Method for designing a robot trajectory tracking controller based on a specified time

Abstract

The application provides a robot trajectory tracking controller design method based on specified time, and relates to the technical field of robot control. The method combines specified time stability theory with non-singular terminal sliding mode control, constructs a non-singular terminal sliding mode surface with specified time convergence characteristics based on a specified time adjusting function, and designs a corresponding sliding mode controller, so as to ensure that the system state converges to the origin within the specified time, and to avoid the singularity problem. The application can effectively deal with the model uncertainty and external disturbance existing in the robot system, ensure that the trajectory tracking error converges with high precision within the specified time, and the convergence time is completely independent of the initial state and parameters of the system.

Description

Technical Field

[0001] This invention relates to the field of robot control technology, and in particular to a design method for a robot trajectory tracking controller based on a specified time. Background Technology

[0002] Robotics plays a crucial role in the automation process of advanced fields such as modern aerospace and intelligent manufacturing. The execution of complex and precise tasks fundamentally depends on the rapid and accurate tracking of the desired motion trajectory, i.e., the trajectory tracking control problem. However, the inherent strong nonlinearity, significant coupling effects, model uncertainties, and external disturbances of robot systems pose enormous challenges to achieving high-precision and robust trajectory tracking.

[0003] Traditional PD control and adaptive control methods primarily achieve asymptotic stability, meaning the tracking error converges asymptotically to zero or a bounded neighborhood as time approaches infinity. Due to this characteristic, such methods struggle to meet the stringent requirements of high-dynamic, high-precision applications regarding transient performance and convergence time. To address this issue, finite-time control methods have been proposed, ensuring the system trajectory converges to an equilibrium point within a finite time. Compared to asymptotic methods, they offer faster convergence, higher control accuracy, and stronger robustness. However, the convergence time of finite-time control typically depends on the system's initial conditions, which are often difficult to accurately obtain in practical applications. Furthermore, as the initial state approaches infinity, the convergence time can increase infinitely, significantly limiting the engineering applications of finite-time control methods.

[0004] Based on finite-time stability theory, fixed-time stability theory has been further proposed, where the system convergence time is independent of the initial state and determined solely by the controller parameters. However, finite-time control methods rely heavily on initial system conditions, and the theoretically derived upper bound of convergence time for fixed-time control methods is often conservative, much larger than the actual convergence time, making it difficult to accurately characterize transient performance. To further reduce the dependence of convergence time on initial conditions and parameter fluctuations, specified-time control methods have received widespread attention in recent years.

[0005] While existing time-defined control methods achieve convergence within a specified time, they often suffer from insufficient robustness, decreased tracking accuracy, and complex control structures. To address these issues, sliding mode control has been introduced due to its simple structure and strong robustness to matching uncertainties. However, existing time-defined sliding mode control methods suffer from reduced robustness during the sliding phase due to the linear sliding surface, and singularities in the sliding surface affecting controller reliability and practicality. Therefore, designing a sliding mode control strategy that simultaneously guarantees time-defined convergence, strong robustness, high tracking accuracy, and is free of singularities remains a crucial and challenging research topic. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a design method for a robot trajectory tracking controller based on a specified time.

[0007] This invention provides a design method for a robot trajectory tracking controller based on a specified time, comprising the following steps:

[0008] Step 1: Based on the robot dynamics under model uncertainties and external disturbances, establish an open-loop robot dynamics system;

[0009] The robot dynamics under the model uncertainty and external disturbances are as follows:

[0010] ;

[0011] in, These are the joint position, velocity, and acceleration vectors, respectively. It is a symmetric positive definite inertial matrix. The matrix contains Coriolis force and centrifugal force. The gravity vector To control the input vector, This is the external interference vector. Represents an n-dimensional real vector space. Represents the n-dimensional real matrix space;

[0012] The open-loop robot dynamics system is as follows:

[0013] ;

[0014] in, The nominal value of the inertia matrix of the robot system; The nominal value of the centrifugal force-Coriolis force matrix; The nominal value representing the gravity vector of the robot system. This represents the uncertainty value of the robot system's inertia matrix. This represents the uncertainty value of the Coriolis force and centrifugal force matrix of the robot system. This represents the uncertainty value of the gravity vector of the robot system; This represents the lumped uncertainty term of the robot system;

[0015] Step 2: Design a time-adjustment function;

[0016] The specified time adjustment function is:

[0017] ;

[0018] ;

[0019] ;

[0020] Where t represents time, T p For the specified convergence time, T p >0, when When specifying a time adjustment function Approaching infinity, used to ensure that the closed-loop system reaches infinity within a specified time. When the internal temperature reaches stability, T p - This is the left limit of the convergence time; , , , , , For the positive constants to be designed.

