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 invention relates to the technical field of mechanical system modeling and dynamics analysis, in particular to a method and a device for modeling and static balancing of a two-wheeled robot based on a towing distance.
Background
The towing distance is a distance between an intersection point of a handle rotation shaft and the ground and a front wheel ground contact point, and has an important influence on balance control of the two-wheeled robot by using handle steering.
To the two-wheeled robot that the distance of dragging is nonzero, the centre of mass height of robot can finely be adjusted to the handlebar that rotates, prevents to topple over. When the speed of the two-wheeled robot reaches a certain degree, the return torque provided by the ground enables the front wheel rotating shaft to automatically right the robot without extra control force; however, when the two-wheeled robot is at ultra-low speed and even is still, the self-stability disappears, and the balance is realized mainly by means of steering of the handlebar. It follows that for more challenging static balancing, the drag distance becomes critical. The existing research on the steering static balance of the handlebar indicates that a controller designed based on the positive towing distance has a small attraction area and poor robustness, so that the influence of the towing distance on the control performance needs to be researched. However, current research models only robots with a fixed tow distance and simplifies the centroid height variation to a linear model. Such a model is not sufficient to analyze the influence of the drag distance, and for this reason, it is urgently required to establish a dynamic model suitable for any drag distance.
Disclosure of Invention
The present invention is directed to solving, at least to some extent, one of the technical problems in the related art.
Therefore, an object of the present invention is to provide a method for modeling and static balancing a two-wheeled robot based on a tow distance, which can reflect a non-linear relationship between a turning angle and a height change of a center of mass of the vehicle handle at different tow distances, and can provide a set of analysis processes for selecting the tow distance to improve a control effect of static balancing.
Another objective of the present invention is to provide a device for modeling and static balancing of a two-wheeled robot based on a towing distance.
In order to achieve the above object, an embodiment of an aspect of the present invention provides a method for modeling and static balancing a two-wheeled robot based on a tow-track, including:
s1, when detecting that the two-wheeled robot system meets the preset equivalent setting condition, setting the two-wheeled robot system as a multi-link multi-joint system, defining a plurality of coordinate systems in the multi-link multi-joint system, and calculating the towing distance range of the two-wheeled robot system according to the geometrical relationship of the coordinate systems;
s2, establishing two constraint equations according to the closed-loop kinematic chain borne by the multi-link multi-joint system and the wheel geometric characteristics of the two-wheel robot system, and establishing a kinematic model according to the two constraint equations;
s3, solving the kinematics model by using a first Lagrange equation to obtain a two-wheeled robot dynamics model, analyzing a singular value of a controllability matrix, an attraction domain of a closed-loop controller and a control cost according to the two-wheeled robot dynamics model, and determining a towing distance meeting a control requirement in the towing distance range according to an analysis result.
According to the modeling and static balancing method of the two-wheeled robot based on the towing distance, firstly, according to the motion constraint, the two-wheeled robot is equivalent to a multi-connecting-rod multi-joint system with front and rear vehicle bodies hinged from the multi-rigid system. And secondly, establishing two constraint equations based on the closed-loop kinematic chain of the system and the contact characteristics of the wheels and the ground to obtain a kinematic model of the system. The first type of lagrangian equation is then used to derive a kinetic model of the system. And finally, analyzing the influence of the towing distance on the static balance from three aspects of the singular value of the system controllability matrix, the attraction domain of the closed-loop controller and the control energy consumption. The nonlinear relation between the handle turning angle and the height change of the mass center of the vehicle at different towing distances can be reflected, and a set of analysis process can be provided for the selection of the towing distances so as to improve the control effect of static balance.
In addition, the two-wheeled robot modeling and static balancing method based on the towing distance according to the above embodiment of the present invention may further have the following additional technical features:
further, in an embodiment of the present invention, the preset equivalent setting condition includes:
setting the rear body center of mass to include only roll and pitch caused by handlebar steering;
setting that the front and the rear wheels are braked and do not move relative to the frame, and the rear wheels roll with the ground;
the thickness and deformation of the tire are neglected, and the front wheel and the rear wheel are regarded as rigid thin discs with the same size.
Further, in one embodiment of the present invention, the plurality of coordinate systems are:
(1) inertial reference system { I }, A0xyz: the origin is fixed at A0Point, x axis from A0Direction E0The z-axis is vertical downwards, and the y-axis, the x-axis and the z-axis form a right-hand system; wherein A is0Contact point of rear wheel with ground when handlebar is turned, E0The contact point between the front wheel and the ground when the handlebar rotates;
(2) rear wheel coordinate system { B }, Bxbybzb: the origin is fixed at point B, xbThe axis being parallel to the x-axis of the inertial reference system, the z-and y-axes being rotatable about the x-axis by the inertial reference systemThe angle is found, then the rotation matrix from { I } to { B } is:
wherein B is the circle center of the rear wheel,the roll angle of the rear body;
(3) coordinate system of handlebar { C }, Cxcyczc: the origin is fixed at point C, ycAxle and rear wheel coordinate system ybThe axes being parallel, the x and z axes being surrounded by { B } around ybThe axis is rotated by an angle θ + η, and the rotation matrix from { B } to { C } is:
c is a connecting point of the handlebar rotating pair and the rear frame, theta is a pitch angle of the rear vehicle body, and eta is a handlebar inclination angle;
the handlebar inclination angle η satisfies the following geometrical constraints:
wherein, theta0Is the pitch angle of the rear frame connecting rod vector when the handlebar turning angle is zero, and epsilon is the rear frame connecting rod mounting angle;
(4) front wheel coordinate system { D }, Dxdydzd: the origin is fixed at point D, zdAxis and handlebar coordinate system zcThe axes being parallel, the x and y axes being defined by the handlebar coordinate systemcThe axis is rotated by an angle δ, and the rotation matrix from { C } to { D } is:
wherein D is the center of the front wheel circle, and delta is the handlebar turning angle;
(5) rear vehicle body coordinate system { G1}，G1x1y1z1: origin fixed to G1The rear vehicle body coordinate system is wound by the rear wheel coordinate system ybThe angle of rotation of the shaft is obtained byWhere η is 0, from { B } system to { G-1Rotation matrix of the system
Wherein G is1Is the center of mass of the rear vehicle body;
(6) front vehicle body coordinate system { G2}，G2x2y2z2: origin fixed to G2The front vehicle body coordinate system is parallel to the front wheel coordinate system，G2Is the front bodywork centroid.
