Mechanical arm switching control transient analysis method based on system response characteristics

Abstract

The invention provides a mechanical arm switching control transient analysis method based on system response characteristics. The mechanical arm switching control transient analysis method comprises the following steps that 1, a mechanical arm input and output response model is constructed; 2, mechanical arm switching input serves as a segmented continuous function, and an input and output relation is established; and step 3, introducing a step function to separate steady-state output from transient-state output, and analyzing a transient-state process. According to the method, a mechanical arm switching input signal is converted into a segmented continuous function through a step function, the root of generation of the mechanical arm switching transient state is researched, and the relation between the mechanical arm transient state and the historical process is established. According to the method, the mechanical arm switching control transient state is analyzed on the basis of system response characteristics in combination with historical dynamic information, the mechanical arm transient state generation mechanism is analyzed, and analysis is more accurate and reliable.

Description

Technical Field

[0001] The present invention belongs to the technical field of robotic arm control, and particularly relates to a transient analysis method for switching control of a robotic arm based on system response characteristics. Background Art

[0002] Robots have broad application prospects in the fields of medical treatment, industrial and agricultural production, national defense, etc. A robotic arm is the main executing mechanism of a robot, and a robot usually controls the movement of the robotic arm by controlling a driving motor.

[0003] According to the working requirements, a robotic arm often needs to switch between different states, and this switching often realizes the switching of the output of the working state of the robotic arm by switching the input of the driving motor. Any system in nature is not an instantaneous control system: when the input changes, the output does not immediately respond to this input change, but has a response process. Different systems have different response characteristics, and under the same input conditions, the response processes of the outputs are also different. Existing research almost always regards the robotic arm as an ideal rigid body, ignores the deformation of the robotic arm caused by the change of the input torque, and ignores the response process of the robotic arm. There is no ideal rigid body in nature, and the change of the input torque will inevitably cause the output deformation. The larger the stiffness of the robotic arm, the smaller the deformation. However, the larger the stiffness also means that the structural size of the robotic arm is larger and the response delay is larger, making it difficult to respond to the rapid changes of the system input.

[0004] In fact, in fields such as medical treatment and aerospace, due to working condition limitations, too large a structural size is not allowed, so it is necessary to consider the output response process caused by the input switching - the transient process. The transient process often caused by the switching of the working state of the robotic arm reduces the execution accuracy of the robotic arm, generates noise, and exacerbates the fatigue fracture of the robotic arm, etc. Studying the transient process of the robotic arm is of great significance.

[0005] The root cause of the transient process of the robotic arm is due to the response characteristics of the robotic arm. Both historical dynamic information and the current input will affect the output, and transient oscillations are generated under the alternating action of historical information and the current input. Therefore, it is necessary to combine the response characteristics of the robotic arm and the dynamic switching process to study the transient process of the robotic arm. Summary of the Invention

[0006] Object of the Invention: Aiming at the steady states such as transient oscillations of the output of the robotic arm caused by input switching, reduction of accuracy, generation of noise, etc., starting from the system response characteristics, the present invention establishes the relationship between the output of the robotic arm and the historical dynamic process, and uses a window function to separate the steady state and the transient state of the output of the robotic arm, and proposes a transient process analysis method for switching control of the robotic arm, including the following steps:

[0007] Step 1, construct a response model of the input and output of the robotic arm;

[0008] Step 2, taking the robot arm switching input as a piecewise continuous function to establish an input-output relationship;

[0009] Step 3: Introduce a step function to separate the steady-state and transient outputs and analyze the transient process.

[0010] Step 1 includes the following steps:

[0011] Step 1.1, establish the following equation based on the robot arm dynamics model:

[0012]

[0013] Where t represents time, q(t) represents the joint angle at time t, and denote the first and second derivatives of q(t), respectively, M(q(t)) is the inertia matrix, is the Coriolis and centrifugal force matrix, is the gravity torque matrix, τ(t) is the driving torque;

[0014] The input and output of the robot arm are represented by the following response model:

[0015] q(t)=h(t)*τ(t) (2),

[0016] Where h(t) represents the system impulse response, and the symbol * represents the convolution operation;

