Flexible finger frictional contact collision dynamic response simulation method based on absolute node coordinate method

Abstract

The invention discloses a flexible finger friction contact collision dynamics simulation method based on an absolute node coordinate method, which comprises the following steps: simplifying a finger of a flexible manipulator into a connecting rod system, taking a tail end knuckle as a flexible beam for consideration, and carrying out unit division on the flexible beam system based on the absolute node coordinate method; a contact collision process is described by using an additional constraint method, meanwhile, switching of a viscous state and a sliding state is considered, a friction-containing flexible finger contact collision dynamic model is established, and a dynamic equation of a system is solved by adopting a double-circulation implicit integral algorithm; and obtaining a curve graph that the flexible finger tail end displacement changes along with time and the finger tail end contact force changes along with time. The invention provides a new frictional contact collision dynamic model for a flexible manipulator system, and provides more perfect data information and image information for researchers in the field.

Description

Technical Field

[0001] The present invention belongs to the field of contact and collision of flexible multi-body system dynamics, and in particular is a flexible finger friction contact and collision dynamic response simulation method based on an absolute node coordinate method. Background Art

[0002] When a flexible manipulator grasps a workpiece, its flexible fingers inevitably experience contact and collision with the workpiece. This grasping action relies on friction between the finger surface and the object being grasped, so the role of friction must be considered when studying this contact and collision process. Furthermore, considering that the flexible fingers may experience significant deformation when subjected to force, the dynamics of the contact interface during contact and collision become even more complex. Existing research on the dynamics of frictional contact and collision of flexible fingers lacks precise and comprehensive descriptions and calculations of friction, and most flexible finger modeling methods are only applicable to situations where the manipulator's fingers experience small deformations. If further breakthroughs are made in the study of frictional contact and collision dynamics of flexible fingers, large computational errors will persist in the structural design and improvement of flexible manipulators and manipulators, as well as in the optimization of their motion control strategies. Therefore, studying the dynamics of frictional contact and collision of flexible fingers has a crucial engineering background and practical significance.

[0003] The dynamic evolution of stress in the contact zone, characterized by the coexistence of motion and deformation, lies at the intersection of solid mechanics and computational multibody dynamics. Currently, research into its evolutionary mechanisms is in its infancy, and understanding of its underlying principles is still insufficient. Therefore, understanding the dynamic behavior of two dissimilar materials at the interface is a core scientific issue in finger system design, and is of great significance to the structural and control design of biomimetic manipulators. Summary of the Invention

[0004] The present invention aims to provide a simulation method for the dynamic response of flexible finger friction contact collision based on the absolute node coordinate method.

[0005] The technical solution for achieving the purpose of the present invention is: a method for simulating the dynamic response of friction contact collision of a flexible finger based on the absolute node coordinate method, comprising the following steps:

[0006] Step 1: Simplify the fingers of the flexible manipulator into a flexible beam system connected by hinges, establish a physical model of the friction, contact and collision of the flexible fingers, which includes a flexible finger model and a rigid collision plane, and set the geometric parameters, material parameters, mesh parameters of the flexible fingers, and motion parameters applied to the model;

[0007] Step 2: Based on the absolute nodal coordinate method, the one-dimensional two-node Euler beam element improved by the selective reduced integration method is used to describe and discretize the coordinates of the flexible finger system, and the kinetic energy and mass matrix of the element are calculated. The flexible beam is considered as an Euler-Bernoulli beam, and the elastic force matrix of the element is derived through the theory of continuum mechanics. The generalized external force matrix of the beam element, such as concentrated force, uniformly distributed force, and external torque, is derived based on the principle of virtual work. Finally, the dynamic equations of the flexible finger system in a collision-free state are assembled.

[0008] Step 3: Based on the additional constraint method, the contact collision process is described. According to the determination of the stick-slip state, the Lagrange multiplier is introduced at the contact node position to release the constraint. According to the Lagrange equation and the principle of virtual work, the contact collision dynamics equations of the flexible finger system in the stick-slip state and the sliding state are obtained respectively.

[0009] Step 4: Analyze the friction calculation methods in the sticky and sliding states respectively. In the sticky state, add a tangential constraint. In the sliding state, describe the sliding friction force using the Coulomb friction model and the LuGre friction model respectively. Substitute them into the contact collision dynamics equation of the flexible finger system to obtain the sticky-sliding friction contact collision dynamics equation of the flexible finger system. By solving the initial collision conditions, avoid the motion incoordination caused by the velocity step.

[0010] Step 5: Use the double-loop implicit integration method to solve the stick-sliding friction contact collision dynamics equation of the flexible finger system. Through iterative calculation, obtain the displacement, velocity, acceleration of each unit node and the constraint force of the end node at each time step. Visualize the obtained data to obtain the motion change image of the finger system model over time during the flexible manipulator grasping the object under different working conditions, the curve graph of the finger end displacement over time, and the curve graph of the normal contact force and tangential contact force over time.

[0011] A flexible finger friction contact collision dynamics response simulation system based on the absolute node coordinate method is provided. Based on the flexible finger friction contact collision dynamics response simulation method based on the absolute node coordinate method, the flexible finger friction contact collision dynamics simulation based on the absolute node coordinate method is realized.

[0012] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, the flexible finger friction contact collision dynamics simulation based on the absolute node coordinate method is realized based on the flexible finger friction contact collision dynamics response simulation method based on the absolute node coordinate method.

[0013] A computer-readable storage medium stores a computer program. When the computer program is executed by a processor, the flexible finger friction contact collision dynamics response simulation method based on the absolute node coordinate method is used to implement the flexible finger friction contact collision dynamics simulation based on the absolute node coordinate method.