[0021] Step 3: Construct a non-singular terminal sliding surface with specified time convergence characteristics by combining a specified time adjustment function;

[0022] First, define the following nonlinear piecewise function:

[0023] ;

[0024] Where r is a positive constant and satisfies r>1, and x is the independent variable. It is a positive integer and satisfies ,symbol For standard symbolic functions, symbolic To take the absolute value;

[0025] The non-singular terminal sliding surface with specified time convergence characteristics is:

[0026] ;

[0027] ;

[0028] in, Represent the non-singular terminal sliding surface; define the position error vector. Its weight The tracking error for the nth joint; the actual joint position vector. , Represents the actual position of the nth joint; the expected joint trajectory vector. ,in Let n be the desired position of the nth joint; superscript This indicates the transpose of the corresponding matrix; For vectors The derivative; parameters It is a positive integer and satisfies and , This refers to the exponential parameter in the adjustment function;

[0029] Step 4: Design controller based on the design of the sliding surface.

[0030] The sliding surface design controller is:

[0031] ;

[0032] ;

[0033] ;

[0034] ;

[0035] ;

[0036] in, Indicates the nominal terms of a nonlinear mechanical region system; The nominal value of the centrifugal force-Coriolis force matrix; It is the gravity vector; The second derivative represents the desired position; , Represents the derivative of a specified time-adjusting function; symbol Represents the vector modulo function; symbol , as well as Represented as:

[0037] ;

[0038] ;

[0039] ; ;

[0040] ;

[0041] in, , Representation matrix The values ​​of the upper and lower bounds; for identity matrix of order 1; symbol This represents the function for constructing a diagonal matrix; , , All are known positive constants, and their coefficients satisfy the following relationship:

[0042] ;

[0043] ;

[0044] ;

[0045] In the formula, For matrix The upper bound coefficient of the norm, For vectors The upper bound coefficient of the norm, For an uncertain matrix The upper bound coefficient of the norm, For vectors The upper bound coefficient of the norm, External interference vector Upper boundary;

[0046] Step 5: Deploy the designed sliding surface design controller to the robot control system. During actual operation, first complete parameter initialization: set the specified convergence time. Select the adjustment function parameters and sliding surface parameters, and specify the desired position trajectory. and its derivative During each control cycle, the actual position is acquired through the joint encoder and speed sensor. and speed Calculate position tracking error and Then, based on the specified time adjustment function and the defined nonlinear piecewise function... Construct a non-singular terminal sliding surface s; then, design a controller based on the sliding surface, calculate the equivalent control term and robust switching term, and obtain the control torque. The output is then sent to the robot joint driver, and the above process is repeated until the preset tracking task is completed.

[0047] The beneficial effects of adopting the above technical solution are as follows:

[0048] This invention provides a design method for a robot trajectory tracking controller based on a specified time, which has the following beneficial effects:

[0049] 1. This invention proposes a non-singular terminal sliding surface based on a specified time adjustment function. The vector field structure of the error dynamics is reconstructed by the specified time adjustment function, which strictly guarantees that the system tracking error converges to the origin within a specified time. The convergence time is completely independent of the initial state and parameters of the system and can be arbitrarily specified within the expected range according to the actual operation requirements.

[0050] 2. The nonlinear function designed in this invention This effectively solves the singularity problem caused by the derivative term appearing in the denominator in traditional terminal sliding mode control, ensuring that the control input always remains bounded, and improving the reliability and practicality of the controller.

