Tension distribution modeling method considering friction and preload

Abstract

The invention relates to a tension distribution modeling method considering friction and preload, belongs to the technical field of pulley and cable friction, and effectively solves the challenges of tension prediction and estimation in a complex pulley-cable system by establishing a node element tension distribution model. The comprehensive analysis comprises tension transmission loss, and dynamic sliding and rolling friction from a Lugre model are combined. The comparative simulation reveals the influence of different friction effects. Experimental verification carried out by using a rolling joint platform proves the accuracy of the model in predicting the tension in a cable and the force on a pulley node in a complex pulley system. The model provides a robust frame for analyzing and designing a cable-driven structure, and is of great significance to cable safety and real-time tension monitoring and prediction. Not only are the modeling speed and quality improved, but also a powerful tool is provided for real-time monitoring and prediction evaluation of the safety of a cable system.

Description

Technical Field

[0001] The invention relates to the technical field of pulley and cable friction, and in particular to a tension distribution modeling method considering friction and preload. Background Art

[0002] In modern engineering, cable-driven mechanisms are revered for their minimal mass, economical nature, and ability to transmit power over large distances. These mechanisms are particularly indispensable for the functionality of surgical robots and interactive manipulators. Nonetheless, due to the inherent limitations of cable stiffness and tensile strength, strategies need to be developed to enhance operational safety and reduce the likelihood of cable failure. As the complexity of structural designs escalates, pulley configurations become increasingly complex, and the increase in pulleys within the system significantly increases the cumulative friction impact. At the same time, in applications where miniaturization is critical, such as surgical robots, there is a strict requirement for meticulous calibration of cable tension. However, due to the limitations of design compactness and budgetary considerations, it is impractical to incorporate supplementary sensors for tension monitoring. Therefore, it is imperative to improve the accuracy of the system's tension prediction and estimation models. Such enhancements are critical not only to the safety of the system, but also to its controllability. The fidelity of tension prediction is critical to the accuracy and reliability of robotic maneuvers, especially in the context of high-risk medical procedures.

[0003] At present, many studies have conducted relevant work on the modeling of pulleys and cable bodies. Eric et al. conducted static and dynamic stiffness analysis on a large-span CDPR (cable-driven parallel robot) mechanism, in which the mass and elasticity of the cable cannot be ignored. Zhaohui et al. developed a motion model for flexible cables, taking into account the change in length, and performed coupled motion analysis between the cable and the winch or pulley. Peng et al. introduced an effective multi-body modeling method for cable-pulley systems with friction and introduced a variable-length cable element with movable nodes. In large-scale rope-driven systems, the mass and deformation of the cable can significantly affect the response of the system. Commonly used cable modeling methods include the finite element method and the absolute node coordinate method. Smith et al. proposed a repeating unit cell (RUC) FE modeling strategy for tensile loads and initial tensile load bending of multi-layer helical chain cables. However, in small rope-driven systems, the cables used are usually small in length and size, and the mass effect during motion can be ignored. In the above cable bearing system, basically only the static or dynamic deformation analysis of the cable is involved, and the friction of the cable pulley and the preload of the cable are hardly considered, which often makes it difficult to make a more accurate correctness assessment of the system status.

[0004] In terms of the contact characteristics of pulleys and cables, two popular friction models are used to describe the friction between pulleys and cables. The first model is Euler's model, which elegantly represents the tension on a pulley or capstan as a decaying exponential function. Ju et al. introduced a parametric superelement model in which the cable passes through multiple pulleys based on the Euler model, mainly for static analysis under heavy loads. The second friction model, which improves the Euler model by dividing the contact arc into adhesion arc and sliding arc, in which the tension remains unchanged while in the sliding arc the tension changes exponentially like the Euler model. The mesh-based model illustrates the associated deformation. The static friction model is used to analyze the existing cable pulley model, which mainly considers the sliding friction. In general mechanical systems, dynamic friction models are widely used to capture more characteristics of the system by using additional state variables. Summary of the invention

[0005] The present invention aims to solve the above technical problems and provides a tension distribution modeling method taking friction and preload into consideration.

[0006] In order to solve the above technical problems, the technical solution provided by the present invention is:

[0007] A method for modeling tension distribution considering friction and preload, comprising the following steps:

[0008] Step 1: Build a multi-pulley cable system;

[0009] Step 1.1, establish the rope drive system and clarify the configuration of different rope drive systems;

[0010] Step 1.2: Identify the characteristics and problems of rolling joints;

[0011] Step 2: Establish the kinematics and dynamics model of the rolling joint;

[0012] Step 2.1, clarify the specific configuration of the rolling joint system;

[0013] Step 2.2, clarify the kinematic characteristics of the rolling joint;

[0014] Step 2.3, clarify the dynamic characteristics of the rolling joint;

[0015] Step 3: Establish the node element model of friction and preload;

[0016] Step 3.1: Cable unit and pulley node force modeling.

[0017] Step 3.2, cable pulley tension transfer modeling;

[0018] Step 4: Establish the friction model of sliding and rolling contact;

[0019] Step 4.1, modeling using sliding friction model;

[0020] Step 4.2: Modeling using rolling friction model;

[0021] Step 5: Based on the numerical solution of the rolling joint characteristics and the tension distribution model, a node element tension distribution model is established.

