Dynamic modeling method for any three-dimensional curved surface motion of spherical robot

Abstract

The invention provides a dynamic modeling method for any three-dimensional curved surface motion of a spherical robot, which comprises the following steps of: describing a motion model of the spherical robot driven by a trolley on any three-dimensional curved surface, defining physical parameters of a system, a reference system and a model assumption condition, and describing the ground and a motion state. Performing parameterization representation on the quaternion describing the attitude; the motion constraint of the trolley-driven spherical robot under generalized coordinates when the trolley-driven spherical robot moves on the three-dimensional curved surface is described and expressed in the form of quasi angular velocity, then unknown constraint force in a Lagrange equation is eliminated through the motion constraint, and a kinetic equation of the trolley-driven spherical robot is obtained. The motion principle of the trolley-driven spherical robot on any three-dimensional curved surface is disclosed, and a theoretical basis is provided for further achieving stable motion of the trolley-driven spherical robot in a complex task scene.

Description

Technical Field

[0001] The invention relates to the technical field of robot dynamics, and in particular to a dynamics modeling method for arbitrary three-dimensional curved surface motion of a spherical robot. Background Art

[0002] With the development of science and technology and modern industry, robotics has been widely used in all walks of life. Spherical robots have structural features and motion characteristics such as full wrapping, never tipping over, lightweight, under-actuated, and rolling motion, which make them have better sealing, balance, low energy consumption, and impact resistance, and can better adapt to various extreme environments and diverse tasks. Among them, the car-driven spherical robot has good climbing and obstacle-crossing capabilities, and the internal car platform has a large space, which is easy to carry sensors and other equipment, and is more suitable for complex application scenarios.

[0003] For spherical robots, dynamic models are the basis for achieving stable motion control of robots. However, spherical robots have the characteristics of non-holonomic constraints, underactuation, and strong coupling, which increase the difficulty of dynamic modeling of spherical robots. Considering the environmental complexity of application scenarios, it is necessary to construct a dynamic model of spherical robots moving on arbitrary surfaces to expand the scope of application of the model and the application scenarios of spherical robots. For the problem of dynamic modeling of spherical robots, the methods currently used mainly include Lagrange equations, Newton-Euler equations, and Kane equations, among which Lagrange equations are the most widely used. From the current research status at home and abroad, it can be seen that existing research mainly focuses on the motion of spherical robots on planes, and has not conducted in-depth analysis of the dynamic model of their motion on three-dimensional surfaces. This results in the common plane motion model being difficult to effectively cope with the new challenges brought about by complex motion constraints and motion characteristics when facing the motion of spherical robots on three-dimensional surfaces. Summary of the invention

[0004] In order to solve the problems existing in the prior art, the present invention provides a dynamic modeling method for the motion of a spherical robot on any three-dimensional curved surface. The Lagrange equation under the non-holonomic constrained generalized coordinates is used to obtain the dynamic model of the motion of a spherical robot driven by a car on any three-dimensional curved surface. Then, the dynamic simulation software Adams is used to establish the dynamic simulation model of the motion of the spherical robot driven by a car on a three-dimensional curved surface. The motion simulation is performed and compared with the Matlab theoretical calculation results of the dynamic model to verify the correctness of the established dynamic model of the spherical robot driven by a car. The present invention provides a theoretical basis for further realizing the stable motion control of the spherical robot driven by a car in complex task scenarios.

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

[0006] A method for dynamic modeling of arbitrary three-dimensional curved surface motion of a spherical robot is provided, comprising:

[0007] Describe the motion model of a spherical robot driven by a car on any three-dimensional surface, define the system physical parameters, reference system, model assumptions, describe the ground and motion state, and parameterize the quaternion describing the posture;

[0008] The motion constraints of the spherical robot driven by the cart in generalized coordinates when moving on a three-dimensional surface are described and expressed in the form of quasi-angular velocity. Then, the motion constraints are used to eliminate the unknown constraint forces in the Lagrange equation to obtain the dynamic equation of the spherical robot driven by the cart.

[0009] As a further explanation of the present invention, the description of the motion model of the spherical robot driven by the car on any three-dimensional surface, the definition of system physical parameters, reference system and model assumptions, specifically includes:

[0010] The physical structure of the car-driven spherical robot consists of a spherical shell and a two-wheel differential-driven car. The two-wheel differential-driven car moves forward and turns in the spherical shell, and drives the overall movement of the car-driven spherical robot system by changing its center of gravity.

