Feedrate Optimization with Contouring Error and Drive Constraints in 5-Axis Machining

INTRODUCTION
Three fundamental branches of industrial fabrication include subtractive machining, additive manufacturing and dimensional inspection of parts. These applications require a tool to be moved relative to a workpiece with multiple axes without violating the part accuracy. However, the servo tracking errors of the drives lead to contouring errors along the tool path which need to be maintained within the tolerance of the part.   Tracking error is the difference in position of the individual axes in time, while contouring error is the point of closest approach between the actual tool position and the reference trajectory.
The importance of a high servo drive bandwidth on tracking performance was first described by Pritschow [1]. Tomizuka [2] employed a ‘Zero Phase Error Tracking Control’ (ZPETC) algorithm to cancel the closed loop poles and zeros to ensure zero phase error. However, ZPETC alone is not robust to plant variation and modelling errors. Koren [3] directly improved axis tracking error and thereby reduced the contouring error with a two-axis coupled feedback controller. More complex controllers may include both feedforward and feedback methods. Erkorkmaz and Altintas [4] worked on a systematic approach for designing a control law which minimizes tracking error in each axis. They used Kalman filter to estimate position, velocity and disturbance, a pole placement controller is implemented to cancel out disturbance, and a ZPETC is implemented to increase the tracking bandwidth. Although tracking error is linked to contouring error, there are cases where small axis tracking error can lead to large contouring error depending on the kinematics of the machine [5], and thus reducing tracking error alone is not sufficient to guarantee contouring error reduction.
The contouring errors are also compensated by modifying the reference toolpath trajectory. Erkorkmaz et al. [6] increased the reference toolpath trajectory spline control point density at low radius corners such that the error between expected and actual toolpaths is reduced. Zhang et al. [7] predicted contouring errors prior to the axis motion control loops and compensated for these errors by modifying axis position commands prior to motion. An online contouring error compensation method was developed by Khoshdarregi et al. [8] where an input shaper was superimposed on the reference input, thereby removing harmonic content causing undesirable vibrations. They implemented a similar compensation strategy to [7] using a different contouring error model which allowed for the contouring error to be deterministically calculated within the chosen sampling time.
Contouring error may also be compensated by estimating the contouring errors and modifying axis drive commands in a feedback controller. Altintas and Sencer [9] developed an analytical contouring error model which takes workpiece coordinate frame (WCF) tracking error and projects it onto the normal direction of the reference trajectory. This model was used in a sliding mode controller which is designed to minimize the calculated contouring error. Although these control and compensation strategies are successful at reducing contouring error, they do not address the problem of reducing cycle time.
Increasing feedrate, the tangential velocity of the tool along the prescribed toolpath, leads to a reduction in cycle time of a machining operation on the machine tool. However, high feedrate can lead to drive saturation which may excite structural modes, distort the tool path and reduce the surface quality of the part. Early work on feedrate optimization was done by Bobrow et al. [10] and Shin and McKay [11] who formulated a time-optimal feed profile, limited by torque constraints for the movement of a robotic manipulator along a defined toolpath. They reduced the multi-degree of freedom problem to a single dimension by parameterizing displacement along the path as a single dimension . The feedrate, , is maximized while ensuring that transformed actuator constraints in the  - plane are not violated. Schiller and Lu [12], [13],  Pfeiffer and Johanni [14], Slotine and Yang [15]    considered a bang-bang style optimization to improve the efficiency of the algorithms proposed in [10], [11]. However, the second time derivative of path displacement  flips between maximum acceleration and maximum deceleration at switching points in bang-bang optimization. Therefore, the results of this optimization strategy is not physically realizable due to drive dynamics and can cause undesirable structural vibrations.
Work by Shiller [16] solved the time-optimization problem using the Pontryagin maximum principle which resulted in smooth optimal control. The addition of jerk limits also eliminates discontinuities in acceleration that lead to structural vibrations. Piazzi and Visoli [17] used cubic splines to ensure continuous position, velocity, and acceleration under their respective constraints. Altintas and Erkorkmaz [18] developed a feedrate optimization algorithm which minimizes cycle time with velocity, torque, and jerk limits to avoid drive saturation. Sencer et al. [19] extended this work to 5-axis machines.
Many 2-axis feedrate scheduling methods have been developed which reduce cycle time without compromising part accuracy. Work by Dong and Stori [20], Lin et al. [21], Dong et al. [22], Jia et al., and Chen et al. [23] presented feedrate scheduling strategies which maximize feedrate while ensuring 2-axis contouring error limits are respected. Although these methods produce promising results in restricting contouring error within limits while minimizing cycle time, they can only be applied to two-axis profiles. Due to the increasing demand of complex freeform surface parts, a 5-axis feedrate scheduling strategy limited by contouring error would be a powerful tool, applicable in a wide-range of industries. Most recently, Yang et al. [24] developed a 5-axis feedrate scheduling algorithm which limits contouring errors under cutting load disturbances but ignored the contribution of tracking errors.