CN116125910A - NURBS curve interpolation speed planning method based on fourth-order polynomial - Google Patents
NURBS curve interpolation speed planning method based on fourth-order polynomial Download PDFInfo
- Publication number
- CN116125910A CN116125910A CN202211217405.1A CN202211217405A CN116125910A CN 116125910 A CN116125910 A CN 116125910A CN 202211217405 A CN202211217405 A CN 202211217405A CN 116125910 A CN116125910 A CN 116125910A
- Authority
- CN
- China
- Prior art keywords
- speed
- curve
- sub
- maximum
- acceleration
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Pending
Links
- 238000000034 method Methods 0.000 title claims abstract description 93
- 230000001133 acceleration Effects 0.000 claims abstract description 114
- 230000036461 convulsion Effects 0.000 claims abstract description 37
- 238000004422 calculation algorithm Methods 0.000 claims abstract description 8
- 238000006073 displacement reaction Methods 0.000 claims description 44
- 238000012937 correction Methods 0.000 claims description 21
- 238000003754 machining Methods 0.000 claims description 12
- 230000014509 gene expression Effects 0.000 claims description 4
- 230000003044 adaptive effect Effects 0.000 claims description 3
- 230000000694 effects Effects 0.000 claims description 2
- 238000012545 processing Methods 0.000 abstract description 28
- 239000013598 vector Substances 0.000 description 6
- 230000009286 beneficial effect Effects 0.000 description 2
- 238000004364 calculation method Methods 0.000 description 2
- 238000006243 chemical reaction Methods 0.000 description 1
- 230000003247 decreasing effect Effects 0.000 description 1
- 230000001627 detrimental effect Effects 0.000 description 1
- 238000011160 research Methods 0.000 description 1
- 230000011218 segmentation Effects 0.000 description 1
- 238000004088 simulation Methods 0.000 description 1
- 238000012360 testing method Methods 0.000 description 1
Images
Classifications
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B19/00—Programme-control systems
- G05B19/02—Programme-control systems electric
- G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
- G05B19/41—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by interpolation, e.g. the computation of intermediate points between programmed end points to define the path to be followed and the rate of travel along that path
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B2219/00—Program-control systems
- G05B2219/30—Nc systems
- G05B2219/34—Director, elements to supervisory
- G05B2219/34083—Interpolation general
Landscapes
- Engineering & Computer Science (AREA)
- Computing Systems (AREA)
- Theoretical Computer Science (AREA)
- Human Computer Interaction (AREA)
- Manufacturing & Machinery (AREA)
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Automation & Control Theory (AREA)
- Complex Calculations (AREA)
- Image Generation (AREA)
Abstract
The invention discloses a NURBS curve interpolation speed planning method based on a fourth-order polynomial, which comprises the following implementation steps of obtaining data of a NURBS curve and dynamic characteristic parameters of equipment, determining the maximum speed allowed by each point on a path according to a bow-height error tolerance, and defining the maximum speed as a speed tolerance; defining a speed tolerance minimum point as a key point, dividing an original curve into a plurality of sub-curves at the key point, and solving the lengths of the sub-curves; and constructing a NURBS curve feed rate curve according to the velocity planning algorithm based on the fourth-order polynomial. The method fully considers the influence of bow-height errors and equipment dynamics performance on the processing speed, and can obtain a smooth continuous jerk section, an acceleration section and a speed section; the flexible vibration in the processing can be effectively avoided, and the processing efficiency is remarkably improved.
Description
Technical Field
The invention relates to NURBS curve segmentation, speed tolerance correction, curve merging and speed planning methods, in particular to a NURBS curve interpolation speed planning algorithm based on a fourth-order polynomial.
Background
NURBS interpolation has a plurality of advantages of small NC program quantity, high machining precision, high machining efficiency and the like, and is widely applied to medium-high end machine tools. However, the machining efficiency and the machining accuracy tend to be mutually exclusive. A balance is made between accuracy and efficiency, and it is difficult to obtain high-efficiency and high-accuracy machining.
In general, by planning an appropriate processing speed profile in accordance with a given processing error margin, efficient processing can be obtained while ensuring processing accuracy.
Bow height errors are important bases for evaluating machining accuracy. Typically, bow-height errors are mainly affected by interpolation period, feed rate and curvature. Thus, some methods of adaptively modifying the feed rate according to curvature have emerged in the early days. However, due to lack of consideration for the dynamics of the equipment, excessive or abrupt acceleration and jerk often occur, which is detrimental to the stability of the processing process and reduces the processing quality. And a series of S-curve acceleration and deceleration control, polynomial acceleration and deceleration control, trigonometric function acceleration and deceleration control and the like which appear in the later stage. At present, a plurality of high-grade numerical control systems use S-curve acceleration and deceleration, the speed curve and the acceleration curve are continuous, and flexible impact is avoided in the acceleration and deceleration process. But its jerk is not continuous and therefore flexibility is limited. The conventional polynomial acceleration and deceleration is controlled as a cubic polynomial, and the obtained jerk is continuous, but the maximum jerk cannot be maintained, and the efficiency is low. The trigonometric function acceleration and deceleration control faces the problem of complex calculation.
Based on the method, the invention provides a NURBS curve interpolation speed planning method based on a fourth-order polynomial to provide smooth continuous jerk, acceleration and speed profile in NURBS interpolation.
Disclosure of Invention
The invention aims to provide a NURBS curve interpolation speed planning method based on a fourth-order polynomial, which can provide an optimal speed profile for an interpolation process by combining the curvature of NURBS and the dynamic characteristics of equipment. The method can ensure smooth and continuous jerk, acceleration and speed, and has higher processing efficiency and processing stability.
