A multi-body system motion state acquisition method based on an implicit double-loop calculation framework
By employing an implicit double-loop computational framework with internal and external iteration methods, the displacement, velocity, and acceleration of a multibody system are solved simultaneously. This addresses the issues of constraint failure and computational divergence in traditional methods, enabling efficient and stable acquisition of the motion state of the multibody system.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- NORTHWESTERN POLYTECHNICAL UNIV
- Filing Date
- 2026-04-20
- Publication Date
- 2026-07-31
AI Technical Summary
Traditional implicit methods cannot simultaneously satisfy constraint equations at the displacement, velocity, and acceleration levels when solving the dynamic equations of multibody systems, resulting in insufficient accuracy and stability of numerical solutions. In particular, constraint failure and computational divergence are prone to occur in long-term simulations.
An implicit double-loop computational framework is adopted, which combines the absolute node coordinate method, Gaussian decomposition method and Newton iteration method to solve the displacement, velocity and acceleration of the multibody system simultaneously, ensuring the accurate satisfaction of the constraint equations.
It achieves precise satisfaction of constraints at all levels, improves computational efficiency and stability, significantly reduces the complexity of iterative computation, adapts to strongly coupled and strongly nonlinear multi-rigid-body and multi-flexible-body systems, avoids constraint failure and numerical oscillation, and improves the accuracy and robustness of the solution.
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Abstract
Description
Technical Field
[0001] The embodiments of this application relate to the field of mechanical dynamics technology, and in particular to a method for obtaining the motion state of a multibody system based on an implicit double-loop computational framework. Background Technology
[0002] A multibody system refers to a system composed of multiple objects connected by certain connection methods or constraints. A characteristic of this type of system is that there is usually a large range of mutual motion between the components. Multibody systems can be broadly divided into two categories: multi-rigid-body systems and multi-flexible-body systems. In a multi-rigid-body system, all components are considered rigid bodies, and their structural deformation is ignored. In a multi-flexible-body system, the structural deformation and stress of the components must be considered.
[0003] With the continuous development of engineering technology, mechanical system components are becoming increasingly lightweight and large-scale. The structural deformation of these components has a significant impact on the mechanical system, especially in high-speed mechanical systems such as aerospace and vehicles. Multibody systems, which require simultaneous consideration of structural flexibility and large-scale component motion, typically exhibit strong coupling and strong nonlinear characteristics. Furthermore, due to the diverse connections and constraints between system components, the system dynamics equations are usually expressed as differential-algebraic equations (DAEs), making analytical solutions nearly impossible. Therefore, numerical methods are needed to solve the system dynamics equations to obtain more accurate numerical solutions. Currently, implicit numerical methods for solving DAEs equations for multibody systems mainly include the Newmark method, the Hilber-Hughes-Taylor (HHT) integral method, and generalized methods. These traditional implicit methods, while applied to the dynamic analysis of some specific systems, require calculating the derivative of the generalized forces to compute the Jacobian matrix. This operation is a major cause of numerical errors and consumes significant computational time. Furthermore, traditional implicit methods cannot simultaneously satisfy constraint equations at the displacement, velocity, and acceleration levels. When solving long-term multibody dynamics problems, this can affect the accuracy of numerical solutions and even lead to solution failure. Summary of the Invention
[0004] In view of this, embodiments of this application propose a method for obtaining the motion state of a multibody system based on an implicit double-loop computational framework, aiming to overcome the limitation of traditional implicit methods that cannot simultaneously satisfy constraint equations at the displacement, velocity, and acceleration levels, and improve the accuracy and stability of numerical solutions to DAEs equations of multibody systems.
[0005] To achieve the above objectives, embodiments of this application propose a method for obtaining the motion state of a multibody system based on an implicit dual-loop computational framework. The implicit dual-loop computational framework includes an inner loop and an outer loop. The method includes: S1, establishing the dynamic equations of the target multibody system using the absolute node coordinate method, representing the target multibody system as a set of differential-algebraic equations; S2, setting various parameters required for numerical integration calculation based on the characteristics and computational requirements of the target multibody system, including simulation deadline, outer loop convergence condition parameters, and inner loop convergence condition parameters; S3, identifying the independent and non-independent coordinates of the target multibody system using Gaussian decomposition, entering the outer loop iteration process, and calculating the generalized position vector and generalized velocity vector of the independent coordinates at the next moment; S4, calculating the independent coordinates at the next moment... The target's generalized position vector and generalized velocity vector are substituted into the position-level constraint equations, entering the inner loop iteration process. The Newton iteration method is used to solve for the generalized position vector of the non-independent coordinates at the next time step. Then, the generalized position vectors of all coordinates at the next time step and the generalized velocity vectors of the independent coordinates are substituted into the velocity-level constraint equations to solve for the generalized velocity vector of the non-independent coordinates at the next time step. In S5, the dynamic equations of the target multibody system are reconstructed based on the generalized position vectors and generalized velocity vectors of all coordinates at the next time step. The acceleration-level constraint equations are solved simultaneously to solve for the generalized acceleration vector and Lagrange multipliers at the next time step. If the outer loop convergence condition is met, the calculation proceeds to the next time step; otherwise, it returns to S3. In S6, S3 to S5 are repeated continuously until the calculation time reaches the simulation deadline, completing the solution calculation and obtaining the cooperative motion state of the target multibody system.
