CN114952831B - Robot milling stability prediction method considering body structure vibration - Google Patents
Robot milling stability prediction method considering body structure vibration Download PDFInfo
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Abstract
The invention belongs to the field of milling, and particularly discloses a robot milling stability prediction method considering body structure vibration, which comprises the following steps: constructing a dynamic cutting force model considering damping force brought by cutter separation and time-varying process damping, and obtaining a discrete diagram of multiple time-lag cutting states according to the dynamic cutting force model and a robot dynamics equation; calculating a transition matrix of all tooth passing periods in a low-frequency vibration period of the robot structure according to the discrete diagram of the multiple time-lag cutting state, and calculating a model of a characteristic value of the transition matrix according to the milling parameters of the robot; if the modes of the characteristic values are all smaller than 1, the system is stable, otherwise, the system is unstable. According to the invention, a cutter workpiece separation model and a time-varying process damping model are established, a cutting-contact state related time-lag coefficient is provided, a milling regeneration flutter stability model considering the structural vibration of the robot body is established based on the cutting-contact state related time-lag coefficient, and the accurate prediction of the milling stability of the robot is realized.
Description
Technical Field
The invention belongs to the field of milling, and particularly relates to a robot milling stability prediction method considering vibration of a body structure.
Background
In the current research reports about the milling stability of robots, the differences of the spindle-tool or robot modes under different stability models and the chatter at different rotation speeds are focused on. The effect of structural low frequency vibrations of a poorly rigid robotic milling system on stability has not been considered. However, the robot milling process has a structure with larger amplitude and low-frequency vibration, and the effect of the component on the processing stability cannot be considered in a conventional dynamics model. Therefore, the action mechanism of the low-frequency vibration on the interaction force and the geometric relation of the tool workpiece is to be researched, and a milling stability prediction model considering the low-frequency vibration of the robot structure is established, so that more accurate prediction of the robot flutter-free cutting parameters is facilitated.
At present, the research on the milling stability of a robot considering low-frequency vibration is very lack, and only Mohammadi et al develop related research on the influence of axial low-frequency vibration on milling regeneration chatter. The axial low-frequency vibration of the tool caused by the vibration of the robot is considered, and the vibration is directly introduced into the stability solving process, so that the corresponding stability modeling and solving are realized. However, since only axial vibration is considered, there is still a large difference between the predicted result of the new model and the stability in actual processing.
Disclosure of Invention
Aiming at the defects or improvement demands of the prior art, the invention provides a robot milling stability prediction method considering the vibration of a body structure, and aims to accurately predict the milling stability of a robot by considering the low-frequency vibration of the robot structure.
In order to achieve the above purpose, the invention provides a method for predicting the milling stability of a robot by considering the vibration of a body structure, which comprises the following steps:
S1, according to a dynamic cutting force model, combining a robot dynamics equation to obtain a discrete diagram of a multi-time-lag cutting state;
The dynamic cutting force model is specifically as follows:
Wherein F x、Fy is the dynamic cutting force in the x and y directions, x (t), y (t) is the dynamic displacement in the x and y directions at time t, the x direction is the feeding direction, and the y direction is the normal direction; c pd,ij (T) is a process damping coefficient, A ij is a dynamic cutting force coefficient, B ij is a dynamic process damping coefficient, tau is a time lag coefficient, i represents an ith cutting element, j represents a jth cutter tooth, T tp is a tooth on period time, M is a total cutting element number, N is a total cutter tooth number, and dz is an axial element height; for the current radial contact angle, A unit step function for the current cutting state;
s2, calculating a transition matrix of all tooth pass periods in a low-frequency vibration period of the robot structure according to a discrete diagram of a multiple time-lag cutting state, and calculating a model of a characteristic value of the transition matrix according to a robot milling parameter; if the modes of the characteristic values are all smaller than 1, the system is stable, otherwise, the system is unstable.
As a further preference, the current cutting state is a unit step functionWherein,A first unit step function used for indicating whether the current cutting edge infinitesimal participates in cutting; /(I)Is a second unit step function for indicating whether the radial cutter workpiece is separated.
As a further preferred feature, the first unit step function is specifically as follows:
wherein, For the current radial contact angle,For the angle of cut,To cut out the angle.
As a further preferred feature, the second unit step function is specifically as follows:
wherein h rv,ij is the thickness cutting change caused by the low-frequency vibration of the robot structure, and h ij (t) is the current cutting thickness.
