Int J Adv Manuf Technol (2009) 40:769–775 DOI 10.1007/s00170-008-1390-z ORIGINAL ARTICLE Variable-period feed interpolation algorithm for high-speed five-axis machining Yuyao Li & Jingchun Feng & Yuhan Wang & Jianguo Yang Received: 7 August 2007 / Accepted: 8 January 2008 / Published online: 4 March 2008 # Springer-Verlag London Limited 2008 Abstract To alleviate the feed fluctuation and to maintain a smooth feed in conventional five-axis machining, an optimal feed interpolation algorithm (look-ahead) is proposed. However, the problem arises where the segment usually cannot be interpolated exactly in an integer period because of the nonzero joint feed at the junction. To overcome this problem, and to achieve faster machining speed and higher quality parts, this paper presents an optimal feed interpolation algorithm for high-speed, fiveaxis machining having the function of “look-ahead”, i.e., variable-period linear interpolation algorithm. In real applications, the proposed algorithm results in: (1) constant speed; (2) high machining accuracy. Moreover, in this paper, an efficient method for acceleration and deceleration control is presented to achieve the highest-quality feed profiles and to shorten machining times. The precision and speed of machining is improved greatly. Experimental results verify the effectiveness of the proposed method. Keywords Five-axis machining . Optimal feed . Look-ahead algorithm . Variable periods . Linear Interpolation Y. Li (*) : J. Feng : Y. Wang : J. Yang School of Mechanical Engineering, Shanghai Jiao tong University, Shanghai 200240, PR China e-mail: liyuyao@sjtu.edu.cn Y. Wang State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao tong University, Shanghai, PR China 1 Introduction Five-axis tool paths are currently generated as a set of discrete data points with commercial CAD/CAM systems. Post processors perform the inverse kinematics algorithm on the tool path points, and then transform the tool path into a series of short linear segments [1]. In the conventional interpolation method, the cutting tool must accelerate and decelerate at each line segment, and the adjacent linear segments contour features are overlooked. This method results in velocity discontinuities at the linear segment junctions which leads to higher accelerations, poor surface finish, lower surface accuracy, and longer machining times [2]. When the linear segments are shorter than the minimum distance required for the acceleration/deceleration, the CNC machining task may never achieve the desired feed; it will slow down the cutting speed. However, the five-axis machining programs generated with the CAD/ CAM system are usually composed of such short linear segments. As a result, these short linear segments not only reduce the cutting speed, but they also influence the machining accuracy because of the feed variation. Therefore, these drawbacks suggest that conventional machining methods fail to meet the requirements of high-speed, highaccuracy machining in modern industry. Considering the above-described problems, the smooth feed optimization scheme has been used in many CNC machines to minimize the start/stop motions and to achieve a constant feed rate. In Sect. 2, we discuss some existing interpolation algorithms and the respective effects. In Sect. 3, we propose an optimal interpolation algorithm, i.e., a variable-period feed interpolation algorithm that can overcome the demerits in the existing interpolation algorithms. In this scheme, the proposed interpolation algorithm is executed in variable-period length instead of the fixed length in the 770 conventional interpolation algorithm. In Sect. 4, the proposed algorithm is demonstrated by the experimental results. We can see that the better machining accuracy is obtained in the proposed interpolation algorithm than in the conventional one. The optimized speed contour shows the efficiency of the proposed interpolation algorithm. 2 Previous work In order to overcome the disadvantages of conventional CNC machining methods, the look-ahead scheme has been used in many CNC machines to minimize the start/stop motions and to maintain a constant feed rate. This scheme is realized by analyzing the contour of the consecutive short linear segments. At the same time, the machining dynamics is taken into account. This means that hundreds of small line blocks are read and analyzed in advance to obtain the smooth continuous motion accommodating the capacity of the machine tool. Instead of accelerating and decelerating within each program segment, a joint feed is calculated at each segment junction to generate a smooth feed profile. Some work has been done about the look-ahead scheme [3–6]. Hu et al. [7] presented an optimal velocity model, and based on it, proposed an algorithm to seek the approximate optimal feedrate by evaluating the tool path ahead. However, this scheme has the