[0017] Step 1.2, establish the system impulse response h(t): Where i is the independent variable, k i is the unknown parameter corresponding to ti, and m represents the modeling order. According to the intermediate process theory, all impulse response models can always be expressed as follows:

[0018]

[0019] Among them, ε and α are unknown constants and power function orders respectively. Only by identifying the parameters ε and α, the input and output response model of the robot arm can be established;

[0020] Step 1.3, transform formula (2) into an algebraic model using Legendre transformation;

[0021] Step 1.4, identify unknown parameters ε and α: q(t) obtained by formula (2) and the joint angle at time t obtained by actual sampling The mean square error is calculated as the cost function, and the parameters ε and α are identified based on the particle swarm optimization algorithm and gradient algorithm that do not rely on gradient information;

[0022] Step 1.5, based on the relationship between fractional-order differentials and convolution operations, use a fractional-order model to represent the input and output of the robot;

[0023] Define the fractional-order derivative:

[0024]

[0025] where \(x(t)\) represents the state variable, represents the \(\nu\) - order CAPUTO fractional - order derivative of the state variable \(x(t)\), represents the first - order derivative, \(u\) s (t) represents the unit step function, \(\Gamma(\cdot)\) represents the gamma function; the fractional - order differential model with differential order \(\nu\) can represent a system with impulse response as the system;

[0026] Combining the parameters \(\varepsilon\) and \(\alpha\) in the impulse response model \(h(t)=\varepsilon t\) α one can always find the corresponding parameters \(\xi\) and differential order \(\nu\) such that the joint angle \(q(t)\) and driving torque \(\tau(t)\) of the robotic arm satisfy:

[0027]

[0028] Step 1.6, establish the switching control model of the robotic arm.

[0029] Step 1.6 includes: introducing an event - triggered mechanism and establishing the following switching control model:

[0030]

[0031] The event - triggered signal \(s(t)\): \(t_0\) represents the initial moment when the signal is activated, \(f\) s(t) (q(t)) represents the function under the action of the switching signal \(s(t)\). When \(s(t)\) is in different time slices, it corresponds to different activation functions. For example, \(f\) k (q(t)) is a non - linear function of the torque with respect to the joint angle, satisfying the Lipschitz condition: \(q(t)f\) k (q(t))\(\leq\rho\) k \(q\) 2 (t), when \(t\in[t_0,t_1)\), the robotic arm control system is in the initial state and \(f_0(q(t))\) is activated; when \(t\in[t_1,t_2)\), \(f_1(q(t))\) is activated; when \(t\in[t_2,t_3)\), \(f_2(q(t))\) is activated, and so on, where \(k\) takes values from 1 to \(N\), \(\rho\) k is the Lipschitz constant corresponding to the function \(f\) k (q(t)).

[0032] Step 2 includes the following steps:

[0033] Step 2.1, represent the switching control input of the robotic arm with a piecewise continuous function: when \(t\in[t_0,t_1)\), the system is in the initial state and \(f_0(q(t))\) is activated; when \(t\in[t_1,t_2)\), \(f_1(q(t))\) is activated; when \(t\in[t_2,t_3)\), \(f_2(q(t))\) is activated, when \(t\in[t n-1 ,t n )\), \(f n-1 (q(t))\) is activated, where \(t n represents the trigger time of the \(n\)th switch;

[0034] Define the unit step function \(u(t)\) and represent the switching input signal \(f(q(t))\) as a piecewise continuous function:

[0035]

[0036] Obtain the robotic arm switching model:

[0037]

[0038] where \(u(t - t n )\) represents delaying \(u(t)\) by \(t n ;

[0039] Step 2.2, construct the following positive definite function \(V(t)\):

[0040]

[0041] Step 2.3, establish the relationship between the positive definite function and the fractional order derivative:

[0042]

[0043] Step 2.4, obtain the relationship of \(V(t)\) according to the comparison principle and the fractional order integral relationship:

[0044]

[0045] where \(\rho n is the Lipschitz constant corresponding to the function \(f n (q(t))\).