[0011] Define the physical parameters of the system, l is the radius of the spherical shell, ρ is the distance between the spherical shell and the center of mass of the two-wheel differential drive car, m s is the mass of the spherical shell, m t is the mass of the two-wheel differential drive car, c t is the first mass moment of the two-wheel differential drive car, J t is the moment of inertia of the two-wheel differential drive vehicle, J s is the moment of inertia of the spherical shell;

[0012] Define four reference frames, inertial coordinate system Σ I , spherical shell coordinate system Σ s , the body coordinate system of the two-wheel differential drive car Σ t , virtual tangent plane coordinate system Σ p , the z-axis of the virtual tangent plane coordinate system is the tangent plane normal vector n at the contact point between the spherical shell and the curved surface, and the x-axis is e1=k I ×n, the y-axis is e2=n×e1;

[0013] The ideal model assumes that the spherical shell only rolls but does not slide on the curved surface, and will not rotate around the normal vector of the tangent plane at the contact point, and the car will not rotate around the forward direction.

[0014] As a further explanation of the present invention, the description of the ground and motion state specifically includes:

[0015] (1) Ground description

[0016] Any three-dimensional surface can be represented as

[0017] f(x g ,y g ,z g )=0 (1)

[0018] The unit normal vector of the tangent plane at the contact point is

[0019]

[0020] in, and '||·||' denotes the magnitude of the gradient and the vector;

[0021] (2) System generalized coordinates

[0022] In order to describe the motion state of the spherical robot driven by a car on an arbitrary three-dimensional surface, the generalized coordinates of the system are defined as

[0023]

[0024] in, represents the position of the center of mass of the spherical shell in the inertial system, q sI ,q tI The quaternions represent the directions of the spherical shell and the two-wheel differential drive car relative to the inertial system.

[0025] As a further explanation of the present invention, the parameterized representation of the quaternion describing the posture specifically includes:

[0026] Define quaternion as

[0027]

[0028] Among them, η ba is the real part, ∈ ba is the imaginary part, is the unit length rotation axis, φ is the rotation angle, and the transformed direction cosine matrix C ba for

[0029]

[0030] Where E3 is the identity matrix, the symbol '×' represents the cross product matrix of the vector, and the quaternion is constrained to unit length

[0031]

[0032] The time derivative of formula (6) is expressed at the rate level

[0033]

[0034] Quaternion Derivatives With angular velocity The conversion relationship is

[0035]

[0036] on the contrary,

[0037]

[0038] in, is the mapping matrix between quaternion derivatives and angular velocity, satisfying

[0039]

[0040] In order to facilitate the derivation of the kinetic equation, the following relationship exists

[0041]

[0042] Among them, a In the reference coordinate system Σ a Vector notation for vectors in .

[0043] As a further explanation of the present invention, the motion constraints of the spherical robot driven by the car in the three-dimensional curved surface motion under the generalized coordinates are described and expressed in the form of quasi-angular velocity, specifically including:

[0044] (1) Quaternion constraints

[0045] Using quaternion q sI and q tI represents the direction of the spherical shell and the two-wheel differential drive car, satisfying the unit length constraint in equation (7), expressed as

[0046]

[0047] (2) Spherical shell constraint

[0048] Assume that the spherical robot driven by the car can only roll on the curved surface without sliding, and the contact point p' (on the spherical shell) is relative to the inertial system Σ I The speed is expressed as

[0049]

[0050] in, Indicates the contact point p' in the inertial system relative to the inertial system Σ I speed, Indicates the spherical shell relative to the inertial system Σ I speed, Indicates the spherical shell relative to the inertial system Σ I The angular velocity of represents the position of the contact point p' relative to the mass center of the spherical shell in the inertial system;

[0051] The spherical robot driven by the car can only roll but not slide, so the center of the spherical shell is relative to the inertial system Σ I The speed is

[0052]

[0053] Among them, C sI Represents the direction cosine matrix of the transformation from the spherical shell coordinate system to the inertial system, Indicates the spherical shell coordinate system relative to the inertial system Σ I Angular velocity;

[0054] Another constraint of the spherical shell is that it does not rotate around the normal direction of the ground, which is expressed as