To achieve the above object, the present invention adopts the following method:
(1): acquiring data of NURBS curves and dynamic characteristic parameters of equipment, determining maximum speeds allowed by points on a path according to the bow-height error tolerance, and defining the maximum speeds as speed tolerance;
(2): defining a speed tolerance minimum point as a key point, dividing an original curve into a plurality of sub-curves at the key point, and solving the lengths of the sub-curves;
(3): and constructing a NURBS curve feed rate curve according to the velocity planning algorithm based on the fourth-order polynomial.
Further, the method comprises the steps of,
in said (1), NURBS curve C (u i ) At parameter u i The speed at which the following two conditions are satisfied:
a) Full arch height error limit to ensure machining accuracy, namely:
wherein ,κi and Vi Respectively the curves are in the parameter u i Curvature and feed speed at delta max Is the maximum allowable bow height error, T s Is the interpolation period. Thereby obtaining the maximum allowable speed under the limit of bow height error
The method comprises the following steps:
b) Satisfying centripetal acceleration limitation and jerk to satisfy the dynamics of the device, i.e
wherein ,
is the limit of the dynamic performance of the equipment on the feeding speed, v f 、/>
and Jmax The maximum feed speed, the centripetal acceleration and the jerk allowed by the device are respectively.
Thus, NURBS curve C (u i ) At parameter u i Maximum speed allowed by the department
The method comprises the following steps:
further, the method comprises the steps of,
in the step (2), according to the speed margin in the step (1), defining a point with a minimum value of the speed margin as a key point, dividing the curve into a plurality of sub-curves at the key point, calculating the length of each sub-curve by adopting an adaptive orthogonal method of a Simpson rule, and storing the characteristic information of the sub-curve in a data buffer { [ v { s ,v e ,S]}. Wherein v is s and ve Is the starting speed tolerance and the ending speed tolerance of the sub-line segment, and S is the arc length of the sub-curve.
Further, the method comprises the steps of,
in the step (3), the most core task is to obtain a speed profile of each sub-curve, which comprises three core parts of sub-curve speed planning, speed tolerance correction and sub-curve self-adaption combination.
The jerk of the proposed speed planning algorithm is a piecewise polynomial constructed from a fourth order polynomial, the jerk expression being:
wherein ,T1 To T 7 Respectively representing the duration time of seven stages of acceleration, uniform acceleration, deceleration, uniform speed, acceleration and deceleration, uniform deceleration and deceleration; τ 1 To tau 7 Representing the time currently spent at t in each segment, respectively.
J (T) has strict symmetry, T 3 ＝T 1 ，T 7 ＝T 5 . J (t) is smooth continuous and 0 at the beginning and end, with a maximum value taken in the middle.
In the speed planning, the end speed is greater than the initial speed and overall presents an acceleration process, and when the end speed is less than the initial speed, the overall presents a deceleration process. The speed planning method of the invention is described by taking the example that the end speed is greater than the initial speed as an example. And the speed planning is carried out according to the initial speed, the final speed and the curve length of each sub-curve, and the duration of each movement stage is obtained by solving.
The speed programming step of the sub-curve is as follows:
step1, judging whether the maximum speed limit of the equipment can be reached.
Assuming that the maximum speed tolerance of the device can be reached, the duration of each phase can be expressed as:
at this time, the minimum limit of the maximum speed of the apparatus can be reachedDisplacement S 1 The method comprises the following steps:
if S is greater than or equal to S 1 It is explained that the maximum speed limit of the apparatus can be reached and the time found in step1 is correct.
If S<S 1 The maximum speed limit of the device cannot be reached, and the process goes to step2.
Step2, judging whether acceleration can be realized at a given displacement distance.
Assuming that the velocity can be exactly determined by v within a given sub-curve length s Increasing to v e Therefore, no deceleration stage exists, and the acceleration and uniform acceleration stages are used as follows:
minimum displacement S of acceleration is achieved 2 The following are provided:
if S=S 2 The description just can realize the acceleration process, and the obtained T 1 and T2 Is correct.
If S<S 2 Indicating that the length of the sub-curve is insufficient to cause the velocity to be varied from v s Increasing to v e . At this time, it is necessary to reduce the speed limit v at the end of the sub-curve by using the speed margin correction section e . The speed margin correction part willAs described later.
If S>S 2 The length of the sub-curve is stated to be sufficient to achieve an acceleration process and there is a deceleration process. At this time, the next step of re-finding the duration of each stage is performed.
Step3, judging whether the maximum deceleration can be reached.
Assuming that the maximum deceleration can be reached just within the displacement distance of the sub-curve length, the maximum speed F that can be reached is as follows:
due to v e >v s There must be a ramp-up phase, the duration and total displacement of each phase being as follows:
wherein ,S3 Is the minimum displacement capable of reaching the maximum deceleration
If S>S 3 It is stated that the maximum deceleration can be achieved and that there may be a uniform deceleration segment. Assuming that the maximum speed achievable at this time is F, the duration and total displacement for each stage is as follows:
by solving the above equation, the duration of each stage can be obtained.
If S<S 3 It is stated that the maximum deceleration is not reached and it is necessary to go to step4 to re-solve the duration of each phase.
Step4, judging whether the maximum acceleration can be reached.
It is assumed first that the maximum acceleration can be reached, but there is no ramp-up phase, at which time the duration and total displacement of each phase are as follows:
wherein ,S4 Is the minimum displacement that can achieve the maximum acceleration.