As a further preferred mode, the current cutting thickness is calculated as follows:
Wherein f t is the feeding amount of each tooth, T is the current time, T tp is the tooth on period time, T rv is the single low-frequency vibration period time, and the symbol "\" represents the remainder operation.
As a further preferable mode, the time lag coefficient τ is determined as follows:
Wherein h dy,ij is the cutting thickness variation, T is the current time, T tp is the tooth pass period time, n is the number of tooth pass periods to be analyzed, T rv is the single low frequency vibration cycle time, signRepresenting a rounding up operation.
As a further preferred feature, the process damping coefficient C pd,ij (t) is determined from time-varying process damping;
The time-varying process damping calculation is as follows:
wherein, Respectively radial and tangential time-varying process damping, S rd,ij (t) is the time-varying total indentation area, K d is the indentation coefficient related to the material, alpha is the scale factor representing the actual indentation amount, and mu represents the ratio of tangential to radial cutting force coefficients.
As a further preferred way of calculating the time-varying total indentation area S rd,ij (t) is as follows:
Srd,ij(t)＝Srst,ij(t)+Srdy,ij(t)
Wherein S rst,ij (t) is a time-varying dynamic indentation area, and S rdy,ij (t) is a time-varying static indentation area; r h is the cutter edge radius, beta rs,ij is the dynamic separation angle, and gamma is the cutter relief angle; l rpd,ij (t) is the dynamic indentation length, r v,ij (t) represents the cutter tooth radial vibration speed, and V ctp represents the cutter tooth tangential speed.
As a further preferable mode, in step S3, a transition matrix in a single time period is constructed through a series of discrete diagrams of multiple time-lag cutting states, and then a transition matrix of all tooth-on periods in a low-frequency vibration period of the robot structure is calculated.
In general, compared with the prior art, the above technical solution conceived by the present invention mainly has the following technical advantages:
1. According to the invention, the low-frequency vibration of two degrees of freedom of the tangential direction of the cutter caused by the low-frequency vibration of the robot structure is considered, the change of the tangential contact state of the cutter workpiece is considered, a dynamic cutting force model which takes damping force caused by cutter separation and time-varying process damping into consideration is constructed, and a milling regeneration flutter stability model which takes the vibration of the robot body structure into consideration is established based on the dynamic cutting force model, so that the accurate prediction of the milling stability of the robot is realized, and a theoretical basis is laid for the optimization of the stability of the robot.
2. The invention establishes a specific radial and tangential cutter workpiece separation model and a time-varying process damping model to describe the change mechanism of the cutting contact state of the cutter workpiece; and providing a time lag coefficient depending on the tangential contact state, and establishing a stability prediction model considering the vibration of the robot body structure based on the time lag coefficient.
3. According to the method, the influence of the structural vibration of the robot on the narrow stable domain under different postures is researched, the robot posture with higher flexibility is found, the more obvious tool-workpiece separation is realized, the more time-varying process damping and the more prominent narrow stable domain are realized, the spindle rotating speed with better stability in the narrow stable domain in the robot milling process can be accurately selected, and the stability optimization of the robot milling process is realized.
Drawings
Fig. 1 is a schematic view of a cutting thickness of a robot structure considering low-frequency vibration according to an embodiment of the present invention, wherein (a) is a cutting thickness of a first three-tooth through period, and (b) to (d) are cutting moments of first to third cutter teeth;
Fig. 2 is an analysis chart of the influence of the low-frequency vibration of the robot structure on the cutting process, wherein (a) is a cutting area, and (b) to (e) respectively show the changes of the cutting edge track and the cutting thickness at the time t=t tp、2Ttp、3Ttp、4Ttp;
FIG. 3 is a process damping model without consideration of low frequency vibration of a robot structure in accordance with an embodiment of the present invention;
FIG. 4 shows a time-varying process damping model that accounts for low frequency vibration of a robot structure, where (a) is the vibration trajectory variation throughout the low frequency vibration period, (b) is the indentation model that does not account for vibration of the robot structure, and (c) - (e) are the indentation models for three typical cutting regions A, B, C that account for vibration of the robot structure;
Fig. 5 is a comparison of process damping indentation area per radial vibration speed for an embodiment of the present invention, where (a) is x rv＝yrv =15 μm and (b) is x rv＝yrv =5 μm;
FIG. 6 is a schematic view of a cutting force model according to an embodiment of the present invention;
fig. 7 is a schematic diagram of stability calculation considering low-frequency vibration of a robot structure according to an embodiment of the present invention.
