CN109079850B

CN109079850B - D-H system and D-H parameter determination method of multi-axis robot based on axis invariance

Info

Publication number: CN109079850B
Application number: CN201810933322.XA
Authority: CN
Inventors: 居鹤华; 石宝钱
Original assignee: Individual
Current assignee: Individual
Priority date: 2018-08-16
Filing date: 2018-08-16
Publication date: 2020-01-07
Anticipated expiration: 2038-08-16
Also published as: WO2020034417A1; CN109079850A

Abstract

The invention discloses a multi-axis robot D-H system and D-H parameter determination method based on axis invariants. Defining a one-dimensional coordinate axis which is composed of an original point and a unit base and is a direction reference line with scales, wherein the scales of the points on the axis are coordinates; the shaft invariants are parameterized natural coordinate bases, are elements of a multi-shaft system, and are equivalent to the translation and rotation of a coordinate system fixedly connected with the fixed shaft invariants; the indexes in the D-H system conform to the father indexes; determining a D-H system through a fixed shaft invariant, determining a middle point and a D-H system origin point, and obtaining a unit coordinate vector; and obtaining the attitude according to the rotation characteristic of the vector. The method is based on the structural parameters of the axis invariants, does not need to establish an intermediate coordinate system, avoids measurement errors caused by introducing the intermediate coordinate system, and ensures the accuracy of determining the D-H system and the D-H parameters. The method has an important function of improving the absolute positioning and attitude determination precision of the robot.

Description

D-H system and D-H parameter determination method of multi-axis robot based on axis invariance

Technical Field

The invention relates to a D-H system and D-H parameter determination method for a multi-axis robot, and belongs to the technical field of robots.

Background

When the robot applies D-H system modeling and D-H parameters to calculate inverse kinematics, the absolute positioning and attitude determination precision of a robot system is far lower than the repeated precision of the system due to the existence of machining and assembling errors; meanwhile, the D-H system establishing and D-H parameter determining processes in the prior art are complicated, and when the degree of freedom of the system is high, the manual completion of the process is low in reliability. Therefore, there is a need to solve the problem of determining the D-H system and D-H parameters by a computer. Meanwhile, the high-precision D-H system and D-H parameters are the basis for the accurate operation of the robot and the development of the Teaching-Playback (Teaching and playing) robot to the autonomous robot.

Disclosure of Invention

The invention aims to solve the technical problem of providing a D-H system and a D-H parameter determination method of a multi-axis robot based on axis invariance, which does not need to establish an intermediate coordinate system, avoids measurement errors caused by introducing the intermediate coordinate system, ensures the accuracy of determining the D-H system and the D-H parameter, and improves the absolute positioning and attitude determination accuracy of the robot.

In order to solve the technical problems, the invention adopts the following technical scheme:

a method for determining D-H system and D-H parameters of a multi-axis robot based on axis invariance is characterized in that,

defining a one-dimensional coordinate axis l from an origin O_lAnd a unit radical e_lIs formed by a scaleThe scale of the point S on the axis l is the coordinate; basis vector e in whole form_lRepresenting an objective unit direction, the component form being noted

Namely, the three-dimensional space-time-domain three-dimensional space-time domain;

is the unit volume of the rod l in the x-axis under the D-H system,

in the same way, the same applies when the rod is l';

base vector e_lIs a reaction of with F^[l]Any vector of consolidation, like the base vector

Is and

any vector, axis invariant of consolidation

Is F^[l]And

a common reference group; the shaft invariants are parameterized natural coordinate bases, are elements of a multi-shaft system, and are equivalent to the translation and rotation of a coordinate system fixedly connected with the fixed shaft invariants;

when the system is in a zero position, the natural coordinate system is used as a reference, and coordinate axis vectors are obtained through measurement

In the kinematic pairAxial vector during motion

Is an invariant; axial vectorAnd joint variables

Uniquely identifying kinematic pairThe rotational relationship of (a);

are respectively a connecting rod piece

Coordinate axis vectors and kinematic pairs of the rod pieces l are similar to those of other rod pieces;

natural coordinate system

Corresponding D-H is described as

F^[l']The same process is carried out; kinematic pair

The corresponding axes are denoted by

That is, the indexes in the D-H system conform to the parent indexes;

determining a D-H system by fixing axis invariants, comprising the steps of:

determining intermediate points

And D-H is origin O_l′，

【1】 Order toAnd z_l′Respectively passing through the axes by constant amountsAndand is