If S is greater than or equal to S 4 It is stated that the maximum acceleration can be reached and that there may be a ramp up phase. Assuming that the maximum speed achievable at this time is F, the duration and total displacement for each stage is as follows:
by solving the above equation, the duration of each stage can be obtained.
If S<S 4 It is indicated that the maximum acceleration cannot be reached, that no uniform acceleration phase exists, and that no uniform deceleration phase exists. Assuming that the maximum speed achievable at this time is F, the duration and total displacement for each stage is as follows:
by solving the above equation, the duration of each stage can be obtained.
The speed margin correction steps are as follows:
assuming that maximum acceleration can be achieved and no ramp up and ramp down phases exist, the duration and total displacement of each phase is as follows:
S 21 is the minimum displacement that can reach the maximum acceleration without the deceleration phase.
If S<S 21 Indicating that the maximum acceleration cannot be reached, T needs to be re-determined 1 Assuming that the end speed that can be reached isv' e The duration and total displacement of each stage is as follows:
solving the above method to obtain T 1 . At this time, the end speed margin of the sub-curve needs to be corrected to v' e . At the same time, the initial velocity v of the next sub-curve s Also need to be corrected to v' e To ensure smooth continuation of the overall velocity profile.
If S>S 21 Indicating that the maximum acceleration can be reached, a uniform acceleration phase exists. Assuming that the end speed that can be reached is v' e The duration and total displacement of each stage is as follows:
solving the above method to obtain T 1 and T2 . At this time, the end speed margin of the sub-curve needs to be corrected to v' e . At the same time, the initial velocity v of the next sub-curve s Also need to be corrected to v' e To ensure smooth continuation of the overall velocity profile.
The self-adaptive interval comprises the following steps:
when the sub-curve C i After the end speed margin of (C) is corrected, try to divide the sub-curve C i And a sub-curve C i+1 Merging, using sub-curve C i Initial speed limit, sub-curve C of (2) i+1 And (3) carrying out speed planning on the sum of the end speed limit of the speed profile and the length of the two sub-curves, and combining the two sub-curves into one sub-curve if the planned speed profile does not exceed the speed tolerance.
When the sub-curve C i After the initial speed margin of (C) is corrected, attempt to make the sub-curve C i And a sub-curve C i-1 Merging, using sub-curve C i-1 Initial speed limit, sub-curve C of (2) i The end speed limit of (2) and the sum of the lengths of the two sub-curvesAnd (3) carrying out speed planning, and if the planned speed profile does not exceed the speed tolerance, merging the two sub-curves into one sub-curve.
The beneficial effects of the invention are as follows:
the invention can obtain smooth and continuous speed profile, acceleration profile and jerk profile in NURBS interpolation, thereby avoiding the problems of discontinuous jerk and limited flexibility in the common S-shaped acceleration and deceleration planning method; the speed limit fully considers the bow height error and the dynamic characteristics of equipment, so that the processing effect can be improved more efficiently and more stably under the condition of ensuring the processing precision; complex trigonometric function operation and the like are avoided in the speed planning, larger jerk can be kept for a long time, and the problems of large operation amount and low efficiency in the acceleration and deceleration planning method based on the trigonometric function are effectively avoided; the interval self-adaptive correction can effectively reduce acceleration and deceleration processes in the NURBS curve processing process, and effectively improve processing efficiency and processing stability.
Drawings
Fig. 1 is a flowchart of a NURBS curve interpolation speed planning method based on a fourth order polynomial according to the present invention.
Fig. 2 is a velocity profile, acceleration profile and jerk profile of a velocity planning method constructed in the present invention.
Fig. 3 is a flow chart of the speed planning method of the present invention.
Fig. 4 shows the velocity profile, acceleration profile and jerk profile of the present invention after planning for different situations.
Fig. 5 is a butterfly curve, curvature, speed tolerance, and key points in an example of the invention.
Fig. 6 shows bow-height error, speed profile, centripetal acceleration profile, tangential acceleration profile and jerk profile, which are planned in the interpolation of butterfly curve NURBS according to the present invention.
Detailed Description
In order that those skilled in the art can better understand the technical solutions of the present invention, the following description will clearly and completely describe the specific technical solutions of the present invention in conjunction with the embodiments to help those skilled in the art to further understand the present invention. It will be apparent that the embodiments described herein are only some, but not all, embodiments of the invention. It should be noted that embodiments and features of embodiments in this application may be combined with each other by those of ordinary skill in the art without departing from the inventive concept and conflict. All other embodiments, which are derived from the embodiments herein without creative effort for a person skilled in the art, shall fall within the disclosure and the protection scope of the present invention.
A NURBS curve interpolation speed planning method based on a fourth-order polynomial comprises the following steps:
(1): acquiring data of NURBS curves and dynamic characteristic parameters of equipment, determining maximum speeds allowed by points on a path according to the bow-height error tolerance, and defining the maximum speeds as speed tolerance;
(2): defining a speed tolerance minimum point as a key point, dividing an original curve into a plurality of sub-curves at the key point, and solving the lengths of the sub-curves;
the purpose of steps (1) and (2) is to divide the complex and diverse NURBS curve into simpler sub-curves and to obtain the main parameters of each sub-curve for speed planning, initial speed, end speed and curve length.
