Task space variable exponential coefficient fixed time compound control method for Stewart parallel mechanism of sand blasting derusting parallel robot

Abstract

The invention discloses a variable exponential coefficient fixed time compound control method for a task space of a Stewart parallel mechanism of a sand blasting derusting parallel robot. Establishing an uncertain Stewart parallel mechanism dynamic model by adopting a Lagrangian method; based on an uncertain Stewart parallel mechanism dynamic model, a variable exponential coefficient fixed time sliding mode disturbance observer is designed, and meanwhile, a variable exponential power adjustment item also enables the observer to avoid generating overlarge integral increment, so that the stability and smoothness of an estimation process are improved; secondly, designing a variable exponential coefficient fixed time sliding mode control algorithm by constructing a sliding mode function with a variable exponential power term and an approaching law, and combining the variable exponential coefficient fixed time sliding mode control algorithm with a variable exponential coefficient fixed time sliding mode disturbance observer, so that rapid convergence of task space tracking control of the uncertain Stewart parallel mechanism is realized within fixed time, and the interference of the variable exponential coefficient fixed time sliding mode control algorithm is avoided; and tracking overshoot and oscillation can be effectively relieved, a stable and continuous tracking process is generated, and excessive control input is avoided.

Description

Technical Field

[0001] The invention belongs to the technical field of parallel mechanism control and relates to a task space variable exponential coefficient fixed time sliding mode composite control method of a Stewart parallel mechanism of a sandblasting and rust removal parallel robot. Background Art

[0002] Currently, sandblasting and rust removal operations on steel box girders of bridges are mostly manual labor, resulting in low efficiency, high safety risks, and potential health hazards for operators. Existing sandblasting and rust removal robots are mostly based on serial mechanism designs, resulting in low stiffness, weak load-bearing capacity, a low deadweight-to-load ratio, and poor mobility. Furthermore, they are difficult to perform sandblasting and rust removal operations on large steel components such as steel box girders with complex curved surfaces and U-shaped ribs. To address this issue, a sandblasting and rust removal parallel robot based on a Stewart parallel mechanism was developed. This robot, consisting of a Stewart parallel mechanism, a lifting mechanism, and a mobile platform, offers high stiffness, strong load-bearing capacity, and excellent mobility, enabling arbitrary movement, free lifting, and precise six-degree-of-freedom motion. However, from a control perspective, the Stewart parallel mechanism is a strongly coupled, nonlinear, multi-input, multi-output system with parameter uncertainty. Furthermore, it is susceptible to external disturbances, such as the jet reaction force during sandblasting. Therefore, the present invention proposes a variable exponential coefficient fixed time sliding mode composite control method to achieve high-performance trajectory tracking control in the task space of the Stewart parallel mechanism of a sandblasting and rust removal parallel robot.

[0003] The article "Zhang Wenjie, Gao Guoqin. Adaptive backstepping sliding mode control of sandblasting and rust removal parallel mechanism against non-matching interference" (Zhang Wenjie, Gao Guoqin, Software Guide, 2023, Issue 11, Volume 22, Pages 111-117) proposes an adaptive backstepping sliding mode control method combined with an extended state observer for the Stewart parallel mechanism of the sandblasting and rust removal parallel robot, so as to enable the Stewart parallel mechanism to maintain good trajectory tracking performance under the interference of the sandblasting jet reaction force. However, the extended state observer usually requires a high observer gain to ensure the dynamic response and estimation accuracy of time-varying interference and uncertainty estimation. Excessive observer gain will amplify the measurement noise into the estimation result. In addition, its sliding mode control method is designed based on the finite time stability theory. Its stabilization time theoretically depends on the initial state information of the system. When the system state is far away from the equilibrium point, the stabilization time will tend to infinity, and it is difficult to truly ensure the control performance of the system. The paper "Sliding Mode Control of Manipulator Terminal Based on Sliding Mode Disturbance Observer" (Han Junqing et al., Journal of Central South University (Natural Science Edition), 2020, No. 10, Vol. 51, pp. 2749-2757) uses a sliding mode disturbance observer to estimate the system's lumped uncertainty. Compared with the extended state observer, the sliding mode observer can improve the dynamic response to time-varying disturbances and uncertainties through the high-frequency switching behavior of the sliding mode switching term. However, the estimation of disturbances and uncertainties by this sliding mode disturbance observer can only achieve finite-time convergence. The paper "Event-triggered Formation Control of AUVs Based on Fixed-time Sliding Mode Disturbance Observer" (Su Bo et al., Control and Decision, 2022, No. 5, Vol. 37, pp. 1116-1126) proposes a fixed-time sliding mode disturbance observer that can estimate the system's lumped uncertainty within a fixed time that is independent of the initial state and uniformly bounded. The paper "Fixed-time non-singular piecewise sliding mode control of a hybrid mechanism with fluid resistance interference" (Zhu Zhiyu, Gao Guoqin, Automation Technology and Applications, 2024, first published online) proposes a fixed-time sliding mode controller that can achieve system stability within a fixed time that is independent of the initial state and uniformly bounded. However, both the fixed-time sliding mode observer and the fixed-time sliding mode controller need to introduce multiple constant exponential power terms in their design. When the constant exponential coefficients of these power terms are selected to be greater than 1 to meet performance requirements, if the system state is far from the equilibrium point, the observer and controller may generate excessive integral increments or control quantities, resulting in oscillation or overshoot in the estimation or control process. When the constant exponential coefficients of these power terms are selected to be less than 1, if the system state is close to the equilibrium point, the integral increments or control quantities will also be amplified, making it difficult to ensure the smoothness or stability of the estimation or control process. Summary of the Invention

