents a user-defined positive definite diagonal inertia matrix, and τ = E(q)τ * represents the equivalent input torque of the mobile manipulator is the generalized uncertain dynamics of the mobile manipulator 5. The adaptive fixed-time motion control method for a mobile manipulator based on a delay observer according to claim 4, wherein The delay observer for the generalized uncertain dynamics of the mobile manipulator designed in Step 1.3 is as follows: In formula (3) is the estimated value, L represents the delay time and is often set as the sampling period of the control system; this observer can make the uncertain dynamic estimation error converge to a bounded region, that is, there exists a constant such that the inequality holds.

6. The adaptive fixed-time motion control method for a mobile manipulator based on a delay observer according to claim 5, characterized in that, The specific steps included in Step 3 are as follows: Deduce the dynamic equation of the delay observer from formula (2) and formula (3) as follows When the controller delay time L is small enough, in formula (8), is bounded; Substitute into Equation (8), we get In formula (9) and are both bounded; therefore, when the user-defined inertia matrix satisfies , the solution of the equivalent differential equation corresponding to formula (9) will asymptotically converge to a bounded region, that is, the inequality holds. Define the Lyapunov function as Take the first-order derivative with respect to time to obtain In formula (10) It can be seen that the nonsingular terminal sliding mode manifold s will converge to a bounded region within a fixed time t, that is In Formula (11) is a constant and satisfies Therefore, it can be seen from formula (4) that the motion error e of the mobile manipulator will converge to a bounded region within a fixed time, that is, the controller stabilization time is only determined by its design parameters and is independent of the initial conditions of the system state. And it can be seen from the proof process that the stability of the entire observation-control system does not depend on the prior knowledge of the supremum of dynamic uncertainties and the supremum of the delay observer.

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