[0022] The preload, friction effect, pulley node parameters and rope unit parameters are calculated through the pulley node force, rope unit force and rolling node in the rope drive system through the node element tension distribution model. The rope length, joint angle, winding angle and tension difference are calculated through kinematics and dynamics.

[0023] Preferably, the rope drive system comprises open chain and closed chain configurations.

[0024] Preferably, the node element tension distribution model can simultaneously consider the preload and friction of the independent rope drive system, use the tension transfer ratio as the iterative variable, know any node or element parameters, and iteratively solve other pulley and cable force, displacement and deformation problems.

[0025] Preferably, the cable unit is a continuous cable that is elastically deformed through a plurality of intermediate pulleys, and the pulleys are fixed on a bracket, including n nodes, n-1 cables and n-2 pulleys, and the transfer length of the pulley is expressed as si (i=2,..., n-1). Cable tension is generated by cable deformation and movement between adjacent pulley groups, and the corresponding tension is t 0,i =k (i) (u 0,i +s 0,i -u 0,i+1 -s 0,i+1 ), the tension distribution and kinematic model are established, and the relationship between the elastic force of each unit, the pulley node displacement and the cable length change can be determined.

[0026] Preferably, the cable pulley tension transfer is modeled as when the tension is transmitted through the pulley, relative movement occurs between the cable and the pulley, resulting in different tensions on both sides, and the friction comes from the sliding friction between the cable and the pulley and the rotational friction of the bearing. By creating iterative equations, when the tension, preload and motion displacement of the end cable at any cable position are known, the deformation of the cable and the tension of each cable are inferred.

[0027] Preferably, the sliding friction model and the rolling friction model calculate the tension transfer coefficient by taking into account the friction loss of the rolling bearing and the sliding friction caused by the contact of the cable pulley.

[0028] Preferably, the sliding friction model takes friction as a whole, combines the cable pulley static friction model to obtain equivalent LuGre model parameters, and expands the LuGre model into a dynamic friction model suitable for calculating cable pulley friction. The calculation formula is as follows:

[0029] Preferably, when using the rolling friction model, the actual tension transmission coefficient η satisfies η r ≤η<η d , sliding friction influence coefficient ρ, η = ρη d +(1-ρ)η r The ρ value can be used to simulate the change of friction coefficient according to different stress states.

[0030] By adopting the above method, the present invention has the following advantages:

[0031] This invention proposes a new node-element tension distribution model that greatly facilitates the analysis and solution of complex cable-driven systems. The model cleverly distills the complex pulley-cable assembly into a series of understandable nodes and elements, and carefully considers the different effects of preload and friction dynamics. It uses an iterative approach to infer the conditions of uncertain node elements based on established parameters. We integrate dynamic sliding and rolling friction based on the LuGre model and conduct a thorough investigation, revealing the significant impact of sliding friction on the system's force transmission efficiency. The inventive results clarify the direct correlation between increased preload and the reduced necessity of driving force in combined drives.

[0032] The above summary is for illustrative purposes only and is not intended to be limiting in any way. In addition to the illustrative aspects, embodiments and features described above, further aspects, embodiments and features of the present invention will be readily apparent by reference to the accompanying drawings and the following detailed description. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the drawings required for use in the embodiments or technical descriptions will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0034] Figure 1 is a comparative schematic diagram of a conventional minimum unit of the rope drive system of the present invention;

[0035] Figure 2 It is a schematic diagram of the applicable fields of the rolling joint of the present invention and the layout of the pulley cable system;

[0036] Figure 3is a schematic diagram of the rolling joint tension amplification characteristic of the present invention;

[0037] Figure 4 is a schematic diagram of the rolling joint motion characteristics of the present invention;

[0038] Figure 5 It is a schematic diagram of the system node unit and tension transmission model of the present invention;

[0039] Figure 6 is a simplified schematic diagram of the rolling joint system of the present invention and a schematic diagram of the winding of the tension amplification mechanism;

[0040] Figure 7 It is a basic principle schematic diagram of the cable and pulley winding of the present invention;

[0041] Figure 8 is a schematic diagram of different cable pulley friction models of the present invention;

[0042] Fig. 9 It is a block diagram of the overall solution of the present invention. DETAILED DESCRIPTION

[0043] Specific embodiments of the present invention will now be mentioned in detail. Although the present invention is described in conjunction with these specific embodiments, it should be appreciated that it is not intended to limit the present invention to these specific embodiments. On the contrary, these embodiments are intended to cover substitutions, changes or equivalent embodiments that may be included in the spirit and scope of the invention defined by the claims. In the following description, a large number of specific details are set forth in order to provide a comprehensive understanding of the present invention. The present invention can be implemented without some or all of these specific details. In other cases, in order not to make the present invention unnecessarily obscure, well-known process operations are not described in detail.

[0044] When used in conjunction with "including", "methods including", or similar language in this specification and the appended claims, the singular forms "a", "an", and "the" include plural references unless the context clearly dictates otherwise. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs.

[0045] The object of the present invention is to provide a method for establishing a tension distribution model of a complex multi-pulley cable system taking friction and preload into consideration, which is used for tension prediction in a pulley rope system and pulley force analysis.

[0046] The present invention is further described in detail below in conjunction with the full text.