[0004] In order to overcome the shortcomings of the existing technology, the present invention adopts the Lagrangian method to establish the dynamic model of the Stewart parallel mechanism of the sandblasting and rust removal parallel robot which is affected by uncertainties such as modeling error, joint friction and sandblasting jet reaction force interference; by introducing an auxiliary system to construct a sliding mode switching term and a variable exponential power adjustment term, a variable exponential coefficient fixed-time sliding mode interference observer is designed, which can improve the dynamic response to interference and uncertainty through the high-frequency switching behavior of the sliding mode switching term, reduce the conservatism of the observer gain selection, and realize the lumped uncertainty estimation of the Stewart parallel mechanism within a uniformly bounded fixed time. The rapid convergence of the estimation is achieved. At the same time, the variable exponential power adjustment term can also prevent the observer from generating excessive integral increments, thereby ensuring the stability and smoothness of the estimation process. Secondly, by constructing a sliding mode function and reaching law with a variable exponential power term, a new variable exponential coefficient fixed-time sliding mode control algorithm is designed, and it is combined with a variable exponential coefficient fixed-time sliding mode disturbance observer, so as to not only achieve rapid convergence of the task space tracking control of the uncertain Stewart parallel mechanism within a fixed time and improve its robustness, but also effectively alleviate tracking overshoot and oscillation, produce a smooth and continuous tracking process, and avoid excessive control input.

[0005] The technical solution of the present invention is: a method for controlling a Stewart parallel mechanism of a sandblasting and rust removal robot with a variable exponential coefficient in a task space and a fixed time, characterized by comprising the following steps:

[0006] 1) The Lagrangian method is used to integrate the uncertainties such as modeling error, joint friction, and jet reaction force interference, and the dynamic model of the Stewart parallel mechanism in the task space is established.

[0007] 2) Based on the uncertain Stewart parallel mechanism dynamic model obtained in step 1), an auxiliary system is introduced to construct the sliding mode switching term and the variable exponent power term, and a variable exponential coefficient fixed-time sliding mode disturbance observer is designed to respond to time-varying disturbances and uncertainties through the high-frequency switching behavior of the sliding mode switching term, and to obtain the lumped uncertainty estimate quickly and accurately within a uniformly bounded time, while ensuring the stability and smoothness of the estimation process.

[0008] 3) Based on the dynamic model of the uncertain Stewart parallel mechanism obtained in step 1), a variable exponential coefficient fixed-time sliding mode control algorithm is designed by constructing a sliding mode function and reaching law with variable exponential power terms. The algorithm is combined with the variable exponential coefficient sliding mode disturbance observer obtained in step 2) to obtain a variable exponential coefficient fixed-time sliding mode composite controller, which can not only achieve rapid convergence of the task space tracking of the uncertain Stewart parallel mechanism within a uniformly bounded fixed time and improve its robustness, but also effectively alleviate tracking overshoot and oscillation, produce a smooth and continuous tracking process, and avoid excessive control input.

[0009] 4) A distributed structure, namely the "upper computer (industrial computer) + lower computer (multi-axis motion controller)" structure, is used to construct a variable exponential coefficient fixed-time composite control system for the uncertain Stewart parallel mechanism task space, realizing the task space tracking control of the Stewart parallel mechanism.

[0010] Furthermore, in step 2), an auxiliary system is introduced based on the uncertain Stewart parallel mechanism dynamic model:

[0011]

[0012] The sliding mode variables are constructed as:

[0013]

[0014] Therefore, in order to improve the dynamic response to disturbance and uncertainty estimation, the sliding mode switching term is constructed as:

[0015] s v =sgn(v) (3)

[0016] In order to achieve fast convergence of the lumped uncertainty estimation of the Stewart parallel mechanism within a fixed time, avoid excessive observer integral increments, and ensure the smoothness or stability of the estimation process, the variable exponent power term is constructed as follows:

[0017]

[0018] in is a function with variable exponential coefficient.

