A tension distribution modeling method considering friction and preload

Abstract

The present application relates to a tension distribution modeling method considering friction and preload, and belongs to the technical field of pulley and cable friction, the present application effectively solves the challenge of tension prediction and estimation in complex pulley-cable system by establishing a node element tension distribution model. The comprehensive analysis of the present application includes tension transmission loss, and combines dynamic sliding and rolling friction from the Lugre model. Comparative simulation reveals the influence of different friction effects. Experimental verification using a rolling joint platform confirms the accuracy of the model of the present application in predicting tension in the cable and force on the pulley nodes in complex pulley systems. The model provides a robust framework for analyzing and designing cable-driven structures, and is of great significance for the safety and real-time tension monitoring and prediction of cables. Not only the speed and quality of modeling are improved, but also a powerful tool is provided for real-time monitoring and prediction evaluation of cable system safety.

Description

TECHNICAL FIELD

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

[0002] In modern engineering, cable-driven mechanisms are respected 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 robotic arms. Despite this, due to the inherent limitations of cable stiffness and tensile strength, strategies need to be formulated to enhance operational safety and reduce the likelihood of cable failure. As the complexity of structural design escalates, the configuration of pulleys becomes increasingly complex, and the increase in pulleys within the system greatly enhances the cumulative frictional impact. At the same time, in applications where miniaturization is of paramount importance, such as surgical robots, there are stringent requirements for the meticulous calibration of cable tension. However, due to the constraints of design compactness and budgetary considerations, the addition of supplementary sensors for tension monitoring is impractical. Therefore, it is imperative to enhance the precision of the system's tension prediction and estimation models. This enhancement is not only crucial for the safety of the system but also for its control capabilities. The fidelity of tension prediction is of paramount importance for the accuracy and reliability of robotic maneuvers, especially in the context of high-risk medical procedures.

[0003] Currently, many studies have made relevant work on the modeling of pulleys and cable bodies. Eric et al. conducted a static and dynamic stiffness analysis of a large-span CDPR (cable-driven parallel robot) mechanism, in which the mass and elasticity of the cable cannot be ignored. Chao-Hui et al. developed a motion model of flexible cables, considering the change in length and performing 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 significantly affect the response of the system. Common cable modeling methods include the finite element method and the absolute nodal coordinate method. Smith et al. proposed a repeating unit cell (RUC) FE modeling strategy for the tensile load and initial tensile load bending of multi-layered spiral chain cables. However, in small-scale rope-driven systems, the cables used are typically small in length and size, and the mass effect during motion can be negligible. Under the above cable bearing systems, only static or dynamic deformation analysis of the cable is basically involved, and little consideration is given to cable pulley friction and cable preload, making it difficult to make more accurate and correct assessments of the system state.

[0004] In terms of the contact characteristics of the pulley and the cable, two prevalent friction models are employed to describe the friction between the pulley and the cable. The first model is Euler's model, which elegantly represents the tension on the pulley or winch as a decaying exponential function. Ju et al. introduced a parametric hyper-element 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 a sticking arc and a sliding arc, the sticking arc maintains a constant tension, while the tension in the sliding arc changes as an exponential function as in Euler's model. The associated deformation is illustrated based on the grid model. The existing cable-pulley model is analyzed using a static friction model, mainly considering 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

[0005] The present application aims to solve the above technical problems, and provides a tension distribution modeling method considering friction and preload.

[0006] To solve the above technical problems, the technical scheme provided by the present application is:

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

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

[0009] Step 1.1, establish a rope drive system, and specify different rope drive system configurations;

[0010] Step 1.2, specify the characteristics and problems of the rolling joint;

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

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

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

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

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

[0016] Step 3.1, cable element 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, model using sliding friction model;

[0020] Step 4.2: Model using a rolling friction model;

[0021] Step 5: Based on the numerical solution of the rolling joint characteristics and tension distribution model, establish the tension distribution model of the nodal elements.