Further, in one embodiment of the present invention, the tow-distance range of the two-wheeled robot system is:
wherein R is the wheel radius, lrIs the length of segment BC, d is the length of segment CC', lfThe length of the front frame line segment C' D, λ is the handlebar fork angle, and η is the handlebar inclination angle.
Further, in an embodiment of the present invention, the establishing two constraint equations according to the closed-loop kinematic chain suffered by the multi-link multi-joint system and the wheel geometry of the two-wheel robot system includes:
constraint 1, closed-loop kinematic chain constraint
Wherein e isz＝[0,0,1]TIs a z-axis unit direction vector, r, of the inertial reference system { I }1Is the rear wheel link vector pointing from a to B; r is2Is directed from B to CA frame link vector; r is3Is the front frame link vector pointing from C to D; r is4Is a front wheel connecting rod vector pointed to E by D, the coordinate system corresponding to the vector is indicated by a superscript,for the rotation matrix from { B } to { I },is a rotation matrix from C to B,is a rotation matrix from { D } to { C };
restraint 2, wheel geometry restraint
Wherein n isyY being the front wheel coordinate system { D }dAxial unit direction vector.
Further, in an embodiment of the present invention, the building a kinematic model according to the two constraint equations includes:
and (3) carrying out derivation on constraint conditions of the constraint 1 and the constraint 2 to obtain:
wherein J is a Jacobian matrix; the Jacobian matrix can be based on generalized coordinatesAnd non-independent coordinatesThe decomposition is carried out in two parts:
the velocity of the non-independent coordinates is represented by the generalized velocity:
the kinematic model can be found as follows:
wherein,is an identity matrix of order 2.
Further, in one embodiment of the present invention, solving the two-wheeled robot dynamics model using the first type of lagrangian equation comprises:
wherein, L is a Lagrange function, T is the total kinetic energy of the two-wheeled robot system, V is the total potential energy of the two-wheeled robot system, gamma is a Lagrange multiplier,the generalized non-conservative external force on the two-wheeled robot system, D and D are respectively the disturbance torque on the rolling and steering channels of the handlebar, taucIs the driving torque of the rotating shaft of the handlebar.
Further, in an embodiment of the present invention, the S3 specifically includes:
and in the dragging distance range, analyzing singular values of the controllability matrix, the attraction domain of the closed-loop controller and the control cost, making a corresponding image, and determining the dragging distance meeting the control requirement according to the corresponding image and the control requirement.
In order to achieve the above object, another embodiment of the present invention provides a device for modeling and static balancing of a two-wheeled robot based on a towing distance, including:
the system comprises an equivalence module, a control module and a control module, wherein the equivalence module is used for setting a two-wheeled robot system into a multi-connecting-rod multi-joint system when detecting that the two-wheeled robot system meets preset equivalence setting conditions, defining a plurality of coordinate systems in the multi-connecting-rod multi-joint system, and calculating the dragging distance range of the two-wheeled robot system according to the geometric relations of the coordinate systems;
the constraint module is used for establishing two constraint equations according to a closed-loop kinematic chain borne by the multi-connecting-rod multi-joint system and the wheel geometric characteristics of the two-wheeled robot system, and establishing a kinematic model according to the two constraint equations;
and the modeling analysis module is used for solving the kinematic model by utilizing a first class of Lagrange equations to obtain a two-wheeled robot dynamic model, analyzing the singular value of a controllability matrix, the attraction domain of the closed-loop controller and the control cost according to the two-wheeled robot dynamic model, and determining the towing distance meeting the control requirement in the towing distance range according to the analysis result.
According to the modeling and static balancing device of the two-wheeled robot based on the towing distance, firstly, the two-wheeled robot is equivalent to a multi-link multi-joint system with front and rear vehicle bodies hinged from the multi-rigid system according to the motion constraint. And secondly, establishing two constraint equations based on the closed-loop kinematic chain of the system and the contact characteristics of the wheels and the ground to obtain a kinematic model of the system. The first type of lagrangian equation is then used to derive a kinetic model of the system. And finally, analyzing the influence of the towing distance on the static balance from three aspects of the singular value of the system controllability matrix, the attraction domain of the closed-loop controller and the control energy consumption. The nonlinear relation between the handle turning angle and the height change of the mass center of the vehicle at different towing distances can be reflected, and a set of analysis process can be provided for the selection of the towing distances so as to improve the control effect of static balance.
In addition, the towing-distance-based two-wheeled robot modeling and static balancing apparatus according to the above-described embodiment of the present invention may further have the following additional technical features:
further, in an embodiment of the present invention, the preset equivalent setting condition includes:
setting the rear body center of mass to include only roll and pitch caused by handlebar steering;
setting that the front and the rear wheels are braked and do not move relative to the frame, and the rear wheels roll with the ground;
the thickness and deformation of the tire are neglected, and the front wheel and the rear wheel are regarded as rigid thin discs with the same size.
