Response characteristic-based mechanical arm pulse load oscillation analysis method

Abstract

The invention discloses a mechanical arm pulse load oscillation analysis method based on response characteristics, and the method comprises the steps: building an equivalent feedback control process of a mechanical arm pulse load according to an equivalent principle; an input and output convolution model is constructed according to mechanical arm dynamic characteristics, an impulse response function form is determined by identifying system parameters, and a mechanical arm input and output impulse response model is constructed; based on the impulse response model, a mathematical model for mechanical arm pulse load feedback control is established; and the output response of the mechanical arm under the pulse load effect is analyzed through the mathematical model. On the basis of the system response characteristics, the mechanical arm pulse load output relation is established, the mechanical arm pulse load steady state and transient state analysis method and the mechanical arm oscillation calculation method are provided, and a basis is provided for mechanical arm pulse load analysis and design.

Description

Technical Field

[0001] The present invention relates to a response characteristic-based pulse load oscillation analysis method for a robotic arm. Background Art

[0002] Robots have broad application prospects in the fields of medicine, industrial and agricultural production, and national defense. The robotic arm is the main actuator of the robot. Due to sudden changes in load due to factors such as loading and unloading, the robotic arm is often subjected to pulse loads.

[0003] No system is ideal; it always has a response process. A robotic arm is no exception. When subjected to pulsed loads, its output undergoes a response process. As a key actuator in a robot, the robotic arm often has high requirements for precision. Output feedback is typically used to ensure control accuracy. This feedback is fed back to the input, and the input is adjusted in real time based on the output error to ensure output control accuracy. However, when the robotic arm is subjected to pulsed loads, its output can jump. The transient process generated by this jump acts on the input through feedback, inevitably causing oscillation. While this oscillation gradually subsides over time, it is a process. If the robotic arm is subjected to a new pulsed load before the previous oscillation has fully subsided, it will inevitably generate a new oscillation, creating one wave after another, with each wave potentially exceeding the previous one. This can severely impact the robotic arm's precision. Applications in defense, medical care, and thin-walled part handling all place higher demands on robotic arms' precision. Therefore, studying the oscillatory behavior of robotic arms under pulsed loads is of great significance. Summary of the Invention

[0004] The present invention aims to solve the above problems in the prior art and provides a method for analyzing pulse load oscillation of a robotic arm based on response characteristics.

[0005] The technical solutions adopted in the present invention are:

[0006] A method for analyzing pulse load oscillation of a robotic arm based on response characteristics includes the following steps:

[0007] (1) Based on the equivalent principle, an equivalent feedback control process of the manipulator pulse load is established;

[0008] (2) Based on the dynamic characteristics of the manipulator, an input-output convolution model is constructed, and the impulse response function form is determined by identifying the system parameters to construct the input-output impulse response model of the manipulator;

[0009] (3) Based on the impulse response model in step (2), a mathematical model of pulse load feedback control of the manipulator is established;

[0010] (4) Using the mathematical model in step (3), analyze the output response of the manipulator under the pulse load. The steps are as follows:

[0011] 1. Before applying the pulse load, obtain the output response expression of the robot arm;

[0012] 2. Calculate the output jump of the robotic arm when the pulse load is triggered;

[0013] 3. Separate the input effects before and after the pulse load is triggered, and divide the output of the robot arm into steady-state output and transient output;

[0014] 4. Recursively establish the transient response analysis process after each pulse load action to obtain the complete output response sequence of the robotic arm.

[0015] Furthermore, in step (1), the equivalent feedback control process of the pulse load of the manipulator is:

[0016] The input control signal acts on the input end of the robot arm, and the output signal of the robot arm is fed back to the controller used to control the movement of the robot arm, forming a closed-loop feedback control system. At the same time, the output end of the robot arm is triggered at irregular time t k Subject to pulse load.

[0017] Furthermore, in step (2), the impulse response model of the manipulator input and output is constructed as follows:

[0018] The input-output relationship of the manipulator is represented by a convolution model, that is, the angular displacement output q(t) of the manipulator is the convolution of the input torque τ(t) and the manipulator impulse response h(t). The mathematical expression is q(t) = h(t)*τ(t);

[0019] The impulse response h(t) satisfies the absolute integrability condition and can be expressed as h(t)=ηt -β , where η and β are parameters to be identified, and the two parameters are identified by particle swarm optimization algorithm combined with gradient algorithm, so as to establish the impulse response model.

