CN110348140A - Based on towing away from two-wheel robot modeling and static balance method and device - Google Patents
Based on towing away from two-wheel robot modeling and static balance method and device Download PDFInfo
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Abstract
The invention discloses it is a kind of based on towing away from two-wheel robot modeling and static balance method and device, wherein, this method comprises: setting multi link multi-joint system for two-wheel robot system, defining multiple coordinate systems in multi link multi-joint system and calculating the towing of two-wheel robot system away from range；Two constraint equations are established according to the wheel geometrical property of closed loop moving chain suffered by multi link multi-joint system and two-wheel robot system and establish kinematics model；Kinematics model is solved using lagrange equations of the first kind to obtain two-wheel Dynamic Models of Robot Manipulators, and the singular value of controllability matrix, the domain of attraction of closed loop controller and control cost are analyzed, based on the analysis results towing away from determine to meet in range the towing of demand for control away from.This method can be reflected in it is different towing away from lower handlebar corner and height of center of mass variation non-linear relations, and can for towing away from selection a set of analysis process is provided, improve the control effect of static balance.
Description
Technical field
The present invention relates to mechanical system modeling and Dynamics analysis technology field, in particular to it is a kind of based on towing away from it is double
Wheel robot modeling and static balance method and device.
Background technique
Pull away from be handlebar shaft and ground intersection point between front-wheel ground contact points at a distance from, to two-wheel robot utilize
The balance control that handlebar turns to has important influence.
For towing away from the two-wheel robot being not zero, the height of center of mass of handlebar energy fine tuned robot is rotated, prevents from toppling over.
When the speed of two-wheel robot reaches a certain level, the restoring moment that ground provides makes front-wheel shaft without additional control force
It can be by robot auto-strengthening；But when two-wheel robot be in Ultra-Low Speed it is even static when, this self-stability disappears, at this time flat
Weighing apparatus relies primarily on handlebar and turns to realization.It follows that for the static balance for having more challenge, towing is away from becoming key.It is existing
Some handlebars turn to static balance research and point out, based on positive controller of the towing away from design, domain of attraction is smaller, robustness is poor,
Therefore towing is probed into away from the influence to control performance with regard to necessary.However, current research all only for fixed towing away from
Robot is modeled, and height of center of mass variation is reduced to linear model.Such model be not enough to analyze towing away from shadow
Ring, thus there is an urgent need to establish be suitable for it is any towing away from kinetic model.
Summary of the invention
The present invention is directed to solve at least some of the technical problems in related technologies.
For this purpose, an object of the present invention is to provide it is a kind of based on towing away from two-wheel robot modeling and static balance
Method, this method can reflect the non-linear relation in different towings away from lower handlebar corner and height of center of mass variation, and can be
Pull away from selection a set of analysis process is provided, to improve the control effect of static balance.
It is another object of the present invention to propose it is a kind of based on towing away from two-wheel robot modeling and static balance fill
It sets.
In order to achieve the above objectives, one aspect of the present invention embodiment propose it is a kind of based on towing away from two-wheel robot modeling
With static balance method, comprising:
S1 sets the two-wheel robot system when detection two-wheel robot system meets default equivalence setting condition
It is set to multi link multi-joint system, and defines multiple coordinate systems in the multi link multi-joint system, according to the multiple seat
The geometrical relationship of mark system calculates the towing of the two-wheel robot system away from range；
S2, according to the wheel of closed loop moving chain suffered by the multi link multi-joint system and the two-wheel robot system
Geometrical property establishes two constraint equations, and establishes kinematics model according to described two constraint equations；
S3 solves the kinematics model using lagrange equations of the first kind to obtain two-wheel robot dynamics
Model, and according to the two-wheel Dynamic Models of Robot Manipulators to the singular value of controllability matrix, the domain of attraction of closed loop controller and
Control cost is analyzed, based on the analysis results the towing away from determine to meet in range the towing of demand for control away from.
The embodiment of the present invention based on towing away from two-wheel robot modeling and static balance method, first according to movement about
Two-wheel robot is equivalent to the hinged multi link multi-joint system of two car body of front and back from the angle of multi-rigid-body system by beam.Secondly
The contact performance of closed loop moving chain and wheel and ground based on system, establishes two constraint equations, obtains the kinematics of system
Model.Followed by the kinetic model of lagrange equations of the first kind derivation system.Finally, from the surprise of system controllability and measurability matrix
Three different value, closed loop controller domain of attraction and control energy consumption aspects are pulled to analyze away from the influence to static balance.It can reflect
It is different towing away from lower handlebar corner and height of center of mass variation non-linear relations, and can for towing away from selection provide it is a set of
Analysis process, to improve the control effect of static balance.
In addition, it is according to the above embodiment of the present invention based on towing away from two-wheel robot modeling and static balance method also
It can have following additional technical characteristic:
Further, in one embodiment of the invention, the default equivalence setting condition, comprising:
Pitching caused by setting aftercarriage mass center only includes rolling and turned to by handlebar；
Setting front and back two-wheeled is checked, without relative motion between vehicle frame, and is pure rolling between rear-wheel and ground；
Tire thickness and deformation are ignored in setting, and front and back two-wheeled is considered as equal-sized rigid thin discs.