(3): a velocity planning algorithm based on a fourth order polynomial is constructed. And calculating the duration time of the acceleration section, the uniform acceleration section, the deceleration section, the uniform speed section, the acceleration and deceleration section, the uniform speed section, the deceleration and deceleration section in the process of processing the sub-curve according to the initial speed, the final speed and the curve length of the sub-curve. For the case that the curve length is insufficient to realize the speed change, the initial or final speed tolerance of the sub-curve is corrected, and the continuity and smoothness of the speed are ensured. And in the speed planning, the adjacent sub-curves are combined into one sub-curve to carry out the speed planning, so that the speed change in the curve processing process is reduced, and the processing efficiency and the processing stability are improved. The following expands the detailed description:
the speed tolerance is the processing of points on NURBS given the processing precision and equipment dynamicsMaximum speed allowed at that time. The speed margin is mainly affected by the interpolation period, the feed speed and the curvature of the curve. NURBS curve at parameter u i Curvature at kappa i Can be expressed as:
wherein C'(ui) and C”(ui ) Respectively the curves are in the parameter u i A first order inverted vector and a second order inverted vector.
NURBS curve C (u) i ) At parameter u i The speed at which the following two conditions are satisfied:
a) Full arch height error limit to ensure machining accuracy, i.e
wherein ,κi and Vi Respectively the curves are in the parameter u i Curvature and feed speed at delta max Is the maximum allowable bow height error, T s Is the interpolation period. Thereby obtaining the maximum allowable speed under the limit of bow height error
The method comprises the following steps:
b) Satisfying centripetal acceleration limitation and jerk to satisfy the dynamics of the device, i.e
wherein 
Is the dynamic performance of the equipmentLimitation of feed speed, v f 、/>
and Jmax The maximum feed speed, the centripetal acceleration and the jerk allowed by the device are respectively.
Thus, NURBS curve C (u i ) At parameter u i Maximum speed allowed by the department
The method comprises the following steps:
in addition, the speed limit at both the beginning and end of the curve needs to be set to 0.
However, the speed tolerance obtained according to the bow-height error and the device performance limit is obtained by considering a single point, if the acceleration is performed according to the speed, the situation that the acceleration and jerk exceed the limit still exists in the process, and the speed, the acceleration and jerk curves are discontinuous. Therefore, a suitable speed planning algorithm is required to perform quadratic programming on the speed.
Defining a point with a minimum value of the speed tolerance as a key point according to the speed tolerance obtained in the step one, dividing the curve into a plurality of sub-curves at the key point, calculating the length of each sub-curve by adopting an adaptive orthogonal method of a Simpson rule, and storing the characteristic information of the sub-curve in a data buffer area { [ v ] s ,v e ,S]}. Wherein v is s and ve Is the starting speed tolerance and the ending speed tolerance of the sub-line segment, and S is the arc length of the sub-curve.
And carrying out speed planning on each sub-curve according to the initial speed limit, the tail speed limit and the curve length of the obtained sub-curve, and finally forming a smooth continuous speed profile is a main task of speed planning.
The jerk of the proposed speed planning algorithm is a piecewise polynomial constructed from a fourth order polynomial, the jerk expression being:
wherein ,T1 To T 7 Respectively representing the duration time of seven stages of acceleration, uniform acceleration, deceleration, uniform speed, acceleration and deceleration, uniform deceleration and deceleration; τ 1 To tau 7 Representing the time currently spent at t in each segment, respectively. J (T) has strict symmetry, T 3 ＝T 1 ，T 7 ＝T 5 . J (t) is smooth continuous and 0 at the beginning and end, with a maximum value taken in the middle.
Integrating the jerk can result in the corresponding acceleration and velocity expressions:
a jerk profile, acceleration profile and velocity profile with a complete seven-segment motion is shown in figure 2.
The flow of speed scheduling for the sub-curves is shown in fig. 3, where the speed planning result is divided into 11 cases according to the duration of each segment. Table 1 shows the motion phases involved in each velocity planning result. Each phase is marked with a value of ∈v, and this phase is necessarily present. And when the label is o, the label may or may not exist, and the label needs to be determined according to the solving result. In addition to M1, if T 2 The notation o indicates that the maximum acceleration must be reached, but that no uniform acceleration segment is known; if T 4 The label o indicates that the maximum deceleration must be achieved, but that no uniform deceleration segment is known. M3 and M8 respectively represent that the length of the sub-curve is insufficient to realize acceleration or deceleration, and the speed gauge cannot be carried outAnd (5) scribing. In this case, correction of the initial speed tolerance or the final speed tolerance of the sub-curve is required. After correction it contains the same motion segments as M2 and M7, respectively.
Table 1 speed planning results
The discrimination of the speed planning situation and the determination of the duration of each phase are described in detail below.
Step1, judging whether the maximum speed limit of the equipment can be reached.
Assuming that the maximum speed tolerance of the device can be reached, the duration of each phase can be expressed as:
at this time, the minimum displacement S capable of achieving the maximum speed limit of the apparatus 1 The method comprises the following steps:
if S is greater than or equal to S 1 It is illustrated that the maximum speed limit of the device can be reached, the speed planning result being M1. The time required in step1 is correct.
If S<S 1 Indicating that the maximum speed limit of the device cannot be reached. The speed plan is divided into two types according to an initial speed limit and an end speed limit. Step2-step4 is performed when the end speed limit is greater than the initial speed limit, otherwise step5-step7 is performed.
Step2, judging whether acceleration can be realized at a given displacement distance.
Assuming that the velocity can be exactly determined by v within a given sub-curve length s Increasing to v e Therefore, no deceleration stage exists, and the acceleration and uniform acceleration stages are used as follows:
minimum displacement S of acceleration is achieved 2 The following are provided:
if S=S 2 The method is used for explaining that the acceleration process can be just realized, the speed planning result is M2, and the obtained T 1 and T2 Is correct.