[0020] Furthermore, in step (3), the process of establishing the mathematical model is as follows:

[0021] Set the angular displacement output of the robot arm to q(t) and the feedback channel output to g c q(t), where g c is the feedback gain;

[0022] If the kth pulse load is expressed as θ(t k ), the output of the pulse load through the feedback channel is: f(θ(t k )), simplified as dθ(t k )=f(θ(tk )), where d k is a constant coefficient;

[0023] Based on the closed-loop feedback control system, a feedback model is established as follows:

[0024] The difference between the angular displacement output q(t) of the robotic arm under the action of the pulse load and the angular displacement q(0) at the initial moment is equal to the convolution of the impulse response h(t) and the input signal. The input signal includes the feedback term g c q(t), the external input torque τ(t), and the output d of each pulse load after passing through the feedback channel k θ(t k )δ(t - t k ), where δ(t - t k ) represents an ideal pulse acting at time t k , that is:

[0025]

[0026] where, "*" represents the convolution operation.

[0027] Furthermore, in step (4), the process of analyzing the output response of the robotic arm under the action of the pulse load includes:

[0028] Before the pulse load is applied, that is, when 0 ≤ t < t1, the input-output time-domain model is obtained according to the mathematical model established in step (3):

[0029] q(t) - q(0) = h(t)*(g c q(t) + τ(t));

[0030] Performing the Laplace transform on the above model, we get:

[0031]

[0032] where Q(s), H(s), and T(s) are the Laplace transforms of q(t), h(t), and τ(t) respectively, and s is the Laplace transform operator;

[0033] According to the Laplace-transformed model, performing the inverse Laplace transform gives:

[0034]

[0035] where denotes the inverse Laplace transform, and each term in the formula corresponds to the output generated by the initial state of the robotic arm and the output generated by the external input torque.

[0036] Furthermore, in step (4), the output jump condition at the pulse load triggering time t1 is calculated according to the expression obtained by the inverse Laplace transform:

[0037]

[0038] in, Indicates the output before the pulse load is triggered. It represents the output after the pulse load is triggered, and d1θ(t1) represents the mutation caused by the first pulse load.

[0039] Furthermore, in step (4), analyzing the response process after the pulse load triggering moment includes:

[0040] Introduce the unit step function μ(t) and construct the window function to separate the input effects before and after the pulse load trigger:

[0041] q(t)-q(0)=h(t)*((g c q(t)+τ(t))μ(t)-μ(t-t1))+h(t)*((g c q(t)+τ(t))μ(t-t1))+d1θ(t1)δ(t—t1)

[0042] According to the convolution characteristics, the output is divided into steady-state output and transient output:

[0043] The output generated by the input before the pulse load trigger time t1 is divided into steady-state output and transient output;

[0044] The output generated by the input after the pulse load trigger time t1 is recorded as the transient output.

[0045] Furthermore, in step (4), the process of recursively establishing the transient response analysis process after each pulse load action includes:

[0046] By using the step-by-step recursive method, the transient response after the k-th pulse load is analyzed:

[0047]

[0048] Among them, q(t k ) represents the output before the kth pulse load is triggered, It represents the output after the kth pulse load is triggered, d k θ(t k ) represents the mutation caused by the kth pulse load;

[0049] H(s) represents the Laplace transform of the manipulator impulse response h(t);

[0050] gc represents feedback gain;

[0051] τ(t) represents the external input torque;

[0052] s represents the Laplace transform operator;

[0053] It represents the transient output generated by the input signal before the i-th pulse load trigger and after the k-1-th pulse output;

[0054] Based on the above recursive relationship, the complete output response of the robotic arm under multiple pulse loads is obtained:

[0055]

[0056] Among them, each convolution term represents the impact of the initial state of the robot arm, the external input torque, and the historical pulse load transient on the current output.

[0057] The present invention has the following beneficial effects:

[0058] (1) The present invention can accurately simulate and predict the dynamic response behavior of the manipulator under the action of pulse loads, especially in terms of transient and steady-state response characteristics. It can take into account the impact of historical pulse load transients on the current manipulator response. For situations involving multiple pulse loads, it can predict the superposition effect of transient responses and the possible continuous oscillation phenomenon. By recursively establishing the transient response analysis process after each pulse load action, a more accurate method is provided for the dynamic characteristics analysis of the manipulator, which helps to optimize the control strategy of the manipulator and improve its stability and accuracy in the pulse load environment.

