Research On Feedrate Scheduling And Contour Error Pre-compensation For Five-axis Machining Based On Constrained Linear Optimization

Complex sculptured surface parts,such as integrated blisk,blade and mould,have been widely used in the fields of aerospace,energy and power,etc.These critical parts working in unconventional condition have strict requirements on surface quality,machining efficiency and accuracy.Five-axis CNC(Computer Numerical Control)milling is an important technique for production of these complex surface parts.However,due to the limited bandwidth of the feed servo systems and the dynamic mismatch among all involved drives,etc,if the feedrate parameter of five-axis machining is improperly selected,it is very prone to cause excessive contour error,making it more difficult to guarantee the machining accuracy.And,it is also likely to cause the abrupt acceleration change of cutter or machine drives when motion direction reverses,thus inducing milling instability,surface quality degradation and other problems.At present,for assurance of machining accuracy,the five-axis feedrate scheduling in practical production is mainly dependent on the experience of machinist.Conservative feedrate selection not only leads to the prolongation of the product manufacturing cycle,but also greatly restrains the production capacity of the five-axis machines.Due to the fact that the geometric and kinematic constraints of five-axis feedrate scheduling is highly nonlinear and coupled,most existing researches employ the nonlinear optimization method to cope with these constraints,which has low computation efficiency and does not consider the contour error constraint.In addition,the research of five-axis contour error pre-compensation is also limited.Therefore,this paper carries out the research on feedrate scheduling and contour error pre-compensation for five-axis machining based on constrained linear optimization,aiming at maximizing the production efficiency while simultaneously satisfying the accuracy requirement of machined parts.The specific research contents are as follows:(1)A parameter identification method considering the influence of friction and gravity is proposed for feed servo system.Firstly,based on the theory of rigid body dynamics,the mathematical model of feed servo system is established,and the position-dependent property between the gravity of drive mechanism and the driving torque of servo system is clarified.By applying the constant feed velocity and acceleration command input,identification equations for the equivalent moment of inertial and the nonlinear friction parameters of the servo system are derived.Then,using z-transform of the velocity loop error transfer function,the relation between the velocity loop error and the driving torque of servo system is established in time domain,and a general velocity loop parameter identification method is given.Finally,the unbiased estimation of the position loop parameter is achieved by using the final value theorem of Laplace transform.On this basis,the influence of system damping ratio on path contouring performance is further investigated.Simulation and experimental results show that,compared with the traditional method,the prediction accuracy of driving torque of the servo system can be improved by 53.37%via the proposed method;and the prediction error of the maximum tracking error of machine drives is also controlled to be less than 0.4%.(2)A novel estimation model of five-axis contour error based on feedrate being expressed explicitly is established.Firstly,based on the steady-state response of the feed servo system and the Jacobian matrix of the five-axis machines,a linear mapping relation of axis tracking errors from the machine coordinate system to the workpiece coordinate system is established,And,an explicit expression of tracking errors relative to the five-axis cutter location command is deduced with respect to the feedrate.Then,with the aid of the kinematic equation of shifted Frenet frame attached on the path curve,it is revealed that there exists an approximate positive linear relation between the five-axis contour error and feedrate.It is also noted that in the local area near the desired point,the Euclidean distance from the actual cutter location to the desired trajectory is a convex function with respect to path parameter and has a unique extreme point.On this basis,an accurate prediction model of tool tip and tool orientation contour error in five-axis machining is established by taking the feedrate as a parameter.Simulation and experimental results show that,the contour errors predicted by the proposed method have good agreement with the measured ones from both shapes and magnitudes,where the prediction errors of the maximum tool tip contour error and tool orientation contour error are less than 6.0%.(3)A five-axis feedrate scheduling method based on piecewise constrained linear optimization was proposed.Considering the position-dependent property among axis displacement,tool path and feedrate,B-spline parametric description of the feedrate curve is given at first,through which the impact of time dimension and five-axis kinematics models on feedrate scheduling is removed.Then,using an inequality scaling strategy,linearized expressions of the constraint conditions in terms of chord error,contour error,drive characteristics of machine and kinematic characteristics of cutter,with regard to the control points of the feedrate spline curve,are given.Using the piecewise linear programming strategy,a multi-constrained adaptive feedrate optimization model and its corresponding solution method are proposed.Finally,by employing the prediction-correction scheme,a high-precision parameter interpolation algorithm based on Steffensen accelerated iteration is proposed,which not only can eliminate the feedrate fluctuation of classical Taylor’s expansion algorithm,but also fulfill the requirement of path interpolation accuracy and real-time performance.Simulation and experimental results show that,compared with the traditional constant feedrate modulation technology,the proposed method can reduce the contour error by 36.36%.Compared with the existing numerical optimization methods,the proposed method can effectively solve the problems of low computational efficiency,poor robustness and constraints violation when dealing with the massive data operation,and perform better in controlling the nonlinear constraints.(4)A contour error pre-compensation method based on minimum correction of the sequence of interpolation commands is proposed.Firstly,by analyzing the transient response of the servo system to the typical input signal,an analytical relation between the tracking error of machine drives and the sequence of interpolation commands is built.Meanwhile,a linear recursive tracking error prediction model is also presented.On this basis,the impact of small correction of interpolation commands,which is conducted in workpiece coordinate system,on the tracking error of machine drives is investigated.Then,considering the influence of tool orientation contour error compensation on the tool tip contour error compensation,a quadratic programming model,which takes the minimum correction of the sequence of interpolation commands as the optimization objective and the zero contouring error as the constraint condition,is proposed.Finally,by introducing Lagrange Multipliers,the complex nonlinear contour error pre-compensation is converted into a simple problem of solving linear equations,and the tool tip contour error and tool orientation contour error are compensated synchronously.Simulation and experimental results show that,compared with the uncompensated condition,the maximum tool tip contour error and tool orientation contour error are reduced by 86.75%and 70.34%,respectively,and significant improvement in path contouring accuracy is achieved.