[0047] Attachment Figure 1-Figure 9 , a tension distribution modeling method considering friction and preload, comprising the following steps:

[0048] Step 1: Build a multi-pulley cable system;

[0049] Step 1.1, establish the rope drive system and clarify the configuration of different rope drive systems;

[0050] Step 1.2: Identify the characteristics and problems of rolling joints;

[0051] Step 2: Establish the kinematics and dynamics model of the rolling joint;

[0052] Step 2.1, clarify the specific configuration of the rolling joint system;

[0053] Step 2.2, clarify the kinematic characteristics of the rolling joint;

[0054] Step 2.3, clarify the dynamic characteristics of the rolling joint;

[0055] Step 3: Establish the node element model of friction and preload;

[0056] Step 3.1: Cable unit and pulley node force modeling.

[0057] Step 3.2, cable pulley tension transfer modeling;

[0058] Step 4: Establish the friction model of sliding and rolling contact;

[0059] Step 4.1, modeling using sliding friction model;

[0060] Step 4.2: Modeling using rolling friction model;

[0061] Step 5: Based on the numerical solution of the rolling joint characteristics and the tension distribution model, a node element tension distribution model is established.

[0062] The preload, friction effect, pulley node parameters and rope unit parameters are calculated through the pulley node force, rope unit force and rolling node in the rope drive system through the node element tension distribution model. The rope length, joint angle, winding angle and tension difference are calculated through kinematics and dynamics.

[0063] Rope drive systems include open chain and closed chain configurations.

[0064] The nodal element tension distribution model can simultaneously consider the preload and friction of the independent rope drive system, use the tension transfer ratio as the iteration variable, know any nodal or element parameters, and iteratively solve other pulley and cable forces, displacements, and deformations.

[0065] The cable unit is the elastic deformation of a continuous cable passing through multiple intermediate pulleys, and the pulleys are fixed on the bracket, including n nodes, n-1 cables and n-2 pulleys. The transfer length of the pulley is expressed as si (i=2,...,n-1). The cable tension is generated by the cable deformation and the movement between adjacent pulley groups. The corresponding tension is t 0,i =k (i) (u 0,i +s 0,i -u 0,i+1 -s 0,i+1 ), the tension distribution and kinematic model are established, and the relationship between the elastic force of each unit, the pulley node displacement and the cable length change can be determined.

[0066] The cable-pulley tension transfer is modeled as the relative motion between the cable and the pulley occurs when the tension is transmitted through the pulley, resulting in different tensions on both sides. The friction comes from the sliding friction between the cable and the pulley and the rotational friction of the bearing. By creating iterative equations, the deformation of the cable and the tension of each cable are inferred when the tension, preload and motion displacement of the end cable at any cable position are known.

[0067] The sliding friction model and rolling friction model calculate the tension transmission coefficient by considering the friction loss of rolling bearing and the sliding friction caused by the cable pulley contact.

[0068] The sliding friction model takes friction as a whole and combines it with the static friction model of the cable pulley to obtain equivalent LuGre model parameters. The LuGre model is expanded into a dynamic friction model suitable for calculating cable pulley friction. The calculation formula is as follows:

[0069] When using the rolling friction model, the actual tension transfer coefficient η satisfies η r ≤η<η d , sliding friction influence coefficient ρ, η = ρη d +(1-ρ)η r The ρ value can be used to simulate the change of friction coefficient according to different stress states.

[0070] In one embodiment, a method for establishing a tension distribution model of a complex multi-pulley cable system considering friction and preload of the present invention is compared with a multi-pulley cable system. Figure 1 As shown, the technical approach of the method is implemented by the following steps:

[0071] Furthermore, the step 1 specifically includes the following steps:

[0072] Step 1.1: Identify different rope drive system configurations;

[0073] Rope drive systems can be divided into open chain and closed chain configurations. Typical unit modules of these two systems are as follows Figure 1 A typical open chain system, such as Figure 1 (a) shows a lifting mechanism where the rotation of the motor provides the output tension. The torque potential depends on the motor's capabilities, such as winch traction and lifting systems. In the case where the terminal is a joint rather than a load, at least two cables are required to facilitate bidirectional rotation of the terminal joint. Without a predefined preload, the cable tension will be load dependent. Conversely, Figure 1 The most basic closed-chain unit shown in (b) works on the basis of friction drive. Here, the maximum output torque at the joint is limited by the friction interaction between the cable and the pulley. Although the load-bearing capacity is limited, bidirectional rotation of the terminal joint can be achieved through the bidirectional operation of a single motor, which is a common feature in the robotic arm joint drive scenario. Transforming the cable guide to a tendon sheath structure leads to tendon sheath drive, as shown in Figure 1(c). This adaptation eliminates the need for a complex pulley layout and provides flexibility in complex environments. However, the internal sliding of the cable within the tendon sheath generates significant friction, reducing energy efficiency. Despite this, tendon sheath drive is widely used in various cable-driven robotic arm applications due to its adaptability. When various open-chain and closed-chain cable drives are arranged in different configurations, they can be combined into series or parallel cable drive systems. The integration of these series and parallel units can form a series-parallel hybrid structure. Within this framework, the combination of open-chain units forms a redundant cable drive system. In such a system, the equilibrium position determines the tension between each cable, which depends on the state of its corresponding cable. In contrast, the smallest unit of the closed-chain system requires an initial preload, which will affect the maximum output torque of the motor and the ultimate load capacity of the system.

