Event triggering sliding mode control method for in-orbit collaborative assembly of space robot

Abstract

The invention provides an event triggering sliding mode control method for in-orbit cooperative assembly of a space robot. The event triggering sliding mode control method comprises the following steps: step 1, defining a coordinate system and parameters of a space robot cooperative system; step 2, adopting quaternion to carry out kinematics description on the structure module; 3, performing dynamic description of the coupling system, and establishing a dynamic equation of the structure module by adopting a Newton-Euler equation; step 4, describing a control error; 5, describing an expected state of the system; and step 6, designing an event-triggered sliding mode control law, and defining a sliding mode surface. The six-degree-of-freedom motion of a robot system is expressed by adopting a unit quaternion and a position vector, coupling dynamics of a robot and an assembly module is further realized, and an event triggering sliding mode control strategy is designed on the basis of constructing tracking error representation. The space robot can drive the module to track the needed track, and meanwhile communication resources are saved.

Description

Technical Field

[0001] The present invention belongs to the field of space technology, and particularly relates to an event-triggered sliding mode control method for on-orbit cooperative assembly of space robots. Background Art

[0002] As is well known, with the development of space technology, the space missions to be completed are becoming increasingly complex, and large space structures are more and more widely used in space missions. Due to the limitations of the size of large structures and the transportation capacity of transporters, they cannot be launched into orbit as a whole. At this time, it is a feasible solution to disperse large space structures into small space modules for launch and transportation and assemble them in space to form large space structures. Using robots to assemble small modules is efficient and convenient. In the space module assembly task, as the task becomes increasingly complex, the coordination and flexibility of single-arm robots can no longer meet the requirements of the task. In this case, multi-robot space robots emerge as the times require, with stronger operability, which can meet the requirements of the task and better complete space missions. Therefore, the present invention discloses a method for dual-arm cooperative assembly module, which models a multi-robot system using quaternions and applies event-triggered sliding mode control to it. On the one hand, the four real numbers of the quaternion are more concise than the Euler angle representation, making it more efficient in terms of storage and transportation. At the same time, the operation of the quaternion only requires four floating-point multiplications and two floating-point additions, which is faster. At the same time, the quaternion representation has no singularity, that is, each quaternion uniquely represents a rotation. On the other hand, considering the limited communication sources in space applications, traditional continuous control methods are difficult to implement in this case. In order to achieve six-degree-of-freedom trajectory tracking of a multi-robot cooperative drive structure module, the present invention studies an event-triggered algorithm to achieve the control target and proves that the tracking error is bounded. Summary of the Invention

[0003] Object of the Invention: The technical problem to be solved by the present invention is to provide an event-triggered sliding mode control method for on-orbit cooperative assembly of space robots in view of the deficiencies of the prior art, including the following steps:

[0004] Step 1, define the coordinate system and parameters of the space robot cooperative system;

[0005] Step 2, kinematically describe the structural module using quaternions;

[0006] Step 3, perform dynamic description of the coupled system, and establish the dynamic equation of the structural module using Newton-Euler equations;

[0007] Step 4, describe the control error;

[0008] Step 5, describe the desired state of the system;

[0009] Step 6: Design the event-triggered sliding mode control law SMC (sliding mode control) and define the sliding surface.

[0010] Step 1 includes: The space robot cooperation system includes a space cooperative robot. The space cooperative robot includes a base and n six-degree-of-freedom robots. The n six-degree-of-freedom robots are connected to the base and jointly control a structural module regarded as a rigid body.

[0011] Step 1 also includes: is the set of positive integers, represents an m×r-dimensional real matrix, represents an r-dimensional real vector, represents the r-dimensional Euclidean space;

[0012] The coordinate system {I} is the inertial coordinate system, {O} is the body coordinate system of the structural module, and {H i} is the coordinate system fixed on the end effector of the i-th robot, where i = 1, 2, …, n, and n is the number of six-degree-of-freedom robots.

[0013] Step 2 includes: The quaternion is used to describe the attitude of the coordinate system {O} fixed on the structural module relative to the inertial coordinate system {I}, where a o and b o are the components of the quaternion ζ o b o = [b1, b2, b3] T , and b1, b2, b3 are the components of b o satisfy the condition T represents matrix transpose. Calculate the derivative of the quaternion ζ o where the intermediate parameter satisfies Φ (ζ T ) Φ(ζ o ) = I o , 3×3 , is the angular velocity of the structural module relative to the inertial coordinate system {I}, and ω1, ω2, ω3 are the components of is the skew-symmetric matrix given based on b1, b2, b3;