[0006] To achieve the above objectives, embodiments of this application also propose an electronic device, including: a processor and a memory, wherein the memory stores instructions executable by the processor, and the processor is configured to execute the instructions such that the electronic device can implement the multibody system motion state acquisition method based on the implicit double-loop computing framework described above.
[0007] To achieve the above objectives, embodiments of this application also propose a computer-readable storage medium storing a computer program that, when executed by a processor, enables a method for obtaining the motion state of a multibody system based on an implicit double-loop computation framework as described above.
[0008] Optionally, the dynamic equations of the target multibody system are established using the absolute nodal coordinate method, representing the target multibody system as a set of differential-algebraic equations, which are achieved through the following formulas: (1) in, , and These are the generalized position vector, generalized velocity vector, and generalized acceleration vector of the target multibody system, respectively. , These are the mass matrix and damping matrix of the target multibody system, respectively. The external forces acting on the target multibody system These are the constraint equations for the target multibody system, used to describe the constraint relationships, connection relationships, control forces, and control torques acting on each component of the target multibody system. Let be the Jacobian matrix of the constraint equation relative to the generalized position vector. Equation (1) is a set of differential-algebraic equations with an index of 3.
[0009] Optionally, the simulation deadline is denoted as... The outer loop convergence condition parameters include the outer loop convergence error threshold. Maximum number of iterations in the outer loop The inner loop convergence condition parameters include the inner loop convergence error threshold. and the maximum number of iterations in the inner loop The parameters required for numerical integration calculations also include the integration step size. and controllable numerical dissipation parameters .
[0010] Optionally, Gaussian decomposition is used to identify the independent and non-independent coordinates of the target multibody system. Then, an outer loop iterative process is initiated to calculate the generalized position vector and generalized velocity vector of the independent coordinates at the next time step, including: Generalized position vector of target multibody system Represented as: (2) in, A generalized position vector with independent coordinates. It is a generalized position vector with non-independent coordinates; constraint equations Taking the second derivative with respect to time, and solving the simultaneous equations, we obtain a differential-algebraic equation with an index of 1: (3) in, Let Lagrange multiplier vectors be used. These are the second-order partial derivatives of the constraint equations; By directly solving equation (3), the generalized acceleration vector of the target multibody system at the current moment can be obtained. ; Entering the outer loop iteration process, using the following formula, a generalized... The algorithm performs integration to obtain the generalized position vector and generalized velocity vector of the independent coordinates at the next time step: (4-1) (4-2) (4-3) (4-4) (4-5) (4-6) Among them, the bottom right corner mark Indicates the current time, indicated by the subscript in the lower right corner. Indicates the next moment, The generalized position vector is the independent coordinate at the current moment. The generalized position vector representing the independent coordinates at the next moment. and They are the generalized meanings of the current moment. Algorithm parameter vector and the generalized next time step Algorithm parameter vector, and In a broad sense The hyperparameters of the algorithm are used to control the discrete moments of acceleration and velocity, respectively.
[0011] Optionally, the generalized position vector and generalized velocity vector of the independent coordinates at the next time step are substituted into the position-level constraint equations to enter the inner loop iteration process. The Newton iteration method is used to solve for the generalized position vector of the non-independent coordinates at the next time step, including: Substituting the generalized position vector and generalized velocity vector of the independent coordinates at the next moment into the position-level constraint equation using the following formula, we enter the inner loop iteration process: (5-1) in, It is the identity matrix. The number of iterations is Newton's iterations. The Newton iterative method is used to solve for the generalized position vector of the non-independent coordinates at the next time step, and the inner loop convergence error is calculated. ,like Less than Or the current inner loop iteration count reaches If the condition for convergence of the inner loop iteration is met, the inner loop iteration process is exited; otherwise, the inner loop iteration continues. The calculation process is as follows: (5-2) in, , The first sequence The generalized position vector of the non-independent coordinates obtained during the next inner loop iteration.