[0074] Step 1.2: Identify the characteristics and problems of rolling joints;

[0075] In order to increase the joint torque and improve energy efficiency without a surplus motor, we worked hard to introduce a rolling joint structure, which, by taking advantage of the opposite characteristics of the cable, ensures that as one side of the cable changes length during movement, the other side will also adjust accordingly. These joints, commonly known as rolling joints, are usually implemented using conjugate gears and antagonistic cables. The symmetry of the pulleys ensures that the slender and contracting cables maintain the same length and speed. The rolling joint mainly consists of two parts, the upper and lower fixed pulleys and the active pulley through independent cable systems. The first ends of both cable systems are fixed to the edge of the motor winch and the two ends are fixed at the rolling joint. When the length of the upper cable decreases, the length of the lower cable increases accordingly, and the joint angle changes. In this process, the dynamic pulley position changes with the change of the joint angle, and the number of turns of the cable wrapped around the dynamic fixed pulley affects the overall operating speed and efficiency. In addition, the different number of turns of the cable can enhance the force on the pulley, producing a combined effect similar to that of the active pulley and the static pulley.

[0076] The rolling joint is driven by tension difference and is divided into two independent upper and lower cable drive systems. Different preloads can be set for the two cable drive systems in the initial static state. During the joint movement, the cable length of the upper and lower cable systems is limited by the cable deformation and the initial total length.

[0077] CDRJ uses a dual cable system (upper and lower) and uses the tension difference between them to generate output torque, as well as the added benefit of tension amplification. While this structure has high output torque, compact form factor, and eliminates redundant motors, it also introduces complexity in system structure, including the dual cable system and the tension amplification mechanism. To my knowledge, there is a lack of research that considers an accurate model of cable pulley drive combined with a rolling joint mechanism.

[0078] There are several differences between this type of drive system and the traditional cable-driven minimum unit system.

[0079] (1) The position of the pulley in a conventional cable drive remains fixed, and the macroscopic cable length between two adjacent pulleys remains unchanged. In contrast, in a rolling joint there is a moving pulley, and the relative position between the pulleys changes.

[0080] (2) In a conventional cable drive unit, the driving force comes from the tension or friction of a single continuous cable, and the preload force is continuous and equal when stationary. In a rolling joint, the driving force comes from the tension difference between the upper and lower cable systems, and the preload forces of the two cable systems are independent of each other in a static state.

[0081] (3) The number of pulleys involved in the traditional cable drive unit is small, and the tension transmission relationship between the cables is relatively simple. On the other hand, as the tension magnification factor n changes, the rolling joint will add more movable and fixed pulleys.

[0082] It can be foreseen that if a rolling joint is used to construct a multi-DOF manipulator, the number of mobile and fixed pulleys contained in the entire system will be many times that of a traditional cable drive mechanism, so it is necessary to consider the overall pulley effect. At the same time, it is necessary to establish the kinematic and dynamic relationship between the upper and lower pulley cable systems and solve the cable tension problem between the upper and lower pulley systems.

[0083] Step 2: Establishment of rolling joint kinematics and dynamics model:

[0084] In pursuit of increasing joint torque and improving energy efficiency without surplus motors, recent academic efforts have introduced rolling joint structures, which are widely used in cable-driven elbow and knee joints of the upper and lower limbs. By utilizing the opposite characteristics of the cable, it can be ensured that when one side of the cable changes length during movement, the other side will also adjust accordingly. This invention proposes a method for realizing a rolling joint based on conjugate gears, such as Figure 2 As shown in (a), this method can improve the overall stiffness and operational stability while ensuring pure rolling characteristics. At the same time, a symmetrical pulley cable arrangement method is proposed, such as Figure 2 (b) as shown.

[0085] Furthermore, the step 2 specifically includes the following steps:

[0086] Step 2.1: Identify different rope drive system configurations:

[0087] The symmetry of the pulley ensures that the slender and retracted cables maintain the same length and speed. The rolling joint mainly consists of two parts, independent cable systems that pass through the upper and lower fixed pulleys and the movable pulley respectively. The first ends of the two cable systems are fixed to the edge of the electric winch, and the two ends are fixed at the rolling joint. When the length of the upper cable decreases, the length of the lower cable increases accordingly, and the angle of the joint changes at the same time. In this process, the position of the movable pulley changes with the change of the joint angle, and the number of cable turns wrapped around the movable and fixed pulleys will affect the overall operating speed and efficiency. In addition, cables with different numbers of turns can enhance the force on the pulley, thereby producing an effect similar to the combination of a movable pulley and a static pulley. Such as Figure 3 As shown, these parameters follow the following relationship:

[0088]

[0089] Step 2.2: Determine the kinematic characteristics of the rolling joint:

[0090] like Figure 4As shown in (a), we can get the overall relationship between the rolling joint motion angle and the cable length. When θ is zero, the lengths of l-up and l-down are the same as ND. According to the simplified winding method, the following relationship can be obtained:

[0091]

[0092] like Figure 4 As shown in (c), considering the size of the guide wheel, assuming that n is an even number (here n = 2), the actual wire length l-total is:

[0093]

[0094] When the bending angle θ changes, only the lengths, la and lc., are adjusted accordingly, while the other lengths remain unchanged. For an even number of turns, the actual wire length l-total. depends entirely on l-left., reflecting the movement of the simplified winding. The extension and contraction of the cable at both ends are symmetrical. In principle, no pretensioning mechanism is required to prevent cable sag, but the total cable length must be adjusted to establish the required cable pretension.