Detailed Description
The present invention will be described in further detail with reference to the drawings and examples, in order to make the objects, technical solutions and advantages of the present invention more apparent. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention. In addition, the technical features of the embodiments of the present invention described below may be combined with each other as long as they do not collide with each other.
According to the robot milling stability prediction method considering body structure vibration, low-frequency vibration of two tangential degrees of freedom of a cutter caused by low-frequency vibration of a robot structure is considered, a cutter workpiece separation model and a time-varying process damping model are established, a cutter workpiece cutting and touching state change is considered, a cutting and touching state related time lag coefficient is proposed, a milling regeneration flutter stability model considering the robot body structure vibration is finally established based on the method, and robot milling stability prediction is realized.
The specific process is as follows:
1.1 modeling of tool-workpiece cut-touch mechanism considering robot structural vibration
The robot structure is easy to generate low-frequency vibration in milling due to the relatively weak rigidity, and the phenomenon of cutter workpiece separation and process damping change can be generated due to the larger amplitude, and the phenomenon can influence the processing stability. To achieve the above objective, first, a tool-workpiece tangent mechanism modeling is required that considers the tool workpiece separation and the process damping variation.
1.1.1 Tool workpiece separation
The robot tool workpiece separation can be classified into Radial Tool Workpiece Separation (RTWS) and Tangential Tool Workpiece Separation (TTWS) according to the difference of the robot low-frequency vibration directions. First, RTWS expansion mechanisms are analyzed and modeled.
The low-frequency vibration of the robot structure has the characteristic of long vibration period, a plurality of tooth through periods T tp exist in a single low-frequency vibration period T rv, and the influence of the low-frequency vibration on the thickness cutting in each tooth through period is greatly different. Therefore, considering the low-frequency vibration of the robot structure, the stability analysis should be performed with the low-frequency vibration cycle as a unit time.
As shown in fig. 1 (a), a schematic diagram of the cutting thickness under the influence of the low-frequency vibration of the robot is shown, the abscissa is the rotation angle (or time history) of the tool, the ordinate is the cutting thickness, and the number of teeth of the tool is 3 (see table 1, the parameters in table 1 are merely examples). When the cutting thickness influenced by the low-frequency vibration of the robot structure is calculated, the cutting lag time, namely the tooth pass cycle time T tp, is needed to be considered. Since the ratio of the period of the low-frequency vibration to the period of the tooth pass is likely to be different from an integer, the last remainder portion of the graph will generally occur. When the low-frequency vibration of the robot structure is considered, the stability change in the whole vibration period needs to be completely analyzed, so that the last remainder part needs to be extended backwards to a complete tooth pass period for calculation. The number of tooth pass periods to be finally analyzed and calculated is as follows:
TABLE 1 parameter settings
Further, for ease of understanding, the first three tooth pass periods in fig. 1 (a), i.e., the first spindle rotation period, will be described in detail. As shown in fig. 1 (b) to (d), the first cutter tooth cutting time is set when the cutter rotation angle θ is 0 ° to 120 °, and the second cutter tooth cutting time and the third cutter tooth cutting time are set when θ is 120 ° to 240 ° and 240 ° to 360 °. Each cutter tooth cutting time is a tooth pass period time T tp, namely a lag time.
It is obvious that the influence of the low-frequency vibration of the robot structure on each tooth pass period is different. Subject to instantaneous radial contact angleThe effect of the low frequency vibration induced radial vibration of the cutter teeth can be expressed as:
wherein X rv (t) and Y rv (t) are low-frequency vibrations in the X and Y directions, respectively. The variation in cutting thickness caused by radial vibration can be expressed as:
hrv,ij(t)＝rrv,ij(t)-rrv,ij(t-Ttp)(0≤t＜Trv) (3)
When the low-frequency vibration of the robot structure is not considered, the cutting thickness is the same in 9 tooth through periods, and the cutting thickness at any moment can be expressed as the feeding quantity f t per tooth and the instantaneous radial contact angle Is a function of:
when the low-frequency vibration of the robot structure is considered, the cutting thickness will change, and the specific expression is as follows:
hdy,ij(t)＝hij(t)+hrv,ij(t)(0≤t＜Trv) (5)
When the thickness cutting change h rv,ij caused by the low-frequency vibration of the robot structure is negative and the absolute value of the thickness cutting change h rv,ij is larger than the current cutting thickness h ij (t), the cutting contact state of the tool workpiece is a radial tool workpiece separation state (RTWS), and RTWS is mainly influenced by the low-frequency vibration of the robot structure and the feeding amount of each tooth, as shown in the formulas (3) and (4). When at RTWS, the current cutting force is zero and no cutting of the current cutter tooth occurs. The judgment index can be expressed by the following unit step function.
