A corner smoothing method with feedrate blending for linear segments under geometric and kinematic constraints

Abstract

To eliminate the disadvantages of linear interpolation and geometry-based smoothing methods, the one-step corner smoothing method with feedrate blending algorithm for linear segments is proposed in this article. In the proposed method, the variable acceleration and jerk optimization method will be first adopted to determine the dynamic performance of each linear segment for feedrate scheduling, which takes the different axial limits into consideration. Then, the feedrate scheduling method based on the S-type acceleration and deceleration algorithm is implemented to generate smooth feedrate profiles. Moreover, the feedrate blending algorithm is adopted to calculate the corner time analytically within the specified corner error and thus to generate the smooth trajectory without inserting a smooth curve at adjacent linear segments. With the presented one-step corner smoothing method, a smoother and more accurate trajectory could be generated with faster machining time. The results of simulations and machining experiments demonstrate the effectiveness of the proposed method.

Keywords

Corner smoothing feedrate blending variable acceleration and jerk corner error kinematic constraints

Introduction

In modern computer-aided design (CAD) system, smooth parametric splines such as nonuniform rational basis spline (NURBS) or B-splines are commonly used to design the free-form surfaces. However, due to the difficulties in parametric interpolation, the direct parametric interpolators have not been widely used in industrial computer numerical control (CNC) systems.^1,2 Moreover, linear segments (G01) are still the most commonly used format to approximate the toolpaths in both the computer-aided manufacturing (CAM) and current industrial CNC systems.^3,4 Nevertheless, the linear segments are only G0 continuity; as a consequence, the motion has to stop at the junctions to completely avoid the discontinuities of feedrate, acceleration, and jerk,^2,5,6 which absolutely lead to longer machining time.

To achieve nonstop continuous motion, thus to improve the machining efficiency and quality, two major techniques—that is, global smoothing and corner smoothing techniques—have been proposed.³ The global smoothing means that a portion of the linear toolpath is fitted to generate the continuous parametric toolpath. The corner smoothing method means that a smooth curve is inserted at the corner of two adjacent linear segments to achieve nonstop continuous motion. Generally, both two methods mentioned above contain two steps, curve fitting followed by feedrate scheduling. First, the linear segments will be smoothed by a parametric curve, such as Bezier,^3,7 B-spline,^4,8,9 or NURBS,¹⁰ and the toolpath becomes parametric curves or a mixture of linear segments and parametric curves. Then, the feedrate scheduling will be applied to generate a smooth feedrate profile for real-time interpolation. However, it is inefficient to separate the smoothing method into curve fitting and feedrate scheduling.² In addition, the feedrate fluctuation is unavoidable due to the discrepancy between the spline parameter and the arc length¹¹ in parametric interpolation. Besides, it is time-consuming to accomplish the complicated computation of the parametric curve,¹² which will affect the real-time performance of CNC systems.

In order to eliminate the shortcomings for two-step smoothing methods, one-step local corner smoothing method,^{2,5,6,13–18} which blends the axial velocities directly, has become the research hotspot in the field of CNC interpolation. For instance, Huang¹³ provided a blending feedrate planning method for both linear and S-curve acceleration and deceleration algorithms, by which the feedrate at both ends of two blocks will automatically be overridden with consideration of maximum corner error. This method can ensure that the feedrate variation of two blocks is continuous, and the end feedrate of previous block does not decelerate to zero. However, only the corner error is considered in the above-mentioned algorithm, and the dynamic constraints of individual axis are not paid attention to. He et al.¹⁴ presented an algorithm to smooth the track and velocity by adding motion vectors of two adjacent segments. The algorithm can smooth linear segments according to the maximum corner error based on the trapezoidal velocity profile. However, only geometrical error was taken into consideration in the proposed method. Zhang et al.⁵ proposed a multi-period algorithm to improve the transition feedrate at the corner based on the linear acceleration and deceleration mode. This method utilized the maximal acceleration capabilities of the CNC machines and significantly improved the cutting efficiency with satisfying the machining accuracy. However, with linear acceleration and deceleration approach, sudden changes of acceleration at the junction of two linear segments will occur.¹⁶ Then, Zhang et al.¹⁶ developed a new multi-period turning interpolation algorithm based on the S-curve control method, which improves the feedrate at the corner by inserting a cubic curve at the junction of two segments. With the proposed method, the machining vibration was decreased, and the machining quality was improved. However, this algorithm ignores the constant acceleration phase of S-curve control algorithm, which is only suitable for some special cases. Zhang et al.¹⁵ used a one-step strategy to generate the transition trajectory based on the linear acceleration algorithm, which aims to minimize the total machining time with the maximal axial accelerations and error tolerance. However, sudden changes of acceleration still exit. Tajima and Sencer^2,17 proposed the kinematic corner smoothing method for long linear segments, which plans the jerk-limited feedrate profiles directly around the segment junctions. This method can generate a continuous feed motion with precisely controlled contouring errors of corners. Then, Tajima and Sencer⁶ and Du et al.¹⁸ came up with a new global toolpath smoothing method for short linear segments with jerk-limited feedrate profile and jounce-limited feedrate profile, respectively, which blends the axis feedrate between consecutive linear toolpaths within limited corner errors and kinematic limits of drives. Both of the two above-mentioned corner smoothing techniques can eliminate the need for two-step geometric corner smoothing methods, and provide better contouring performance. However, the proposed methods are based on the assumption that cornering motion starts and ends with identical cornering feedrate and acceleration to generate symmetric trajectories around the corners, which limits the application in one-step corner smoothing for linear segments.