[0014] Let represent the position vector of the {O} coordinate system relative to the inertial coordinate system {I}, and r1, r2, r3 are the three components of the position vector ; represents the translational velocity of the {O} coordinate system relative to the inertial coordinate system {I}; then the generalized position of the structural module and the generalized velocity V o is expressed as

[0015] By integrating the formulas, the position derivative is obtained and the module velocity V o The relationship is

[0016]

[0017] Set the rigid grasping structure module of the robot to obtain the velocity V of the end effector of the i-th robot Hi and the structure module velocity V o The relationship is where V Hi is the generalized velocity of the end effector of the i-th robot relative to the inertial coordinate system {I}, and respectively represent the position velocity and angular velocity of the coordinate system {Hi} fixed to the end of the i-th robot relative to the inertial system {I}; is the grasping matrix of the {O} coordinate system relative to the {Hi} coordinate system, where is the position vector of the {O} coordinate system relative to the {Hi} coordinate system. For the sake of simplicity of expression, use the intermediate parameter E i to represent

[0018] The expression of the robot joint velocity is given as

[0019]

[0020] where is the first derivative of the joint angle of the i-th robot with respect to time, is the six components of, is the inverse matrix of the Jacobian matrix J i of the i-th robot.

[0021] Step 3 includes: The dynamic equation of the structure module is

[0022]

[0023] where D o and C o are respectively the inertia matrix and damping matrix of the structure module, is the first derivative of the generalized velocity V o of the structure module with respect to time; is the angular velocity of the structure module the first derivative of with respect to time; is the inertia matrix of the structural module with respect to the inertial coordinate system {I}; is the external force vector acting on the center of gravity of the structural module;

[0024] For the robot, applying the Lagrange equation, the dynamic equation of the i-th robot in the task space is obtained:

[0025]

[0026] where D i ∈R 6×6 , C i ∈R 6×6 , D i and C i are the inertia matrix and damping matrix of the i-th robot respectively, is the first derivative with respect to time of the generalized velocity V Hi of the end effector of the i-th robot; is the control action on the end effector caused by the joint torque of the i-th robot; is the force or torque generated at the contact point between the structural module and the i-th end effector;

[0027] The module acting force F o has the following relationship with the contact force F of the robot end effector:

[0028] F o =(E) T F

[0029] where, is the grasping matrix of the n-th robot, and E is the integration matrix of the grasping matrices of n robots; is the force or torque generated at the contact point between the structural module and the end effector of the n-th robot; F is the integration vector of the acting forces of n robots and the structural module;

[0030] Integrating all the formulas, the dynamic model of the coupled system is obtained:

[0031]

[0032] where,

[0033] D a =D o +E T DE, D a is the inertia matrix of the coupled system;

[0034] c a is the damping matrix of the coupled system; is the first derivative with respect to time of the integrated grasping matrix E;

[0035] u is the input vector of the system, u = E T f, where f is the control action on the end effector caused by the torques of n robot joints, is the control action on the end effector caused by the torque of the nth robot joint;

[0036] D is the integrated inertia matrix composed of the inertia matrices of n robots, D n is the inertia matrix of the nth robot;

[0037] C is the integrated damping matrix composed of the damping matrices of n robots, C n is the damping matrix of the nth robot; d represents the disturbance vector in space;

[0038] It is assumed that there exist scalars η1 and η2 satisfying η2 > η1 > 0 such that η1I 6×6 ≤ D a ≤ η2I 6×6 .

[0039] Step 4 includes: the desired position r d and attitude ζ d of a d , are the components of the desired attitude ζ d . The position error r e and the attitude error ζ e are expressed as:

[0040]

[0041] where, is the position vector of the structure module, is the conjugate quaternion of the structure module attitude ζ o , is the quaternion product, ζ e is the attitude error, a e and b e are the components of ζ e ;

[0042] Differentiating gives:

[0043]

[0044] where, and respectively represent the error translational velocity and error angular velocity of the structure module, and are the translational velocity and angular velocity of the structural module, respectively, and ω d are the desired translational velocity and desired angular velocity of the structural module, respectively;

[0045] The goal of the control method is to make the structural module reach the desired position and attitude at a moment t from time t by adding an appropriate control law, expressed as: f when reaching the desired position and attitude, ‖()‖ represents taking the norm of the matrix within the parentheses.

