CN112818542A - Motion behavior analysis method of asymmetric double-wheel rimless wheel model in spatial inclined plane - Google Patents

Motion behavior analysis method of asymmetric double-wheel rimless wheel model in spatial inclined plane Download PDF

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CN112818542A
CN112818542A CN202110138734.6A CN202110138734A CN112818542A CN 112818542 A CN112818542 A CN 112818542A CN 202110138734 A CN202110138734 A CN 202110138734A CN 112818542 A CN112818542 A CN 112818542A
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贾文川
李文捷
马书根
袁建军
俞佳程
孙翊
蒲华燕
鲍晟
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University of Shanghai for Science and Technology
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Abstract

The invention discloses an analysis method of motion of an asymmetric double-wheel rimless wheel model in a spatial inclined plane, which comprises six steps of establishing a double-wheel rimless wheel model and a fixed coordinate system, establishing a motion coordinate system and a state space, analyzing two rotation collision modes, calculating a kinetic equation of single-leg support three-dimensional motion, calculating a kinetic equation of double-leg support three-dimensional motion, analyzing stability and analyzing a motion relation. The asymmetric double-wheel borderless wheel model is formed by two plane single-wheel borderless wheels which are arranged in a staggered mode in space, and the two plane single-wheel borderless wheels are connected through a cross beam to simulate the distance between two legs of a human. The invention can simulate the motion of the lower limbs of human more truly and provides an effective basis for the research and development of the foot type robot and the realization of high-energy-efficiency motion performance.

Description

Motion behavior analysis method of asymmetric double-wheel rimless wheel model in spatial inclined plane
Technical Field
The invention relates to a motion behavior analysis method of a rimless wheel, in particular to a motion behavior and stability analysis method of an asymmetric double-wheel rimless wheel model in a three-dimensional space slope.
Background
The passive stable walking action of the rimless wheel on the spatial inclined plane is similar to the gait of a person walking on the ground, the movement of the rimless wheel is spontaneous and energy-saving, so the rimless wheel is considered as a basic model for researching foot type walking, and relevant research conclusions have important references for the research and development of the foot type robot.
The rimless wheel model mainly comprises a plane single-wheel rimless wheel, a plane double-wheel rimless wheel and the like, and the motion environment is usually simplified into an inclined plane in a plane coordinate system for modeling and calculation. In order to simulate and use the human lower limb movement more truly, the method is a feasible idea for researching and analyzing the movement behavior of the spatial double-wheel rimless wheel on the spatial inclined plane. This is because for a conventional rimless wheel in planar space, all the legs are distributed in the same plane; however, the human legs are not located on the longitudinal section of the human body, and the hip distance exists between the legs, and the important characteristic is that the edgeless wheel in the plane space does not have.
Therefore, how to provide a motion behavior analysis method of a symmetric double-wheel rimless wheel model on a spatial slope is a technical problem which needs to be solved urgently by a person skilled in the art.
Disclosure of Invention
The invention aims to provide a motion behavior analysis method of an asymmetric double-wheel rimless wheel model in a spatial inclined plane, which divides the motion state of a system into two motion modes of single-leg support three-dimensional motion and double-leg support three-dimensional motion according to the contact quantity of legs and inclined planes of double-wheel rimless wheels in the spatial inclined plane, establishes a kinetic equation under the two motion modes according to a motion rule of 'rotation-collision' during motion, analyzes the motion stability, and further provides a method for solving and verifying the kinetic model.
In order to achieve the purpose, the invention adopts the following technical scheme:
a motion behavior analysis method of an asymmetric double-wheel rimless wheel on a spatial slope comprises a double-wheel rimless wheel model and fixed coordinate system establishing step, a motion coordinate system and state space establishing step, two rotation collision mode analysis steps, a single-leg support three-dimensional motion kinetic equation calculating step, a double-leg support three-dimensional motion kinetic equation calculating step and a stability analysis and motion relation analysis step.
The double-wheel rimless wheel model comprises a plane single-wheel rimless wheel L, a plane single-wheel rimless wheel R and a cross beam, wherein two ends of the cross beam are respectively connected with the geometric centers of the plane single-wheel rimless wheel L and the plane single-wheel rimless wheel R; the double-wheel edgeless wheel model is positioned on the spatial inclined plane, and the inclination angle of the spatial inclined plane is marked as alpha; the double-wheel edgeless wheel model moves downwards along the spatial inclined plane under the action of gravity.
And a step of creating a double-wheel rimless wheel model and a fixed coordinate system, which is used for constructing and describing the double-wheel rimless wheel model and the fixed coordinate system, setting an inertia tensor relative to a self geometric center, calculating the inertia tensor relative to the geometric center of the unilateral plane rimless wheel, and calculating the inertia tensor relative to an indirect contact point between the unilateral plane rimless wheel and a spatial inclined plane, wherein the unilateral plane rimless wheel refers to the plane single-wheel rimless wheel L or the plane single-wheel rimless wheel R.
And a motion coordinate system and state space establishing step, which is used for establishing the motion coordinate system fixedly connected on the double-wheel rimless wheel model, describing two transformation forms of a projection type and a rotational symmetry type in the motion process of the double-wheel rimless wheel model of the motion coordinate system, and establishing and describing the system state space of the double-wheel rimless wheel.
And two rotating collision mode analysis steps, which are used for describing and analyzing two rotating collision modes in the motion of the double-wheel rimless wheel model in the space inclined plane, wherein the two rotating collision modes are recorded as single-leg support three-dimensional motion and double-leg support three-dimensional motion.
And calculating a dynamic equation of the single-leg support three-dimensional motion, calculating the dynamic equation of the system according to the theorem of angular momentum of the rotating contact point of the system, and expressing the dynamic equation by using a state space.
And calculating a kinetic equation of the three-dimensional motion of the double-leg support, namely calculating the kinetic equation according to the angular momentum theorem of the rotating contact shaft of the system and expressing the kinetic equation by using a state space.
And a stability analysis and motion relation analysis step, wherein the stability analysis is carried out on the motion behavior of the double-wheel rimless wheel model in the spatial inclined plane, and the motion behavior relation between the double-wheel rimless wheel model and the plane single-wheel rimless wheel model in the spatial inclined plane is analyzed.
Preferably, the length of the cross beam in the double-wheel rimless wheel model is 2h, and the mass of the cross beam is ignored; the plane single-wheel edgeless wheel L and the plane single-wheel edgeless wheel R are both composed of a group of legs with the length of L and the number of n/2, the plane single-wheel edgeless wheel L and the plane single-wheel edgeless wheel R are arranged in a staggered mode, the included angle beta between adjacent legs is 2 pi/n, the effect of asymmetrical arrangement between the two wheels is achieved, and the wheel width leg length ratio is defined as rho 2 h/L; the mass of the plane single-wheel edgeless wheel L and the mass of the plane single-wheel edgeless wheel R are both m/2, and the respective masses are only distributed in the respective longitudinal tangent planes and are centrosymmetric relative to the respective geometric center points;
and (3) recording the geometric center point of the double-wheel rimless wheel as D and the geometric center point of the plane single-wheel rimless wheel R as C, and calculating the inertia tensor of the double-wheel rimless wheel model in a fixed coordinate system { F } as follows:
for the two-wheeled rimless wheel model, the inertia tensor I relative to point DDComprises the following steps:
Figure BDA0002927822290000021
with respect to the inertia tensor I at point CCComprises the following steps:
Figure BDA0002927822290000022
the inertia tensor of the contact point of the plane single-wheel edgeless wheel L or the plane single-wheel edgeless wheel R and the space inclined plane is recorded as IAThe value is:
Figure BDA0002927822290000023
wherein D is a constant associated with the rimless wheel mass distribution and has a value of D > 0.
