A method for fixed-time compound control of a sand-blasting and rust-removing parallel robot stewart parallel mechanism in a task space with variable exponential coefficients

Abstract

The application discloses a kind of sand blasting derusting and parallel robot Stewart parallel mechanism task space variable exponential coefficient fixed time compound control method.Lagrange method is used to establish the dynamics model of uncertain Stewart parallel mechanism;Based on the dynamics model of uncertain Stewart parallel mechanism, a variable exponential coefficient fixed time sliding mode disturbance observer is designed, at the same time, the variable exponential power adjustment term also enables the observer to avoid excessive integral increment, thereby improving the stability and smoothness of the estimation process;Secondly, by constructing a sliding mode function with a variable exponential power term and an approach law, a variable exponential coefficient fixed time sliding mode control algorithm is designed, and it is compounded with the variable exponential coefficient fixed time sliding mode disturbance observer, not only to realize the rapid convergence of the task space tracking control of the uncertain Stewart parallel mechanism in a fixed time, but also to effectively alleviate the tracking overshoot and oscillation, produce a smooth and continuous tracking process, and avoid excessive control input.

Description

TECHNICAL FIELD

[0001] The application 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 for a sand blasting derusting parallel robot Stewart parallel mechanism. BACKGROUND

[0002] At present, sand blasting derusting operation on the surface of a bridge steel box girder is mainly performed by manpower, which is low in efficiency, high in safety risk and easy to harm the health of operators. Existing sand blasting derusting robots are mostly designed based on a serial mechanism, and have the disadvantages of low rigidity, weak bearing capacity, low self-weight load ratio, poor mobility and the like, and are difficult to perform sand blasting derusting operation on large steel members such as the steel box girder with a U-shaped rib complex surface. Therefore, a sand blasting derusting parallel robot based on a Stewart parallel mechanism is developed, which is composed of a Stewart parallel mechanism, a lifting mechanism and a moving platform and has the advantages of high rigidity, strong bearing capacity and good mobility, and can realize arbitrary movement, free lifting, six-degree-of-freedom pose precise motion operation and the like. However, from the control aspect, the Stewart parallel mechanism is a strongly coupled nonlinear multi-input multi-output system with parameter uncertainty, and in addition, the Stewart parallel mechanism is also affected by external disturbances, such as jet reaction force interference when performing sand blasting operation. Therefore, the application provides a variable exponential coefficient fixed time sliding mode composite control method to realize high-performance trajectory tracking control of the task space of the sand blasting derusting parallel robot Stewart parallel mechanism.

[0003] Zhang W, Gao G. Anti-mismatch disturbance adaptive backstepping sliding mode control for sandblasting parallel mechanism. Journal of Software, 2023, 11, 22, 111-117. An adaptive backstepping sliding mode control method combined with extended state observer is proposed for the sandblasting parallel robot Stewart parallel mechanism to maintain good trajectory tracking performance under the disturbance of sandblasting jet reaction force. However, the extended state observer usually needs high observer gain to ensure the dynamic response and estimation accuracy of the time-varying disturbance and uncertainty. The high observer gain will amplify the measurement noise into the estimation result. In addition, the sliding mode control method is designed based on finite time stability theory, and the stability time theoretically depends on the initial state information of the system. When the system state is far from the equilibrium point, the stability time tends to infinity, making it difficult to truly guarantee the control performance of the system. The paper "Terminal sliding mode control of manipulator based on sliding mode disturbance observer" (Han J, et al. Journal of Central South University (Science Edition), 2020, 10, 51, 2749-2757) uses a sliding mode disturbance observer to estimate the system 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 disturbance and uncertainty by the sliding mode disturbance observer can only achieve finite time convergence. The paper "AUVs event-triggered formation control based on fixed-time sliding mode disturbance observer" (Su B, et al. Control and Decision, 2022, 5, 37, 1116-1126) proposes a fixed-time sliding mode disturbance observer that can estimate the system lumped uncertainty within a fixed time that is independent of the initial state and uniformly bounded. The paper "Fixed-time nonsingular segmented sliding mode control for hybrid serial-parallel mechanism with liquid resistance disturbance" (Zhu Z, Gao G. Automation Technology and Application, 2024, online first) 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. If the constant exponential coefficients of these power terms are selected to be greater than 1 to meet the performance requirements, if the system state is far from the equilibrium point, the observer and controller may produce excessive integral increments or control amounts, causing the estimation or control process to oscillate or overshoot. If 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 increment or control amount will also be amplified, making it difficult to ensure the smoothness or stability of the estimation or control process. SUMMARY

