Mechanical arm trajectory tracking control method based on preset time disturbance observer and all-drive system method

Abstract

The application discloses a mechanical arm trajectory tracking control method based on a preset time disturbance observer and a full drive system method, adopts a nominal dynamics model of a mechanical arm, considers lumped disturbance (including model uncertainty and external environment disturbance), introduces an existing preset time disturbance observer (PTDO) to estimate the lumped disturbance, utilizes a high-order full drive system method (HOFA) to perform strict error transformation and dynamics full drive, combines observation output and residual error boundary to construct a composite control law containing nominal full drive control and robust compensation, and thus realizes accurate trajectory tracking of the mechanical arm within preset time of a user. The method effectively improves transient performance and robustness of a controller, and ensures that trajectory tracking of the mechanical arm is strictly constrained within a preset envelope boundary.

Description

Technical Field

[0001] This invention belongs to the field of nonlinear system control and relates to a robotic arm trajectory tracking control method, specifically a robotic arm trajectory tracking control method based on a Prescribed-Time Disturbance Observer (PTDO) and a High-Order Fully Actuated (HOFA) method. Background Technology

[0002] Robotic arms are highly nonlinear, strongly coupled, and difficult to model accurately. In real-world working environments, robotic arms not only face uncertainties in model parameters such as friction and payload variations, but are also subject to unknown environmental disturbances. These factors pose significant challenges to the high-precision trajectory tracking and control of robotic arms.

[0003] While traditional sliding mode control and other methods offer some robustness, they often only guarantee system convergence within an infinite or finite time, and the convergence time typically depends on the initial state. In recent years, preset-time control has attracted widespread attention because it can achieve system convergence within any user-specified preset time, and the convergence time is independent of the system's initial conditions.

[0004] However, existing preset time control methods still have the following shortcomings when applied to robotic arm systems with unknown disturbances: First, conventional preset time observers, when dealing with systems with integral correction terms, are prone to divergence or numerical singularity problems due to the time-varying gain tending to infinity, and lack rigorous boundedness proofs; Second, in controller design, conventional scaling variable methods or error transformation methods generate extremely complex nonlinear residual terms during the differentiation process, leading to difficulties in controller design; Third, how to effectively connect the disturbance estimation results given by existing preset time disturbance observers with the all-drive system control framework, and rigorously characterize the impact of observation residuals on the closed-loop system mathematically, thereby achieving preset time stable tracking under actual physical constraints, remains a problem that needs further improvement.

[0005] Therefore, considering the model uncertainty, external disturbances, transient performance and steady-state accuracy requirements, there is an urgent need for a robotic arm preset time trajectory tracking control method that is structurally clear, theoretically rigorous and engineering-feasible. Summary of the Invention

[0006] This invention provides a robotic arm trajectory tracking control method based on a preset time disturbance observer and a full-drive system method. This method effectively improves the transient performance and robustness of the controller and ensures that the robotic arm trajectory tracking is strictly constrained within a preset envelope boundary.

[0007] The objective of this invention is achieved through the following technical solution:

[0008] A robotic arm trajectory tracking control method based on a preset time disturbance observer and an all-drive system includes the following steps:

[0009] Step 1: For models considering uncertainties and external disturbances... The problem of trajectory tracking control for a rigid manipulator with multiple degrees of freedom is addressed by establishing a nominal dynamic model of the system and error state equations.

[0010]

[0011] in, , Let the joint position vector be... For joint velocity vectors, For the nominal dynamic model, To control the input, This refers to the lumped disturbance that includes both model uncertainty and external perturbations.

[0012] Step 2: Use a preset time interference observer (PTDO) to monitor lumped interference. Make an estimate:

[0013]

[0014] in, For speed estimation error, and To set the gain, For time-varying gain, , Set a convergence time for the observer. To switch the gain;

[0015] Step 3: Define the trajectory tracking error of the joint position. ,in This represents the actual joint trajectory. Let the desired joint trajectory be defined; let the initial time regularization variable be defined. ,in For time, To avoid the small normal number of the initial singularity, a pre-defined time performance function in the form of exponential decay is designed. To explicitly specify the controller preset time and steady-state accuracy ;

[0016] ;

[0017] Based on the theory of all-drive systems, component-wise intermediate variables are introduced. and composite function :

[0018]

[0019] in , and For design parameters, For the robotic arm joint position tracking error; the normalized generalized transform state is defined as... , ;

[0020] Step 4: Perform higher-order differentiation and full-drive modeling on the normalized generalized transformation state in Step 3 to derive the second-order full-drive nonlinear system model:

[0021]

[0022] in, for Regarding time The second derivative, The transformation gain matrix, The nonlinear remainder term is known for the kinematics and dynamics after aggregation. The vector of the kinematic nonlinear remainder term;

[0023] The second-order total drive model is obtained by expanding the normalized transformation state using the first and second total derivatives:

[0024]

[0025] Define the transform gain , And write it in vector form: ;

[0026] Step 5: Based on the second-order all-drive nonlinear system model from Step 4 and the observer output from Step 2. The design incorporates a composite control law that includes nominal all-wheel drive control and robust compensation terms:

[0027]

[0028] Among them, the nominal control law Used to construct the desired closed-loop properties, where The constant matrix to be designed; robust compensation control law Used to eliminate residuals caused by gain cutoff in practical applications of the observed instrument, where This is the upper bound of the observer residual. Any positive design parameters set by the user, matrix satisfy , To expand the state vector, As a constant, this enables the robotic arm to achieve consistent, ultimately bounded stability and practical preset time trajectory tracking.

[0029] Compared with the prior art, the present invention has the following advantages:

[0030] 1. This invention, based on the introduction of a preset time disturbance observer, transforms its error analysis results into a residual upper bound that can be directly used by the subsequent all-drive controller. Furthermore, it employs a differential inequality analysis method based on integral factors combined with L'Hôpital's rule to address the observation error within a preset time... The convergence process within the system is rigorously characterized, thus providing a clear basis for the robust compensation term in the composite control law.

[0031] 2. Based on the obtained disturbance estimation results, this invention constructs the controller design main line around HOFA error transformation and full-drive modeling. By expanding the second derivative in detail and simplifying it using Clairaut's theorem, the source of nonlinear variables is fully presented, avoiding the problem of difficult controller design in conventional error transformation methods.

[0032] 3. To address the gain overflow problem in digital control in practical engineering, this invention defines the residual for the gain cutoff condition of the adopted observer and embeds this residual boundary into the robust compensation control law. This eliminates uncertain cross terms that disrupt stability and ensures that the tracking error is within acceptable limits at any user-specified time. Converging and maintaining steady-state accuracy Inside. Attached Figure Description

[0033] Figure 1 This is a flowchart of the robotic arm trajectory tracking control method based on a preset time disturbance observer and an all-drive system method according to the present invention.

[0034] Figure 2 This is a logical architecture diagram of the robotic arm trajectory tracking control system based on the preset time disturbance observer and all-drive system method of the present invention;

[0035] Figure 3 The example shows the trajectory tracking and error curves of a two-degree-of-freedom robotic arm.

[0036] Figure 4 The two-degree-of-freedom robotic arm perturbation observer and error curve are shown in the embodiment.

[0037] Figure 5 The control torque curve of the two-degree-of-freedom robotic arm in the embodiment is shown.

[0038] Figure 6 The example shows the trajectory tracking curve of the five-degree-of-freedom robotic arm.

[0039] Figure 7 The example shows the trajectory tracking error curve of the five-degree-of-freedom robotic arm.

[0040] Figure 8 The figure shows the disturbance observer curve of the five-DOF robotic arm in the embodiment;

[0041] Figure 9 The control torque curve of the five-degree-of-freedom robotic arm in the embodiment is shown. Detailed Implementation

[0042] The technical solution of the present invention will be further described below with reference to the accompanying drawings, but it is not limited thereto. Any modifications or equivalent substitutions to the technical solution of the present invention that do not depart from the spirit and scope of the technical solution of the present invention should be covered within the protection scope of the present invention.

[0043] This invention provides a robotic arm trajectory tracking control method based on a preset time disturbance observer and a high-order homing system method. The method employs the nominal dynamics model of the robotic arm, considering lumped disturbances (including model uncertainties and external environmental disturbances). Based on estimating the lumped disturbances using an existing preset time disturbance observer (PTDO), it utilizes the high-order homing system method (HOFA) for rigorous error transformation and homing dynamics, and combines the observed output and its residual boundaries to construct a composite control law including nominal homing control and robust compensation terms, thereby achieving accurate trajectory tracking of the robotic arm within a user-preset time. Figure 1 As shown, the specific steps are as follows:

[0044] Step 1: System Modeling and Problem Description

[0045] consider The dynamic equations of a rigid manipulator with degrees of freedom:

[0046] (1)

[0047] in, Let the joint position vector be... and These are the joint velocity vector and the joint acceleration vector, respectively. The inertia matrix, The matrix represents the Coriolis force and the centrifugal force. For gravity, To control the input, This refers to the external disturbance torque.