[0055]

[0056] in, The same applies to 11 and 12; Represents the spherical shell relative to the inertial system Σ in the virtual tangent plane coordinate system I The angular velocity, C sp Represents the direction cosine matrix of the transformation from the spherical shell coordinate system to the virtual tangent plane coordinate system;

[0057] (3) Car restraint

[0058] Assuming that the two-wheel differential drive car will not rotate around the forward direction, it can be expressed as

[0059]

[0060] in, represents the angular velocity of the two-wheel differential drive car body relative to the spherical shell in the body coordinate system, C st Represents the direction cosine matrix of the transformation from the spherical shell coordinate system to the body coordinate system of the two-wheel differential drive car;

[0061] (4) Motion constraint set

[0062] Combining equations (12), (14), (15), and (16), the motion constraint is shown in equation (17):

[0063]

[0064] The generalized coordinates of the system are simplified using express

[0065]

[0066] Further simplified derivatives of generalized coordinates Expressed in the form of quasi-angular velocity

[0067]

[0068] in, Pick The first and second elements of Pick The first and third elements;

[0069] Combining equations (17), (18), and (19), the motion constraint is expressed as angular velocity Expressed as

[0070]

[0071] Quasi angular velocity Independent of each other,

[0072] ΞΥΛ=0 (21).

[0073] As a further explanation of the present invention, the use of motion constraints to eliminate the unknown constraint forces in the Lagrange equation to obtain the dynamic equation of the spherical robot driven by the car specifically includes:

[0074] (1) Kinetic energy and potential energy

[0075] Define the velocity matrix of the car-driven spherical robot system as

[0076]

[0077] in, Indicates the spherical shell relative to the inertial system Σ I speed, Indicates the spherical shell coordinate system relative to the inertial system Σ I The angular velocity of It represents the coordinate system of the two-wheel differential drive car body relative to the inertial system Σ I Angular velocity;

[0078] The system kinetic energy is

[0079]

[0080] in,

[0081]

[0082] Among them, c t =m t r t st , the system potential energy is

[0083]

[0084] (2) Lagrangian equations for nonholonomic systems

[0085] The Lagrangian equation in nonholonomic constrained generalized coordinates is

[0086]

[0087] in,

[0088]

[0089] Multiply (26) on the left Using equations (21) and (11), we can obtain the dynamic equation of the spherical robot driven by the car on any three-dimensional surface:

[0090]

[0091] in,

[0092]

[0093] In the formula, C pI Represents the direction cosine matrix of the virtual tangent plane coordinate system transformed to the inertial system, C ps The direction cosine matrix, C, represents the transformation from the virtual tangent plane coordinate system to the spherical shell coordinate system. pt The direction cosine matrix, C, represents the transformation of the virtual tangent plane coordinate system to the body coordinate system of the two-wheel differential drive car. tI Represents the direction cosine matrix of the two-wheel differential drive car body coordinate system transformed to the inertial system;

[0094] The velocity matrix of the car-driven spherical robot system is further expressed as a quasi-angular velocity form:

[0095]

[0096] Therefore, the dynamic equation of the spherical robot driven by the car on any three-dimensional surface is:

[0097]

[0098] in, is the generalized external force in the form of quasi-angular velocity, and its expression is

[0099]

[0100] Among them, u f is the forward force exerted by the car, which is proportional to the average wheel torque, u τ is the rotational torque applied by the cart, which is proportional to the difference in wheel torques.

[0101] Compared with the prior art, the present invention has the following beneficial technical effects:

[0102] Aiming at the complex problem of dynamic modeling of a spherical robot with the characteristics of non-holonomic constraints, under-actuation, strong coupling, etc., the present invention utilizes the Lagrangian equations under non-holonomic constraints generalized coordinates, uses quaternions to describe the posture of the spherical robot, expresses the motion constraints of the spherical robot driven by a cart in a generalized coordinate system when it moves on a three-dimensional surface, utilizes the motion constraints to eliminate the unknown constraint forces in the Lagrangian equations, and obtains the dynamic model of the spherical robot driven by a cart moving on an arbitrary three-dimensional surface. The present invention reveals the motion principle of the spherical robot driven by a cart on an arbitrary three-dimensional surface, and provides a theoretical basis for further realizing the stable motion of the spherical robot driven by a cart in complex task scenarios.