For convenience of expression, the cutting contact state of the tool workpiece is classified into two types, i.e., a cutting state in which RTWS of the tool workpiece is separated from the tool workpiece, according to whether the radial tool workpiece separation occurs. In the cutting state, the cutting state is subdivided by judging whether radial cutter workpiece separation has occurred in the preceding cutter tooth and several times of radial cutter workpiece separation have occurred in succession. The cutting state in which the radial tool workpiece separation does not occur in the preceding tooth is referred to as a single time lag cutting State (STDC), and the cutting state in which the radial tool workpiece separation occurs in the preceding tooth or the radial tool workpiece separation occurs in the preceding tooth continuously is referred to as a multiple time lag cutting state (MTDC).
The states of the cutting contact of the cutter workpiece with different tooth through periods are represented by adopting the expression mode in fig. 1, and are shown in fig. 2. Wherein the vertical axis is the cutter tooth cutting time history, and the horizontal axis is the cutter rotation angle theta. And analyzing and modeling the cutting contact state of the cutter workpiece considering the low-frequency vibration of the robot structure by recording the cutting thickness changes of different tooth through periods in the same low-frequency vibration period. Fig. 2 (b) to (e) show changes in the cutting edge locus and the cutting thickness at time t=t tp～4Ttp, respectively, and correspond to the cutting regions 1 to 4 in fig. 2 (a), respectively. In the figure, a black solid line and a black dot line respectively represent a current blade track and a nominal blade track, a black dotted line represents a blade track at a historical moment, an orange dotted line represents a workpiece surface contour, a shaded part represents a cut area, and a grid part is a cut area at the current moment. For ease of understanding, in fig. 2 (b) to (e), the points of entry and points of entry of the same tooth cutting path are represented by points of the same color, and different colors represent different tooth cutting paths. When not at RTWS, the tool workpiece is in a cutting state at this time, and the cutting trajectories of the two cutter teeth constituting the cutting region are marked below the cutting state annotation in the drawing. As shown in fig. 2 (c), the green and purple tooth cutting paths constitute the cutting thickness at this time.
The cutting area 1 is STDC, and the cutting thickness model is shown in fig. 2 (b). The k-th cutter tooth and the k+1-th cutter tooth are not separated from each other by radial cutter workpieces, the current cutting force is not zero, the cutting thickness is affected by vibration of two adjacent cutter teeth (the k-th cutter tooth and the k+1-th cutter tooth), and the time lag time is a single-time tooth pass period. This state only needs to consider the influence of the low frequency vibration on the cutting thickness.
The cutting region 2 has two cutting states, STDC and RTWS, and a cutting thickness model is shown in fig. 2 (c). The radial cutter workpiece separation occurs in the latter half of the cutting process for the cutter tooth k+2, where the cutting force is zero.
The cutting area 3 is RTWS, and the cutting thickness model is shown in fig. 2 (d). The k+3 cutter tooth is separated from the radial cutter workpiece, and the cutting force is zero.
The cutting area 4 is MTDC, and the cutting thickness model is shown in fig. 2 (e). The k+4 cutter tooth is not separated from the radial cutter workpiece, the k+3 cutter tooth in the first half is separated from the radial cutter workpiece, the cutting surface of the k+4 cutter tooth is the cutting surface reserved for the k+2 cutter tooth which is nearest to the radial cutter workpiece and is not separated from the radial cutter workpiece, the cutting thickness is affected by the vibration of the k+4 cutter tooth and the k+2 cutter tooth, and the time lag time is ((k+4) - (k+2)). T tp＝2Ttp. Radial cutter workpiece separation occurs in both the k+3 cutter teeth and the k+2 cutter teeth in the second half stroke, the cutting surface of the k+4 cutter teeth is the cutting surface reserved by the k+1 cutter teeth which are closest to the cutting surface of the k+4 cutter teeth and are not subjected to radial cutter workpiece separation, the cutting thickness is affected by vibration of the k+4 cutter teeth and the k+1 cutter teeth, and the time lag time is 3T tp. Similarly, when the j-th cutter tooth is not separated from the radial cutter workpiece, and the j-1-th cutter tooth is separated from the j-n-th cutter tooth, the current cutting force is not zero, the cutting thickness is influenced by the vibration of the j-th cutter tooth and the j-n-1-th cutter tooth, and the time lag time is changed into (j- (j-n-1)). T tp＝(n+1)Ttp.