[0059] (2) Based on the response characteristics, this paper constructs a complete model of the manipulator's pulse load response process, providing a basis for manipulator pulse load analysis and design, and helping to improve the manipulator's control accuracy and stability. A window function is used to separate the input effect and divide it into steady-state output and transient output. A transient output calculation method is established, and the relationship between the system's steady-state output, input, and historical pulse transients is also established.

[0060] (3) Based on the transient process analysis, a method for calculating the pulse load oscillation of the robotic arm was established. This method is based on all historical transients and can accurately describe the mechanism of pulse load generation of the robotic arm, providing an important basis for the pulse load oscillation analysis, design and working condition restrictions of the robotic arm. BRIEF DESCRIPTION OF THE DRAWINGS

[0061] Figure 1 This is the structure diagram of the robotic arm pulse load.

[0062] Figure 2 This is the flowchart of the invention patent for pulse load oscillation analysis of robotic arms.

[0063] Figure 3 This is the flow chart of the impulse response modeling of the robotic arm.

[0064] Figure 4 This is the flow chart of the pulse load oscillation analysis of the robotic arm.

[0065] Figure 5 This is a simulation diagram of ΔT=0.05 in Example 1.

[0066] Figure 6 This is a simulation diagram of ΔT=0.2 in Example 1.

[0067] Figure 7 This is a simulation diagram of ΔT=0.5 in Example 1. DETAILED DESCRIPTION

[0068] The present invention will be further described below with reference to the accompanying drawings.

[0069] like Figure 1 As shown, the present invention provides a method for analyzing pulse load oscillation of a robotic arm based on response characteristics, which includes the following steps.

[0070] Step 1: Based on the equivalent principle, establish the pulse load equivalent feedback control process;

[0071] According to the robot control process, the robot pulse load system can be as follows Figure 1 The structure shown in the figure is studied. The input control signal acts on the robot arm to obtain the robot arm output, and the output signal is fed back to the input end. The output end of the robot arm is triggered at irregular time t k Subject to pulse load, acting on the output.

[0072] Step 2: Establish the input and output impulse response model of the robot arm;

[0073] Step 2.1 Build the input-output convolution model.

[0074] Typically, a manipulator dynamics model is constructed based on its inertia matrix, Coriolis force matrix, centrifugal force matrix, gravitational moment matrix, and driving torque. However, these matrices all vary with the manipulator's angle, making modeling extremely difficult. Furthermore, the model assumes ideal input and output conditions, ignoring factors such as damping forces. When the input changes, the output changes immediately; when the input stops changing, the output stops changing immediately. This makes it impossible to study the oscillations of the manipulator's angular displacement under the action of pulsed loading.

[0075] However, any system has a response process, and the same is true for the robotic arm. Considering the response process of the robotic arm, if the impulse response of the robotic arm is set to h(t), the input and output of the robotic arm can be expressed as:

[0076] q(t)=h(t)*τ(t) (1)

[0077] The symbol * represents the convolution operation.

[0078] Step 2.2 Establish the time domain expression of the impulse response h(t):

[0079] For any manipulator, it is a stable system with bounded input and bounded output, and its impulse response satisfies the absolute integrability condition. Regardless of the model, the impulse response of the stable system can always be expressed as:

[0080] h(t)=ηt -β (2)

[0081] η and β are unknown parameters. Therefore, once the parameters η and β are identified, the input-output response model of the robot arm can be established.

[0082] Step 2.3: Use Legendre transformation to transform the formula (1).

[0083] Step 2.4 combines the particle swarm optimization algorithm and the gradient algorithm to identify the parameters η and β.

[0084] Step 3: Establish a mathematical model for the pulse load feedback control of the robotic arm.

[0085] Assume that the output of the robot arm is q(t) and the output of the feedback channel is: g c q(t),g c is the feedback gain. If the kth pulse load is θ(t k )(k=1,2,3,…), we can get the pulse load output through the feedback channel: f(θ(t k )), for simplicity, dθ(t k )=f(θ(t k )), where d is a constant, which can be expressed by dθ(t k )=f(θ(t k ))calculate.