[0019] Therefore, based on equations (1)-(4) and the uncertain Stewart parallel mechanism dynamic model, a variable exponential coefficient fixed-time sliding mode disturbance observer is designed as follows:

[0020]

[0021] In formulas (1)-(5), γ1,γ2,γ3,m o ,ρ o ,λ o ,η o >0 is an adjustable parameter; is the estimated value of the lumped uncertainty; q is the end position of the Stewart parallel mechanism; M(q), and G(q) are the nominal inertia matrix, Coriolis and centrifugal force matrices, and gravity term of the Stewart parallel mechanism dynamic model, respectively; J(q) is the Jacobian matrix; is the tracking error of the end position of the Stewart parallel mechanism; q dis the desired pose of the end. e =2πζ e / l s ,ζ e is the motor mechanical efficiency, l s is the motor screw lead; τ is the servo motor driving torque. v is the sliding mode switching item; is the variable exponential power term; v is the sliding mode variable; is a variable exponential power function; sgn(·) is a sign function.

[0022] Under the action of the designed variable exponential coefficient fixed-time sliding mode disturbance observer, the estimation error can converge within a fixed time, and the convergence time is:

[0023]

[0024] where γ2,ρ o >0 is an adjustable parameter; γ o =(m o +1+λ o tanh(1)) / 2 is the upper bound of the variable exponential coefficient function.

[0025] Furthermore, in step 3), in order to make the end position of the Stewart parallel mechanism converge quickly to the desired position within a fixed time, a sliding mode function with a variable exponential power term is constructed as follows:

[0026]

[0027] where λ s ,η s ,ρ s >0 is an adjustable parameter; is the tracking error of the end position of the Stewart parallel mechanism, q d is the desired pose of the end; is a variable exponential coefficient function; is a variable exponent power term; is a variable exponential power function; tanh(·) is a hyperbolic tangent function.

[0028] To achieve sliding mode variables in fixed time In order to achieve rapid convergence and avoid excessive control input, and ensure the stability of the control process, the sliding mode reaching law with variable exponential power term is constructed as follows:

[0029]

[0030] where λ c ,ρ c ,m c ,k1,k2>0 are adjustable parameters; is a variable exponential power function; is a variable exponential coefficient function; tanh(·) is a hyperbolic tangent function.

[0031] Based on equations (5), (7), (8) and the uncertain Stewart parallel mechanism dynamics model, a variable exponential coefficient fixed-time sliding mode composite controller is designed as follows:

[0032]

[0033] Where τ1 is the variable exponential coefficient fixed time sliding mode control term, τ2 is the variable exponential coefficient fixed time sliding mode disturbance observer compensation term; M(q), and G(q) are the nominal inertia matrix, Coriolis and centrifugal force matrices, and gravity term of the Stewart parallel mechanism dynamic model, respectively; J(q) is the Jacobian matrix; Q s =J -T (q)M(q) is the inverse matrix term; and is an auxiliary variable; is the sliding mode variable; e =2πζ e / l s ,ζ e is the motor mechanical efficiency, l s is the lead of the motor screw; τ is the driving torque of the servo motor. Under the action of the variable exponential coefficient fixed time sliding mode compound control proposed, the sliding mode variable It can converge within a fixed time, and the convergence time is:

[0034]

[0035] Where T s is the sliding mode variable convergence time; γ c =(m c +1+λ c tanh(1)) / 2 is the upper bound of the variable exponential coefficient; exp(·) is an exponential function with a natural constant as the base. The tracking error can be converged within a fixed time, and the convergence time is:

[0036]

[0037] where γ s =(1+λ s tanh(1)) / 2 is the upper bound of the variable exponential coefficient; T c is the control system convergence time.

[0038] The present invention proposes for the first time a variable exponential coefficient fixed time sliding mode composite control method for the Stewart parallel mechanism of a sandblasting and rust removal parallel robot to achieve high-performance trajectory tracking control in the Stewart parallel mechanism task space. The method has the following characteristics and beneficial effects:

[0039] 1. A variable exponential coefficient fixed-time sliding mode disturbance observer is designed to estimate the uncertainty in a Stewart parallel mechanism system, including modeling errors, joint friction, and sandblasting jet reaction force interference. Its unique feature is that it introduces an auxiliary system to construct a sliding mode switching term and a variable exponential power adjustment term. This beneficial effect is that the variable exponential coefficient fixed-time sliding mode disturbance observer can respond to time-varying disturbances and uncertainties through the high-frequency switching behavior of the sliding mode switching term, and quickly and accurately obtain the lumped uncertainty estimate within a uniformly bounded time. Furthermore, with the help of the variable exponential power adjustment term, the variable exponential coefficient fixed-time sliding mode disturbance observer can avoid generating excessively large integral increments, thereby ensuring the stability and smoothness of the estimation process.