[0012] Optionally, the generalized position vectors of all coordinates at the next moment and the generalized velocity vectors of the independent coordinates are substituted into the velocity level constraint equations to solve for the generalized velocity vectors of the non-independent coordinates at the next moment, which is achieved through the following formula: (6) in, Let the generalized velocity vector of the target multibody system at the next moment be . Let be the generalized velocity vector of the independent coordinates at the current moment. This is the partial derivative of the constraint equation with respect to time.
[0013] Optionally, based on the generalized position vector and generalized velocity vector of all coordinates at the next moment, the dynamic equations of the target multibody system are reconstructed, the acceleration level constraint equations are solved simultaneously, and the generalized acceleration vector and Lagrange multipliers at the next moment are obtained, including: According to the broad sense The algorithm's difference scheme reconstructs the dynamic equations of the target multibody system based on the generalized position and velocity vectors of all coordinates at the next time step. It then solves for the acceleration level constraint equations, obtaining the generalized acceleration vector and Lagrange multipliers at the next time step. (7) in, , These are the partial derivative matrices of the constraint equations with respect to the generalized position vector at the current and next time steps, respectively. , These are the generalized external force vectors at the current moment and the next moment, respectively. , Let these be the generalized acceleration vectors of the previous moment and the next moment, respectively. , These can be considered as the Lagrange multipliers of the previous moment and the next moment, respectively.
[0014] Optionally, after obtaining the generalized acceleration vector and Lagrange multipliers at the next time step, the outer loop convergence error is calculated based on the generalized acceleration vector and Lagrange multipliers at the next time step. ,like Less than Or the current outer loop iteration count reaches If the outer loop convergence condition is met, the outer loop iteration process is exited, and the calculation of the next time step begins; otherwise, the process returns to S3 to continue the outer loop iteration.
[0015] The proposed method for obtaining the motion state of a multibody system based on an implicit double-loop computational framework has the following advantages compared with the traditional DAEs equation solving method.
[0016] First, it achieves precise satisfaction of constraints at all levels. This application uses an internal and external dual-loop iterative calculation framework to solve simultaneously at the three dynamic levels of displacement, velocity, and acceleration. This overcomes the shortcomings of traditional DAE equation solving methods, such as constraint drift and constraint error accumulation, ensuring the geometric and dynamic consistency of the numerical solutions of the multibody system. It also avoids the computational divergence problem caused by constraint failure during long-term simulations from the root cause.
[0017] Second, it significantly improves computational efficiency and solution speed. This application eliminates the need to solve for the Jacobian matrix of the generalized forces of the system, avoiding numerical errors and high computational costs caused by the differentiation of the Jacobian matrix. This greatly reduces the complexity of iterative calculations and the computation time per time step. Combined with the Gaussian decomposition method, it quickly identifies independent and non-independent coordinates, simplifies the solution dimension of the equations, and significantly improves the overall computational efficiency of large-scale, long-term multibody dynamics simulations.
[0018] Third, it improves the stability and robustness of the solution. The outer loop of this application adopts an implicit generalized... The algorithm ensures the controllability of integration accuracy and numerical dissipation. The inner loop uses the Newton iteration method to guarantee the convergence of solutions for non-independent coordinates. The dual iteration mechanism can be adapted to strongly coupled and highly nonlinear multi-rigid-body and multi-flexible-body systems, effectively suppressing the numerical oscillation and solution failure problems of traditional implicit methods, and improving the convergence and robustness of computation.
[0019] Fourth, the adaptability of solving high-index DAEs equations is optimized. This application can efficiently transform high-index DAEs equations of multibody systems into low-index equations for solution, reducing the difficulty of solving high-index differential-algebraic equations. It is perfectly compatible with large-scale, lightweight, flexible multibody systems modeled by the absolute node coordinate method, breaking through the limitations of traditional methods in solving flexible components and high-speed operating systems. Attached Figure Description
[0020] To more clearly illustrate the technical solutions in the embodiments or related technologies of this application, the accompanying drawings used in the description of the embodiments or related technologies of this application will be briefly introduced below. Obviously, the following drawings are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. The drawings described herein are only used to explain this application and are not intended to limit this application.
[0021] Figure 1This is a flowchart of a method for obtaining the motion state of a multibody system based on an implicit double-loop computation framework, provided in one embodiment of this application; Figure 2 This is a detailed schematic diagram of a method for obtaining the motion state of a multibody system based on an implicit double-loop computation framework, provided in one embodiment of this application; Figure 3 This is a schematic diagram of the structure of a multibody system provided in one embodiment of this application; Figure 4 This method, display algorithm, and generalized method are provided in one embodiment of this application. A schematic diagram of the displacement curve of the pendulum endpoint in the X direction calculated by the algorithm; Figure 5 This method is provided in one embodiment of this application and is generalized. A schematic diagram of the change curve of the position of the pivot joint hinge point in the X direction calculated by the algorithm; Figure 6 This method is provided in one embodiment of this application and is generalized. A schematic diagram of the velocity variation curve in the X direction of the pivot joint hinge point calculated by the algorithm; Figure 7 This is a schematic diagram of the structure of an electronic device provided in another embodiment of this application. Detailed Implementation
[0022] To make the objectives, technical solutions, and advantages of the embodiments of this application clearer, the various embodiments of this application will be described in detail below with reference to the accompanying drawings. Those skilled in the art will understand that many technical details have been provided in the embodiments of this application to facilitate better understanding. However, the technical solutions claimed in this application can be implemented even without these technical details and various variations and modifications based on the following embodiments. The division of the following embodiments is for ease of description and should not constitute any limitation on the specific implementation of this application. The following embodiments can be combined with and referenced by each other without contradiction.