Further, in one embodiment of the invention, the multiple coordinate system are as follows:
(1) inertial reference system { I }, A0Xyz: origin is fixed on A0Point, x-axis is by A0It is directed toward E0, z-axis straight down, y-axis and x
Axis and z-axis form right-handed system；Wherein, A0Rear wheel and ground contact points when being rotated for handlebar, E0Front vehicle wheel when being rotated for handlebar
With the contact point on ground；
(2) rear-wheel coordinate system { B }, Bxbybzb: origin is fixed on B point, xbAxis is parallel with inertial reference system x-axis, z and y two
Axis can be rotated around x axis by inertial reference systemAngle obtains, then from { I } to the spin matrix of { B } are as follows:
Wherein, B is the rear wheel center of circle,For the roll angle of aftercarriage；
(3) handlebar coordinate system { C }, Cxcyczc: origin is fixed on C point, ycAxis and rear-wheel coordinate system ybAxis is parallel, x and z two
Axis can be by { B } around ybThe axis rotation angle θ+η obtains, then from { B } to the spin matrix of { C } are as follows:
Wherein, C is the tie point of handlebar revolute and Rear frame, and θ is the pitch angle of aftercarriage, and η is handlebar inclination angle；
Handlebar inclination angle η meets following geometrical constraint:
Wherein, θ0The pitch angle of Rear frame connecting rod vector when be handlebar corner being zero, ε is Rear frame connecting rod established angle；
(4) front-wheel coordinate system { D }, Dxdydzd: origin is fixed on D point, zdAxis and handlebar coordinate system zcAxis is parallel, x and y two
Axis can be by handlebar coordinate system around zcThe axis rotation angle δ obtains, then from { C } to the spin matrix of { D } are as follows:
Wherein, D is the front vehicle wheel center of circle, and δ is handlebar corner；
(5) aftercarriage coordinate system { G1, G1x1y1z1: origin is fixed on G1, aftercarriage coordinate system is by rear-wheel coordinate system around yb
The axis rotation angle θ obtains, and enablesMiddle η=0 can be obtained from { B } system to { G1System spin matrix
Wherein, G1For aftercarriage mass center；
(6) front vehicle body coordinate system { G2, G2x2y2z2: origin is fixed on G2, front vehicle body coordinate system is parallel with front-wheel coordinate system,
G2For front vehicle body mass center.
Further, in one embodiment of the invention, the towing of the two-wheel robot system is away from range are as follows:
Wherein, R is radius of wheel, lrFor the length of line segment BC, d is the length of line segment CC ', lfFor front frame line segment C ' D's
Length, λ are handlebar front fork angles, and η is handlebar inclination angle.
Further, in one embodiment of the invention, described to be closed according to suffered by the multi link multi-joint system
The wheel geometrical property of ring kinematic chain and the two-wheel robot system establishes two constraint equations, comprising:
Constraint 1, the constraint of closed loop moving chain
Wherein, ez=[0,0,1]TFor the z-axis unit direction vector of inertial reference system { I }, r1It is the rear wheel that B is directed toward by A
Connecting rod vector；r2It is the Rear frame connecting rod vector that C is directed toward by B；r3It is the front frame connecting rod vector that D is directed toward by C；r4It is to be directed toward by D
The front vehicle wheel connecting rod vector of E, subscript indicate coordinate system corresponding to the vector,For from { B } to the spin matrix of { I },
For from { C } to the spin matrix of { B },For from { D } to the spin matrix of { C }；
Constraint 2, wheel geometrical constraint
Wherein, nyFor the y of front-wheel coordinate system { D }dAxis unit direction vector.
Further, in one embodiment of the invention, described to establish kinematics mould according to described two constraint equations
Type, comprising:
Derivation is carried out to the constraint condition of constraint 1 and constraint 2, is obtained:
Wherein, J is Jacobian matrix；Jacobian matrix again can be according to generalized coordinatesWith dependent coordinateResolve into two parts:
The speed of dependent coordinate is indicated by generalized velocity:
It can obtain, the kinematics model are as follows:
Wherein,For 2 rank unit matrixs.
Further, in one embodiment of the invention, the twin turbo is solved using lagrange equations of the first kind
Device human occupant dynamic model includes:
Wherein, L=T-V is Lagrangian, T is the total kinetic energy of two-wheel robot system, V is two-wheel robot system
Total potential energy, γ is Lagrange multiplier,The broad sense being subject to for two-wheel robot system is non-conservative
External force, D and d are respectively that rolling and handlebar turn to disturbing moment suffered by channel, τcFor handlebar shaft driving moment.
Further, in one embodiment of the invention, the S3 is specifically included:
In the towing away from range, to the singular value of the controllability matrix, the domain of attraction of the closed loop controller and
The control cost is analyzed, and makes correspondence image, is determined for compliance with the control according to correspondence image and the demand for control
The towing of demand processed away from.
In order to achieve the above objectives, another aspect of the present invention embodiment propose it is a kind of based on towing away from two-wheel robot build
Mould and static balance device, comprising:
Equivalent modules are used for when detection two-wheel robot system meets default equivalence setting condition, by the twin turbo
Device people's system is set as multi link multi-joint system, and defines multiple coordinate systems in the multi link multi-joint system, according to
The geometrical relationship of the multiple coordinate system calculates the towing of the two-wheel robot system away from range；
Constraints module, for the closed loop moving chain according to suffered by the multi link multi-joint system and the two-wheel robot
The wheel geometrical property of system establishes two constraint equations, and establishes kinematics model according to described two constraint equations；
Modeling analysis module, it is double for being solved to obtain to the kinematics model using lagrange equations of the first kind
Wheel robot kinetic model, and the singular value according to the two-wheel Dynamic Models of Robot Manipulators to controllability matrix, closed loop control
The domain of attraction of device processed and control cost are analyzed, based on the analysis results in the towing away from determining that meeting control needs in range
The towing asked away from.
The embodiment of the present invention based on towing away from two-wheel robot modeling and static balance device, first according to movement about
Two-wheel robot is equivalent to the hinged multi link multi-joint system of two car body of front and back from the angle of multi-rigid-body system by beam.Secondly
The contact performance of closed loop moving chain and wheel and ground based on system, establishes two constraint equations, obtains the kinematics of system
Model.Followed by the kinetic model of lagrange equations of the first kind derivation system.Finally, from the surprise of system controllability and measurability matrix
Three different value, closed loop controller domain of attraction and control energy consumption aspects are pulled to analyze away from the influence to static balance.It can reflect
It is different towing away from lower handlebar corner and height of center of mass variation non-linear relations, and can for towing away from selection provide it is a set of
Analysis process, to improve the control effect of static balance.