[0046] Step 5 includes: The desired state of the system is expressed as where is the desired generalized velocity of the structural module; Define the system state vector System state vector Obtain the first-order derivatives of the system state vectors x1 and x2 with respect to time The expression of:

[0047]

[0048] where, a e is an intermediate parameter;

[0049] M(x) is the transformation matrix between the state derivative and the state x2;

[0050] G(x) is the transformation matrix between the state derivative and the desired velocity V d ;

[0051] k1(x, x d ) is the damping term of the state equation, is the inverse matrix of D a ;

[0052] is the perturbation term of the state equation;

[0053] Let the total state variable Define the transformation term h2(x, x ) of the system state derivative d = M(x)x2 + G(x)V d , then the error dynamics equation is expressed as:

[0054]

[0055] where is the first-order derivative of the state variable x with respect to time;

[0056] k(x,x d ) is the state term in the state equation;

[0057] R = [0 6×6 , I 6×6 T , where R is the disturbance and reference regulation matrix;

[0058] n(t) is the input regulation matrix;

[0059] Set the disturbance term to be a bounded quantity, where d1 < 0 is a constant;

[0060] Set the existence of positive scalars c1, c2, c3,

[0061] Set the inequality ‖k(v1, w1) - k(v2, w2)‖ ≤ L‖v1 - v2‖ + L‖w1 - w2‖ to hold for a selected constant L, where v1, v2, w1, w2 are vectors constrained in a compact domain Δ,

[0062] Step 6 includes: Define the sliding mode surface v as:

[0063] δ := {x ∈ Δ: s = gx = 0}

[0064] where s is the sliding mode variable, g = [g1, I 6×6 , and g is a sliding mode variable regulation matrix,

[0065] Combining the error dynamics equation and the definition of the sliding mode surface, the expression of the first-order derivative of the sliding mode variable s with respect to time is obtained:

[0066]

[0067] Let denote the sequence of triggering instants for sampling the system state x and calculating the control input u. For event-triggered control, once the input u is determined at time t j , it remains unchanged until time t j+1 . It is expressed as: The system input u(t) at time t = u(t j ), t ∈ [t j , t j+1 )).

[0068] ​​Step 6 further includes: selecting a Lyapunov candidate function In a time interval, L‖g‖‖e x (t)‖ < β always holds, where β is a range parameter, e x (t) = x(t j ) - x(t), t ∈ [t j , t j+1 ), e x (t) is the system state error, and thus the time derivative of the Lyapunov candidate function V s is calculated as: As follows:

[0069]

[0070] To make the system achieve global uniform stability, the control law is taken as:

[0071]

[0072] where λ is a positive constant, the intermediate parameter ρ = η2 / η1, ε(||x(t j )||) = ε1 + ε2||x(t j )||, ε1, ε2 are designed parameters, and ε(||x(t j )||) represents the adjustment parameter determined by the norm of the system state x(t j ) at time t j );

[0073] The time derivative becomes:

[0074]

[0075] where the intermediate parameter γ = gn(t)(gn(t j )) -1 , gR = I 6×6 ;

[0076] It is obtained that ‖Υ‖ ≤ ρ, ‖I - Υ‖ ≤ 1 - ρ -1 , ‖ρI - βγ‖ ≤ ρ - βρ -1 ;

[0077] Given positive scalar values c1, c2, β, ρ, μ, λ, D1, L, the Lyapunov candidate function adjustment parameter Lyapunov candidate function adjustment parameter Adding and subtracting terms on the right - hand side of the above formula, we get:

[0078]

[0079] Consider the case starting from ‖s‖>β / L. When sgn(s(t j )) = sgn(s(t)), the function sgn() represents returning the sign of the value inside the parentheses, and we get:

[0080]

[0081] When sgn(s(t j )) = sgn(s(t)), the system trajectory moves towards the sliding surface. Subsequently, when sgn(s(t j )) ≠ sgn(s(t)), the trajectory may cross the sliding surface and move away from it. The deviation between s(t j ) and s(t) is restricted by the following formula:

[0082] ||s(t j ) - s(t)|| = ||gx(t j ) - gx(t)|| ≤ ‖g‖‖e x ‖ ≤ β / L.

[0083] Step 6 also includes: calculating the convergence intervals of the position error x1 and the velocity error x2:

[0084] Given positive scalars c3, L, β, positive definite matrices P and Q, satisfying Q = (Mg1) T P + P(Mg1) > 0; M is the transformation matrix between the state derivative and the state x2, which is a positive definite matrix;

[0085] Select the Lyapunov function V in the form of Find the time derivative of V

[0086]

[0087] where χ min {} represents taking the minimum eigenvalue of the matrix inside the parentheses. To make the time derivative Select the boundary

[0088] It means that the position error x1 finally enters the Ξ range and stays there, and the velocity error x2 is also bounded:

[0089]

[0090] Beneficial effects: It is challenging to control multiple space robots to transport structural modules along a predetermined trajectory to a predetermined position. Due to the complexity of multi-module transportation, high requirements are imposed on the dynamic modeling of the system. In addition, due to various disturbances, the tracking performance during transportation will also deteriorate. The coupled dynamic modeling of the system in the present invention can well describe the actual situation, and a reasonable event-triggered sliding mode control algorithm is adopted, which can always limit the tracking error within an interval and has good control performance. Description of the Drawings

[0091] The following further specific description of the present invention will be made in conjunction with the drawings and specific embodiments, and the above and / or other advantages of the present invention will become clearer.