[0023] One embodiment of this application proposes a method for obtaining the motion state of a multibody system based on an implicit dual-loop computation framework. The implicit dual-loop computation framework includes two loops: an inner loop computation framework and an outer loop computation framework. The implementation details of the method for obtaining the motion state of a multibody system based on an implicit dual-loop computation framework proposed in this embodiment are described below. The following implementation details are provided for ease of understanding and are not necessary for implementing this solution.
[0024] The specific process of the multibody system motion state acquisition method based on an implicit double-loop computation framework proposed in this embodiment can be described as follows: Figure 1As shown, its implementation details are as follows: Figure 2 As shown, the method includes: S1. For the target multibody system to be analyzed, the dynamic equations of the target multibody system are established by using the absolute nodal coordinate method, and the target multibody system is represented as a set of differential-algebraic equations.
[0025] In practical implementation, to obtain the motion state of the target multibody system to be analyzed, it is first necessary to use the Absolute Nodal Coordinate Formulation (ANCF) method to establish the dynamic equations of the target multibody system, thereby representing the target multibody system as a set of differential-algebraic equations.
[0026] In one example, the dynamic equations of the target multibody system are expressed as: (1) in, , and These are the generalized position vector, generalized velocity vector, and generalized acceleration vector of the target multibody system, respectively. , These are the mass matrix and damping matrix of the target multibody system, respectively. The external forces acting on the target multibody system These are the constraint equations for the target multibody system, used to describe the constraint relationships, connection relationships, control forces, and control torques acting on each component of the target multibody system. Let Jacobi be the constraint equation relative to the generalized position vector. Equation (1) is a set of differential-algebraic equations with an index of 3.
[0027] S2 sets the parameters required for numerical integration calculation based on the characteristics and computational requirements of the target multibody system, including simulation deadline, outer loop convergence condition parameters, and inner loop convergence condition parameters.
[0028] In practical implementation, in addition to establishing the dynamic equations of the target multibody system, we also need to set various parameters required for numerical integration calculation based on the characteristics and computational requirements of the target multibody system, including simulation deadline, outer loop convergence condition parameters, and inner loop convergence condition parameters.
[0029] In one example, the simulation deadline is denoted as... The outer loop convergence condition parameters include the outer loop convergence error threshold. Maximum number of iterations in the outer loop The inner loop convergence condition parameters include the inner loop convergence error threshold. and the maximum number of iterations in the inner loop The parameters required for numerical integration calculations also include the integration step size. and controllable numerical dissipation parameters .
[0030] S3 uses Gaussian decomposition to identify the independent and non-independent coordinates of the target multibody system, enters the outer loop iteration process, and calculates the generalized position vector and generalized velocity vector of the independent coordinates at the next moment.
[0031] In the specific implementation, the first step in solving the problem is to use Gaussian decomposition to identify the independent and non-independent coordinates of the target multibody system. Then, the outer loop iterative process is entered to calculate the generalized position vector and generalized velocity vector of the independent coordinates at the next moment.
[0032] In one example, let the generalized position vector of the target multibody system be... Represented as: (2) in, A generalized position vector with independent coordinates. It is a generalized position vector with non-independent coordinates.
[0033] constraint equations Taking the second derivative with respect to time, and solving the simultaneous equations, we obtain a differential-algebraic equation with an index of 1: (3) in, Let Lagrange multiplier vectors be used. This is the second-order partial derivative term of the constraint equation.
[0034] By directly solving equation (3), the generalized acceleration vector of the target multibody system at the current moment can be obtained. Then, the outer loop iteration process begins, using the following formula, with generalized... The algorithm performs integration to obtain the generalized position vector and generalized velocity vector of the independent coordinates at the next time step: (4-1) (4-2) (4-3) (4-4) (4-5) (4-6) Among them, the bottom right corner mark Indicates the current time, indicated by the subscript in the lower right corner. Indicates the next moment, The generalized position vector is the independent coordinate at the current moment. The generalized position vector representing the independent coordinates at the next moment. and They are the generalized meanings of the current moment. Algorithm parameter vector and the generalized next time step Algorithm parameter vector, and In a broad sense The hyperparameters of the algorithm are used to control the discrete moments of acceleration and velocity, respectively.
