# Smooth Interpolation Design with Consideration of Corner Tolerance Constraints for Robotics
## 1 Introduction
To avoid frequent ACC/DEC that affects the machining efficiency and causes vibration of the machine, thus damaging the surface quality, employing the B-spline method to fit line segments prevents this issue [14; 15]. In addition, [PERSON] et al. [16] and [PERSON] et al. [17] presented a real-time lookahead interpolation methodology involving a B-spline scheme for short line segments. Because of the flexibility of NURBS curves, the methodology has been applied to fit any irregularly shaped curves in robotics applications [19; 20]. However, the Bezier, B-spline, and NURBS curves developed for applications in machine tools are more complicated and time consuming, which might hinder them from being widely used in robot computer-aided manufacturing systems. Another approach involves designing a finite impulse response (FIR) filter [21; 22; 23; 24] to convolve the velocity profile and achieve corner smoothing. This method provides a one-step solution for corner transition. [PERSON] et al. [23] proposed a one-step corner transition solution that reduces the influence of corner errors by adjusting the feed rate. However, the FIR filtering method involves setting a time constant of the FIR filter to limit acceleration and jerk. A larger time constant results in a longer cycle time and an increase in corner tolerance. In this study, a novel velocity profile overlap (VPO) algorithm for a robot interpolator is presented. The algorithm employs the S-shaped profile approach to design the velocity profile of line segments, and then overlaps each segment using overlapping time. The proposed algorithm requires only adjusting the overlapping time between the two velocity profiles to ensure that the kinematic constraints, velocity continuity, and corner smoothing criteria can be achieved. The approach is very easy to implement. Furthermore, the proposed algorithm can individually specify corner tolerances at each corner, which has not been reported in the current literature. The experimental results indicate that the reduction of cycle time, tracking error, and contour error outperforms those of other approaches. This paper is organized into six sections. Section 2 introduces the S-shaped profile used to generate a trajectory line segment and the orientation of the robot in the task space. In Section 3, the VPO algorithm is introduced that can ensure that velocity is continuous between line segments and that the resulting corners are smooth. Moreover, corner trajectory equations were derived to establish relationships among the overlapping time, machining efficiency, and trajectory accuracy. A corner tolerance constraint (CTC) equation was then derived, as described in Section 4, to enable recalculating the overlapping time and satisfy the corner tolerance constraint. Section 5 presents the results of simulations and experiments conducted on the HIWIN RT605 robot manipulator to validate the effectiveness of the VPO method. Finally, the VPO method was compared with the ADAI method [5] to demonstrate its advantages. The conclusions are presented in Section 6.