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

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

[0024] Preferably, 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 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 elastically deformed by multiple intermediate pulleys, with the pulleys fixed to a support. It includes n nodes, n-1 cables, and n-2 pulleys. The transfer length of the pulleys is denoted as si (i = 2, ..., n-1). Cable tension is generated through cable deformation and movement between adjacent pulley groups, with the corresponding tension being t. 0,i =k (i) (u 0,i +s 0,i -u 0,i+1 -s 0,i+1 By establishing tension distribution and kinematic models, the relationship between elastic force, pulley node displacement and cable length variation in each unit can be determined.

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

[0027] Preferably, the sliding friction model and rolling friction model are used to calculate the force transmission coefficient by considering 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 treats friction as a whole, combines it with the static friction model of the cable wheel to obtain equivalent LuGre model parameters, and extends the LuGre model into a dynamic friction model suitable for calculating cable wheel friction. The calculation formula is as follows:

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

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

[0031] This invention proposes a novel tension distribution model for nodal elements, greatly facilitating the analysis and solution of complex cable-driven systems. This model ingeniously distills the complex pulley-cable assembly into a series of understandable nodes and elements, carefully considering the different effects of preload and tribodynamics. It employs an iterative method to infer the conditions of uncertain nodal elements based on established parameters. We thoroughly investigated dynamic sliding and rolling friction by integrating them based on the LuGre model, revealing the significant impact of sliding friction on the system's force transmission efficiency. The results of this invention elucidate the direct correlation between increased preload and reduced driving force necessity in combined drives.

[0032] The above overview 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 invention will become readily apparent from the accompanying drawings and the following detailed description. Attached Figure Description

[0033] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below 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.

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

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

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

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

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

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

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

[0041] Figure 8 These are schematic diagrams of different cable pulley friction models of the present invention;

[0042] Figure 9 This is a block diagram illustrating the overall solution of the present invention. Detailed Implementation

[0043] Specific embodiments of the invention will now be described in detail. Although the invention is described in conjunction with these specific embodiments, it should be understood that the invention is not intended to be limited to these specific embodiments. Rather, these embodiments are intended to cover alternative, modified, or equivalent embodiments that may be included within the spirit and scope of the invention as defined by the claims. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the invention. The invention may be practiced without some or all of these specific details. In other instances, well-known processes have not been described in detail so as not to unnecessarily obscure the invention.

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

[0045] The purpose of this invention is to provide a method for establishing a tension distribution model of a complex multi-pulley cable system that considers friction and preload, for tension prediction and pulley force analysis in pulley rope systems.

[0046] The present invention will now be described in further detail with reference to the full text.

[0047] Attachment Figures 1-9 A method for modeling tension distribution considering friction and preload includes the following steps:

[0048] Step 1: Construct 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: Clarify the characteristics and problems of rolling joints;

[0051] Step 2: Establish the kinematic and dynamic model of the rolling joint;

[0052] Step 2.1: Determine the specific configuration of the rolling joint system;

[0053] Step 2.2: Clarify the kinematic characteristics of the rolling joint;

[0054] Step 2.3: Determine the dynamic characteristics of the rolling joint;

[0055] Step 3: Establish the nodal element model for friction and preload;

[0056] Step 3.1: Model the forces of the cable unit and pulley node.

[0057] Step 3.2: Modeling the tension transfer of the cable pulley;

[0058] Step 4: Establish friction models for sliding and rolling contacts;

[0059] Step 4.1: Model using a sliding friction model;

[0060] Step 4.2: Model using a rolling friction model;

[0061] Step 5: Based on the numerical solution of the rolling joint characteristics and tension distribution model, establish the tension distribution model of the nodal elements.

[0062] By using the pulley node force, rope element force, and rolling node in the rope drive system, the preload, friction effect, pulley node parameters, and rope element parameters are calculated through the node element tension distribution model. The rope length, joint angle, wrapping 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 an independent rope drive system, using the tension transfer ratio as the iterative variable. Knowing any node or element parameter, it can iteratively solve other pulley and cable force, displacement, and deformation problems.

[0065] The cable unit consists of a continuous cable elastically deformed by multiple intermediate pulleys fixed to a support. It includes n nodes, n-1 cables, and n-2 pulleys. The transfer length of each pulley is denoted as si (i = 2, ..., n-1). Cable tension is generated through cable deformation and movement between adjacent pulley groups, with the corresponding tension being t. 0,i =k (i) (u 0,i +s 0,i -u 0,i+1 -s 0,i+1 By establishing tension distribution and kinematic models, the relationship between elastic force, pulley node displacement and cable length variation in each unit can be determined.