[0004] The present application is to overcome the deficiencies of the prior art, aiming at the sandblasting derusting parallel robot Stewart parallel mechanism affected by modeling errors, joint friction and sandblasting jet reaction force interference and other uncertainties, a dynamics model is established by using Lagrange method; by introducing auxiliary system to construct sliding mode switching term and variable exponent power adjustment term, a variable exponent coefficient fixed time sliding mode disturbance observer is designed, which can improve the dynamic response to disturbance 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 fast convergence of the lumped uncertainty estimation of the Stewart parallel mechanism in a uniformly bounded fixed time, at the same time, the variable exponent power adjustment term can also make the observer avoid generating excessive integral increment, so as to ensure the stability and smoothness of the estimation process; secondly, by constructing a sliding mode function with a variable exponent power term and a reaching law, a new variable exponent coefficient fixed time sliding mode control algorithm is designed, and it is compounded with the variable exponent coefficient fixed time sliding mode disturbance observer, so as to not only realize the fast convergence of the task space tracking control of the uncertain Stewart parallel mechanism in a fixed time, but also improve its robustness, and effectively alleviate the tracking overshoot and oscillation, produce a smooth and continuous tracking process, and avoid excessive control input.

[0005] The technical scheme of the present application is: a sandblasting derusting robot Stewart parallel mechanism task space variable exponent coefficient fixed time composite control method, characterized by comprising the following steps:

[0006] 1) The modeling error, joint friction, jet reaction force disturbance and other uncertainties are lumped by using Lagrange method, and the dynamics model of the Stewart parallel mechanism in the task space is established.

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

[0008] 3) Based on the uncertain Stewart parallel mechanism dynamics model obtained in step 1), a variable exponent coefficient fixed time sliding mode control algorithm is designed by constructing a sliding mode function with a variable exponent power term and a reaching law, and the variable exponent coefficient fixed time sliding mode disturbance observer obtained in step 2) is compounded to obtain a variable exponent coefficient fixed time sliding mode composite controller, so as to not only realize the fast convergence of the task space tracking of the uncertain Stewart parallel mechanism within a uniformly bounded fixed time, but also improve its robustness, effectively alleviate the tracking overshoot and oscillation, produce a smooth and continuous tracking process, and avoid excessive control input.

[0009] 4) Adopting distributed structure, i.e. "upper computer (industrial computer) + lower computer (multi-axis motion controller)", to construct the task space variable exponential coefficient fixed-time compound control system of uncertain Stewart parallel mechanism, and realize the task space tracking control of Stewart parallel mechanism.

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

[0011]

[0012] The sliding mode variable is constructed as:

[0013]

[0014] To improve the dynamic response of the estimation of disturbance and uncertainty, the sliding mode switching term is constructed as:

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

[0016] To realize the fast convergence of the lumped uncertainty estimation of Stewart parallel mechanism in fixed time, avoid excessive observer integral increment, and ensure the smoothness or stability of the estimation process, the variable exponential power term is constructed as:

[0017]

[0018] wherein is the variable exponential coefficient function.

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

[0020]

[0021] In equations (1)-(5), γ1, γ2, γ3, m o , ρ o , λ o , η o > 0 are adjustable parameters; is the lumped uncertainty estimation value; q is the end pose of Stewart parallel mechanism; M(q), and G(q) are the nominal inertia matrix, Coriolis and centrifugal force matrix, and gravity term of the dynamic model of Stewart parallel mechanism, respectively; J(q) is the Jacobian matrix; is the end pose tracking error of Stewart parallel mechanism; q dThis represents the desired pose at the end point. ξ e =2πζ e / l s , ζ e For the mechanical efficiency of the motor, l s τ is the lead of the motor leadscrew; τ is the driving torque of the servo motor. v This is a sliding mode switching option; The term is a variable exponent; v is a sliding mode variable; is a function of variable exponent; sgn(·) is a sign function.

[0022] Under the design of a 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 indicates 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), to ensure that the end-effector pose of the Stewart parallel mechanism converges quickly to the desired pose within a fixed time, a sliding mode function with a variable exponential term is constructed as follows:

[0026]

[0027] Where λ s ,η s ,ρ s >0 indicates an adjustable parameter; For the end effector pose tracking error of the Stewart parallel mechanism, q d The desired pose of the end effector; It is a function with variable exponential coefficients; For terms with variable exponents; is a function of variable exponent; tanh(·) is the hyperbolic tangent function.

