CN111002308A - Industrial mechanical arm path planning method based on segmented multistage polynomial interpolation - Google Patents

Industrial mechanical arm path planning method based on segmented multistage polynomial interpolation Download PDF

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CN111002308A
CN111002308A CN201911237104.3A CN201911237104A CN111002308A CN 111002308 A CN111002308 A CN 111002308A CN 201911237104 A CN201911237104 A CN 201911237104A CN 111002308 A CN111002308 A CN 111002308A
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mechanical arm
polynomial
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point
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李策
单梁
黄天宇
汪萌
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Nanjing University of Science and Technology
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
    • B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning

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Abstract

The invention discloses an industrial mechanical arm path planning method based on segmented multistage polynomial interpolation, which comprises the steps of establishing a mechanical arm mathematical model according to mechanical arm standard D-H parameters; determining an initial point, a termination point and a path intermediate point of the mechanical arm; carrying out segmentation processing on the path passing points; path planning is carried out on different sections by applying different-order polynomial interpolation algorithms to obtain planning curves; and combining the curves to obtain the whole path planning curve. According to the invention, aiming at different requirements on the motion characteristics of the mechanical arm in different environments, different-order polynomial interpolation algorithms are selected for segment path planning, so that under the condition of ensuring the operation precision of the mechanical arm, the operation efficiency of the algorithms is improved, the rigid impact at the joint of the mechanical arm is reduced, and the service life of the mechanical arm is prolonged.