If S<S 2 Indicating that the length of the sub-curve is insufficient to cause the velocity to be varied from v s Increasing to v e The speed planning result is M3. At this time, it is necessary to reduce the speed limit v at the end of the sub-curve by using the speed margin correction section e The speed planning result will then become M2. The speed margin correction section will be described later.
If S>S 2 The length of the sub-curve is stated to be sufficient to achieve an acceleration process and there is a deceleration process. At this time, the process is performedThe next step is to re-find the duration of each phase.
Step3, judging whether the maximum deceleration can be reached.
Assuming that the maximum deceleration can be reached just within the displacement distance of the sub-curve length, the maximum speed F that can be reached is as follows:
due to v e >v s There must be a ramp-up phase, the duration and total displacement of each phase being as follows:
wherein ,S3 Is the minimum displacement that can achieve the maximum deceleration.
If S>S 3 It is explained that the maximum deceleration can be reached and there may be a uniform deceleration segment, the speed planning result being M4. Assuming that the maximum speed achievable at this time is F, the duration and total displacement for each stage is as follows:
by solving the above equation, the duration of each stage can be obtained.
If S<S 3 It is stated that the maximum deceleration is not reached and it is necessary to go to step4 to re-solve the duration of each phase.
Step4, judging whether the maximum acceleration can be reached.
It is assumed first that the maximum acceleration can be reached, but there is no ramp-up phase, at which time the duration and total displacement of each phase are as follows:
wherein ,S4 Is the minimum displacement that can achieve the maximum acceleration.
If S is greater than or equal to S 4 It is stated that the maximum acceleration can be reached and there may be a ramp up phase, with a speed plan result of M5. Assuming that the maximum speed achievable at this time is F, the duration and total displacement for each stage is as follows:
by solving the above equation, the duration of each stage can be obtained.
If S<S 4 The maximum acceleration cannot be reached, the uniform acceleration stage does not exist, the uniform deceleration stage does not exist, and the speed planning result is M6. Assuming that the maximum speed achievable at this time is F, the duration and total displacement for each stage is as follows:
by solving the above equation, the duration of each stage can be obtained.
When the end speed margin is less than the initial speed margin, the deceleration process may be considered a reverse acceleration process. Therefore, the calculation process of step5-step7 for each time period is completely identical to that of step2-step4, and will not be described in detail here.
For the M3 case, the end speed margin needs to be corrected as follows:
assuming that maximum acceleration can be achieved and no ramp up and ramp down phases exist, the duration and total displacement of each phase is as follows:
S 21 is capable of reaching maximum acceleration in the absence of a deceleration stageMinimum displacement of velocity.
If S<S 21 Indicating that the maximum acceleration cannot be reached, T needs to be re-determined 1 Assuming that the end speed that can be reached is v' e The duration and total displacement of each stage is as follows:
solving the above method to obtain T 1 . At this time, the end speed margin of the sub-curve needs to be corrected to v' e . At the same time, the initial velocity v of the next sub-curve s Also need to be corrected to v' e To ensure smooth continuation of the overall velocity profile.
If S>S 21 Indicating that the maximum acceleration can be reached, a uniform acceleration phase exists. Assuming that the end speed that can be reached is v' e The duration and total displacement of each stage is as follows:
solving the above method to obtain T 1 and T2 。
After the speed tolerance correction, the length of the sub-curve can just achieve acceleration, and the speed planning result is consistent with M2.
For the M8 case, the initial speed margin needs to be corrected, and the correction method is almost consistent with the correction of the final speed margin. After the speed tolerance correction, the length of the sub-curve can just realize deceleration, and the speed planning result is consistent with M7.
The various speed curves, acceleration curves and jerk curves obtained using this speed planning method are shown in fig. 4. The speed planning method can obtain a smooth continuous speed profile, an acceleration profile and a jerk profile in various conditions, can effectively reduce flexible vibration in the processing process, and improves the processing stability. In addition, the obtained jerk can be kept for a long time near the maximum jerk, and the reaction capacity and the processing efficiency of the equipment can be effectively improved.
Whereas for a continuous NURBS curve, after correction of the end speed margin of the sub-curve, the initial speed v of the next sub-curve s Also need to be corrected to v' e To ensure smooth continuation of the overall velocity profile. If the initial speed tolerance of the sub-curve is modified, the last speed tolerance v of the last sub-curve e Also need to be corrected to v' s 。
If the initial speed or end speed tolerance of a sub-curve is modified, it has the ability to merge with an adjacent sub-curve into one sub-curve. The speed planning method ensures that the jerk and the acceleration at the starting point and the end point of the sub-curve are zero, so that the machining efficiency is reduced due to excessive sub-curves, and the acceleration and deceleration process is reduced through sub-curve combination, so that a more stable speed profile is provided. The specific implementation is as follows:
when the sub-curve C i After the end speed margin of (C) is corrected, try to divide the sub-curve C i And a sub-curve C i+1 Merging, using sub-curve C i Initial speed limit, sub-curve C of (2) i+1 And (3) carrying out speed planning on the sum of the end speed limit of the speed profile and the length of the two sub-curves, and combining the two sub-curves into one sub-curve if the planned speed profile does not exceed the speed tolerance.
When the sub-curve C i After the initial speed margin of (C) is corrected, attempt to make the sub-curve C i And a sub-curve C i-1 Merging, using sub-curve C i-1 Initial speed limiter, curve C i And (3) carrying out speed planning on the sum of the end speed limit of the speed profile and the length of the two sub-curves, and combining the two sub-curves into one sub-curve if the planned speed profile does not exceed the speed tolerance.