[0086] Therefore, according to Figure 1 The structure shown in the figure establishes the following feedback model:

[0087]

[0088] δ(tt k ) indicates that at t k The ideal pulse at time q(0) represents the angular displacement of the robot arm at the initial time.

[0089] Step 4: Analysis of the robot arm pulse load output.

[0090] The analysis was performed before and after pulse load loading.

[0091] Step 4.1: Before the pulse load is applied, i.e., when \(0\leq t\lt t_1\), the input-output time-domain model is obtained from Equation (3):

[0092] \(q(t)-q(0)=h(t)*(g c q(t)+\tau(t))\ (4)

[0093] Performing Laplace transform gives:

[0094]

[0095] where \(Q(s)\), \(H(s)\), and \(T(s)\) are the Laplace transforms of \(q(t)\), \(h(t)\), and \(\tau(t)\) respectively, and \(s\) represents the Laplace operator.

[0096] According to Equation (5), performing the inverse Laplace transform gives:

[0097]

[0098] where denotes the inverse Laplace transform, and the symbol \(*\) represents the convolution operation.

[0099] Equation (6) represents the output within the time period \(0\leq t\lt t_1\) of the input torque within the time period \(0\leq t\lt t_1\).

[0100] In the equation reflects the output generated by the initial state of the robotic arm. In the equation represents the output generated by the external input torque.

[0101] Step 4.2: Analyze the pulse cargo triggering moment.

[0102] According to Equation (6), the load pulse characteristics can be calculated

[0103]

[0104] Step 4.3: Take the transient generated before the pulse trigger as an additional input and analyze the response process after the pulse cargo trigger moment \(t_1\).

[0105] Introduce the unit step function \(\mu(t)\) to construct a window function and separate the input before the pulse cargo trigger:

[0106]

[0107] where, \((t)*((g c q(t)+\tau(t))\mu(t)-\mu(t - t_1))\) represents the output generated by the input before the pulse cargo trigger moment \(t_1\), and \(h(t)*((g cq(t)+τ(t))μ(t-t1)) represents the output generated by the input after the pulse cargo trigger time t1.

[0108] The output generated by the input before time t1 is divided into steady-state output and transient output.

[0109] According to the convolution characteristics, the system has memory characteristics. The input signal is cut off, but the output signal is not cut off.

[0110] (t)*((g c q(t)+τ(t))μ(t)-μ(t-t1)) still has output after time t1, and this process continues. The output before t1 is called steady-state output, and the output after t1 is called transient output.

[0111]

[0112] Combining formula (9) and formula (10), we can get:

[0113]

[0114] remember Indicates the transient output generated after the first pulse output of the input signal before the first pulse trigger after the start time.

[0115] Simplifying, we get:

[0116]

[0117] The transient output generated at time t1 is used as an additional input, and together with the robot input, the input and output equations of the robot after t1 are established:

[0118]

[0119] Similar to formula (6), the output response of the robot arm after pulse loading can be obtained:

[0120]

[0121] in the formula express Indicates the impact of the transient state of the input signal generated during the first pulse trigger at the start time on the current robot output.

[0122] One step of recursion to establish the output after the second pulse load.

[0123] The second pulse load triggering time is recorded as: t2, similar to formula (7), we can get:

[0124]

[0125] and

[0126]

[0127] The last two terms in formula (14) will generate new transients when t>t2. Similar to formulas (10) and (11), we can get the value of each term when t>t2. Note:

[0128] Indicated by Transients generated when t>t2;

[0129] Indicated by Transients generated when t>t2;

[0130] The above two transient outputs will affect the output when t>t2, and formula (12) shows that Affected by previous pulse loading.

[0131] Step 4.4: Recursively establish the k-th pulse load transient analysis process.

[0132] By recursion step by step, we can get

[0133]

[0134] The subsequent convolution terms in formula (16) will generate new transient states, which are recorded as

[0135] Further we get:

[0136]

[0137] In formula (18), It represents the angular displacement output of the manipulator caused by the initial state immediately after the pulse loading is triggered. Represents the angular displacement output of the manipulator generated by the external torque,

[0138] Represents the output generated by the robot's previous historical pulse loading transients.