Thus, in this article, the one-step corner smoothing algorithm is proposed, where the linear segments are smoothed by blending the tangential feedrate from one segment to the other, based on the S-type acceleration and deceleration algorithm. The control of the smoothing corner error, axis acceleration, and axis jerk are tackled by simple and analytical methods. The advantages of the proposed algorithm are as follows: (1) with the one-step method, the corner error is directly constrained in an analytical way, which will also determine the corner time when the blending feedrate starts; (2) variable acceleration and jerk optimizations determine the jerk and acceleration for feedrate scheduling, which can generate continuous axial acceleration and limited axial jerk in corner smoothing process; (3) compared with the two-step corner smoothing method (TCM), a faster and smoother motion can be obtained around the corner without inserting a blending curve. Besides, the curve scanning, parameter calculation, and derivative computation of the parametric curve are eliminated by the proposed algorithm, which is more convenient to apply in real-time CNC systems. The rest of this article is organized in detail as follows. In section “Corner smoothing method with feedrate blending for linear segments,” the feedrate scheduling method and the feedrate blending algorithm for corner smoothing are presented. In section “Simulation and experiment,” simulation and experiment are conducted to compare different methods with the proposed algorithm. Finally, the conclusion of this article is described in section “Conclusion.”

Corner smoothing method with feedrate blending for linear segments

The whole architecture of the proposed corner smoothing method is shown in Figure 1, which is divided into two blocks. The former offline module aims to determine the parameters of constraints in feedrate scheduling, according to the target feedrate, limited axial acceleration, axial jerk, and corner error. Then, the latter real-time module will generate the smooth S-type feedrate profiles and determine the corner time under corner error constraint, and thus calculate the commanded position and generate the smoothed trajectory at the corner.

Figure 1.

The flowchart of the proposed method.

Feedrate scheduling based on S-type acceleration and deceleration algorithm

The S-type acceleration and deceleration algorithm is widely used in modern CNC systems,^{2,6,16,17,19–22} due to its stability and low computational burden.²³ As shown in Figure 2(a), the complete S-type acceleration and deceleration profile is usually divided into seven phases,^19–22 including increasing acceleration (Phase I), constant acceleration (Phase II), decreasing acceleration (Phase III), constant feedrate (Phase IV), increasing deceleration (Phase V), constant deceleration (Phase VI), and decreasing deceleration (Phase VII). Furthermore, Phases I, II, and III are defined as the acceleration phase, meanwhile, Phases V, VI, and VII are defined as the deceleration phase.

Figure 2.

Four types of feedrate profiles: (a) type I, (b) type II, (c) type III, and (d) type IV.