Experimental example
With the invented method, analytical simulation of butterfly curves is performed in this example to evaluate the performance of the proposed speed planning method.
Butterfly curve with complexThe mixed curvature is widely applied to test cases of NURBS interpolation research. Parameters of the butterfly curve: including the order, control points, node vectors and weight vectors are provided in detail in appendix 1. The equipment performance parameters adopted by the butterfly curve are as follows: limited maximum jerk of 15000mm/s 3 The method comprises the steps of carrying out a first treatment on the surface of the Limited maximum tangential acceleration of 2500mm/s 2 The method comprises the steps of carrying out a first treatment on the surface of the Limited maximum normal acceleration of 2500mm/s 2 The method comprises the steps of carrying out a first treatment on the surface of the The limited maximum feed speed is 120mm/s; the interpolation period is 1ms. The maximum bow height error allowed in the speed planning is 1 μm.
The speed tolerance curve and the sub-curve division results obtained by the method of the invention are shown in fig. 5 (a) -5 (d). The speed profile and bow height error in the parameter dimension, the speed profile, the centripetal acceleration profile, the tangential acceleration profile and the jerk profile in the time dimension, which are obtained by the method, are shown in fig. 6 (a) -6 (f). The invention can effectively limit the bow height error, and the obtained speed, tangential acceleration and jerk are smooth and continuous.
In summary, the invention provides a NURBS curve interpolation speed planning method based on a fourth-order polynomial, which has the advantages that:
1. the maximum running speed is limited by using bow height errors and equipment dynamics characteristics, and the machining precision and the machining efficiency are ensured.
2. The constructed speed planning method can provide a smooth continuous cutting high-efficiency speed profile, an acceleration profile and a jerk profile, and is beneficial to improving the track tracking precision and efficiency in the processing process.
3. The sub-curves are combined, so that the speed increasing and decreasing times in NURBS curve processing are effectively reduced, and the processing stability is improved.
Appendix: parameters of butterfly curves
Order: p=3;
control point (mm): p= [ (36.33,34.76), (37,34.76), (37.39,33.08), (37.85,29.98), (46.38,34.24), (51.86,39.05), (60.35,44.72), (70.65,42.53), (66.93,31.55), (63.04,26.61), (61.58,20.32), (55.63,22.5), (61.26,19.01), (59.63,13.6), (55.48,10.3), (58.41,3.22), (53.96,6.18), (53.22,9.69), (50.72,5.68), (46.79,8.37), (42.78,11.24), (40,14.75), (37.12,24.24), (37.95,16.66), (39.84,13.22), (36.33,9.96), (32.81,13.22), (34.71,16.66), (35.54,24.24), (32.66,14.75), (29.88,11.24), (25.87,8.37), (21.94,5.68), (19.43,9.69), (18.69,6.18), (14.24,3.22), (17.18,10.3), (13.03,13.59), (11.4,19.01), (17.02,22.5), (11.07,20.33), (9.47,26.54), (5.78,31.61), (2,42.53), (12.31,44.72), (20.8,39.05), (26.27,34.24), (34.8,29.98), (35.27,33.08), (35.65,34.76), (36.33,34.76) ];
node vector, u= [0,0,0,0,0.0083,0.015,0.0361,0.0855,0.1293,0.1509,0.1931,0.2273,0.2435,0.2561,0.2692,0.2889,0.317,0.3316,0.3482,0.3553,0.3649,0.3837,0.4005,0.4269,0.451,0.466,0.4891,0.5,0.5109,0.534,0.5489,0.5731,0.5994,0.6163,0.6351,0.6447,0.6518,0.6683,0.683,0.7111,0.7307,0.7439,0.7565,0.7729,0.8069,0.8491,0.8707,0.9145,0.9639,0.985,0.9917,1,1,1,1];
weight vector w= [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0.5,1,1,2,1,1,5,3,3,3,5,1,1,2,1,1,0.5,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].
Claims (9)
1. The NURBS curve interpolation speed planning method based on the fourth-order polynomial is characterized by comprising the following steps of:
acquiring data of NURBS curves and dynamic characteristic parameters of equipment, determining maximum speeds allowed by points on a path according to the bow-height error tolerance, and defining the maximum speeds as speed tolerance;
defining a speed tolerance minimum point as a key point, dividing an original curve into a plurality of sub-curves at the key point, and solving the lengths of the sub-curves;
and constructing a NURBS curve feed rate curve based on a velocity planning algorithm of a fourth-order polynomial.
2. A method for NURBS curve interpolation speed planning based on a fourth order polynomial according to claim 1, characterized in that the maximum feed speed acceptable for each point on the NURBS curve requires the following two conditions to be met:
a) Full arch height error limit to ensure machining accuracy, namely:
wherein ,κi and Vi Respectively the curves are in the parameter u i Curvature and feed speed at delta max Is the maximum allowable bow height error, T s Is the interpolation period; the maximum allowable speed under the limit of bow height error is thus obtained as:
b) The centripetal acceleration limit and jerk are satisfied to satisfy the dynamics of the device, namely:
wherein ,
is the limit of the dynamic performance of the equipment on the feeding speed, v f 、/>
and Jmax The maximum feeding speed, the centripetal acceleration and the jerk allowed by the equipment are respectively;
thus, NURBS curve C (u i ) At parameter u i Maximum speed allowed by the department
The method comprises the following steps:
3. as claimed in claim 1The NURBS curve interpolation speed planning method based on the four-time polynomial is characterized in that points with minimum speed tolerance are defined as key points according to the speed tolerance, the curve is divided into a plurality of sub-curves at the key points, the length of each sub-curve is calculated by adopting an adaptive orthogonal method of a Simpson rule, and the characteristic information of the sub-curve is stored in a data buffer area { [ v ] s ,v e ,S]-a }; wherein v is s and ve Is the starting speed tolerance and the ending speed tolerance of the sub-line segment, and S is the arc length of the sub-curve.