[0046] Step 3 includes the following steps:

[0047] Step 3.1, analyze the output according to the convolution operation;

[0048] Step 3.2, separate the output into the steady state and the transient state;

[0049] Step 3.3, calculate the steady state component and the transient state component respectively;

[0050] Step 3.4, establishing the stability condition: The convergence condition jointly established by the transient component and the steady-state component is the system stability condition;

[0051] Step 3.5, estimating the peak value of the transient process;

[0052] Step 3.6, estimating the transient oscillation process of the robotic arm switching.

[0053] Step 3.1 includes:

[0054] When t ∈ [t0, t1), we get:

[0055]

[0056] When t ∈ [t1, t2), we get:

[0057]

[0058] When t ∈ [t n , t n+1 ) we get:

[0059]

[0060] Step 3.2 includes:

[0061] Input the output generated when t ∈ [t0, t1):

[0062]

[0063] When t ∈ [t1, t2), we get:

[0064]

[0065] When t ∈ [t n , t n+1 ) we get:

[0066]

[0067] Define the steady-state output and the transient output as V s (t) and V t (t) respectively, and we get V(t) = V s (t) + V t (t).

[0068] Step 3.3 includes: When t ∈ (t n , t n+1 , calculate the steady-state output:

[0069] First, perform step-by-step calculations to get:

[0070] Vs (t) ≤ V(0)(E ν,1 (ρ0(t1 - t0) ν )(E ν,1 (ρ1(t2 - t1) ν )…(E ν,1 (ρ n-1 (t n - t n-1 ) ν )(E ν,1 (ρ n (t - t n ) ν ) (18);

[0072] Let V k t (t) represent the transient generated by the k - th switching. It is calculated as:

[0073]

[0074] By successive recursion, the transient output generated by each switching is obtained. The transient output at time t ∈ (t n , t n+1 is the superposition of the transients generated by previous level switchings:

[0075] V t (t) = V1 t (t) + V2 t (t) +, …, + V n-1 t (t)(21).

[0076] Step 3.5 includes: calculating the transient peak value V t (t) using the following formula:

[0077]

[0078] where represents the transient generated by the j - th switching, and L is the number of switchings.

[0079] Step 3.6 includes: estimating the transient oscillation process of the robotic arm switching based on the transients generated by the previous L switchings before the current moment:

[0080]

[0081] Beneficial effects: 1. The present invention proposes a transient analysis method for robotic arm switching control based on system response characteristics. This method analyzes the transient of robotic arm switching control based on system response characteristics, combines historical dynamic information, and analyzes from the transient generation mechanism of the robotic arm, with more accurate and reliable analysis.

[0082] 2. The present invention uses a step function to convert the switching input signal of the robotic arm into a piecewise continuous function, studies the root cause of the switching transient of the robotic arm, and establishes the relationship between the transient state of the robotic arm and the historical process.

[0083] 3. According to the convolution property, the present invention separates the steady state and transient state of the robotic arm output, analyzes the steady state component and the transient state component respectively, and establishes a stability analysis method combining the transient process of the system.

[0084] 4. Based on the system response characteristics and the historical dynamic process, the present invention proposes a method for analyzing the transient peak value and transient process of the switching control of the robotic arm, provides a new basis for the transient analysis, and provides a basis for the selection of switching control parameters. BRIEF DESCRIPTION OF THE DRAWINGS

[0085] The following further describes the present invention in detail with reference to the drawings and specific embodiments, and the above and / or other advantages of the present invention will become clearer.

[0086] Figure 1 It is a schematic diagram of the robotic arm switching control system.

[0087] Figure 2 It is a flow chart of the transient analysis of the robotic arm switching control system.

[0088] Figure 3 It is a flow chart of the impulse response modeling of the robotic arm.

[0089] Figure 4 It is a flow chart of the relationship analysis of the robotic arm.

[0090] Figure 5 It is a flow chart of the transient process analysis of the robotic arm switching control.

[0091] Figure 6 It is a simulation diagram when ν = 0.3 in the embodiment of the present invention.

[0092] Figure 7 It is a simulation diagram when ν = 0.6 in the embodiment of the present invention.

[0093] Figure 8 It is a simulation diagram when ν = 0.8 in the embodiment of the present invention.