Wherein,is that

The unit coordinate vector of (a) is,

the expression form of (1) is a projector, and represents a projection vector or a projection sequence of a vector or a second-order tensor to a reference base, namely a coordinate vector or a coordinate array;

【2】

is defined as

To

The male vertical line of (a);

【3】 To obtain

Wherein

Is that

A unit coordinate vector of (a);

【4】 Rotation characteristics according to vectorTo obtainIs represented by

To F^[l']In an attitude of, i.e. by the rod member

Rotate to the rod l'.

In the step (3),

in the formula,is axis invariant

The cross-multiplication matrix of (a) is,

is a shaft

The unit coordinate vector of (a) is,

is a shaft

The unit coordinate vector of (2).

In the step (4),

is that

Unit seatThe vector of the standard is marked on the basis of the standard vector,

is represented by

To F^[l']In a posture of being composed of

Rotate to l'.

In step [ 2 ], ifThen

In the formula,is a shaft

The unit coordinate vector of (a) is,

is that

A unit coordinate vector of (a);

is a rod piece

From the origin to the rod

OfTranslation vector of the point.

In step [ 2 ], if

And is

Then

In the formula,

is a rod piece

The origin of the rod l to the origin of the rod l;

is axis invariant

A cross-product matrix of; 0₃＝[0 0 0]^T。

In step [ 2 ], if

And is

Then

And is

Is represented by the formula (12) andto obtain

In the formula,is a rod piece

The origin of the rod l to the origin of the rod l;

is a rod piece

To the i' origin of the rod.

And the method also comprises a D-H parameter determination step based on the fixed axis invariant.

Determining D-H parameters

And

and

are respectively rod pieces

To the rod piece

The wheelbase and the offset of the wheel,

in the formula,

is a shaft

A unit coordinate vector of (a);

is a rod piece

The origin of the rod l to the origin of the rod l;is a rod piece

From the origin to the rod

The translation vector of the origin of (a);

is a rod piece

From the origin to the rod

The origin of the translation vector.

Order to

Defined by the angle of twist of the shaft

In the formula,

is a shaft

To the shaft

The twist angle of (c).

Order toDefined by the angle of rotation of the joint

In the formula,

is a shaft

Zero position of (a).

The invention achieves the following beneficial effects:

the invention provides and analyzes a D-H system and a D-H parameter determination method based on fixed axis invariants. The model engineering application of the CE3 rover demonstrates the correctness of this principle. The invention has simple chain symbol system and expression form of axis invariant, has pseudo code function, accurate physical meaning, and ensures reliability of engineering realization; based on the structural parameters of the axis invariant, an intermediate coordinate system does not need to be established, so that the measurement error caused by introducing the intermediate coordinate system is avoided, and the accuracy of determining the D-H system and the D-H parameters is ensured. The method has an important function of improving the absolute positioning and attitude determination precision of the robot.

Drawings

FIG. 1 a natural coordinate system and axis chain;

FIG. 2 is a fixed axis invariant;

FIG. 3 is a diagram illustrating the relationship between the natural coordinate system and the D-H system.

Detailed Description

The invention is further described below with reference to the accompanying drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.

Define 1 natural coordinate axes: a unit reference axis having a fixed origin, referred to as being coaxial with the axis of motion or measurement, is a natural coordinate axis, also referred to as the natural reference axis.

Defining 2 a natural coordinate system: as shown in fig. 1, if the multi-axis system D is located at the zero position, the directions of all cartesian body coordinate systems are the same, and the origin of the body coordinate system is located on the axis of the moving shaft, the coordinate system is a natural coordinate system, which is simply referred to as a natural coordinate system.

The natural coordinate system has the advantages that: (1) the coordinate system is easy to determine; (2) the joint variable at zero is zero; (3) the system postures at the zero position are consistent; (4) and accumulated errors of measurement are not easily introduced.