Additional aspects and advantages of the invention will be set forth in part in the description which follows and, in part, will be obvious from the description, or may be learned by practice of the invention.
Drawings
The foregoing and/or additional aspects and advantages of the present invention will become apparent and readily appreciated from the following description of the embodiments, taken in conjunction with the accompanying drawings of which:
FIG. 1 is a flow chart of a method for modeling and static balancing a two-wheeled robot based on a tow-track according to one embodiment of the present invention;
FIG. 2 is a schematic view of a multi-link multi-joint equivalent vehicle body according to one embodiment of the present invention;
FIG. 3 is a chart illustrating a drag distance controllability analysis according to one embodiment of the present invention;
fig. 4 is a schematic structural diagram of a tow-track-based two-wheeled robot modeling and static balancing apparatus according to an embodiment of the present invention.
Detailed Description
Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like or similar reference numerals refer to the same or similar elements or elements having the same or similar function throughout. The embodiments described below with reference to the drawings are illustrative and intended to be illustrative of the invention and are not to be construed as limiting the invention.
The following describes a method and an apparatus for modeling and static balancing a two-wheeled robot based on a tow-track according to an embodiment of the present invention with reference to the accompanying drawings.
First, a proposed tow-distance-based two-wheeled robot modeling and static balancing method according to an embodiment of the present invention will be described with reference to the accompanying drawings.
Fig. 1 is a flowchart of a tow-range-based two-wheeled robot modeling and static balancing method according to an embodiment of the present invention.
As shown in fig. 1, the method for modeling and static balancing of a two-wheeled robot based on a tow distance comprises the following steps:
and step S1, when the two-wheeled robot system is detected to meet the preset equivalent setting condition, setting the two-wheeled robot system as a multi-link multi-joint system, defining a plurality of coordinate systems in the multi-link multi-joint system, and calculating the dragging distance range of the two-wheeled robot system according to the geometric relationship of the coordinate systems.
Further, presetting equivalent setting conditions, including:
setting the rear body center of mass to include only roll and pitch caused by handlebar steering;
setting that the front and the rear wheels are braked and do not move relative to the frame, and the rear wheels roll with the ground;
the thickness and deformation of the tire are neglected, and the front wheel and the rear wheel are regarded as rigid thin discs with the same size.
Specifically, in the process of deriving the model, some assumptions need to be made to facilitate establishing and deriving the model, such as:
(1) assuming that the rear body center of mass has no motion other than roll and pitch caused by handlebar steering;
(2) the front and the rear wheels are braked, do not move relative to the frame, and roll purely with the ground;
(3) neglecting tire thickness and deformation, both wheels are considered as equally sized thin rigid disks.
After the above-mentioned conditions that can be assumed are satisfied, the two-wheeled robot system can be equivalent to a multi-link multi-joint system, as shown in fig. 2, which is a multi-link multi-joint equivalent vehicle body schematic diagram. Wherein A is0And A is the contact point of the rear wheel and the ground when the handlebar rotates and does not rotate respectively. B is the rear wheel circleThe center of the handlebar is the connecting point of the handlebar rotating pair and the rear frame, the center of the front wheel is D, and the contact point of the front wheel and the ground is E.Delta and theta respectively represent the roll angle, the handlebar rotation angle and the pitch angle of the rear vehicle body, theta1The angular displacement of point E on the front wheel.Is the front wheel roll angular velocity. Δ xrIs a pi along the rear car body plane caused by the pitching motion of the rear wheels1Translation distance of the intersection with the ground. Δ xfAnd Δ yfIs point E relative to initial point E due to handlebar rotation0Offset coordinates on the ground. Table 1 is a summary of equivalent joints for an embodiment of the present invention, such that the above translational and rotational movements are achieved by five joints. r isi(i is 1-4) is an equivalent connecting rod vector linking the five joints, wherein r is1Is the rear wheel link vector pointing from a to B; r is2Is the rear frame link vector pointing from B to C; r is3Is the front frame link vector pointing from C to D; r is4Is the front wheel link vector pointing from D to E. Further, θ0The pitch angle of the rear frame connecting rod vector when the handlebar rotation angle is zero; epsilon is the mounting angle of the connecting rod of the rear frame and is r2The included angle between the handle bar and the rotating shaft; eta is the inclination angle of the handlebar, which is the included angle between the rotating shaft of the handlebar and the plane of the rear vehicle body in the vertical direction; lambda is the handlebar front fork angle, which is the angle between the front frame C' D and the handlebar turning axis. G2And G1Respectively front and rear vehicle body mass centers.
TABLE 1
After setting the two-wheeled robot system as a multi-link multi-joint system, the following six coordinate systems are defined in the multi-link multi-joint system:
(1) inertial reference system { I }, A0xyz: the origin is fixed at A0Point, x axis from A0Point of directionE0The z-axis is vertical downwards, and the y-axis, the x-axis and the z-axis form a right-hand system; wherein A is0Contact point of rear wheel with ground when handlebar is turned, E0The contact point between the front wheel and the ground when the handlebar rotates;
(2) rear wheel coordinate system { B }, Bxbybzb: the origin is fixed at point B, xbThe axis being parallel to the x-axis of the inertial reference system, the z-and y-axes being rotatable about the x-axis by the inertial reference systemThe angle is found, then the rotation matrix from { I } to { B } is:
wherein B is the circle center of the rear wheel,the roll angle of the rear body;
(3) coordinate system of handlebar { C }, Cxcyczc: the origin is fixed at point C, ycAxle and rear wheel coordinate system ybThe axes being parallel, the x and z axes being surrounded by { B } around ybThe axis is rotated by an angle θ + η, and the rotation matrix from { B } to { C } is:
c is a connecting point of the handlebar rotating pair and the rear frame, theta is a pitch angle of the rear vehicle body, and eta is a handlebar inclination angle;
the handlebar inclination angle η satisfies the following geometrical constraints:
wherein, theta0Is the pitch angle of the rear frame connecting rod vector when the handlebar turning angle is zero, and epsilon is the rear frame connecting rod mounting angle;
(4) front wheel coordinate system { D }, Dxdydzd: the origin is fixed at point D, zdAxis and handlebar coordinate system zcThe axes being parallel, the x and y axes being defined by the handlebar coordinate systemcThe axis is rotated by an angle δ, and the rotation matrix from { C } to { D } is:
wherein D is the center of the front wheel circle, and delta is the handlebar turning angle;
(5) rear vehicle body coordinate system { G1}，G1x1y1z1: origin fixed to G1The rear vehicle body coordinate system is wound by the rear wheel coordinate system ybThe angle of rotation of the shaft is obtained byWhere η is 0, from { B } system to { G-1Rotation matrix of the system
Wherein G is1Is the center of mass of the rear vehicle body;
(6) front vehicle body coordinate system { G2}，G2x2y2z2: origin fixed to G2The front body coordinate system is parallel to the front wheel coordinate system, G2Is the front bodywork centroid.