[0028] To achieve sliding mode variable within a fixed time period To achieve rapid convergence and avoid excessive control input, ensuring the stability of the control process, a sliding mode reaching law with a variable exponent term is constructed as follows:

[0029]

[0030] Where λ c ,ρ c ,m c k1, k2 > 0 are adjustable parameters; It is a function with a variable exponent; is a function with variable exponential coefficients; tanh(·) is the hyperbolic tangent function.

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

[0032]

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

[0034]

[0035] Where T s γ is the convergence time of the sliding mode variable; c =(m c +1+λ c tanh(1)) / 2 is the upper bound of the variable exponential coefficient; exp(·) is an exponential function with the natural constant as its base. The tracking error can converge in 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 This refers to the convergence time of the control system.

[0038] The application first proposes a sandblasting derusting parallel robot Stewart parallel mechanism task space variable exponential coefficient fixed time sliding mode composite control method to realize high performance trajectory tracking control of the Stewart parallel mechanism task space.

[0039] 1. A variable exponential coefficient fixed time sliding mode disturbance observer is designed to estimate the uncertainty in the Stewart parallel mechanism system including modeling error, joint friction, sandblasting jet reaction force disturbance, etc. The variable exponential coefficient fixed time sliding mode disturbance observer introduces an auxiliary system to construct a sliding mode switching term and a variable exponential power adjustment term. The beneficial effect is that the variable exponential coefficient fixed time sliding mode disturbance observer can respond to time-varying disturbance and uncertainty through the high-frequency switching behavior of the sliding mode switching term, and quickly and accurately obtain the lumped uncertainty estimate value within a uniformly bounded time. With the help of the variable exponential power adjustment term, the variable exponential coefficient fixed time sliding mode disturbance observer can also avoid generating excessive integral increment, thereby ensuring the stability and smoothness of the estimation process.

[0040] 2. A variable exponential coefficient fixed time sliding mode control algorithm is designed to realize task space tracking control of the Stewart parallel mechanism. The variable exponential coefficient fixed time sliding mode control algorithm is based on the sliding mode function with variable exponential power term and the reaching law design. The beneficial effect is that the variable exponential coefficient fixed time sliding mode control algorithm can realize the fast convergence of the task space tracking control of the uncertain Stewart parallel mechanism within a uniformly bounded fixed time, and can avoid excessive control input. The combination of the variable exponential coefficient fixed time sliding mode control algorithm and the variable exponential coefficient fixed time sliding mode disturbance observer can effectively improve the robustness of the task space tracking control of the Stewart parallel mechanism and produce a smooth and continuous tracking process. BRIEF DESCRIPTION OF DRAWINGS

[0041] Figure 1 is a sandblasting derusting parallel robot prototype.

[0042] Figure 2 is a Stewart parallel mechanism structure diagram.

[0043] Figure 3 is a bridge steel box girder structure diagram.

[0044] Figure 4 is a Stewart parallel mechanism task space variable exponential coefficient fixed time sliding mode composite control method principle diagram.

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

[0046] Figure 6 are the end pose trajectory tracking curves and tracking error curves of the Stewart parallel mechanism under asymptotic sliding mode composite control (AsSMCC), finite time sliding mode composite control (FnTSMCC), fixed time sliding mode composite control (FxTSMDO) and variable-exponent coefficient fixed time sliding mode composite control (VEC-FxTSMCC) proposed respectively.

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

[0048] Figure 8 is the control system block diagram of the sandblasting derusting parallel robot. DETAILED DESCRIPTION

[0049] The specific embodiments of the present application are further illustrated below in conjunction with the accompanying drawings.