[0066] The tension transfer model of the cable pulley is based on the relative motion between the cable and the pulley when 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 can be inferred when the tension, preload and displacement of the end cable at any cable position are known.

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

[0068] The sliding friction model treats friction as a whole, combining it with the static friction model of the cable wheel to obtain equivalent LuGre model parameters. The LuGre model is then extended to a dynamic friction model suitable for calculating cable wheel friction, and the calculation formula is as follows:

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

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

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

[0072] Step 1.1: Define the configuration of different rope drive systems;

[0073] Rope-driven systems can be divided into open-chain and closed-chain configurations. Typical unit modules for these two systems include...Figure 1 As shown. A typical open-chain system, such as Figure 1 As shown in (a), this is a lifting mechanism where the rotation of a motor provides the output tension. The torque potential depends on the motor's capabilities, such as a winch traction and hoisting system. When 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 depend on the load. Conversely, Figure 1 (b) shows the most basic closed-chain unit, which operates on the basis of friction drive. Here, the maximum output torque at the joint is limited by the frictional interaction between the cable and the pulley. Although the load capacity is limited, bidirectional rotation of the end joint can be achieved through the bidirectional operation of a single motor, a common feature in robotic arm joint drive scenarios. Converting the cable guide to a tendon sheath structure results in tendon sheath drive, as shown in Figure 1(c). This adaptation eliminates the need for complex pulley layouts and provides flexibility in complex environments. However, the internal sliding of the cable within the tendon sheath generates significant friction, reducing energy efficiency. Nevertheless, 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 rope 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 rope drive system. In such a system, the equilibrium position determines the tension between each cable, which depends on the state of its corresponding cable. Conversely, the smallest unit of the closed-chain system requires an initial preload, which affects the maximum output torque of the motor and the final load capacity of the system.

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

[0075] To increase joint torque and improve energy efficiency without the need for a spare motor, we have introduced a rolling joint structure. By utilizing the opposing characteristics of the cables, it ensures that as one side of the cable changes length during movement, the other side adjusts accordingly. These joints, commonly referred to as rolling joints, are typically implemented using conjugate gears and opposing cables. The symmetry of the pulleys ensures that the slender and contracting cables maintain the same length and speed. A rolling joint consists primarily of two parts: upper and lower fixed pulleys and a movable pulley, connected by independent cable systems. The first end of each cable system is fixed to the edge of the motor winch, and both ends are fixed at the rolling joint. As the upper cable length decreases, the lower cable length increases accordingly, and the joint angle changes. During this process, the position of the movable pulley changes with the joint angle, and the number of cable turns wound on the movable fixed pulley affects the overall operating speed and efficiency. Furthermore, different numbers of cable turns can enhance the force on the pulleys, producing an effect similar to a combination of movable and static pulleys.

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

[0077] The CDRJ employs a dual-cable system (upper and lower) and utilizes the tension difference between them to generate output torque, along with the additional benefit of tension amplification. While this structure offers high output torque, a compact form factor, and eliminates redundant motors, it also introduces system structural complexity, including the dual-cable system and the tension amplification mechanism. To my knowledge, there is currently a lack of research considering an accurate cable pulley drive model combined with a rolling joint mechanism.

[0078] This type of drive system differs from the traditional cable-driven minimum unit system in several ways.

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

[0080] (2) In a traditional 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) Traditional cable drive units involve fewer pulleys, and the tension transmission relationship between cables is relatively simple. On the other hand, as the tension amplification factor n changes, the rolling joint will require an additional number of movable and fixed pulleys.

[0082] It is foreseeable that if a rolling joint is used to construct a multi-degree-of-freedom robotic arm, the number of moving and fixed pulleys in the entire system will be many times that of a traditional cable-driven mechanism. Therefore, 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 the kinematic and dynamic model of the rolling joint:

[0084] To increase joint torque and improve energy efficiency without the need for a spare motor, 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 as one side of the cable changes length during movement, the other side adjusts accordingly. This invention proposes a method for implementing a rolling joint based on conjugate gears, such as... Figure 2 As shown in (a), this method can improve overall stiffness and operational stability while ensuring pure rolling characteristics. Simultaneously, a symmetrical pulley cable arrangement method is proposed, such as... Figure 2 As shown in (b).