[0014] Compared with the prior art, the present invention has the following significant advantages: (1) The absolute node coordinate method is used to describe and discretize the coordinates of the flexible finger. The absolute node coordinate method has good unit discretization characteristics, can better adapt to flexible bodies of different shapes and structures, and realize accurate modeling of large deformations of complex structures. It not only simplifies the calculation of constraint equations and generalized forces, but also can effectively deal with geometric nonlinear problems while accurately describing the motion of rigid bodies. When dealing with large deformation problems, it can improve the calculation efficiency while ensuring the calculation accuracy. (2) The use of the additional constraint method to model the contact collision process can accurately describe the dynamic behavior of the system during the collision process, avoid the assumption that the collision bodies are embedded in each other, and conform to the actual situation that the two objects do not penetrate each other when they are in contact. (3) The switching between the sticky state and the sliding state caused by friction during the contact process is taken into account, simulating the real physical process, and can better study the dynamic characteristics of the flexible manipulator when it relies on friction to complete the grasping action. (4) The double-loop implicit numerical integration method is used to solve the dynamic equations. This method has unique advantages in solving differential-algebraic equations. The calculation results of each time step have double accuracy guarantees, and can complete the solution work with high quality and high efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 Flowchart of the present invention.

[0016] Figure 2 Schematic diagram of the cross section of a flexible manipulator grasping an object.

[0017] Figure 3 Schematic diagram of the simplified model of the flexible finger and the established coordinate system.

[0018] Figure 4 Schematic diagram of the flexible beam element.

[0019] Figure 5 This is the flow chart for determining and switching the collision state of stick-sliding contact.

[0020] Figure 6 This is the flow chart of the double-loop algorithm structure.

[0021] Figure 7 These are images of the finger system model's motion changes over time during the flexible manipulator's grasping process under different working conditions, graphs of the finger tip displacement changing over time, and graphs of the normal contact force and tangential contact force changing over time. DETAILED DESCRIPTION

[0022] In order to make the purpose, technical solutions and advantages of this application more clear, the following further describes this application in detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not intended to limit this application.

[0023] like Figure 1 As shown, the flexible finger friction contact collision dynamic response simulation method based on the absolute node coordinate method of the present invention includes the following steps:

[0024] Step 1: Establish a physical model of friction, contact and collision of flexible fingers, set the geometric parameters, material parameters, mesh parameters of the flexible finger and the motion parameters applied to the model. The specific method is as follows:

[0025] (1) Physical model of flexible finger friction contact collision

[0026] The manipulator finger has two knuckles, which are connected to the palm and each other by hinge joints. A motor is installed at the hinge joint to provide driving torque to enable it to rotate. Below it is the surface of the grasped object. Compared with the flexible finger, the surface of the grasped object is regarded as a rigid plane. The finger moves under the action of gravity or driven by the motor. When the end of the finger is consistent with the vertical coordinate of the plane, the contact collision process is considered to have begun.

[0027] (2) Geometric parameters

[0028] The length of the flexible finger is L, the contact plane is at a height h0 below its initial position, and the cross section of the finger is rectangular with a width and height of w and h respectively.

[0029] (3) Material parameters

[0030] Material parameters of the flexible finger: elastic modulus E, density ρ;

[0031] Material parameters of the rigid plane: the friction coefficient is f s ;

[0032] (4) Grid parameters

[0033] Mesh parameters of the flexible system: Divide the distal knuckle of the flexible finger into n units along the x-direction, with the length of each unit being l = L / n;

[0034] (5) Motion parameters

[0035] The gravitational acceleration during vertical fall is g, and the driving torque at the finger joint during horizontal rotation is τ.

[0036] Step 2: Based on the absolute nodal coordinate method, the one-dimensional two-node Euler beam element improved by the selective reduced integration method is used to describe and discretize the coordinates of the flexible finger system, and the kinetic energy and mass matrix of the element are calculated. The flexible beam is considered as an Euler-Bernoulli beam, and the elastic force matrix of the element is derived through the theory of continuum mechanics. The generalized external force matrix of the beam element, such as concentrated force, uniformly distributed force, and external torque, is derived based on the principle of virtual work. Finally, the dynamic equation of the flexible finger system in the non-collision state is assembled. The specific method is as follows:

[0037] (1) Based on the absolute nodal coordinate method, a one-dimensional two-node Euler beam element is selected to describe and discretize the system, and the kinetic energy and mass matrix of the element are calculated.

[0038] Based on the modeling theory of the absolute node coordinate method, the flexible finger system is discretized according to the grid parameters, the finger end joint is considered as a two-dimensional flexible beam, and the geometric center of the left end joint of the beam is used as the origin to establish a Figure 3 The Cartesian coordinate system shown.

[0039] Discretize the two-dimensional flexible beam into one-dimensional two-node Euler beam elements, such as Figure 4 Shown is a deformed flexible beam element in the absolute coordinate system.

[0040] The global position vector of any point P on the midline of the beam element can be expressed as:

[0041] r=[r1 r2] T =S(x)q e (t)(1) Where S(x) is the element shape function matrix, which is used to perform displacement interpolation within the element:

[0042]

[0043] The expressions of the components in the unit shape function matrix are as follows:

[0044]

[0045] Where ξ=x / l, x is the coordinate of any point P on the center axis of the beam unit in the beam unit coordinate system when the flexible beam is not deformed, and l is the length of the beam unit when the flexible beam is not deformed.

[0046] The beam element has two nodes, A and B. Each node has four node coordinates, including two position coordinates and two slope coordinates of the node. Therefore, the absolute coordinate array of the element is as follows:

[0047]

[0048] By taking the time derivative of formula (1), we can get the absolute velocity of any point on the beam as follows:

[0049]

[0050] Therefore, the kinetic energy expression of the beam element is:

[0051]

[0052] Where V is the volume of the beam element, ρ is the material density of the beam, and the element mass matrix M is e It can be expressed as:

[0053]

[0054] (2) Considering the flexible beam as an Euler-Bernoulli beam, the elastic force matrix of the unit is derived through the theory of continuum mechanics.

[0055] Assume that the cross section perpendicular to the beam axis in the undeformed state remains flat and perpendicular to the deformed beam axis after the beam is deformed. That is, the flexible beam is considered as an Euler-Bernoulli beam. The work done by the elastic force of the beam element is:

[0056] δW ke =-∫ V δε x σ x dV=-∫ V Eδε x ε x dV (8)

[0057] Where E is the elastic modulus of the material, V is the volume of the beam element, and ε x is the strain at any point other than the midline:

[0058] ε x =ε x0 -yκ (9) where ε x0 is the strain at the corresponding point on the midline, and κ is the curvature.