In summary, if the radial tool workpiece is separated continuously in the previous n (n is greater than or equal to 1) cycles of the current cutting cycle, the workpiece surface of the current cutting is caused by the cutting process of n+1 cycles, and the lag time of the cutting is n+1 times of the tooth through cycle, namely (n+1) T tp. The ratio of the current cutting lag time to the tooth pass time is expressed as a time lag coefficient tau, and the corresponding judging function is as follows:
According to the formulas (6) and (7), when g 2 =0, the radial cutter workpiece separation phenomenon occurs at this time, and the cutting force is zero; when g 2 =1, τ is equal to or greater than 1, the cutting thickness is determined by the cutting track of the current cutting edge and the closest cutting edge without radial tool workpiece separation, the time lag time is determined by formula (7), and the corresponding stability modeling and solving process is detailed in the following section 1.2.
1.1.2 Time-varying Process damping
The process damping is a damping force generated when a cutter contacts with vibration marks on the surface of a workpiece in low-speed milling processing, and generally shows a suppression effect on processing vibration. Since the magnitude of process damping is inversely proportional to the tangential velocity of the cutter teeth, it is generally only considered in low speed machining. However, there is a low frequency vibration in the robot process which will cause a change in the process damping setback area, thereby affecting the magnitude of the process damping. As a result, low frequency vibration of the robot structure may cause a process damping phenomenon in processing at a higher rotational speed.
In the process damping model, the process damping force is modeled as a function of the setback region, and the radial and tangential process damping expressions are as follows:
Wherein S d,ij (t) is the volume of the pressed-in area and is the product of the area of the pressed-in area and the cutting width; k d is the indentation coefficient associated with the material; alpha is a scale factor representing the actual amount of indentation; μ represents the tangential to radial cutting force coefficient ratio, where K d, α, μ are all constant terms related to experimental conditions.
The indentation area S d,ij is divided into a static indentation area S st (t) and a dynamic indentation area S dy,ij (t), as shown in fig. 3. Both regions have an effect on the process stability. When the low-frequency vibration is not considered, the separation point SP is only determined by the cutter structure, and the calculation formulas of the static indentation area and the dynamic indentation area are as follows:
Sd,ij＝Sst+Sdy,ij (11)
Wherein r v,ij represents the radial vibration speed of the cutter tooth, L pd is the indentation length, and as shown in fig. 3, the specific calculation expression is as follows:
Lpd＝rh[sinβs+sinγ+(cosγ-cosβs)/tanγ] (13)
Where r h is the cutter edge radius, β s is the separation angle that defines the Separation Point (SP), γ is the cutter relief angle, and V ctp is the tangential velocity of the cutter teeth.
When the low-frequency vibration of the robot structure is considered, the separation point SP will change along with the change of the separation angle, and at the moment, the indentation area will change dynamically along with the change, and the change is different in different cutter tooth periods. As shown in fig. 4, the influence of the robot structure low-frequency vibration on the separation point SP and the dynamic indentation area in different tooth pass periods is described by describing the vibration locus variation in the entire low-frequency vibration period. Wherein the gray dotted line indicates a blade trajectory taking no consideration of the low-frequency vibration of the robot structure, the black solid line indicates a blade trajectory taking consideration of the low-frequency vibration of the robot structure, and the orange dotted line indicates a blade trajectory taking consideration of the low-frequency vibration of the robot structure. Three typical cutting areas were selected A, B, C for analysis, the details of which are shown in fig. 4 (b) - (e).