Preferably, the fixed coordinate system { F } is a spatial coordinate system fixed on the spatial inclined plane, and the coordinate axes thereof are if、jf、kfWherein ifDown a spatial ramp, kfVertical spatial slope upwards, jfPerpendicular to ifkfA plane; i.e. ifkfThe plane is a longitudinal symmetrical plane of the double-wheel edgeless wheel model.
The motion coordinate system { W } is fixed on the double-wheel rimless wheel model and is obtained by performing Euler transformation of a type 3-1-2 on the fixed coordinate system { F }, namely, the { F } winds around kfAfter the shaft rotates by an angle phi, a coordinate system { H } is formed, and a coordinate system { H } winds around ihAnd after the shaft is rotated by the angle psi, a coordinate system { B } is formed, and after the coordinate system { B } is rotated by the angle theta around the jb shaft, a motion coordinate system { W } is formed.
The motion coordinate system (W) has two expression modes; the first type is called a rotational symmetry type, and the expression mode is that when the double-wheel borderless wheel model rotates on a spatial inclined plane, the origin of a motion coordinate system { W } is always on a contact point of the double-wheel borderless wheel model and the spatial inclined plane, and the establishment mode of the motion coordinate system is helpful for improving the performance of kinetic equation calculation; the second is called projection type, and its expression is called: when the plane single-wheel edgeless wheel R is in contact with the space inclined plane, the origin of a motion coordinate system { W } is a contact point of the plane single-wheel edgeless wheel R, when the plane single-wheel edgeless wheel L is in contact with the space inclined plane, the origin is changed into a projection point of the contact point of the plane single-wheel edgeless wheel L on the plane of the plane single-wheel edgeless wheel R, and the establishment mode of the motion coordinate system is helpful for describing and analyzing the motion behavior process of the double-wheel edgeless wheel model (11);
the system state space, expressed as a system of nonlinear second order differential equations, i.e.
Figure BDA0002927822290000031
Wherein
Figure BDA0002927822290000032
M, C, N is a functional relation.
The system state space is defined as
Figure BDA0002927822290000033
Wherein
Figure BDA0002927822290000034
Respectively, phi, psi, theta, as a function of time.
Preferably, for the double-wheel rimless wheel model, the legs colliding with the spatial inclined plane in the motion process are alternately subordinate to the planar single-wheel rimless wheel L and the planar single-wheel rimless wheel R, and the motion cycle of the double-wheel rimless wheel model comprises two groups of rotation and collision processes; during the motion of the planar single-wheel rimless wheel L in contact with the spatial ramp, the system state space is represented as qL(τ); during motion in which the flat single-wheel rimless wheel R is in contact with the spatial ramp, the system state space is represented as qR(τ); according to the number of the legs which are simultaneously contacted with the space inclined plane, the two collision modes are divided into a single-leg support three-dimensional motion mode and a double-leg support three-dimensional motion mode.
The definition "-" represents the moment before the collision of the wheel leg with the inclined plane, "+" represents the moment after the collision of the wheel leg with the inclined plane, and the definition
Figure BDA0002927822290000035
Representing the system state space at the instant after the ith collision of the leg of the planar edgeless wheel R;
Figure BDA0002927822290000036
represents the system state space at the moment before the i +1 th collision of the leg of the plane single-wheel rimless wheel L,
Figure BDA0002927822290000037
after collision becomes
Figure BDA0002927822290000038
Represents the system state space at the instant before the i +2 th collision of the leg of the planar rimless wheel R,
Figure BDA0002927822290000039
after collision becomes
Figure BDA00029278222900000310
In the single-leg support three-dimensional motion mode, the legs and the inclined plane perform continuous contact and separation cyclic motion of 'rotation around a planar single-wheel edgeless wheel R leg, L leg collision around a planar single-wheel edgeless wheel L leg, rotation around a planar single-wheel edgeless wheel L leg, and R leg collision around a planar single-wheel edgeless wheel R leg', only two legs are in contact with the spatial inclined plane at the moment of collision, and only one leg of the system is in contact with the inclined plane at other moments.
In a double-leg support three-dimensional motion mode, the legs and the inclined plane perform continuous contact and separation cyclic motion of 'rotating around AB', -colliding with legs L-rotating around AB-colliding with legs R ', wherein A represents a contact point of the ith leg of the plane single-wheel edgeless wheel L and the space inclined plane, and B' represent contact points of the ith-1 leg and the (i + 1) th leg of the plane single-wheel edgeless wheel R and the space inclined plane respectively; the double-wheel rimless wheel model has two legs on the inclined plane to contact with the inclined plane at other moments except the collision moment.