In addition, it is according to the above embodiment of the present invention based on towing away from two-wheel robot modeling and static balance device also
It can have following additional technical characteristic:
Further, in one embodiment of the invention, the default equivalence setting condition, comprising:
Pitching caused by setting aftercarriage mass center only includes rolling and turned to by handlebar；
Setting front and back two-wheeled is checked, without relative motion between vehicle frame, and is pure rolling between rear-wheel and ground；
Tire thickness and deformation are ignored in setting, and front and back two-wheeled is considered as equal-sized rigid thin discs.
The additional aspect of the present invention and advantage will be set forth in part in the description, and will partially become from the following description
Obviously, or practice through the invention is recognized.
Detailed description of the invention
Above-mentioned and/or additional aspect and advantage of the invention will become from the following description of the accompanying drawings of embodiments
Obviously and it is readily appreciated that, in which:
Fig. 1 be according to one embodiment of the invention based on towing away from two-wheel robot modeling and static balance method stream
Cheng Tu；
Fig. 2 is the equivalent car body schematic diagram of multi link multi-joint according to one embodiment of the invention；
Fig. 3 is the towing according to one embodiment of the invention away from degree of controllability analysis flow chart diagram；
Fig. 4 be according to one embodiment of the invention based on towing away from two-wheel robot modeling and static balance device knot
Structure schematic diagram.
Specific embodiment
The embodiment of the present invention is described below in detail, examples of the embodiments are shown in the accompanying drawings, wherein from beginning to end
Same or similar label indicates same or similar element or element with the same or similar functions.Below with reference to attached
The embodiment of figure description is exemplary, it is intended to is used to explain the present invention, and is not considered as limiting the invention.
Describe to propose according to embodiments of the present invention with reference to the accompanying drawings based on towing away from two-wheel robot modeling with it is quiet
Only balance method and device.
Describe to propose according to embodiments of the present invention with reference to the accompanying drawings first based on towing away from two-wheel robot modeling with
Static balance method.
Fig. 1 be according to one embodiment of the invention based on towing away from two-wheel robot modeling and static balance method stream
Cheng Tu.
As shown in Figure 1, should based on towing away from two-wheel robot modeling and static balance method the following steps are included:
Step S1 sets two-wheel robot system when detection two-wheel robot system meets default equivalence setting condition
It is set to multi link multi-joint system, and defines multiple coordinate systems in multi link multi-joint system, according to the several of multiple coordinate systems
What relationship calculates the towing of two-wheel robot system away from range.
Further, equivalence setting condition is preset, comprising:
Pitching caused by setting aftercarriage mass center only includes rolling and turned to by handlebar；
Setting front and back two-wheeled is checked, without relative motion between vehicle frame, and is pure rolling between rear-wheel and ground；
Tire thickness and deformation are ignored in setting, and front and back two-wheeled is considered as equal-sized rigid thin discs.
Specifically, during deriving model, need to carry out some it is assumed that facilitating foundation and deriving model, such as:
(1) aftercarriage mass center is assumed other than rolling and the pitching caused by handlebar steering, and no other move；
(2) front and back two-wheeled is checked, without relative motion between vehicle frame, and is pure rolling between rear-wheel and ground；
(3) ignore tire thickness and deformation, two wheels are considered as equal-sized rigid thin discs.
After meeting the above-mentioned condition that assume that, two-wheel robot system can be equivalent to multi link multi-joint system
System, as shown in Fig. 2, being the equivalent car body schematic diagram of multi link multi-joint.Wherein, A0When being respectively handlebar rotation and do not rotate with A
Rear wheel and ground contact points.B is the rear wheel center of circle, and C is the tie point of handlebar revolute and Rear frame, and D is front vehicle wheel circle
The heart, E are the contact point of front vehicle wheel and ground.δ, θ respectively represent bowing for the roll angle of aftercarriage, handlebar corner and aftercarriage
The elevation angle, θ1For angular displacement of the E point on front-wheel.It is front-wheel angular velocity in roll.ΔxrIt is rear-wheel as caused by pitching movement
Along aftercarriage plane Π1With the translation distance of the intersection on ground.ΔxfWith Δ yfIt is E point since handlebar is rotated relative to initial point
E0Offset coordinates on the ground.Table 1 is the equivalent joint conclusive table of the embodiment of the present invention, make it is above translation and rotary motion by
It realizes in five joints.ri(i=1~4) are the equivalent connecting rod vector for linking this five joints, wherein r1It is the rear car that B is directed toward by A
Take turns connecting rod vector；r2It is the Rear frame connecting rod vector that C is directed toward by B；r3It is the front frame connecting rod vector that D is directed toward by C；r4It is to be referred to by D
To the front vehicle wheel connecting rod vector of E.In addition, θ0The pitch angle of Rear frame connecting rod vector when be handlebar corner being zero；ε is Rear frame
Connecting rod established angle is r2Angle between handlebar shaft；η is handlebar inclination angle, is handlebar shaft and aftercarriage plane vertical direction
Angle；λ is handlebar front fork angle, the angle between front frame C ' D and handlebar shaft.G2And G1Respectively front and back car body mass center.