[0092] Figure 1 It is a schematic diagram of a space robot cooperative system.

[0093] Figure 2 It is a schematic diagram of the module position trajectory.

[0094] Figure 3 It is a schematic diagram of the module attitude tracking. Specific Embodiments

[0095] In the embodiments of the present invention, an event-triggered sliding mode control method for on-orbit cooperative assembly of space robots is provided. For space cooperative robots, its composition is as Figure 1 shown, including a base and n six-degree-of-freedom robots of the same model, all of which are (RM-P60-RNH). The n six-degree-of-freedom robots are connected to the base and jointly control a structural module that can be regarded as a rigid body. The specific steps of this method are as follows:

[0096] Step 1, for the Figure 1 shown space robot cooperative system, the following definitions of coordinate systems and parameters are first made:

[0097] is the set of positive integers, represents an m×r-dimensional real matrix, represents an r-dimensional real vector, represents the r-dimensional Euclidean space.

[0098] The coordinate system {I} is the inertial coordinate system, {O} is the body coordinate system of the structural module, and {H i} is the coordinate system fixed on the end effector of the i-th robot, where i = 1, 2,..., n, and n is the number of robots.

[0099] Step 2, use quaternions to describe the kinematics of the structural module. The quaternion Used to describe the attitude of the coordinate system {O} fixedly connected to the structural module with respect to the inertial coordinate system {I}, where a o and b o =[b1, b2, b3] T are the components of the quaternion ζ o , b1, b2, b3 are the components of b o , satisfying the condition T represents matrix transpose, calculating the derivative of the quaternion ζ o where satisfies Φ T (ζ o )Φ(ζ o ) = I 3×3 , is the angular velocity of the structural module with respect to the inertial coordinate system {I}, ω1, ω2, ω3 are its components. is the skew-symmetric matrix given based on b1, b2, b3.

[0100] Let represent the position vector of the coordinate system {O} with respect to the inertial coordinate system {I}, r1, r2, r3 are the three components of the position vector . represents the translational velocity of the coordinate system {O} with respect to the inertial coordinate system {I}; then the generalized position and the generalized velocity V o are expressed as

[0101] Formula integration can obtain the relationship between the position derivative and the velocity.

[0102]

[0103] Suppose the robot rigidly grasps the structural module, obtaining the velocity V Hi of the i-th robot end effector and the velocity V o of the structural module. The relationship is: where is the generalized velocity of the i-th robot end effector with respect to the inertial coordinate system {I}, and are the position velocity and angular velocity of the coordinate system {Hi} fixedly connected to the i-th robot end with respect to the inertial system {I}. is the grasping matrix of the coordinate system {O} with respect to the coordinate system {Hi}, where is the position vector of the coordinate system {O} with respect to the coordinate system {Hi}. For simplicity of expression, use E i to represent

[0104] ​Give the expression of the robot joint velocity:

[0105]

[0106] where is the first derivative of the joint angle of the i-th robot with respect to time, is six components of is the inverse matrix of the Jacobian matrix J i of the i-th robot.

[0107] Step 3, perform the dynamic description of the coupled system, adopt the Newton-Euler equation, and establish the dynamic equation of the structural module:

[0108]

[0109] where, D o and C o are respectively the inertia matrix and damping matrix of the structural module, is the first derivative of the generalized velocity V o of the structural module with respect to time; is the angular velocity of the structural module the first derivative of with respect to time; is the inertia matrix of the structural module relative to the inertial coordinate system {I}; is the external force vector acting on the center of gravity of the structural module.

[0110] For the robot, apply the Lagrange equation to obtain the dynamic equation of the i-th robot in the task space:

[0111]

[0112] where D i ∈R 6×6 , C i ∈R 6×6 , D i and C i are respectively the inertia matrix and damping matrix of the i-th robot, is the first derivative of the generalized velocity V Hi of the end effector of the i-th robot with respect to time; is the control action on the end effector caused by the joint torque of the i-th robot; is the force or torque generated at the contact point between the structural module and the end effector of the i-th robot.

