CN113478479A - Acceleration selection method based on industrial robot quintic polynomial trajectory planning intermediate point - Google Patents

Acceleration selection method based on industrial robot quintic polynomial trajectory planning intermediate point Download PDF

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CN113478479A
CN113478479A CN202110669069.3A CN202110669069A CN113478479A CN 113478479 A CN113478479 A CN 113478479A CN 202110669069 A CN202110669069 A CN 202110669069A CN 113478479 A CN113478479 A CN 113478479A
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acceleration
industrial robot
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CN113478479B (en
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李富平
张栋杰
赵永胜
程强
李德威
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Beijing University of 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
    • 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
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/1651—Programme controls characterised by the control loop acceleration, rate control
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
    • Y02P90/00—Enabling technologies with a potential contribution to greenhouse gas [GHG] emissions mitigation
    • Y02P90/02—Total factory control, e.g. smart factories, flexible manufacturing systems [FMS] or integrated manufacturing systems [IMS]

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Abstract

The invention discloses an acceleration selection method based on an intermediate point of a five-degree polynomial track planning of an industrial robot. And (4) combining the relation between the jerk and the step function of the adjacent three points in the track planning process to obtain a step expression of the jerk function at the middle point. And preliminarily solving the speed and the acceleration of the middle point of the path according to a planning method in the process of planning the track of the industrial robot. And (5) verifying by using matlab simulation to obtain the change images of the joint angle, the angular velocity, the angular acceleration and the jerk of the industrial robot in the track planning process before and after optimization. Compared with the result of a seventh polynomial in the trajectory planning process, the method greatly reduces the operation time of path planning, and provides an optimization scheme with simple and convenient solving process for the polynomial trajectory planning problem.