[0050] Firstly, the Lagrange method is adopted, and the effects of uncertainties such as modeling errors, joint friction, and fluidic reaction force disturbances are lumped to establish the dynamic model of the sandblasting derusting parallel robot uncertain Stewart parallel mechanism in task space. Based on the dynamic model of the uncertain Stewart parallel mechanism, a variable exponent coefficient fixed-time sliding mode disturbance observer is designed by introducing auxiliary systems to construct sliding mode switching terms and variable exponent power terms, so as to 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 value within a uniformly bounded time. At the same time, the variable exponent power adjustment term can also prevent the observer from generating excessive integral increments, thereby ensuring the stability and smoothness of the estimation process. By constructing a sliding mode function with a variable exponent power term and a reaching law, a variable exponent coefficient fixed-time sliding mode control algorithm is designed, which is combined with the variable exponent coefficient fixed-time sliding mode disturbance observer to not only achieve the 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. Finally, a distributed structure, i.e., "upper computer (industrial computer) + lower computer (multi-axis motion controller)", is used to construct the task space variable exponent coefficient fixed-time compound control system of the uncertain Stewart parallel mechanism, and the end of the Stewart parallel mechanism moves according to the expected trajectory. The specific method is as follows:

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

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

[0053]

[0054] where M(q) is a positive definite inertia matrix, is the Coriolis force and centrifugal force term, and G(q) is the gravity term; q, are the end pose vector, velocity vector, and acceleration vector of the Stewart parallel mechanism, respectively; ζ e is the mechanical efficiency; l s is the lead screw pitch, and 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 errors, joint friction, and fluidic reaction force disturbances during operation, the dynamic model of the uncertain Stewart parallel mechanism in task space is established:

[0056]

[0057] where ΔM(q), ΔL(q) and ΔG(q) are modeling errors; d r is the jet reaction force disturbance; d f is the joint friction.

[0058] The effects of modeling errors, joint friction and jet reaction force disturbance are lumped, then equation (13) can be rewritten as

[0059]

[0060] where d is the lumped uncertainty, and its specific form is:

[0061]

[0062] The end tracking error dynamic equation of the Stewart parallel mechanism can be obtained from equation (15) as:

[0063]

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

[0065] 2) Based on the uncertain dynamics model of the Stewart parallel mechanism, an auxiliary system

[0066]

[0067] The sliding mode variable is constructed as:

[0068]

[0069] where η o is a tunable parameter; and are auxiliary variables; is the lumped uncertainty estimate.

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

[0071]

[0072] To achieve fast convergence of the lumped uncertainty estimation of the Stewart parallel mechanism in a fixed time, avoid excessive observer integral increment, ensure the smoothness or stability of the estimation process, the variable exponential power term is constructed as:

[0073] ​

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

[0075] 4) is to obtain the lumped uncertainty estimation value in a uniformly bounded fixed time At the same time, ensure the smoothness and stability of the estimation process, based on formula (16)-(20), design the variable exponent coefficient fixed time sliding mode disturbance observer as:

[0076]

[0077] In the formula, γ1, γ2, γ3>0 are adjustable parameters.

[0078] Under the action of the designed variable exponent coefficient fixed time sliding mode disturbance observer, the estimation error can realize convergence in 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) is to make the end pose of Stewart parallel mechanism converge to the desired pose in a fixed time, construct the sliding mode function with variable exponent power term as

[0082]

[0083] Where, λ s ,η s ,ρ s >0 are adjustable parameters; is the end pose tracking error of Stewart parallel mechanism, q d is the desired end pose; is the variable exponent power term; is the variable exponent power function; is the variable exponent coefficient function; tanh(·) is the hyperbolic tangent function.

[0084] 6) is to realize the rapid convergence of the sliding mode variable in a fixed time, and avoid excessive control input, ensure the smoothness and stability of the control process, construct the sliding mode reaching law with variable exponent power term as:

[0085]

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

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

[0088]

[0089] where, τ1 is the variable exponent coefficient fixed-time sliding mode control term, τ2 is the variable exponent coefficient fixed-time sliding mode disturbance observer compensation term; is the lumped uncertainty estimation value, q is the end pose of the Stewart parallel mechanism, is the end pose tracking error of the Stewart parallel mechanism, q d is the end desired pose; M(q), and G(q) are the nominal inertia matrix, Coriolis and centrifugal force matrix, and gravity term of the dynamics model of the Stewart parallel mechanism, respectively; J(q) is the Jacobian matrix; and are auxiliary variables; v is the sliding mode variable; ξ 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; is a variable exponent power function; sgn(·) is the sign function; Q s = J -T (q)M(q) is the inverse matrix term. Under the action of the proposed variable exponent coefficient fixed-time sliding mode composite control, the sliding mode variable can achieve convergence in 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 exponent coefficient function; the tracking error can achieve convergence in 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, realize the task space variable exponential coefficient fixed time sliding mode composite control of Stewart parallel mechanism of sandblasting and rust removal robot.