[0059] Substituting formula (9) into formula (8), we have:

[0060]

[0061] Assuming that the beam element is symmetrical about the midline, then ∫ A ydA=0,I z =∫ A y 2 dA is the moment of inertia of the cross section.

[0062] According to the theory of continuum mechanics, ε x0 can be expressed as the Green-Lagrange strain tensor:

[0063]

[0064] The exact expression for the curvature of a beam element is:

[0065]

[0066] in,

[0067]

[0068] Substituting formula (1) into (11) and (12) respectively, we obtain:

[0069]

[0070] By varying equation (14), we can get:

[0071] δε x0 =δq e T S' T S'q e (16)

[0072] Square both sides of equation (15) to obtain:

[0073]

[0074] According to the Lagrange identity, equation (17) is transformed as follows:

[0075]

[0076] By varying equation (18), we can get:

[0077]

[0078] Substituting equations (14), (16) and (19) into equation (10), the virtual work done by the elastic force of the beam element is:

[0079] δW ke =δq e T Q ke (20)

[0080] Among them, Q ke is the generalized elastic force matrix of the beam element, which is expressed as follows:

[0081]

[0082] (3) The generalized external force matrix of the beam unit, such as the concentrated force, uniformly distributed force, and external torque, is derived from the principle of virtual work, and the dynamic equation of the flexible finger system in the non-collision state is finally assembled.

[0083] For concentrated force, when there is an external force matrix F=(F x F y ) T When acting on point P of the beam element, the principle of virtual work yields:

[0084] δW pe =δr p T F=δq e T S T F=δq e T Q pe (twenty two)

[0085] Among them, S is the function of the unit coordinates of point P, r p is the global position coordinate of point P.

[0086] Therefore, the generalized force form of concentrated force is:

[0087] Q pe =S T F(23)

[0088] For uniform force, taking gravity along the negative direction of y axis as an example, the acceleration due to gravity can be expressed as g=(g x g y ) T , the unit gravity matrix is a constant matrix, then the gravity matrix of the beam element is:

[0089] Q ge =∫ V ρS T gdV (24)

[0090] As for the external moment, when there is a moment τ acting on the first node of the i-th element of the beam, its virtual work can be expressed as:

[0091] δW τe =τδθ (25)

[0092] Where θ is the cross-sectional rotation angle of the beam element.

[0093] The coordinate rotation matrix can be obtained:

[0094]

[0095] in,

[0096]

[0097] Then the virtual angle generated by the external torque is:

[0098]

[0099] The generalized moment of the i-beam element node can be expressed as:

[0100]

[0101] By linearly superimposing all three generalized external force matrices, we can obtain the generalized external force matrix Q of the unit. fe .

[0102] Q fe =Q pe +Q ge +Q τe (31)

[0103] Assume B e is the transformation matrix of the beam element node coordinate array corresponding to the overall node coordinate array, then the beam element node coordinate array q e The transformation matrix B can be used to connect the beam's overall node coordinate array q e To convert between:

[0104] q e =B e q(32)

[0105] The calculated unit mass matrix M e , unit generalized elastic force matrix Q ke and unit generalized external force matrix Q fe Assemble them separately to obtain the overall mass matrix M and overall elastic force matrix Q of the beam k and the overall external force matrix Q f :

[0106]

[0107] According to the overall mass matrix M obtained above, the overall elastic force matrix Q k and the overall external force matrix Q f , the dynamic equation of the system in the unconstrained state is established as follows:

[0108]

[0109] Step 3: Based on the additional constraint method, the contact collision process is described. According to the determination of the stick-slip state, the Lagrange multiplier is introduced at the contact node position to release the constraint. According to the Lagrange equation and the principle of virtual work, the contact collision dynamics equations of the flexible finger system in the sticky state and the sliding state are obtained respectively. The specific method is as follows:

[0110] The contact collision process is modeled using the additional constraint method. Before contact occurs, constraints are only added at the knuckle joints. After contact occurs, Lagrange multipliers are introduced at the contact nodes to release the constraints based on the stick-slip state. The general form of the constraint equation is:

[0111] Φ(q,t)=0 (37)

[0112] According to the Lagrange equation and the principle of virtual work, the dynamic equation of the flexible beam system is obtained:

[0113]

[0114] Where q is the generalized coordinate of the system, Φ q is the derivative matrix of the constraint equation Φ(q,t) with respect to generalized coordinates, λ is the Lagrange multiplier matrix corresponding to the constraint equation, and its components in each direction are the magnitudes of the corresponding contact forces.

[0115] For the stick-slip contact collision process, the dynamic equations are different when sticking and sliding. When in the sticky state, there are constraints on both the normal and tangential directions of the finger tip:

[0116]

[0117] Among them, h0 is the vertical coordinate of the sticking position, and x0 is the horizontal coordinate of the sticking position. Since each slide will change the horizontal coordinate of the finger end, x0 needs to be recalculated at the initial moment of each sticking contact. N and Φ T All are row arrays:

[0118]

[0119] When in the sliding state, the finger end is only constrained in the normal direction:

[0120] Φ(q,t)=Φ N q-h0 (42)

[0121] Among them, h0 is the vertical coordinate of the sliding position, which is actually the same as the vertical coordinate of the sticking position, both of which are the vertical coordinates of the contact collision plane position, and Φ N The same remains unchanged.

[0122] At the same time, the generalized sliding friction force array needs to be added to the dynamic equation of the flexible beam system:

[0123]

[0124] in,

[0125] Q s =Φ T T Fs (44)

[0126] F s is the sliding friction, and its calculation method depends on the friction model used.

[0127] When solving dynamics, it is necessary to determine the contact state of the current time step. The specific process is as follows: Figure 5 shown.