[0051] 3. This invention combines specified time control with sliding mode control, which retains the advantage of strong robustness of sliding mode control to uncertainty, and realizes the specification of convergence time. Compared with existing finite time control and fixed time control methods, it has significant advantages in terms of settable convergence time and tracking accuracy. Attached Figure Description

[0052] Figure 1 Overall flowchart of the robot trajectory tracking controller design method according to an embodiment of the present invention;

[0053] Figure 2 Trajectory tracking position error curves of a dual-joint robot system under different initial conditions according to an embodiment of the present invention;

[0054] Among them, (a) is the trajectory tracking position error diagram of joint 1, and (b) is the trajectory tracking position error diagram of joint 2;

[0055] Figure 3 Trajectory tracking position error curves of a dual-joint robot system with different specified convergence times according to an embodiment of the present invention;

[0056] Among them, (a) is the trajectory tracking position error diagram of joint 1, and (b) is the trajectory tracking position error diagram of joint 2. Detailed Implementation

[0057] The specific implementation methods of this application will be further described in detail below with reference to the accompanying drawings and embodiments.

[0058] Example 1:

[0059] On the one hand, this invention provides a design method for a robot trajectory tracking controller based on a specified time, such as... Figure 1 As shown, it includes the following steps:

[0060] Step 1: Based on the robot dynamics under model uncertainties and external disturbances, establish an open-loop robot dynamics system;

[0061] The robot dynamics under the model uncertainty and external disturbances are as follows:

[0062] ;

[0063] in, These are the joint position, velocity, and acceleration vectors, respectively. It is a symmetric positive definite inertial matrix. The matrix contains Coriolis force and centrifugal force. The gravity vector To control the input vector, This is the external interference vector. Represents an n-dimensional real vector space. Represents the n-dimensional real matrix space;

[0064] The open-loop robot dynamics system is as follows:

[0065] ;

[0066] in, The nominal value of the inertia matrix of the robot system; The nominal value of the centrifugal force-Coriolis force matrix; The nominal value representing the gravity vector of the robot system. This represents the uncertainty value of the robot system's inertia matrix. This represents the uncertainty value of the Coriolis force and centrifugal force matrix of the robot system. This represents the uncertainty value of the gravity vector of the robot system; This represents the lumped uncertainty term of the robot system;

[0067] Step 2: Design a time-adjustment function;

[0068] The specified time adjustment function is:

[0069] ;

[0070] ;

[0071] ;

[0072] Where t represents time, T p For the specified convergence time, T p >0, when When specifying a time adjustment function Approaching infinity, used to ensure that the closed-loop system reaches infinity within a specified time. When the internal temperature reaches stability, T p - This is the lower bound limit of the convergence time; , , , , , For the positive constants to be designed.

[0073] Step 3: Construct a non-singular terminal sliding surface with specified time convergence characteristics by combining a specified time adjustment function;

[0074] First, define the following nonlinear piecewise function:

[0075] ;

[0076] Where r is a positive constant and satisfies r>1, and x is the independent variable. It is a positive integer and satisfies To avoid strange issues with the controller, symbol For standard symbolic functions, symbolic To take the absolute value;

[0077] The non-singular terminal sliding surface with specified time convergence characteristics is:

[0078] ;

[0079] ;

[0080] in, Represent the non-singular terminal sliding surface; define the position error vector. Its weight The tracking error for the nth joint; the actual joint position vector. , Represents the actual position of the nth joint; the expected joint trajectory vector. ,in Let n be the desired position of the nth joint; superscript This indicates the transpose of the corresponding matrix; For vectors The derivative; parameters It is a positive integer and satisfies and , This refers to the exponential parameter in the adjustment function;

[0081] Step 4: Design controller based on the design of the sliding surface.

[0082] The sliding surface design controller is:

[0083] ;

[0084] ;

[0085] ;

[0086] ;

[0087] ;

[0088] in, Indicates the nominal terms of a nonlinear mechanical region system; The nominal value of the centrifugal force-Coriolis force matrix; It is the gravity vector; The second derivative represents the desired position; , Represents the derivative of a specified time-adjusting function; symbol Represents the vector modulo function; symbol , as well as Represented as:

[0089] ;

[0090] ;

[0091] ; ;

[0092] ;

[0093] in, , Representation matrix The values ​​of the upper and lower bounds; for identity matrix of order 1; symbol This represents the function for constructing a diagonal matrix; , , All are known positive constants, and their coefficients satisfy the following relationship:

[0094] ;

[0095] ;

[0096] ;

[0097] In the formula, For matrix The upper bound coefficient of the norm, For vectors The upper bound coefficient of the norm, For an uncertain matrix The upper bound coefficient of the norm, For vectors The upper bound coefficient of the norm, External interference vector The upper bound is defined above all positive numbers.