[0094] Figure 9 It is a simulation diagram when ν = 1 in the embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0095] The embodiment of the present invention provides a method for analyzing the transient state of the robotic arm switching control based on the system response characteristics (as shown in Figure 2 ), including the following steps:

[0096] Step 1, construct the input-output response model of the robotic arm;

[0097] Step 2, regard the switched input of the robotic arm as a piecewise continuous function, and establish the input-output relationship;

[0098] Step 3, introduce a step function to separate the steady-state and transient outputs, and analyze the transient process.

[0099] Step 1 includes the following steps (as Figure 3 shown):

[0100] Step 1.1, establish the following equation according to the dynamic model of the robotic arm:

[0101]

[0102] where t represents time, q(t) represents the joint angle at time q(t), and respectively represent the first derivative and the second derivative of q(t), M(q(t)) is the inertia matrix, is the Coriolis force and centrifugal force matrix, is the gravity torque matrix, and τ(t) is the driving torque.

[0103] Obviously, the model in formula (1) is an ideal impulse response model: the input change immediately causes the output change, and when the input stops, the output also stops immediately. Factors such as the deformation of the robotic arm and various dampings are ignored.

[0104] There is no ideal impulse response system in nature. Otherwise, the instantaneous power would be infinite and any mechanical system would be destroyed.

[0105] Generally, the input-output of the robotic arm is represented by the following response model:

[0106] q(t) = h(t) * τ(t) (2),

[0107] h(t) represents the system impulse response, and the symbol * represents the convolution operation. The model in formula (1) can be regarded as a special case of the model in formula (2).

[0108] Step 1.2, establish the system impulse response h(t):

[0109] The input-output of the robotic arm can be regarded as a bounded-input bounded-output system. Its impulse response system can be represented by where i is the independent variable, k i is an unknown parameter, m represents the modeling order, and the larger m is, the higher the modeling accuracy, but there are also problems such as an increase in the number of parameters to be identified and difficulty in parameter identification.

[0110] According to the intermediate process theory, any impulse response model can always be represented by the following formula:

[0111]

[0112] Therefore, as long as the identification parameters ε and α are known, the input-output response model of the robotic arm can be established;

[0113] Step 1.3: Use the Legendre transform to convert the convolution model into an algebraic model;

[0114] Step 1.4: Identify the unknown parameters ε and α: The parameter α is the exponent of the function and is insensitive to gradient information. Therefore, construct a cost function, and use a particle swarm optimization algorithm and a gradient algorithm that do not rely on gradient information to identify the parameters ε and α.

[0115] Step 1.5: According to the relationship between fractional-order differentiation and convolution operation, use the fractional-order model to represent the input and output of the robotic arm;

[0116] Define fractional-order differentiation:

[0117]

[0118] where u s (t) represents the unit step function, and Γ(·) represents the gamma function. According to the fact that fractional-order differentiation is a convolution model, the fractional-order differential model with a differential order of ν can represent the impulse response as for the system.

[0119] Combining the parameters ε and α in the system impulse response h(t) = εt α , the corresponding parameters ξ and differential order ν can always be found such that the joint angle q(t) and the driving torque τ(t) of the robotic arm satisfy:

[0120]

[0121] Step 1.6: Establish a switching control model for the robotic arm: The robotic arm often needs to adjust the control input according to the output feedback. Discontinuous feedback is a commonly used feedback control method, and its control schematic diagram is as Figure 1 shown.

[0122] That is, adjust the input driving torque according to the joint output angle. Introduce an event-triggering mechanism and establish the following switching control model:

[0123]

[0124] Event-triggering signal s(t): f k (q(t)) reflects the non-linear function of the torque with respect to the joint angle and satisfies the Lipschitz condition: q(t)f k (q(t)) ≤ ρ k q 2(t). That is, when \(t\in[t_0,t_1)\), the system is in the initial state and \(f_0(q(t))\) is activated; when \(t\in[t_1,t_2)\), \(f_1(q(t))\) is activated; when \(t\in[t_2,t_3)\), \(f_2(q(t))\) is activated, and so on.

[0125] Step 2 includes the following steps (as Figure 4 shown):

[0126] Step 2.1, represent the switching control input of the robotic arm with a piecewise continuous function: when \(t\in[t_0,t_1)\), the system is in the initial state and \(f_0(q(t))\) is activated; when \(t\in[t_1,t_2)\), \(f_1(q(t))\) is activated; when \(t\in[t_2,t_3)\), \(f_2(q(t))\) is activated.