[0095] Step 2.3: Determine the rolling joint dynamics:

[0096] Building a dynamic model for a cable can be more complex, mainly because of its stiffness. To simplify the analysis process in the control and modeling of cable-driven manipulators, cables are usually treated as rigid cables and their mass and deformation are neglected. Given the relatively small size of the cable-driven joint system and the limited inner diameter and length of the cables used, the mass of the cable is negligible compared to the joint and the load. Therefore, only the effect of its deformation is considered.

[0097] Following the principles of Newtonian and Lagrangian mechanics, the basic dynamic model of the robotic arm joint is as follows:

[0098]

[0099] τ is the joint bending moment, D(q) is the n×n mass matrix of the joint, are the n×1 centrifugal and Coriolis force vectors, and G(q) is the n×1 gravity vector.

[0100] like Figure 4As shown in (a), the pure rolling meshing process of the conjugate gears results in the angular velocity of the joint being twice the angular velocity of the gear rotation. This design method effectively expands the range of joint motion within the limited rotation range of the gears. Under load, the joint can achieve a range of motion within 45 degrees, and the range of motion of the end load can reach 180 degrees. Assuming that the rolling component can be approximated as a circle, and the mass and radius of the pulley and cable, as well as the deformation and friction of the end cable are not considered, the Lagrangian dynamic equations can be used to describe the joint dynamics. The rotational kinetic energy and moment of inertia can be determined using the following formulas:

[0101]

[0102] When the initial horizontal position is regarded as the zero point of potential energy, the potential energy of the joint can be calculated as:

[0103] P joint =m joint g·(2R)sinθ (6)

[0104] Likewise, the kinetic and potential energies of the end load can be determined as follows, where Ld. represents the distance between the end load and the center of the joint:

[0105]

[0106] P load =m load g(2Rsinθ+L d sin2θ) (8)

[0107] By substituting into the Lagrangian dynamics equation, the driving torque τ-θ.joint is equal to the product of the tension difference T between the upper and lower sets of cables and the moment arm:

[0108]

[0109] Considering that the tension difference at the tension amplification pulley forms the active moment of the driving joint, it can be expressed as:

[0110]

[0111] Therefore, it can be seen that the output torque of the joint is related to the tension difference between the upper and lower rope systems and the angle of the joint. The tension difference between the upper and lower rope systems can be inversely solved by the dynamic equation in (9).

[0112] Step 3: Nodal element model considering friction and preload;

[0113] Although the layout and application of rope drive systems are diverse, their basic components mainly consist of winches, cables, pulleys and fixed anchors. Therefore, using the concept of node elements in the finite element method, complex systems can be simplified and solved through an iterative method. The current pulley node finite element analysis scenarios are mainly aimed at cable bearing systems such as cranes, transmission lines, and parachutes. Most of the existing models only analyze statics without considering dynamics. At present, this sliding cable system does not involve cable pretensioning issues.

[0114] Since the CDRJ system uses a tension amplification joint, the number of pulleys included in the system far exceeds that of a traditional pulley system. In order to fully analyze the impact of the entire pulley cable system, the finite element iteration method can more accurately capture the complex mechanical and motion characteristics of the system. Especially in the case of multiple factors such as cable length changes, preload, tension, friction, and pulley system, the finite element iteration method provides more refined simulation results through segmented simulation and stepwise approximation. To the best of our knowledge, this invention is the first to apply the finite element iteration method to a similar rolling joint cable drive analysis scenario.

[0115] Furthermore, the step 3 specifically includes the following steps:

[0116] Step 3.1: Modeling of cable elements and pulley node forces;

[0117] Figure 4 The elastic deformation of a continuous cable passing through multiple intermediate pulleys fixed to a support is shown. The entire system consists of n nodes, n-1 cables and n-2 pulleys. The cable can transition from one side of the pulley to the other due to pulley rotation, cable sliding friction, or a combination of both. It can be assumed that when the length of the cable between two nodes greatly exceeds the winding length of the pulley, the cable elongation is mainly concentrated in the middle part (i) of the cable and is affected by the previous and next pulleys. The transfer length of the pulley is denoted as, si (i = 2, ····, n-1), where, s-1.=, sn.=0. The entire pulley system may consist of fixed pulleys and movable pulleys, resulting in node displacements ui. (i = 1, 2, ···n), especially when the adjacent pulley is a fixed pulley or a fixed anchor point, where, ui = 0. In the model construction and experimental process, in order to simplify the system structure and analysis process, we made the following assumptions: the cable exhibits linear elongation under small deformation. During the movement, the changes in the elastic modulus and the cross-sectional area of ​​the cable are negligible. The mass of the cable is negligible and therefore not considered in the calculations.

[0118] Meanwhile, the cable force acting on each node is recorded as, where the subscript i represents the i-th node and the superscript (i) represents the i-th section of the cable. Many previous studies have developed dynamic models for cable-driven robotic manipulators with elastic cables. In these models, the elastic cable connecting the two pulleys is usually represented by a linear spring. However, a linear spring can exert both tensile and compressive forces at the same time, which may not accurately reflect the actual properties of the tendon. In this study, we proposed a method to simulate unidirectional forces in tendons.