[0035] S4. Substitute the generalized position vector and generalized velocity vector of the independent coordinates at the next moment into the position-level constraint equations and enter the inner loop iteration process. Use the Newton iteration method to solve for the generalized position vector of the non-independent coordinates at the next moment. Then, substitute the generalized position vector of all coordinates and the generalized velocity vector of the independent coordinates at the next moment into the velocity-level constraint equations to solve for the generalized velocity vector of the non-independent coordinates at the next moment.
[0036] In the specific implementation, after solving for the generalized position vector and generalized velocity vector of the independent coordinates at the next moment, the generalized position vector and generalized velocity vector of the independent coordinates at the next moment can be substituted into the position-level constraint equations to enter the inner loop iteration process. The Newton iteration method is used to solve for the generalized position vector of the non-independent coordinates at the next moment. Then, the generalized position vectors of all coordinates and the generalized velocity vector of the independent coordinates at the next moment are substituted into the velocity-level constraint equations to solve for the generalized velocity vector of the non-independent coordinates at the next moment.
[0037] In one example, the generalized position vector and generalized velocity vector of the independent coordinates at the next moment are substituted into the position-level constraint equations using the following formula, thus entering the inner loop iteration process: (5-1) in, It is the identity matrix. This is the number of Newton iterations.
[0038] Next, the Newton iteration method is used to solve for the generalized position vector of the non-independent coordinates at the next time step, and the inner loop convergence error is calculated. ,like Less than Or the current inner loop iteration count reaches If the condition is met, the inner loop iteration convergence condition is satisfied, and the inner loop iteration process is exited; otherwise, the inner loop iteration continues.
[0039] In one example The calculation process is as follows: (5-2) in, , The first sequence The generalized position vector of the non-independent coordinates obtained during the next inner loop iteration.
[0040] By substituting the generalized position vectors of all coordinates and the generalized velocity vectors of independent coordinates into the velocity level constraint equations at the next time step using the following formula, the generalized velocity vectors of the non-independent coordinates at the next time step can be solved: (6) in, Let the generalized velocity vector of the target multibody system at the next moment be . Let be the generalized velocity vector of the independent coordinates at the current moment. This is the partial derivative of the constraint equation with respect to time.
[0041] S5: Reconstruct the dynamic equations of the target multibody system based on the generalized position vector and generalized velocity vector of all coordinates at the next time step, solve the acceleration level constraint equations simultaneously, and solve for the generalized acceleration vector and Lagrange multipliers at the next time step. If the outer loop convergence condition is met, proceed to the calculation of the next time step; otherwise, return to S3.
[0042] In the specific implementation, after solving for the generalized position vector and generalized velocity vector of all coordinates at the next time step, the dynamic equations of the target multibody system can be reconstructed based on the generalized position vector and generalized velocity vector of all coordinates at the next time step. The acceleration level constraint equations are solved simultaneously, and the generalized acceleration vector and Lagrange multiplier at the next time step are solved. If the outer loop convergence condition is met, the calculation of the next time step is entered; otherwise, the process returns to S3.
[0043] In one example, according to the generalized The algorithm's difference scheme reconstructs the dynamic equations of the target multibody system based on the generalized position and velocity vectors of all coordinates at the next time step. It then solves for the acceleration level constraint equations, obtaining the generalized acceleration vector and Lagrange multipliers at the next time step. (7) in, , These are the partial derivative matrices of the constraint equations with respect to the generalized position vector at the current and next time steps, respectively. , These are the generalized external force vectors at the current moment and the next moment, respectively. , Let these be the generalized acceleration vectors of the previous moment and the next moment, respectively. , These can be considered as the Lagrange multipliers of the previous moment and the next moment, respectively.
[0044] After obtaining the generalized acceleration vector and Lagrange multipliers at the next time step, the outer loop convergence error is calculated based on the generalized acceleration vector and Lagrange multipliers at the next time step. ,like Less than Or the current outer loop iteration count reaches If the outer loop convergence condition is met, the outer loop iteration process is exited, and the calculation of the next time step begins; otherwise, the process returns to S3 to continue the outer loop iteration.
[0045] S6, repeat S3 to S5 continuously until the calculation time reaches the simulation deadline, complete the solution calculation, and obtain the target multi-body cooperative motion state.
[0046] In the specific implementation, under the implicit double-loop calculation framework, S3 to S5 are repeated continuously until the calculation time reaches the simulation deadline, and the solution calculation is completed to obtain the target multi-body cooperative motion state.