[0048] Using the nominal dynamics model Rewrite the system and define lumped interference. Includes model uncertainties and external disturbances, the nominal dynamic model of the system, and the error state equation:

[0049] (2)

[0050] Furthermore, in the following disturbance estimation analysis, the following assumptions are established: lumped disturbance It is continuously differentiable and its first derivative is bounded, meaning there exists a constant. , making .

[0051] Step 2: Introduction of the preset time interference observer and controller design:

[0052] In this invention, a type of Preset Time Interference Observer (PTDO) is used to estimate lumped interference, which can be written in the following form:

[0053] (3)

[0054] In the above structure, For velocity estimation error; the parameter is defined as follows: , where parameters , Preset convergence time for the observer; time-varying gain is Its time Time tends towards infinity; To switch the gain, the following must be met: In engineering implementation, the time-varying gain saturation cutoff is: , This is a preset constant threshold.

[0055] Due to practical physical limitations preventing the gain from reaching infinity, the observer used will inevitably have residuals. (Definition) Let be the transformation gain matrix derived from the first-order derivative mapping of the all-drive model (its specific form will be derived in detail later). Assume the observer has a preset convergence time. and residual upper bound This makes the following practical stability conclusions hold:

[0056] (4)

[0057] In the design of a preset time controller based on HOFA, the trajectory tracking error of the joint position is first defined. ,in Let be the desired joint trajectory. ,in To avoid the small normal numbers of initial singularity, a pre-defined time performance function in the form of exponential decay is designed. Explicitly specify the controller preset time and steady-state accuracy :

[0058] (5)

[0059] The control objective here is to ensure that the error is strictly within the preset envelope, i.e.

[0060] Based on the above performance function, HOFA error transformation is performed, and component-wise intermediate variables are introduced. and composite function ,in , and Design parameters for the controller:

[0061] (6)

[0062] Therefore, the normalized transformation state is defined as:

[0063] (7)

[0064] Step 3: Error analysis and rigorous convergence proof of the PTDO used:

[0065] To facilitate the subsequent design of the all-drive controller, this step performs a rigorous error dynamics analysis on the preset time disturbance observer (PTDO). This step employs a differential inequality analysis method based on the integral factor, combined with L'Hôpital's rule, to determine the observation error within the preset time... The proof process of exact convergence and bounded residuals.

[0066] (1) Derivation of the dynamics of the observer error

[0067] The velocity estimation error is obtained by subtracting the system dynamics equation (1) from the observer equation. The differential equation is:

[0068] (8)

[0069] Substitute into the lumped disturbance estimation law of the observer We can obtain:

[0070] (9)

[0071] For ease of analysis, we define an auxiliary variable for uncompensated interference residuals. Therefore, the error system of the observer used can be transformed into the following cascaded form:

[0072] (10)

[0073] (2) Auxiliary variables Finite-time boundedness proof

[0074] Before proving the convergence of the core error, it is necessary to prove the external forcing term. It is uniformly bounded before the preset time arrives. Based on the aforementioned bounded setting regarding the rate of change of external disturbances, there exists a constant. , making Consider equation (10), for any component Taking the absolute value, we have:

[0075] (11)

[0076] Within a limited time interval Integrating the above equation, we get:

[0077] (12)

[0078] Because the analysis interval is limited Therefore The components of this function will not escape in finite time. Transforming to the L2 norm, there must exist a time-independent... constant , so that:

[0079] (13)

[0080] This conclusion indicates that, due to the switching item The existence of auxiliary variables It is strictly limited to a definite upper bound within a finite, pre-defined time interval. within the tight set.

[0081] (3) Proof of rigorous pre-defined time convergence based on the integral factor method

[0082] Next, we will demonstrate the state estimation error. At the preset time point It converges precisely to zero. A positive definite, radially unbounded Lyapunov function is chosen. .