Description

Industrial mechanical arm path planning method based on segmented multistage polynomial interpolation
Technical Field
The invention relates to an industrial mechanical arm path planning method based on segmented multistage polynomial interpolation.
Background
With the development of robotics, industrial robots are becoming increasingly prominent in the manufacturing industry. Path planning is one of the hot topics in the field of robotics, and plays an important role in the development of industrial robots. The path planning problem mainly aims at that in a robot working space, a mechanical arm can stably and stably operate according to a preset middle path point. Compared with the traditional path planning algorithm, the path planning method has many calculation and analysis methods, and the balance between the operation cost and the performance is difficult to obtain.
Disclosure of Invention
The invention aims to provide an industrial mechanical arm path planning method based on segmented multi-order polynomial interpolation.
The technical scheme for realizing the purpose of the invention is as follows: an industrial mechanical arm path planning method based on segmented multistage polynomial interpolation comprises the following steps:
step 1, segmenting a starting point, an end point and a middle path node of a mechanical arm, and performing inverse solution on the starting point, the end point and the middle path node to obtain joint angle values corresponding to the starting point, the end point and the middle path node;
and 2, carrying out path planning on the sections obtained in the step 1 by using different-order polynomial interpolation functions to obtain different sections of planning curves, and arranging and integrating the different sections of planning curves according to the interval sequence to obtain a joint angular motion law diagram of the sampling points in the whole track.
Preferably, the specific steps of step 1 are:
step 1-1, establishing a mathematical model of a mechanical arm in a three-dimensional space according to a standard D-H parameter of a robot;
step 1-2, determining space coordinate points of the mechanical arm, wherein the space coordinate points comprise a starting point, an end point and a necessary middle path node in the motion process;
1-3, segmenting all spatial coordinate points;
and 1-4, reversely solving a joint angle value corresponding to the space coordinate point according to the space coordinate point.
Preferably, the number of segments is 3 or 4.
Preferably, the polynomial order selects the order of the first and last segments to be higher than the order of the remaining segments.
Preferably, the specific steps of step 2 are:
step 2-1, selecting polynomial interpolation functions of different orders for different sections, and selecting the angle q of each sectiondThe polynomial formula with respect to time t is:
qd＝a0+a1t+a2t2+...antn
order 2N +1 selected according to a polynomial, angle q for each segmentdPerforming a N-order derivation with respect to the polynomial expression at time t;
combining the constraint conditions:
Figure BDA0002305174740000021
Figure BDA0002305174740000022
solving the polynomial general formula parameter a0，a1，a2...an；
Step 2-2, describing the polynomial as a function of the angle with respect to time t according to the obtained parameters and the polynomial order to obtain a time-varying rule graph of the M-section motion trail;
and 2-3, arranging and integrating the time-varying law graphs of the joint angular motion of each section according to the time sequence to generate a law graph of the joint angular motion of the sampling point in the whole track.
Compared with the prior art, the invention has the following remarkable advantages: according to the invention, through segmented analysis and respective processing, different polynomial interpolation algorithms are selected for calculation according to performance requirements, the calculation amount is reduced for the path section with low precision requirement, and the precision requirement can be fully met for the path section with high precision requirement.
Drawings
FIG. 1 is a flow chart of the present invention.
Figure 2 is a diagrammatic representation of a robotic arm of the present invention.
Fig. 3 is a characteristic diagram of the robot joint 1 according to the present invention with respect to each angle.
Detailed Description
As shown in fig. 1 to 3, a method for planning a path of an industrial mechanical arm based on piecewise multi-order polynomial interpolation includes the following steps:
step 1, segmenting a starting point, a termination point and a middle path node of the mechanical arm, and performing inverse solution on the starting point, the termination point and the middle path node to obtain joint angle values corresponding to the starting point, the termination point and the middle path node, wherein the specific steps are as follows:
step 1-1, establishing a mathematical model of the mechanical arm in a three-dimensional space according to standard D-H parameters of the robot, and converting the mathematical model of the three-dimensional space into a mathematical model only based on a joint angle theta from the D-H parameters of a serial mechanical arm with joints all being rotary jointsiAn equation of motion for a variable;
step 1-2, determining a space coordinate point of the mechanical arm, including a starting point, a terminating point and a necessary middle path node in the motion process, which are respectively expressed as: initial point (x)0,y0,z0) Intermediate path node (x)1,y1,z1)...(xn-1,yn-1,zn-1) End point (x)n,yn,zn)；
Step 1-3, segmenting all spatial coordinate points according to an actual system, where the number of segments is M, and usually M is 3 or 4, for example: taking M as 3; segment 3 is represented as:
part 1 is section 1, i.e. (x)0,y0,z0) To (x)1,y1,z1) (ii) a Part 2 is from paragraph 2 to paragraph n-2, i.e. (x)1,y1,z1) To (x)n-1,yn-1,zn-1) (ii) a Part 3 is a section n-1, i.e. (x)n-1,yn-1,zn-1) To (x)n,yn,zn)；
And 1-4, reversely solving the joint angle value corresponding to the spatial coordinate point according to the spatial coordinate point, and in some embodiments, calling an ikine () function library in Matlab software to realize reverse solution so as to obtain the joint angle value.
Step 2, path planning is carried out on the sections obtained in the step 1 by applying polynomial interpolation functions of different orders to obtain planning curves of different sections, the polynomial order selection is determined by precision requirements and mechanical characteristics, the order of the selected section 1 and the last section is higher than that of the other sections, and the specific method comprises the following steps:
step 2-1, selecting polynomial interpolation functions of different orders for different sections, and selecting the angle q of each sectiondThe polynomial formulae with respect to time t can each be represented as:
qd＝a0+a1t+a2t2+...antn
order 2N +1 selected according to a polynomial, angle q for each segmentdThe polynomial expression with respect to time t is subjected to an N-th order derivation.
Combining the constraint conditions:
Figure BDA0002305174740000031
Figure BDA0002305174740000032
can find a0，a1，a2...an。
Step 2-2, describing the polynomial as an angle q according to the parameters and the polynomial order obtained in the step,
Figure BDA0002305174740000033
Figure BDA0002305174740000041
the function of time t is equal, and a time-varying rule graph of M motion tracks can be obtained;
and 2-3, arranging and integrating the time-varying law graphs of the joint angular motion of each section according to the time sequence to generate a law graph of the joint angular motion of the sampling point in the whole track.
According to the invention, aiming at different requirements on the motion characteristics of the mechanical arm in different environments, different-order polynomial interpolation algorithms are selected for segment path planning, so that under the condition of ensuring the operation precision of the mechanical arm, the operation efficiency of the algorithms is improved, the rigid impact at the joint of the mechanical arm is reduced, and the service life of the mechanical arm is prolonged.
Examples
The present embodiment is a path planning method for a seven-degree-of-freedom robot, as shown in fig. 1, where parameters of a mechanical arm DH of the present embodiment are shown in table 1:
TABLE 1D-H PARAMETERS
Figure BDA0002305174740000042
Wherein theta is a joint angle, d is the length of the connecting rod, α is the joint torsion, and a is the joint offset;
step 1-1, establishing a mathematical model of a mechanical arm in a three-dimensional space according to a robot standard D-H parameter, as shown in figure 1;
step 1-2, determining a starting point and a terminating point of the mechanical arm and a necessary middle path node in the motion process according to actual requirements, specifically expressed as:
the initial point is represented as (x)0,y0,z0) The middle point represents (x)1,y1,z1)...(xn-1,yn-1,zn-1) The end point is expressed as (x)n,yn,zn)；
1-3, segmenting all points according to the characteristic requirements of an actual system on the mechanical arm, wherein the number of the segments is M, and M is taken as 3 in the embodiment; and take (x)0,y0,z0) To (x)1,y1,z1) Is the 1 st section; (x)1,y1,z1) To (x)3,y3,z3) Is the 2 nd paragraph; (x)3,y3,z3) To (x)4,y4,z4) Is paragraph 3;
step 1-4, reversely solving a joint angle value corresponding to the space point according to the space coordinate point, wherein the embodiment mainly considers a first joint and a second joint as follows:
q1＝[0 2π 0 0 0 0]，q2＝[π/2 π/5 0 0 0 0]，
q3＝[π/3 π 0 0 0 0]，q4＝[π π/2 0 0 0 0]，
q5＝[π/5 π/4 0 0 0 0]；
step 2-1, selecting different-order polynomial interpolation algorithm for different sections, and obtaining an angle qdThe polynomial with respect to time t can be expressed as:
in this embodiment, the requirement on the mechanical arm performance is high for the precision of the start section and the end section, and the requirement on the mechanical arm pulsation characteristic is high, a seventh-order polynomial is selected for planning, the requirement on the middle section is low, and a third-order polynomial is selected for planning, and the specific details are implemented as follows:
the selection rule is that the middle section selects low order, the two ends select high order, and the order selected at the two ends is the middle section order plus two:
establishing a cubic polynomial interpolation function, comprising: the function expressions of angular displacement and angular speed are respectively:
qd＝a0+a1t+a2t2+a3t3
Figure BDA0002305174740000051
the constraints are as follows:
Figure BDA0002305174740000052
Figure BDA0002305174740000053
solving to obtain:
Figure BDA0002305174740000054
where t is time, a is a polynomial coefficient, q is a joint angle vector, t0And t1The start and end time points of the joint interpolation;
establishing a seventh order polynomial interpolation function, comprising: the functional expressions of angular displacement, angular velocity, angular acceleration, and pulsation characteristics (angular jerk) are respectively:
qd＝a0+a1t+a2t2+a3t3+a4t4+a5t5+a6t6+a7t7
Figure BDA0002305174740000055
Figure BDA0002305174740000061
Figure BDA0002305174740000062
the constraints are as follows:
Figure BDA0002305174740000063
Figure BDA0002305174740000064
the following equation can be solved:
Figure BDA0002305174740000065
where t is time, a is a polynomial coefficient, q is a joint angle vector, t0And t1For the starting and ending time points of the joint interpolation, q0To be t0Substitution qdThe resulting polynomial, q, for t1To be t1Substitution qdResulting polynomial, v, about t0To be t0Substitution into
Figure BDA0002305174740000066
Resulting multiple terms for tFormula (v)1To be t1Substitution into
Figure BDA0002305174740000067
The resulting polynomial for t, a0To be t0Substitution into
Figure BDA0002305174740000068
The resulting polynomial for t, a1To be t1Substitution into
Figure BDA0002305174740000069
The resulting polynomial for t, j0To be t0Substitution into
Figure BDA00023051747400000610
The resulting polynomial for t, j1To be t1Substitution into
Figure BDA00023051747400000611
The obtained polynomial about t, and the time is set according to mechanical characteristics of the mechanical arm;
step 2-2, obtaining a time-varying rule graph of 3 sections of motion tracks according to the parameters and polynomial orders obtained in the step;
and 2-3, arranging and integrating the time-varying law graphs of the joint angular motion of each section according to the time sequence to generate a law graph of the joint angular motion of the sampling point in the whole track.
The invention adopts a piecewise polynomial interpolation algorithm to plan the path of the mechanical arm, divides the section composed of all path intermediate points into three sections, wherein the first section and the last section adopt seventh-order polynomial interpolation to plan the path points, the rest intermediate sections adopt fifth-order polynomial interpolation to plan the path points, and obtains the planning result of the joint angle of the joint changing along with time. The method has important significance for prolonging the service life of the mechanical arm.

Claims (5)

1. An industrial mechanical arm path planning method based on segmented multistage polynomial interpolation is characterized by comprising the following steps:
step 1, segmenting a starting point, an end point and a middle path node of a mechanical arm, and performing inverse solution on the starting point, the end point and the middle path node to obtain joint angle values corresponding to the starting point, the end point and the middle path node;
and 2, carrying out path planning on the sections obtained in the step 1 by using different-order polynomial interpolation functions to obtain different sections of planning curves, and arranging and integrating the different sections of planning curves according to the interval sequence to obtain a joint angular motion law diagram of the sampling points in the whole track.
2. The method for planning the path of the industrial mechanical arm based on the piecewise multi-order polynomial interpolation as claimed in claim 1, wherein the specific steps of the step 1 are as follows:
step 1-1, establishing a mathematical model of a mechanical arm in a three-dimensional space according to a standard D-H parameter of a robot;
step 1-2, determining space coordinate points of the mechanical arm, wherein the space coordinate points comprise a starting point, an end point and a necessary middle path node in the motion process;
1-3, segmenting all spatial coordinate points;
and 1-4, reversely solving a joint angle value corresponding to the space coordinate point according to the space coordinate point.
3. The method as claimed in claim 1, wherein the number of segments is 3 or 4.
4. The method as claimed in claim 1, wherein the order of the first and last segments of the polynomial order is higher than the order of the other segments.
5. The method for planning the path of the industrial mechanical arm based on the piecewise multi-order polynomial interpolation as claimed in claim 1, wherein the step 2 comprises the following steps:
step 2-1, selecting polynomial interpolation functions of different orders for different sections, and selecting the angle q of each sectiondThe polynomial formula with respect to time t is:
qd＝a0+a1t+a2t2+...antn
order 2N +1 selected according to a polynomial, angle q for each segmentdPerforming a N-order derivation with respect to the polynomial expression at time t;
combining the constraint conditions:
Figure FDA0002305174730000021
Figure FDA0002305174730000022
solving the polynomial general formula parameter a0，a1，a2...an；
Step 2-2, describing the polynomial as a function of the angle with respect to time t according to the obtained parameters and the polynomial order to obtain a time-varying rule graph of the M-section motion trail;
and 2-3, arranging and integrating the time-varying law graphs of the joint angular motion of each section according to the time sequence to generate a law graph of the joint angular motion of the sampling point in the whole track.
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CN112045397A (en) * 2020-09-07 2020-12-08 中铁工程装备集团有限公司 Steel arch splicing device and working method thereof
CN112045685A (en) * 2020-09-07 2020-12-08 中铁工程装备集团有限公司 Multi-oil-cylinder cooperative control device and working method thereof
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