Table 1
After setting multi link multi-joint system for two-wheel robot system, in multi link multi-joint system define with
Lower six coordinate systems:
(1) inertial reference system { I }, A0Xyz: origin is fixed on A0Point, x-axis is by A0It is directed toward E0, z-axis straight down, y-axis and x
Axis and z-axis form right-handed system；Wherein, A0Rear wheel and ground contact points when being rotated for handlebar, E0Front vehicle wheel when being rotated for handlebar
With the contact point on ground；
(2) rear-wheel coordinate system { B }, Bxbybzb: origin is fixed on B point, xbAxis is parallel with inertial reference system x-axis, z and y two
Axis can be rotated around x axis by inertial reference systemAngle obtains, then from { I } to the spin matrix of { B } are as follows:
Wherein, B is the rear wheel center of circle,For the roll angle of aftercarriage；
(3) handlebar coordinate system { C }, Cxcyczc: origin is fixed on C point, ycAxis and rear-wheel coordinate system ybAxis is parallel, x and z two
Axis can be by { B } around ybThe axis rotation angle θ+η obtains, then from { B } to the spin matrix of { C } are as follows:
Wherein, C is the tie point of handlebar revolute and Rear frame, and θ is the pitch angle of aftercarriage, and η is handlebar inclination angle；
Handlebar inclination angle η meets following geometrical constraint:
Wherein, θ0The pitch angle of Rear frame connecting rod vector when be handlebar corner being zero, ε is Rear frame connecting rod established angle；
(4) front-wheel coordinate system { D }, Dxdydzd: origin is fixed on D point, zdAxis and handlebar coordinate system zcAxis is parallel, x and y two
Axis can be by handlebar coordinate system around zcThe axis rotation angle δ obtains, then from { C } to the spin matrix of { D } are as follows:
Wherein, D is the front vehicle wheel center of circle, and δ is handlebar corner；
(5) aftercarriage coordinate system { G1, G1x1y1z1: origin is fixed on G1, aftercarriage coordinate system is by rear-wheel coordinate system around yb
The axis rotation angle θ obtains, and enablesMiddle η=0 can be obtained from { B } system to { G1System spin matrix
Wherein, G1For aftercarriage mass center；
(6) front vehicle body coordinate system { G2, G2x2y2z2: origin is fixed on G2, front vehicle body coordinate system is parallel with front-wheel coordinate system,
G2For front vehicle body mass center.
According to above-mentioned coordinate system, each vector in Fig. 2 be may be expressed as:
In formula: subscript indicates that coordinate system corresponding to the vector, R are radius of wheel, lrFor the length of line segment BC, d is line
The length of section CC ', lfFor the length of line segment C ' D.
By geometrical relationship can obtain two-wheel robot towing away from are as follows:
Step S2, according to the wheel geometry of closed loop moving chain suffered by multi link multi-joint system and two-wheel robot system
Characteristic establishes two constraint equations, and establishes kinematics model according to two constraint equations.
Further, in one embodiment of the invention, the closed loop moving chain according to suffered by multi link multi-joint system
Two constraint equations are established with the wheel geometrical property of two-wheel robot system, comprising:
Constraint 1, the constraint of closed loop moving chain
Wherein, ez=[0,0,1]TFor the z-axis unit direction vector of inertial reference system { I }, r1It is the rear wheel that B is directed toward by A
Connecting rod vector；r2It is the Rear frame connecting rod vector that C is directed toward by B；r3It is the front frame connecting rod vector that D is directed toward by C；r4It is to be directed toward by D
The front vehicle wheel connecting rod vector of E, subscript indicate coordinate system corresponding to the vector,For from { B } to the spin matrix of { I },
For from { C } to the spin matrix of { B },For from { D } to the spin matrix of { C }；
Constraint 2, wheel geometrical constraint
Wherein, nyFor the y of front-wheel coordinate system { D }dAxis unit direction vector.
Further, in one embodiment of the invention, kinematics model is established according to two constraint equations, comprising:
Derivation is carried out to the constraint condition of constraint 1 and constraint 2, is obtained:
Wherein, J is Jacobian matrix；Jacobian matrix again can be according to generalized coordinatesWith dependent coordinateResolve into two parts:
The speed of dependent coordinate is indicated by generalized velocity:
It can obtain, kinematics model are as follows:
Wherein,For 2 rank unit matrixs.
Specifically, system generalized coordinates and holonomic constriants are defined first,
In above-mentioned movement, the only roll angle of aftercarriageSteering angle sigma with handlebar is independent, therefore defines it
Be system generalized coordinates:
Two-wheeled is both needed to land before and after vehicle during static balance, forms a closed loop moving chain.Closed loop moving chain
In the presence of enabling any point having in two paths arrival systems from ground.In view of the dependent that front-wheel is related to is sat
Mark more, therefore the present invention is with the contact point A of rear-wheel and ground0As path starting point.Only need the coordinate of half can table in this way
Show position of the front-wheel contact point under { I } system.Therefore, new system coordinates are enabled are as follows:
The closed loop moving chain according to suffered by system and wheel geometrical property construct following two holonomic constriants:
Constraint 1, the constraint of closed loop moving chain
Wherein, ez=[0,0,1]TFor the z-axis unit direction vector of inertial reference system { I }.
Constraint 2, wheel geometrical constraint
Wherein, nyFor the y of front-wheel coordinate system { D }dAxis unit direction vector and front vehicle body plane Π2Normal vector.
As δ=θ=0, θ1It should meet:
θ1=-η |δ=θ=0 (11)
In addition, θ0Also the following constraint that should meet:
Robot kinematics' model is established according to constraint equation, to obtain non-linear height of center of mass variation model,
Holonomic constriants (9) and (10) derivation can be obtained:
In formula: J is Jacobian matrix.The matrix again can be according to qiWith dependent coordinateResolve into two
Point:
Based on above formula, the speed of dependent coordinate can be indicated by generalized velocity:Therefore, it can get
System kinematics model are as follows:
In formula:For 2 rank unit matrixs.
Step S3 solves kinematics model using lagrange equations of the first kind to obtain two-wheel robot dynamics
Model, and according to two-wheel Dynamic Models of Robot Manipulators to the singular value of controllability matrix, the domain of attraction of closed loop controller and control
Cost is analyzed, based on the analysis results towing away from determine to meet in range the towing of demand for control away from.