[0103] Other features and advantages of the technical solution will be described in the subsequent description, and partly become apparent from the description, or understood by implementing the technical solution. The purpose and other advantages of the technical solution can be achieved and obtained through the structures specifically pointed out in the written description and the drawings.

[0104] The technical solution of the present technical solution is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0105] The accompanying drawings are used to provide a further understanding of the technical solution and constitute a part of the specification. Together with the embodiments of the technical solution, they are used to explain the technical solution and do not constitute a limitation of the technical solution. In the accompanying drawings:

[0106] Figure 1 This is a schematic diagram of the arbitrary three-dimensional curved surface motion model of a spherical robot driven by a trolley provided by the present invention.

[0107] Figure 2 A comparison diagram of the trajectory of the spherical shell provided by the present invention that only applies forward force to the vehicle.

[0108] Figure 3 A comparison diagram of the internal trolley rotation angular velocity provided by the present invention in which only rotational torque is applied to the trolley.

[0109] Figure 4 A comparison diagram of the spherical shell trajectory for applying forward force and rotational torque to the vehicle provided by the present invention. DETAILED DESCRIPTION

[0110] The preferred embodiments of the present technical solution are described below in conjunction with the accompanying drawings. It should be understood that the preferred embodiments described herein are only used to illustrate and explain the present technical solution, and are not used to limit the present technical solution.

[0111] In order to reveal the dynamic principle of a spherical robot driven by a car on an arbitrary three-dimensional surface and provide a theoretical basis for further realizing the stable motion control of a spherical robot driven by a car in complex task scenarios, this invention patent proposes a dynamic modeling method for the motion of a spherical robot on an arbitrary three-dimensional surface, which includes the following steps:

[0112] 1. System description and quaternion parameterization. Describe the motion model of a spherical robot driven by a car on an arbitrary three-dimensional surface, define system parameters, reference system and model assumptions, and parameterize the quaternion that describes the posture.

[0113] 1.1 System Description, the schematic diagram of the spherical robot motion model driven by a car on any three-dimensional surface is as follows Figure 1 As shown in the figure, the physical structure of the robot consists of a spherical shell and a two-wheel differential drive car. The car moves forward and turns in the spherical shell, and drives the overall movement of the spherical robot system by changing its center of gravity. Define the physical parameters of the system, l is the radius of the spherical shell, ρ is the distance between the spherical shell and the center of mass of the car, m s is the mass of the spherical shell, m t is the mass of the trolley, c t is the first mass moment of the car, J t is the moment of inertia of the car, J s is the moment of inertia of the spherical shell; four reference systems are defined, the inertial coordinate system Σ I , spherical shell coordinate system Σ s , vehicle body coordinate system Σ t , virtual tangent plane coordinate system Σ p , the z-axis of the virtual tangent plane coordinate system is the tangent plane normal vector n at the contact point between the spherical shell and the curved surface, and the x-axis is e1=k I ×n, the y-axis is e2=n×e1; the ideal model assumes that the spherical shell only rolls but does not slide on the curved surface, and will not rotate around the normal vector of the tangent plane at the contact point, and the car will not rotate around the forward direction.

[0114] (1) Ground description

[0115] Any three-dimensional surface can be represented as

[0116] f(x g ,y g ,z g )=0 (1)

[0117] The unit normal vector of the tangent plane at the contact point is

[0118]

[0119] in, and '||·||' denote the gradient and the magnitude of the vector.

[0120] (2) System generalized coordinates

[0121] In order to describe the motion state of the spherical robot on any three-dimensional surface, the generalized coordinates of the system are defined as

[0122]

[0123] in, represents the position of the center of mass of the spherical shell in the inertial system, q sI ,q tI The quaternions represent the orientation of the spherical shell and the car relative to the inertial system.

[0124] 1.2 Quaternion Parameterization

[0125] Define quaternion as

[0126]

[0127] Among them, η ba is the real part, ∈ ba is the imaginary part, is the unit length rotation axis, φ is the rotation angle. The transformed direction cosine matrix C ba for

[0128]

[0129] Where E3 is the identity matrix and the symbol '×' represents the cross product matrix of the vector. Quaternions are subject to the unit length constraint

[0130]

[0131] The time derivative of equation (6) can be expressed at the rate level as

[0132]

[0133] Quaternion Derivatives With angular velocity The conversion relationship is

[0134]

[0135] on the contrary,

[0136]

[0137] in, is the mapping matrix between quaternion derivatives and angular velocity, satisfying

[0138]

[0139] In order to facilitate the derivation of the kinetic equation, the following relationship exists

[0140]

[0141] Among them, a In the reference coordinate system Σ a Vector notation for vectors in .