From definition 2, it can be seen that the natural coordinate system of all the rods coincides with the orientation of the base or world system when the system is in the zero position. With the system in zero position

Time, natural coordinate system

Vector around axis

Angle of rotationWill be provided with

Go to F^[l]；

In that

Coordinate vector of

At F^[l]Coordinate vector of

Is constant, i.e. has

According to the formula, the method has the advantages that,

or

Independent of adjacent coordinate systems

And F^[l](ii) a So it is called

Or

Is axis invariant. When invariance is not emphasized, the method can be called a coordinate axis vector (axis vector for short).

OrCharacterized by being a bodyCoordinate vector of reference unit common to body l, and reference point

And O_lIs irrelevant. Body

The body l is a rod or a shaft.

The axis invariants are essentially different from coordinate axes:

(1) the coordinate axis is a reference direction with a zero position and unit scales, and can describe the position of translation along the direction, but cannot completely describe the rotation angle around the direction, because the coordinate axis does not have a radial reference direction, namely, the zero position representing rotation does not exist. In practical applications, the radial reference of the shaft needs to be supplemented. For example: in the Cartesian system F^[l]In the rotation around lx, ly or lz is used as a reference zero position. The coordinate axes themselves are 1D, with 3 orthogonal 1D reference axes constituting a 3D cartesian frame.

(2) The axis invariant is a 3D spatial unit reference axis, which is itself a frame. It itself has a radial reference axis, i.e. a reference null. The spatial coordinate axes and their own radial reference axes may define cartesian frames. The spatial coordinate axis may reflect three basic reference properties of the motion axis and the measurement axis.

The axis vector of the chainless index is recorded in the literature

And is called the Euler Axis (Euler Axis), and the corresponding joint Angle is called the Euler Angle. The present application is no longer followed by the euler axis, but rather is referred to as the axis invariant because the axis invariant has the following properties:

【1】 Given rotation transformation array

Since it is a real matrix whose modes are unity, it has a real eigenvalue λ₁And two complex eigenvalues λ conjugated to each other₂＝e^iφAnd lambda₃＝e^-iφ(ii) a Wherein: i is a pure imaginary number. Therefore, | λ₁|·||λ₂||·||λ₃1, to obtain lambda₁1. Axial vector

Is a real eigenvalue λ₁1, is an invariant;

【2】 Is a 3D reference shaft, not only having an axial reference direction, but also having a radial reference null, as will be described in section 3.3.1.

【3】 Under a natural coordinate system:i.e. axial invariant

Is a very special vector, has invariance to the derivative of time and has very good mathematical operation performance;

for an axis invariant, its absolute derivative is its relative derivative. Since the axis invariant is a natural reference axis with invariance, its absolute derivative is always a zero vector. Thus, the axis invariants have invariance to the time differential. Comprises the following steps:

【4】 In a natural coordinate system, passing an axial vector

And joint variables

Can directly describe the rotating coordinate array

It is not necessary to establish a separate system for the rods other than the roots. Meanwhile, the measurement precision of the system structure parameters can be improved by taking the only root coordinate system to be defined as reference;

【5】 Using axial vectors

The method has the advantages that a completely parameterized unified multi-axis system kinematics and dynamics model comprising a topological structure, a coordinate system, a polarity, a structure parameter and a mechanics parameter is established.

Factor vector e_lIs a reaction of with F^[l]Any vector, base vector, of consolidation

Is and

any vector of consolidation, in turn

Is F^[l]And

a common unit vector, therefore

Is F^[l]And

a common basis vector. Thus, the axis is invariantIs F^[l]And

common reference base. The axis invariants are parameterized natural coordinate bases, and are primitives of the multi-axis system. The translation and rotation of the fixed shaft invariant are equivalent to the translation and rotation of a coordinate system fixedly connected with the fixed shaft invariant.

In the kinematic pair

Axial vector during motionIs not changedAn amount; axial vector

And joint variables

Uniquely identifying kinematic pair

The rotational relationship of (1).

Thus, with the natural coordinate system, only a common reference frame need be determined when the system is in the null position, rather than having to determine individual body coordinate systems for each rod in the system, as they are uniquely determined by the axis invariants and the natural coordinates. When performing system analysis, the other natural coordinate systems, apart from the base system, to which the bars are fixed, only occur conceptually, and are not relevant to the actual measurement. The theoretical analysis and engineering functions of a natural coordinate system on a multi-axis system (MAS) are as follows:

(1) the measurement of the structural parameters of the system needs to be measured by a uniform reference system; otherwise, not only is the engineering measurement process cumbersome, but the introduction of different systems introduces greater measurement errors.