[0119] Figure 5 The process of generating cable tension through cable deformation and movement between adjacent pulley blocks is illustrated. We assume that the deformation caused by the initial preload on each section is expressed as, and the corresponding tension is t 0,i =k (i) (u 0,i +s 0,i -u 0,i+1 -s 0,i+1 ). When a preload is applied to hold the joint in the initial position, where ui.=0 (i=1,2,···n), only the pulley segments will be slightly displaced. These accumulated small displacements are then constrained by the end anchors. If a positive rotation of the pulley is defined as a counterclockwise rotation of the pulley, then for a small rotation angle of the pulley, the tension in the i-th cable is, ti.:

[0120] t i =Γ + {t 0,i -k (i) (u i +s i -u i+1 -s i+1 )} (11)

[0121] It can be noted that Γ + (x) returns its value only if the argument is positive. Otherwise, Γ + (x) gives a zero value. Since the cable can only bear axial tension, the forces on both sides of the same cable unit are equal but in opposite directions, so the relationship between cable deformation and force can be established as:

[0122]

[0123] Substituting the expression and transforming the matrix we get:

[0124]

[0125] Among them, f i (i) , ui., si. and u-i+1, s-i+1 are the axial force, displacement and envelope length at nodes i and i+1 respectively. The nominal stiffness of subunit k-(i). It is defined as:

[0126]

[0127] where E-(i), A-(i) and L-(i) are Young's modulus, cross-sectional area and length of the sub-element. If the changes in elastic modulus and cross-sectional area of ​​the cable during motion are negligible, then the stiffness of each cable segment will also change when the cable length between adjacent nodes changes.

[0128] Transforming the nodal forces and displacements from local subelement coordinates to global coordinates yields:

[0129]

[0130] where {F i (i)}, {Δ i},{Δ i+1} are the nodal forces and displacement vectors in the global coordinate system, such as Figure 4 As shown, the definition is as follows:

[0131]

[0132] where λ (i) is the transformation coefficient from the local coordinate system to the global coordinate system, corresponding to the change in the node displacement ui. Δi , and the cable length si. is transmitted by the pulley winding along the direction of cable movement.

[0133] By substituting (15) and (16) into (13) and using the orthogonal property of the transformation matrix, the following element matrix equation is obtained:

[0134]

[0135] Considering the force balance condition at each node, the following node element equation can be obtained:

[0136]

[0137] where {TF} 3n×1 and {Δ} 3n×1 are the conventional nodal forces and displacements of the cable system, [K ΔΔ ] 3n×3n and [K Δs ] 3n×(n-2) Assembled from (18) and (19), {S} (n-2)×1 for:

[0138] {S} (n-2)×1 =[s2 s3 ... s n-2 s n-1 ] T (twenty one)

[0139] The system consists of two separate cable systems, such as Figure 6 (a) As shown. At the rolling joint, the cable is wound multiple times through the tension amplification mechanism, as shown in Figure 6 (b) is shown. By integrating the established tension distribution and kinematic models, the relationship between the elastic force of each cable unit, the pulley node displacement, and the cable length change can be determined. The rolling joint system discussed represents a more complex configuration in cable-driven mechanisms. Here, a single motor drives two independent preloaded cable systems, and the pulleys bring about significant changes in the length of the cable segments. The proposed model is applicable to most existing cable-driven mechanisms. For example, in a traditional lifting system without preload, the model can be applied by setting the preload to zero. Similarly, in a traditional serial or parallel cable-driven robot, each motor typically drives one cable and the guide pulley is fixed, and the model can be simplified to a scenario with a single preload and zero pulley node displacement. Therefore, the model is widely applicable to mechanisms containing pulley-cable systems, such as cable-driven dexterous hands, robotic arms, and micro-surgical robots.

[0140] Step 3.2: Modeling of cable pulley tension transfer;

[0141] When tension is transferred through the pulley, relative motion occurs between the cable and the pulley. Friction affects this process, resulting in different tensions on both sides. The actual friction comes from the sliding friction between the cable and the pulley and the rotational friction of the bearings. The difference in tension is quantified by the transmission coefficient η, as Figure 7 (a) shown.

[0142] T2=ηT1 (22)

[0143] Consider the force at node i, where cable elements i-1 and i are connected to the i-th pulley, as Figure 7 (b). Substituting, the node axial force can be expressed as:

[0144]

[0145] Where j = i and i-1, corresponding to two adjacent cable elements respectively. Based on equation (22). Then the axial force relationship between the two cable elements at node i can be expressed as:

[0146] f i (i) =-η i f i(i-1) (twenty four)

[0147] where the negative sign is due to the force constraint in the rope element, and η-i depends on the friction coefficient and the contact angle of the pulley. Substituting f i (i) and f i (i-1) To (24). We get:

[0148]

[0149] Summing for i=2 to n-1 gives:

[0150]

[0151] Therefore, (26) defines the relationship between the node displacement and the pulley motion, which is satisfied by the cables passing through all the pulleys in the system. The specific expressions for the defined stiffness coefficients can be found in the Appendix. Integrating the tension transfer relationship between the cable elements and the two sides of the pulley node, (27) gives the final relationship between the general node displacement and force:

[0152]

[0153] Therefore, by creating iterative equations, the cable deformation and the tension in each cable can be inferred when the tension, preload and the motion displacement of the end cable are known at any cable location.