[0127] Define the unit step function \(u(t)\) and represent the switching signal as a piecewise continuous function:

[0128]

[0129] Obtain the robotic arm switching model:

[0130]

[0131] Step 2.2, to study the stability and transient process of the robotic arm switching control, construct a positive definite function:

[0132]

[0133] Step 2.3, establish the relationship between the positive definite function and the fractional-order differential:

[0134]

[0135] Step 2.4, obtain the relationship of \(V(t)\) according to the comparison principle and the fractional-order integral relationship:

[0136]

[0137] Step 3 includes the following steps (as Figure 5 shown):

[0138] Step 3.1, analyze the output according to the convolution operation;

[0139] 1) When \(t\in[t_0,t_1)\), obtain:

[0140]

[0141] 2) When \(t\in[t_1,t_2)\), obtain:

[0142]

[0143] 3) When \(t\in[t n ,t n+1 ), the following is obtained:

[0144]

[0145] Since \((tu(t)) ν-1 is a function that decays with time, but this function is never zero. According to the convolution operation, the length of the output signal is equal to the sum of the lengths of the input signal and the impulse response. When the signal input is cutoff, its output never cuts off, there is also a transient state, and this transient state always exists. Since the impulse response decays with time, this transient state also decays with time.

[0146] When the manipulator input switches between different states, its output is not only affected by the current input, but also affected by various historical transients. Under the interaction of these effects, there must be a transient oscillation process.

[0147] Step 3.2, separate the steady state and transient states of the output:

[0148] 1) The output generated when the input is in \(t\in[t_0,t_1)\):

[0149]

[0150] 2) When \(t\in[t_1,t_2)\), the following is obtained:

[0151]

[0152] 3) When \(t\in[t n ,t n+1 ), the following is obtained:

[0153]

[0154] The first part represents the steady-state output of the system: the part where the input is cutoff and the output is cutoff. The end point of the steady-state output is the starting point of the next input switch.

[0155] The second part represents the transient output of the system: the part where the input is cutoff and the output does not cutoff. Since the transient part never cuts off, the transient formed by each switch will affect all subsequent processes.

[0156] Therefore, it is necessary to calculate the steady-state output and the transient output step by step. Define the steady-state output and the transient output as: \(V s (t)\) and \(V t (t)\). There is \(V(t)=V s (t)+V t (t)\).

[0157] Step 3.3, calculate the steady-state component and the transient component respectively:

[0158] When \(t\in(t n ,t n+1 \), calculate the steady-state output:

[0159] Calculate step by step to obtain:

[0160] V s (t)\(\leq V(0)(E ν,1 (\rho_0(t_1 - t_0) ν )(E ν,1 (\rho_1(t_2 - t_1) ν )\cdots(E ν,1 (\rho n-1 (t n -t n-1 ) ν )(E ν,1 (\rho n (t - t n ) ν ) (18);

[0162] Calculate the transient output: Each switching will generate a transient, and each generated transient will affect all subsequent processes.

[0163] Denote V k t (t) as the transient generated by the \(k\)-th switching. Calculate to obtain:

[0164]

[0165] Recursively calculate step by step to obtain the transient output generated by each switching. At time \(t\in(t n ,t n+1 \), the transient output is the superposition of the transients generated by the previous levels of switching:

[0166] V t (t)=V_1 t (t)+V_2 t (t)+\cdots+V n-1 t (t)(21);

[0167] Step 3.4, establish the stability condition: Since the manipulator output includes a steady-state component and a transient component, the convergence of both the steady-state component and the transient component is required to ensure the convergence of the control output.

[0168] As long as the steady-state component decreases step by step in each period, the convergence of the steady-state component can be ensured.

[0169] Although each switch will generate a new transient state, and any moment is affected by all previous transient states, due to the transient decay characteristic, the transient state effects beyond a certain time can be ignored. Therefore, it is only necessary to calculate a finite number of transient components. For a finite number of transient components, as long as each switch can make the transient components converge, the convergence of the transient components can be ensured.

[0170] The convergence condition established by the transient components and the steady-state components together is the system stability condition.