In the cutting area A, the low-frequency vibration of the robot structure causes the cutter to be pressed into the surface of the workpiece, the separation angle beta s is increased, the separation point SP moves upwards, and the indentation area is increased. At this point the process damping increases and the indentation model pair is shown, for example, in fig. 4 (b) and (c). In the cutting region B, the vibration amplitude of the robot structure is smaller, and the influence on the area of the indentation region is smaller. The process damping can be considered approximately constant at this time, with the indentation model pairs such as those shown in fig. 4 (b) and (d). In the cutting region C, the robot structure low frequency vibration causes the tool to move away from the workpiece surface, the separation angle β s becomes small, the separation point SP moves down, and the indentation region becomes small. At this point the process damping is reduced and the indentation model pair is shown, for example, in fig. 4 (b) and (e).
With the parameters in table 1, the indentation area was calculated for each of the cases of taking into account and not taking into account the low frequency vibration of the robot structure at the unit radial vibration velocity, and the calculation results are shown in fig. 5 (a). The area of the indentation area is shown as a function of the period of low frequency vibration (T rv). Corresponding to fig. 4 (a), in the first half of T rv, the robot structure low-frequency vibration causes the tool to press into the workpiece surface, the indentation area increases, and the process damping increases. In the latter half of T rv, the robot structure low-frequency vibration causes the cutter to be far away from the surface of the workpiece, the area of an indentation area is reduced, and the process damping is reduced. When the low-frequency vibration amplitude is reduced to 5 μm, the influence of the low-frequency vibration on the area of the indentation area is also reduced as shown in fig. 5 (b). Therefore, the magnitude of the time-varying process damping is related to the magnitude of the low-frequency vibration amplitude of the robot, and when the low-frequency vibration amplitude is large, the time-varying process damping is large, and the stability boundary is also more obviously affected.
In summary, the separation angle β s and the indentation length L pd will be changed according to the low-frequency vibration of the robot structure. The dynamic separation angle β rs,ij (t) and the dynamic indentation length L rpd,ij (t) are calculated as follows:
Lrpd,ij(t)＝rh[sin(βrs,ij(t))+sinγ+(cosγ-cos(βrs,ij(t)))/tanγ] (15)
Wherein r rv,ij (t) represents radial vibration of the cutter teeth caused by low-frequency vibration, as shown in formula (2). The influence of the low-frequency vibration of the robot structure on the damping of the milling process is related to the size and the direction of the radial vibration of the robot, and the area expression of the damping indentation area of the time-varying process of the low-frequency vibration of the robot structure is considered as follows:
Srd,ij(t)＝Srst,ij(t)+Srdy,ij(t) (18)
Wherein S rst,ij (t) is a time-varying dynamic indentation area, S rdy,ij (t) is a time-varying static indentation area, and S rd,ij (t) is a time-varying total indentation area. R v,ij (t) denotes the tooth radial vibration velocity, V ctp denotes the tooth tangential velocity (V ctp =2pi nR/60, n is the rotational speed, R is the tool radius). Therefore, the time-varying process damping calculation expression considering the robot structure low frequency vibration is as follows:
1.2 stability modeling of structural vibration influence of robots
1.2.1 Dynamic cutting force modeling
As shown in fig. 6, when the robot structure low frequency vibration is not considered, the dynamic thickness-cutting model is as follows:
wherein, For the instantaneous radial contact angle, x (T), y (T) represents the dynamic displacement in x and y directions, respectively, and T tp represents the tooth on-period time.
As in section 1.1.1, the occurrence of tool separation will change the way dynamic cutting forces are calculated. When the cutter separation occurs, the cutting force of the current infinitesimal is 0, the infinitesimal expressions of the radial and axial dynamic cutting forces are as follows:
wherein, Is a unit step function used for indicating whether the current cutting edge infinitesimal participates in cutting. The expression is as follows:
wherein, AndRespectively representing the cut-in and cut-out angles. On the other hand, in calculating the dynamic thickness cut, it is necessary to consider the influence of the time lag coefficient on the cutting state dependence caused by the tool separation. The dynamic thickness cutting expression taking the low-frequency vibration of the robot structure into consideration is as follows:
Wherein τ is a time lag coefficient, and the judgment function is shown as formula (7). In summary, the dynamic cutting force infinitesimal calculation expression considering the low-frequency vibration of the robot structure is as follows:
As shown in the analysis of section 1.1.2, the magnitude of the damping force of the time-varying process will be changed along with the change of the low-frequency vibration of the robot structure in the cutting process, and considering the damping force brought by the damping of the time-varying process as shown in the formula (19), the radial and axial dynamic cutting force infinitesimal expressions in the formula (24) are modified as follows:
through coordinate transformation, the dynamic cutting force components in a rectangular coordinate system can be obtained by integrating along the axial direction and summing each cutter tooth, and the components are as follows:
the combined type (23), formula (25) and formula (26) final dynamic cutting force can be expressed as follows:
Wherein:
For ease of expression, formula (27) is rewritten as follows:
wherein w represents a cutting width:
1.2.2 stability modeling
The kinetic equation taking into account the effect of the low frequency vibration of the robot structure is as follows:
where m ij (i, j=x, y) represents the modal mass of the system excitation in the j direction, the i direction responding; c ij (i, j=x, y) represents the modal damping of the system excitation in the j direction, i direction response; k ij (i, j=x, y) represents the modal stiffness of the system excitation in the j direction, the i direction response.