[0040] 2. A variable exponential coefficient fixed-time sliding mode control algorithm was designed to achieve task-space tracking control of a Stewart parallel mechanism. Its unique feature is that it is based on a sliding mode function with a variable exponential power term and a reaching law. This beneficial effect is that it can achieve rapid convergence of task-space tracking control of an uncertain Stewart parallel mechanism within a uniformly bounded fixed time while avoiding excessive control input. Combining this algorithm with a variable exponential coefficient fixed-time sliding mode disturbance observer effectively improves the robustness of the Stewart parallel mechanism's task-space tracking control and produces a smooth and continuous tracking process. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 This is a prototype diagram of a parallel robot for sandblasting and rust removal.

[0042] Figure 2 This is a simplified diagram of the Stewart parallel mechanism structure.

[0043] Figure 3 It is a schematic diagram of the steel box girder structure of a bridge.

[0044] Figure 4 This is the principle diagram of the Stewart parallel mechanism task space variable exponential coefficient fixed time sliding mode compound control method.

[0045] Figure 5These are the estimation result curves and estimation error curves of the asymptotic sliding mode disturbance observer (AsSMDO), finite time sliding mode disturbance observer (FnTSMDO), fixed time sliding mode disturbance observer (FxTSMDO) and the proposed variable exponential coefficient fixed time sliding mode disturbance observer (VEC-FxTSMDO) for lumped uncertainty.

[0046] Figure 6 These are the terminal posture trajectory tracking curves and tracking error curves of the Stewart parallel mechanism under asymptotic sliding mode compound control (AsSMCC), finite time sliding mode compound control (FnTSMCC), fixed time sliding mode compound control (FxTSMDO) and the proposed variable exponential coefficient fixed time sliding mode compound control (VEC-FxTSMCC).

[0047] Figure 7 These are the driving torque curves of the active joint servo motor of the Stewart parallel mechanism under asymptotic sliding mode compound control (AsSMCC), finite time sliding mode compound control (FnTSMCC), fixed time sliding mode compound control (FxTSMDO) and the proposed variable exponential coefficient fixed time sliding mode compound control (VEC-FxTSMCC).

[0048] Figure 8 This is the block diagram of the control system of the sandblasting and rust removal parallel robot. DETAILED DESCRIPTION

[0049] The specific embodiments of the present invention are further described below with reference to the accompanying drawings.

[0050] Firstly, the Lagrangian method is adopted to lump the effects of uncertainties such as modeling error, joint friction, and jet reaction force interference, and then an uncertain Stewart parallel mechanism dynamic model of the sandblasting and rust removal parallel robot is established in the task space. Based on the uncertain Stewart parallel mechanism dynamic model, an auxiliary system is introduced to construct a sliding mode switching term and a variable exponent power term, and a variable exponential coefficient fixed-time sliding mode disturbance observer is designed. The observer responds to time-varying disturbances and uncertainties through the high-frequency switching behavior of the sliding mode switching term, and the lumped uncertainty estimate is obtained quickly and accurately within a uniformly bounded time. At the same time, the variable exponent power adjustment term can also prevent the observer from generating excessively large integral increments, thereby ensuring the stability and smoothness of the estimation process. By constructing a sliding mode function and reaching law with variable exponential power terms, a variable exponential coefficient fixed-time sliding mode control algorithm is designed. This is combined with a variable exponential coefficient fixed-time sliding mode disturbance observer to achieve rapid convergence of the task space tracking of the uncertain Stewart parallel mechanism within a uniformly bounded fixed time and improve its robustness. It can also effectively alleviate tracking overshoot and oscillation, produce a smooth and continuous tracking process, and avoid excessive control input. Finally, a distributed structure, namely the "host computer (industrial computer) + slave computer (multi-axis motion controller)" structure, is used to construct a variable exponential coefficient fixed-time composite control system for the task space of the uncertain Stewart parallel mechanism to control the end of the Stewart parallel mechanism to move according to the desired trajectory. The specific method is as follows:

[0051] 1) The Lagrangian method is used to establish the dynamic model of the uncertain Stewart parallel mechanism in the task space:

[0052] According to the Lagrangian method, the standard dynamic equation of the Stewart parallel mechanism in the task space is established:

[0053]

[0054] Where M(q) is the positive definite inertia matrix, are the Coriolis force and centrifugal force terms, G(q) is the gravity term; q, are the end position vector, velocity vector and acceleration vector of the Stewart parallel mechanism respectively; ζ e is the mechanical efficiency; l s Screw lead; τ is the driving torque of the active joint servo motor.