[0142] 2. Dynamic modeling. Since the spherical robot is a non-holonomic system, the Lagrange equations under non-holonomic constrained generalized coordinates are used to construct the dynamic model. First, the motion constraints of the spherical robot under generalized coordinates when it moves on the surface are described and expressed in the form of quasi-angular velocity. Then, the motion constraints are used to eliminate the unknown constraint forces in the Lagrange equations, and the dynamic equations of the spherical robot driven by the car are obtained.

[0143] 2.1 Motion Constraints

[0144] (1) Quaternion constraints

[0145] Using quaternion q sI and q tI represents the direction of the spherical shell and the car, satisfying the unit length constraint in equation (7), which can be expressed as

[0146]

[0147] (2) Spherical shell constraint

[0148] Assume that the spherical robot can only roll but not slide on the curved surface. The contact point p' (on the spherical shell) is relative to the inertial system Σ I The speed is expressed as

[0149]

[0150] in, Indicates the contact point p' in the inertial system relative to the inertial system Σ I speed, Indicates the spherical shell relative to the inertial system Σ I speed, Indicates the spherical shell relative to the inertial system Σ I The angular velocity of Represents the position of the contact point p' relative to the center of mass of the spherical shell in the inertial system.

[0151] The spherical robot can only roll but not slide, so the center of the spherical shell is relative to the inertial system Σ I The speed is

[0152]

[0153] Among them, C sI Represents the direction cosine matrix of the transformation from the spherical shell coordinate system to the inertial system, Indicates the spherical shell coordinate system relative to the inertial system ΣI angular velocity.

[0154] Another constraint of the spherical shell is that it does not rotate around the normal direction of the ground, which is expressed as

[0155]

[0156] in, The same applies to 11 and 12. Represents the spherical shell relative to the inertial system Σ in the virtual tangent plane coordinate system I The angular velocity, C sp Represents the direction cosine matrix of the transformation from the spherical shell coordinate system to the virtual tangent plane coordinate system.

[0157] (3) Car restraint

[0158] Assuming that the car will not rotate around the forward direction, it can be expressed as

[0159]

[0160] in, represents the angular velocity of the two-wheel differential drive car body relative to the spherical shell in the body coordinate system, C st Represents the direction cosine matrix of the transformation from the spherical shell coordinate system to the body coordinate system of the two-wheel differential drive car.

[0161] (4) Motion constraint set

[0162] Combining equations (12), (14), (15), and (16), the motion constraint is shown in equation (17):

[0163]

[0164] The generalized coordinates of the system are simplified using express

[0165]

[0166] Furthermore, the derivative of the simplified generalized coordinates It can be expressed in the form of quasi-angular velocity

[0167]

[0168] in, Pick The first and second elements of Pick The first and third elements.

[0169] Combining equations (17), (18), and (19), the motion constraint is expressed as angular velocity Expressed as

[0170]

[0171] Quasi angular velocity Independent of each other,

[0172] ΞΥΛ=0 (21)

[0173] 2.2 Kinetic equation

[0174] (1) Kinetic energy and potential energy

[0175] Define the velocity matrix of the spherical robot system as

[0176]

[0177] in, Indicates the spherical shell relative to the inertial system Σ I speed, Indicates the spherical shell coordinate system relative to the inertial system Σ I The angular velocity of It represents the coordinate system of the two-wheel differential drive car body relative to the inertial system Σ I angular velocity.