The combination of formula (32) and formula (34) can be obtained:
Wherein q (t) = [ x (t) y (t) ] T, M, C, K is a modal mass, damping, stiffness matrix, τ is a time lag coefficient.
Based on a fully discrete solving idea, the dynamic equation is rewritten into a state space form, and meanwhile, the tau time lag phenomenon caused by cutter separation is considered, and the formula (35) can be rewritten into the following form:
x(t)＝A0x(t)+A(t)x(t)-A(t)x(t-τTtp) (36)
Wherein:
Discretizing a kinetic equation, adopting a direct integration method and a linear approximation method, equally dividing a tooth pass period T tp into m discrete times with tau as an interval, and obtaining the expression of x k+1 by using the following expression in each time interval, wherein kτ is less than or equal to T (k+1) tau (k=0, …, m):
Wherein:
Since a (T) is a periodic function, i.e., a (T) =a (T-T tp)＝A(t-τTtp), equation F m-1＝Fτm-1,Fm＝Fτm exists in the equation. From equation (40), the definition of the discrete map is:
wherein, Discrete diagram representing single time lag cut state,A discrete diagram representing a multiple time-lapse cut state, the corresponding expression is as follows:
yk＝col(xk xk-1 … xk+1-m xk-m … xk+1-2m xk-2m … xk+1-τm xk-τm) (42)
and constructing a transition matrix phi in a single time period through a series of discrete graphs, and judging the stability of the system in the current tooth passing period through judging whether the modulus of the characteristic value of the transition matrix is smaller than 1. The transition matrix expression is as follows:
As shown in fig. 1, since the period time of the low-frequency vibration of the robot structure is longer than the period time of the tooth-on period, the transition matrix of the single tooth-on period cannot reflect the influence of the vibration of the robot structure on the dynamic cutting force. Therefore, as shown in fig. 7, when the vibration of the robot structure is considered, the transition matrix Φ η of all the tooth-on periods in the low-frequency vibration period of the robot structure is calculated according to the formula (45), and the system stability is judged according to the eigenvalues of all the transition matrices.
The invention further discloses a process for establishing the milling stability prediction model of the vibration robot with the body structure, and a milling stability judgment formula convenient for the model to use. Specifically, when stability prediction is performed, firstly, robot milling parameters are collected or preset, and then the following steps are performed:
S1, calculating a dynamic cutting force taking damping force brought by cutter separation and time-varying process damping into consideration according to a formula (27);
s2, taking dynamic cutting force into consideration, establishing a dynamic equation taking the influence of low-frequency vibration of a robot structure into consideration, and further deriving a specific formula of a discrete diagram of a multiple time-lag cutting state
S3, according to the discrete diagramThe specific formulas and the formula (46) of the system are used for calculating transition matrixes phi η of all tooth passing periods in the low-frequency vibration period of the robot structure, calculating the modes of the characteristic values of all the transition matrixes, and according to the Floquet theory, if the modes of the characteristic values are smaller than 1, the system is stable, otherwise, the system is unstable.
In summary, the invention considers the low-frequency vibration of the robot structure, establishes a cutter workpiece separation model and a time-varying process damping model, proposes a time-lag coefficient related to the cutting contact state, and analyzes the influence of the low-frequency vibration of the robot structure on the regeneration flutter based on the time-lag coefficient. And the milling regeneration flutter stability model considering the structural vibration of the robot body is established based on the related time lag coefficient of the cutting contact state by considering the cutting contact state change of the cutter workpiece. And the milling experiment is combined, the established stability model is subjected to experimental verification and analysis, and the stability lobe diagrams of different postures are compared by combining the pose dependence of the structural vibration of the robot, so that guidance is provided for the rotation speed posture selection of the stability optimization of the robot.