[0128] Contact determination is required at the beginning of each time step. If no contact occurs, the dynamic equations for the collision-free state are solved. If contact occurs, the contact position is first assumed to be in a sticky state. After solving the dynamic equations for the sticky state, the calculated result of the tangential contact force is compared with the maximum static friction force to determine whether the contact position is actually in a sticky state. If the tangential contact force is less than the maximum static friction force, the contact position is indeed in a sticky state. The current calculation result is stored and the calculation of the next time step begins directly. Conversely, if the contact is actually in a sliding state, the constraints and generalized external force matrix corresponding to the sliding state are added, the sliding state dynamic equations are solved, and the velocity, acceleration, and normal and tangential contact forces for that time step are recalculated.

[0129] Step 4: Analyze the friction calculation methods in the sticky state and sliding state respectively. Add tangential constraint in the sticky state, and describe the sliding friction force in the sliding state by using Coulomb friction model and LuGre friction model respectively. Then substitute them into the contact collision dynamics equation of the flexible finger system to obtain the sticky-sliding friction contact collision dynamics equation of the flexible finger system. By solving the initial collision conditions, the motion incoordination caused by velocity step is avoided. The specific method is as follows:

[0130] (1) Analyze the friction calculation methods in the viscous state and the sliding state respectively

[0131] When in a viscous state, the system has tangential constraints, and the static friction force is actually the corresponding tangential constraint force, while the friction force here mainly refers to the sliding friction force.

[0132] If the Coulomb friction model is used to describe the friction force, the friction force at the contact point can be expressed as follows:

[0133] F s =-sgn(v τ )μ d λ N (45)

[0134] Among them, v τ is the tangential velocity of the contact point, μ d is the coefficient of kinetic friction, λ N is the normal contact force at the contact point.

[0135] If the LuGre friction model is used to describe friction, the real-time friction coefficient calculation needs to be performed at every moment:

[0136]

[0137] Among them, σ0 represents the bristle stiffness coefficient, σ1 represents the bristle damping coefficient, σ2 represents the viscous friction coefficient, and z and are the average bristle deformation and deformation rate of the contact pair respectively:

[0138]

[0139] g(v τ ) is about v τ The function is always greater than 0, describing the stribeck phenomenon, and its expression is as follows:

[0140]

[0141] Among them, μ s is the static friction coefficient, v s is the stribeck rate.

[0142] When solving practical problems, we can consider the specific situation and needs and choose a suitable friction model to calculate the dynamic friction force.

[0143] (2) Avoid motion incoordination caused by velocity step by solving the initial collision conditions

[0144] Directly adding contact constraints at the initial moment of each collision will lead to inconsistent motion, making subsequent solutions difficult. To avoid this, the impulse-momentum method is used to solve the initial conditions of the collision at the initial moment of the collision.

[0145] The collision dynamics equation of the impulse-momentum method of the system is as follows:

[0146]

[0147] Where, is the generalized velocity increment array caused by the collision, I is the collision impulse, e is the restitution coefficient, v F1 、v F2 are the velocity vector arrays of the two colliding points at the moment before the collision, Δv F1 , Δv F2 are the velocity increment vector arrays of the two colliding points during the collision process, and D is the array representing the position of the impulse.

[0148] For the dynamic model established in this paper, the position of the collision plane does not change, and Equation (2.49) can be expressed as:

[0149]

[0150] Where v0 is the velocity of the collision point along the collision normal at the moment before the collision. Assume that the finger tip collides with the rigid surface in a completely inelastic manner, i.e., the coefficient of restitution e = 0. At the moment of collision, the generalized coordinates of the system are continuous and smooth, but the generalized velocity is not smooth. Therefore, the following motion state transformation is performed:

[0151]

[0152] From then on until the contact between the collision points ends, their relative displacement, relative velocity and relative acceleration are always zero.

[0153] Step 5: Use the double-loop implicit integration method to solve the stick-sliding friction contact collision dynamics equation of the flexible finger system. Through iterative calculation, obtain the displacement, velocity, acceleration of each unit node and the constraint force of the end node at each time step. Visualize the obtained data to obtain the motion change image of the finger system model over time during the flexible manipulator grasping the object under different working conditions, the curve graph of the finger end displacement over time, and the curve graph of the normal contact force and tangential contact force over time. The specific method is as follows:

[0154] (1) Constructing the state space based on LU decomposition

[0155] First, the Jacobian matrix of the constraint equation Φ q Perform LU decomposition:

[0156] P1Φ q P2=LU(52)

[0157] Assume Φ q It is an s×n matrix, the constraint matrix has s rows, and there are s non-independent variables, resulting in an s×s row transformation matrix P1 and an n×n column transformation matrix P2.

[0158] Then substitute the obtained matrix transformation into the system coordinate q and rearrange it to obtain the ordinary differential equations in the state space:

[0159]

[0160] Separate the rearranged ordinary differential equations to obtain:

[0161]

[0162]

[0163] By transforming Equation (55), the Lagrange multiplier can be expressed as:

[0164]

[0165] Substituting equation (56) into equation (54) eliminates the Lagrange multiplier:

[0166]

[0167] Among them, the dependent acceleration can be expressed in terms of independent acceleration as:

[0168]

[0169] Substituting Equation (58) into Equation (57) can eliminate the dependent acceleration:

[0170]

[0171] Let the matrix Arranging formula (54) yields:

[0172]

[0173] Equation (60) is a set of ordinary differential equations in the state space defined based on LU decomposition.

[0174] (2) Solving using a double-loop algorithm structure

[0175] like Figure 6 As shown in the figure, at the beginning of the calculation of each time step, the acceleration is first solved in the outer loop based on the position and velocity at the previous moment, and then the displacement and velocity at the next moment are estimated, and the acceleration at the next moment is estimated based on this, and then the differential equation is integrated.

[0176] After that, the inner loop is entered to solve the constraint equations, updating the velocity and position coordinates each time the loop is repeated until the inner loop calculation results meet the accuracy requirements, and then exiting the inner loop. Finally, the acceleration and Lagrange multipliers are updated until the outer loop calculation results meet the accuracy requirements.

[0177] This algorithm structure can ensure that the calculation results of each time step have double accuracy.