According to the above coordinate system, each vector in fig. 2 can be represented as:
in the formula: the superscript indicates the coordinate system to which the vector corresponds, R is the wheel radius, lrIs the length of segment BC, d is the length of segment CC', lfIs the length of segment C' D.
The geometrical relationship can obtain the following towing distance of the two-wheeled robot:
and step S2, establishing two constraint equations according to the closed-loop kinematic chain borne by the multi-link multi-joint system and the wheel geometric characteristics of the two-wheel robot system, and establishing a kinematic model according to the two constraint equations.
Further, in one embodiment of the present invention, two constraint equations are established according to the closed-loop kinematic chain suffered by the multi-link multi-joint system and the wheel geometry of the two-wheel robot system, including:
constraint 1, closed-loop kinematic chain constraint
Wherein e isz＝[0,0,1]TIs a z-axis unit direction vector, r, of the inertial reference system { I }1Is the rear wheel link vector pointing from a to B; r is2Is the rear frame link vector pointing from B to C; r is3Is the front frame link vector pointing from C to D; r is4Is the front wheel link vector pointed to E by D, the superscript indicating this vectorThe coordinate system of the corresponding coordinate system is provided with a coordinate system,for the rotation matrix from { B } to { I },is a rotation matrix from C to B,is a rotation matrix from { D } to { C };
restraint 2, wheel geometry restraint
Wherein n isyY being the front wheel coordinate system { D }dAxial unit direction vector.
Further, in one embodiment of the present invention, the kinematic model is built according to two constraint equations, including:
and (3) carrying out derivation on constraint conditions of the constraint 1 and the constraint 2 to obtain:
wherein J is a Jacobian matrix; the Jacobian matrix can be based on generalized coordinatesAnd non-independent coordinatesThe decomposition is carried out in two parts:
speed of non-independent coordinates is defined by generalized speedTo show that:
the kinematic model can be found as:
wherein,is an identity matrix of order 2.
Specifically, first, system generalized coordinates and complete constraints are defined,
in the above-described motions, only the roll angle of the rear vehicle bodySteering angle δ is independent of the handlebar, so they are defined as the generalized coordinates of the system:
in the static balance process, the front and the rear wheels of the vehicle need to be grounded to form a closed-loop kinematic chain. The existence of a closed-loop kinematic chain allows two paths from the ground to reach any point on the system. Considering that the front wheel has more non-independent coordinates, the invention has the contact point A between the rear wheel and the ground0As a starting point for the path. Thus only half the coordinates are needed to represent the position of the front wheel contact point in the I system. Thus, let the new system coordinates be:
the following two complete constraints are constructed according to the closed-loop kinematic chain and wheel geometry that the system is subjected to:
constraint 1, closed-loop kinematic chain constraint
Wherein e isz＝[0,0,1]TIs the z-axis unit direction vector of the inertial reference system { I }.
Restraint 2, wheel geometry restraint
Wherein n isyY being the front wheel coordinate system { D }dUnit direction vector of axis, also front body plane Π2The normal vector of (2).
When δ is 0, θ1It should satisfy:
θ1＝-η|δ＝θ＝0 (11)
further, θ0The following constraints should also be satisfied:
establishing a robot kinematic model according to a constraint equation to obtain a nonlinear centroid height variation model,
the derivation of the integrity constraints (9) and (10) can be:
in the formula: j is the Jacobian matrix. The matrix may be further based on qiAnd non-independent coordinatesThe decomposition is carried out in two parts:
based on the above equation, the velocity of the dependent coordinates can be represented by a generalized velocity:thus, a kinematic model of the system can be obtained as:
in the formula:is an identity matrix of order 2.
And S3, solving the kinematic model by using a first class of Lagrange equations to obtain a two-wheeled robot dynamic model, analyzing the singular value of the controllability matrix, the attraction domain of the closed-loop controller and the control cost according to the two-wheeled robot dynamic model, and determining the towing distance meeting the control requirement in the towing distance range according to the analysis result.
Further, in one embodiment of the present invention, solving the two-wheeled robot dynamics model using the first type of lagrangian equation comprises:
wherein, L is a Lagrange function, T is the total kinetic energy of the two-wheeled robot system, V is the total potential energy of the two-wheeled robot system, gamma is a Lagrange multiplier,the generalized non-conservative external force on the two-wheeled robot system, D and D are respectively the disturbance torque on the rolling and steering channels of the handlebar, taucIs the driving torque of the rotating shaft of the handlebar.
