# Smooth Interpolation Design with Consideration of Corner Tolerance Constraints for Robotics
## 1 Introduction  
Robots are widely used in various industrial applications, such as pick and place, welding, grinding, polishing, and processing. Appropriate trajectory planning is critical in these applications to prevent extremely high acceleration in the velocity profile, thereby avoiding excessive torque output, which can cause machine vibration and damage components. An acceleration/deceleration (ACC/DEC) constraint should therefore be applied when designing interpolators to ensure smooth robot movements. An interpolator generates position inputs for the servo drives in each sampling period for a given trajectory command. Robot interpolation methods are usually divided into joint space and task space planning approaches. In the joint space planning method, inverse kinematics are used to compute moving angles for each joint. Each joint angle is then simultaneously interpolated to generate appropriate command positions to cause each joint to reach the task position. This approach is used in pick-and-place applications in which only the initial and final positions and the posture of the end-effector are relevant factors. In this approach, the continuity of the velocity and posture of the end-effector during the motion trajectory are not considered; thus, this approach may be unsuitable for grinding, welding, or polishing processes. In the task space method of planning the end-effector trajectory, both the continuity of velocity and posture rotation are considered. The trajectory of an end-effector typically comprises many line segments, and designing the velocity profile at the intersections of these line segments is the key challenge in the design of an interpolator. Several approaches have been used to achieve smooth motion along a trajectory [1], such as designing trapezoidal or bell-shaped velocity profiles by using cubic, quartic, or quintic polynomials [2]. These profiles constrain the level of acceleration and jerk, allowing the vibrations of the robot to be used for useful work [3; 4; 5; 6; 7]. [PERSON] et al. [4] proposed thata trajectory could be computed by using an envelope of tangents for small line segments; this method can describe a given path with high accuracy. [PERSON] et al. [5] combined the ACC/DEC before interpolation (ADBI) and the ACC/DEC after interpolation (ADAI) methods in a dynamic servo system to effectively reduce the contour error, but contour error is difficult to control accurately. Interpolating a robot's orientation is another challenge in trajectory planning [8; 9; 10]. The Euler angle and quaternion approaches are the most used approaches in orientation interpolation. However, the Euler angle approach can result in a singularity (gimbal lock), which can cause robots to malfunction. The quaternion approach can avoid this singularity by clearly defining the unit vector and the angle parameters [8; 9]. [PERSON] et al. [10] presented a logarithmic quaternion interpolation method with a cubic B-spline curve that guaranteed the continuity of the rotation angle, angular velocity, angular acceleration, and path. The literature has rarely considered corner smoothing control between consecutive line segments, which may lead to a poor surface finish and high cycle time. Corner smoothing interpolation methods were investigated in [11; 12; 13]; these methods were applied to three- or five-axis machine tools. [PERSON] et al. [11] and [PERSON] et al. [12] proposed similar methods for designing a finite impulse response filter to convolve the speed command, transform discontinuous speed to continuous speed, and generate a smooth corner path for both three-axis and five-axis machine tools. Other approaches have employed the B-spline [14; 15; 16; 17; 18] or nonuniform rational B-spline (NURBS) [19; 20] methods to fit small line segments, which can reduce frequent ACC/DECs during machine movement.