[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 is sent to the corresponding servo driver of the motor to drive the Stewart parallel mechanism moving platform to move along the desired trajectory.

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

[0097] Example 1

[0098] like Figure 1 As shown, the sandblasting and rust removal parallel robot mainly consists of an upper computer (industrial control computer), a lower computer (multi-axis motion controller), a lifting platform, a moving platform, a sandblasting gun, and a Stewart parallel mechanism.

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

[0100] 1) Based on the Lagrange method, and by incorporating the uncertainties such as modeling errors, joint friction, and jet reaction force disturbances, an uncertain Stewart parallel mechanism dynamic model is established in the task space:

[0101]

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

[0103]

[0104] In the formula,

[0105]

[0106] In the formula, m p To improve platform quality, I x ,I y ,I zis the moment of inertia; cos(·) is the cosine function; sin(·) is the sine function.

[0107] The terms are Coriolis force and centrifugal force, and their specific forms are as follows:

[0108]

[0109] In the formula,

[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] In the formula, s i =L i / l i Let L be the unit vector of the i-th branch. i =x t +R γβα A i -B i Let l be the position vector of the i-th branch. i =||L i || is the length of the i-th branch, A i Let B be the coordinates of the connection point between the i-th branch and the moving platform. i Let A be the coordinates of the connection point between the i-th branch and the fixed base. i and B i See the specific location Figure 2 The diagram shown is a simplified representation of the Stewart parallel mechanism; x t =[x,y,z] T R represents the position coordinates of the end effector of the Stewart parallel mechanism. γβα It is a rotation matrix, and its specific form is:

[0116]

[0117] c and s represent the cosine and sinine functions, respectively, and α, β, and γ are the end-effector attitudes of the Stewart parallel mechanism.

[0118] 2) From equation (28), the dynamic equation for the end tracking error of the Stewart parallel mechanism can be obtained as follows:

[0119]

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

[0121] 3) Based on the uncertain dynamics model of the Stewart parallel mechanism and the dynamic equation of the end-effector tracking error, an auxiliary system

[0122]

[0123] The sliding mode variable is constructed as:

[0124]

[0125] where η o is a tunable parameter; and are auxiliary variables; is the lumped uncertainty estimate.

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

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

[0128] The variable exponential term is constructed as:

[0129]

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

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

[0132]

[0133] where γ1, γ2, γ3 > 0 are tunable parameters.

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

[0135]

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

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

[0138]

[0139] where, is the end pose tracking error of Stewart parallel mechanism, q d is the end desired pose; is the variable exponential coefficient function; λ s , η s , ρ s > 0 are adjustable parameters.

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

[0141]

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

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

[0144]

[0145] In the equation, 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 can achieve convergence in fixed time, and the convergence time is:

[0146]

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

[0148]

[0149] where γ s = (1 + λ stanh(1)) / 2.

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

[0151] The Stewart parallel mechanism task space variable exponential coefficient fixed time sliding mode control system has higher flexibility and reliability based on the "upper computer + lower computer" distributed control structure, is suitable for large-scale and complex control systems and scenes with high requirements for system response speed and real-time performance, and the control system structure is as shown in Figure 8 The control system is composed of an upper computer (industrial computer), a lower computer (multi-axis motion controller), a servo system and a sand blasting and derusting parallel robot. The lower computer communicates with the upper computer through an Ethernet port; the motion controller collects the encoder information of the Stewart parallel mechanism active joint servo motor through the expansion board, and sends the analog quantity obtained by digital-to-analog conversion of the motor drive control amount of each active joint of the Stewart parallel mechanism to the corresponding servo driver, to drive the parallel mechanism to move according to the expected trajectory.

[0152] Firstly, 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) through MATLAB simulation and sand blasting and derusting parallel robot Stewart parallel mechanism prototype experiment; then, the proposed variable exponential coefficient fixed time sliding mode compound control method (VEC-FxTSMCC) is compared with the existing asymptotic sliding mode compound control (AsSMCC), finite time sliding mode compound control (FnTSMCC) and fixed time sliding mode compound control (FxTSMCC); finally, the lumped uncertainty estimation results and error curves of the Stewart parallel mechanism as shown in Figure 5 , the end position and posture tracking error curves as shown in Figure 6 , and the servo motor driving torque curves as shown in Figure 7 .