[0083] Taking its derivative, we get:

[0084] (14)

[0085] Using the Cauchy-Schwarz inequality and The upper bound (13) is obtained by scaling the cross terms:

[0086] (15)

[0087] Define the state of a scalar system .when hour, Substituting this into the above equation, we obtain the standard first-order time-varying linear differential inequality:

[0088] (16)

[0089] To solve this time-varying differential inequality, an integration factor is introduced:

[0090] (17)

[0091] Substitute the preset time gain Calculate the integral and obtain the integration factor. The explicit parsing expression is:

[0092] (18)

[0093] Multiply both sides of inequality (16) by a non-negative factor. :

[0094] (19)

[0095] In the time interval Integrating and rearranging, we get The upper bound of closed analytic expression:

[0096] (20)

[0097] At this point, a rigorous analysis is required. hour The limiting evolution behavior: For the limit of the initial value decay term of the first term, since when Time gain effect makes Therefore:

[0098] (twenty one)

[0099] For the cumulative limit of the second forced term, which constitutes a typical indeterminate form of infinite ratio to infinite, we can use L'Hôpital's rule for differentiation analysis:

[0100] (twenty two)

[0101] By definition of the integrating factor, its derivative is exactly... Substituting into the limit expression, we get:

[0102] (twenty three)

[0103] Combining the two limits mentioned above, we can obtain a rigorous convergence conclusion:

[0104] (twenty four)

[0105] This indicates that regardless of the initial system error or lumped disturbances... How it works in time-varying gain Driven by this, the velocity estimation error of the observer used It will definitely be at the exact preset time. It converges completely to zero.

[0106] (4) Practical stability analysis and observation residual definition

[0107] In practical digital control, time-varying gain exist This could potentially lead to singularities. Therefore, in engineering implementations, it is truncated to a certain extremely large preset constant threshold. ,Right now:

[0108] (25)

[0109] when Approaching hour, Saturation At this point, the differential inequality (16) transforms into an inequality with constant coefficients:

[0110] (26)

[0111] This means that the estimation error will be exponentially confined to a very small bound, i.e.:

[0112] (27)

[0113] because Determined by the designer and typically set to a large value, observation errors can be suppressed to a negligible level. Correspondingly, the final residuals of the disturbance estimation... It will also be confined to the neighborhood of the origin.

[0114] Based on the control law design requirements in step 5, consider the HOFA transformation gain. There must exist a controllable upper bound for the positive real residual. , so that:

[0115] (28)

[0116] This definitive conclusion eliminates the ambiguity of the ideal assumption and will directly serve as the basis for the robust compensation boundary in the subsequent design of the HOFA closed-loop controller.

[0117] Step 4: Rigorous derivation and full-drive transformation of HOFA:

[0118] This step rigorously derives the transformed second-order dynamic equations.

[0119] Step 1: Find the first total derivative:

[0120] To achieve full-drive capability, it is necessary to obtain and The explicit relationship between them. Note It's about error. and time A bivariate function. Using the rule of total differentials, find its first total derivative:

[0121] (29)

[0122] Step 2: Find the second total derivative:

[0123] Again with respect to the first derivative with respect to time Differentiating by applying the multiplication rule and the chain rule, we get:

[0124] (30)

[0125] Let's continue expanding the total derivative within the parentheses. The first term expands to:

[0126] (31)

[0127] The third item expands to:

[0128] (32)

[0129] Substituting the above expansion back... The expression is derived using Clairaut's theorem (i.e., the mixed partial derivatives are equal). After sorting, we get:

[0130] (33)

[0131] Define the transform gain And define the kinematic nonlinear remainder term as:

[0132] (34)

[0133] Rewriting the above equation in vector form, we obtain the rigorous total drive transformation equation:

[0134] (35)

[0135] Wherein, the transform gain matrix is The kinematic nonlinear remainder vector is .

[0136] Step 3: Dynamics modeling of the all-drive system:

[0137] use Substituting the system dynamics equation (1) into the vector form equation (35):

[0138] (36)

[0139] The resulting second-order all-drive nonlinear system model is:

[0140] (37)

[0141] Here, all known nonlinear terms within the system are defined as follows:

[0142] (38)

[0143] Step 5: HOFA Controller Design and Stability Analysis:

[0144] This step is based on the parameterized paradigm of the high-order all-drive system, and designs the controller by combining the disturbance estimation results obtained in step 2 and the residual boundary given in step 3. To overcome nonlinear uncertainties and observer residuals, a composite control law including nominal all-drive control and robust compensation terms is constructed, and a rigorous and practical proof of its time stability is provided.

[0145] (1) Construction of closed-loop error system and design of control law

[0146] To facilitate writing and ensure the rigor of matrix dimensions, an extended system state vector is defined. For the second-order all-drive model (37), a composite control law with the following structure is designed:

[0147] (39)

[0148] Substituting the control law (39) into the system dynamics and canceling out the known nonlinear terms, we get:

[0149] (40)

[0150] make The sub-control laws are designed as follows:

[0151] Nominal all-wheel drive control law The property used to construct the desired linear closed-loop characteristic is defined as:

[0152] (41)

[0153] in This is the constant matrix to be designed.