[0139] H(s) represents the Laplace transform of the manipulator impulse response h(t);

[0140] gc represents feedback gain;

[0141] τ(t) represents the external input torque;

[0142] s represents the Laplace transform operator;

[0143] Denote the transient output generated by the input signal before the $i$-th pulse load trigger after the $(k - 1)$-th pulse output.

[0144] Equation (16) shows that the steady state of the robotic arm is affected by the transient of the pulsed load, and Equation (18) also shows that the current transient of the robotic arm is affected by all historical pulse transients. The current transient is the result of the superposition of all pulse load transients of the robotic arm. If the previous transient does not decay below a certain level and is subjected to a new pulse load, one wave after another, the robotic arm will produce continuous oscillations.

[0145] Example: The parameters related to the pulsed load of the robotic arm are selected as follows:

[0146] $\beta = 0.5$, the switching signal trigger time $t$ k $= k\Delta T$, $\Delta T$ is the pulse load interval time, corresponding to the pulse load $d$ k $\theta(t_k$ ) $= 0.1$, the initial angular displacement of the robotic arm $\tau(t)$ is 2, the impulse response of the robotic arm: $\tau(t)=0.1t$.

[0147] According to the Laplace transform, $H(s)=\Gamma(1 - \beta)s$ β-1 , $\Gamma(\cdot)$ represents the gamma function.

[0148] That is, when $0\leq t\lt t_1$,

[0149]

[0150] Where:

[0151] <s

[0152] Where $E$ β-1，β-1 represents the Mittag-Leffler function with two parameters $\beta - 1$. Therefore, the expression of $q(t)$ when $0\leq t\lt t_1$ can be calculated.

[0153] Substituting the parameters can calculate At the first pulse trigger time

[0154] According to the expression of $q(t)$ when $0\leq t\lt t_1$, it can be calculated

[0155] Proceed step by step, and according to Equation (18), the expressions of $q(t)$ in each time period can be obtained. The specific steps are shown in Step 4 and will not be listed one by one. The simulation diagrams of $\Delta T = 0.05$, $0.2$, and $0.5$ are respectively as Figure 5 、 Figure 6 and Figure 7The simulation results show that under the action of pulse load, due to the response characteristics and the influence of historical information, the robot arm has a rebound process. If this process has not recovered and a new pulse load is received, it will produce continuous oscillation. This puts forward requirements for the pulse load interval of the robot arm. If the pulse interval is small, it will cause continuous oscillation. Figure 5 shown.

[0156] The above description is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements can be made without departing from the principles of the present invention. These improvements should also be regarded as the scope of protection of the present invention.

Claims

1. A method for analyzing pulse load oscillation of a robotic arm based on response characteristics, characterized by comprising the following steps: (1) Based on the equivalent principle, an equivalent feedback control process of the manipulator pulse load is established; (2) Based on the dynamic characteristics of the manipulator, an input-output convolution model is constructed, and the impulse response function form is determined by identifying the system parameters to construct the input-output impulse response model of the manipulator; (3) Based on the impulse response model in step (2), a mathematical model of pulse load feedback control of the manipulator is established; (4) Using the mathematical model in step (3), analyze the output response of the manipulator under the action of pulse load. The steps are as follows:

1. Before the pulse load is applied, obtain the output response expression of the robotic arm; 2. Calculate the output jump of the robotic arm at the moment when the pulse load is triggered; 3. Separate the input effects before and after the pulse load is triggered, and divide the output of the robotic arm into steady-state output and transient output; 4. Recursively establish the transient response analysis process after each pulse load acts, and obtain the complete output response sequence of the robotic arm.

2. The method for analyzing pulse load oscillation of a robotic arm based on response characteristics according to claim 1, wherein: In step (1), the equivalent feedback control process of the robotic arm pulse load is as follows: The input control signal acts on the input end of the robot arm, and the output signal of the robot arm is fed back to the controller used to control the movement of the robot arm, forming a closed-loop feedback control system. At the same time, the output end of the robot arm is triggered at irregular time t k Subject to pulse load.