[0154] At the same time, due to the friction between the cable and the pulley, the stiffness matrix [K] is usually asymmetric. ΔΔ ,K Δs ,K sΔ ,K ss The specific results are shown in the Appendix. If the friction effect can be ignored, the matrix will become symmetrical because η=1.

[0155] Step 4: Friction model establishment considering sliding and rolling contact

[0156] The classical Euler formula assumes that the pulley remains fixed and does not rotate, resulting in the cable contacting all areas of the capstan edge, resulting in pure sliding. In this case, the friction loss reaches a maximum and the transmission coefficient reaches a minimum, as shown in Figure 8 (b). However, when the center pulley can rotate, the area where sliding friction occurs between the cable and the edge of the pulley gradually decreases due to the presence of the rolling bearing. Ideally, there is no slippage between the cable and the pulley, resulting in friction losses in the bearing movement, and the transmission coefficient reaches a maximum value, as shown in Figure 8 (c) The present invention takes into account comprehensively.

[0157] Furthermore, the step 4 specifically includes the following steps:

[0158] Step 4.1: Modeling using sliding friction model;

[0159] Consider a pulley fixed to the ground and the surrounding tendons, such as Figure 7 (a) shows that one end of the tendon is pulled with a tension of T-1 while the other end maintains a tension of T-2. When the tendon is about to move, the differential element is subjected to two tensions T+dT and T on both sides, a normal force dN and a friction force df-s.=, μ-cdN, where μ-c. is the Columb friction coefficient. A force balance is generated on the tangential component and the normal component.

[0160]

[0161] Where β is the contact angle. The friction force generated by the sliding part is:

[0162]

[0163] When there is no rolling friction, the pulley is fixed and does not rotate. The passive holding tension, T-2, is completely determined by the active pulling force, T-1, the friction coefficient, μ-c, and the contact angle, β. The amount of tension lost from T-1 to T-2 is due to the friction force, fs, applied along the contact path.

[0164]

[0165] The cable pulley friction model obtained by the equation does not consider the case where the relative velocity is equal to zero. When the velocity direction changes, the friction force is discontinuous. It cannot handle the friction changes during the joint reverse motion. The LuGre model is a dynamic friction model that can more realistically describe the changes in friction.

[0166]

[0167] Equation (33) is the basic expression of the LuGre model, where f is the friction force and z is the state variable representing the average bristle deflection. v is the relative velocity of the two contacting surfaces. This function simulates the Stribeck effect and g(v)>0. ,-0.,,-1. and,σ-2. are the ElasticStiffness, damping and Viscousfriction parameters respectively. ,fc. and,fs. are the Coulomb friction and the maximum static friction respectively. ,vs. is the Stribeck velocity.

[0168] The cable and the pulley are in linear contact, and the calculation of friction needs to be integrated along the pulley path. In order to extend the LuGre model to the linear contact dynamic friction model, some improvements are made to the LuGre model given in. The basic idea is to take friction as a whole, combine the cable pulley static friction model to obtain equivalent LuGre model parameters, and thus extend the LuGre friction model to a dynamic friction model suitable for calculating cable pulley friction. We have made some improvements to convert static friction into a dynamic friction process.

[0169] Improvement 1: Combine the equations to obtain the equivalent Coulomb friction, and add a sufficiently small damping term ε to avoid numerical problems.

[0170]

[0171] Improvement 2: Since static friction is difficult to obtain, the friction coefficient is determined by the materials of the two contact surfaces. The equivalent maximum static friction is defined as follows:

[0172]

[0173] Where μ-s is the static friction coefficient and μ-c is the Coulomb friction coefficient. Considering steady-state sliding, the system satisfies z.=0 and When viscous friction is neglected, we have:

[0174]

[0175] Then we can get:

[0176]

[0177] The parameters θ and v are related to the pulley-cable configuration and the motor drive, with an emphasis on the LuGre model parameters: μ-s., μ-c. and vs. Given that pulley-cable systems typically use metal cables and operate at relatively low speeds, the ranges of μ-s. and μ-c. are identified as [0.2, 0.4] and [0.1, 0.3], respectively, while vs. is typically below 0.1 mm / s. Sensitivity analysis of these parameters shows that within these specified ranges, the dynamic tension transmission coefficient shows an approximately linear relationship with parameter variations, always remaining above 0.8. This analysis shows that despite the influence of the LuGre model parameters μ-s., μ-c. and vs., the model remains robust and reliable, thus validating its applicability and effectiveness in accurately representing the friction dynamics in pulley-cable systems.

[0178] Step 4.2: Modeling using rolling friction model;

[0179] When there is no sliding friction, the following equation is valid:

[0180]

[0181] (T1-T2)r p =f d dN (39)

[0182] from Figure 8 (c) shows that φ = π - β, and set α = r p / (f d d), we then substitute these two parameters into equations (38) and (39):

[0183] (α 2 -1)η 2 -2(α 2 -cosβ)η+α 2 -1=0 (40)

[0184]

[0185] When the sliding friction between the pulley and the cable and the rolling friction of the bearing are considered simultaneously, the following moment equilibrium equation is satisfied:

[0186] (T1-T2)r p =f d dN+f s r p (42)

[0187] The actual tension transfer coefficient η satisfies:

[0188] η r ≤η<η d (43)

[0189] Different weighting factors are assigned to different friction effects, and ρ is defined as the sliding friction influence coefficient. Definition η = ρη d +(1-ρ)η r , different ρ values ​​can be selected to simulate the variation of friction coefficient according to different stress states.