(2) A natural coordinate system is applied, and except for the root rod piece, the natural coordinate systems of other rod pieces are naturally determined by the structural parameters and the joint variables, so that the kinematics and dynamics analysis of the MAS system is facilitated.

(3) In engineering, the method can be applied to optical measurement equipment such as a laser tracker and the like to realize the accurate measurement of the invariable of the fixed shaft.

(4) As the kinematic pairs R and P, the spiral pair H and the contact pair O are special cases of the cylindrical pair C, the MAS kinematics and dynamics analysis can be simplified by applying the cylindrical pair.

Definition 3 invariant: the quantities that are not measured in dependence on a set of coordinate systems are called invariant.

Define 4 rotational coordinate vectors: vector around coordinate axis

Rotated to an angular position

Coordinate vector ofIs composed of

Define 5 translation coordinate vectors: vector along coordinate axis

Translation to linear position

Coordinate vector of

Is composed of

Define 6 natural coordinates: taking the vector of the natural coordinate axis as a reference direction, and marking the angular position or linear position of the zero position of the system as q_lCalled natural coordinates; weighing the quantity mapped one by one with the natural coordinate as a joint variable; wherein:

define 7 mechanical zero: for kinematic pairAt an initial time t₀Zero position of time, joint absolute encoder

Not necessarily zero, which is called mechanical zero;

hence the joint

Control amount of

Is composed of

Defining 8 natural motion vectors: will be represented by natural coordinate axis vectors

And natural coordinate q_lDetermined vectorReferred to as natural motion vectors. Wherein:

the natural motion vector realizes the unified expression of the translation and rotation of the shaft. Vectors to be determined from natural coordinate axis vectors and joints, e.g.

Called free motion vector, also called free helix. Obviously, axial vector

Is a specific free helix.

Define 9 the joint space: by joint natural coordinates q_lThe space represented is called joint space.

Define a 10-bit shape space: a cartesian space expressing a position and a posture (pose for short) is called a configuration space, and is a dual vector space or a 6D space.

Defining 11 a natural joint space: with reference to natural coordinate system and by joint variables

Indicating that there must be at system zero

Is called the natural joint space.

As shown in FIG. 2, a given linkOrigin O_lPosition-dependent vector

Constrained axis vector

Is a fixed axis vector, is denoted asWherein:

axial vector

Is the natural reference axis for the natural coordinates of the joint. Due to the fact thatIs an axis invariant, so it is calledFor the invariants of fixed axes, it characterizes kinematic pairs

The natural coordinate axis is determined. Fixed shaft invariant

Is a chain link

Natural description of structural parameters.

Defining 12 a natural coordinate axis space: the fixed axis invariant is used as a natural reference axis, and a space represented by corresponding natural coordinates is called a natural coordinate axis space, which is called a natural axis space for short. It is a 3D space with 1 degree of freedom.

As shown in figure 2 of the drawings, in which,

and

without rod omega_lIs a constant structural reference.Determines the axis l relative to the axis

Five structural parameters of (a); and joint variable q_lTogether, the rod omega is expressed completely_lThe 6D bit shape. Given aThe natural coordinate system of the fixed rod can be determined by the structural parameters

And joint variables

And (4) uniquely determining. Balance shaft invariant

Fixed shaft invariant

Variation of joint

And

is naturally invariant. Obviously, invariant by a fixed axis

And joint variables

Natural invariance of constituent jointsAnd from a coordinate system

To F^[l]Determined spatial configuration

Having a one-to-one mapping relationship, i.e.

Given a multi-axis system D ═ T, a, B, K, F, NT }, in the system null position, only the base or inertial frame is established, as well as the reference points O on the axes_lOther rod coordinate systems are naturally determined. Essentially, only the base or inertial frame need be determined.

Given a structural diagram with a closed chain connected by kinematic pairs, any kinematic pair in a loop can be selected, and a stator and a mover which form the kinematic pair are divided; thus, a loop-free tree structure, called Span tree, is obtained. T represents a span tree with direction to describe the topological relation of tree chain motion.