[0171] Step 3.5, peak value estimation of the transient process: Although all processes will affect the transient state, it is impossible to analyze the transient state based on all historical processes in engineering. It is only necessary to study the historical information within a certain length range, and then estimate the lost historical information with the estimation error within a certain range. According to the stability condition, both the steady-state components and the transient components will gradually decay. Therefore, the peak value of the transient components appears some time after the start of the switch. According to the decay characteristic of the impulse response of the robotic arm, an appropriate L can be selected to calculate the transient peak value:

[0172]

[0173] Step 3.6, estimation of the transient oscillation process of the robotic arm switch:

[0174] Similarly, the transient process estimation cannot be analyzed based on all historical processes. The transient state can be estimated based on the transients generated by the previous L switches before the current moment:

[0175]

[0176] In the embodiment of the present invention, the robotic arm joint angle parameters are selected as follows:

[0177] The trigger times t1, t2, t3 of the switching signal are successively selected as subsystem 1, subsystem 2, and subsystem 3. The corresponding parameters of each subsystem are f1(q(t)) = q(t), f2(q(t)) = -3q(t), f3(q(t)) = q(t). Different differential orders ν represent different response characteristics of different robotic arms, and different ν are selected for simulation.

[0178] When ν = 0.3, ν = 0.6, ν = 0.8, and ν = 1, the simulation results are respectively as Figure 6 , Figure 7 , Figure 8 , Figure 9 shown ( Figures 6 to 9 In, the abscissa represents time, and the ordinate represents the joint angle). The simulation results show that: under the same conditions, different systems have different transient processes due to different response characteristics. It is necessary to analyze the transient process in combination with the response characteristics of the robotic arm and study the control strategy on this basis.

[0179] The present invention provides a transient analysis method for manipulator switching control based on system response characteristics. There are many methods and approaches to specifically implement this technical solution. The above description 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 implemented by using the prior art.

Claims

1. A transient analysis method for robotic arm switching control based on system response characteristics, characterized in that It includes the following steps: Step 1, construct the input-output response model of the robotic arm; Step 2, take the switched input of the robotic arm as a piecewise continuous function and establish the input-output relationship; Step 3, introduce the step function to separate the steady-state and transient outputs and analyze the transient process.

2. The method according to claim 1, wherein Step 1 includes the following steps: Step 1.1, establish the following equation according to the dynamic model of the robotic arm: where t represents time, and q(t) represents the joint angle at time t, and represent the first derivative and the second derivative of q(t) respectively, M(q(t)) is the inertia matrix, is the Coriolis and centrifugal force matrix, is the gravity torque matrix, and τ(t) is the driving torque; The input-output of the robotic arm is represented by the following response model: q(t) = h(t) * τ(t) (2), where h(t) represents the system impulse response, and the symbol * represents the convolution operation; Step 1.2, establish the system impulse response h(t): where i is the independent variable, k i is the unknown parameter corresponding to t - i, and m represents the modeling order; according to the intermediate process theory, all impulse response models can always be expressed by the following formula: where ε and α are the unknown constant and the order of the power function respectively. Only by identifying the parameters ε and α can the input-output response model of the robotic arm be established; Step 1.3, use the Legendre transform to transform formula (2) into an algebraic model; Step 1.4, identify unknown parameters ε and α: The joint angle at time t obtained from q(t) derived from formula (2) and the actual sampling Calculate the mean square error as the cost function, and identify parameters ε and α based on the particle swarm optimization algorithm and the gradient algorithm that do not rely on gradient information; Step 1.5, according to the relationship between fractional-order differentiation and convolution operation, represent the input-output of the robotic arm using the fractional-order model; Define the fractional-order differentiation: where \(x(t)\) represents the state variable, denotes the \(\nu\) - th order CAPUTO fractional - order derivative of the state variable \(x(t)\), denotes the first - order derivative, \(u\) s (t) represents the unit - step function, \(\Gamma(\cdot)\) represents the gamma function; the fractional - order differential model with the differential order \(\nu\) can represent a system with an impulse response of the system; Combined with the impulse response model \(h(t)=\varepsilon t\) α For the parameters \(\varepsilon\) and \(\alpha\) in it, the corresponding parameters \(\xi\) and differential order \(\nu\) can always be found such that the robotic arm joint angle \(q(t)\) and driving torque \(\tau(t)\) satisfy: Step 1.6, establish the switched control model of the robotic arm.