Preferably, in the single-leg rotation stage, q in the single-leg support three-dimensional motion is calculated according to the system angular momentum theorem on the rotation contact pointRThe kinetic equation for the (τ) process is:
Figure BDA00029278222900000311
dimensionless qRThe equation for the (τ) process is as follows:
Figure BDA00029278222900000312
Figure BDA0002927822290000041
Figure BDA0002927822290000042
Figure BDA0002927822290000043
dimensionless qLThe equation for the (τ) process is as follows:
Figure BDA0002927822290000044
Figure BDA0002927822290000045
Figure BDA0002927822290000046
the moment of inertia D and the time t are subjected to dimensionless transformation as follows:
J＝D/ml2，
Figure BDA0002927822290000051
Figure BDA0002927822290000052
qR(τ) and qLThe equation of dynamics of the (tau) process can be expressed in a state space q,
qLkinetic method of (tau) processThe process is obtained by the same method;
the state space when rotating around the planar single-wheel rimless wheel R is expressed as: s1R(q,λ2,α)，
The state space when rotating around the planar single-wheel rimless wheel L is expressed as: s1L(q,λ2,α)。
Preferably, during the two-legged turning phase, the two legs support a three-dimensional movement, for qR(τ) and qL(tau) calculating the kinetic equation of the rotary contact shaft according to the angular momentum theorem of the system; the force acting on contact A includes gravity mg and vertically upward supporting force F1And a supporting force F of the vertical space inclined plane upward2Q after dimensionless transformationRThe equation for the (τ) process is as follows:
Figure BDA0002927822290000053
Figure BDA0002927822290000054
Figure BDA0002927822290000055
dimensionless qLThe equation for the (τ) process is as follows:
Figure BDA0002927822290000056
Figure BDA0002927822290000061
Figure BDA0002927822290000062
Figure BDA0002927822290000063
and the system motion limit equation is:
Figure BDA0002927822290000064
Figure BDA0002927822290000065
qR(τ) and qLThe equation of dynamics of the (tau) process can be expressed in a state space q,
the state space when rotating around the planar single-wheel rimless wheel R is expressed as: s2R(q,λ2,α)，
The state space when rotating around the planar single-wheel rimless wheel L is expressed as: s2L(q,λ2,α)。
Preferably, the motion coordinate system { W } winds j before and after the collision phasewThe axis being rotated by an angle-2 pi/n, i.e. the rotational movement angle theta is defined by theta-Reset to thetatThe angle relationship of the front and rear systems of the double-leg collision stage is-pi/n
Figure BDA0002927822290000066
Wherein
Figure BDA0002927822290000067
Obtaining q from the theorem of conservation of angular momentumR(τ) angular velocity relationship of the system before and after the two-leg impact phase of the process
Figure BDA0002927822290000068
Figure BDA0002927822290000069
qL(τ) angular velocity relationship of the system before and after the two-leg impact phase of the process
Figure BDA00029278222900000610
Figure BDA00029278222900000611
The kinetic equation can be expressed in terms of state space q in the two-leg collision phase as
q+＝TR(q-,λ2)·q-，
q+＝TL(q-,λ2)·q-Wherein, in the step (A),
Figure BDA0002927822290000071
Figure BDA0002927822290000072
preferably, during the stability analysis, the pacing amplitude function is F for qR(τ) procedure, expressed as a step function
i+1q+＝FR(iq+)，
Figure BDA0002927822290000073
Wherein the function DRIs formed by qRCalculation of the equation of rotation in the (T) process, function TRIs qRCalculation of collision equation, sign, in (tau) process
Figure BDA0002927822290000074
The representative F is formed by combining a function D and a function T through a certain relation; for theqL(τ) procedure, expressed as a step function
i+2q+＝FL(i+1q+)，
Figure BDA0002927822290000075
Wherein the function DLIs formed by qRCalculation of the equation of rotation in the (T) process, function TLIs qL(tau) the collision equation in the process is calculated, and the complete step function of the three-dimensional motion of the double-wheel rimless wheel model on the spatial inclined plane is
i+2q+＝F2(iq+)＝FL(FR(iq+))，
Wherein the square represents that the stride function F is mapped twice;
suppose qkIs a stationary point q*The k-th iteration value of (2), then the (k + 1) -th iteration value calculates the formula qk+1Comprises the following steps:
Figure BDA0002927822290000076
wherein DE2(qk) Is F2At qkJacobian, the iteration cutoff condition can be defined as:
F2(qk)-qk＝qk+1-qk≤qaccuracy，
wherein q isaccuracyFor the calculation accuracy set in advance, when the difference between the state space values obtained by two adjacent calculations is less than the accuracy, it is considered that q isk+1≈qk≈q*。
Preferably, the stability analysis is used for analyzing and verifying the motion condition of the double-wheel rimless wheel model after being subjected to micro interference on the inclined plane after single-leg support three-dimensional motion; the final stable motion of the single-leg support three-dimensional motion of the double-wheel rimless wheel model is obtained by superposing a vibration effect on the basis of the planar single-wheel rimless wheel inclined plane three-dimensional motion;
the projection of the single-leg supported three-dimensional motion of the double-wheel rimless wheel model on the longitudinal section is the inclined plane three-dimensional motion of the plane single-wheel rimless wheel model, and after the two-wheel rimless wheel model is subjected to small interference in stability, the rimless wheel returns to the initial motion direction;
analyzing and verifying the motion condition of the double-wheel rimless wheel model after being subjected to micro interference and colliding for a plurality of times on the inclined plane under the three-dimensional motion of the double-leg support; the two-wheeled rimless model moves back to the original steady state after being disturbed.
Preferably, the motion behavior relation is that for single-leg support three-dimensional motion, the projection of a double-wheel rimless wheel model on a longitudinal section is a planar single-wheel rimless wheel, and the projection of a spatial double-wheel rimless wheel model on the single-leg support three-dimensional motion of a spatial inclined plane on the longitudinal section is the inclined plane three-dimensional motion of the planar single-wheel rimless wheel; the inertia tensor of the winding contact point of the double-wheel borderless wheel model only differs from the inertia tensor of the winding contact point of the plane borderless wheel by a component of-hlm; differential equation of spatial slope motion of a two-wheel rimless wheel model, comprising qR(τ) and qL(τ) the difference from the differential equation of the motion of the spatial ramp of the flat rimless wheel by a number of components with a parameter h;
q of a single motion cycle in a two-wheeled rimless wheel modelRDuring (tau), the gravitational moment is associated with qLThe gravity moments in the process (tau) are the same in size and opposite in direction, the motion effects of the space borderless wheels are approximately mutually offset, and the offset effect proves that the three-dimensional motion of the space bevel of the plane borderless wheel is approximately the projection of the three-dimensional motion of the space bevel on the longitudinal section of the double-wheel borderless wheel model in the process of carrying out single-leg supporting three-dimensional stable motion on the bevel;
for the motion behavior relationship, the final stable motion parameter of the spatial endless wheel in the three-dimensional motion supported by the two legs of the spatial inclined plane is a stable state oscillating back and forth around a certain fixed value; when the number of legs of the double-wheel rimless wheel model approaches infinity, the rimless wheel is equivalent to a cylinder moving on the inclined plane; when the number n of the legs is larger, the speed curve of the system is closer to a straight line, and the system is more equivalent to a cylindrical inclined plane motion model.
In conclusion, the beneficial technical effects obtained by the invention are as follows:
(1) a structural composition scheme of a double-wheel rimless wheel model is provided, and a basic theoretical model closer to geometric characteristics and motion characteristics of human-shaped legs is constructed. The method specifically comprises the steps of establishing an expression mode of the pose and motion state of the model in a spatial inclined plane, providing a calculation method of an inertia tensor of the double-wheel borderless wheel model relative to a geometric center of the model, a calculation method of an inertia tensor of the geometric center of the borderless wheel relative to a unilateral plane, and a calculation method of an inertia tensor of an indirect contact point of the borderless wheel relative to the unilateral plane and the spatial inclined plane, and establishing a state variable of the double-wheel borderless wheel model.
(2) The motion of the double-wheel borderless wheel model on the spatial slope is divided into two motion modes of single-leg support three-dimensional motion and double-leg support three-dimensional motion, the two motion modes are further subdivided into a rotation stage and a collision stage, system kinetic equations under the two motion modes are calculated respectively and serve as a basic model for describing the motion behavior of the double-wheel borderless wheel model. The subdivision processing mode of the motion mode lays a foundation for accurately describing and discussing the motion behaviors of the spatial borderless wheel and deeply researching the conversion among different motion behaviors.
(3) A complete scheme for analyzing the motion behavior of the double-wheel rimless wheel in the space slope is provided, and the scheme is divided into a plurality of operable specific steps and can be used for accurately simulating and calculating the motion behavior of the double-wheel rimless wheel.
(4) A specific method for analyzing the motion stability of the double-wheel rimless wheel on the spatial inclined plane is provided, the motion equivalence is analyzed, and a corresponding physical explanation is provided for the evolution of the dynamic model.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings in the embodiments will be briefly introduced below, the drawings in the following description are only some embodiments of the present invention, and other drawings can be obtained by those skilled in the art without creative efforts.