I is a structural parameter; a is an axis sequence, F is a rod reference system sequence, B is a rod body sequence, K is a kinematic pair type sequence, and NT is a sequence of constraint axes, i.e., a non-tree.

For taking an axis sequence

Is a member of (1). The revolute pair R, the prismatic pair P, the helical pair H and the contact pair O are special cases of the cylindrical pair C.

The basic topological symbol and operation for describing the kinematic chain are the basis for forming a kinematic chain topological symbol system, and are defined as follows:

【1】 The kinematic chain is identified by a partially ordered set (].

【2】A^[l]Is a member of the axis-taking sequence A; since the axis name l has a unique number corresponding to A^[l]Number of (2), therefore A^[l]The computational complexity is O (1).

【3】

Is a father axis of the taking axis l; shaft

The computational complexity of (2) is O (1). The computation complexity O () represents the number of operations of the computation process, typically referred to as the number of floating point multiplies and adds. The calculation complexity is very complicated by the expression of the times of floating point multiplication and addition, so the main operation times in the algorithm circulation process are often adopted; such as: joint pose, velocity, acceleration, etc.

【4】

For taking an axis sequenceA member of (a);

the computational complexity is O (1).

【5】^ll_kTo take the kinematic chain from axis l to axis k, the output is represented as

And is

Cardinal number is recorded as $^ll_k|。^ll_kThe execution process comprises the following steps: executeIf it is

Then execute

Otherwise, ending.^ll_kThe computational complexity is O (# |)^ll_k|)。

【6】^ll is a child of axis l. The operation is represented in

Finding the address k of the member l; thus, a sub-A of the axis l is obtained^[k]. Due to the fact that

Has no off-order structure, therefore^lThe computational complexity of l is

【7】^lL denotes obtaining a closed sub-tree consisting of the axis L and its sub-tree, ^l l is a subtree containing no L; recursive execution^ll, the computational complexity is

【8】 Adding and deleting operations of branches, subtrees and non-tree arcs are also necessary components; thus, the variable topology is described by a dynamic Span tree and a dynamic graph. In the branch^ll_kIn, if

Then remember

Namely, it isRepresenting the child of member m taken in the branch.

The following expression or expression form is defined:

the shafts and the rod pieces have one-to-one correspondence; quantity of property between axes

And the amount of attribute between the rods

Has the property of order bias.

Appointing:

representing attribute placeholders; if the attribute P or P is location-related, then

Is understood to be a coordinate system

To F^[l]The origin of (a); if the property P or P is directional, thenIs understood to be a coordinate systemTo F^[l]。

And

are to be understood as a function of time t, respectively

And

and is

And

is t₀A constant or array of constants at a time. But in the body

And

should be considered a constant or an array of constants.

In the present application, the convention: in a kinematic chain symbolic operation system, attribute variables or constants with partial order include indexes representing partial order in name; or the upper left corner and the lower right corner, or the upper right corner and the lower right corner; the direction of the parameters is always from the upper left corner index to the lower right corner index of the partial order index or from the upper right corner index to the lower right corner index. For example:can be briefly described as (representing k to l) translation vectors;

represents the line position (from k to l);^kr_lrepresents a translation vector (from k to l); wherein: r represents a "translation" attribute symbolAnd the other attribute symbols correspond to: the attribute token phi represents "rotate"; the attribute symbol Q represents a "rotation transformation matrix"; the attribute symbol l represents "kinematic chain"; attribute character u represents a "unit vector"; the attribute symbol ω represents "angular velocity"; the angle index i represents an inertial coordinate system or a geodetic coordinate system; other corner marks can be other letters and can also be numbers.

The symbolic specification and convention of the application are determined according to the principle that the sequence bias of the kinematic chain and the chain link are the basic unit of the kinematic chain, and reflect the essential characteristics of the kinematic chain. The chain index represents the connection relation, and the upper right index represents the reference system. The expression of the symbol is simple and accurate, and is convenient for communication and written expression. Meanwhile, the data are structured symbolic systems, which contain elements and relations for forming each attribute quantity, thereby facilitating computer processing and laying a foundation for automatic modeling of a computer. The meaning of the index needs to be understood through the context of the attribute symbol; such as: if the attribute symbol is of a translation type, the index at the upper left corner represents the origin and the direction of a coordinate system; if the attribute is of the pivot type, the top left indicator represents the direction of the coordinate system.