3. The method according to claim 2, wherein Step 1.6 includes: introduce the event-triggering mechanism and establish the following switched control model: Event trigger signal t0 represents the initial moment when the signal is activated, f s(t) (q(t)) represents the function under the action of the switching signal s(t). When s(t) is in different time slices, it corresponds to different activation functions, such as f k (q(t)) should be a non-linear function of the torque with respect to the joint angle and satisfy the Lipschitz condition: q(t)f k (q(t)) ≤ ρ k q 2 (t), when t ∈ [t0, t1), the manipulator control system is in the initial state and f0(q(t)) is activated; when t ∈ [t1, t2), f1(q(t)) is activated; when t ∈ [t2, t3), f2(q(t)) is activated, and so on. Among them, k takes values from 1 to N, ρ k is the Lipschitz constant corresponding to the function f k (q(t)).

4. The method according to claim 3, wherein Step 2 includes the following steps: Step 2.1, represent the switching control input of the robotic arm with a piecewise continuous function: when t ∈ [t0, t1), the system is in the initial state and f0(q(t)) is activated; when t ∈ [t1, t2), f1(q(t)) is activated; when t ∈ [t2, t3), f2(q(t)) is activated, and when t ∈ [t n-1 , t n ), f n-1 (q(t)) is activated, where t n represents the triggering time of the nth switch; Define the unit step function u(t) and represent the switched input signal f(q(t)) as a piecewise continuous function: Obtain the switched model of the robotic arm: where u(t - t n ) represents the delay of u(t) by t n ; Step 2.2, construct the following positive definite function V(t): Step 2.3, establish the relationship between the positive definite function and the fractional-order differentiation; Step 2.4, obtain the relationship of V(t) according to the comparison principle and the fractional-order integral relationship; where ρ n is the Lipschitz constant corresponding to the function f n (q(t)).

5. The method according to claim 4, wherein Step 3 includes the following steps: Step 3.1, analyze the output according to the convolution operation; Step 3.2, separate the output into the steady-state and transient parts; Step 3.3, calculate the steady-state component and the transient component respectively; Step 3.4, establish the stability condition: the convergence condition jointly determined by the transient component and the steady-state component is the system stability condition; Step 3.5, estimate the peak value of the transient process; Step 3.6, estimate the transient oscillation process of the robotic arm switching.

6. The method according to claim 5, characterized in that Step 3.1 includes: When t ∈ [t0, t1), obtain: When t ∈ [t1, t2), obtain: When t ∈ [t n , t n+1 ), the following is obtained:

7. The method according to claim 6, characterized in that, Step 3.2 includes: the output generated by the input when t ∈ [t0, t1): When t ∈ [t1, t2), obtain: When \(t\in[t n ,t n+1 ), the following is obtained: Define the steady-state output and the transient output as V s (t) and V t (t), and we get V(t) = V s (t) + V t (t).

8. The method according to claim 7, wherein Step 3.3 includes: when t ∈ (t n , t n+1 , calculate the steady-state output: First, perform step-by-step calculations to obtain: V s V(t) ≤ V(0)(E ν,1 (ρ0(t1 - t0) ν )(E ν,1 (ρ1(t2 - t1) ν )…(E ν,1 (ρ n-1 (t n -t n-1 ) ν )(E ν,1 (ρ n (t - t n ) ν ) (18)； Denote V k t (t) represents the transient generated by the k-th switching, and it is calculated as follows: By successive recursion, the transient output generated by each switch is obtained. The transient output at \(t\in(t n ,t n+1 \) is the superposition of the transients generated by the previous level switches: V t V(t) = V1 t V(t) + V2 t V(t) + … + V n-1 t V(t) (21).

9. The method according to claim 8, wherein Step 3.5 includes: calculating the transient peak value V t (t) using the following formula: Among them represents the transient generated by the j-th switching, and L is the number of switchings.

10. The method according to claim 9, characterized in that, Step 3.6 includes: estimate the transient oscillation process of the robotic arm switching according to the transients generated by the previous L switches before the current moment:

Images