The expression of the jerk, the acceleration, the velocity, and the position are summarized below

J (t) = {\begin{matrix} J_{max} & t_{0} \leq t \leq t_{1} \\ \begin{matrix} 0 \\ - J_{max} \\ 0 \\ - J_{max} \\ 0 \\ J_{max} \end{matrix} & \begin{matrix} t_{1} \leq t \leq t_{2} \\ t_{2} \leq t \leq t_{3} \\ t_{3} \leq t \leq t_{4} \\ t_{4} \leq t \leq t_{5} \\ t_{5} \leq t \leq t_{6} \\ t_{6} \leq t \leq t_{7} \end{matrix} \end{matrix}

(1)

a (t) = {\begin{matrix} \begin{matrix} J_{max} τ_{1} \\ a_{max} \\ a_{max} - J_{max} τ_{3} \\ 0 \\ - J_{max} τ_{5} \\ - a_{max} \\ - a_{max} + J_{max} τ_{7} \end{matrix} & \begin{matrix} t_{0} \leq t \leq t_{1} \\ t_{1} \leq t \leq t_{2} \\ t_{2} \leq t \leq t_{3} \\ t_{3} \leq t \leq t_{4} \\ t_{4} \leq t \leq t_{5} \\ t_{5} \leq t \leq t_{6} \\ t_{6} \leq t \leq t_{7} \end{matrix} \end{matrix}

(2)

v (t) = {\begin{matrix} v_{s} + \frac{1}{2} J_{max} τ_{1}^{2} & t_{0} \leq t \leq t_{1} \\ v_{1} + a_{max} τ_{2} & t_{1} \leq t \leq t_{2} & v_{1} = v_{s} + \frac{1}{2} J_{max} T_{1}^{2} \\ v_{2} + a_{max} τ_{3} - \frac{1}{2} J_{max} τ_{3}^{2} & t_{2} \leq t \leq t_{3} & v_{2} = v_{1} + a_{max} T_{2} \\ v_{3} & t_{3} \leq t \leq t_{4} & v_{3} = v_{t} = v_{2} + \frac{1}{2} J_{max} T_{1}^{2} \\ v_{4} - \frac{1}{2} J_{max} τ_{5}^{2} & t_{4} \leq t \leq t_{5} & v_{4} = v_{3} = F \\ v_{5} - a_{max} τ_{6} & t_{5} \leq t \leq t_{6} & v_{5} = F - \frac{1}{2} J_{max} T_{5}^{2} \\ v_{6} - a_{max} τ_{7} + \frac{1}{2} J_{max} τ_{7}^{2} & t_{6} \leq t \leq t_{7} & v_{6} = v_{5} - a_{max} T_{6} \end{matrix}

(3)

s (t) = {\begin{matrix} v_{s} τ_{1} + \frac{1}{6} J_{max} τ_{1}^{3} & t_{0} \leq t \leq t_{1} \\ s_{1} + v_{1} τ_{2} + \frac{1}{2} a_{max} τ_{2}^{2} & t_{1} \leq t \leq t_{2} & s_{1} = v_{s} T_{1} + \frac{1}{6} J_{max} T_{1}^{3} \\ s_{2} + v_{2} τ_{3} + \frac{1}{2} a_{max} τ^{2} - \frac{1}{6} J_{max} τ_{3}^{3} & t_{2} \leq t \leq t_{3} & s_{2} = s_{1} + v_{1} T_{2} + \frac{1}{2} a_{max} T_{2}^{2} \\ s_{3} + v_{3} τ_{4} & t_{3} \leq t \leq t_{4} & s_{3} = s_{2} + v_{2} T_{j} + \frac{1}{3} J_{max} T_{1}^{3} \\ s_{4} + v_{4} τ_{5} - \frac{1}{6} J_{max} τ_{5}^{3} & t_{4} \leq t \leq t_{5} & s_{4} = s_{3} + v_{3} T_{4} \\ s_{5} + v_{5} τ_{6} - \frac{1}{2} a_{max} τ_{6}^{2} & t_{5} \leq t \leq t_{6} & s_{5} = s_{4} + v_{4} T_{5} - \frac{1}{6} J_{max} T_{5}^{3} \\ s_{6} + v_{6} τ_{7} - \frac{1}{2} a_{max} τ^{2} + \frac{1}{6} J_{max} τ_{7}^{3} & t_{6} \leq t \leq t_{7} & s_{6} = s_{5} + v_{5} T_{6} - \frac{1}{2} a_{max} T_{6}^{2} \end{matrix}