Fig. 1 is a schematic diagram of a two-wheel rimless wheel model and a coordinate system according to the present invention, in which fig. 1(a) shows a right side view of the two-wheel rimless wheel model, fig. 1(b) shows a top view of the two-wheel rimless wheel model, fig. 1(c) shows a simplified schematic diagram of the two-wheel rimless wheel model, and fig. 1(d) shows a posture description of the two-wheel rimless wheel model on a spatial slope.
Fig. 2 is a schematic flow chart of a method for analyzing the motion behavior and stability of a double-wheel rimless wheel on a spatial slope according to the present invention.
Fig. 3 is a schematic diagram of two transformation forms, a projection type and a rotation symmetry type, of the motion coordinate system { W }, according to the present invention, in which fig. 3(a) shows a rotation symmetry type coordinate expression, and fig. 3(b) shows a projection type coordinate expression.
FIG. 4 is a schematic view of the "rotation-collision" process of the single-leg support three-dimensional motion according to the present invention, in which FIG. 4(a) shows a state
Figure BDA0002927822290000091
FIG. 4(b) shows a state qR(τ), FIG. 4(c) shows the state
Figure BDA0002927822290000092
FIG. 4(d) shows a state qL(τ), FIG. 4(e) shows the state
Figure BDA0002927822290000093
FIG. 5 is a schematic view of the "rotation-collision" process of the two-leg support three-dimensional motion according to the present invention, in which FIG. 5(a) shows a state
Figure BDA0002927822290000094
FIG. 5(b) shows a state qR(τ), FIG. 5(c) shows the state
Figure BDA0002927822290000095
FIG. 5(d) shows a state qL(τ), FIG. 5(e) shows the state
Figure BDA0002927822290000096
Fig. 6 is a schematic diagram of external force moments in two groups of rotation processes of the double-wheel rimless wheel model, wherein fig. 6(a) shows the schematic diagram of the external force moments when the double-wheel rimless wheel model rotates around AB ', and fig. 6(b) shows the schematic diagram of the external force moments when the double-wheel rimless wheel model rotates around AB'.
Fig. 7 is a schematic diagram of a state of a two-wheel borderless wheel model before and after collision during a spatial inclined plane motion, according to the present invention, where fig. 7(a) shows a position of a motion coordinate system { W } of the two-wheel borderless wheel model at an instant before a collision at point B, and fig. 7(B) shows a position of the motion coordinate system { W } of the two-wheel borderless wheel model at an instant after a collision at point B.
FIG. 8 is a graphical representation of the results of a stability analysis of a two-wheel rimless wheel model in accordance with the present invention.
FIG. 9 is a schematic diagram of an equivalent projection of a spatial double-wheel borderless model when a single-leg support of the double-wheel borderless wheel model according to the present invention is stable in three-dimensional motion.
Fig. 10 is a cylindrical equivalent schematic diagram of a two-wheel borderless wheel model when three-dimensional motion is stabilized by two-leg support, according to the present invention, where fig. 10(a) shows the change of angular velocity of rotational motion with the number of collisions when the two-wheel borderless wheel model continues to accelerate, and fig. 10(b) shows a cylindrical equivalent schematic diagram of three-dimensional motion of two-leg support of the spatial borderless wheel model.
Detailed Description
The present invention will be described in detail with reference to the drawings and specific embodiments, and technical solutions in the embodiments of the present invention will be clearly and completely described below. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the scope of the present invention.
As shown in fig. 1, the dual-wheel rimless wheel model 11 is formed by fixedly connecting a planar single-wheel rimless wheel L111 and a planar single-wheel rimless wheel R112 with a cross beam, and two ends of the cross beam are respectively connected with the geometric centers of the two planar single-wheel rimless wheels; the double-wheel rimless wheel model 11 is positioned on a spatial inclined plane, the inclination angle of the spatial inclined plane is marked as alpha, the length of a cross beam of the double-wheel rimless wheel model 11 is 2h, the mass of the cross beam is ignored, the mass of the double-wheel rimless wheel model 11 is m, and the total number of legs is n; the plane single-wheel rimless wheel L111 and the plane single-wheel rimless wheel R112 are arranged in a staggered manner, the included angle beta between adjacent legs is 2 pi/n, the number of the legs is n/2, the length is L, and the wheel width-leg length ratio rho is defined; the mass of the plane single-wheel edgeless wheel L111 and the mass of the plane single-wheel edgeless wheel R112 are both m/2, and the respective masses are only distributed in the respective longitudinal tangential planes and are centrosymmetric relative to the respective geometric center points;
if the geometric center point of the two-wheel rimless wheel model 11 is denoted by D and the geometric center point of the planar single-wheel rimless wheel R112 is denoted by C, the inertia tensor of the two-wheel rimless wheel model 11 is calculated in the fixed coordinate system { F }12 as follows:
two-wheeled model of endless wheel 11, inertia tensor I with respect to point DDComprises the following steps:
Figure BDA0002927822290000101
with respect to the inertia tensor I at point CCComprises the following steps:
Figure BDA0002927822290000102
the tensor of inertia of the contact point of the edgeless wheel leg with the spatial inclined plane is recorded as IAThe value is:
Figure BDA0002927822290000103
wherein D is a constant associated with the rimless wheel mass distribution and has D > 0;
the fixed coordinate system { F }12 is a space coordinate system and is fixed at a space slantOn the surface, the coordinate axes are if、jf、kfWherein ifDown a spatial ramp, kfVertical spatial slope upwards, jfPerpendicular to ifkfA plane; i.e. ifkfThe plane is a longitudinal symmetrical plane of the double-wheel rimless wheel model 11;
the moving coordinate system { W }21 is fixed on the double-wheel rimless wheel model 11 and is obtained by the fixed coordinate system { F }12 through the Euler transformation of 3-1-2 type, namely, the { F }12 winds kfAfter the shaft rotates by an angle phi, a coordinate system { H } is formed, and a coordinate system { H } winds around ihAfter the shaft rotates by the angle psi, a coordinate system { B }, and after the coordinate system { B } rotates by the angle theta around the jb shaft, a motion coordinate system { W }21 is formed;
the system state space 22, expressed as a system of nonlinear second order differential equations, i.e.
Figure BDA0002927822290000104
Wherein
Figure BDA0002927822290000105
M, C, N is a functional relation.