(1)l_S-a point S in the bar l; and S denotes a point S in space.

(2)

Origin O of bar k_kTo the origin O of the rod l_lA translation vector of (a);

^kr_l－

in a natural coordinate system F^[k]The coordinate vector from k to l;

(3)

-origin O_kTo point l_SA translation vector of (a);

－

at F^[k]A lower coordinate vector;

(4)

-origin O_kA translation vector to point S;

^kr_S－

at F^[k]A lower coordinate vector;

(5)

-a connecting rod member

And a kinematic pair of the rod piece l;

kinematic pair

An axis vector of (a);

and

－

are respectively at

And F^[l]A lower coordinate vector;is an axis invariant, being a structural constant;

as rotation vector, rotation vector/angle vector

Is a free vector, i.e., the vector is free to translate;

(6)

along the axisThe linear position (translational position) of (c),

-about an axis

The angular position of (a), i.e. joint angle, joint variable, is a scalar;

(7) when the index of the lower left corner is 0, the index represents a mechanical zero position; such as:

-a translation shaft

The mechanical zero position of the magnetic field sensor,

-a rotating shaftMechanical zero position of (a);

(8) 0-three-dimensional zero matrix; 1-a three-dimensional identity matrix;

(9) appointing: "\ indicates continuationSymbol;

representing attribute placeholders; then

Power symbolTo represent

To the x-th power of; the right upper corner is marked with ^ or

A representation separator; such as:

or

Is composed of

To the x power of.

To represent

The transpose of (1) indicates transposing the set, and no transpose is performed on the members; such as:

the projection symbol is a projection vector or a projection sequence of a vector or a second-order tensor to a reference base, namely a coordinate vector or a coordinate array, and the projection is dot product operation "·"; such as: position vector

In a coordinate system F^[k]The projection vector in (1) is recorded as

Is a cross multiplier; such as:

is axis invariant

A cross-product matrix of; given any vector

Is cross-multiplication matrix of

The cross-multiplication matrix is a second order tensor.

Cross-multiplier operations have a higher priority than projecters

The priority of (2). Projecting sign

Is higher priority than the member access character

Or

Member access signPriority higher than power symbol

(10) Projection vector of unit vector in geodetic coordinate systemUnit zero vector

(11)Zero position by origin

To the origin O_lIs translated by the vector of

Representing the location structure parameter.

(12)ⁱQ_lA rotation transformation matrix in relative absolute space;

(13) taking the vector of the natural coordinate axis as a reference direction, and marking the angular position or linear position of the zero position of the system as q_lCalled natural coordinates; variation of jointNatural joint coordinate phi_l；

(14) For a given ordered set r ═ 1,4,3,2]^TRemember r^[x]The representation takes the x-th row element of the set r. Frequently remembered [ x ]]、[y]、[z]And [ omega ]]This is shown with the elements in columns 1, 2, 3 and 4.

(15)ⁱl_jRepresents a kinematic chain from i to j;^ll_ktaking a kinematic chain from an axis l to an axis k;

given kinematic chain

If n represents a Cartesian rectangular system, it is calledIs a Cartesian axis chain; if n represents a natural reference axis, then call

Is a natural axis chain.

(16) Rodrigues quaternion expression form:

euler quaternion expression:

quaternion (also called axis quaternion) representation of invariants

Such as position vectors

The projection vector on three Cartesian coordinate axes is

Definition of

Due to the fact that^lr_lSThe top left indicator indicates the reference frame,^lr_lSnot only indirectly represent displacement vector

And the displacement coordinate vector is directly expressed, namely, the dual functions of the vector and the coordinate vector are realized.

1. D-H system modeling method based on fixed axis invariant

As shown in fig. 3, a ═ 0,1, …, k]，F＝{F^[l]|l∈A}，Wherein: f^[l]Is a natural coordinate system, F^[l′]Is a D-H series; and is provided with

A is an axis sequence, and F is a rod reference system sequence;

under the adjacent natural coordinate system, the adjacent rods l and

have the same coordinates;

i-fixed axis invariants;

one-dimensional coordinate axis l is defined by the origin O_lAnd a unit radical e_lThe structure is that the direction reference line is provided with scales; which are the primitives that make up the reference frame. The scale of point S on axis i is the coordinate. Basis vector e in whole form_lRepresenting an objective unit direction; its component form is described as

I.e., composed of three independent ordered symbols, representing three independent degrees of freedom.