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

[0086] Step 2.1: Define the configuration of different rope drive systems:

[0087] 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: independent cable systems passing through upper and lower fixed pulleys and a movable pulley, respectively. The first end of each cable system is fixed to the edge of the electric winch, and both ends are fixed to the rolling joint. When the upper cable length decreases, the lower cable length increases accordingly, and the joint angle changes. During this process, the position of the movable pulley changes with the joint angle, and the number of cable turns wound on the fixed and movable pulleys affects the overall operating speed and efficiency. Furthermore, cables with different numbers of turns can increase the force on the pulleys, thus producing an effect similar to a combination of movable and static pulleys. 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 obtain 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. Based on 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 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 other lengths remain constant. For an even number of turns, the actual conductor length l-total. depends entirely on l-left., reflecting the movement of the simplified winding. The elongation and contraction at both ends of the cable are symmetrical. In principle, a preload mechanism is not required to prevent cable slack, but the total cable length must be adjusted to establish the required cable preload.

[0095] Step 2.3: Define the dynamic characteristics of the rolling joint:

[0096] Developing a dynamic model for a rope can be more complex, primarily due to its stiffness. To simplify the analysis process in the control and modeling of cable-driven manipulators, cables are typically treated as rigid cables, neglecting their mass and deformation. Given the relatively small size of the cable-driven joint system and the finite 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 Lagrange mechanics, the basic dynamic model of the robotic arm joint is as follows:

[0098]

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

[0100] like Figure 4As shown in (a), the pure rolling meshing process of the conjugate gears results in the joint's angular velocity being twice the gear's rotational angular velocity. This design effectively expands the joint's range of motion within the gear's finite rotational range. Under load, the joint can achieve a range of motion up to 45 degrees, and the range of motion under end load can reach 180 degrees. Assuming the rolling component can be approximated as circular, and neglecting the mass and radius of the pulleys and cables, as well as the deformation and friction of the end cables, the joint dynamics can be described using the Lagrange equations. The rotational kinetic energy and moment of inertia can be determined using the following formulas:

[0101]

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

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

[0104] Similarly, the kinetic and potential energy 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 Lagrange dynamics equations, the driving torque τ-θ is equal to the product of the tension difference T between the upper and lower sets of cables and the torque arm:

[0108]

[0109] Considering that the tension difference at the tension amplification pulley forms the active torque 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 solved by inversely solving the dynamic equation in (9).

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

[0113] Although the layout and application of rope-driven systems vary widely, their basic components mainly consist of winches, cables, pulleys, and fixed anchor points. Therefore, using the nodal element concept in the finite element method, complex systems can be simplified and solved iteratively. Current finite element analysis scenarios for pulley nodes primarily target cable bearing systems such as cranes, power transmission lines, and parachutes; existing models mostly only analyze statics and do not consider dynamics. Currently, this type of sliding cable system does not involve cable pretensioning issues.

[0114] Because the CDRJ system uses a tension amplification joint, the number of pulleys in the system far exceeds that of a traditional pulley system. To comprehensively analyze the impact of the entire pulley cable system, the finite element iterative method can more accurately capture the complex mechanical and kinematic characteristics of the system. Especially under conditions involving multiple factors such as cable length variation, preload, tension, friction, and the pulley system itself, the finite element iterative method provides more refined simulation results through piecewise simulation and successive approximation. To our knowledge, this invention is the first to apply the finite element iterative method to a similar rolling joint cable drive analysis scenario.