Further, in one embodiment of the invention, two-wheel robot is solved using lagrange equations of the first kind
Kinetic model includes:
Wherein, L=T-V is Lagrangian, T is the total kinetic energy of two-wheel robot system, V is two-wheel robot system
Total potential energy, γ is Lagrange multiplier,The broad sense being subject to for two-wheel robot system is non-conservative
External force, D and d are respectively that rolling and handlebar turn to disturbing moment suffered by channel, τcFor handlebar shaft driving moment.
Further, in one embodiment of the invention, S3 is specifically included:
In towing away from range, the singular value of controllability matrix, the domain of attraction of closed loop controller and control cost are carried out
Analysis, and makes correspondence image, according to correspondence image and demand for control be determined for compliance with the towing of demand for control away from.
Specifically, two-wheel Dynamic Models of Robot Manipulators is established according to kinematics model, firstly, deriving two-wheel robot system
Total kinetic energy, by the mass center translational kinetic energy T of front and back car bodytiWith the rotational kinetic energy T around mass centerriComposition:
Kinetic energy 1, the mass center G by aftercarriage1Coordinate is expressed at inertial system { I }:
In formula:xG1=xg1cosθ0+zg1sinθ0、zG1=zg1cosθ0-xg1sinθ0, xg1With
zg1Coordinate for rear car mass center relative to Rear frame.It is translation of the rear-wheel caused by pitching in x-axis, it can according to hypothesis 2
It acquiresRear frame angular speed is projected into aftercarriage coordinate system { G again1In:
Therefore, it is as follows that aftercarriage kinetic energy can be obtained:
In formula: m1For rear car weight, { G is enabled1Be aftercarriage principal axis of inertia system, then
It is aftercarriage in body coordinate system { G1In moment of inertia matrix.
Kinetic energy 2, the mass center G by front vehicle body2Coordinate is expressed at inertial system { I }:
In formula:xg2And zg2Coordinate for front truck mass center relative to front frame, xG2=d+
xg2cosλ+zg2sinλ、zG2=zg2cosλ-xg2sinλ.Similarly, then by front frame angular speed front vehicle body coordinate system is projected
{G2In:
Therefore, it is as follows that front vehicle body kinetic energy can be obtained:
In formula: m2For front truck weight, { G is enabled2Front vehicle body principal axis of inertia system, thenFor
Front vehicle body is in body coordinate system { G2In moment of inertia matrix.
It is zero potential energy level with ground, then takes z-axis component of the two car body mass centers at inertial system { I }, the total potential energy of vehicle can be obtained
Are as follows:
In formula: g is earth's surface acceleration of gravity, ()zFor amount of orientation z-axis components operation symbol.
With the kinetic model of lagrange equations of the first kind solving system:
In formula: L=T-V is Lagrangian, T is total kinetic energy, V is total potential energy, and γ is Lagrange multiplier,For broad sense non-conserved external force suffered by system, D and d are respectively that rolling and handlebar turn to channel
Suffered disturbing moment, τcFor handlebar shaft driving moment.
Formula (23) is written as the general Euler-Lagrange form of following robot:
In formula: M is general mass matrix, and V is centrifugal force and coriolis force matrix, and E is gravity matrix, and expression exists
It is presented in annex.Model above can be converted to using kinematics model (15) ODE for containing only 2 equations.First
System coordinates acceleration can be obtained to formula (15) derivation:
Above formula and kinematical equation (15) are substituted into formula (25), then both ends are the same as multiplied by GTIt can obtain:
According to GTJT=0, γ can be eliminated, dimensionality reduction kinetic model is obtained:
In formula: Mi=GTMG is dimensionality reduction mass of system matrix,It is dimensionality reduction system damping matrix,
Qiext=GTQextFor dimensionality reduction system generalized external force matrix.
By dimensionality reduction kinetic model (28) in equalization pointPlace's linearisation:
In formula:It is minor shifts amount of the generalized coordinates relative to equilbrium position；ΔτcIt is control deviation
Amount；Easily cardAndIgnoreThe linearly invariant kinetics equation of following state space form can be obtained.
According to above process analysis towing away from the influence for turning to static balance control to handlebar, dragged as shown in figure 3, illustrating
It drags away from degree of controllability analysis flow chart diagram, in the range of towing is away from permission, from the suction of the singular value, closed loop controller of controllability matrix
Draw domain and control cost three in terms of come analyze towing away from influence.
Firstly, analysis degree of controllability, constructs the controllability matrix Q of linear model (30)c:
Sytem matrix A and B substitution above formula can be obtained:
In formula:It indicatesThe 2nd column.To QcCarry out singular value decomposition Qc=U ∑ V, makes minimum singular value
About towing away from image.
Then, the domain of attraction of closed loop controller is analyzed.Design following Linear-Quadratic Problem performance indicator:
In formula:WithFor weight coefficient matrix undetermined, can be adjusted according to specific control effect
Examination.By solving Riccati algebraic equation as follows:
Matrix P can be obtained, and then can be in the hope of the linear Feedback Control Laws of handlebar torque are as follows:
Definition handlebar steering moment maximum value is τc max, then the reachable state space of system meets inequality:
|K|∞|x|∞≤||τc| |=| | Kx | |≤τc max (36)
In formula: | |∞Indicate the Infinite Norm of amount of orientation.Therefore, estimate domain of attraction are as follows:
Make domain of attraction (37) about towing away from image.
Finally, analysis towing is away from the influence to control cost.Definition control cost are as follows:
In formula: integral subscript T indicates control end time.Make control cost about towing away from image.
, can be according to specific demand for control, such as high controllability according to the above three width image, big domain of attraction and low control generation
Valence come select suitably pull away from.
The above-mentioned formula used is supplemented below.