[0047] In summary, the multibody system motion state acquisition method based on the implicit double-loop computation framework proposed in this embodiment has the following advantages compared with the traditional DAE equation solving method.
[0048] First, it achieves precise satisfaction of constraints at all levels. This embodiment uses a dual-loop iterative calculation framework to solve the problem simultaneously at the three dynamic levels of displacement, velocity, and acceleration. This overcomes the shortcomings of traditional DAE equation solving methods, such as constraint drift and constraint error accumulation, ensuring the geometric and dynamic consistency of the numerical solution of the multibody system. It also avoids the computational divergence problem caused by constraint failure during long-term simulations from the root cause.
[0049] Second, it significantly improves computational efficiency and solution speed. This embodiment eliminates the need to solve for the Jacobian matrix of the generalized forces of the system, avoiding numerical errors and high computational costs caused by the differentiation of the Jacobian matrix. This greatly reduces the complexity of iterative calculations and the computation time per time step. Combined with Gaussian decomposition, it quickly identifies independent and non-independent coordinates, simplifies the solution dimension of equations, and significantly improves the overall computational efficiency of large-scale, long-term multibody dynamics simulations.
[0050] Third, it improves the stability and robustness of the solution. The outer loop in this embodiment employs an implicit generalized... The algorithm ensures the controllability of integration accuracy and numerical dissipation. The inner loop uses the Newton iteration method to guarantee the convergence of solutions for non-independent coordinates. The dual iteration mechanism can be adapted to strongly coupled and highly nonlinear multi-rigid-body and multi-flexible-body systems, effectively suppressing the numerical oscillation and solution failure problems of traditional implicit methods, and improving the convergence and robustness of computation.
[0051] Fourth, the adaptability of solving high-index DAEs equations is optimized. This embodiment efficiently transforms the high-index DAEs equations of multibody systems into low-index equations, reducing the difficulty of solving high-index differential-algebraic equations. It is perfectly compatible with large-scale, lightweight, flexible multibody systems modeled by the absolute node coordinate method, breaking through the limitations of traditional methods in solving flexible components and high-speed operating systems.
[0052] The steps described above are merely for clarity in describing the technical solution. In actual implementation, they can be combined into one step, or certain steps can be broken down into multiple steps, as long as they involve the same logical relationship, they are all within the scope of protection of this application. Any insignificant modifications or designs added to the algorithm or process, as long as they do not change the core of the algorithm or process, are also within the scope of protection of this application.
[0053] In one embodiment, to verify the proposed method for obtaining the motion state of a multibody system based on an implicit dual-loop computational framework (hereinafter referred to as TLISMNI-), this application is used. To assess the effectiveness of (or TLISMNI-alpha), we conducted relevant simulation experiments. The multibody system selected for this simulation experiment consists of a revolute joint and a flexible pendulum, which reciprocates under the influence of gravity. Its specific structure is as follows: Figure 3 As shown in Table 1, the flexible pendulum is modeled using ANCF fully parametric beam elements. The physical and material parameters of the pendulum are shown in Table 1.
[0054] Table 1: Physical and material parameters of a simple pendulum
[0055] The solution is calculated according to the aforementioned calculation steps. This application is compared with traditional explicit algorithms and implicit generalized algorithms. The results of the algorithm (implicit generalized alpha algorithm) are compared. Figure 4 The curves showing the variation of the displacement of the pendulum endpoint in the X direction calculated by three methods are shown below. Figure 4 As can be seen from this, TLISMNI- The calculation results are consistent with those of the traditional algorithm.
[0056] To verify the constraint-preserving solution capability of this application for the constraint equations at the displacement, velocity, and acceleration levels, the curves of the position and velocity variation of the pivot joint hinge point obtained by this application are compared with those of the implicit generalized equations. The calculation results of the algorithms are compared. Figure 5 and Figure 6 The curves showing the X-direction position and velocity changes of the pivot joint hinge point are shown respectively. Both this application and the generalized alpha algorithm can satisfy the constraint equations at the acceleration level. Figure 5 , Figure 6 As can be seen, this method can simultaneously satisfy the constraint equations at the displacement, velocity, and acceleration levels, and can accurately and efficiently solve the dynamic equations of DAEs for multibody systems.
[0057] Another embodiment of this application provides an electronic device, such as Figure 7 As shown, it includes: a processor M1 and a memory M2. The memory M2 stores instructions that the processor M1 can execute. When the processor M1 is configured to execute the instructions, the electronic device can realize a method for obtaining the motion state of a multibody system based on an implicit double-loop computing framework as described in the above method embodiment.