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

[0116] Step 3.1: Model the forces at the cable unit and pulley node;

[0117] Figure 4 This illustrates the elastic deformation of a continuous cable passing through multiple intermediate pulleys fixed to a support. The entire system comprises n nodes, n-1 cables, and n-2 pulleys. The cable can transition from one side of a pulley to the other due to pulley rotation, cable sliding friction, or a combination of both. It can be assumed that when the cable length between two nodes significantly exceeds the pulley winding length, the cable elongation is mainly concentrated in the middle portion (i) of the cable and is influenced by the preceding and following 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 and movable pulleys, resulting in node displacements ui (i = 1, 2, ..., n), especially when adjacent pulleys are fixed pulleys or fixed anchor points, where ui = 0. In the modeling and experimental process, to simplify the system structure and analysis, we made the following assumption: the cable exhibits linear elongation under small deformations. During the motion, the changes in the elastic modulus and the cross-sectional area of ​​the cable are negligible. The mass of the cable is also negligible and therefore not considered in the calculations.

[0118] Meanwhile, the cable force acting on each node is recorded as , where the subscript i denotes the i-th node and the superscript (i) denotes the i-th segment 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 two pulleys is typically represented by a linear spring. However, linear springs can apply both tensile and compressive forces simultaneously, which may not accurately reflect the actual characteristics of tendons. In this study, we propose a method to simulate unidirectional forces in tendons.

[0119] Figure 5 This illustrates the process by which cable tension is generated through cable deformation and movement between adjacent pulley blocks. We assume that the deformation caused by the initial preload at each cross-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 its initial position, where ui. = 0 (i = 1, 2, ..., n), only the pulley segment experiences slight displacement. These accumulated small displacements are then constrained by the end anchor points. If we define the forward rotation of the pulley as the counterclockwise rotation, 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] Note that Γ + (x) Returns its value only if the parameter is positive. Otherwise, Γ + (x) gives a zero value. Since a cable can only withstand axial tension, the force values ​​on both sides of the same cable element are equal but opposite in direction. Therefore, the relationship between cable deformation and force can be established as follows:

[0122]

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

[0124]

[0125] Among them, f i (i) ,ui.,si. and u-i+1, s-i+1, represent the axial force, displacement, and envelope length at nodes i and i+1, respectively. The nominal stiffness of sub-element k-(i) 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, respectively. If the changes in elastic modulus and cross-sectional area of ​​the cable are negligible during motion, then the stiffness of each cable segment will also change when the cable length between adjacent nodes changes.

[0128] Converting nodal forces and displacements from local sub-element coordinates to global coordinates yields:

[0129]

[0130] Where {F i (i)}, {Δ i},{Δ i+1} represents 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) These are the transformation coefficients from the local coordinate system to the global coordinate system, corresponding to the changes in the nodal displacement ui. (This refers to the transformation coefficients along the cable movement direction and the global displacement.) Δi The cable length si is transmitted by the pulley winding, always in the direction of cable movement.

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

[0134]

[0135] Considering the force balance conditions at each node, the following nodal element equations can be obtained:

[0136]

[0137] Where {TF} 3n×1 and {Δ} 3n×1 These are the conventional nodal forces and displacements of the cable system, [K ΔΔ ] 3n×3n and [K] Δs ] 3n×(n-2) It is assembled from (18) and (19) respectively, {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 independent cable systems, such as Figure 6 As shown in (a). At the rolling joint, the cable is wound multiple times by a tension amplification mechanism, as shown in... Figure 6 As shown in (b), by integrating the established tension distribution and kinematic model, the relationship between the elastic force, pulley node displacement, and cable length variation of each cable element can be determined. The rolling joint system discussed represents a more complex configuration in cable-driven mechanisms. Here, a single motor drives two independent pre-tensioned cable systems, and the pulleys cause significant variations in cable segment length. The proposed model is applicable to most existing cable-driven mechanisms. For example, in conventional lifting systems without pre-tension, the model can be applied by setting the pre-tension to zero. Similarly, in conventional serial or parallel cable-driven robots, where each motor typically drives one cable and the guide pulley is fixed, the model can be simplified to a scenario with a single pre-tension 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 microsurgical robots.