The system state space 22 is defined as
Figure BDA0002927822290000106
Wherein
Figure BDA0002927822290000107
The differential of phi, psi and theta to time;
for the double-wheel borderless wheel model 11, the legs colliding with the spatial inclined plane in the motion process are alternately from the plane single-wheel borderless wheel L111 and the plane single-wheel borderless wheel R112, and the motion cycle of the double-wheel borderless wheel model comprises two groups of rotation and collision processes; the system state space 22 of the contact leg movement process around the plane single-wheel endless wheel L111 uses qL(τ) represents; the contact leg movement process around the plane single-wheel endless wheel R112 uses q as the system state space 22R(τ) represents; according to whichThe two collision modes 31 can be divided into two motion modes of single-leg support three-dimensional motion 32 and double-leg support three-dimensional motion 33 according to the number of legs in contact with the inclined plane;
the definition "-" represents the moment before the collision of the wheel leg with the inclined plane, "+" represents the moment after the collision of the wheel leg with the inclined plane, and the definition
Figure BDA0002927822290000108
A system state space 22 representing the moment after the ith impact of the leg of the planar rimless wheel R;
Figure BDA0002927822290000109
representing the system state space 22 immediately prior to the i +1 st impact of the leg of the planar single-wheeled rimless wheel L111,
Figure BDA00029278222900001010
after collision becomes
Figure BDA00029278222900001011
Representing the system state space 22 at the moment before the i +2 th collision of the leg of the planar rimless wheel R,
Figure BDA0002927822290000111
after collision becomes
Figure BDA0002927822290000112
As shown in fig. 2, the schematic flow chart of the analysis method for the motion behavior and stability of the dual-wheel rimless wheel model 11 on the spatial slope is shown, which includes six steps, namely a dual-wheel rimless wheel model and fixed coordinate system creation step, a motion coordinate system and state space creation step, two rotation collision mode analysis steps, a single-leg supported three-dimensional motion kinetic equation calculation step, a dual-leg supported three-dimensional motion kinetic equation calculation step, and a stability analysis and motion relationship analysis step; wherein the two steps of analysis of the rotating collision mode comprise analysis of single-leg support three-dimensional motion 32 and analysis of double-leg support three-dimensional motion 33; the single-leg support three-dimensional motion mechanical equation calculation step comprises the steps of single-leg rotation stage 411 dynamic equation calculation and collision stage 51 dynamic equation calculation; the calculation step of the two-leg support three-dimensional motion mechanical equation is divided into a two-leg rotation stage 412 and a dynamic equation calculation collision stage 51.
As shown in FIG. 3, the motion coordinate system { W }21 has two expressions; the method shown in fig. 3(a) is called a rotational symmetry formula, and is expressed in such a way that when the dual-wheel rimless wheel model 11 rotates on the spatial inclined plane, the origin of the motion coordinate system { W }21 is always on the contact point of the dual-wheel rimless wheel model 11 and the spatial inclined plane; fig. 3(b) shows a method called a projection method, which is expressed as: when the plane single-wheel edgeless wheel R112 is in contact with the space inclined plane, the origin of the motion coordinate system { W }21 is the contact point, and when the plane single-wheel edgeless wheel L111 is in contact with the space inclined plane, the origin is changed into the projection point of the contact point of the plane single-wheel edgeless wheel L111 on the plane where the plane single-wheel edgeless wheel R112 is located.
As shown in fig. 4, the single-leg support three-dimensional motion 32, except for the moment of collision, only one leg of the system is in contact with the inclined plane at other moments, and the wheel leg and the inclined plane perform continuous contact separation cyclic motion of 'rotating around a planar single-wheel edgeless wheel R112 leg-colliding around a planar single-wheel edgeless wheel L111 leg-rotating around a planar single-wheel edgeless wheel L111 leg-colliding around a planar single-wheel edgeless wheel R112 leg';
in the one-leg rotation phase 411, for qR(τ) and qL(tau) calculating the dynamic equation of the system according to the theorem of angular momentum of the system to the rotating contact point, and obtaining the equation of the system for qR(τ) process, its kinetic equation is:
Figure BDA0002927822290000113
dimensionless qRThe equation for the (τ) process is as follows:
Figure BDA0002927822290000114
Figure BDA0002927822290000115
Figure BDA0002927822290000116
Figure BDA0002927822290000121
the moment of inertia D and the time t are subjected to dimensionless transformation as follows:
J＝D/ml2，
Figure BDA0002927822290000122
Figure BDA0002927822290000123
qR(τ) and qLThe equation of dynamics of the (tau) process can be expressed in a state space q,
the state space when rotating around the planar single-wheel rimless wheel R (112) is expressed as: s1R(q,λ2,α)，
The state space when rotating around the planar single-wheel rimless wheel L (111) is expressed as: s1L(q,λ2,α)。
As shown in fig. 5, the two legs support the three-dimensional motion 33, except for the moment of collision, the two-wheel rimless wheel model 11 has two legs on the inclined plane to contact with the inclined plane at other times, and the wheel legs and the inclined plane perform continuous contact and separation cyclic motion of ' rotating around AB ', -leg L collision ', -rotating around AB ', -leg R collision '; wherein A represents the contact point of the ith leg of the planar single-wheel rimless wheel L111 with the spatial ramp, and B' represent the contact points of the (i-1) th leg and the (i + 1) th leg of the planar single-wheel rimless wheel R112 with the spatial ramp, respectively.
As shown in FIG. 6, during the two-legged turn phase 412, for qR(τ) and qL(tau) calculating the kinetic equation of the rotary contact shaft according to the angular momentum theorem of the system; the force acting on contact A includes gravity mg and vertically upward supporting force F1And a supporting force F of the vertical space inclined plane upward2Q after dimensionless transformationRThe equation for the (τ) process is as follows:
Figure BDA0002927822290000124
Figure BDA0002927822290000125
Figure BDA0002927822290000126
Figure BDA0002927822290000131
and the system motion limit equation is:
Figure BDA0002927822290000132
Figure BDA0002927822290000133
qR(τ) and qLThe equation of dynamics of the (tau) process can be expressed in a state space q,
the state space when rotating around the planar single-wheel rimless wheel R112 is expressed as: s2R(q,λ2,α)，
The state space when rotating around the planar single-wheel rimless wheel L111 is expressed as: s2L(q,λ2,α)。
As shown in FIG. 7, the motion coordinate system { W }21 wraps around j before and after the collision phase 51wThe axis being rotated by an angle-2 pi/n, i.e. the rotational movement angle theta is defined by theta-Reset to theta+The angular relationship of the system before and after the two-leg collision phase 51 is-pi/n
Figure BDA0002927822290000134
Wherein
Figure BDA0002927822290000135
Obtaining q from the theorem of conservation of angular momentumR(τ) course the angular velocity of the system before and after the two-leg impact phase 51 is related to
Figure BDA0002927822290000136
Figure BDA0002927822290000137
qL(τ) course angular velocity relationship of the front and rear systems during the two-leg impact phase (51) is
Figure BDA0002927822290000138
Figure BDA0002927822290000139
The kinetic equation in the two-leg collision phase 51 can be expressed in terms of a state space q as
q+＝TR(q-,λ2)·q-，
q+＝TL(q-,λ2)·q-，
Wherein the content of the first and second substances,
Figure BDA00029278222900001310
Figure BDA00029278222900001311
as shown in fig. 8, it is the change situation of the state space q of the dual-wheel borderless wheel model 11 after the dual-wheel borderless wheel model 11 is calculated by a numerical iteration method in the motion process;
for stability analysis 61, the pacing amplitude function is F, for qR(τ) procedure, expressed as a step function
i+1q+＝FR(iq+)，
Figure BDA0002927822290000141
Wherein the function DRIs formed by qRCalculation of the equation of rotation in the (T) process, function TRIs qRCalculation of collision equation, sign, in (tau) process
Figure BDA0002927822290000142
The representative F is formed by combining a function D and a function T through a certain relation; for qL(τ) procedure, expressed as a step function
i+2q+＝FL(i+1q+)，
Figure BDA0002927822290000143