4. The NURBS curve interpolation speed planning method of claim 1, wherein a piecewise fourth-order polynomial is used to construct a jerk function:
wherein ,T1 To T 7 Respectively representing the duration time of seven stages of acceleration, uniform acceleration, deceleration, uniform speed, acceleration and deceleration, uniform deceleration and deceleration; τ 1 To tau 7 Respectively representing the time spent currently at each segment at t;
j (T) has strict symmetry, T 3 ＝T 1 ，T 7 ＝T 5 The method comprises the steps of carrying out a first treatment on the surface of the J (t) is smooth continuous and 0 at the beginning and end, with a maximum value taken in the middle.
5. The NURBS curve interpolation speed planning method based on a fourth-order polynomial as claimed in claim 1, wherein the acceleration and speed expressions are:
6. the NURBS curve interpolation speed planning method based on a fourth order polynomial according to claim 1, characterized in that the speed planning result of the sub-curve is divided into 11 forms, and the following is solved for the specific judgment and each motion segment respectively:
step1, judging whether the maximum speed limit of the equipment can be reached;
assuming that the maximum speed tolerance of the device can be reached, the duration of each phase is expressed as:
at this time, the minimum displacement capable of achieving the maximum speed limit of the apparatus is S 1 ：
If S is greater than or equal to S 1 Form 1, which illustrates the maximum speed limit that can be reached by the device; the time required in step1 is correct;
if S<S 1 Description of failure to reach the deviceMaximum speed limit of (2); dividing the speed plan into two types according to an initial speed limit and an end speed limit; step2-step4 is performed when the end speed limit is greater than the initial speed limit, otherwise step5-step7 is performed;
step2, judging whether acceleration can be realized at a given displacement distance;
assuming that the velocity can be exactly determined by v within a given sub-curve length s Increasing to v e The deceleration stage does not exist, and the acceleration and uniform acceleration stages are as follows:
minimum displacement S of acceleration is achieved 2 The following are provided:
if S=S 2 Form 2, which shows that the acceleration process can be just realized, the obtained T 1 and T2 Is correct;
if S<S 2 Indicating that the length of the sub-curve is insufficient to cause the velocity to be varied from v s Increasing to v e I.e., form 3; at this time, it is necessary to reduce the speed limit v at the end of the sub-curve by using the speed margin correction section e The speed planning result will then be changed to form 2; the speed margin correction section will be described later;
if S>S 2 The length of the sub-curve is sufficient to achieve an acceleration process and there is a deceleration process; at this time, the duration of each stage is re-calculated in the next step;
step3, judging whether the maximum deceleration can be reached;
assuming that the maximum deceleration can be achieved just within the displacement distance of the sub-curve length, the maximum speed that can be achieved is as follows:
due to v e >v s There must be a ramp-up phase, the duration and total displacement of each phase being as follows:
wherein ,S3 Is the minimum displacement that can achieve the maximum deceleration;
if S>S 3 Indicating that the maximum deceleration can be achieved and that there may be a uniform deceleration segment, form 4; assuming that the maximum speed achievable at this time is F, the duration and total displacement for each stage is as follows:
solving the above formula to obtain the duration of each stage;
if S<S 3 Indicating that the maximum deceleration is not reached, the step4 needs to be switched to solve the duration of each stage again;
step4, judging whether the maximum acceleration can be reached;
it is assumed first that the maximum acceleration can be reached, but there is no ramp-up phase, at which time the duration and total displacement of each phase are as follows:
wherein ,S4 Is the minimum displacement that can reach the maximum acceleration;
if S is greater than or equal to S 4 Indicating that the maximum acceleration can be reached and that there may be a ramp up phase, form 5; assuming that it can be reached at this timeF, the duration and total displacement of each stage are as follows:
solving the above formula to obtain the duration time of each stage;
if S<S 4 The maximum acceleration cannot be reached, the uniform acceleration stage does not exist, and the uniform deceleration stage, namely form 6, does not exist; assuming that the maximum speed achievable at this time is F, the duration and total displacement for each stage is as follows:
solving the above formula to obtain the duration of each stage;
when the end speed tolerance is less than the initial speed tolerance, treating the deceleration process as a reverse acceleration process; step5-step7 is then completed according to the above steps to obtain form 7 to form 11 results and the duration of each stage; form 7 is a curve length just capable of achieving a deceleration process, form 8 is a curve length insufficient to achieve a deceleration process, form 9 is a curve length capable of achieving maximum acceleration, and there may be a ramp up phase, form 10 is a curve length capable of achieving maximum deceleration, and there may be a ramp up phase, form 11 is a curve length insufficient to achieve non-maximum deceleration, there are no ramp up phase and ramp up phase.