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

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

[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 shown below. Figure 7 As shown in (b). Substituting this, the nodal axial force can be expressed as:

[0144]

[0145] Where j = i and i-1 correspond to two adjacent cable elements. Based on equation (22), 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] The negative sign is generated by the force constraints in the rope element, and η-i depends on the coefficient of friction and the contact angle of the pulley. (Replace f) i (i) and f i (i-1) To (24)... We get:

[0148]

[0149] Summing over i = 2 to n-1, we get:

[0150]

[0151] Therefore, (26) the relationship between nodal displacement and pulley motion is defined, which is satisfied by the cable passing through all pulleys in the system. The specific expression for the defined stiffness coefficient can be found in the appendix. Combining the tension transfer relationship between the cable elements and both sides of the pulley nodes, (27) the final relationship between general nodal displacement and force can be obtained:

[0152]

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

[0154] Meanwhile, due to friction between the cable and the pulley, the stiffness matrix [K] is usually asymmetric. ΔΔ ,K Δs ,K sΔ ,K ss See the appendix for a detailed representation of the results. If the frictional effect is negligible, this matrix will become symmetric because η = 1.

[0155] Step 4: Establish a friction model considering sliding and rolling contacts

[0156] Classical Euler's formula assumes the pulley remains fixed and does not rotate, causing the cable to contact all areas of the winch edge, resulting in pure slippage. In this case, frictional losses reach their maximum and the transmission coefficient reaches its minimum, such as... Figure 8 As shown in (b). However, when the central pulley can rotate, the area of ​​sliding friction between the cable and the pulley edge gradually decreases due to the presence of the rolling bearing. Ideally, there is no slippage between the cable and the pulley, resulting in frictional losses during bearing movement, and the transmission coefficient reaches its maximum value, as shown in (b). Figure 8 As shown in (c), this invention takes into account all aspects.

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

[0158] Step 4.1: Model using a sliding friction model;

[0159] Consider a pulley fixed to the ground and the surrounding tendons, such as Figure 7 As shown in (a), the end of the tendon is pulled under tension T-1, while the other end remains under tension T-2. As movement approaches, the differential element of the tendon is subjected to two tensions T+dT and T on both sides, a normal force dN, and a frictional force df-s = μ-cdN, where μ-c is the Columb coefficient of friction. The forces are generated by the balance of the tangential and normal components.

[0160]

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

[0162]

[0163] When there is no rolling friction, the pulley is stationary and does not rotate. It passively maintains tension, T-2. This is entirely determined by the active pulling force T-1, the coefficient of friction μ-c, and the contact angle β. The tension loss from T-1 to T-2 is due to the frictional force fs applied along the contact path.

[0164]

[0165] The cable pulley friction model derived from the equations does not consider the case where the relative velocity is zero. When the velocity direction changes, the frictional force is discontinuous. It cannot handle frictional changes during the reverse motion of the joint. The LuGre model is a dynamic friction model that more realistically describes the changes in friction.

[0166]

[0167] Equation (33) is the basic expression of the LuGre model, where f is the frictional force, z is the state variable representing the average bristle deflection, and v is the relative velocity between the two contact 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 maximum static friction, respectively. ,vs. is the Stribeck velocity.

[0168] The cable and pulley are in linear contact, and the calculation of friction requires integration along the pulley path. To extend the LuGre model to a line-contact kinetic friction model, some improvements were made to the LuGre model presented in [reference needed]. The basic idea is to treat friction as a whole, combining it with the cable-pulley static friction model to obtain equivalent LuGre model parameters, thereby extending the LuGre friction model into a kinetic friction model suitable for calculating cable-pulley friction. We made some improvements, transforming 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 coefficient of friction 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 obtain:

[0176]

[0177] Parameters θ and v are related to the pulley-cable configuration and motor drive, with emphasis on the LuGre model parameters: μs, μc, and vs. . Given that pulley-cable systems typically use metal cables and operate at relatively low speeds, the ranges for μs and μc 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 exhibits an approximately linear relationship with parameter variations, consistently remaining above 0.8. This analysis demonstrates that the model remains robust and reliable despite the influence of LuGre model parameters μs, μc, and vs. , thus validating its applicability and effectiveness in accurately representing the tribodynamics of pulley-cable systems.

[0178] Step 4.2: Model using a 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 α = r p / (f d d), and then we 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 considering both the sliding friction between the pulley and the cable and the rolling friction of the bearing, the following moment balance equation is satisfied:

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

[0187] The actual tension transmission coefficient η satisfies:

[0188] η r ≤η<η d (43)

[0189] Different weighting factors were assigned to different friction effects, and ρ was defined as the sliding friction influence coefficient. η was defined as ρη. d +(1-ρ)η r Different ρ values ​​can be selected to simulate the change in friction coefficient according to different stress states.