Wherein the function DLIs formed by qRCalculation of the equation of rotation in the (T) process, function TLIs qL(tau) the complete step function of the three-dimensional motion of the double-wheel rimless wheel model (11) on the spatial slope is calculated as
i+2q+＝F2(iq+)＝FL(FR(iq+))，
Wherein the square represents that the stride function F is mapped twice;
suppose qkIs a stationary point q*The k-th iteration value of (2), then the (k + 1) -th iteration value calculates the formula qk+1Comprises the following steps:
Figure BDA0002927822290000144
wherein, DF2(qk) Is F2At qkJacobian, the iteration cutoff condition can be defined as:
F2(qk)-qk＝qk+1-qk≤qaccuracy，
wherein q isaccuracyFor the calculation accuracy set in advance, when the difference between the state space values obtained by two adjacent calculations is less than the accuracy, it is considered that q isk+1≈qk≈q*。
As shown in fig. 9, stability analysis 61 is performed under the single-leg supported three-dimensional motion 32, and the motion situation that the double-wheel rimless wheel model 11 undergoes several times of collision on the inclined plane after being slightly disturbed under the single-leg supported three-dimensional motion 32 is verified; the final stable motion of the single-leg supported three-dimensional motion 32 of the dual-wheel rimless wheel model 11 is the superposition of the vibration effect on the basis of the planar single-wheel rimless wheel inclined plane three-dimensional motion; the projection of the single-leg supported three-dimensional motion 32 of the double-wheel rimless wheel model 11 on the longitudinal section is the inclined plane three-dimensional motion of the plane single-wheel rimless wheel model, and after small interference during stabilization, the rimless wheel returns to the initial motion direction;
the motion behavior relation 62 is that for the single-leg supported three-dimensional motion 32, the projection of the double-wheel rimless wheel model 11 on the longitudinal section is a planar single-wheel rimless wheel through geometric analysis, and the projection of the spatial double-wheel rimless wheel model 11 on the single-leg supported three-dimensional motion of the spatial inclined plane on the longitudinal section is the inclined plane three-dimensional motion of the planar single-wheel rimless wheel through kinematic analysis; the inertia tensor of the winding contact point of the double-wheel borderless wheel model 11 only differs from the inertia tensor of the winding contact point of the plane borderless wheel by a component of-hlm; two wheels withoutDifferential equation of spatial slope motion of edge wheel model 11, comprising qR(τ) and qL(τ) the process differs from the differential equation of the spatial ramp motion of the flat rimless wheel by a number of components with a parameter h.
As shown in fig. 10(a), for fig. 10(a), the abscissa indicates the number of times of collision between a leg and an inclined plane, the ordinate table shows that the spatial borderless wheel model is on the spatial inclined plane, the number of leg parameters n is 50, 55 and 60, the inclination angle parameter α ═ pi/10, the width-leg length ratio ρ of the borderless wheel is 0.8, the dimensionless moment of inertia 2J is 0.5, and the dimensionless angular acceleration of motion of the cylinder on the inclined plane α ═ pi/10 is calculated by the ideal cylindrical inclined plane motion model as follows:
Figure BDA0002927822290000151
the angular acceleration of the spatial edgeless wheel slope motion is estimated as follows:
Figure BDA0002927822290000152
for the stability analysis 61, the motion situation that the double-wheel rimless wheel model 11 undergoes a plurality of times of collision on the inclined plane after being slightly interfered under the double-leg supported three-dimensional motion 33 is verified; the two-wheeled rimless wheel model 11 moves back to the original steady state after being disturbed.
Q of a single motion cycle in the two-wheeled rimless wheel model 11RDuring (tau), the gravitational moment is associated with qLThe gravity moments in the process (tau) are the same in size and opposite in direction, the motion effects of the spatial borderless wheels can be considered to be mutually offset, and the offset effect proves that the spatial slope three-dimensional motion of the planar borderless wheels can be considered to be the projection of the spatial slope three-dimensional motion of the dual-wheel borderless wheel model 11 on the longitudinal section in the process of single-leg supporting three-dimensional stable motion of the spatial borderless wheels on the slope;
for the motion behavior relationship 62, the final stable motion parameter of the spatial borderless wheel 11 in the two-leg supported three-dimensional motion 33 of the spatial slope is a stable state oscillating back and forth around a certain fixed value; when the number of legs of the double-wheel rimless wheel model 11 approaches infinity, the rimless wheel approaches a cylinder moving on the inclined plane; when the number n of the legs is larger, the speed curve of the system is closer to a straight line, and the system is more equivalent to a cylindrical inclined plane motion model.
The principle and the implementation of the present invention are explained by applying specific examples in the present specification, the above description of the embodiments is only used to help understand the method and the core idea of the present invention, and the content of the present specification should not be construed as limiting the present invention.

Claims (10)

1. A motion behavior analysis method of an asymmetric double-wheel rimless wheel model on a spatial inclined plane is characterized by comprising the following steps: the method comprises the following steps:
step 1, establishing a double-wheel rimless wheel model and a fixed coordinate system: constructing and describing a double-wheel rimless wheel model in a fixed coordinate system { F }, wherein the double-wheel rimless wheel model comprises a plane single-wheel rimless wheel L, a plane single-wheel rimless wheel R and a cross beam, and two ends of the cross beam are respectively connected with the geometric centers of the plane single-wheel rimless wheel L and the plane single-wheel rimless wheel R; the double-wheel rimless wheel model is positioned on the spatial inclined plane, the inclination angle of the spatial inclined plane is recorded as alpha, the double-wheel rimless wheel model moves downwards along the spatial inclined plane under the action of gravity, and the inertia tensor I of the double-wheel rimless wheel model relative to the geometric center of the double-wheel rimless wheel model is calculatedDCalculating the inertia tensor I relative to the geometric center of the unilateral plane rimless wheelCCalculating the inertia tensor I of the indirect contact point between the edgeless wheel and the spatial inclined plane relative to the unilateral planeAThe unilateral plane rimless wheel refers to a plane single-wheel rimless wheel L or a plane single-wheel rimless wheel R;
step 2, establishing a motion coordinate system and a state space: establishing a motion coordinate system { W } fixedly connected to the double-wheel rimless wheel model, describing two definition forms of a projection type and a rotational symmetry type of the motion coordinate system { W } in the motion process of the double-wheel rimless wheel model, establishing a system state space of the double-wheel rimless wheel and recording the system state space as q, and winding the plane single-wheel rimless wheel L and the plane single-wheel rimless wheel L according to the double-wheel rimless wheel modelThe difference of the motion states of the wheels R represents the system state space as qL(τ) and qR(τ)；
Step 3, analyzing two rotation collision modes: two rotation collision modes of the double-wheel rimless wheel model moving in the spatial inclined plane are researched and respectively recorded as single-leg support three-dimensional movement and double-leg support three-dimensional movement;
step 4, calculating a dynamic equation of the single-leg support three-dimensional motion: respectively calculating a system dynamic equation of the single-leg rotation stage and a system dynamic equation of the single-leg collision stage in the single-leg support three-dimensional motion of the double-wheel rimless wheel model in a spatial inclined plane;
step 5, calculating a kinetic equation of the two-leg support three-dimensional motion: respectively calculating a system dynamic equation of the double-leg rotation stage and a system dynamic equation of the double-leg collision stage in the double-leg support three-dimensional motion of the double-wheel rimless wheel model in the spatial inclined plane;
step 6, stability analysis and motion relation analysis: and carrying out stability analysis on the motion behavior of the double-wheel rimless wheel model in the spatial inclined plane, and analyzing the motion behavior relation between the double-wheel rimless wheel model and the planar single-wheel rimless wheel model in the spatial inclined plane.