[0178] (3) Data visualization

[0179] The time array, displacement array, normal contact force array and tangential contact force array are visualized in MATLAB to obtain the motion change image of the entire flexible finger system model over time, the curve graph of the finger end displacement changing with time, and the curve graph of the normal contact force and tangential contact force changing with time.

[0180] Example

[0181] In order to verify the effectiveness of the solution of the present invention, the following examples are carried out for verification.

[0182] This embodiment simulates the dynamic response of the flexible finger friction contact collision based on the absolute node coordinate method. The specific method is as follows:

[0183] Step 1. In this embodiment, under the first working condition, a horizontal plane at height 0 is set as the collision plane. The flexible finger is initially located within the horizontal plane at height h0. It is then allowed to freely fall from its initial position within the vertical plane. Contact occurs when the tip of the finger coincides with the collision plane. The parameters of the flexible finger system are shown in Table 1 (initial parameters are in parentheses).

[0184] Table 1 Parameters of the free-falling flexible finger system under gravity

[0185]

[0186] In the second operating condition, the flexible finger moves in a horizontal plane, eliminating the effect of gravity. The initial position h0 is corrected to a minimum, meaning that contact between the finger and the grasped object begins very soon after the calculation begins. Material and volume parameters are still selected according to Table 1, and the contact surface friction coefficient is always maintained at 0.3.

[0187] Step 2. Based on the absolute nodal coordinate method, the one-dimensional two-node Euler beam element improved by the selective reduced integration method is used to describe and discretize the coordinates of the flexible finger system, and the kinetic energy and mass matrix of the element are calculated; the flexible beam is considered as an Euler-Bernoulli beam, and the elastic force matrix of the element is derived through the theory of continuum mechanics. The generalized external force matrix of the beam element, such as concentrated force, uniformly distributed force and external torque, is derived from the principle of virtual work. Finally, the dynamic equation of the flexible finger system in a collision-free state is assembled and the process proceeds to step 3.

[0188] Step 3: Describe the contact collision process based on the additional constraint method. According to the determination of the stick-slip state, introduce the Lagrange multiplier at the contact node position to release the constraint. According to the Lagrange equation and the principle of virtual work, obtain the contact collision dynamics equation of the flexible finger system in the sticky state and the sliding state respectively, and then proceed to step 4.

[0189] Step 4: Analyze the friction calculation methods in the sticky and sliding states respectively. In the sticky state, add a tangential constraint, and in the sliding state, use the Coulomb friction model to describe the sliding friction. Substitute it into the contact collision dynamics equation of the flexible finger system to obtain the sticky-sliding friction contact collision dynamics equation of the flexible finger system. By solving the initial collision condition, avoid the motion incoordination caused by the velocity step, and then proceed to step 5.

[0190] Step 5. Change the elastic modulus of the flexible finger under the first working condition and the driving torque at the flexible finger joint under the second working condition, and use the double-loop implicit integration method to solve the viscous-sliding friction contact collision dynamics equation of the flexible finger system. Through iterative calculation, the displacement, velocity, acceleration of each unit node and the constraint force of the end node at each time step are obtained. The obtained data are visualized to obtain the motion change image of the finger system model over time in the process of grasping objects under the two working conditions, the curve graph of the finger end displacement over time, and the curve graph of the normal contact force and tangential contact force over time.

[0191] like Figure 7 As shown in (a), the configuration diagram of the flexible finger under the original parameters during the falling process is drawn. Since one end of the flexible finger is constrained by the joint and the other parts fall freely, it actually produces a large deformation before the contact collision occurs. After the contact collision occurs, since the end is also constrained, the overall deformation trend of the finger will change. During the calculation process, the deformation of the flexible finger is very large, and after the contact ends, the end will bounce upward slightly, and after a few seconds it will fall again to cause a second contact collision.

[0192] Figure 7 (b) shows a comparison of the longitudinal displacements of the flexible finger tips with different elastic moduli under the first working condition. The contact collision occurs at around 0.175s. In the free fall stage before the contact collision, the displacements of the flexible finger tips with different elastic moduli are different, and the final collision moments of the flexible finger and the rigid plane are also different. The finger tip with a larger elastic modulus falls faster at first, but the moment of contact and collision with the plane is the latest. The finger with a smaller elastic modulus is just the opposite. It falls slower at first but contacts and collides with the plane the earliest. This is because the fingers with different elastic moduli undergo different deformations, which leads to different changes in their center of gravity positions, and the laws of the end movement are naturally different.

[0193] Figure 7(c) shows a comparison of the calculated normal contact force at the end of the flexible finger under the first working condition, with different elastic moduli. The normal contact force fluctuates over time, but generally tends to increase. This clearly indicates that sliding is the primary process during the contact collision, with a relatively short period of sticking, or rather, that sticking occurs at some point during the continuous sliding. For example, comparing the calculated results with elastic moduli of 0.5 MPa and 1.0 MPa, it is found that an increase in the elastic modulus leads to an increase in the contact force, while the result with an elastic modulus of 1.5 MPa shows a delayed contact collision. This not only indicates that the increased elastic modulus reduces the deformation of the finger during free fall, but also suggests that if the timing is also measured from the moment of collision, the contact force at the same moment is still greater than the result with an elastic modulus of 1.0 MPa. This indicates that the elastic modulus of the flexible finger should not be too small, as otherwise, a low tangential contact force may result in only sliding and a failed grasp. However, it should also not be too large, as excessive normal contact force may induce plastic deformation or even damage to the structure.

[0194] Figure 7 (d) and (e) show the comparison of the longitudinal displacement and tangential contact force of the flexible finger tip under different motor drive torques under the second working condition, respectively. It can be observed that within the first 0.03 seconds, the finger tip with a drive torque of 0.1 N·m maintains continuous contact with the grasped object. However, after 0.03 seconds, the finger tip tilts up due to deformation in other parts of the finger and is no longer in contact. The finger tip with a drive torque of 0.5 N·m briefly contacts the grasped object and then bounces up. However, due to the continued action of the torque, the finger tip will contact the object again within a short period of time and then bounce up again. This process may be repeated many times. The deformation and vibration of the flexible finger throughout the process will cause its tip to shake slightly while in the air.