[0154] Robust Compensation Control Law Used to eliminate the adverse effects of the residuals of the observed instruments, defined as:

[0155] (42)

[0156] in This is the upper bound of the observer residual. Any positive design parameters set by the user.

[0157] Define the state-space system matrix With input matrix for:

[0158] (43)

[0159] Therefore, the system can be strictly converted into the first-order closed-loop state-space form of the HOFA standard:

[0160] (44)

[0161] (2) Stable configuration conditions of closed-loop matrix

[0162] To ensure the closed-loop system has the desired convergence rate, the gain matrix can be selected using the pole placement method. and This makes the closed-loop system matrix All eigenvalues ​​satisfy , where constant This represents the expected convergence rate. According to Lyapunov stability theory, there exists a symmetric positive definite matrix at this point. This makes the following inequality hold:

[0163] (45)

[0164] (3) Practical proof of final boundedness of closed-loop system

[0165] Choose a standard quadratic Lyapunov function:

[0166] (46)

[0167] Taking its derivative, we get:

[0168] (47)

[0169] Substituting inequality (45) into the above closed-loop matrix stable configuration condition, we get:

[0170] (48)

[0171] Substituting the robust control law into the second term of the above equation, we get:

[0172] (49)

[0173] For the third interference coupling term, the Cauchy-Schwarz inequality and the observer residual limit are used. ,have:

[0174] (50)

[0175] Next, we apply the bounding rule of algebraic inequalities. For any real number... and parameters Hengyou Unfold to obtain .make Then we have:

[0176] (51)

[0177] Substitute all expansions back In the inequality, due to the robust compensation design, the uncertain cross terms that might violate stability are completely canceled out, resulting in:

[0178] (52)

[0179] Solving this first-order linear differential inequality, we obtain The evolutionary boundary is:

[0180] (53)

[0181] when hour, Will be restricted to Within the neighborhood of [the system], thus proving the state of the closed-loop system. It is uniformly bounded. There must exist a constant. This makes it possible for all All have .

[0182] (4) Proof of state boundedness to preset time tracking

[0183] prove The boundedness is that the physical tracking error satisfies the preset performance constraints. Sufficient condition.

[0184] Based on the definition of HOFA normalization transformation Suppose at some point Tracking error Touching the performance boundary ,Right now .

[0185] According to composite variables The definition at this time Further examination of the mapping function The limit at this point is:

[0186] (54)

[0187] if ,because This will inevitably lead to the divergence of the transformed states, that is:

[0188] (55)

[0189] This divergent conclusion is consistent with the rigorous proof above. This creates a direct contradiction. Therefore: for all ,error It must be strictly limited to the boundaries Inside, it's impossible to touch the boundary. That is:

[0190] (56)

[0191] At the same time, according to the performance function The definition, in back, This ensures that the tracking error is within the preset time specified by the user. Converges and maintains the preset accuracy Inside.

[0192] In summary, for cases involving unknown nonlinear dynamics and external disturbances... The degree-of-freedom robotic arm system (1) adopts a closed-loop system (44) containing PTDO observation output and HOFA parameterized composite control law (39). Under the condition of satisfying the above-mentioned closed-loop matrix stable configuration, it can guarantee that: First, all internal signals of the closed-loop system, including , All of these are uniformly bounded; secondly, the tracking error of the robotic arm joint positions. Never exceed the preset performance envelope and at the preset time The accuracy shall not be lower than the preset precision. The standard achieves stable tracking, thereby enabling practical preset time trajectory tracking control.

[0193] Example

[0194] Two simulation examples, a two-degree-of-freedom robotic arm and a five-degree-of-freedom robotic arm, are given below to further illustrate the feasibility of the method of the present invention.

[0195] Example 1: Two-DOF robotic arm

[0196] The actual parameters of the two-degree-of-freedom robotic arm are: kg, kg, m、 m、 m、 m、 kg , kg , m / The nominal model parameters are set as follows: kg, kg, m、 m、 m / .

[0197] The observer preset time is taken in the simulation. s, controller preset time s, gain limiting Control torque limit N m, buffer time The parameters used for PTDO are: , , , The HOFA controller parameters are set as follows: , , , , , , , .