Further, in an embodiment of the present invention, S3 specifically includes:
and in the range of the dragging distance, analyzing the singular value of the controllability matrix, the attraction domain of the closed-loop controller and the control cost, making a corresponding image, and determining the dragging distance meeting the control requirement according to the corresponding image and the control requirement.
Specifically, a two-wheeled robot dynamic model is established according to a kinematic model, and first, the total kinetic energy of the two-wheeled robot system is deduced and translated from the mass center translation kinetic energy T of the front and rear vehicle bodiestiAnd rotational kinetic energy T around the center of massriConsists of the following components:
kinetic energy 1, center of mass G of the rear vehicle body1The coordinates are expressed under the inertial system { I }:
in the formula:xG1＝xg1cosθ0+zg1sinθ0、zG1＝zg1cosθ0-xg1sinθ0，xg1and zg1Is the coordinate of the mass center of the rear vehicle relative to the rear vehicle frame.Is the translation of the rear wheel in the x-axis due to pitch, which can be found from hypothesis 2Then the angular velocity of the rear frame is projected to a rear frame coordinate system { G }1In the method, the following steps:
thus, the available rear body kinetic energy is as follows:
in the formula: m is1For rear body mass, let { G1Is the inertia main axis of the rear vehicle body, thenIn-body coordinate system { G for rear vehicle body1The moment of inertia matrix in (1).
Kinetic energy 2, center of mass G of front vehicle body2The coordinates are expressed under the inertial system { I }:
in the formula:xg2and zg2Is the coordinate of the center of mass of the front vehicle relative to the front frame, xG2＝d+xg2cosλ+zg2sinλ、zG2＝zg2cosλ-xg2sin lambda. Similarly, the angular velocity of the front frame is projected to a front frame coordinate system { G }2In the method, the following steps:
thus, the available front body kinetic energy is as follows:
in the formula: m is2Order { G } for front body mass2Inertia main shafting of front body, thenIn-vivo coordinate system { G for front vehicle body2The moment of inertia matrix in (1).
And taking the ground as a zero potential energy surface, and taking the z-axis component of the mass centers of the two vehicle bodies under the inertial system { I }, wherein the obtained total potential energy of the vehicle is as follows:
in the formula: g is the surface gravity acceleration (.)zThe orientation quantity z-axis component operator.
Solving the dynamic model of the system by applying a first type of Lagrange equation:
in the formula: L-T-V is the lagrange function, T is the total kinetic energy, V is the total potential energy, γ is the lagrange multiplier,d and D are respectively the disturbance torque borne by the rolling and the handlebar steering channels, taucIs the driving torque of the rotating shaft of the handlebar.
Equation (23) is written as the following robot-generic euler-lagrange form:
in the formula: m is a generalized mass matrix, V is a centrifugal force and Coriolis force matrix, and E is a gravity matrix, specific expressions of which are presented in the appendix. The above model can be converted into ordinary differential equations containing only 2 equations by using a kinematic model (15). The system coordinate acceleration can be derived by first deriving equation (15):
substituting the above equation and kinematic equation (15) into equation (25), and multiplying both ends by GTThe following can be obtained:
according to GTJTWhen 0, γ can be eliminated, yielding a dimensionality reduction kinetic model:
in the formula: mi＝GTThe MG is a dimension-reduced system quality matrix,is a dimension-reducing system damping matrix, Qiext＝GTQextIs a generalized external force matrix of a dimension reduction system.
The dimension reduction dynamic model (28) is arranged at an equilibrium pointAnd (3) linearization:
in the formula:is the slight offset of the generalized coordinate relative to the equilibrium position; delta taucIs a control deviation amount;easy certificateAnd isIgnoreA linear time-invariant kinetic equation in the form of a state space can be obtained as follows.
The influence of the drag distance on the handlebar steering static balance control is analyzed according to the process, and as shown in fig. 3, a drag distance controllability analysis flow chart is shown, and the influence of the drag distance is analyzed from three aspects of the singular value of the controllability matrix, the attraction domain of the closed-loop controller and the control cost within the allowable range of the drag distance.
First, the controllability is analyzed to construct a controllability matrix Q of the linear model (30)c：
Substituting the system matrices a and B into the above equation yields:
in the formula:to representColumn 2. To QcPerforming singular value decomposition QcAnd (5) making an image of the minimum singular value with respect to the dragging distance.
The attraction domain of the closed-loop controller is then analyzed. The following linear quadratic performance index is designed:
in the formula:andthe weight coefficient matrix to be determined can be debugged according to specific control effects. Solving the Riccati algebraic equation by:
a matrix P can be obtained, and then the handlebar torque linear feedback control law can be obtained as follows:
defining the maximum value of handlebar steering moment as tauc maxThen the reachable state space of the system satisfies the inequality:
|K|∞|x|∞≤||τc||＝||Kx||≤τc max (36)
in the formula: l. capillary∞Representing taking the infinite norm of the vector. Thus, the estimated attraction domain is:
an image of the attraction field (37) is made with respect to the drag distance.
And finally, analyzing the influence of the dragging distance on the control cost. The control cost is defined as:
in the formula: the integral superscript T indicates the control termination time. An image of the control cost with respect to the drag distance is made.
According to the three images, the appropriate dragging distance can be selected according to specific control requirements, such as high controllability, large attraction area and low control cost.
The above-mentioned formulas are supplemented below.