[0055] Considering that the Stewart parallel mechanism system is affected by uncertainties such as modeling error, joint friction, and jet reaction force disturbance during operation, the dynamic model of the uncertain Stewart parallel mechanism in the task space is established:

[0056]

[0057] Where, ΔM(q), and ΔG(q) is the modeling error; d r is the jet reaction force interference; d f For joint friction.

[0058] By summing up the effects of modeling error, joint friction, and jet reaction force interference, Equation (13) can be rewritten as

[0059]

[0060] Where d is the lumped uncertainty, which is in the form of:

[0061]

[0062] From formula (15), the dynamic equation of the terminal tracking error of the Stewart parallel mechanism can be obtained as follows:

[0063]

[0064] where q d is the desired end position of the Stewart parallel mechanism, is the tracking error of the Stewart parallel mechanism end, d l =M -1 (q)d is the lumped uncertainty.

[0065] 2) Based on the uncertain Stewart parallel mechanism dynamic model, by introducing the auxiliary system

[0066]

[0067] The sliding mode variables are constructed as:

[0068]

[0069] Among them, η o is an adjustable parameter; and is an auxiliary variable; is the aggregate uncertainty estimate.

[0070] 3) To improve the dynamic response to disturbance and uncertainty estimation, based on (18), the sliding mode switching term is constructed as:

[0071]

[0072] In order to achieve fast convergence of the lumped uncertainty estimation of the Stewart parallel mechanism within a fixed time, avoid excessive observer integral increments, and ensure the smoothness or stability of the estimation process, the variable exponent power term is constructed as follows:

[0073]

[0074] in is a variable exponential coefficient function, m c ∈(0,1),λ o ,ρ o >0.

[0075] 4) To obtain the aggregate uncertainty estimate in a uniformly bounded fixed time At the same time, to ensure the stability and smoothness of the estimation process, based on equations (16)-(20), the variable exponential coefficient fixed-time sliding mode disturbance observer is designed as follows:

[0076]

[0077] Where γ1,γ2,γ3>0 are adjustable parameters.

[0078] Under the action of the designed variable exponential coefficient fixed-time sliding mode disturbance observer, the estimation error can be converged within a uniformly bounded fixed time, and the convergence time is:

[0079]

[0080] where γ o =(m o +1+λ o tanh(1)) / 2.

[0081] 5) In order to make the end position of the Stewart parallel mechanism converge quickly to the desired position within a fixed time, a sliding mode function with a variable exponential power term is constructed as follows:

[0082]

[0083] Among them, λ s ,η s ,ρ s >0 is an adjustable parameter; is the tracking error of the end position of the Stewart parallel mechanism, q d is the desired pose of the end; is a variable exponent power term; is a variable exponential power function; is a variable exponential coefficient function; tanh(·) is a hyperbolic tangent function.

[0084] 6) To achieve sliding mode variables within a fixed time In order to achieve rapid convergence and avoid excessive control input, and ensure the stability and smoothness of the control process, the sliding mode reaching law with variable exponential power term is constructed as follows:

[0085]

[0086] in, is a variable exponential coefficient function, λ c ,ρ c ,m c ,k1,k2>0 are adjustable parameters.

[0087] 7) Based on equations (21), (23), (24) and the uncertain Stewart parallel mechanism dynamic model, a variable exponential coefficient fixed-time sliding mode composite controller is designed as follows:

[0088]

[0089] Where τ1 is the variable exponential coefficient fixed time sliding mode control term, τ2 is the variable exponential coefficient fixed time sliding mode disturbance observer compensation term; is the estimated value of lumped uncertainty, q is the end pose of the Stewart parallel mechanism, is the tracking error of the end position of the Stewart parallel mechanism, q d is the desired position of the end; M(q), and G(q) are the nominal inertia matrix, Coriolis and centrifugal force matrices, and gravity term of the Stewart parallel mechanism dynamic model, respectively; J(q) is the Jacobian matrix; and is the auxiliary variable; v is the sliding mode variable; ξ e =2πζ e / l s ,ζ e is the motor mechanical efficiency, l s is the motor screw lead; τ is the servo motor driving torque; is a variable exponential power function; sgn(·) is a sign function; Q s =J -T (q)M(q) is the inverse matrix term. Under the action of the proposed variable exponential coefficient fixed time sliding mode compound control, the sliding mode variable It can converge within a fixed time, and the convergence time is:

[0090]

[0091] where γ c =(m c +1+λ c tanh(1)) / 2 is the upper bound of the variable exponential coefficient function; the tracking error can converge within a fixed time, and the convergence time is:

[0092]

[0093] where γ s=(1+λ s tanh(1)) / 2; tanh(·) is the hyperbolic tangent function.