2. The method for analyzing the motion behavior of the asymmetric double-wheel rimless wheel model on the spatial inclined plane according to claim 1, characterized in that:
the length of a cross beam in the double-wheel rimless wheel model is 2h, and the mass of the cross beam is ignored; the plane single-wheel rimless wheel L and the plane single-wheel rimless wheel R are both composed of a group of legs with the length of L and the number of n/2, the plane single-wheel rimless wheel L and the plane single-wheel rimless wheel R are arranged in an asymmetric staggered mode, the included angle beta between adjacent legs is 2 pi/n, and the wheel width-leg length ratio is defined as rho 2 h/L; the mass of the plane single-wheel rimless wheel L and the mass of the plane single-wheel rimless wheel R are both m/2, and the respective masses are only distributed in the respective longitudinal cutting planes and are centrosymmetric relative to the respective geometric center points;
and the geometric center point of the double-wheel rimless wheel model is D, and the geometric center point of the plane single-wheel rimless wheel L or the plane single-wheel rimless wheel R is C, then the inertia tensor of the double-wheel rimless wheel model is calculated in the fixed coordinate system { F } as follows:
inertia tensor I of the double-wheel non-edge wheel model relative to the D pointDIs composed of
Figure FDA0002927822280000011
The two-wheel rimless model has an inertia tensor I relative to the point CCIs composed of
Figure FDA0002927822280000021
Inertia tensor I of contact point of the plane single-wheel rimless wheel L or the plane single-wheel rimless wheel R and the spatial inclined plane in the double-wheel rimless wheel modelAIs composed of
Figure FDA0002927822280000022
Wherein D is a constant associated with the mass distribution of the two-wheeled rimless model, and has a value of D > 0.
3. The method for analyzing the motion behavior of the asymmetric double-wheel rimless wheel model on the spatial inclined plane according to claim 2, characterized in that:
the fixed coordinate system { F } is a space coordinate system and is fixed on the space inclined plane, and the coordinate axes are i respectivelyf、jf、kfWherein ifDown the spatial ramp, kfPerpendicular to the inclined plane of the space, upward, jfPerpendicular to ifkfA plane; i.e. ifkfThe plane is a longitudinal symmetrical plane of the double-wheel rimless wheel model;
the motion coordinate system { W } is fixed on the double-wheel rimless wheel model and passes through the fixedFixed coordinate system { F } winding kfAfter the shaft rotates by an angle phi, a coordinate system { H } is formed, and the coordinate system { H } winds around ihForming a coordinate system { B } after the shaft is rotated by the angle psi, and forming the motion coordinate system { W } after the coordinate system { B } is rotated by the angle theta around the jb shaft;
the motion coordinate system { W } comprises two expression modes, wherein the first expression mode is a rotational symmetry mode, and the expression mode is as follows: when the double-wheel rimless wheel model rotates on the spatial inclined plane, the origin of the motion coordinate system { W } is always the contact point of the double-wheel rimless wheel model and the spatial inclined plane, and the establishment mode of the motion coordinate system is beneficial to improving the performance of kinetic equation calculation; the second is a projection type, which is expressed as follows: when the planar single-wheel rimless wheel R is in contact with the space inclined plane, the origin of the motion coordinate system { W } is the contact point, when the planar single-wheel rimless wheel L is in contact with the space inclined plane, the origin is changed into a projection point of the contact point of the planar single-wheel rimless wheel L on the plane where the planar single-wheel rimless wheel R is located, and the establishment mode of the motion coordinate system is helpful for describing and analyzing the motion behavior process of the double-wheel rimless wheel model;
the system state space is expressed in the form of a nonlinear second order differential equation system, i.e.
Figure FDA0002927822280000023
Wherein
Figure FDA0002927822280000024
M, C, N is a functional relation;
the system state space is defined as
Figure FDA0002927822280000025
Wherein
Figure FDA0002927822280000026
Respectively, phi, psi, theta, as a function of time.
4. The method for analyzing the motion behavior of the asymmetric double-wheel rimless wheel model on the spatial inclined plane according to claim 3, wherein:
in the double-wheel rimless wheel model, legs colliding with the spatial inclined plane in the motion process are alternately subordinate to the planar single-wheel rimless wheel L and the planar single-wheel rimless wheel R; during the motion process that the plane single-wheel edgeless wheel L is in contact with the spatial inclined plane, the system state space is expressed as qL(τ) said system state space is represented as q during the motion of said planar single-wheel rimless wheel R in contact with said spatial rampR(τ)；
Defining the superscript symbol "-" as the moment before the leg collides with the inclined plane, and the superscript symbol "+" as the moment after the leg collides with the inclined plane, then
Figure FDA0002927822280000031
Represents the system state space of the plane single-wheel rimless wheel R at the moment after the ith collision,
Figure FDA0002927822280000032
the system state space representing the moment before the i +1 th collision of the L-th edgeless wheel of the plane single wheel has
Figure FDA0002927822280000033
After collision becomes
Figure FDA0002927822280000034
The system state space representing the instant before the i +2 th collision of the planar single-wheel rimless wheel R is provided
Figure FDA0002927822280000035
After collision becomes
Figure FDA0002927822280000036
The motion cycle of the double-wheel rimless wheel model comprises two groups of rotation and collision processes;
according to the number of legs simultaneously contacted with the space inclined plane, the two collision modes are divided into single-leg support three-dimensional motion and double-leg support three-dimensional motion;
in the single-leg support three-dimensional motion mode, the legs and the inclined plane perform continuous contact separation cyclic motion of 'rotating around the planar single-wheel edgeless wheel R leg, colliding around the planar single-wheel edgeless wheel L leg, rotating around the planar single-wheel edgeless wheel L leg, colliding around the planar single-wheel edgeless wheel R leg', and only two legs contact with the spatial inclined plane at the moment of collision, while only one leg of the system contacts with the spatial inclined plane at other moments;
in the double-leg support three-dimensional motion mode, the legs and the inclined plane perform continuous contact separation cyclic motion of rotating around AB ', colliding with a L leg of a plane single-wheel endless wheel, rotating around AB, colliding with an R leg of the plane single-wheel endless wheel, wherein A represents a contact point of the ith leg of the plane single-wheel endless wheel L and the space inclined plane, and B' represent contact points of the ith-1 leg and the (i + 1) th leg of the plane single-wheel endless wheel R and the space inclined plane respectively; and at other moments except the moment of collision, the double-wheel rimless wheel model has two legs simultaneously contacted with the space inclined plane.