The jacobian matrix in equation (14) is defined as follows:
in the formula:
in the formula:
the generalized mass matrix in equation of dynamics (25) is defined as follows:
in the formula:
M23＝m2RxG2sinδcos(θ+η)+m2lrxG2sinδsin(θ0-η)-m2xG2zG2sinδ
M32＝m2RxG2sinδcos(θ+η)+m2lrxG2sinδsin(θ0-η)-m2xG2zG2sinδ
the damping matrix in equation (25) of dynamics is defined as follows:
in the formula:
V22＝0
the gravity matrix in equation (25) of dynamics is defined as follows:
in the formula:
according to the modeling and static balancing method of the two-wheeled robot based on the towing distance, which is provided by the embodiment of the invention, firstly, according to the motion constraint, the two-wheeled robot is equivalent to a multi-link multi-joint system with front and rear vehicle bodies hinged from the multi-rigid system. And secondly, establishing two constraint equations based on the closed-loop kinematic chain of the system and the contact characteristics of the wheels and the ground to obtain a kinematic model of the system. The first type of lagrangian equation is then used to derive a kinetic model of the system. And finally, analyzing the influence of the towing distance on the static balance from three aspects of the singular value of the system controllability matrix, the attraction domain of the closed-loop controller and the control energy consumption. The nonlinear relation between the handle turning angle and the height change of the mass center of the vehicle at different towing distances can be reflected, and a set of analysis process can be provided for the selection of the towing distances so as to improve the control effect of static balance.
Next, a proposed tow-distance-based two-wheeled robot modeling and static balancing apparatus according to an embodiment of the present invention will be described with reference to the drawings.
Fig. 4 is a schematic structural diagram of a tow-track-based two-wheeled robot modeling and static balancing apparatus according to an embodiment of the present invention.
As shown in fig. 4, the apparatus for modeling and static balancing a two-wheeled robot based on a tow distance includes: an equivalence module 100, a constraint module 200, and a modeling analysis module 300.
And the equivalent module 100 is used for setting the two-wheeled robot system into a multi-link multi-joint system when detecting that the two-wheeled robot system meets preset equivalent setting conditions, defining a plurality of coordinate systems in the multi-link multi-joint system, and calculating the towing distance range of the two-wheeled robot system according to the geometric relationship of the plurality of coordinate systems.
And the constraint module 200 is used for establishing two constraint equations according to the closed-loop kinematic chain borne by the multi-link multi-joint system and the wheel geometric characteristics of the two-wheel robot system, and establishing a kinematic model according to the two constraint equations.
And the modeling analysis module 300 is configured to solve the kinematic model by using a first class of lagrangian equations to obtain a two-wheeled robot dynamic model, analyze the singular value of the controllability matrix, the attraction domain of the closed-loop controller, and the control cost according to the two-wheeled robot dynamic model, and determine a towing distance meeting the control requirement in a towing distance range according to an analysis result.
Further, in an embodiment of the present invention, the presetting of the equivalent setting condition includes:
setting the rear body center of mass to include only roll and pitch caused by handlebar steering;
setting that the front and the rear wheels are braked and do not move relative to the frame, and the rear wheels roll with the ground;
the thickness and deformation of the tire are neglected, and the front wheel and the rear wheel are regarded as rigid thin discs with the same size.
Further, in one embodiment of the present invention, the plurality of coordinate systems are:
(1) inertial reference system { I }, A0xyz: the origin is fixed at A0Point, x axis from A0Direction E0The z-axis is vertical downwards, and the y-axis, the x-axis and the z-axis form a right-hand system; wherein A is0Contact point of rear wheel with ground when handlebar is turned, E0The contact point between the front wheel and the ground when the handlebar rotates;
(2) rear wheel coordinate system { B }, Bxbybzb: the origin is fixed at point B, xbThe axis being parallel to the x-axis of the inertial reference system, the z-and y-axes being rotatable about the x-axis by the inertial reference systemThe angle is found, then the rotation matrix from { I } to { B } is:
wherein B is the circle center of the rear wheel,the roll angle of the rear body;
(3) coordinate system of handlebar { C }, Cxcyczc: the origin is fixed at point C, ycAxle and rear wheel coordinate system ybThe axes being parallel, the x and z axes being surrounded by { B } around ybThe axis is rotated by an angle θ + η, and the rotation matrix from { B } to { C } is:
c is a connecting point of the handlebar rotating pair and the rear frame, theta is a pitch angle of the rear vehicle body, and eta is a handlebar inclination angle;
the handlebar inclination angle η satisfies the following geometrical constraints:
wherein, theta0Is the pitch angle of the rear frame connecting rod vector when the handlebar turning angle is zero, and epsilon is the rear frame connecting rod mounting angle;
(4) front wheel coordinate system { D }, Dxdydzd: the origin is fixed at point D, zdAxis and handlebar coordinate system zcThe axes being parallel, the x and y axes being defined by the handlebar coordinate systemcThe axis is rotated by an angle δ, and the rotation matrix from { C } to { D } is:
wherein D is the center of the front wheel circle, and delta is the handlebar turning angle;
(5) rear vehicle body coordinate system { G1}，G1x1y1z1: origin fixed to G1The rear vehicle body coordinate system is wound by the rear wheel coordinate system ybThe angle of rotation of the shaft is obtained byWhere η is 0, from { B } system to { G-1Rotation matrix of the system
Wherein G is1Is the center of mass of the rear vehicle body;
(6) front vehicle body coordinate system { G2}，G2x2y2z2: origin fixed to G2The front body coordinate system is parallel to the front wheel coordinate system, G2Is the front bodywork centroid.
Further, in one embodiment of the present invention, the tow-throw range of the two-wheeled robotic system is:
wherein R is the wheel radius, lrIs the length of segment BC, d is the length of segment CC', lfThe length of the front frame line segment C' D, λ is the handlebar fork angle, and η is the handlebar inclination angle.