Description

Acceleration selection method based on industrial robot quintic polynomial trajectory planning intermediate point
Technical Field
The invention belongs to the field of robot trajectory planning research, and relates to a speed and acceleration selection method for a fifth-order polynomial trajectory planning intermediate point of an industrial robot. The method can help a user to automatically select the speed and the acceleration of the middle point so as to reduce the flexible impact of the motor and achieve the effect of seven-degree polynomial track planning.
Background
With the continuous development of industrial automation in China, a large number of repetitive, high-risk and precise manual operations in the industrial industry in China are replaced by industrial robots, and by means of the characteristics of high efficiency, high accuracy, flexibility, low price and the like of the industrial robots, the productivity of enterprises is greatly improved, the burden of workers is reduced, and even more dangerous or human-unfinishable work can be completed. The 6-degree-of-freedom industrial robot is the most common in the industrial field, and the track of the industrial robot is particularly important in the motion control of the 6R industrial robot. The joint space planning describes the track of the robot by using a function of a joint angle, the track planning is carried out in a joint variable space, firstly, the joint variable value corresponding to the end effector on each path point is solved according to an inverse kinematics equation of the robot, and then, for each joint variable, a smooth time function representing the joint track is fitted by using the joint variable value. Meanwhile, as the joint space and the Cartesian coordinate space are not continuously corresponding, the singularity problem of the mechanism can not occur. This patent optimizes the phenomenon that can produce flexible impact to joint space quintic polynomial trajectory planning in-process, turns into the problem of solving step function equation set to the step phenomenon that jerk image produced to make the jerk curve continuous, reduce the flexible impact of quintic polynomial trajectory planning in-process by a wide margin. And this patent can help the user to provide the acceleration value scheme of a midpoint, makes industrial robot orbit planning process become simple.
Disclosure of Invention
The invention aims to provide a speed and acceleration selection method for planning a middle point based on a fifth-order polynomial locus of an industrial robot. The method is mainly characterized in that the minimum value of a function is found out by analyzing the step function of two sections of quintic polynomial jerk functions of adjacent intermediate points, so that the two adjacent sections of the quintic polynomial jerk functions are continuous, and a value-taking scheme is provided for the intermediate point speed and the acceleration in the quintic polynomial trajectory planning problem of the industrial robot.
The invention is realized by adopting the following technical means:
step 1, according to a quintic polynomial track planning method of an industrial robot, a jerk step function relation of two adjacent points in the track planning process is established.
And 2, deducing according to a formula in the step 1, and obtaining a step expression of the jerk function at the middle point by combining the relationship between the jerks of the adjacent three points and the step function in the track planning process.
And 3, preliminarily solving the speed and the acceleration of the middle point of the path according to the result of the step 2 and a planning method in the process of planning the track of the industrial robot.
Step 4, according to the angle change of a certain joint of the industrial robot with the set path point and the speed of the path point in the step 3, substituting a step function relationship, and solving the accelerated speed of the optimized path point; and (5) verifying by using matlab simulation to obtain the change images of the joint angle, the angular velocity, the angular acceleration and the jerk of the industrial robot in the track planning process before and after optimization.
The method is characterized in that the acceleration selecting process of the middle point of the path of the industrial robot in the track planning process is converted into an equation set solving problem, and a user does not need to designate the speed and the acceleration of the middle point in actual operation, so that the method is convenient for the user to operate. In addition, the acceleration value of the path point is solved by using the improved quintic polynomial algorithm in the track planning process, so that the acceleration effect of solving the path point in the seven-degree polynomial track planning can be achieved. Compared with the result of a seventh polynomial in the trajectory planning process, the method greatly reduces the operation time of path planning, and provides an optimization scheme with simple and convenient solving process for the polynomial trajectory planning problem.
Drawings
Fig. 1 is a diagram of the ith path in the motion path.
FIG. 2 is a path diagram of the ith and (i +1) th segments adjacent to the motion path.
Fig. 3 is an image of the joint angle, velocity, acceleration, jerk of the robot initially optimized in trajectory planning.
Fig. 4 is an image of the joint angle, velocity, acceleration, jerk of the robot after optimization in trajectory planning.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and examples.
The technical scheme adopted by the invention is an acceleration selection method for planning a middle point based on a fifth-order polynomial locus of an industrial robot, which comprises the following steps,
step (1) according to a fifth-order polynomial track planning method of an industrial robot, establishing a step function relation of jerks of two adjacent points in the track planning process
The function relationship of the fifth-order polynomial can be obtained:
θ(t)＝c0+c1t+c2t2+c3t3+c4t4+c5t5 (1)
Figure BDA0003118339620000021
Figure BDA0003118339620000031
Figure BDA0003118339620000032
in the above formula, theta is the angle of the path point in the moving process of the industrial robot,
Figure BDA0003118339620000033
is the angular velocity of the path point or points,
Figure BDA0003118339620000034
is the angular acceleration of the path point(s),
Figure BDA0003118339620000035
angular acceleration as a waypoint; c. C0To c5Constant coefficients in the process of the quintic polynomial trajectory planning are obtained; and t is the time between adjacent motion points in the trajectory planning process.
As shown in FIG. 1, (subscript i represents the path point position) at Pointi, the angle is θiAngular velocity viAngular acceleration of ai. The angle at Point +1 is θi+1Angular velocity vi+1Angular acceleration of ai+1. The movement time is T. The distance theta between the pointi and the pointi +1 can be obtainedi(t) expression:
when t is 0, the following results are obtained:
Figure BDA0003118339620000036
when T is T, the following is obtained:
Figure BDA0003118339620000037
wherein
Figure BDA0003118339620000038
In the above formula, A is set for convenience of calculationi，Bi，CiAnd constant variables of the ith motion point in the quintic polynomial trajectory planning process are represented.
Step (2) according to the relation of the adjacent three points in the track planning process, obtaining a step expression of the jerk function at the middle point
As shown in FIG. 2, the angle at Pointi +2 is θi+2Angular velocity vi+2Angular acceleration of ai+2. Theta of two adjacent segmentsi(t) th stage and thetai+1The path times of (T) are all T.
Step expression of jerk function at Point i +1
Figure BDA0003118339620000041
From equations (4), (5), (6), the following equations are obtained:
Figure BDA0003118339620000042
in the above formula, k is set for convenience of calculationi，ki+1And constant variables representing the ith motion point in the quintic polynomial trajectory planning process.
Therefore FiThe expression is as follows:
Figure BDA0003118339620000043
wherein:
Figure BDA0003118339620000044
step (3) preliminarily solving the speed and the acceleration of the middle point of the path according to the result of the step (2) and a planning method in the process of planning the track of the industrial robot
Assume that there are (n +1) nodes (the initial node is Point0, the second node is Point 1, and so on, the end node is Point n) in total during path planning, and F1To Fn-1A step function for each waypoint; the interval time of each section is equal to T; initial point velocity v0Acceleration a ═ 000; velocity v at end pointnAcceleration a ═ 0n0. The simultaneous equations are as follows:
Figure BDA0003118339620000045
wherein, let F1To Fn-1Is 0, the only a is obtained1To an-1And the continuous acceleration curve of each intermediate node is satisfied. The equation is as follows:
Figure BDA0003118339620000051
written in matrix form Aa ═ K as follows:
Figure BDA0003118339620000052
wherein:
Figure BDA0003118339620000053
wherein: order to
Figure BDA0003118339620000061
It is known from matrix analysis knowledge that a is reversible, so that the equation Aa has a unique solution when K is equal to K, each intermediate point has a uniquely determined acceleration value, and the step expression of the jerk function of each intermediate point is 0. Step (4) according to the angle change of a certain joint of the industrial robot with the set path point and the speed of the path point in the step (3), substituting a step function relationship, and solving the accelerated speed of the optimized path point; verification is carried out by using matlab simulation, and the angle, angular velocity, angular acceleration and jerk change images of the joints of the industrial robot in the track planning process before and after optimization are obtained
1. Certain joint parameters of the industrial robot are as follows:
Figure BDA0003118339620000062
wherein, theta is an angle value matrix of different path points of the same joint from Point0 to Point 4; v is a velocity value matrix of different path points of the same joint from Point0 to Point 4; a is an acceleration value matrix of different path points of the same joint from Point0 to Point 4.
Wherein, each interval time is assumed to be T ═ 3s, and a is an initial Point and an end Point because Point0 and Point 4 are the initial Point and the end Point0＝0,v0＝0；v4＝0,a40; the velocity of the intermediate point is determined by the following equation:
Figure BDA0003118339620000071
wherein d isk＝(θk-θk-1) T, sign (·) is a sign function.
The following velocity and acceleration values can be obtained by equation (14):
Figure BDA0003118339620000072
2. applying the data after the initial optimization 1, and performing path simulation by using matlab, as shown in fig. 3:
it is apparent from the jerk curve in fig. 3 that there is a sudden change in jerk, resulting in a soft impact on the motor.
3. Acceleration optimization using the invention
The acceleration is solved by adopting the equation (11), and the acceleration a is obtained as [ 0-3.2539-2.81161.760 ] and the result of the trajectory planning after the acceleration is optimized by adopting the method is shown in fig. 4:
it can be obviously seen from the jerk change curve in fig. 4 that the jerk curve becomes a smooth curve and the jerk curve becomes a continuous curve, thereby greatly reducing the flexible impact of the motor.

Claims (5)

1. An acceleration selection method based on an industrial robot quintic polynomial trajectory planning intermediate point is characterized by comprising the following steps: the method comprises the following steps of,
step 1, establishing a jerk step function relationship between two adjacent points in a track planning process according to a quintic polynomial track planning method of an industrial robot;
step 2, combining the relation between the jerk and the step function of the adjacent three points in the track planning process to obtain a step expression of the jerk function at the middle point;
step 3, preliminarily solving the speed and the acceleration of the middle point of the path according to the result of the step 2 and a planning method in the path planning process of the industrial robot;
step 4, according to the angle change of a certain joint of the industrial robot with the set path point and the speed of the path point in the step 3, substituting a step function relationship, and solving the accelerated speed of the optimized path point; and (5) verifying by using matlab simulation to obtain the change images of the joint angle, the angular velocity, the angular acceleration and the jerk of the industrial robot in the track planning process before and after optimization.
2. The acceleration selection method based on the industrial robot quintic polynomial trajectory planning intermediate point as claimed in claim 1, characterized in that: step 1, establishing a step function relation of jerks of two adjacent points in the track planning process according to a quintic polynomial track planning method of an industrial robot
The function relationship of the fifth-order polynomial can be obtained:
θ(t)＝c0+c1t+c2t2+c3t3+c4t4+c5t5 (1)
Figure FDA0003118339610000011
Figure FDA0003118339610000012
Figure FDA0003118339610000013
in the above formula, theta is the angle of the path point in the moving process of the industrial robot,
Figure FDA0003118339610000014
is the angular velocity of the path point or points,
Figure FDA0003118339610000015
is the angular acceleration of the path point(s),
Figure FDA0003118339610000016
angular acceleration as a waypoint; c. C0To c5Constant coefficients in the process of the quintic polynomial trajectory planning are obtained; t is the time between adjacent motion points in the trajectory planning process;
the subscript i represents the path point location; angle at Pointi is θiAngular velocity viAngular acceleration of ai(ii) a The angle at Point +1 is θi+1Angular velocity vi+1Angular acceleration of ai+1(ii) a The movement time is T; the distance theta between the pointi and the pointi +1 can be obtainedi(t) expression:
when t is 0, the following results are obtained:
Figure FDA0003118339610000021
when T is T, the following is obtained:
Figure FDA0003118339610000022
wherein
Figure FDA0003118339610000023
In the above formula, A is set for convenience of calculationi，Bi，CiAnd constant variables of the ith motion point in the quintic polynomial trajectory planning process are represented.
3. The acceleration selection method based on the industrial robot quintic polynomial trajectory planning intermediate point as claimed in claim 1, characterized in that: step 2, according to the relation of the adjacent three points in the track planning process, obtaining a step expression of the jerk function at the middle point
Angle at Point +2 is θi+2Angular velocity vi+2Angular acceleration of ai+2(ii) a Theta of two adjacent segmentsi(t) th stage and thetai+1(T) the path times are all T;
step expression of jerk function at Point i +1
Figure FDA0003118339610000024
From equations (4), (5), (6), the following equations are obtained:
Figure FDA0003118339610000025
in the above formula, k is set for convenience of calculationi，ki+1Constant variables representing the ith motion point in the quintic polynomial trajectory planning process;
therefore FiThe expression is as follows:
Figure FDA0003118339610000031
wherein:
Figure FDA0003118339610000032
Figure FDA0003118339610000035
4. the acceleration selection method based on the industrial robot quintic polynomial trajectory planning intermediate point as claimed in claim 1, characterized in that: in step 3, the speed and the acceleration of the middle point of the path are preliminarily solved according to the result of the step 2 and a planning method in the process of planning the track of the industrial robot
Assume that there are (n +1) nodes (the initial node is Point0, the second node is Point 1, and so on, the end node is Point n) in total during path planning, and F1To Fn-1A step function for each waypoint; the interval time of each section is equal to T; initial point velocity v0Acceleration a ═ 000; velocity v at end pointnAcceleration a ═ 0n0; the simultaneous equations are as follows:
Figure FDA0003118339610000033
wherein, let F1To Fn-1Is 0, the only a is obtained1To an-1And the continuous acceleration curve of each intermediate node is satisfied; the equation is as follows:
Figure FDA0003118339610000034
written in matrix form Aa ═ K as follows:
Figure FDA0003118339610000041
wherein:
Figure FDA0003118339610000042
wherein: order to
Figure FDA0003118339610000043
It is known from matrix analysis knowledge that a is reversible, so that the equation Aa has a unique solution when K is equal to K, each intermediate point has a uniquely determined acceleration value, and the step expression of the jerk function of each intermediate point is 0.
5. The acceleration selection method based on the industrial robot quintic polynomial trajectory planning intermediate point as claimed in claim 1, characterized in that: step 4, according to the angle change of a certain joint of the industrial robot with the set path point and the speed of the path point in the step 3, substituting a step function relationship, and solving the accelerated speed of the optimized path point; and (5) verifying by using matlab simulation to obtain the angle, angular velocity, angular acceleration and jerk change images of the joints of the industrial robot in the track planning process before and after optimization.
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CN101733749A (en) * 2009-12-22 2010-06-16 哈尔滨工业大学 Multidomain uniform modeling and emulation system of space robot
CN102328315A (en) * 2010-07-12 2012-01-25 深圳华强游戏软件有限公司 Control method and device for action simulation of robot
DE102017129665B3 (en) * 2017-12-12 2019-01-24 Pilz Gmbh & Co. Kg Collision-free motion planning with closed kinematics
CN109765887A (en) * 2018-12-21 2019-05-17 费子偕 A kind of automatic Pilot control method
CN109910013A (en) * 2019-04-04 2019-06-21 江南大学 A kind of PTP method for planning track of the continuous bounded of SCARA robot acceleration
CN112405519A (en) * 2019-08-23 2021-02-26 贵州大学 Motion trajectory planning method for loading and unloading robot

Patent Citations (8)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101100059A (en) * 2007-07-27 2008-01-09 北京工业大学 Flexible double-wheel self-balancing robot attitude detecting method
US20090037021A1 (en) * 2007-07-31 2009-02-05 Rockwell Automation Technologies, Inc. Blending algorithm for trajectory planning
CN101733749A (en) * 2009-12-22 2010-06-16 哈尔滨工业大学 Multidomain uniform modeling and emulation system of space robot
CN102328315A (en) * 2010-07-12 2012-01-25 深圳华强游戏软件有限公司 Control method and device for action simulation of robot
DE102017129665B3 (en) * 2017-12-12 2019-01-24 Pilz Gmbh & Co. Kg Collision-free motion planning with closed kinematics
CN109765887A (en) * 2018-12-21 2019-05-17 费子偕 A kind of automatic Pilot control method
CN109910013A (en) * 2019-04-04 2019-06-21 江南大学 A kind of PTP method for planning track of the continuous bounded of SCARA robot acceleration
CN112405519A (en) * 2019-08-23 2021-02-26 贵州大学 Motion trajectory planning method for loading and unloading robot

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