[0113] The module acting force F o has the following relationship with the contact force F of the robot end effector:

[0114] Fo =(E) T F

[0115] Among them, is the grasping matrix of the nth robot, and E is the integrated matrix of the grasping matrices of n robots is the force or torque generated at the contact point between the structural module and the end effector of the nth robot; F is the integrated vector of the forces exerted by n robots on the structural module.

[0116] Integrating all the formulas, the dynamic model of the coupled system is obtained:

[0117]

[0118] Among them:

[0119] D a =D o +F T DE is the inertia matrix of the coupled system;

[0120] is the damping matrix of the coupled system; is the first derivative of the integrated grasping matrix E with respect to time;

[0121] u = E T f is the input vector of the system, is the control action on the end effector caused by the joint torques of n robots, is the control action on the end effector caused by the joint torque of the nth robot;

[0122] is the integrated inertia matrix composed of the inertia matrices of n robots, D n is the inertia matrix of the nth robot;

[0123] is the integrated damping matrix composed of the inertia matrices of n robots, C n is the damping matrix of the nth robot;

[0124] d represents the disturbance vector in space.

[0125] It is assumed that there exist scalars η1 and η2 satisfying η2 > η1 > 0 such that η1I 6×6 ≤ D a ≤ η2I 6×6 .

[0126] Step 4, the desired position r d and attitude ζ d of the structural module are is the desired attitude ζ d of the components, the position error re and the attitude error ζ e is expressed as:

[0127]

[0128] wherein, is the position vector of the structural module, is the conjugate quaternion of the attitude ζ o of the structural module, is the quaternion product, is the attitude error, a e , b e are its components.

[0129] Taking the differential gives:

[0130]

[0131] wherein, and respectively represent the error translational velocity and error angular velocity of the structural module, and are respectively the translational velocity and angular velocity of the structural module, and τ d are respectively the desired translational velocity and desired angular velocity of the structural module.

[0132] The goal of the control method is to make the structural module reach the desired position and attitude at a certain moment t of time t f , which is expressed as: ‖()‖ represents taking the norm of the matrix inside the parentheses, and the same applies hereinafter.

[0133] Step 5, the desired state of the system is expressed as wherein is the desired generalized velocity of the structural module; define the system state vector System state vector Obtain the first derivative of the system state vectors x1 and x2 with respect to time expression:

[0134]

[0135] wherein:

[0136] is an intermediate parameter,

[0137] is the state derivative and the transformation matrix between the state x2;

[0138] is the state derivative The conversion matrix with the desired speed V d ;

[0139] is the damping term of the state equation, is D a inverse matrix;

[0140] is the perturbation term of the state equation.

[0141] Let the total state variable Define k2(x,x d ) = M(x)x2 + G(x)V d , which is the conversion term of the system state derivative , then the error dynamics equation is expressed as:

[0142]

[0143] where, is the first-order derivative of the state variable x with respect to time;

[0144] is the state term in the state equation;

[0145] R = [0 6×6 , I 6×6 T , which is the perturbation and reference regulation matrix;

[0146] is the input regulation matrix.

[0147] Set the perturbation term as a bounded quantity, where d1 < 0 is a constant;

[0148] Set the existence of positive scalars c1, c2, c3,

[0149] Set the inequality ‖k(v1, w1) - k(v2, w2)‖ ≤ L‖v1 - v2‖ + L‖w1 - w2‖ holds for the selected constant L, where v1, v2, w1, w2 are vectors constrained in a compact domain Δ,

[0150] Step 6, define the sliding mode surface δ as:

[0151] δ := {x ∈ Δ: s = gx = 0}

[0152] where, s is the sliding mode variable, g = [g1, I 6×6 , is a sliding mode variable regulation matrix, is a positive definite matrix; ​

[0153] Based on the comprehensive error dynamics equation and the definition of the sliding mode surface, the first derivative of the sliding mode variable s with respect to time is obtained The expression of:

[0154]

[0155] Let denote the sequence of triggering moments for sampling the system state x and calculating the control input u. For event-triggered control, once the input u is determined at time t j , it remains unchanged until time t j+1 . The system input u(t) at time t is expressed as: u(t) = u(t j ), t ∈ [t j , t j+1 ).