[0098] Step 5: Deploy the designed sliding surface design controller to the robot control system. During actual operation, first complete parameter initialization: set the specified convergence time. Select the adjustment function parameters and sliding surface parameters, and specify the desired position trajectory. and its derivative During each control cycle, the actual position is acquired through the joint encoder and speed sensor. and speed Calculate position tracking error and Then, based on the specified time adjustment function and the defined nonlinear piecewise function... Construct a non-singular terminal sliding surface s; then, design a controller based on the sliding surface, calculate the equivalent control term and robust switching term, and obtain the control torque. The output is then sent to the robot joint actuator, and the above process is repeated until the preset tracking task is completed. Throughout the control process, due to the forced convergence characteristic of the specified time adjustment function, the system state will... Previously, the robot reached the sliding surface and converged to its minimum neighborhood along the sliding surface, with the convergence time being independent of the initial joint position and velocity. This implementation process can be directly applied to various industrial robots, collaborative robots, or aerospace robotic arms to achieve precise trajectory tracking control within any specified time.

[0099] Example 2:

[0100] This embodiment uses a planar dual-joint rigid robot as an example to verify the effectiveness of the specified-time non-singular end sliding mode trajectory tracking control method proposed in this invention. The simulation period of the system is set to 0.01s, and the relevant parameters of this uncertain robot system are set as follows:

[0101] ;

[0102] ;

[0103] ;

[0104] In the formula, the system model parameters The definitions and values ​​are as follows:

[0105] ;

[0106] ;

[0107] In the formula, Let the mass of robot lever 1 be... For the mass of robot lever 2, The length of robot lever 1, The length of robot rod 2, Let the moment of inertia of robot lever 1 be... Let the moment of inertia of robot lever 2 be... This is the acceleration due to gravity.

[0108] In addition, external interference includes:

[0109] ;

[0110] The desired trajectory of the two-jointed robot is set as follows:

[0111] ;

[0112] Finally, the other control parameters are shown in Table 1:

[0113] Table 1 Control Parameter Setting Table:

[0114] Example Analysis:

[0115] This example comprehensively verifies the effectiveness of the proposed time-based non-singular terminal sliding mode control method from two aspects: different initial conditions and a specified time. This method can achieve precise pre-setting of the convergence time, exhibits strong robustness to initial conditions, high steady-state accuracy, and avoids singularity issues, providing an effective solution for high-performance trajectory tracking control of robot systems.

[0116] Figure 2 (a) and (b) are the position tracking error curves of joint 1 and joint 2 of the robot system at different specified times, respectively. The initial state of the robot system is set to... , Set the convergence time to As can be seen from the graph, for each specified time, the tracking error is within the set range. It converges to the neighborhood of zero before a certain time, and the convergence time closely matches the preset value. At different specified times, the error accuracy after entering steady state is consistently within a certain range. The results, measured in rads, indicate that changes in the convergence time do not affect the steady-state control accuracy. These results fully demonstrate that the method of this invention can flexibly preset the convergence time according to actual needs, meeting the convergence speed requirements of different application scenarios, and the convergence process is smooth and reliable.

[0117] Figure 3 (a) and (b) show the position tracking error curves of joint 1 and joint 2 of the robot system under different initial conditions, respectively. The specified convergence time is set to... The initial conditions for the three operating conditions are shown in Table 2. As can be seen from the figure, although the initial errors of the three operating conditions differ significantly, the tracking errors under all conditions are within a certain range. Within the neighborhood of zero, the convergence time is independent of the initial conditions. After entering the steady-state phase, the steady-state accuracy is unaffected by the initial conditions. These results demonstrate that the method of this invention is highly robust to different initial conditions and can achieve high-precision trajectory tracking within a specified time, meeting the needs of practical applications where the initial state is difficult to obtain accurately.

[0118] Table 2 Initial conditions for three operating conditions:

[0119] The various embodiments in this application are described in a progressive manner. The same or similar parts between the various embodiments can be referred to each other. Each embodiment focuses on describing the differences from other embodiments.

[0120] The scope of protection of this application is not limited to the embodiments described above. Obviously, those skilled in the art can make various modifications and variations to this disclosure without departing from the scope and spirit of this disclosure. If such modifications and variations fall within the scope of the methods disclosed herein and their equivalents, then the intent of this disclosure also includes such modifications and variations.