problem of having some short linear segments that cannot be discretized exactly in an integer interpolation period because of the nonzero start velocity of the segment. So the small remnant that is caused in the end will cause a position error or speed fluctuation. Han et al. [8] developed a high-speed machining algorithm based on the look-ahead interpolation technique for the machining of a 3D surface obtained by CAD/CAM system. The proposed algorithm improves the machining speed without any hardware support and develops the lowcost CNC controller having high-speed machining ability. However, his algorithm alters the machining path to keep feed at a constant level. Although the algorithm is able to maintain a constant feed, it is unfavorable for many applications since it modifies the design of the product. Guo [9] introduce a high-speed interpolation algorithm for consecutive minute linear segments. Firstly, this algorithm optimizes the manufacturing data without losing accuracy. Then thinking of the segments’ continuity, geometric relationship between adjacent segments, acceleration and deceleration capability and manufacturing precision are considered to realize the smooth transition between segments, limiting the maximum transition feed at the turning points. Thus the machine tool is avoided from flexible impacts at its startup, stop or velocity changing. The feed is controlled smoothly and reasonably. However, the joint velocity limit has to be very low when a tight Int J Adv Manuf Technol (2009) 40:769–775 contour error is needed. His algorithm is hard to apply in finish machining, for the joint velocity is nearly constrained to zero. According to Wang et al. [10], when the start velocity of the segment satisfying some conditions, we can adjust the feed profile of the conventional interpolation method to overcome the feed fluctuation. So, A new constrain to the start velocity of the segment is obtained and the corresponding adjusting method is proposed. The details are as follows: As described in Fig. 1, we can drop the velocity higher than Va to the velocity Va, that is: Vij ¼ Va ; when Vij > Va : ð1Þ By adjusting these Vij to Va, the length of the segment in the first (na +nc +nd) interpolation periods will decrease by Sa. Obviously, we can find a suitable Va, such that: Sa ð V a Þ ¼ Se ð2Þ In which Se ¼ Viþ1 ð1 f ÞT ð3Þ In this method, we know that Sa has a upper bound denoted by Sam. So we must ensure that: Sam  Se ð4Þ To ensure that the method works, the start feed introduce the new constrain:  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   li þ 12 16 24p þ 25 76 16 24p þ 25 76 þ 1 aT 2  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7   Vi  6 24p þ 25 6 þ2 T ð5Þ In which: p ¼ aimliT 2 Then, we can obtain a suitable feed profile such that the segment can be interpolated in integer period, namely, t= (na+nc+nd+1)T, accommodating the machine tool dynamics. However, because the new constrains are introduced, Fig. 1 The speed adjusting method Int J Adv Manuf Technol (2009) 40:769–775 771 the speed programming is more complicated and the adjusting method is very complicated. So the scheme requires much more computation. In this paper, a variable-period feed interpolation algorithm based on the look-ahead interpolation technique is proposed to solve the above-mentioned disadvantage. 3 Proposed algorithm In this paper, to overcome this drawback in the smooth feed optimization scheme and to achieve faster machining speed and higher-quality parts, it takes the following constraints into account and proposed an optimal feed interpolation algorithm, namely, the variable-period linear interpolation algorithm based on look-ahead interpolation technique for high-speed, five-axis machining. The proposed algorithm will not induce any contour error and the joint velocity is allowed to a considerable value. 3.1 Machining dynamics and tool path contour constrains To implement smooth motion during the machining process, it is necessary to read and analyze a number of segments in advance. The optimal feed for each segment and the joint feed for every segment junction should be calculated according to the contour information of the segments and the machining dynamics. At the same time, the machining dynamics limit the speed and acceleration for each axis that can be realized physically during the machining. These limits obtained from experiments are called the system performance envelope representing the machining dynamics of a specified machine tool. The system performance envelope is usually simplified as a rectangle zone in realtime interpolation for alleviating calculation burden. Also, in this paper, the machining dynamics is represented by a fixed acceleration limit aΛm and a fixed velocity limit VΛm available to each axis. In the following, the joint feed is calculated according to the machining dynamics and the analysis