[0058] The memory and processor are connected via a bus, which includes any number of interconnecting buses and bridges. The bus can connect various circuits of one or more processors and memories, as well as other circuits such as peripherals, voltage regulators, and power management circuits—all well-known in the art and therefore not described further herein. The bus interface provides an interface between the bus and the transceiver. The transceiver can be a single component or multiple components, such as multiple receivers and transmitters, providing a unit for communicating with various other devices over a transmission medium. Data processed by the processor is transmitted over the wireless medium via an antenna, which also receives and transmits data to the processor.
[0059] The processor manages the bus and handles general processing, providing various functions, including but not limited to timing, peripheral interfaces, voltage regulation, power management, and other control functions. Memory, on the other hand, is used to store data used by the processor during operation.
[0060] Another embodiment of this application proposes a computer-readable storage medium storing a computer program that, when executed by a processor, can implement a method for obtaining the motion state of a multibody system based on an implicit double-loop computation framework as described in the above method embodiments.
[0061] That is, those skilled in the art will understand that all or part of the steps in the above method embodiments can be implemented by a program instructing related hardware. The program is stored in a storage medium and includes several instructions to cause a device (such as a microcontroller, chip, etc.) or processor to execute all or part of the steps of the method described in the method embodiments of this application. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory, random access memory, magnetic disks, or optical disks.
[0062] It will be understood by those skilled in the art that the above embodiments are specific implementations of this application, and various changes in form and detail can be made in practical applications without departing from the spirit and scope of this application. For those skilled in the art, several improvements and modifications can be made without departing from the principles of this application, and these improvements and modifications are also considered to be within the scope of protection of this application.
Claims
1. A method for obtaining the motion state of a multibody system based on an implicit dual-loop computational framework, wherein the implicit dual-loop computational framework includes an inner loop and an outer loop, characterized in that, The method includes: S1. For the target multibody system to be analyzed, the dynamic equations of the target multibody system are established by the absolute nodal coordinate method, and the target multibody system is represented as a set of differential-algebraic equations. S2, based on the characteristics and computational requirements of the target multibody system, set the parameters required for numerical integration calculation, including simulation deadline, outer loop convergence condition parameters, and inner loop convergence condition parameters; S3, using Gaussian decomposition to identify the independent and non-independent coordinates of the target multibody system, enters the outer loop iteration process to calculate the generalized position vector and generalized velocity vector of the independent coordinates at the next moment; S4, substitute the generalized position vector and generalized velocity vector of the independent coordinates at the next moment into the position-level constraint equation, enter the inner loop iteration process, use the Newton iteration method to solve the generalized position vector of the non-independent coordinates at the next moment, and then substitute the generalized position vector of all coordinates at the next moment and the generalized velocity vector of the independent coordinates into the velocity-level constraint equation to solve the generalized velocity vector of the non-independent coordinates at the next moment. S5: Reconstruct the dynamic equations of the target multibody system based on the generalized position vector and generalized velocity vector of all coordinates at the next time step, solve the acceleration level constraint equations, and solve for the generalized acceleration vector and Lagrange multiplier at the next time step. If the outer loop convergence condition is met, proceed to the calculation of the next time step; otherwise, return to S3. S6, repeat S3 to S5 continuously until the calculation time reaches the simulation deadline, complete the solution calculation, and obtain the target multi-body cooperative motion state.
2. The method for obtaining the motion state of a multibody system based on an implicit double-loop computational framework according to claim 1, characterized in that, The dynamic equations of the target multibody system are established using the absolute nodal coordinate method, representing the target multibody system as a set of differential-algebraic equations, which are achieved through the following formulas: ；（1） in, , and These are the generalized position vector, generalized velocity vector, and generalized acceleration vector of the target multibody system, respectively. , These are the mass matrix and damping matrix of the target multibody system, respectively. The external forces acting on the target multibody system These are the constraint equations for the target multibody system, used to describe the constraint relationships, connection relationships, control forces, and control torques acting on each component of the target multibody system. Let be the Jacobian matrix of the constraint equation relative to the generalized position vector. Equation (1) is a set of differential-algebraic equations with an index of 3.
3. The method for obtaining the motion state of a multibody system based on an implicit double-loop computational framework according to claim 1, characterized in that, Let the simulation deadline be... The outer loop convergence condition parameters include the outer loop convergence error threshold. Maximum number of iterations in the outer loop The inner loop convergence condition parameters include the inner loop convergence error threshold. and the maximum number of iterations in the inner loop The parameters required for numerical integration calculations also include the integration step size. and controllable numerical dissipation parameters .