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

[0191] It is foreseeable that if a rolling joint is used to construct a multi-degree-of-freedom robotic arm, the number of moving and fixed pulleys in the entire system will be many times that of a traditional cable-driven mechanism. Therefore, it is necessary to consider the overall pulley effect. Simultaneously, 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. This invention proposes a nodal element tension distribution model, which can simultaneously consider the preload and friction of independent cable systems, using the tension transfer ratio as an iterative variable. Knowing any node or element parameter, it can iteratively solve other problems related to pulley and cable forces, displacements, and deformations. The solution block diagram of the entire system is as follows: Figure 9 As shown, this invention mainly considers the tension distribution model solution and the joint dynamics model for the integral solution.

[0192] The present invention and its embodiments have been described above. This description is not restrictive, and the embodiments shown throughout are only one of the embodiments of the present invention. The actual structure is not limited to this. In conclusion, if those skilled in the art are inspired by this description and design similar structures and embodiments without departing from the spirit of the present invention, they should all fall within the protection scope of the present invention.

Claims

1. A method for modeling tension distribution considering friction and preload, characterized in that, Includes the following steps: Step 1: Construct 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: Clarify the characteristics and problems of rolling joints; Step 2: Establish the kinematic and dynamic model of the rolling joint; Step 2.1: Determine the specific configuration of the rolling joint system; Step 2.2: Clarify the kinematic characteristics of the rolling joint; Step 2.3: Determine the dynamic characteristics of the rolling joint; Step 3: Establish the nodal element model for friction and preload; Step 3.1: Model the forces at the cable unit and pulley node; Step 3.2: Modeling the tension transfer of the cable pulley; Step 4: Establish friction models for sliding and rolling contacts; Step 4.1: Model using a sliding friction model; Step 4.2: Model using a rolling friction model; Step 5: Based on the numerical solution of the rolling joint characteristics and tension distribution model, establish the tension distribution model of the nodal elements. By using the pulley node force, rope element force, and rolling node in the rope drive system, the preload, friction effect, pulley node parameters, and rope element parameters are calculated through the node element tension distribution model. The rope length, joint angle, wrapping angle, and tension difference are calculated through kinematics and dynamics. Rope drive systems include open-chain and closed-chain configurations; The nodal element tension distribution model can simultaneously consider the preload and friction of an independent rope drive system, use the tension transfer ratio as an iterative variable to calculate any nodal or element parameters, and can iteratively solve other pulley and cable force, displacement, and deformation problems.

2. The tension distribution modeling method considering friction and preload according to claim 1, characterized in that: The cable unit consists of a continuous cable elastically deformed by multiple intermediate pulleys fixed to a support. It includes n nodes, n-1 cables, and n-2 pulleys. The transfer length of each pulley is denoted as si (i = 2, ..., n-1). Cable tension is generated through cable deformation and movement between adjacent pulley groups, with the corresponding tension being t. 0,i =k (i) (u 0,i +s 0,i -u 0,i+1 -s 0,i+1 By establishing tension distribution and kinematic models, the relationship between elastic force, pulley node displacement and cable length variation in each unit can be determined.

3. The tension distribution modeling method considering friction and preload according to claim 1, characterized in that: The tension transfer model of the cable pulley is based on the relative motion between the cable and the pulley when 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 can be inferred when the tension, preload and displacement of the end cable at any cable position are known.

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

5. The tension distribution modeling method considering friction and preload according to claim 1, characterized in that: The sliding friction model treats friction as a whole, combining it with the static friction model of the cable wheel to obtain equivalent LuGre model parameters. The LuGre model is then extended to a dynamic friction model suitable for calculating cable wheel friction. The calculation formula is as follows:

6. The tension distribution modeling method considering friction and preload according to claim 5, characterized in that: When using a rolling friction model, the actual tension transmission coefficient η satisfies η r ≤η<η d The coefficient of influence of sliding friction is ρ, η = ρη d +(1-ρ)η r The change in the friction coefficient can be simulated by using the ρ value according to different stress states.

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