of the segments contour. the velocity of each axis should be constrained in the acceleration limits of each axis.  1  Ni  V ViΛ ð8Þ  aΛm T iþ1Λ 4. Two adjacent joint feeds Vi, Vi+1 are constrained by the linear length li, the upper bound of the translational composite acceleration aim, the upper bound of the joint feed at the junction constrained by li and ai is written as follows:  1 Viþ1 1 þ Vi  Viþ1 þ 2 aim T 2aim T qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð2Viþ1 aim T Þ2 þ 8aim li 1 aim T ð9Þ the lower bound of the joint feed at the junction constrained by li and aim is written as follows  1 Viþ1 1 þ Vi  Viþ1 2 aim T 2aim T qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð2Vi þaim T Þ2 8aim li 1 aim T ð10Þ 3.2 Variable-period interpolation algorithm 3.2.1 The principle of proposed algorithm Figure 2 shows the Variable-period interpolation algorithm. In Fig. 2, Vi is the start velocity of the segment (the feed in the first interpolation period); Vi+1 is the end velocity (the feed in the last interpolation period); Vc is the command feed. Parameter t shows the total interpolation time, The duration of interpolation is t=(na+nc+nd+f )*T, where the f *T means the none integer interpolation period (last portion less than one period), 0 ≤ f ≤1. Parameter f ranges from 0 and 1. So, there will make a small remnant in the 1. The velocity of each axis VijΛ during the machining should be lower than the velocity limits VΛm: j ViΛ  VΛm ð6Þ ajiΛ 2. The acceleration of each axis during the machining should be lower than the acceleration limits aΛm: ajiΛ  aΛm ð7Þ 3. Although the feed stays constant at the junction of the segments, the velocity of each axis changes according to the contour of the linear segments. The changes of Fig. 2 The proposed algorithm 772 last interpolation period. Parameter Sf expresses this remainder length. Sf =Vi+1 *f *T. If we want to obtain an expected velocity, namely, Vi+1, it is less than one period (f *T); However, in the previous interpolation scheme, the amount of the periods must be an integer, then the velocity of the last interpolation is V ¶i+1 =f *Vi+1, which would cause big fluctuation, when f is very small. In the proposed algorithm, we adjust the length of the last interpolation period to solve the problems. Here, we called it a Variable-period interpolation algorithm. It means the length of the interpolation period is variable. The length of the last interpolation period is f *T instead of T. The detail is as follows: First: When f *T<0.5T, add the none integer interpolation period f *T to the previous interpolation. So the last interpolation period T′ range between T with 1.5*T. Secondly: When 0.5T ≤ f *T<T, regard the f *T as a signal interpolation. So the last interpolation T ′ range between 0.5*T with T. Fig. 3 The flowchart of the speed programming scheme for the proposed algorithm Int J Adv Manuf Technol (2009) 40:769–775 In conclusion, the new interpolation period T′ range between 0.5*T with 1.5*T. 3.2.2 Deciding the deceleration zone of the proposed algorithm To achieve faster machining speed and higher-quality parts, a smooth feed should be maintained throughout the motion. An efficient method for acceleration and deceleration control of servomotors always bring out the maximum operating capability of a machine equipped with servomotors, such as a robot, NC machine tool, etc. As we know, the allowable maximum acceleration (am) can generate a more efficient velocity profile if only the deceleration stage wound not be performed. For the long segment, it has the process: acceleration– constant–deceleration, but for the other ones, only acceleration–deceleration. So, it’s a pivotal factor for acceleration and Int J Adv Manuf Technol (2009) 40:769–775 773 The present project is made in consideration of the disadvantages mentioned above. Compared with the conventional algorithm, the approach is more feasible. On all accounts, we have a clear understanding of the proposed algorithm. Figure 3 shows the flowchart of the speed programming scheme for the Variable-period interpolation algorithm and describes the proposed algorithm in detail. Fig. 4 The tool path for a blade generated with UGNX3.0 4 Simulation result deceleration control to how to forecast the deceleration point for the varied distance movement. The conventional speed layout is to calculate the acceleration time, constant time, and the deceleration time in advance. Here we adopt a more flexible scheme. In this scheme we decide whether to perform the deceleration stage by comparing the Sd with Sr in every interpolation period. Here, Sd expresses the needful length decelerating from current velocity to the end velocity, and Sr expresses the remainder length of the machining segment. The characteristics equation is as follows. p ¼ SR ð11Þ Sd The interpolation performs the deceleration stage, when parameter p is negative. Sd