5. The method for analyzing the motion behavior of the asymmetric double-wheel rimless wheel model on the spatial inclined plane according to claim 3, wherein:
in the single-leg rotation stage, q in the single-leg support three-dimensional motion is calculated according to the angular momentum theorem of the system on the rotation contact pointRThe kinetic equation for the (τ) process is:
Figure FDA0002927822280000037
dimensionless qRThe equation for the (τ) process is as follows:
Figure FDA0002927822280000038
Figure FDA0002927822280000039
Figure FDA0002927822280000041
Figure FDA0002927822280000042
the moment of inertia D and the time t are subjected to dimensionless transformation as follows:
J＝D/ml2，
Figure FDA0002927822280000043
Figure FDA0002927822280000044
qR(τ) and qLThe equation of dynamics of the (tau) process can be expressed in a state space q,
the state space when rotating around the planar single-wheel rimless wheel R is expressed as: s1R(q，λ2，α)，
The state space when rotating around the planar single-wheel rimless wheel L is expressed as: s1L(q，λ2，α)。
6. The method for analyzing the motion behavior of the asymmetric double-wheel rimless wheel model on the spatial inclined plane according to claim 3, wherein:
during the two-leg rotation phase, the two legs support three-dimensional motion for qR(τ) and qL(τ) process, the kinetic equations of which are all based on the system's theorem on angular momentum of rotating contact axisLine calculation; the force acting on the contact point A with moment includes gravity mg, and supporting force F vertically upward1And a supporting force F directed upward perpendicular to the inclined plane2Q after dimensionless transformationRThe equation for the (τ) process is as follows:
Figure FDA0002927822280000045
Figure FDA0002927822280000046
Figure FDA0002927822280000051
and the system motion limit equation is:
Figure FDA0002927822280000052
Figure FDA0002927822280000053
qR(τ) and qLThe equation of dynamics of the (tau) process can be expressed in a state space q,
the state space when rotating around the planar single-wheel rimless wheel R is expressed as: s2R(q，λ2，α)，
The state space when rotating around the planar single-wheel rimless wheel L is expressed as: s2L(q，λ2，α)。
7. The method for analyzing the motion behavior of the asymmetric double-wheel rimless wheel model on the spatial inclined plane according to claim 4, wherein:
before and after the collision stage, the motion coordinate system { W } winds around jwThe axis being rotated by an angle-2 pi/n, i.e. the rotational movement angle theta is defined by theta-Reset to theta+The angle relationship of the front and rear systems of the double-leg collision stage is-pi/n
Figure FDA0002927822280000054
Wherein
Figure FDA0002927822280000055
Obtaining q from the theorem of conservation of angular momentumR(τ) angular velocity relationship of the system before and after the two-leg impact phase is
Figure FDA0002927822280000056
Figure FDA0002927822280000057
qL(τ) angular velocity relationship of the system before and after the two-leg impact phase is
Figure FDA0002927822280000058
Figure FDA0002927822280000059
The kinetic equation in the two-leg collision phase is expressed by a state space q
q+＝TR(q-，λ2)·q-，
q+＝TL(q-，λ2)·q-，
Wherein the content of the first and second substances,
Figure FDA0002927822280000061
Figure FDA0002927822280000062
8. the method for analyzing the motion behavior of the asymmetric double-wheel rimless wheel model on the spatial inclined plane according to claim 4, wherein:
the stability analysis, step-and-width function is F, for qR(τ) procedure, expressed as a step function
i+1q+＝FR(iq+)，
Figure FDA0002927822280000063
Wherein the function DRIs formed by qRCalculation of the equation of rotation in the (T) process, function TRIs qRCalculation of collision equation, sign, in (tau) process
Figure FDA0002927822280000064
The representative F is formed by combining a function D and a function T through a certain relation; for qL(τ) procedure, expressed as a step function
i+2q+＝FL(i+1q+)，
Figure FDA0002927822280000065
Wherein the function DLIs formed by qRCalculation of the equation of rotation in the (T) process, function TLIs qL(τ) calculation of the collision equation for a complete step function of the three-dimensional motion of the two-wheel rimless wheel model over the spatial rampIs composed of
i+2q+＝F2(iq+)＝FL(FR(iq+))，
Wherein the square represents that the stride function F is mapped twice;
suppose qkIs a stationary point q*The k-th iteration value of (2), then the (k + 1) -th iteration value calculates the formula qk+1Comprises the following steps:
Figure FDA0002927822280000066
wherein, DF2(qk) Is F2At qkJacobian, the iteration cutoff condition can be defined as:
F2(qk)-qk＝qk+1-qk≤qaccuracy，
wherein q isaccuracyFor the calculation accuracy set in advance, when the difference between the state space values obtained by two adjacent calculations is less than the accuracy, it is considered that q isk+1≈qk≈q*。
9. The method for analyzing the motion behavior of the asymmetric double-wheel rimless wheel model on the spatial inclined plane according to claim 1, characterized in that:
the stability analysis is used for analyzing and verifying the motion condition that the double-wheel rimless wheel model is subjected to micro interference and then is collided for a plurality of times on the inclined plane under the single-leg support three-dimensional motion; the final stable motion of the single-leg support three-dimensional motion of the double-wheel rimless wheel model is obtained by superposing a vibration effect on the basis of the planar single-wheel rimless wheel inclined plane three-dimensional motion;
the projection of the single-leg supported three-dimensional motion of the double-wheel rimless wheel model on the longitudinal section is the inclined plane three-dimensional motion of the plane single-wheel rimless wheel model, and after the three-dimensional motion is slightly interfered during stabilization, the rimless wheel returns to the initial motion direction;
and the stability analysis is to analyze and verify the motion condition that the double-wheel rimless wheel model is subjected to a plurality of times of collision on the inclined plane after being subjected to micro interference under the double-leg support three-dimensional motion, and the double-wheel rimless wheel model is subjected to interference and then moves back to the initial stable state.
10. The method for analyzing the motion behavior of the asymmetric double-wheel rimless wheel model on the spatial inclined plane according to claim 4, wherein:
the motion behavior relation is analyzed from the aspect of geometry, the projection of the double-wheel rimless wheel model on a longitudinal section is a plane single-wheel rimless wheel, and the projection of the double-wheel rimless wheel model on the single-leg support three-dimensional motion of a spatial inclined plane on the longitudinal section is the inclined plane three-dimensional motion of the plane single-wheel rimless wheel; the inertia tensor of the winding contact point of the double-wheel borderless wheel model and the inertia tensor of the winding contact point of the plane borderless wheel model only have a difference of-hlm components; the differential equation of the spatial slope motion of the double-wheel rimless wheel model comprises qR(τ) and qL(τ) the difference from the differential equation of the motion of the spatial ramp of the flat rimless wheel by a number of components with a parameter h;
q of a single motion cycle of the two-wheeled rimless wheel modelRDuring (tau), the gravitational moment is associated with qLThe gravity moments in the process of (tau) are the same in size and opposite in direction, the action effects on the double-wheel rimless wheel model are approximately mutually offset, and the offset effect shows that the projection of the double-wheel rimless wheel model on a longitudinal section is approximately the three-dimensional motion of a spatial inclined plane of a plane rimless wheel in the process of carrying out single-leg supporting three-dimensional stable motion on the inclined plane;
for the motion behavior relationship, the final stable motion of the spatial endless wheel in the three-dimensional motion supported by the two legs of the spatial inclined plane is in a stable state oscillating back and forth around a certain fixed value; when the number of legs of the double-wheel borderless wheel model approaches infinity, the borderless wheel is equivalent to a cylinder moving on an inclined plane, and the larger the number n of legs is, the closer the system moves to the inclined plane of the cylinder.
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