Jacobian matrix in formula (14) is defined as follows:
In formula:
In formula:
General mass matrix in kinetics equation (25) is defined as follows:
In formula:
M23=m2RxG2sinδcos(θ+η)+m2lrxG2sinδsin(θ0-η)-m2xG2zG2sinδ
M32=m2RxG2sinδcos(θ+η)+m2lrxG2sinδsin(θ0-η)-m2xG2zG2sinδ
Damping matrix in kinetics equation (25) is defined as follows:
In formula:
V22=0
Gravity matrix in kinetics equation (25) is defined as follows:
In formula:
It is proposed according to embodiments of the present invention based on towing away from two-wheel robot modeling and static balance method, first root
According to kinematic constraint, two-wheel robot is equivalent to the hinged multi link multi-joint system of two car body of front and back from the angle of multi-rigid-body system
System.Secondly based on the closed loop moving chain and wheel of system and the contact performance on ground, two constraint equations are established, obtain system
Kinematics model.Followed by the kinetic model of lagrange equations of the first kind derivation system.Finally, from system controllability and measurability square
Three singular value, closed loop controller domain of attraction and control energy consumption aspects of battle array are pulled to analyze away from the influence to static balance.It can
Be reflected in it is different towing away from lower handlebar corner and height of center of mass variation non-linear relations, and can for pull away from selection mention
For a set of analysis process, to improve the control effect of static balance.
Referring next to attached drawing description propose according to embodiments of the present invention based on towing away from two-wheel robot modeling with it is quiet
Only balancing device.
Fig. 4 be according to one embodiment of the invention based on towing away from two-wheel robot modeling and static balance device knot
Structure schematic diagram.
As shown in figure 4, should based on towing away from two-wheel robot modeling and static balance device include: equivalent modules 100,
Constraints module 200 and modeling analysis module 300.
Equivalent modules 100 are used for when detection two-wheel robot system meets default equivalence setting condition, by two-wheel machine
People's system is set as multi link multi-joint system, and defines multiple coordinate systems in multi link multi-joint system, according to multiple seats
The geometrical relationship of mark system calculates the towing of two-wheel robot system away from range.
Constraints module 200, for the closed loop moving chain according to suffered by multi link multi-joint system and two-wheel robot system
Wheel geometrical property establish two constraint equations, and establish kinematics model according to two constraint equations.
Modeling analysis module 300, it is double for being solved to obtain to kinematics model using lagrange equations of the first kind
Wheel robot kinetic model, and the singular value according to two-wheel Dynamic Models of Robot Manipulators to controllability matrix, closed loop controller
Domain of attraction and control cost analyzed, based on the analysis results towing away from the towing for determining to meet demand for control in range
Away from.
Further, in one embodiment of the invention, equivalence setting condition is preset, comprising:
Pitching caused by setting aftercarriage mass center only includes rolling and turned to by handlebar；
Setting front and back two-wheeled is checked, without relative motion between vehicle frame, and is pure rolling between rear-wheel and ground；
Tire thickness and deformation are ignored in setting, and front and back two-wheeled is considered as equal-sized rigid thin discs.
Further, in one embodiment of the invention, multiple coordinate systems are as follows:
(1) inertial reference system { I }, A0Xyz: origin is fixed on A0Point, x-axis is by A0It is directed toward E0, z-axis straight down, y-axis and x
Axis and z-axis form right-handed system；Wherein, A0Rear wheel and ground contact points when being rotated for handlebar, E0Front vehicle wheel when being rotated for handlebar
With the contact point on ground；
(2) rear-wheel coordinate system { B }, Bxbybzb: origin is fixed on B point, xbAxis is parallel with inertial reference system x-axis, z and y two
Axis can be rotated around x axis by inertial reference systemAngle obtains, then from { I } to the spin matrix of { B } are as follows:
Wherein, B is the rear wheel center of circle,For the roll angle of aftercarriage；
(3) handlebar coordinate system { C }, Cxcyczc: origin is fixed on C point, ycAxis and rear-wheel coordinate system ybAxis is parallel, x and z two
Axis can be by { B } around ybThe axis rotation angle θ+η obtains, then from { B } to the spin matrix of { C } are as follows:
Wherein, C is the tie point of handlebar revolute and Rear frame, and θ is the pitch angle of aftercarriage, and η is handlebar inclination angle；
Handlebar inclination angle η meets following geometrical constraint:
Wherein, θ0The pitch angle of Rear frame connecting rod vector when be handlebar corner being zero, ε is Rear frame connecting rod established angle；
(4) front-wheel coordinate system { D }, Dxdydzd: origin is fixed on D point, zdAxis and handlebar coordinate system zcAxis is parallel, x and y two
Axis can be by handlebar coordinate system around zcThe axis rotation angle δ obtains, then from { C } to the spin matrix of { D } are as follows:
Wherein, D is the front vehicle wheel center of circle, and δ is handlebar corner；
(5) aftercarriage coordinate system { G1, G1x1y1z1: origin is fixed on G1, aftercarriage coordinate system is by rear-wheel coordinate system around yb
The axis rotation angle θ obtains, and enablesMiddle η=0 can be obtained from { B } system to { G1System spin matrix
Wherein, G1For aftercarriage mass center；
(6) front vehicle body coordinate system { G2, G2x2y2z2: origin is fixed on G2, front vehicle body coordinate system is parallel with front-wheel coordinate system,
G2For front vehicle body mass center.
Further, in one embodiment of the invention, the towing of two-wheel robot system is away from range are as follows:
Wherein, R is radius of wheel, lrFor the length of line segment BC, d is the length of line segment CC ', lfFor front frame line segment C ' D's
Length, λ are handlebar front fork angles, and η is handlebar inclination angle.