[0156] Select the Lyapunov candidate function In a time interval, L‖g‖‖e x (t)‖ < β always holds, where β is a range parameter, and e x (t) = x(t j ) - x(t), t ∈ [t j , t j+1 ), which is the system state error. From this, the time derivative of the Lyapunov candidate function V s is calculated as: For the system to achieve global uniform stability, the control law is taken as:

[0157]

[0158] where λ is a positive constant,

[0159]

[0160] The intermediate parameter ρ = η2 / η1, ε(||x(t )||) = ε1 + ε2||x(t j )||, ε1 and ε2 are designed parameters, and ε(||x(t j )||) represents the adjustment parameter determined by the norm of the system state x(t j ) at time t j . j )

[0161] The time derivative becomes:

[0162]

[0163] where the intermediate parameter Υ = gn(t)(gn(t j ))-1 , gR = I 6×6 ;

[0164] Get ‖Υ‖ ≤ ρ, ‖I - Υ‖ ≤ 1 - ρ -1 , ‖ρI - βΥ‖ ≤ ρ - βρ -1 ;

[0165] Given positive scalar c1, c2, β, ρ, μ, λ, d1, L, Lyapunov candidate function adjustment parameters Lyapunov candidate function adjustment parameters Add and subtract terms on the right - hand side of the above formula to get:

[0166]

[0167] Consider the case starting with ‖s‖ > β / L. When sgn(s(t j )) = sgn(s(t)), sgn() returns the sign of the value inside the parentheses, and the above formula can be written as:

[0168]

[0169] When sgn(s(t j )) = sgn(s(t)), the system trajectory moves towards the sliding surface. Subsequently, when sgn(s(t j )) ≠ sgn(s(t)), the trajectory may cross the sliding surface and move away from the sliding surface. The deviation between s(t j ) and s(t) is restricted by the following formula:

[0170] ||s(t j ) - s(t)|| = ||gx(t j ) - gx(t)|| ≤ ‖g‖‖e x ‖ ≤ β / L.

[0171] Calculate the convergence intervals of the position error x1 and the velocity error x2:

[0172] Given positive scalar c3, L, β, positive - definite matrices P and Q, satisfying Q = (Mg1) T P + P(Mg1) > 0; M is the transformation matrix between the state derivative and the state x2,

[0173] Select the form of the Lyapunov function V as Find the time derivative of V

[0174]

[0175] χ min {} represents taking the minimum eigenvalue of the matrix within the brackets. To make the time derivative Select the boundary

[0176] It means that the position error x1 finally enters the Ξ range and stays there, and the velocity error x2 is also bounded:

[0177]

[0178] Numerical verification: In this embodiment, numerical simulation is carried out on the cooperative operation of two identical six-degree-of-freedom robots in an on-orbit assembly task to verify the rationality of the proposed event-triggered controller.

[0179] The mass of the selected structural module m = 3 kg, I o = diag{1, 1, 1} kg·m 2 .

[0180] The expected position of the module:

[0181] The expected attitude quaternion is expressed as:

[0182] where: z d = [0, 0, 1] T , ψ d (t) = 0.2cos(0.5t) is the expected attitude angle relative to the z d axis.

[0183] The initial conditions of the module:

[0184] The initial conditions of the joint angles:

[0185] Applying the event-triggered controller to the two-arm system, the position and attitude tracking effects of the module are as shown in Figure 2 , Figure 3 It can be seen that the designed controller has good trajectory tracking performance.

[0186] The present invention provides an event-triggered sliding mode control method for on-orbit cooperative assembly of space robots. There are many methods and ways to specifically implement this technical solution. The above is only the preferred embodiment of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention. Each component not clearly defined in this embodiment can be realized by existing technologies.

Claims

1. An event-triggered sliding mode control method for on-orbit cooperative assembly of a space robot, characterized in that, It includes the following steps: Step 1: Define the coordinate system and parameters of the space robot cooperation system; Step 2: Use quaternions to describe the kinematics of the structural module; Step 3: Conduct the dynamic description of the coupled system, use the Newton-Euler equation to establish the dynamic equation of the structural module; Step 4: Describe the control error; Step 5: Describe the desired state of the system; Step 6: Design the event-triggered sliding mode control law SMC (sliding mode control) and define the sliding surface.

2. The method according to claim 1, wherein Step 1 includes: The space robot cooperation system includes a space cooperation robot, and the space cooperation robot includes a base and n six-degree-of-freedom robots. The n six-degree-of-freedom robots are connected to the base and jointly control a structural module regarded as a rigid body.

3. The method according to claim 2, wherein Step 1 further includes: is the set of positive integers, represents an m×r dimensional real matrix, represents an r dimensional real vector, represents an r dimensional Euclidean space; The coordinate system {I} is an inertial coordinate system, {O} is the body coordinate system of the structural module, and {H i} is the coordinate system fixedly connected to the end effector of the i-th robot, where i = 1, 2, …, n, and n is the number of six-degree-of-freedom robots.