[0190] Step 5: Numerical solution based on rolling joint characteristics and tension distribution model;

[0191] It can be foreseen that if a rolling joint is used to construct a multi-degree-of-freedom robotic arm, the number of moving and fixed pulleys contained in the entire system will be many times that of a traditional cable-driven mechanism, so it is necessary to consider the overall pulley effect. At the same time, it is necessary to establish the kinematic and dynamic relationship between the upper and lower pulley cable systems to solve the cable tension problem between the upper and lower pulley systems. The present invention proposes a node element tension distribution model, which can simultaneously consider the preload and friction of an independent cable system, use the tension transfer ratio as an iterative variable, know any node or element parameter, and iteratively solve other pulley and cable forces, displacements, deformations, and other problems. The solution block diagram of the entire system is shown in the figure. Fig. 9 As shown, the present invention mainly considers the tension distribution model solution, mainly for the integral solution of the tension distribution model and the joint dynamics model.

[0192] The present invention and its embodiments are described above, and such description is not restrictive. What is shown in the full text is only one of the embodiments of the present invention, and the actual structure is not limited thereto. In short, if ordinary technicians in the field are inspired by it, without departing from the purpose of the invention, they can design a structure and embodiment similar to the technical solution without creativity, which should belong to the protection scope of the present invention.

Claims

1. A tension distribution modeling method considering friction and preload, characterized in that: The following steps are involved: Step 1: Build a multi-pulley cable system; Step 1.1, establish the rope drive system and clarify the configuration of different rope drive systems; Step 1.2: Identify the characteristics and problems of rolling joints; Step 2: Establish the kinematics and dynamics model of the rolling joint; Step 2.1, clarify the specific configuration of the rolling joint system; Step 2.2, clarify the kinematic characteristics of the rolling joint; Step 2.3, clarify the dynamic characteristics of the rolling joint; Step 3: Establish the node element model of friction and preload; Step 3.1, cable unit and pulley node force modeling; Step 3.2, cable pulley tension transfer modeling; Step 4: Establish the friction model of sliding and rolling contact; Step 4.1, modeling using sliding friction model; Step 4.2: Modeling using rolling friction model; Step 5: Based on the numerical solution of the rolling joint characteristics and the tension distribution model, a node element tension distribution model is established. The preload, friction effect, pulley node parameters and rope unit parameters are calculated through the pulley node force, rope unit force and rolling node in the rope drive system through the node element tension distribution model. The rope length, joint angle, winding angle and tension difference are calculated through kinematics and dynamics.

2. A tension distribution modeling method considering friction and preload according to claim 1, characterized in that: Rope drive systems include open chain and closed chain configurations.

3. A tension distribution modeling method considering friction and preload according to claim 2, characterized in that: The nodal element tension distribution model can simultaneously consider the preload and friction of the independent rope drive system, use the tension transfer ratio as the iteration variable, know any nodal or element parameters, and iteratively solve other pulley and cable forces, displacements, and deformations.

4. A tension distribution modeling method considering friction and preload according to claim 1, characterized in that: The cable unit is the elastic deformation of a continuous cable passing through multiple intermediate pulleys, and the pulleys are fixed on the bracket, including n nodes, n-1 cables and n-2 pulleys. The transfer length of the pulley is expressed as si (i=2,...,n-1). The cable tension is generated by the cable deformation and the movement between adjacent pulley groups. The corresponding tension is t 0,i =k (i) (u 0,i +s 0,i -u 0,i+1 -s 0,i+1 ), the tension distribution and kinematic model are established, and the relationship between the elastic force of each unit, the pulley node displacement and the cable length change can be determined.

5. A tension distribution modeling method considering friction and preload according to claim 1, characterized in that: The cable-pulley tension transfer is modeled as the relative motion between the cable and the pulley occurs when the tension is transmitted through the pulley, resulting in different tensions on both sides. The friction comes from the sliding friction between the cable and the pulley and the rotational friction of the bearing. By creating iterative equations, the deformation of the cable and the tension of each cable are inferred when the tension, preload and motion displacement of the end cable at any cable position are known.

6. A tension distribution modeling method considering friction and preload according to claim 1, characterized in that: The sliding friction model and rolling friction model calculate the tension transmission coefficient by considering the friction loss of rolling bearing and the sliding friction caused by the cable pulley contact.

7. A tension distribution modeling method considering friction and preload according to claim 1, characterized in that: The sliding friction model takes friction as a whole and combines it with the static friction model of the cable pulley to obtain equivalent LuGre model parameters. The LuGre model is expanded into a dynamic friction model suitable for calculating cable pulley friction. The calculation formula is as follows:

8. A tension distribution modeling method considering friction and preload according to claim 7, characterized in that: When using the rolling friction model, the actual tension transfer coefficient η satisfies η r ≤η<η d , sliding friction influence coefficient ρ, η = ρη d +(1-ρ)η r The ρ value can be used to simulate the change of friction coefficient according to different stress states.

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