4. The method for obtaining the motion state of a multibody system based on an implicit double-loop computational framework according to claim 3, characterized in that, The Gaussian decomposition method is used to identify the independent and non-independent coordinates of the target multibody system. Then, an outer loop iterative process is initiated to calculate the generalized position vector and generalized velocity vector of the independent coordinates at the next time step, including: Generalized position vector of target multibody system Represented as: ；（2） in, A generalized position vector with independent coordinates. It is a generalized position vector with non-independent coordinates; constraint equations Taking the second derivative with respect to time, and solving the simultaneous equations, we obtain a differential-algebraic equation with an index of 1: ；（3） in, Let Lagrange multiplier vectors be used. These are the second-order partial derivatives of the constraint equations; By directly solving equation (3), the generalized acceleration vector of the target multibody system at the current moment can be obtained. ; Entering the outer loop iteration process, using the following formula, a generalized... The algorithm performs integration to obtain the generalized position vector and generalized velocity vector of the independent coordinates at the next time step: ；（4-1） ；（4-2） ；（4-3） ；（4-4） ；（4-5） ；（4-6） Among them, the bottom right corner mark Indicates the current time, indicated by the subscript in the lower right corner. Indicates the next moment, The generalized position vector is the independent coordinate at the current moment. The generalized position vector representing the independent coordinates at the next moment. and They are the generalized meanings of the current moment. Algorithm parameter vector and the generalized next time step Algorithm parameter vector, and In a broad sense The hyperparameters of the algorithm are used to control the discrete moments of acceleration and velocity, respectively.
5. The method for obtaining the motion state of a multibody system based on an implicit double-loop computational framework according to claim 4, characterized in that, Substituting the generalized position vector and generalized velocity vector of the independent coordinates at the next time step into the position-level constraint equations, we enter the inner loop iteration process. The Newton iteration method is used to solve for the generalized position vector of the non-independent coordinates at the next time step, including: Substituting the generalized position vector and generalized velocity vector of the independent coordinates at the next moment into the position-level constraint equation using the following formula, we enter the inner loop iteration process: ；（5-1） in, It is the identity matrix. The number of iterations is Newton's iterations. The Newton iterative method is used to solve for the generalized position vector of the non-independent coordinates at the next time step, and the inner loop convergence error is calculated. ,like Less than Or the current inner loop iteration count reaches If the condition for convergence of the inner loop iteration is met, the inner loop iteration process is exited; otherwise, the inner loop iteration continues. The calculation process is as follows: ；（5-2） in, , The first sequence The generalized position vector of the non-independent coordinates obtained during the next inner loop iteration.
6. The method for obtaining the motion state of a multibody system based on an implicit double-loop computational framework according to claim 5, characterized in that, Substituting the generalized position vectors of all coordinates at the next moment and the generalized velocity vectors of the independent coordinates into the velocity level constraint equations, we can solve for the generalized velocity vectors of the non-independent coordinates at the next moment using the following formula: ；（6） in, Let the generalized velocity vector of the target multibody system at the next moment be . Let be the generalized velocity vector of the independent coordinates at the current moment. This is the partial derivative of the constraint equation with respect to time.
7. The method for obtaining the motion state of a multibody system based on an implicit double-loop computational framework according to claim 6, characterized in that, Reconstruct the dynamic equations of the target multibody system based on the generalized position vector and generalized velocity vector of all coordinates at the next time step, solve the acceleration level constraint equations simultaneously, and solve for the generalized acceleration vector and Lagrange multipliers at the next time step, including: According to the broad sense The algorithm's difference scheme reconstructs the dynamic equations of the target multibody system based on the generalized position and velocity vectors of all coordinates at the next time step. It then solves for the acceleration level constraint equations, obtaining the generalized acceleration vector and Lagrange multipliers at the next time step. ；（7） in, , These are the partial derivative matrices of the constraint equations with respect to the generalized position vector at the current and next time steps, respectively. , These are the generalized external force vectors at the current moment and the next moment, respectively. , Let these be the generalized acceleration vectors of the previous moment and the next moment, respectively. , These can be considered as the Lagrange multipliers of the previous moment and the next moment, respectively.
8. The method for obtaining the motion state of a multibody system based on an implicit double-loop computational framework according to claim 6, characterized in that, After obtaining the generalized acceleration vector and Lagrange multipliers at the next time step, the outer loop convergence error is calculated based on the generalized acceleration vector and Lagrange multipliers at the next time step. ,like Less than Or the current outer loop iteration count reaches If the outer loop convergence condition is met, the outer loop iteration process is exited, and the calculation of the next time step begins; otherwise, the process returns to S3 to continue the outer loop iteration.
9. An electronic device, characterized in that, include: The processor and memory, wherein the memory stores instructions that the processor can execute, and the processor is configured to, when executing the instructions, enable the electronic device to implement a method for obtaining the motion state of a multibody system based on an implicit double-loop computing framework as described in any one of claims 1 to 8.
10. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it can implement a method for obtaining the motion state of a multibody system based on an implicit double-loop computation framework as described in any one of claims 1 to 8.