[0094] 8) Through software programming, the variable exponential coefficient fixed time sliding mode compound control of the Stewart parallel mechanism of the sandblasting and rust removal parallel robot is realized.

[0095] The motor drive control quantity of each active joint of the Stewart parallel mechanism is calculated according to formula (25), and the analog quantity obtained by digital-to-analog conversion of the control quantity is sent to the servo driver corresponding to the motor to drive the Stewart parallel mechanism moving platform to move according to the desired trajectory.

[0096] The following provides embodiments of the present invention:

[0097] Example 1

[0098] like Figure 1 As shown in the figure, the sandblasting and rust removal parallel robot is mainly composed of a host computer (industrial computer), a slave computer (multi-axis motion controller), a lifting platform, a mobile platform, a sandblasting gun and a Stewart parallel mechanism.

[0099] The present invention mainly focuses on a variable exponential coefficient fixed time sliding mode composite control technology to achieve high performance control of the task space trajectory tracking of the Stewart parallel mechanism of the sandblasting and rust removal parallel robot. The structural diagram of the Stewart parallel mechanism is as follows Figure 2 The principle block diagram of the Stewart parallel mechanism task space variable exponential coefficient fixed time sliding mode compound control system is shown in the figure. Figure 4 As shown, the specific implementation of the control method is as follows:

[0100] 1) Based on the Lagrangian method, the uncertain effects of modeling errors, joint friction, and jet reaction force disturbance are aggregated to establish the dynamic model of the uncertain Stewart parallel mechanism in the task space:

[0101]

[0102] Among them, M(q) is the positive definite inertia matrix, and its specific form is:

[0103]

[0104] Where,

[0105]

[0106] Where m p is the mass of the dynamic platform, I x ,I y ,I zis the moment of inertia; cos(·) is the cosine function; sin(·) is the sine function.

[0107] are the Coriolis force and centrifugal force terms, and their specific forms are:

[0108]

[0109] Where,

[0110]

[0111] G(q) is the gravity term, and its specific form is:

[0112] G(q)=[0 m p g 0000] T (32)

[0113] J(q) is the Jacobian matrix, and its specific form is:

[0114]

[0115] Where s i =L i / l i is the unit vector of the i-th branch, L i =x t +R γβα A i -B i is the position vector of the i-th branch, l i =||L i || is the length of the i-th branch, A i is the coordinate of the connection point between the i-th branch and the moving platform, B i is the coordinate of the connection point between the i-th branch and the fixed base, A i and B i The specific location is shown in Figure 2 The schematic diagram of the Stewart parallel mechanism shown; x t =[x,y,z] T is the end position coordinate of the Stewart parallel mechanism; R γβα is the rotation matrix, and its specific form is:

[0116]

[0117] c and s represent the cosine and sin functions respectively, and α, β, and γ are the end postures of the Stewart parallel mechanism.

[0118] 2) According to formula (28), the dynamic equation of the terminal tracking error of the Stewart parallel mechanism is:

[0119]

[0120] where q d is the desired end position of the Stewart parallel mechanism, is the tracking error of the Stewart parallel mechanism end, d l =M -1 (q)d is the lumped uncertainty.

[0121] 3) Based on the uncertain Stewart parallel mechanism dynamic model and the terminal tracking error dynamic equation, an auxiliary system is introduced

[0122]

[0123] The sliding mode variables are constructed as:

[0124]

[0125] Among them, η o is an adjustable parameter; and is an auxiliary variable; is the aggregate uncertainty estimate.

[0126] 4) Based on (37), the sliding mode switching term is constructed as:

[0127] s v =sgn(v) (38)

[0128] The construction variable exponential power term is:

[0129]

[0130] in is a variable exponential coefficient function, m c ∈(0,1),λ o ,ρ o >0.

[0131] 5) Based on equations (35)-(39), the variable exponential coefficient fixed-time sliding mode disturbance observer is designed as:

[0132]

[0133] Where γ1,γ2,γ3>0 are adjustable parameters.

[0134] Under the action of the designed variable exponential coefficient fixed-time sliding mode disturbance observer, the estimation error can converge within a fixed time, and the convergence time is:

[0135]

[0136] where γo =(m o +1+λ o tanh(1)) / 2.

[0137] 6) Construct a sliding mode function with variable exponential power term as

[0138]

[0139] in, is the tracking error of the end position of the Stewart parallel mechanism, q d is the desired pose of the end; is a variable exponential coefficient function; λ s ,η s ,ρ s >0 is an adjustable parameter.

[0140] 7) Construct the sliding mode reaching law with variable exponential power term as follows:

[0141]

[0142] in, is a variable exponential coefficient function, λ c ,ρ c ,m c ,k1,k2>0 are adjustable parameters.