[0195] Obviously, when the driving torque is 0.5N·m, the maximum normal contact force and the maximum tangential contact force are both relatively large. However, when a driving torque of 0.1N·m is applied, the contact force is more stable and easier to achieve sustained action. This is because a large contact force will also cause a large deformation of the finger, and the tip will tend to tilt up more quickly. In other words, the larger the motor driving torque, the better. A larger motor driving torque can obtain a larger normal contact force and tangential contact force at a certain moment, but it will reduce the duration of the contact force. Therefore, it is very important to comprehensively consider the requirements of grasping force and grasping time. In actual engineering, it is necessary to select the appropriate motor driving torque according to different working conditions.

[0196] In summary, based on previous research, the present invention has carried out dynamic modeling of the friction contact and collision process of the flexible finger system, and used MATLAB to write an algorithm program according to the obtained dynamic equation. It can accurately output a series of dynamic responses of each unit node of the structure before and after the contact, including velocity, displacement, acceleration and end contact force. At the same time, the changes in the overall motion trajectory of the system can be intuitively observed, verifying the correctness of the dynamic model, and calculating the changes in the end displacement and end contact force of the flexible finger with different parameters under different working conditions, providing researchers in this field with more complete data and image data, which is of great value for the study of the friction contact and collision problems of flexible fingers.

[0197] The technical features of the above embodiments can be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0198] The above-described embodiments merely represent several implementation methods of the present application. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the present application. It should be noted that a person of ordinary skill in the art may make various modifications and improvements without departing from the spirit of the present application, and these modifications and improvements fall within the scope of protection of the present application. Therefore, the scope of protection of the present application shall be determined by the appended claims.

Claims

1. A simulation method for the dynamic response of flexible finger friction contact collision based on the absolute node coordinate method, characterized in that: The following steps are involved: Step 1: Establish a physical model of friction, contact and collision of a flexible finger. The physical model includes a flexible finger model and a rigid collision plane, and sets geometric parameters, material parameters, mesh parameters of the flexible finger model, and motion parameters applied to the flexible finger model. Step 2: Based on the absolute nodal coordinate method, the Euler beam element is used to describe and discretize the coordinates of the flexible finger model, and the kinetic energy and mass matrix of the Euler beam element are calculated. The flexible beam is treated as an Euler-Bernoulli beam, and the elastic force matrix of the element is derived through the theory of continuum mechanics. The generalized external force matrix of the beam element, such as concentrated force, uniformly distributed force, and external torque, is derived based on the principle of virtual work. Finally, the dynamic equations of the flexible finger system in a collision-free state are assembled. Step 3: Based on the additional constraint method, the contact collision process is described. According to the determination of the stick-slip state, the Lagrange multiplier is introduced at the contact node position to release the constraint. According to the Lagrange equation and the principle of virtual work, the contact collision dynamics equations of the flexible finger system in the stick-slip state and the sliding state are obtained respectively. Step 4: Determine the friction calculation model in the sticky and sliding states. Add a tangential constraint in the sticky state. In the sliding state, describe the sliding friction force using the Coulomb friction model and the LuGre friction model respectively. Substitute these into the contact collision dynamics equation of the flexible finger system to obtain the sticky-sliding friction contact collision dynamics equation of the flexible finger system. By solving the initial collision conditions, avoid motion incoordination caused by velocity steps. Step 5: Use the double-loop implicit integration method to solve the stick-sliding friction contact collision dynamics equation of the flexible finger system. Through iterative calculation, obtain the displacement, velocity, acceleration of each unit node and the constraint force of the end node at each time step. Visualize the obtained data to obtain the motion change image of the finger system model over time during the flexible manipulator grasping the object under different working conditions, the curve graph of the finger end displacement over time, and the curve graph of the normal contact force and tangential contact force over time.

2. The method for simulating the dynamic response of friction contact collision of a flexible finger based on the absolute node coordinate method according to claim 1 is characterized in that: The physical model of flexible finger friction contact collision is as follows: The physical model of flexible finger friction contact collision includes a flexible finger model and a rigid collision plane; The robot finger has two knuckles, which are connected to the palm and to each other through hinge joints. A motor is installed at the hinge joint to provide driving torque to make it rotate. The robot finger is a flexible finger model; the surface of the grasped object is a rigid collision plane. The finger moves under the action of gravity or driven by a motor. When the vertical coordinate of the finger end is consistent with the rigid plane, the contact collision process is considered to begin.

3. The method for simulating the dynamic response of flexible finger friction contact collision based on the absolute node coordinate method according to claim 1 is characterized in that: The set geometric parameters include the length of the flexible finger L, the contact plane at a height h0 below the initial position of the flexible finger, and the finger cross section is a rectangle with a width and height of w and h respectively; The set material parameters include those of the flexible finger: elastic modulus is E, density is ρ; Material parameters of the rigid plane: the friction coefficient is f s ; The specific method of setting the mesh parameters of the flexible finger model is as follows: the end joint of the flexible finger is divided into n units along the x direction, and the length of each unit is l = L / n; The set motion parameters include: gravitational acceleration during vertical fall is g, and driving torque at the finger joint during horizontal rotation is τ.

4. The method for simulating the dynamic response of friction contact collision of a flexible finger based on the absolute node coordinate method according to claim 1 is characterized in that: The specific method of step 2 is: Step 2.1: Based on the absolute nodal coordinate method, a one-dimensional two-node Euler beam element is used to describe and discretize the flexible finger model, and the kinetic energy and mass matrix of the beam element are calculated; Step 2.2: Treat the flexible beam as an Euler-Bernoulli beam and derive the elastic force matrix of the beam element using the theory of continuum mechanics. Step 2.3: Derive the generalized external force matrix of the beam unit, including the concentrated force, uniformly distributed force, and external moment, based on the principle of virtual work, and assemble them to obtain the dynamic equation of the flexible finger system in a collision-free state.

5. The method for simulating the dynamic response of friction contact collision of a flexible finger based on the absolute node coordinate method according to claim 4 is characterized in that: Based on the absolute nodal coordinate method, a one-dimensional two-node Euler beam element is used to describe and discretize the flexible finger model. The specific method for calculating the kinetic energy and mass matrix of the beam element is as follows: Based on the modeling theory of the absolute node coordinate method, the flexible finger model is discretized according to the grid parameters. The finger joint at the end is regarded as a two-dimensional flexible beam, and a Cartesian coordinate system is established with the geometric center of the left joint of the beam as the origin. Discretize the two-dimensional flexible beam into one-dimensional two-node Euler beam elements; The global position vector of any point P on the midline of the beam element is expressed as: r=[r1 r2] T =S(x)q e (t) Among them, S(x) is the unit shape function matrix; q e is the absolute coordinate array of the beam element; By taking the time derivative of the global position vector of any point P on the centerline of the beam element, the absolute velocity of any point on the beam is obtained as follows: The kinetic energy expression of the beam element is: Where V is the volume of the beam element and ρ is the material density of the beam; Unit mass matrix M e for: Where l is the length of the beam element when the flexible beam is not deformed, and x is the coordinate of any point P on the center axis of the beam element in the beam element coordinate system when the flexible beam is not deformed.

6. The method for simulating the dynamic response of friction contact collision of a flexible finger based on the absolute node coordinate method according to claim 5 is characterized in that: Taking the flexible beam as an Euler-Bernoulli beam, the specific method of deriving the elastic force matrix of the beam element through the theory of continuum mechanics is as follows: Assume that the cross section perpendicular to the beam axis in the undeformed state remains flat and perpendicular to the deformed beam axis after the beam is deformed. That is, the flexible beam is regarded as an Euler-Bernoulli beam. The work done by the elastic force of the beam element is: δW ke =-∫ V Here x e x dV Where E is the elastic modulus of the material, V is the volume of the beam element, and ε x is the strain at any point other than the midline, specifically: e x =e x0 -yk where ε x0 is the strain at the corresponding point on the midline, κ is the curvature, and y is the distance between any point on the non-midline and the corresponding point on the midline; The work done by the elastic force of the beam element is: Where A = w·h, is the cross-sectional area of the beam element, I z is the moment of inertia of the cross section; Assume that the beam element is symmetrical about the central axis, then ∫ A ydA=0,I z =∫ A y 2 dA According to the theory of continuum mechanics, ε x0 As the Green-Lagrange strain tensor: The expression for the curvature of a beam element is: in, Substituting the global position vector of any point P on the midline of the beam element into the expressions of the Green-Lagrange strain tensor and the curvature of the beam element, we obtain: right Perform variation, and we get: δε x0 =δq e T S' T S'q e Will Squaring both sides, we get: According to the Lagrange identity, Perform transformation and variation to obtain: The virtual work done by the elastic force of the beam element is obtained as: δW ke =δq e T Q ke Among them, Q ke is the generalized elastic force matrix of the beam element, which is expressed as follows:

7. The method for simulating the dynamic response of friction contact collision of a flexible finger based on the absolute node coordinate method according to claim 6 is characterized in that: The specific method of deriving the generalized external force matrix of the beam element, such as the concentrated force, uniformly distributed force, and external moment, from the principle of virtual work and assembling them to obtain the dynamic equation of the flexible finger system in the non-collision state is as follows: For concentrated force, when there is an external force matrix F=(F x F y ) T When acting on point P of the beam element, the principle of virtual work yields: δW pe =δr p T F=δq e T S T F=δq e T Q pe Among them, S is the function of the unit coordinates of point P, r p is the global position coordinate of point P; The generalized force form of concentrated force is: Q pe =S T F For uniform force, gravity along the negative direction of y axis, gravitational acceleration is expressed as g=(g x g y ) T , the unit gravity matrix is a constant matrix, then the gravity matrix of the beam element is: Q ge =∫ V ρS T gdV For external moment, when there is a moment τ acting on the first node of the i-th element of the beam, its virtual work can be expressed as: δW τe =τδθ Where θ is the cross-sectional rotation angle of the beam element; The coordinate rotation matrix is obtained: in, Then the virtual angle generated by the external torque is: The generalized moment of the i-beam element node is expressed as: Linearly superimpose all three generalized external force matrices to obtain the generalized external force matrix Q of the unit fe : Q fe =Q pe +Q ge +Q τe Assume B e is the transformation matrix of the beam element node coordinate array corresponding to the overall node coordinate array, then the beam element node coordinate array q e The transformation matrix B is used to connect the beam's overall node coordinate array q e To convert between: q e =B e q The calculated unit mass matrix M e , unit generalized elastic force matrix Q ke and unit generalized external force matrix Q fe Assemble them separately to obtain the overall mass matrix M and overall elastic force matrix Q of the beam k and the overall external force matrix Q f : According to the overall mass matrix M, the overall elastic force matrix Q k and the overall external force matrix Q f , the dynamic equation of the system in the unconstrained state is established as follows:

8. The method for simulating the dynamic response of friction contact collision of a flexible finger based on the absolute node coordinate method according to claim 7 is characterized in that: Step 3 describes the contact collision process based on the additional constraint method. According to the determination of the stick-slip state, the Lagrange multiplier is introduced at the contact node position to release the constraint. According to the Lagrange equation and the principle of virtual work, the contact collision dynamics equations of the flexible finger system in the sticky state and the sliding state are obtained respectively. The specific method is as follows: The contact collision process is modeled using the additional constraint method. Before contact occurs, constraints are only added at the knuckle joints. After contact occurs, Lagrange multipliers are introduced at the contact node positions to release the constraints based on the stick-slip state. The general form of the constraint equation is: Φ(q,t)=0 According to the Lagrange equation and the principle of virtual work, the dynamic equation of the flexible beam system is obtained: Where q is the generalized coordinate of the system, Φ q is the derivative matrix of the constraint equation Φ(q,t) with respect to generalized coordinates, and λ is the Lagrange multiplier matrix corresponding to the constraint equation; For the stick-slip contact collision process, the dynamic equations are different when sticking and sliding. When in the sticky state, there are constraints on both the normal and tangential directions of the finger tip: Where h0 is the vertical coordinate of the sticking position, and x0 is the horizontal coordinate of the sticking position. Since each slide will change the horizontal coordinate of the finger end, x0 needs to be recalculated at the initial moment of each sticking contact; Φ N and Φ T All are row arrays: When in the sliding state, the finger end is only constrained in the normal direction: Φ(q,t)=Φ N q-h0 Among them, h0 is the vertical coordinate of the sliding position, Φ N The same remains unchanged; At the same time, the generalized sliding friction force array is added to the dynamic equation of the flexible beam system: in, Q s =Φ T T F s F s is the sliding friction; Contact judgment is performed at the beginning of each time step. If no contact occurs, the dynamic equations under the collision-free state are solved. If contact occurs, it is first assumed that the contact position is in a sticky state. After solving the dynamic equations under the sticky state, the calculated results of the tangential contact force are compared with the maximum static friction force to determine whether the contact position is actually in a sticky state. If the tangential contact force is less than the maximum static friction force, the contact position is indeed in a sticky state. The current calculation results are stored and the calculation of the next time step is started directly; otherwise, if the contact is actually in a sliding state, the constraints and generalized external force matrix corresponding to the sliding state are added, the sliding state dynamic equations are solved, and the velocity, acceleration, and normal and tangential contact forces of the time step are recalculated.

9. The method for simulating the dynamic response of friction contact collision of a flexible finger based on the absolute node coordinate method according to claim 8, characterized in that: Step 4 analyzes the friction calculation methods in the sticky state and sliding state respectively. In the sticky state, a tangential constraint is added. In the sliding state, the sliding friction force is described by the Coulomb friction model and the LuGre friction model respectively. Then, the equations of the contact collision dynamics of the flexible finger system are substituted to obtain the sticky-sliding friction contact collision dynamics equation of the flexible finger system. The motion incoordination caused by the velocity step is avoided by solving the initial collision conditions. The specific method is as follows: Step 4.1: Determine the friction calculation model in the sticking state and sliding state: When in a sticky state, the contact point at the end of the finger is subject to tangential constraints. The static friction force is actually the corresponding tangential constraint force. The friction force here refers to the sliding friction force, which is described by the Coulomb friction model or the LuGre friction model. If the Coulomb friction model is used to describe the friction force, the friction force at the contact point can be expressed as follows: F s =-sgn(v τ )m d l N Among them, v τ is the tangential velocity of the contact point, μ d is the coefficient of kinetic friction, λ N is the normal contact force at the contact point; If the LuGre friction model is used to describe friction, the real-time friction coefficient calculation needs to be performed at every moment: Among them, σ0 represents the bristle stiffness coefficient, σ1 represents the bristle damping coefficient, σ2 represents the viscous friction coefficient, and z and are the average bristle deformation and deformation rate of the contact pair respectively: g(v τ ) is about v τ The function of is always greater than 0; Step 4.2: Avoid motion incoordination caused by velocity step by solving the initial collision conditions: At the initial moment of each collision, the impulse-momentum method is used to solve the initial conditions of the collision, specifically: The impulse-momentum collision dynamics equation of the flexible finger model is as follows: Where, is the generalized velocity increment array caused by the collision, I is the collision impulse, e is the restitution coefficient, v F1 、v F2 are the velocity vector arrays of the two points that collided at the moment before the collision, Δv F1 , Δv F2 are the velocity increment vector arrays of the two colliding points during the collision process, is the unit direction vector, and D is the array representing the position of the impulse; For the physical model of flexible finger friction contact collision, the position of the collision plane does not change, and the above formula is expressed as: Where v0 is the velocity of the collision point along the collision normal at the moment before the collision; Assume that the finger tip collides completely inelastically with a rigid plane, i.e., the coefficient of restitution e = 0. At the moment of collision, the generalized coordinates of the system are continuous and smooth, but the generalized velocity is not smooth. Therefore, the following motion state transformation is performed: From then on until the contact between the collision points ends, their relative displacement, relative velocity and relative acceleration are always zero.

10. The method for simulating the dynamic response of friction contact collision of a flexible finger based on the absolute node coordinate method according to claim 9, characterized in that: The specific method of step 5 is: (1) Constructing the state space based on LU decomposition First, the Jacobian matrix of the constraint equation Φ q Perform LU decomposition: P1Φ q P2=LU Assume Φ q It is an s×n matrix, and the constraint matrix has s rows, so there are s non-independent variables, and we get an s×s row transformation matrix P1 and an n×n column transformation matrix P2; Substitute the obtained row-column transformation form into the system coordinate q and rearrange it to obtain the ordinary differential equations in the state space: Separate the rearranged ordinary differential equations to obtain: right Perform the transformation and express the Lagrange multiplier as: Will Substitution Eliminate the Lagrange multipliers: Among them, the dependent acceleration is expressed as independent acceleration: Eliminate dependent accelerations: Let the matrix We obtain a set of ordinary differential equations in the state space defined based on the LU decomposition: (2) Solving using a double-loop algorithm structure At the beginning of the calculation of each time step, the acceleration is solved in the outer loop based on the position and velocity of the previous moment, and then the displacement and velocity of the next moment are estimated, and the acceleration of the next moment is estimated based on this. Then, the ordinary differential equations in the state space defined by the LU decomposition are integrated; After that, the inner loop is entered to solve the constraint equation Φ(q, t) = 0. The velocity and position coordinates are updated in each loop until the inner loop calculation results meet the accuracy requirements, and then the inner loop is exited. Finally, the acceleration and Lagrange multiplier are updated until the outer loop calculation results meet the accuracy requirements. (3) Data visualization The time array, displacement array, normal contact force array and tangential contact force array are visualized in MATLAB to obtain the motion change image of the entire flexible finger system model over time, the curve graph of the finger end displacement changing with time, and the curve graph of the normal contact force and tangential contact force changing with time.

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