[0178] The system kinetic energy is

[0179]

[0180] in,

[0181]

[0182] Among them, c t =m t r t st The potential energy of the system is

[0183]

[0184] (2) Lagrangian equations for nonholonomic systems

[0185] The Lagrangian equation in nonholonomic constrained generalized coordinates is

[0186]

[0187] in,

[0188]

[0189] Multiply (26) on the left Using equations (21) and (11), we can obtain the dynamic equation of the spherical robot driven by the car on any three-dimensional surface:

[0190]

[0191] in,

[0192]

[0193] In the formula, C pI Represents the direction cosine matrix of the virtual tangent plane coordinate system transformed to the inertial system, C ps The direction cosine matrix, C, represents the transformation from the virtual tangent plane coordinate system to the spherical shell coordinate system. pt The direction cosine matrix, C, represents the transformation of the virtual tangent plane coordinate system to the body coordinate system of the two-wheel differential drive car. tI Represents the direction cosine matrix of the two-wheel differential drive car's body coordinate system transformed to the inertial system.

[0194] Further, the velocity matrix of the spherical robot system is expressed as a quasi-angular velocity form

[0195]

[0196] Therefore, the dynamic equation of the spherical robot driven by the car on any three-dimensional surface is:

[0197]

[0198] in, is the generalized external force in the form of quasi-angular velocity, and its expression is

[0199]

[0200] Among them, u f is the forward force exerted by the car (proportional to the average wheel torque), u τ is the rotational torque applied by the cart (proportional to the difference in wheel torques).

[0201] 3. Construction of Adams dynamics simulation model and comparison and verification with MATLAB theoretical calculation results

[0202] 3.1 Build an Adams dynamics simulation model and establish a dynamics simulation platform for the movement of a spherical robot driven by a car on a three-dimensional surface.

[0203] (1) Initial settings. First, set up the system working environment, including units, gravitational acceleration, etc. Then use the objects provided by Adams to build the spherical shell, and import the 3D ground and car generated by external 3D software; finally, edit the initial status and mass information of each component.

[0204] (2) Define the kinematic pairs, including the fixed pair between the ground and the earth, and the rotation pair between the two wheels and the vehicle body.

[0205] (3) Apply loads. First, several contact forces are set, including the contact forces between the spherical shell and the ground, between the two wheels and the spherical shell, and between the vehicle body and the spherical shell. The corresponding collision and friction force parameters are set according to the situation. Then, the torque along the normal direction of the contact between the spherical shell and the ground is applied. This torque has the effect of limiting the rotation of the spherical shell along the normal direction. Finally, the torque is applied at the two wheels as the input of the simulation system.

[0206] (4) Simulation calculation.

[0207] 3.2 Input multiple sets of wheel torques in Adams, run the results, observe the motion trajectory of the center of the spherical shell, output the data results, and compare them with the Matlab calculation results of the dynamic model to verify the correctness of the established car-driven spherical robot dynamic model.

[0208] The present invention is described below in conjunction with specific embodiments.

[0209] In this embodiment, a spherical robot driven by a 0.08m diameter cart moving on a curved surface z=-0.04cos2y is taken as an example, reasonable system parameters are set, the dynamic model is calculated using Matlab, and the system parameters in Adams are uniformly set to perform dynamic simulation calculations, and the results of the two systems are compared to verify the correctness of the dynamic model.

[0210] Table 1 shows the system parameters of the dynamic model. Table 2 shows the Adams simulation parameters.

[0211] Table 1 System parameters of the dynamic model of the spherical robot driven by a car

[0212]

[0213]

[0214] Table 2 Adams dynamics simulation platform system parameters

[0215]

[0216] Figure 2-Figure 4 This is a comparison chart of the Matlab theoretical calculation results and Adams simulation results of the arbitrary three-dimensional surface dynamics model of the spherical robot driven by the car. The simulation time is 5s. Figure 2 This is a comparison diagram of the spherical shell trajectory when only a forward force of 0.1N is applied to the robot (the torque of both wheels in Adams is 0.00125N·m), showing that the spherical robot is moving on the curved surface along the direction of the forward force; Figure 3This is a comparison of the rotational angular velocities of the internal car when only a rotational torque of 0.0001 N·m (the torque of wheel 1 in Adams is 0.000026 N·m, and the torque of wheel 2 is -0.000026 N·m) is applied to the car. It shows that the car inside the spherical robot rotates inside the spherical shell, while the spherical shell remains stationary. Figure 4 The spherical shell trajectory comparison diagram shows that the trolley applies a forward force of 0.1N and a rotational torque of 0.0001N·m (the torque of wheel 1 in Adams is 0.001224N·m, and the torque of wheel 2 is 0.001276N·m), which shows that the spherical robot moves along a certain curved trajectory on the surface. The results show that the Matlab theoretical calculation results of the dynamic model of the trolley driving the spherical robot on any three-dimensional curved surface are basically completely coincident with the output curve of the Adams dynamic simulation results, proving that the dynamic modeling method based on the Lagrange equation of the non-holonomic system designed by the present invention can correctly establish the dynamic model of the trolley driving the spherical robot on any three-dimensional curved surface.

