# Smooth Interpolation Design with Consideration of Corner Tolerance Constraints for Robotics
## 3 Corner Smoothing  
In addition to single-line segment motion in positioning applications, corner interpolation must also be considered for tracking multi-line segment motion in polishing and grinding applications. Corner smoothing techniques can be employed to improve machining efficiency and reduce vibration at junctions. The motion of end-effector translation and orientation should be designed to ensure smoothness and continuity throughout the entire trajectory. Conventional methods for generating a smooth trajectory without excessive ACC/DEC can be categorized as ADBI and ADAI methods [5]. The ADBI method constrains the acceleration and the jerk of the trajectory; this prevents specifying the corner tolerance. However, the ADBI method can cause velocity discontinuities at block junctions, which can result in vibration. As shown in Figure 5, the feed rate (tangential velocity) is distributed across \(x\)- and \(y\)-axes in accordance with the direction of movement. The axis velocities \(V_{x}\) and \(V_{y}\) at the junction are not continuous. The ADAI method, in which digital convolution is used eliminate velocity discontinuities on each axis, can resolve this problem (Figure 6). Corner tolerance occurs because the velocities of blocks N\({}_{1}\) and N\({}_{2}\) overlap. Digital convolution is performed using  
Figure 4: Planning result of a single program command (**a**) Translation. (**b**) Orientation. Figure 5: ADBI method. Equation (20), where \(N=\tau/T_{s}\), \(T_{s}\) represents the sample time and \(\tau\) represents the time constant of the ACC/DEC. The input signal \(x(kT_{s})\) represents a velocity command. \[y(kT_{s})=\frac{1}{N}\sum_{i=0}^{N-1}x[(k-i)T_{s}] \tag{20}\]  
Although the ADAI method is commonly used for interpolators, its applicability is limited because it can only be applied to the entire trajectory; the tolerance for each corner cannot be specified. To achieve both corner smoothing and a specified tolerance function, this paper proposes a VPO algorithm that calculates the overlapping time of line segments to produce blended S-shaped ACC/DEC profiles. This approach improves cycle times and enables the trajectory to be systematically designed to meet different accuracy requirements. An example of the overlapping time of line segments is presented in Figure 7, which depicts two identical blocks with velocity blending. First, blocks 1 and 2 are planned using S-shaped ACC/DEC. Block 2 is then blended into block 1, and \(T_{ol}\) is the overlapped area of the two blocks; this represents as overlapping time. To calculate the overlapping time, the _OVLP_ parameter is defined as a percentage. If _OVLP_ is 100%, the overlapping time is \(2T_{rr}\), and the junction of the two blocks occurs exactly at half of \(V_{m}\). Therefore, \(2V_{c}\) is equal to \(V_{m}\). The relationship between the _OVLP_ parameter and the velocity can be expressed as follows:  
\[OVLP=\frac{2V_{c}}{V_{m}}\times 100\% \tag{21}\]  
where \(V_{c}\) is the velocity junction between the two blocks. The relationship between \(V_{c}\) and the overlapping time (\(T_{ol}\)) can be calculated using the \(A\)-\(T\) diagram in Figure 7. For an S-shaped velocity profile, \(A_{m}=V_{m}/T_{rr}\), where \(A_{m}\) represents the maximum acceleration and \(T_{rr}\) is equal to the time of acceleration from 0 to \(A_{m}\) within the ACC/DEC period. \(A_{c}\) represents \(V_{c}\) at the junction and can be calculated using Equation (22). \[A_{c}=\frac{A_{m}T_{ol}}{2T_{rr}}=\frac{V_{m}T_{ol}}{2T_{rr}^{2}} \tag{22}\]  
The parameter \(V_{c}\) can be calculated by integrating the area of the \(A\)-\(T\) diagram and is given as follows:  
\[V_{c}=\frac{A_{c}T_{ol}}{4} \tag{23}\]  
Figure 6: The ADAI method.