[0143] 8) Based on equations (21), (23), (24) and the uncertain Stewart parallel mechanism dynamic model, a variable exponential coefficient fixed-time sliding mode composite controller is designed as follows:

[0144]

[0145] Where τ1 is the variable exponential coefficient fixed time sliding mode control term, τ2 is the variable exponential coefficient fixed time sliding mode disturbance observer compensation term. Under the action of the proposed variable exponential coefficient fixed time sliding mode composite control, the sliding mode variable It can converge within a fixed time, and the convergence time is:

[0146]

[0147] where γ c =(m c +1+λ c tanh(1)) / 2. The tracking error can be converged within a fixed time, and the convergence time is:

[0148]

[0149] where γ s =(1+λ stanh(1)) / 2.

[0150] 9) Through software programming, the variable exponential coefficient fixed-time sliding mode compound control of the uncertain Stewart parallel mechanism task space is realized.

[0151] The Stewart parallel mechanism task space variable exponential coefficient fixed time sliding mode control system is based on the "host computer + slave computer" distributed control structure, which has high flexibility and reliability. It is suitable for large-scale, complex control systems and scenarios with high requirements for system response speed and real-time performance. The control system structure is as follows: Figure 8 As shown in the figure, the control system consists of a host computer (industrial computer), a slave computer (multi-axis motion controller), a servo system, and a parallel robot for sandblasting and rust removal. The slave computer communicates with the host computer via an Ethernet port. The motion controller collects encoder information from the active shutdown servo motor of the Stewart parallel mechanism via an expansion card. It then converts the calculated motor drive control quantity for each active joint of the Stewart parallel mechanism into analog signals, which are then sent to the corresponding servo driver, driving the parallel mechanism to move along the desired trajectory.

[0152] Firstly, through MATLAB simulation and prototype experiment of Stewart parallel mechanism of sandblasting and rust removal parallel robot, the proposed variable exponential coefficient fixed time sliding mode disturbance observer (VEC-FxTSMDO) is compared with the existing asymptotic sliding mode disturbance observer (AsSMDO), finite time sliding mode disturbance observer (FnTSMDO) and fixed time sliding mode disturbance observer (FxTSMDO); then, the proposed variable exponential coefficient fixed time sliding mode composite control method (VEC-FxTSMCC) is compared with the existing asymptotic sliding mode composite control (AsSMCC), finite time sliding mode composite control (FnTSMCC) and fixed time sliding mode composite control (FxTSMCC); finally, it is obtained Figure 5 The lumped uncertainty estimation results and error curves of the Stewart parallel mechanism are shown, as well as Figure 6 The end pose tracking error curve shown and Figure 7 The servo motor drive torque curve is shown.

[0153] Depend on Figure 5 It can be seen that compared with the asymptotic sliding mode disturbance observer (AsSMDO) and the finite time sliding mode disturbance observer (FnTSMDO), the proposed variable exponential coefficient fixed time sliding mode disturbance observer (VEC-FxTSMDO) has a faster convergence speed and higher convergence accuracy for the estimation of lumped uncertainty. Compared with the fixed time sliding mode disturbance observer (FxTSMDO), the proposed variable exponential coefficient fixed time sliding mode disturbance observer (VEC-FxTSMDO) can ensure the stability and smoothness of the estimation process. Figure 6 and Figure 7 It can be seen that compared with the asymptotic sliding mode composite control (AsSMCC), finite time sliding mode composite control (FnTSMCC) and fixed time sliding mode composite control (FxTSMCC), the proposed variable exponential coefficient fixed time sliding mode composite control method (VEC-FxTSMCC) can not only effectively improve the robustness of the task space tracking control of the uncertain Stewart parallel mechanism and achieve rapid convergence of its trajectory tracking within a fixed time, but also effectively alleviate tracking overshoot and oscillation, produce a smooth and continuous tracking process, and avoid excessive control input.

[0154] It should be understood that the above embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, various equivalent modifications of the present invention made by those skilled in the art fall within the scope defined by the claims attached to this application.