[0217] Obviously, those skilled in the art can make various changes and modifications to the technical solution without departing from the spirit and scope of the technical solution. Thus, if these modifications and variations of the technical solution fall within the scope of the claims of the technical solution and their equivalents, the technical solution is also intended to include these modifications and variations.

Claims

1. A dynamic modeling method for the motion of a spherical robot on any three-dimensional surface, characterized in that: include: Describe the motion model of a spherical robot driven by a car on any three-dimensional surface, define the system physical parameters, reference system, model assumptions, describe the ground and motion state, and parameterize the quaternion describing the posture; The motion constraints of the spherical robot driven by the cart in generalized coordinates when moving on a three-dimensional surface are described and expressed in the form of quasi-angular velocity. Then, the motion constraints are used to eliminate the unknown constraint forces in the Lagrange equation to obtain the dynamic equation of the spherical robot driven by the cart.

2. The dynamic modeling method for the spherical robot's arbitrary three-dimensional curved surface motion as claimed in claim 1, characterized in that: The description of the motion model of the spherical robot driven by the car on any three-dimensional surface defines the system physical parameters, reference system and model assumptions, specifically including: The physical structure of the car-driven spherical robot consists of a spherical shell and a two-wheel differential-driven car. The two-wheel differential-driven car moves forward and turns in the spherical shell, and drives the overall movement of the car-driven spherical robot system by changing its center of gravity. Define the physical parameters of the system, l is the radius of the spherical shell, ρ is the distance between the spherical shell and the center of mass of the two-wheel differential drive car, m s is the mass of the spherical shell, m t is the mass of the two-wheel differential drive car, c t is the first mass moment of the two-wheel differential drive car, J t is the moment of inertia of the two-wheel differential drive vehicle, J s is the moment of inertia of the spherical shell; Define four reference frames, inertial coordinate system Σ I , spherical shell coordinate system Σ s , the body coordinate system of the two-wheel differential drive car Σ t , virtual tangent plane coordinate system Σ p , the z-axis of the virtual tangent plane coordinate system is the tangent plane normal vector n at the contact point between the spherical shell and the curved surface, and the x-axis is e1=k I ×n, the y-axis is e2=n×e1; The ideal model assumes that the spherical shell only rolls but does not slide on the curved surface, and will not rotate around the normal vector of the tangent plane at the contact point, and the car will not rotate around the forward direction.

3. The method for dynamic modeling of a spherical robot's arbitrary three-dimensional curved surface motion as claimed in claim 1 or 2, characterized in that: The description of the ground and motion state specifically includes: (1) Ground description Any three-dimensional surface can be represented as f(x g ,y g ,z g )=0 (1) The unit normal vector of the tangent plane at the contact point is Among them, '▽' and '||·||' represent the gradient and the magnitude of the vector; (2) System generalized coordinates In order to describe the motion state of the spherical robot driven by a car on an arbitrary three-dimensional surface, the generalized coordinates of the system are defined as in, represents the position of the center of mass of the spherical shell in the inertial system, q sI ,q tI The quaternions represent the directions of the spherical shell and the two-wheel differential drive car relative to the inertial system.

4. The method for dynamic modeling of a spherical robot moving on any three-dimensional surface as claimed in claim 3, characterized in that: The parameterized representation of the quaternion describing the posture specifically includes: Define quaternion as Among them, η ba is the real part, ∈ ba is the imaginary part, is the unit length rotation axis, φ is the rotation angle, and the transformed direction cosine matrix C ba for Where E3 is the identity matrix, the symbol '×' represents the cross product matrix of the vector, and the quaternion is constrained to unit length The time derivative of formula (6) is expressed at the rate level Quaternion Derivatives With angular velocity The conversion relationship is on the contrary, in, is the mapping matrix between quaternion derivatives and angular velocity, satisfying In order to facilitate the derivation of the kinetic equation, the following relationship exists Among them, a In the reference coordinate system Σ a Vector notation for vectors in .