Further, in one embodiment of the present invention, two constraint equations are established according to the closed-loop kinematic chain suffered by the multi-link multi-joint system and the wheel geometry of the two-wheel robot system, including:
constraint 1, closed-loop kinematic chain constraint
Wherein e isz＝[0,0,1]TIs the z-axis unit direction vector of the inertial reference system { I },r1is the rear wheel link vector pointing from a to B; r is2Is the rear frame link vector pointing from B to C; r is3Is the front frame link vector pointing from C to D; r is4Is a front wheel connecting rod vector pointed to E by D, the coordinate system corresponding to the vector is indicated by a superscript,for the rotation matrix from { B } to { I },is a rotation matrix from C to B,is a rotation matrix from { D } to { C };
restraint 2, wheel geometry restraint
Wherein n isyY being the front wheel coordinate system { D }dAxial unit direction vector.
Further, in one embodiment of the present invention, the kinematic model is built according to two constraint equations, including:
and (3) carrying out derivation on constraint conditions of the constraint 1 and the constraint 2 to obtain:
wherein J is a Jacobian matrix; the Jacobian matrix can be based on generalized coordinatesAnd non-independent coordinatesThe decomposition is carried out in two parts:
the velocity of the non-independent coordinates is represented by the generalized velocity:
the kinematic model can be found as:
wherein,is an identity matrix of order 2.
Further, in one embodiment of the present invention, solving the two-wheeled robot dynamics model using the first type of lagrangian equation comprises:
wherein, L is a Lagrange function, T is the total kinetic energy of the two-wheeled robot system, V is the total potential energy of the two-wheeled robot system, gamma is a Lagrange multiplier,the generalized non-conservative external force on the two-wheeled robot system, D and D are respectively the disturbance torque on the rolling and steering channels of the handlebar, taucIs the driving torque of the rotating shaft of the handlebar.
Further, in an embodiment of the present invention, the modeling analysis module is specifically configured to analyze the singular value of the controllability matrix, the attraction domain of the closed-loop controller, and the control cost in the range of the towing distance, make a corresponding image, and determine the towing distance meeting the control requirement according to the corresponding image and the control requirement.
It should be noted that the foregoing explanation of the embodiment of the method for modeling and static balancing a two-wheeled robot based on a tow-track is also applicable to the apparatus of this embodiment, and will not be described herein again.
According to the modeling and static balancing device of the two-wheeled robot based on the towing distance, firstly, the two-wheeled robot is equivalent to a multi-link multi-joint system with front and rear vehicle bodies hinged from the multi-rigid system according to the motion constraint. And secondly, establishing two constraint equations based on the closed-loop kinematic chain of the system and the contact characteristics of the wheels and the ground to obtain a kinematic model of the system. The first type of lagrangian equation is then used to derive a kinetic model of the system. And finally, analyzing the influence of the towing distance on the static balance from three aspects of the singular value of the system controllability matrix, the attraction domain of the closed-loop controller and the control energy consumption. The nonlinear relation between the handle turning angle and the height change of the mass center of the vehicle at different towing distances can be reflected, and a set of analysis process can be provided for the selection of the towing distances so as to improve the control effect of static balance.
Furthermore, the terms "first", "second" and "first" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one such feature. In the description of the present invention, "a plurality" means at least two, e.g., two, three, etc., unless specifically limited otherwise.
In the description herein, references to the description of the term "one embodiment," "some embodiments," "an example," "a specific example," or "some examples," etc., mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above are not necessarily intended to refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples. Furthermore, various embodiments or examples and features of different embodiments or examples described in this specification can be combined and combined by one skilled in the art without contradiction.
Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention, and that variations, modifications, substitutions and alterations can be made to the above embodiments by those of ordinary skill in the art within the scope of the present invention.
Claims (10)
1. A method for modeling and static balancing of a two-wheeled robot based on a towing distance is characterized by comprising the following steps:
s1, when detecting that the two-wheeled robot system meets the preset equivalent setting condition, setting the two-wheeled robot system as a multi-link multi-joint system, defining a plurality of coordinate systems in the multi-link multi-joint system, and calculating the towing distance range of the two-wheeled robot system according to the geometrical relationship of the coordinate systems;
s2, establishing two constraint equations according to the closed-loop kinematic chain borne by the multi-link multi-joint system and the wheel geometric characteristics of the two-wheel robot system, and establishing a kinematic model according to the two constraint equations;
s3, solving the kinematics model by using a first Lagrange equation to obtain a two-wheeled robot dynamics model, analyzing a singular value of a controllability matrix, an attraction domain of a closed-loop controller and a control cost according to the two-wheeled robot dynamics model, and determining a towing distance meeting a control requirement in the towing distance range according to an analysis result.
2. The method according to claim 1, wherein the preset equivalent setting condition comprises:
setting the rear body center of mass to include only roll and pitch caused by handlebar steering;
setting that the front and the rear wheels are braked and do not move relative to the frame, and the rear wheels roll with the ground;
the thickness and deformation of the tire are neglected, and the front wheel and the rear wheel are regarded as rigid thin discs with the same size.