Further, in one embodiment of the invention, the closed loop moving chain according to suffered by multi link multi-joint system
Two constraint equations are established with the wheel geometrical property of two-wheel robot system, comprising:
Constraint 1, the constraint of closed loop moving chain
Wherein, ez=[0,0,1]TFor the z-axis unit direction vector of inertial reference system { I }, r1It is the rear wheel that B is directed toward by A
Connecting rod vector；r2It is the Rear frame connecting rod vector that C is directed toward by B；r3It is the front frame connecting rod vector that D is directed toward by C；r4It is to be directed toward by D
The front vehicle wheel connecting rod vector of E, subscript indicate coordinate system corresponding to the vector,For from { B } to the spin matrix of { I },
For from { C } to the spin matrix of { B },For from { D } to the spin matrix of { C }；
Constraint 2, wheel geometrical constraint
Wherein, nyFor the y of front-wheel coordinate system { D }dAxis unit direction vector.
Further, in one embodiment of the invention, kinematics model is established according to two constraint equations, comprising:
Derivation is carried out to the constraint condition of constraint 1 and constraint 2, is obtained:
Wherein, J is Jacobian matrix；Jacobian matrix again can be according to generalized coordinatesWith dependent coordinateResolve into two parts:
The speed of dependent coordinate is indicated by generalized velocity:
It can obtain, kinematics model are as follows:
Wherein,For 2 rank unit matrixs.
Further, in one embodiment of the invention, two-wheel robot is solved using lagrange equations of the first kind
Kinetic model includes:
Wherein, L=T-V is Lagrangian, T is the total kinetic energy of two-wheel robot system, V is two-wheel robot system
Total potential energy, γ is Lagrange multiplier,The broad sense being subject to for two-wheel robot system is non-conservative
External force, D and d are respectively that rolling and handlebar turn to disturbing moment suffered by channel, τcFor handlebar shaft driving moment.
Further, in one embodiment of the invention, modeling analysis module is specifically used for, in towing away from range,
The singular value of controllability matrix, the domain of attraction of closed loop controller and control cost are analyzed, and make correspondence image, according to
Correspondence image and demand for control be determined for compliance with the towing of demand for control away from.
It should be noted that it is aforementioned to based on towing away from two-wheel robot modeling and static balance embodiment of the method solution
The device that explanation is also applied for the embodiment is released, details are not described herein again.
It is proposed according to embodiments of the present invention based on towing away from two-wheel robot modeling and static balance device, first root
According to kinematic constraint, two-wheel robot is equivalent to the hinged multi link multi-joint system of two car body of front and back from the angle of multi-rigid-body system
System.Secondly based on the closed loop moving chain and wheel of system and the contact performance on ground, two constraint equations are established, obtain system
Kinematics model.Followed by the kinetic model of lagrange equations of the first kind derivation system.Finally, from system controllability and measurability square
Three singular value, closed loop controller domain of attraction and control energy consumption aspects of battle array are pulled to analyze away from the influence to static balance.It can
Be reflected in it is different towing away from lower handlebar corner and height of center of mass variation non-linear relations, and can for pull away from selection mention
For a set of analysis process, to improve the control effect of static balance.
In addition, term " first ", " second " are used for descriptive purposes only and cannot be understood as indicating or suggesting relative importance
Or implicitly indicate the quantity of indicated technical characteristic.Define " first " as a result, the feature of " second " can be expressed or
Implicitly include at least one this feature.In the description of the present invention, the meaning of " plurality " is at least two, such as two, three
It is a etc., unless otherwise specifically defined.
In the description of this specification, reference term " one embodiment ", " some embodiments ", " example ", " specifically show
The description of example " or " some examples " etc. means specific features, structure, material or spy described in conjunction with this embodiment or example
Point is included at least one embodiment or example of the invention.In the present specification, schematic expression of the above terms are not
It must be directed to identical embodiment or example.Moreover, particular features, structures, materials, or characteristics described can be in office
It can be combined in any suitable manner in one or more embodiment or examples.In addition, without conflicting with each other, the skill of this field
Art personnel can tie the feature of different embodiments or examples described in this specification and different embodiments or examples
It closes and combines.
Although the embodiments of the present invention has been shown and described above, it is to be understood that above-described embodiment is example
Property, it is not considered as limiting the invention, those skilled in the art within the scope of the invention can be to above-mentioned
Embodiment is changed, modifies, replacement and variant.
Claims (10)
1. it is a kind of based on towing away from two-wheel robot modeling and static balance method, which comprises the following steps:
S1 sets the two-wheel robot system to when detection two-wheel robot system meets default equivalence setting condition
Multi link multi-joint system, and multiple coordinate systems are defined in the multi link multi-joint system, according to the multiple coordinate system
Geometrical relationship calculate the towing of the two-wheel robot system away from range；
S2, according to the wheel geometry of closed loop moving chain suffered by the multi link multi-joint system and the two-wheel robot system
Characteristic establishes two constraint equations, and establishes kinematics model according to described two constraint equations；
S3 solves the kinematics model using lagrange equations of the first kind to obtain two-wheel robot kinetic simulation
Type, and according to the two-wheel Dynamic Models of Robot Manipulators to the singular value of controllability matrix, the domain of attraction of closed loop controller and control
Cost processed is analyzed, based on the analysis results the towing away from determine to meet in range the towing of demand for control away from.
2. the method according to claim 1, wherein the default equivalence setting condition, comprising:
Pitching caused by setting aftercarriage mass center only includes rolling and turned to by handlebar；
Setting front and back two-wheeled is checked, without relative motion between vehicle frame, and is pure rolling between rear-wheel and ground；
Tire thickness and deformation are ignored in setting, and front and back two-wheeled is considered as equal-sized rigid thin discs.