4. The method according to claim 3, wherein Step 2 includes: quaternion used to describe the attitude of the coordinate system {O} fixed to the structural module relative to the inertial coordinate system {I}, where a o and b o are the components of the quaternion ζ o ; b o = [b1, b2, b3] T , and b1, b2, b3 are the components of b o , satisfying the condition T represents matrix transpose, and calculate the derivative of the quaternion ζ o where the intermediate parameter satisfies v satisfies v T (ζ o )v(ζ o ) = I 3×3 , is the angular velocity of the structural module relative to the inertial coordinate system {I}, and ω1, ω2, ω3 are components of is the skew-symmetric matrix given based on b1, b2, b3; Let represent the position vector of the {O} coordinate system relative to the inertial coordinate system {i}, and r1, r2, r3 be the three components of the position vector ; represent the translational velocity of the {O} coordinate system relative to the inertial coordinate system {I}; then the generalized position and the generalized velocity V o are expressed as Perform formula integration to obtain the position derivative and the module speed V o The relationship is as follows: Set the rigid grasping structure module of the robot to obtain the velocity V of the end effector of the i-th robot Hi and the velocity V of the structure module o Relationship: Where V Hi is the generalized velocity of the end effector of the i-th robot relative to the inertial coordinate system {i}, and respectively represent the position velocity and angular velocity of the coordinate system {Hi} fixedly connected to the end of the i-th robot relative to the inertial system {I}; is the grasping matrix of the {O} coordinate system relative to the {Hi} coordinate system, where is the position vector of the {o} coordinate system relative to the {Hi} coordinate system. For the sake of simplicity of expression, use the intermediate parameter E i to represent Give the expression of the robot joint velocity: where is the first derivative of the joint angle of the \(i\)-th robot with respect to time, is six components of, is the inverse matrix of the Jacobian matrix \(J\) of the \(i\)-th robot i .

5. The method according to claim 4, characterized in that, Step 3 includes: The dynamic equation of the structural module is: Among them, D o and C o are the inertia matrix and damping matrix of the structural module respectively, is the first-order derivative of the generalized velocity V o of the structural module with respect to time; is the angular velocity of the structural module and its first-order derivative with respect to time; is the inertia matrix of the structural module relative to the inertial coordinate system {I}; is the external force vector acting on the center of gravity of the structural module; For the robot, apply the Lagrange equation to obtain the dynamic equation of the i-th robot in the task space: where D i ∈ R 6×6 , C i ∈ R 6×6 , D i and C i are respectively the inertia matrix and damping matrix of the i-th robot, is the first derivative with respect to time of the generalized velocity V Hi of the end effector of the i-th robot; is the control action on the end effector caused by the joint torque of the i-th robot; is the force or torque generated at the contact point between the structural module and the i-th end effector; Module acting force F o It has the following relationship with the contact force F of the robot end effector: F o = (E) T F Among them, is the grasping matrix of the nth robot, and E is the integrated matrix of the grasping matrices of n robots; is the force or torque generated at the contact point between the structural module and the end effector of the nth robot; F is the integrated vector of the forces exerted by n robots on the structural module; Integrate all the formulas to obtain the dynamic model of the coupled system: Where, D a = D o + E T DE, D a is the inertia matrix of the coupling system; C a is the damping matrix of the coupling system; is the first derivative of the integrated grasping matrix E with respect to time; u is the input vector of the system, u = E T f, where f is the control action on the end effector caused by the torques of n robot joints, and is the control action on the end effector caused by the torque of the nth robot joint; D is the integrated inertia matrix composed of the inertia matrices of n robots, D n is the inertia matrix of the nth robot; C is the integrated damping matrix composed of the inertia matrices of n robots, C n is the damping matrix of the nth robot; d represents the disturbance vector in space; Let there be scalars η1 and η w such that η w > η1 > 0, so that η1I 6×6 ≤ D a ≤ η2I 6×6 .

6. The method according to claim 5, wherein Step 4 includes: the desired position r of the structural module d and the attitude ζ d are a d , is the component of the desired attitude ζ d The position error r e and the attitude error ζ e are expressed as: Among them, is the position vector of the structural module, is the conjugate quaternion of the attitude ζ o of the structural module, is the quaternion product, ζ e is the attitude error, a e and b e are the components of ζ e ; Conduct differentiation to obtain: Among them, and respectively represent the error translational velocity and error angular velocity of the structural module, and are respectively the translational velocity and angular velocity of the structural module, and τ d are respectively the desired translational velocity and desired angular velocity of the structural module; The goal of the control method is to make the structural module reach the desired position and attitude at a moment \(t'\) starting from time \(t\) by adding an appropriate control law, which is expressed as: f when it reaches the desired position and attitude, which is expressed as: \(\|\left(\right)\|\) represents the norm of the matrix inside the parentheses.