Is the unit product of the x-axis of the axis l under the D-H system,the same is true.

Is andany vector of consolidation, in turn

Is F^[l]And

a common unit vector, therefore

Is F^[l]And

a common basis vector. Thus, the axis is invariantIs F^[l]And

When the system is in a zero position, the natural coordinate system is used as a reference, and coordinate axis vectors are obtained through measurementIn the kinematic pair

Axial vector during motion

Is an invariant; axial vector

And joint variables

Uniquely identifying kinematic pair

The rotational relationship of (1).

The D-H system plays an important role in the inverse kinematics calculation of the robot. Given a multi-axis system, any state is taken as the zero position of the system. Natural coordinate system

Corresponding D-H is described as

According to the numbering habit of the D-H coordinate system, the kinematic pair

The corresponding axes are denoted by

I.e. the indices in the D-H system are used to follow parent indices, unlike the numbering of the natural coordinate system to follow child indices.

First, an intermediate point is determined

And D-H is origin O_l′。

【1】 Order toAnd z_l′Respectively passing through the axes by constant amounts

And

and is

【2】Is defined as

To n_lThe male vertical line of (1) includes three cases.

[ 2.1 ] ifThen ⁰r_l′Available axis invariantAnd

and (4) showing.

Due to the fact thatAnd is

To obtain

Resolution of formula (3) to

Bringing formula (4) into formula (1)

For a natural coordinate system, there are

Rewriting the formula (7), the formula (5) and the formula (6) as follows:

and is

The formula (7) and the formula (9) can be expressed as follows

In general,

for presentation

The null direction.

Is a shaftA unit coordinate vector of (a);

[ 2.2 ] obviously, ifAnd is

To obtainAnd is

[ 2.3 ] if

And isThen

And is

Is represented by the formula (12) and

to obtain

【3】 To obtain

Is a shaft

A unit coordinate vector of (a);

【4】 From formula (15) to

Based on rotation of the vector, the base vector

And base vector e_lIs defined as the outer dot product of

Is a shaftA unit coordinate vector of (a);

is represented by

To F^[l']In a posture of being composed of

Rotate to l'.

Thus, the D-H series is determined by fixing the axis invariant.

2. D-H parameter determination method based on fixed shaft invariant

As shown in figure 3 of the drawings,

is a shaft

The unit coordinate vector of (2).

Order toDefined by the angle of twist of the shaft

Order to

Defined by the angle of rotation of the joint

Wherein: a is_lAnd c_lAre respectively a shaft

Wheelbase and offset, α, to the axis l_lIs a shaft

The angle of twist to the axis l is,is a shaft

Zero position of (a).

In conclusion, the fixed shaft does not change

And

can conveniently express the D-H parameter

And

can express zero position at the same time

The D-H system and D-H parameter determination based on the fixed axis invariant has the following functions:

【1】 The problem that the D-H system and the D-H parameters are difficult to realize in engineering is solved; since the determination of the D-H system and D-H parameters requires the use of optical features, which are usually located inside and outside the rod, it is not possible to measure them accurately in engineering. Whereas the fixed axis invariants can be measured indirectly by means of an optical measuring device such as a laser tracker.

【2】 The accuracy of the D-H system and the D-H parameters is ensured; the process of determining the D-H system and the D-H parameters needs to meet the orthogonality requirement, and is difficult to meet in engineering. When a multi-axis system is designed, the D-H system and the D-H parameters determined according to a drawing are greatly different from the engineering D-H system and the engineering D-H parameters, and errors caused by machining and system assembly need to be considered. The fixed shaft invariant measured by engineering can obtain accurate structural parameters represented by the fixed shaft invariant on the premise of ensuring the precision of the measuring equipment, thereby ensuring the precision of a D-H system and D-H parameters.

The above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, several modifications and variations can be made without departing from the technical principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.