[0153] It can be seen from Figure 5 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 faster convergence speed and higher convergence accuracy in estimating the 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. FromFigure 6 and Figure 7 It can be seen that compared with asymptotic sliding mode composite control (AsSMCC), finite time sliding mode composite control (FnTSMCC) and fixed time sliding mode composite control (FxTSMCC), the variable exponential coefficient fixed time sliding mode composite control method (VEC-FxTSMCC) proposed can not only effectively improve the robustness of the task space tracking control of the uncertain Stewart parallel mechanism, but also realize the rapid convergence of its trajectory tracking in fixed time, effectively alleviate the 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 application and are not used to limit the scope of the present application, and various equivalent modifications of the present application made by those skilled in the art after reading the present application all fall within the scope defined by the claims attached hereto.

Claims

1. A method for task space variable exponential coefficient fixed time hybrid control of a sandblasting derusting and parallel robot Stewart parallel mechanism, characterized in that, Comprising the following steps: 1) Adopting Lagrange method, and collecting the influences of modeling error, joint friction, and disturbance of sandblasting jet reaction force uncertainty, an uncertain dynamics model of Stewart parallel mechanism in task space is established; 2) Based on the uncertain dynamics model of Stewart parallel mechanism in task space obtained in step 1), a variable-exponent coefficient fixed-time sliding mode disturbance observer is designed by introducing an auxiliary system to construct a sliding mode switching term and a variable-exponent power adjustment term, wherein the auxiliary system is constructed as: The sliding mode switching term is constructed as: （1）； The variable-exponent power adjustment term is constructed as: （2）； 3) Based on the uncertain dynamics model of Stewart parallel mechanism in task space obtained in step 1), a variable-exponent coefficient fixed-time sliding mode compound controller is designed by constructing a sliding mode function with a variable-exponent power term and a reaching law, and then combining the variable-exponent coefficient sliding mode disturbance observer obtained in step 2), wherein the sliding mode function with a variable-exponent power term is constructed as: （3）； in formulas (1)-(3), are tunable parameters; is a lumped uncertainty estimate, q is the end-effector position and orientation of the Stewart parallel mechanism, is the end-effector position and orientation tracking error of the Stewart parallel mechanism, is the end-effector desired position and orientation; , and are the nominal inertia matrix, the Coriolis and centrifugal force matrix, and the gravity term of the Stewart parallel mechanism dynamics model, respectively; is the Jacobian matrix; and are auxiliary variables; v is the sliding mode variable; , is the motor mechanical efficiency, is the motor lead screw lead; is the servo motor driving torque; is the variable exponent power term; is the variable exponent power function; is the sign function; is the variable exponent coefficient function; is the hyperbolic tangent function; the variable exponent coefficient fixed-time sliding mode disturbance observer is: （4）； wherein is an adjustable parameter; The reaching law with a variable-exponent power term is constructed as: 4) An uncertain Stewart parallel mechanism task space variable-exponent coefficient fixed-time compound control system is constructed by adopting a distributed structure, i.e., "upper computer + lower computer" structure, wherein the upper computer is an industrial computer, and the lower computer is a multi-axis motion controller, so as to realize task space tracking control of the Stewart parallel mechanism; （6）； The variable-exponent coefficient fixed-time sliding mode compound controller is constructed as: （7）； in formula (6) and formula (7), , , , , , , , is an adjustable parameter; is a sliding mode variable; is a variable exponent power term; and are variable exponent power functions; and are variable exponent coefficient functions; In step 2), under the action of the variable-exponent coefficient fixed-time sliding mode disturbance observer, the estimation error of the collected uncertainty can converge within a uniformly bounded fixed time, and the convergence time is: ​ （8）； In the formula, is a variable exponent coefficient fixed time sliding mode control term, is a variable exponent coefficient fixed time sliding mode disturbance observer compensation term; is an inverse matrix term; is a variable exponent power matrix term, which has the specific form: （9）； Under the action of the proposed variable-index coefficient fixed-time sliding mode composite controller, the tracking error can converge in a uniformly bounded fixed time, and the convergence time is: （10）； for tracking error convergence time; and is a variable exponent coefficient function upper bound.

2. The method of claim 1, wherein: ​ （5）； wherein , is an adjustable parameter; ; is the estimation error convergence time; is an exponential function with base the natural constant.

Images