Claims

1. A design method for a robot trajectory tracking controller based on a specified time, characterized in that, Includes the following steps: Step 1: Based on the robot dynamics under model uncertainties and external disturbances, establish an open-loop robot dynamics system; Step 2: Design a time-adjustment function; Step 3: Construct a non-singular terminal sliding surface with specified time convergence characteristics by combining a specified time adjustment function; Step 4: Design a controller based on the designed sliding surface; Step 5: Deploy the designed sliding surface controller to the robot control system to achieve robot control.

2. The design method for a robot trajectory tracking controller based on a specified time according to claim 1, characterized in that, The robot dynamics under model uncertainty and external disturbances mentioned in step 1 are as follows: ； in, These are the joint position, velocity, and acceleration vectors, respectively. It is a symmetric positive definite inertial matrix. The matrix contains Coriolis force and centrifugal force. The gravity vector To control the input vector, This is the external interference vector. Represents an n-dimensional real vector space. Represents the n-dimensional real matrix space; The open-loop robot dynamics system is as follows: ； in, The nominal value of the inertia matrix of the robot system; The nominal value of the centrifugal force-Coriolis force matrix; The nominal value representing the gravity vector of the robot system. This represents the uncertainty value of the robot system's inertia matrix. This represents the uncertainty value of the Coriolis force and centrifugal force matrix of the robot system. This represents the uncertainty value of the gravity vector of the robot system; This represents the lumped uncertainty term of the robot system.

3. The design method for a robot trajectory tracking controller based on a specified time according to claim 1, characterized in that, The specified time adjustment function mentioned in step 2 is: ； ； ； Where t represents time, T p For the specified convergence time, T p >0, when When specifying a time adjustment function Approaching infinity, used to ensure that the closed-loop system reaches infinity within a specified time. When the internal temperature reaches stability, T p - This is the left limit of the convergence time; , , , , , For the positive constants to be designed.

4. The design method for a robot trajectory tracking controller based on a specified time according to claim 1, characterized in that, Step 3 specifically involves defining the following nonlinear piecewise function: ； Where r is a positive constant and satisfies r>1, and x is the independent variable. It is a positive integer and satisfies ,symbol For standard symbolic functions, symbolic To take the absolute value; The non-singular terminal sliding surface with specified time convergence characteristics is: ； ； in, Represent the non-singular terminal sliding surface; define the position error vector. Its weight The tracking error for the nth joint; the actual joint position vector. , Represents the actual position of the nth joint; the expected joint trajectory vector. ,in Let n be the desired position of the nth joint; superscript This indicates the transpose of the corresponding matrix; For vectors The derivative; parameters It is a positive integer and satisfies and , This is the exponential parameter in the adjustment function.

5. The design method for a robot trajectory tracking controller based on a specified time according to claim 1, characterized in that, In step 4, the sliding surface design controller is: ； ； ； ； ； in, Indicates the nominal terms of a nonlinear mechanical region system; The nominal value of the centrifugal force-Coriolis force matrix; It is the gravity vector; The second derivative represents the desired position; , Represents the derivative of a specified time-adjusting function; symbol Represents the vector modulo function; symbol , as well as Represented as: ； ； ； ； ； in, , Representation matrix The values ​​of the upper and lower bounds; for identity matrix of order 1; symbol This represents the function for constructing a diagonal matrix; , , All are known positive constants, and their coefficients satisfy the following relationship: ； ； ； In the formula, For matrix The upper bound coefficient of the norm, For vectors The upper bound coefficient of the norm, For an uncertain matrix The upper bound coefficient of the norm, For vectors The upper bound coefficient of the norm, External interference vector The upper realm.

6. The design method for a robot trajectory tracking controller based on a specified time according to claim 1, characterized in that, Step 5 specifically involves: during actual runtime, first completing parameter initialization: setting a specified convergence time. Select the adjustment function parameters and sliding surface parameters, and specify the desired position trajectory. and its derivative During each control cycle, the actual position is acquired through the joint encoder and speed sensor. and speed Calculate position tracking error and Then, based on the specified time adjustment function and the defined nonlinear piecewise function... Construct a non-singular terminal sliding surface s; then, design a controller based on the sliding surface, calculate the equivalent control term and robust switching term, and obtain the control torque. The output is then sent to the robot joint driver, and the above process is repeated until the preset tracking task is completed.

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