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.
Claims (9)
1. The robot milling stability prediction method considering the vibration of the body structure is characterized by comprising the following steps of:
S1, according to a dynamic cutting force model, combining a robot dynamics equation to obtain a discrete diagram of a multi-time-lag cutting state;
The dynamic cutting force model is specifically as follows:
Wherein F x、Fy is the dynamic cutting force in the x and y directions, x (t), y (t) is the dynamic displacement in the x and y directions at time t, the x direction is the normal direction, and the y direction is the feeding direction; c pd,ij (T) is a process damping coefficient, A ij is a dynamic cutting force coefficient, B ij is a dynamic process damping coefficient, tau is a time lag coefficient, i represents an ith cutting element, j represents a jth cutter tooth, T tp is a tooth on period time, M is a total cutting element number, N is a total cutter tooth number, and dz is an axial element height; For the current radial contact angle,/> A unit step function for the current cutting state;
s2, calculating a transition matrix of all tooth pass periods in a low-frequency vibration period of the robot structure according to a discrete diagram of a multiple time-lag cutting state, and calculating a model of a characteristic value of the transition matrix according to a robot milling parameter; if the modes of the characteristic values are all smaller than 1, the system is stable, otherwise, the system is unstable.
2. The method for predicting milling stability of robot considering vibration of body structure as claimed in claim 1, wherein the current cutting state is a unit step functionWhereinA first unit step function used for indicating whether the current cutting edge infinitesimal participates in cutting; /(I)Is a second unit step function for indicating whether the radial cutter workpiece is separated.
3. The method for predicting the milling stability of a robot in consideration of vibration of a body structure according to claim 2, wherein the first unit step function is specifically as follows:
wherein, For the current radial contact angle,For the angle of cut,To cut out the angle.
4. The method for predicting the milling stability of a robot in consideration of vibration of a body structure according to claim 2, wherein the second unit step function is specifically as follows:
wherein h rv,ij is the thickness cutting change caused by the low-frequency vibration of the robot structure, and h ij (t) is the current cutting thickness.
5. The method for predicting the stability of milling of a robot in consideration of vibration of a body structure according to claim 4, wherein the current cutting thickness is calculated as follows:
Wherein f t is the feeding amount of each tooth, T is the current time, T tp is the tooth on period time, T rv is the single low-frequency vibration period time, and the symbol "\" represents the remainder operation.
6. The method for predicting the milling stability of a robot considering vibration of a body structure according to claim 1, wherein the time lag coefficient τ is determined as follows:
Wherein h dy,ij is the cutting thickness variation, T is the current time, T tp is the tooth pass period time, n is the number of tooth pass periods to be analyzed, T rv is the single low frequency vibration cycle time, signRepresenting a rounding up operation.
7. The method for predicting the stability of milling of a robot in consideration of vibration of a body structure according to claim 1, wherein the process damping coefficient C pd,ij (t) is determined according to time-varying process damping;
The time-varying process damping calculation is as follows:
wherein, Respectively radial and tangential time-varying process damping, S rd,ij (t) is the time-varying total indentation area, K d is the indentation coefficient related to the material, alpha is the scale factor representing the actual indentation amount, and mu represents the ratio of tangential to radial cutting force coefficients.
8. The method for predicting the stability of milling by a robot in consideration of vibration of a body structure according to claim 7, wherein the time-varying total indentation area S rd,ij (t) is calculated as follows:
Srd,ij(t)＝Srst,ij(t)+Srdy,ij(t)
Wherein S rst,ij (t) is a time-varying dynamic indentation area, and S rdy,ij (t) is a time-varying static indentation area; r h is the cutter edge radius, beta rs,ij is the dynamic separation angle, and gamma is the cutter relief angle; l rpd,ij (t) is the dynamic indentation length, r v,ij (t) represents the cutter tooth radial vibration speed, and V ctp represents the cutter tooth tangential speed.
9. The method for predicting the milling stability of the robot considering the vibration of the body structure according to any one of claims 1 to 8, wherein in the step S3, a transition matrix in a single time period is constructed through a series of discrete diagrams of multiple time-lapse cutting states, and then the transition matrix of all tooth-through periods in the low-frequency vibration period of the robot structure is calculated.
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