7. A method of NURBS curve interpolation speed planning based on a fourth order polynomial according to claim 1, wherein the initial or final speed limit of the sub-curve is modified when the sub-curve length is insufficient to effect a speed change:
for the form 2 case, assuming that maximum acceleration can be reached and no ramp up and ramp down phases exist, the duration and total displacement for each phase is as follows:
wherein ,S21 The minimum displacement which can reach the maximum acceleration when no deceleration stage exists;
if S<S 21 Indicating that the maximum acceleration cannot be reached, T needs to be re-determined 1 Assuming that the end speed that can be reached is v' e The duration and total displacement of each stage is as follows:
solving the above to obtain T 1 The method comprises the steps of carrying out a first treatment on the surface of the At this time, the end speed margin of the sub-curve needs to be corrected to v' e The method comprises the steps of carrying out a first treatment on the surface of the At the same time, the initial velocity v of the next sub-curve s Also need to be corrected to v' e To ensure smooth and continuous overall speed profile;
if S>S 21 Indicating that the maximum acceleration can be reached, and a uniform acceleration stage exists; assuming that the end speed that can be reached is v' e The duration and total displacement of each stage is as follows:
solving the above to obtain T 1 and T2 ；
After the speed tolerance correction, the length of the sub-curve can just achieve acceleration, and the speed planning result is consistent with the form 2;
for the case of form 8, the initial speed margin needs to be corrected, and the correction method is almost consistent with the correction of the final speed margin; after the speed tolerance correction, the length of the sub-curve will just be able to achieve deceleration, and the speed planning result is consistent with form 7.
8. A method of NURBS curve interpolation speed planning based on a fourth order polynomial according to claim 1, characterized in that when the initial speed limit of a sub-curve is modified, the end speed limit of the last sub-curve adjacent to it needs to be modified; when the end speed limit of a sub-curve is corrected, the initial speed limit of the next sub-curve adjacent thereto needs to be corrected.
9. A method for NURBS curve interpolation speed planning based on a fourth order polynomial according to claim 1, characterized in that the parameters combine the sub-curve with the adjacent sub-curve when the initial speed limit of the sub-curve is modified, in particular as follows:
when the sub-curve C i After the end speed margin of (C) is corrected, try to divide the sub-curve C i And a sub-curve C i+1 Merging, using sub-curve C i Initial speed limit, sub-curve C of (2) i+1 Carrying out speed planning on the sum of the end speed limit of the (2) and the length of the two sub-curves, and merging the two sub-curves into one sub-curve if the planned speed profile does not exceed the speed tolerance;
when the sub-curve C i After the initial speed margin of (C) is corrected, attempt to make the sub-curve C i And a sub-curve C i-1 Merging, using sub-curve C i-1 Initial speed limiter, curve C i And (3) carrying out speed planning on the sum of the end speed limit of the speed profile and the length of the two sub-curves, and combining the two sub-curves into one sub-curve if the planned speed profile does not exceed the speed tolerance.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202211217405.1A CN116125910A (en) | 2022-10-03 | 2022-10-03 | NURBS curve interpolation speed planning method based on fourth-order polynomial |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202211217405.1A CN116125910A (en) | 2022-10-03 | 2022-10-03 | NURBS curve interpolation speed planning method based on fourth-order polynomial |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CN116125910A true CN116125910A (en) | 2023-05-16 |
Family
ID=86310632
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202211217405.1A Pending CN116125910A (en) | 2022-10-03 | 2022-10-03 | NURBS curve interpolation speed planning method based on fourth-order polynomial |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN116125910A (en) |
-
2022
- 2022-10-03 CN CN202211217405.1A patent/CN116125910A/en active Pending
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN108549328B (en) | Self-adaptive speed planning method and system | |
| Lai et al. | On the development of a parametric interpolator with confined chord error, feedrate, acceleration and jerk | |
| CN111158318B (en) | Flexible acceleration and deceleration planning method for asymmetric quartic curve | |
| CN112965443B (en) | High-precision interpolation control method for corner trajectory tracking of cutting bed | |
| CN112486101B (en) | NURBS curve self-adaptive look-ahead interpolation method | |
| CN107710084A (en) | A kind of method of machining path planning, machining path device for planning and Digit Control Machine Tool | |
| CN110874082A (en) | Method, apparatus and storage medium for determining cutting path of workpiece corner | |
| CN114077226A (en) | S-shaped curve speed planning method, control terminal and computer readable storage medium | |
| CN116300698A (en) | Thread cutting machining method based on dynamic programming | |
| JP2018073097A (en) | Tool route production system, tool route production method, and tool route production program | |
| CN113311783A (en) | S-shaped motion curve planning method and system | |
| CN116125910A (en) | NURBS curve interpolation speed planning method based on fourth-order polynomial | |
| CN111630461B (en) | Numerical control machining method and system and device with storage function | |
| CN116330299B (en) | Industrial robot compliant motion control method based on acceleration smoothing | |
| CN112650147B (en) | Maximum speed parameter limiting method and device under curvature limitation | |
| CN112506143A (en) | High-quality processing quintic polynomial speed planning method based on S-shaped curve | |
| CN111381565B (en) | Acceleration recursive calculation limiting method for cubic polynomial S curve acceleration and deceleration | |
| CN110879572A (en) | Numerical control device and numerical control method | |
| CN113791585B (en) | Self-adaptive control method for circular arc track speed | |
| CN111045461B (en) | Continuous small line segment discrete speed control method for cutting bed | |
| CN115562183A (en) | Motion control system high-precision circular interpolation method based on direct function method | |
| JP2003076410A (en) | Speed control device, speed control method and program for carrying the method with computer | |
| CN114839927A (en) | S-shaped curve acceleration and deceleration planning method and system and storage medium | |
| CN112764393B (en) | Scanning method and device for determining speed sensitive interval | |
| CN113253772A (en) | Servo positioning system and method based on error estimation and compensation |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination |