Modeling method for foot-ground acting force of hexapod robot in soft slope environment based on discrete element theory

Abstract

Modeling is carried out on the foot-ground interaction force of the hexapod robot walking in the soft slope environment. In order to accurately describe the dynamic process of slippage and collapse during mutual contact of foot and ground, the foot and ground acting force is analyzed in the normal direction and the tangential direction, in the normal direction, a shape theoretical model is established for irregular soil particles, modeling is conducted on the shape of the irregular soil particles based on the Fourier theory, and the shape of the irregular soil particles is obtained. A soil particle Fourier shape description mathematical model is obtained, and the relation between normal contact force and shape parameters in the irregular soil particle contact process is constructed on the basis of a DMT contact model and in combination with an irregular soil particle shape theoretical model based on the Fourier theory; in the tangential direction, due to the fact that the shape of the sand grains is irregular, the local shear rate is nonlinear, and the angle of the shape of the grains causes a remarkable non-local effect, the boundary conditions of grain rolling between different layers are considered when the grains roll, and the boundary conditions are determined according to the rolling characteristics of the grains between the adjacent layers; obtaining different integral equations between the suspension layer and the transition layer and between the transition layer and the bottom layer, and assuming that the local flow velocity is integrated in depth to obtain a progressive solution of the average flow velocity when the bottom does not slip, so as to obtain a constitutive equation of particle rolling. According to the foot-ground interaction force and the tangential particle rolling equation, the phenomena of sinking, sliding, falling and the like of the foot end of the robot in the soft slope environment can be well described, and then the gait of the robot is optimized to cope with the complex environment.

Description

Technical Field

[0001] The invention belongs to the technical field of foot-type robots, and in particular relates to a method for modeling foot-ground interaction forces of a hexapod robot in a soft slope environment based on discrete element theory. Background Art

[0002] With the development of aviation and exploration industries, mobile robots play an important role in terrains or environments that humans cannot enter (such as fire scenes, planetary exploration, deep-sea exploration, etc.). Compared with wheeled and tracked robots, hexapod robots have redundant limb structures, more joint degrees of freedom, and rich gaits, so they have stronger adaptability to unstructured environments. However, in actual application scenarios, complex terrain, harsh environment, external damage and other factors will cause the hexapod robot to be unable to operate according to the expected plan. Therefore, conducting dynamic foot-ground mechanics test research will help improve the mobility of hexapod robots in complex outdoor environments and is an effective way to guide foot-end design. Conducting multi-physical feature terrain mechanics modeling research is a prerequisite for realizing the motion planning of hexapod robots, and can effectively optimize the control strategy of hexapod robots. Summary of the invention

[0003] The purpose of the present invention is to propose a method for modeling the foot-ground interaction force of a hexapod robot in a soft slope environment based on discrete element theory.

[0004] The present invention can solve the problem of large prediction deviation of the force between the hexapod robot's feet and the ground, and significantly improve the accuracy of force prediction. The method of the present invention can simulate in detail the influence of factors such as different slopes and particle shapes on the foot-ground force, and can systematically analyze and optimize the robot's terrain adaptability.

[0005] The present invention is achieved through the following technical solutions:

[0006] Step 1: First, establish a theoretical shape model for irregular soil particles, collect the morphological characteristics of soil particles, model the shape of irregular soil particles based on Fourier theory, and obtain a mathematical model describing the Fourier shape of soil particles. Based on the contact mechanics of irregular soil particles described by shape, simplify the boundary of the theoretical shape model of soil particles into a contact problem between multiple convex bodies and a plane. Based on the DMT contact model and combined with the theoretical shape model of irregular soil particles based on Fourier theory, construct the relationship between the normal contact force and the shape parameters in the contact process of irregular soil particles.

[0007] Step 2: Aiming at the tangential force problem between different layers of soft slopes formed by multiple particles when the foot contacts the ground, a tangential shear stress model between multiple irregular soil particles in the contact process between different layers of soft slopes was established. By calculating the average shear rate of the particles at the foot end and the stopping height when rolling, the foot end settlement and the movement state of the particles at the foot end can be obtained, and the rolling and force state between the particles can be obtained, which provides a basis for the motion control and gait planning of the hexapod robot.

[0008] The beneficial effects of the present invention are as follows: Improve model accuracy: Discrete element theory can simulate the collision, friction and rolling effects between soil particles, as well as their contact mechanics with the robot's feet in detail. Compared with the traditional continuous medium assumption or simplified model, this method can more realistically reflect the nonlinear mechanical properties of soil in soft slope environments, such as particle flow, uneven stress distribution, local shear and other phenomena, thereby improving the accuracy of foot-ground force prediction. Enhance terrain adaptability: For complex terrains such as soft slopes, discrete element models can capture the interaction details between the robot's feet and the soil at different positions and postures, including toe insertion, sole sliding, and foot side squeezing. This helps to deeply understand and quantify the stability and walking efficiency of the robot under the influence of factors such as different slopes, soil moisture, and particle size, and provides key data support for optimizing gait planning and preventing slipping or sinking. Optimize gait design and control strategies: Accurate foot-ground force models can guide the design of more robust and efficient gait patterns, such as adjusting parameters such as stride length, step frequency, and pitch angle to minimize energy consumption, maximize propulsion, or prevent robot instability. In addition, real-time foot-ground force feedback can be used to develop advanced control algorithms, such as adaptive force control and terrain compliance control, so that the robot can dynamically adapt to terrain changes and improve its walking performance and autonomous navigation capabilities on soft slopes. BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 Flowchart of Fourier shape description mathematical model Figure 2 Schematic diagram of DMT single bump multi-bump model Figure 3 Schematic diagram of the hexapod robot's foot contacting the sand slope Figure 4 Flowchart of particle rolling constitutive equation DETAILED DESCRIPTION

[0009] The embodiments of the present invention are further described in detail below in conjunction with the accompanying drawings and examples.

[0010] An embodiment of the present invention: A method for modeling the foot-ground interaction force of a hexapod robot in a soft slope environment based on discrete element theory, mainly comprising the following contents:

[0011] (1): Use Fourier theory to describe the two-dimensional particle contour curve. First, preprocess the particle image to obtain the two-dimensional projection contour of the particle. Then collect the contour points. Through a series of analysis and calculation, the coordinates of the contour points, the position of the centroid and other parameters are obtained. Then the contour curve is drawn. Finally, the Fourier series is used for fitting, and the corresponding Fourier function is obtained. Fitting the discrete points of the particle contour using the 8th-order Fourier series can roughly restore the contour information of the particle. The coefficient values ​​of the sine and cosine components of the Fourier series, that is, the Fourier function fitting value, can also be directly calculated through MATLAB software to establish the Fourier shape description mathematical model. The process is as follows: Figure 1 shown.

[0012] For any irregular particle, the Fourier series of its contour curve can be obtained by expanding the contour curve, and the frequency and amplitude of each item in the Fourier series can be used to characterize the irregularity of the particle.

[0013] (2): Based on the description of the particle shape, the Fourier shape description mathematical model is established to complete the establishment of the normal contact mechanical model of irregular particles. According to the mathematical description of the irregular particle shape, the ordinate in the Fourier description curve is the distance from the particle contour point to the particle centroid, and the abscissa is the angle between the particle contour point and the centroid and the horizontal direction. The counterclockwise direction is positive, and the polar coordinate system of the Fourier description curve is established. In order to determine the contact point when the irregular particle contacts the plate, the straight line representing the position of the plate in the plane rectangular coordinate system is transformed into the same polar coordinate system. When the particle contacts the contour point and deforms, the normal overlap is generated, and the contact point is solved. Then, each contact point is transformed from the polar coordinate system back to the rectangular coordinate system. According to the DMT theory, each contact area can be regarded as a circle, and the radius of the contact area is obtained. In the contact problem between the irregular particle and the plate, the radius of curvature of the plate can be considered to be infinite, so the equivalent radius of the contact process is the radius of curvature of the particle at the contact point. For the study of the mechanical properties of particle contact, irregular particles are simplified to spherical particles for calculation, so it is a single convex contact. In the contact problem between a rough surface and a smooth plane, the actual contact only occurs on the micro-convex body, which can be simplified to the contact problem between multiple convex bodies and a plane, that is, it can be calculated by superimposing multiple single convex body contacts. The schematic diagram of the multi-convex point and single convex point model is shown in Figure 2 As shown. The sum of the forces on multiple convex bodies is obtained by superimposing the forces on the single convex body, i.e., the normal contact force Where: E * is the effective elastic modulus, Δγ is the surface free energy of the particle, δ is the normal overlap between the two particles, and a is the contact surface radius

[0014] (3): For regular spherical particles, the flow law between the local shear rate and the flow start and end heights follows the following relationship:

[0015] in, is the average shear rate, h is the rolling height, h stop (θ) is the critical height as a function of θ, below which no collapse will occur.

[0016] The particle flow generated when gravel rolls can be simplified into three basic layers: suspension layer, transition layer, and bottom layer. Factors such as particle size, shape, density, and fluid velocity between different layers will affect the specific behavior of these layers. Between different layers of particle flow, the boundary conditions between different layers determine the behavior of particles between different layers and the changes in fluid dynamics. The shape of sand particles is irregular and the local shear rate is nonlinear. The angle of the particle shape causes significant non-local effects. The present invention proposes to calculate according to the different boundary conditions corresponding to different rolling layers when particles roll on a slope, calculate and correct the nonlinear deviation caused by non-local effects, and theoretically establish the relative movement of gravel particles when the foot end contacts the gravel particles and the flow rules along the rough slope, such as Figure 4 Flowchart of the particle rolling constitutive equation.

[0017] When the foot lands, the sand particles roll. By treating the sand particles as a two-phase system undergoing a solid-liquid phase transition, the equation describing the fluidization parameter is expressed as:

[0018] The parameter λ describes the solid-liquid stress ratio, and the power exponent 1+b is an undetermined parameter;

[0019] In a steady shear flow the following relationship arises:

[0020] Consider a rough inclined plane such as Figure 3 Schematic diagram of the foot end of the hexapod robot contacting the sand slope, where the flow height h is uniform and θ is the angle between the center of mass of the particle when it starts to roll and the horizontal line when it stops. The Cartesian coordinates are set somewhere upstream of the bottom, where x points down the slope and z is perpendicular to the plane. Assuming that the sand particles are incompressible and the solid volume fraction φ is constant, the shear stress τ and pressure P can be derived from the steady-state momentum balance equation, giving the following formula τ=ρφgsinθ(hz) (6) P=ρφgcosθ(hz) (7)

[0021] According to the above coordinate formula (4), it is simplified to: Since there is no fluidization entering or leaving the suspension layer and the transition layer, it is usually assumed that the suspension layer is flux-free with dI / dz = 0, z = h. For irregular particles flowing down the rough surface described here, it can be assumed that the bottom layer is highly dissipative, indicating a pure solid phase, with I = 1, Z = 0 between the bottom layer and the transition layer. Equation (8) is derived as:

[0022] Among them I S Represents the inertia number on the suspension layer, the variable in unified form is:

[0023] Will I loc Substituting in:

[0024] in

[0025] Between the bottom layer and the transition layer * =1,Z * =0, we can get an integral equation:

[0026] Between the suspension layer and the transition layer * =1, Z * =1The integral equation at this time is:

[0027] By substituting equation (4) into equation (14), the stopping height is expressed as r(θ):

[0028] The stopping height is the depth to which the foot sinks into soft ground, such as sand or soil. When the foot applies a vertical force, the ground particles compress and rearrange until they reach a state of equilibrium, known as the stopping height. An exact solution to this process provides detailed information about the vertical mechanical relationship between the foot and the ground. By accounting for particle interactions, compression behavior, and the force applied by the foot, an exact solution to the stopping height allows the exact calculation of how deep the foot sinks into the ground.

[0029] During the sinking process, the vertical force compresses the particles and forms a load-bearing platform, while during the sliding process, the lateral thrust at the foot end causes the particles to move sideways, affecting the robot's stability and grip on the ground.

[0030] Through the accurate solution of the stop height, we can understand the compression and sinking process of the particles under the foot end in the vertical direction, and know the force under which the foot end will stop sinking. The average velocity asymptotic solution provides the average velocity of the particles in tangential motion, which helps analyze the motion state of the foot end when it slides. The motion state of the particles during sliding determines whether the foot end can maintain a stable force during the process of moving, preventing slippage and enhancing stability.

[0031] The local average velocity is integrated over the depth, and the no-slip condition at the bottom is assumed by partial integration, when u=0,z=0.

[0029] The asymptotic solution for the mean velocity is obtained:

[0032] When the robot foot slips, particles on the soft ground will move sideways. This movement can be described by the mean velocity asymptotic solution, which reflects the flow velocity and strain of the particles in the tangential direction. The asymptotic solution is suitable for describing the gradual movement and sliding of particles when horizontal thrust is applied after the foot contacts the ground. This solution can predict the particle's motion trajectory, velocity distribution, and the change in friction between the particle and the foot.

[0033] Through the relationship between particle flow rate and friction, we can deduce how the force applied by the foot end in the tangential direction is transmitted to the ground through the movement of particles, and further affects the degree of slippage of the foot end. Real-time monitoring of the force condition at the foot end adjusts the gait of each step and the mechanical distribution of the foot end to avoid excessive sinking and slippage. Especially on slopes, the direction and size of the force applied by the foot end can be adjusted in time according to the inclination and hardness of the ground. The analysis of the asymptotic solution of the particle flow rate can adjust the robot's walking strategy, select the appropriate step length and step frequency, avoid generating too much thrust on overly soft ground, and thus reduce the risk of slippage.

[0034] Finally, the constitutive equation of particle rolling is obtained:

[0035] The method of calculating the asymptotic solution of the stopping height and average flow velocity through the constitutive equation of particle rolling can more accurately analyze the ground settlement and slippage that occurs during the robot's landing process, especially the dynamic changes of particulate matter when the foot end contacts the ground, thereby more accurately reflecting the changes in the contact force between the hexapod robot and the ground. When facing soft or inclined terrain, the robot's real-time calculation of the foot end force and the amount of ground settlement and slippage can optimize the robot's gait control algorithm. The robot can optimize the placement of the foot and the distribution of force according to the interaction with the soft slope under different gaits, thereby improving its stability. The above shows and describes the basic principles, main features and scope of application of this patent.

Claims

1. A method for modeling the foot-ground interaction force of a hexapod robot in a soft slope environment based on discrete element theory, characterized in that Use the following steps: Step 1: A mathematical model of the shape of irregular soil particles is established based on the change of normal force during the normal contact between a single irregular particle and the foot. The normal mechanics model is approximated as a multi-convex body formed by multiple single particles in contact with the foot for analysis. This model can construct the relationship between the normal contact force and shape parameters during the contact process of irregular soil particles based on the contact mechanics of irregular soil particles described by the shape; Step 2: Aiming at the tangential force problem between different layers of soft slopes formed by multiple particles when the foot contacts the ground, a tangential shear stress model between multiple irregular soil particles in the contact process between different layers of soft slopes was established. By calculating the average shear rate of the particles at the foot end and the stopping height when rolling, the foot end settlement and the movement state of the particles at the foot end can be obtained, and the rolling and force state between the particles can be obtained, which provides a basis for the motion control and gait planning of the hexapod robot.

2. The relationship between normal contact force and shape parameters in the process of constructing the contact of irregular soil particles according to step 1 of claim 1, characterized in that: Firstly, the two-dimensional particle contour curve is described by Fourier theory to obtain the contour shape characteristics, and the particle contour expansion curve is expanded by Fourier series to obtain the Fourier contour shape descriptor, which is the Fourier shape description mathematical model of soil particles. Among them, N is the number of contour collection points, Z n are the contour boundary sampling points. Secondly, according to the mathematical description of the irregular particle shape, the Fourier description curve polar coordinate system is established, and the contact problem between the foot and the ground is simplified to the contact problem between multiple convex bodies and planes. Based on the DMT contact model, combined with the Fourier theory irregular soil particle shape theoretical model, the relationship between the normal contact force and the shape parameters in the contact process of irregular soil particles is constructed, that is, the normal contact force is Among them, E * is the effective elastic modulus.

3. The constitutive equation of local particle rolling according to step 2 of claim 1 describes the rolling phenomenon of soil particles when the sole of the hexapod robot sinks when walking on a non-rigid contact slope, characterized in that: Consider a rough inclined plane according to the Cartesian coordinates. Assuming that the sand is incompressible and the solid volume fraction φ is constant, the shear stress τ and pressure P can be derived from the steady-state momentum balance equation, giving the following formula τ=ρφgsinθ(hz) (3) P=ρφgcosθ(hz) (4) According to the different boundary conditions corresponding to different rolling layers when particles roll on a slope, a method is proposed to consider the relative motion when irregular particles contact each other, in order to correct the nonlinear deviation caused by non-local effects and increase the offset for correction calculation, so as to better describe the force between the robot foot and the ground. Based on the above calculation method, the constitutive equation of local particle rolling is proposed. The specific formula is as follows: in, is the average shear rate, h is the rolling height, h stop (θ) is the critical height as a function of θ, below which no collapse will occur.

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