3. the method according to claim 1, wherein the multiple coordinate system are as follows:
(1) inertial reference system { I }, A0Xyz: origin is fixed on A0Point, x-axis is by A0It is directed toward E0, z-axis straight down, y-axis and x-axis and z
Axis forms right-handed system；Wherein, A0Rear wheel and ground contact points when being rotated for handlebar, E0Front vehicle wheel and ground when being rotated for handlebar
The contact point in face；
(2) rear-wheel coordinate system { B }, Bxbybzb: origin is fixed on B point, xbAxis is parallel with inertial reference system x-axis, and z and two axis of y can be by
Inertial reference system rotates around x axisAngle obtains, then from { I } to the spin matrix of { B } are as follows:
Wherein, B is the rear wheel center of circle,For the roll angle of aftercarriage；
(3) handlebar coordinate system { C }, Cxcyczc: origin is fixed on C point, ycAxis and rear-wheel coordinate system ybAxis is parallel, and x and two axis of z can
By { B } around ybThe axis rotation angle θ+η obtains, then from { B } to the spin matrix of { C } are as follows:
Wherein, C is the tie point of handlebar revolute and Rear frame, and θ is the pitch angle of aftercarriage, and η is handlebar inclination angle；
Handlebar inclination angle η meets following geometrical constraint:
Wherein, θ0The pitch angle of Rear frame connecting rod vector when be handlebar corner being zero, ε is Rear frame connecting rod established angle；
(4) front-wheel coordinate system { D }, Dxdydzd: origin is fixed on D point, zdAxis and handlebar coordinate system zcAxis is parallel, and x and two axis of y can
By handlebar coordinate system around zcThe axis rotation angle δ obtains, then from { C } to the spin matrix of { D } are as follows:
Wherein, D is the front vehicle wheel center of circle, and δ is handlebar corner；
(5) aftercarriage coordinate system { G1, G1x1y1z1: origin is fixed on G1, aftercarriage coordinate system is by rear-wheel coordinate system around ybAxis rotation
The angle θ obtains, and enablesMiddle η=0 can be obtained from { B } system to { G1System spin matrix
Wherein, G1For aftercarriage mass center；
(6) front vehicle body coordinate system { G2, G2x2y2z2: origin is fixed on G2, front vehicle body coordinate system is parallel with front-wheel coordinate system, G2For
Front vehicle body mass center.
4. according to the method described in claim 3, it is characterized in that, the towing of the two-wheel robot system is away from range are as follows:
Wherein, R is radius of wheel, lrFor the length of line segment BC, d is the length of line segment CC ', lfFor the length of front frame line segment C ' D
Degree, λ is handlebar front fork angle, and η is handlebar inclination angle.
5. according to the method described in claim 4, it is characterized in that, described close according to suffered by the multi link multi-joint system
The wheel geometrical property of ring kinematic chain and the two-wheel robot system establishes two constraint equations, comprising:
Constraint 1, the constraint of closed loop moving chain
Wherein, ez=[0,0,1]TFor the z-axis unit direction vector of inertial reference system { I }, r1It is the rear wheel connecting rod that B is directed toward by A
Vector；r2It is the Rear frame connecting rod vector that C is directed toward by B；r3It is the front frame connecting rod vector that D is directed toward by C；r4It is that E is directed toward by D
Front vehicle wheel connecting rod vector, subscript indicate coordinate system corresponding to the vector,For from { B } to the spin matrix of { I },For from
{ C } arrives the spin matrix of { B },For from { D } to the spin matrix of { C }；
Constraint 2, wheel geometrical constraint
Wherein, nyFor the y of front-wheel coordinate system { D }dAxis unit direction vector.
6. according to the method described in claim 5, it is characterized in that, described establish kinematics mould according to described two constraint equations
Type, comprising:
Derivation is carried out to the constraint condition of constraint 1 and constraint 2, is obtained:
Wherein, J is Jacobian matrix；Jacobian matrix again can be according to generalized coordinatesWith dependent coordinateResolve into two parts:
The speed of dependent coordinate is indicated by generalized velocity:
It can obtain, the kinematics model are as follows:
Wherein,For 2 rank unit matrixs.
7. according to the method described in claim 5, it is characterized in that, solving the twin turbo using lagrange equations of the first kind
Device human occupant dynamic model includes:
Wherein, L=T-V is Lagrangian, T is the total kinetic energy of two-wheel robot system, V is the total of two-wheel robot system
Potential energy, γ are Lagrange multiplier,The broad sense being subject to for two-wheel robot system is non-conservative outer
Power, D and d are respectively that rolling and handlebar turn to disturbing moment suffered by channel, τcFor handlebar shaft driving moment.
8. the method according to claim 1, wherein the S3 is specifically included:
In the towing away from range, to the singular value of the controllability matrix, the domain of attraction of the closed loop controller and described
Control cost is analyzed, and makes correspondence image, is determined for compliance with the control need according to correspondence image and the demand for control
The towing asked away from.
9. it is a kind of based on towing away from two-wheel robot modeling and static balance device characterized by comprising
Equivalent modules are used for when detection two-wheel robot system meets default equivalence setting condition, by the two-wheel robot
System is set as multi link multi-joint system, and defines multiple coordinate systems in the multi link multi-joint system, according to described
The geometrical relationship of multiple coordinate systems calculates the towing of the two-wheel robot system away from range；
Constraints module, for the closed loop moving chain according to suffered by the multi link multi-joint system and the two-wheel robot system
Wheel geometrical property establish two constraint equations, and establish kinematics model according to described two constraint equations；
Modeling analysis module, for being solved to obtain twin turbo to the kinematics model using lagrange equations of the first kind
Device human occupant dynamic model, and the singular value according to the two-wheel Dynamic Models of Robot Manipulators to controllability matrix, closed loop controller
Domain of attraction and control cost analyzed, based on the analysis results in the towing away from determining to meet demand for control in range
Towing away from.
10. device according to claim 9, which is characterized in that the default equivalence setting condition, comprising:
Pitching caused by setting aftercarriage mass center only includes rolling and turned to by handlebar；
Setting front and back two-wheeled is checked, without relative motion between vehicle frame, and is pure rolling between rear-wheel and ground；
Tire thickness and deformation are ignored in setting, and front and back two-wheeled is considered as equal-sized rigid thin discs.
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| CN110348140B (en) | 2021-06-25 |
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