[0198] The initial conditions are set as follows: , Observer initial state Initial value of the integral term The expected trajectory is:

[0199] (57)

[0200] External disturbances are taken as:

[0201] (58)

[0202] The simulation duration is set to 16 seconds. Figure 2 The flow chart of the control method of the present invention is given. Figures 3 to 5The trajectory tracking and error curves, disturbance observation effects, and control torque curves of the two-degree-of-freedom manipulator are presented respectively.

[0203] Example 2: Five-DOF robotic arm

[0204] The nominal inertial parameter vector of the five-degree-of-freedom robotic arm is taken as:

[0205] (59)

[0206] The nominal gravity parameter vector is taken as The actual parameters are the superposition of the nominal inertial parameters. Gaussian random perturbation, superposition of nominal gravity parameters The random seed is fixed at 42 and obtained by Gaussian random perturbation.

[0207] The observer preset time is taken in the simulation. s, controller preset time s, gain limiting Control torque limit N m, buffer time The parameters used for PTDO are: , , , The HOFA controller parameters are set as follows: , , , , , , , .

[0208] The initial conditions are set as follows: , , , . No. The desired trajectory of each joint is taken as External disturbances are taken as:

[0209] (60)

[0210] The simulation duration is set to 8 seconds. Figures 6 to 9 The trajectory tracking curve, trajectory tracking error curve, disturbance observer curve, and control torque curve of the five-degree-of-freedom robotic arm are presented respectively.

[0211] Depend on Figures 2 to 9It can be seen that, based on the disturbance estimation results provided by PTDO, the all-drive composite control law constructed in this invention can achieve high-precision trajectory tracking within a preset time, and the control input remains smooth, indicating that the method of this invention has good robustness and engineering feasibility.

Claims

1. A robotic arm trajectory tracking control method based on a preset time disturbance observer and a full-drive system method, characterized in that... The method includes the following steps: Step 1: For models considering uncertainties and external disturbances... The problem of trajectory tracking control for a rigid manipulator with multiple degrees of freedom is addressed by establishing a nominal dynamic model of the system and error state equations. in, , Let the joint position vector be... For joint velocity vectors, For the nominal dynamic model, To control the input, This refers to the lumped disturbance that includes both model uncertainty and external perturbations. Step 2: Use a preset time interference observer (PTDO) to monitor lumped interference. Make an estimate: in, For speed estimation error, and To set the gain, For time-varying gain, To switch the gain; Step 3: Define the trajectory tracking error of the joint position. ,in This represents the actual joint trajectory. Let the desired joint trajectory be defined; let the initial time regularization variable be defined. ,in For time, To avoid small normal numbers for singularity at the initial time; Design a preset time performance function in the form of exponential decay. To explicitly specify the controller preset time and steady-state accuracy ; Based on the theory of all-drive systems, component-wise intermediate variables are introduced. and composite function : in , and For design parameters, For the robotic arm joint position tracking error; the normalized generalized transform state is defined as follows: , ; Step 4: Perform higher-order differentiation and full-drive modeling on the normalized generalized transformation state in Step 3 to derive the second-order full-drive nonlinear system model: in, for Regarding time The second derivative, The transformation gain matrix, , The nonlinear remainder term is known for the kinematics and dynamics after aggregation. , ; The second-order total drive model is obtained by expanding the normalized transformation state using the first and second total derivatives: Define the transform gain , And write it in vector form: , The vector of the kinematic nonlinear remainder term; Step 5: Based on the second-order all-drive nonlinear system model from Step 4 and the observer output from Step 2. The design incorporates a composite control law that includes nominal all-wheel drive control and robust compensation terms: in, For the nominal control law, This is a robust compensation control law.

2. The robotic arm trajectory tracking control method based on a preset time disturbance observer and a full-drive system method according to claim 1, characterized in that... In step 2, , , The time-varying gain saturation cutoff is set to a preset constant threshold as follows: , Set a convergence time for the observer.

3. The robotic arm trajectory tracking control method based on a preset time disturbance observer and a full-drive system method according to claim 1, characterized in that... In step 2, .

4. The robotic arm trajectory tracking control method based on a preset time disturbance observer and a full-drive system method according to claim 1, characterized in that... In step 5 ,in This is the constant matrix to be designed.

5. The robotic arm trajectory tracking control method based on a preset time disturbance observer and a full-drive system method according to claim 1, characterized in that... In step 5 ,in This is the upper bound of the observer residual. Any positive design parameters set by the user, matrix satisfy , To expand the state vector, It is a constant. For the state-space system matrix, The input matrix is ​​denoted as .

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