5. The method for dynamic modeling of a spherical robot moving on any three-dimensional surface as claimed in claim 4, characterized in that: The description of the motion constraints of the spherical robot driven by the car in the three-dimensional curved surface under the generalized coordinates is expressed in the form of quasi-angular velocity, specifically including: (1) Quaternion constraints Using quaternion q sI and q tI represents the direction of the spherical shell and the two-wheel differential drive car, satisfying the unit length constraint in equation (7), expressed as (2) Spherical shell constraint Assume that the spherical robot driven by the car can only roll on the curved surface without sliding, and the contact point p' (on the spherical shell) is relative to the inertial system Σ I The speed is expressed as in, Indicates the contact point p' in the inertial system relative to the inertial system Σ I speed, Indicates the spherical shell relative to the inertial system Σ I speed, Indicates the spherical shell relative to the inertial system Σ I The angular velocity of represents the position of the contact point p' relative to the mass center of the spherical shell in the inertial system; The spherical robot driven by the car can only roll but not slide, so the center of the spherical shell is relative to the inertial system Σ I The speed is Among them, C sI Represents the direction cosine matrix of the transformation from the spherical shell coordinate system to the inertial system, Indicates the spherical shell coordinate system relative to the inertial system Σ I Angular velocity; Another constraint of the spherical shell is that it does not rotate around the normal direction of the ground, which is expressed as in, The same applies to 11 and 12. Represents the spherical shell relative to the inertial system Σ in the virtual tangent plane coordinate system I Angular velocity, C sp Represents the direction cosine matrix of the transformation from the spherical shell coordinate system to the virtual tangent plane coordinate system; (3) Car restraint Assuming that the two-wheel differential drive car will not rotate around the forward direction, it can be expressed as in, represents the angular velocity of the two-wheel differential drive car body relative to the spherical shell in the body coordinate system, C st Represents the direction cosine matrix of the transformation from the spherical shell coordinate system to the body coordinate system of the two-wheel differential drive car; (4) Motion constraint set Combining equations (12), (14), (15), and (16), the motion constraint is shown in equation (17): The generalized coordinates of the system are simplified using express Further simplified derivatives of generalized coordinates Expressed in the form of quasi-angular velocity in, Pick The first and second elements of Pick The first and third elements; Combining equations (17), (18), and (19), the motion constraint is expressed as angular velocity Expressed as Quasi angular velocity Independent of each other, ΞΥΛ=0 (21).

6. The method for dynamic modeling of the spherical robot's arbitrary three-dimensional curved surface motion as claimed in claim 5, characterized in that: The use of motion constraints to eliminate the unknown constraint forces in the Lagrange equation to obtain the dynamic equation of the spherical robot driven by the car specifically includes: (1) Kinetic energy and potential energy Define the velocity matrix of the car-driven spherical robot system as in, Indicates the spherical shell relative to the inertial system Σ I speed, Indicates the spherical shell coordinate system relative to the inertial system Σ I The angular velocity of It represents the coordinate system of the two-wheel differential drive car body relative to the inertial system Σ I Angular velocity; The system kinetic energy is in, Among them, c t =m t r t st , the system potential energy is (2) Lagrangian equations for nonholonomic systems The Lagrangian equation in nonholonomic constrained generalized coordinates is in, Multiply (26) on the left Using equations (21) and (11), we can obtain the dynamic equation of the spherical robot driven by the car on any three-dimensional surface: in, In the formula, C pI Represents the direction cosine matrix of the virtual tangent plane coordinate system transformed to the inertial system, C ps The direction cosine matrix, C, represents the transformation of the virtual tangent plane coordinate system to the spherical shell coordinate system. pt The direction cosine matrix, C, represents the transformation of the virtual tangent plane coordinate system to the body coordinate system of the two-wheel differential drive car. tI Represents the direction cosine matrix of the two-wheel differential drive car body coordinate system transformed to the inertial system; The velocity matrix of the car-driven spherical robot system is further expressed as a quasi-angular velocity form: Therefore, the dynamic equation of the spherical robot driven by the car on any three-dimensional surface is: in, is the generalized external force in the form of quasi-angular velocity, and its expression is Among them, u f is the forward force exerted by the car, which is proportional to the average wheel torque, u τ is the rotational torque applied by the cart, which is proportional to the difference in wheel torques.

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