7. The method according to claim 6, wherein Step 5 includes: the desired state of the system is expressed as where is the desired generalized velocity of the structural module; define the system state vector System state vector Obtain the first-order derivatives of the system state vectors x1 and x2 with respect to time The expression of: Among them, a e is an intermediate parameter; M(x) is the state derivative and the transition matrix between states x2; G(x) is the state derivative and the conversion matrix d between the desired velocity V k1(x,x d ) is the damping term of the state equation, is the D a inverse matrix; is the perturbation term of the state equation; Let the total state variable Define the derivative of the system state The conversion term h2(x, x d ) = M(x)x2 + G(x)V d , then the error dynamics equation is expressed as: wherein is the first derivative of the state variable x with respect to time; k(x,x d ) is the state term in the state equation; R = [0 6×6 , I 6×6 T , where R is the disturbance and reference adjustment matrix;​ n(t) is the input adjustment matrix; Set the disturbance term as a bounded quantity, where d1 < 0 is a constant; There exist positive scalars c1, c2, c3 such that ‖M(x)‖ ≤ c3; The inequality ‖k(v1,W1)-k(v2,w2)‖≤L‖v1-v2‖+L‖w1-w2‖ is set to hold for a selected constant L, where v1, v2, w1, w2 are vectors constrained in a compact domain Δ.

8. The method according to claim 7, wherein Step 6 includes: Define the sliding surface δ as: δ := {x ∈ Δ: s = gx = 0} where s is the sliding mode variable, g = [g1, I 6×6 , g is a sliding mode variable adjustment matrix, is a positive definite matrix; the symbol δ := means that δ is assigned a specific expression; Combining the comprehensive error dynamics equation and the definition of the sliding mode surface, the expression of the first derivative of the sliding mode variable s with respect to time is obtained as follows: Let denote the sequence of triggering instants for sampling the system state x and calculating the control input u. For event-triggered control, once the input u is determined at time t j , it remains unchanged until time t j+1 . It is expressed as: the system input u(t) at time t = u(t j ), where t ∈ [t j , t j+1 ).

9. The method according to claim 8, wherein Step 6 further includes: selecting a Lyapunov candidate function In a time interval, L‖g‖‖e x (t)‖ < β always holds, where β is a range parameter, and e x (t) = x(t j ) - x(t), t ∈ [t j , t j+1 ), e x (t) is the system state error, and from this, the time derivative of the Lyapunov candidate function V s is calculated as follows: That is: To make the system achieve global uniform stability, take the control law as: where λ is a positive constant, the intermediate parameter ρ = η2 / η1, ε(||x(t j )||) = ε1 + ε2||x(t j )||, ε1, δ2 are the designed parameters, and ε(||x(t j )||) represents the regulation parameter determined by the norm of the system state x(t j at time t j ); The time derivative becomes: Among them, the intermediate parameter γ = gn(t)(gn(t J )) -1 , gR = I 6×6 ; Obtain ‖Υ‖ ≤ ρ, ‖I - Υ‖ ≤ 1 - ρ -1 , ‖ρI - βΥ‖ ≤ ρ - βρ -1 ; Given positive scalars \(c_1\), \(c_2\), \(\beta\), \(\rho\), \(\mu\), \(\lambda\), \(d_1\), \(L\), Lyapunov candidate function tuning parameters Lyapunov candidate function tuning parameters Adding and subtracting terms on the right - hand side of the above equation, we get: Consider the case starting from ‖s‖>β / L. When sgn(s(t j )) = sgn(s(t)), the function sgn() represents returning the sign of the value inside the parentheses, and we get: When sgn(s(t j )) = sgn(s(t)), the system trajectory moves towards the sliding surface. Subsequently, when sgn(s(t j )) ≠ sgn(s(t)), the trajectory may cross the sliding surface and move away from it. The deviation between s(t j ) and s(t) is restricted by the following formula: ||s(t j ) - s(t)|| = ||gx(t j ) - gx(t)|| ≤ ‖g‖‖e x ‖ ≤ β / L。 10. The method according to claim 9, wherein Step 6 also includes: Calculate the convergence interval of the position error x1 and the velocity error x2: Given positive scalars c3, L, β, positive definite matrices P and Q, satisfying Q = (Mg1) T P + P(Mg1) > 0; M is the conversion matrix between the state derivative and the state x2, which is a positive definite matrix; Select the form of the Lyapunov function V as Find the time derivative of V where χ min {} represents the minimum eigenvalue of the matrix within the brackets. To make the time derivative select the boundary It is shown that the position error x1 finally enters the Ξ range and stays there, and the velocity error x2 is also bounded:

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