3. The method of claim 1, wherein the plurality of coordinate systems are:
(1) inertial reference system { I }, A0xyz: the origin is fixed at A0Point, x axis from A0Direction E0The z-axis is vertical downwards, and the y-axis, the x-axis and the z-axis form a right-hand system; wherein A is0Contact point of rear wheel with ground when handlebar is turned, E0The contact point between the front wheel and the ground when the handlebar rotates;
(2) rear wheel coordinate system { B }, Bxbybzb: the origin is fixed at point B, xbThe axis being parallel to the x-axis of the inertial reference system, the z-and y-axes being rotatable about the x-axis by the inertial reference systemThe angle is found, then the rotation matrix from { I } to { B } is:
wherein B is the circle center of the rear wheel,the roll angle of the rear body;
(3) coordinate system of handlebar { C }, Cxcyczc: the origin is fixed at point C, ycAxle and rear wheel coordinate system ybThe axes being parallel, the x and z axes being surrounded by { B } around ybThe axis is rotated by an angle θ + η, and the rotation matrix from { B } to { C } is:
c is a connecting point of the handlebar rotating pair and the rear frame, theta is a pitch angle of the rear vehicle body, and eta is a handlebar inclination angle;
the handlebar inclination angle η satisfies the following geometrical constraints:
wherein, theta0Is the pitch angle of the rear frame connecting rod vector when the handlebar turning angle is zero, and epsilon is the rear frame connecting rod mounting angle;
(4) front wheel coordinate system { D }, Dxdydzd: the origin is fixed at point D, zdAxis and handlebar coordinate system zcThe axes being parallel, the x and y axes being defined by the handlebar coordinate systemcThe axis is rotated by an angle δ, and the rotation matrix from { C } to { D } is:
wherein D is the center of the front wheel circle, and delta is the handlebar turning angle;
(5) rear vehicle body coordinate system { G1}，G1x1y1z1: origin fixed to G1The rear vehicle body coordinate system is wound by the rear wheel coordinate system ybThe angle of rotation of the shaft is obtained byWhere η is 0, from { B } system to { G-1Rotation matrix of the system
Wherein G is1Is the center of mass of the rear vehicle body;
(6) front vehicle body coordinate system { G2}，G2x2y2z2: origin fixed to G2The front body coordinate system is parallel to the front wheel coordinate system, G2Is the front bodywork centroid.
4. The method of claim 3, wherein the tow-throw range of the two-wheeled robotic system is:
wherein R is the wheel radius, lrIs the length of segment BC, d is the length of segment CC', lfThe length of the front frame line segment C' D, λ is the handlebar fork angle, and η is the handlebar inclination angle.
5. The method of claim 4, wherein establishing two constraint equations based on the closed-loop kinematic chain experienced by the multi-link multi-joint system and the wheel geometry of the two-wheeled robotic system comprises:
constraint 1, closed-loop kinematic chain constraint
Wherein e isz＝[0,0,1]TIs a z-axis unit direction vector, r, of the inertial reference system { I }1Is a rear wheel link pointing from A to BVector quantity; r is2Is the rear frame link vector pointing from B to C; r is3Is the front frame link vector pointing from C to D; r is4Is a front wheel connecting rod vector pointed to E by D, the coordinate system corresponding to the vector is indicated by a superscript,for the rotation matrix from { B } to { I },is a rotation matrix from C to B,is a rotation matrix from { D } to { C };
restraint 2, wheel geometry restraint
Wherein n isyY being the front wheel coordinate system { D }dAxial unit direction vector.
6. The method of claim 5, wherein said building a kinematic model according to said two constraint equations comprises:
and (3) carrying out derivation on constraint conditions of the constraint 1 and the constraint 2 to obtain:
wherein J is a Jacobian matrix; the Jacobian matrix can be based on generalized coordinatesAnd non-independent coordinatesThe decomposition is carried out in two parts:
the velocity of the non-independent coordinates is represented by the generalized velocity:
the kinematic model can be found as follows:
wherein,is an identity matrix of order 2.
7. The method of claim 5, wherein solving the two-wheeled robot dynamics model using a first class of lagrangian equations comprises:
wherein, L is a Lagrange function, T is the total kinetic energy of the two-wheeled robot system, V is the total potential energy of the two-wheeled robot system, gamma is a Lagrange multiplier,the generalized non-conservative external force on the two-wheeled robot system, D and D are respectively the disturbance torque on the rolling and steering channels of the handlebar, taucIs the driving torque of the rotating shaft of the handlebar.
8. The method according to claim 1, wherein the S3 specifically includes:
and in the dragging distance range, analyzing singular values of the controllability matrix, the attraction domain of the closed-loop controller and the control cost, making a corresponding image, and determining the dragging distance meeting the control requirement according to the corresponding image and the control requirement.
9. A tow-distance-based two-wheeled robot modeling and static balancing device is characterized by comprising:
the system comprises an equivalence module, a control module and a control module, wherein the equivalence module is used for setting a two-wheeled robot system into a multi-connecting-rod multi-joint system when detecting that the two-wheeled robot system meets preset equivalence setting conditions, defining a plurality of coordinate systems in the multi-connecting-rod multi-joint system, and calculating the dragging distance range of the two-wheeled robot system according to the geometric relations of the coordinate systems;
the constraint module is used for establishing two constraint equations according to a closed-loop kinematic chain borne by the multi-connecting-rod multi-joint system and the wheel geometric characteristics of the two-wheeled robot system, and establishing a kinematic model according to the two constraint equations;
and the modeling analysis module is used for solving the kinematic model by utilizing a first class of Lagrange equations to obtain a two-wheeled robot dynamic model, analyzing the singular value of a controllability matrix, the attraction domain of the closed-loop controller and the control cost according to the two-wheeled robot dynamic model, and determining the towing distance meeting the control requirement in the towing distance range according to the analysis result.
10. The apparatus of claim 9, wherein the preset equivalent setting condition comprises:
setting the rear body center of mass to include only roll and pitch caused by handlebar steering;
setting that the front and the rear wheels are braked and do not move relative to the frame, and the rear wheels roll with the ground;
the thickness and deformation of the tire are neglected, and the front wheel and the rear wheel are regarded as rigid thin discs with the same size.
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN114146363A (en) * | 2021-12-14 | 2022-03-08 | 国家康复辅具研究中心 | Walking aid training system and integrated control method thereof |
| CN116184860A (en) * | 2023-04-27 | 2023-05-30 | 北京国领智能科技有限公司 | Magnetic attraction crawler type wall climbing robot dynamics modeling and control method |
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