Claims

1. A method for controlling the Stewart parallel mechanism of a sandblasting and rust removal parallel robot with variable exponential coefficients in task space and fixed time, characterized in that: The steps include: 1) The Lagrangian method is used to aggregate the uncertain effects of modeling error, joint friction, sandblasting jet reaction force interference, and establish the dynamic model of the Stewart parallel mechanism in the task space; 2) Based on the uncertain Stewart parallel mechanism dynamic model obtained in step 1), an auxiliary system is introduced to construct a sliding mode switching term and a variable exponential power adjustment term, and a variable exponential coefficient fixed-time sliding mode disturbance observer is designed to respond to time-varying disturbances and uncertainties through the high-frequency switching behavior of the sliding mode switching term, and quickly estimate the lumped uncertainty within a fixed time of a uniformly bounded time. At the same time, the variable exponential power adjustment term can also prevent the observer from generating an excessively large integral increment, thereby ensuring the stability and smoothness of the estimation process; 3) Based on the uncertain Stewart parallel mechanism dynamics model obtained in step 1), a variable exponential coefficient fixed-time sliding mode control algorithm is designed by constructing a sliding mode function and reaching law with variable exponential power terms, and it is combined with the variable exponential coefficient sliding mode disturbance observer obtained in step 2), so as to not only achieve fast convergence of the uncertain Stewart parallel mechanism task space tracking control within a uniformly bounded fixed time and improve its robustness, but also effectively alleviate tracking overshoot and oscillation, produce a smooth and continuous tracking process, and avoid excessive control input; 4) A distributed structure, namely "host computer (industrial computer) + slave computer (multi-axis motion controller)" is used to construct a variable exponential coefficient fixed-time composite control system for the uncertain Stewart parallel mechanism task space, and realize the task space tracking control of the Stewart parallel mechanism.

2. The method according to claim 1, characterized in that: In step 2), based on the uncertain Stewart parallel mechanism dynamics model, an auxiliary system is introduced as follows: The sliding mode switching term is constructed as: The construction variable exponential power adjustment term is: In formulas (1)-(3), m o ,ρ o ,λ o ,η o >0 is an adjustable parameter; is an auxiliary variable; is the estimated value of lumped uncertainty, q is the end position of the Stewart parallel mechanism, is the end position tracking error of the Stewart parallel mechanism, q d is the desired position of the end; M(q), and G(q) are the nominal inertia matrix, Coriolis and centrifugal force matrices, and gravity term of the Stewart parallel mechanism dynamic model, respectively; J(q) is the Jacobian matrix; and is the auxiliary variable; v is the sliding mode variable; ξ e =2πζ e / l s , e is the motor mechanical efficiency, l s is the lead of the motor screw; τ is the driving torque of the servo motor; is a variable exponent power term; is a variable exponential power function; sgn(·) is a sign function; ξ(v) is a variable exponential coefficient function; tanh(·) is a hyperbolic tangent function.

3. The method according to claim 2, characterized in that: In step 2), in order to obtain the estimated value of the lumped uncertainty of the Stewart parallel mechanism system By constructing sliding mode switching terms and variable exponential power adjustment terms, a variable exponential coefficient fixed-time sliding mode disturbance observer is designed as follows: Among them, γ1,γ2,γ3>0 are adjustable parameters; is the estimated value of the system aggregate uncertainty; Under the action of the variable exponential coefficient fixed-time sliding mode disturbance observer, the estimation error of the lumped uncertainty can be converged within a uniformly bounded fixed time, and the convergence time is: where γ2,ρ o >0 is an adjustable parameter; γ o =(m o +1+λ o tanh(1)) / 2;T d is the estimated error convergence time; exp(·) is an exponential function with a natural constant as the base.

4. The method according to claim 2, characterized in that: In the step 3), the sliding mode function with variable exponential power term is constructed as: The reaching law of the power term with variable exponent is constructed as In formula (6) and formula (7), ρ s ,λ s ,λ c ,k1,k2,m c ,η s ,ρ c >0 is an adjustable parameter; is the sliding mode variable; is a variable exponent power term; and is a variable exponential power function; and is a variable exponential coefficient function; tanh(·) is a hyperbolic tangent function; is the tracking error of the end position of the parallel mechanism.

5. The method according to claim 2, characterized in that: In the step 3), for the uncertain Stewart parallel mechanism, a variable exponential coefficient fixed time sliding mode composite controller is designed by constructing a sliding mode function and reaching law with a variable exponential power term: In the formula, λ s ,λ c ,k1,k2,m c ,η s >0 is an adjustable parameter; is a variable exponential power function; is a variable exponential coefficient function; is the sliding mode variable; is the estimated value of the system lumped uncertainty; τ1 is the variable exponential coefficient fixed-time sliding mode control term, τ2 is the variable exponential coefficient fixed-time sliding mode disturbance observer compensation term; Q s =J -T (q)M(q) is the inverse matrix term; is a variable exponential coefficient function; The variable exponent power matrix item has the following specific form: λ s ,ρ s >0 is an adjustable parameter; Under the action of the proposed variable exponential coefficient fixed-time sliding mode compound control, the tracking error can converge within a uniformly bounded fixed time, and the convergence time T c for: Where T c is the tracking error convergence time; γ c =(m c +1+λ c tanh(1)) / 2 and γ s =(1+λ s tanh(1)) / 2 is the upper bound of the variable exponential coefficient function; exp(·) is an exponential function with a natural constant as the base; tanh(·) is the hyperbolic tangent function.

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