Claims

1. A method for determining D-H system and D-H parameters of a multi-axis robot based on axis invariance is characterized in that,

defining a one-dimensional coordinate axis l from an origin O_lAnd a unit radical e_lThe method comprises the following steps that a one-dimensional coordinate axis l is a direction reference line with scales, and the scales of a point S on the coordinate axis l are coordinates; basis vector e in whole form_lRepresenting an objective unit direction, the component form being noted

I.e. by three independent ordersSymbolic composition, representing three independent degrees of freedom;

is a unit base of the bar member l on the x-axis under the D-H system,

in the same way, the same applies when the rod is l';

Is andany vector, axis invariant of consolidation

Is F^[l]Anda common reference group; the shaft invariants are parameterized natural coordinate bases, are elements of a multi-shaft system, and are equivalent to the translation and rotation of a coordinate system fixedly connected with the fixed shaft invariants; f^[l]Is a natural coordinate system;

when the system is in a zero position, the axis invariant is measured by taking the natural coordinate system as a referenceIn the kinematic pair

The axes are not changed when in motion

Is an invariant; axial invariant

And joint variables

Uniquely identifying kinematic pair

The rotational relationship of (a);

are respectively a connecting rod piece

The shaft invariance and the kinematic pair of the rod piece I are the same as those of other rod pieces;

natural coordinate system

Corresponding D-H is described as

F^[l']The same process is carried out; kinematic pair

The corresponding axes are denoted by

That is, the indexes in the D-H system conform to the parent indexes;

determining a D-H system by fixing axis invariants, comprising the steps of:

determining intermediate points

And D-H is origin O_l′，

【1】 Order to

And z_l′Respectively passing through the axes by constant amounts

And

and is

Wherein,is that

The unit coordinate vector of (a) is,

【2】

is defined as

To

The male vertical line of (a);

【3】 To obtain

WhereinIs that

A unit coordinate vector of (a);

【4】 According to the rotation characteristic of the vector, theIs represented by

To F^[l']In an attitude of, i.e. by the rod memberRotating to a rod l';

the method also comprises a D-H parameter determination step based on the fixed shaft invariant;

determining D-H parametersAnd

and

are respectively rod pieces

To the rod piece

The wheelbase and the offset of the wheel,

in the formula,

is a shaft

A unit coordinate vector of (a);

is a rod piece

The origin of the rod l to the origin of the rod l;is a rod piece

From the origin to the rod

The translation vector of the origin of (a);

is a rod piece

From the origin to the rodThe origin of the translation vector.

2. The method for determining D-H system and D-H parameters of an axis-invariant multi-axis robot as claimed in claim 1,

in the step (3),

in the formula,

is axis invariant

The cross-multiplication matrix of (a) is,is a shaft

The unit coordinate vector of (a) is,

is a shaftThe unit coordinate vector of (2).

3. The method for determining D-H system and D-H parameters of an axis-invariant multi-axis robot as claimed in claim 1,

in the step (4),

is that

The unit coordinate vector of (a) is,

is represented byTo F^[l']In a posture of being composed of

Rotate to l'.

4. The method for determining D-H system and D-H parameters of an axis-invariant multi-axis robot as claimed in claim 1,

in step [ 2 ], if

Then

In the formula,

is a shaft

The unit coordinate vector of (a) is,

is that

A unit coordinate vector of (a);

is a rod piece

From the origin to the rod

The origin of the translation vector.

5. The method for determining D-H system and D-H parameters of an axis-invariant multi-axis robot as claimed in claim 1,

in step [ 2 ], if

And is

Then

In the formula,

is a rod piece

The origin of the rod l to the origin of the rod l;

is axis invariant

A cross-product matrix of; 0₃＝[0 0 0]^T。

6. The method for determining D-H system and D-H parameters of an axis-invariant multi-axis robot as claimed in claim 1,

in step [ 2 ], if

And is

Then

And is

Is represented by the formula (12) and

to obtain

In the formula,

is a rod piece

The origin of the rod l to the origin of the rod l;is a rod piece

To the i' origin of the rod,

is a rod piece

From the origin to the rodThe translation vector of the origin of the image,^lr_l′is a translation vector from the origin of the rod l to the origin of the rod l', 0₃＝[0 0 0]^T。

7. The method for determining D-H system and D-H parameters of an axis-invariant multi-axis robot as claimed in claim 1,

order toDefined by the angle of twist of the shaft