is an important parameter in the whole speed programming. How to derrive Sd is a key point in the whole speed programming. The conventional method [11] shows as follows: Vf2 Sdown ¼ Ve2 2a þ ΔS ð12Þ Here, Vf is the allowable maximal velocity, Ve is the end velocity (or joint federate) and ΔS is the length setting aside beforehand. This scheme requires some computation. However, the method has an obvious drawback. Not every segment can perform the maximal velocity because of the length of a segment. So, we proposed an optimal scheme. In the scheme, we discussed two instances to compute Sd: In this section, we show the experimental results of the presented optimal feed interpolation algorithm for the fiveaxis path of a blade in finish machining. In the simulation experiment, the part is made of aluminum and the cutting tool is a 10-mm carbide ball-end cutter. The five-axis path of the blade generated by the UG NX3 was run through the simulation software Cimatron as described in Fig. 4. In the simulation experiment, the maximum velocity is 6,000 mm/ min, the maximum acceleration is 1,800,000 mm/min*min, and the sampling period T is 4 ms. Figures 5 and 6 show the feed profile of the same example parts simulated by the two different methods. Figure 5 shows the feed profile obtained with conventional interpolation algorithm. It performed acceleration/deceleration in each unit segment. The command feedrate is 6,000 mm/ min. From Fig. 5, we can see that the feed fluctuates severely and cannot even reach the command feedrate in some short segments. Figure 6 shows the feed profile obtained with the proposed algorithm when the command feed is 6,000 mm/min for the same path. When the command feed increases to 6,000 mm/min, the fluctuation of the feed ranges from 3 to 5% (Fig. 6). Compared with the feed profiles obtained with conventional interpolation algorithm, this feed profile is much smoother and closer to the command feed. Thus, the 1. when Vi >Vi+1: ni ¼ Vinþ1 Viþ1 aT Sdown ¼ Vinþ1  ni  T ð13Þ 1  ni  ð ni 2 1Þ  aT ð14Þ 2 when Vi >Vi+1 a: if Vin+1>Vi+1 Sdown ¼ 0 ð15Þ b: if Vin+1>Vi+1, the calculation is same with instance 1. Fig. 5 Feed profile with the conventional algorithm 774 Int J Adv Manuf Technol (2009) 40:769–775 Fig. 6 Feed profile with the proposed algorithm proposed algorithm can generate the more efficient velocity profile for the varied distance movement and cut down the machining time greatly than the traditional five-axis methods can. Figure 7 shows the propeller being machined on a fiveaxis CNC machine at Shanghai Jiao Tong University. The test is performed to validate the efficiency of the proposed algorithm. In the machining test, the five-axis machine tool, which consists of three axes of motion and two rotational axes, is used as a test bed for the CNC system with the proposed algorithm and the roughcast is aluminum. The manufacturing process includes rough machining, semifinish machining, and finish machining. The corresponding cutting tools are the 10-mm carbide flat-bottomed cutter, the 10-mm carbide ball-end cutter, and the 6-mm carbide ball-end cutter. The command feedrate changes in the range of 4,000–6,000 mm/min and the sampling period T is also 4 ms. Figure 8 represents the surface parts after finish machining based on the proposed algorithms. Compared Fig. 7 Machining of a propeller on five-axis machining tool Fig. 8 A propeller after finish machining with the conventional interpolation algorithm, the proposed algorithm reduces the machining time by one-third. The test results show that the proposed algorithm not only achieves constant speed and shortens machining time but also obtains high machining accuracy. 5 Conclusions Currently, a Look-Ahead scheme is normally used in many advanced CNC machines to overcome the disadvantage of the conventional interpolation method. It can generate a smoother feed profile on one side, but, on the other hand, it is likely to make some short linear segment not be discretized exactly in an integer interpolation period because of the nonzero start velocity of the segment. A Variable-period interpolation algorithm is proposed to solve this disadvantage of the look-ahead scheme. First, this paper analyzed two aspects of constrains in five-axis machining to generate an effective speed profile. Then, to solve the problem of the non-integral interpolation periods when the joint feed is nonzero, new constraints to the joint feed are deduced and the corresponding adjusted interpolation algorithm is presented. In this scheme, the interpolation period is not fixed. The length of the last period in each segment ranges between 0.5T and 1.5T. These constraints and algorithm ensure that each linear segment can be interpolated exactly in an integer period without any contour error. Last, we validate the scheme with a five-axis bed. 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