3. The method for analyzing pulse load oscillation of a robotic arm based on response characteristics according to claim 2, wherein: In step (2), the process of constructing the impulse response model of the robotic arm input and output is as follows: The input-output relationship of the robotic arm is represented by a convolution model, that is, the angular displacement output q(t) of the robotic arm is the convolution of the input torque τ(t) and the impulse response h(t) of the robotic arm. The mathematical expression is q(t) = h(t) * τ(t); The impulse response h(t) satisfies the absolute integrability condition and can be expressed as h(t)=ηt -β , where η and β are parameters to be identified, and the two parameters are identified by particle swarm optimization algorithm combined with gradient algorithm, so as to establish the impulse response model.

4. The method for analyzing pulse load oscillation of a robotic arm based on response characteristics according to claim 3, wherein: In step (3), the process of establishing the mathematical model is as follows: Set the angular displacement output of the robot arm to q(t) and the feedback channel output to g c q(t), where g c is the feedback gain; If the kth pulse load is expressed as θ(t k ), the output of the pulse load through the feedback channel is: f(θ(t k )), simplified as dθ(t k )=f(θ(t k )), where d k is a constant coefficient; Based on the closed-loop feedback control system, establish the following feedback model: The difference between the angular displacement output q(t) of the manipulator under the action of the pulse load and the initial angular displacement q(0) is equal to the convolution of the impulse response h(t) and the input signal, which includes the feedback term g c q(t), external input torque τ(t), and the output d after each pulse load passes through the feedback channel k θ(t k )δ(tt k ), where δ(tt k ) represents the action at time t k The ideal pulse is: where, "*" represents the convolution operation.

5. The method for analyzing pulse load oscillation of a robotic arm based on response characteristics according to claim 4, wherein: In step (4), the process of analyzing the output response of the robotic arm under the action of the pulse load includes: Before the pulse load is applied, that is, when 0 ≤ t < t1, obtain the input-output time-domain model according to the mathematical model established in step (3): q(t)-q(0)=h(t)*(g c q(t)+τ(t)); Perform Laplace transform on the above model to obtain: where, Q(s), H(s), and T(s) are the Laplace transforms of q(t), h(t), and τ(t) respectively, and s is the Laplace transform operator; According to the model after Laplace transform, perform inverse Laplace transform to obtain: Among them, represents the inverse Laplace transform, and each term in the formula corresponds to the output generated by the initial state of the robot arm and the output generated by the external input torque, respectively.

6. The method for analyzing pulse load oscillation of a robotic arm based on response characteristics according to claim 5, wherein: In step (4), According to the expression obtained by inverse Laplace transform, calculate the output jump at the moment t1 when the pulse load is triggered: in, Indicates the output before the pulse load is triggered. It represents the output after the pulse load is triggered, and d1θ(t1) represents the mutation caused by the first pulse load.

7. The method for analyzing pulse load oscillation of a robotic arm based on response characteristics according to claim 6, wherein: In step (4), the process of analyzing the response after the pulse load is triggered includes: Introduce the unit step function μ(t), construct a window function, and separate the input effects before and after the pulse load is triggered: q(t)-q(0)=h(t)*((g c q(t)+τ(t))μ(t)-μ(t-t1))+h(t)*((g c q(t)+τ(t))μ(t-t1))+d1θ(t1)δ(t-t1) According to the convolution property, divide the output into steady-state output and transient output: Divide the output generated by the input before the moment t1 when the pulse load is triggered into steady-state output and transient output; Record the output generated by the input after the moment t1 when the pulse load is triggered as the transient output.

8. The method for analyzing pulse load oscillation of a robotic arm based on response characteristics according to claim 7, wherein: In step (4), the process of recursively establishing the transient response analysis process after each pulse load acts includes: By the method of step-by-step recursion, analyze the transient response after the kth pulse load acts: Among them, q(t k ) represents the output before the kth pulse load is triggered, It represents the output after the kth pulse load is triggered, d k θ(t k ) represents the mutation caused by the kth pulse load; H(s) represents the Laplace transform of the impulse response h(t) of the robotic arm; gc represents the feedback gain; τ(t) represents the external input torque; s represents the Laplace transform operator; It represents the transient output generated by the input signal before the i-th pulse load trigger and after the k-1-th pulse output; Based on the above recurrence relationship, obtain the complete output response of the robotic arm under multiple pulse loads: where, each convolution term represents the influence of the initial state of the robotic arm, the external input torque, and the transient of the historical pulse load on the current output.

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