(4)

In the feedrate scheduling, the maximum acceleration $(a_{max})$ and the maximum jerk $(J_{max})$ are determined according to the machine tools. The maximum jerk time $(T_{j})$ can be expressed as equation (5), which means that the maximum durations of Phases I, III, V, and VII are $T_{j}$

T_{j} = \frac{a_{max}}{J_{max}}

(5)

If the initial values of the length of the segment $(S)$ , starting feedrate $(v_{s})$ , ending feedrate $(v_{e})$ , programmed feedrate $(v_{t}),$ $a_{max} and J_{max}$ are specified. The feedrate scheduling will determine the type of feedrate profile and calculate the duration of each phase $(T_{i}, i = 0, 1, \dots, 7)$ . Then, the critical time $(t_{i}, i = 0, 1, \dots, 7)$ together with the critical feedrate $(v_{i}, i = 0, 1, \dots, 7)$ , the acceleration $(a_{i}, i = 0, 1, \dots, 7)$ , the jerk $(J_{i}, i = 0, 1, \dots, 7)$ , and the critical length $(s_{i}, i = 0, 1, \dots, 7)$ of each phase will be generated.

As mentioned above, to avoid the sudden change of acceleration, the machine usually comes to a full stop at junctions of linear segments with only position (G0) continuity, which means that in feedrate scheduling, starting feedrate and ending feedrate are equal to zero $(v_{s} = v_{e} = 0)$ . Since the maximum acceleration and the maximum jerk together with the starting and ending feedrate are the same, the scheduled feedrate profile is symmetrical. In this condition, the durations of Phases I, III, V, and VII are equal $(T_{1} = T_{3} = T_{5} = T_{7})$ , and the durations of Phases II and VI are equal $(T_{2} = T_{6})$ . However, according to the different initial conditions, the feedrate profile will be less than seven phases.²¹ Therefore, feedrate profiles could be classified into four types, under the condition that both the starting feedrate and the ending feedrate are zero, as shown in Figure 2.

The criterion lengths for type determination and the durations of phases for each forward scheduling type are listed in Tables 1 and 2, respectively. Then, according to the relationship between $v_{t} and a_{max}^{2} / J_{max} and S and s_{cl}$ , the type of feedrate profile could be determined. Furthermore, the duration of each phase is calculated, as listed in Table 2. Then, the critical time, the critical feedrate, the critical acceleration, the critical jerk, and the critical length can be obtained according to equations (1)–(4).

Table 1.

Criterion length for feedrate scheduling.

Criterion length	Detailed derivation
$s_{cl 1}$	If $v_{t} \leq \frac{a_{max}^{2}}{J_{max}}$ , let $T_{1} = T_{3} = T_{5} = T_{7} = \sqrt{\frac{v_{t}}{J_{max}}}$ , $T_{2} = T_{6} = 0$ , $s_{cl 1} = 2 (\frac{1}{6} J_{max} T_{1}^{3} + \frac{1}{2} J_{max} T_{1}^{2} T_{3} + \frac{1}{2} J_{max} T_{1} T_{3}^{2} - \frac{1}{6} J_{max} T_{3}^{3}) = 2 J_{max} T_{1}^{3}$
$s_{cl 2}$	If $v_{t} > \frac{a_{max}^{2}}{J_{max}}$ , let $T_{1} = T_{3} = T_{5} = T_{7} = T_{j} = \frac{a_{max}}{J_{max}}$ , $T_{2} = T_{6} = \frac{v_{t}}{J_{max} \cdot T_{j}} - T_{j}$ , $s_{cl 2} = 2 (s_{2} + \frac{1}{2} J_{max} T_{1}^{3} + J_{max} T_{1}^{2} T_{2} + \frac{1}{2} J_{max} T_{1}^{3} - \frac{1}{6} J_{max} T_{1}^{3})$ , where $s_{2} = \frac{1}{6} J_{max} T_{1}^{3} + \frac{1}{2} J_{max} T_{1}^{2} T_{2} + \frac{1}{2} J_{max} T_{1} T_{2}^{2}$ . Therefore, $s_{cl 2} = \frac{3}{2} J_{max} T_{1}^{2} T_{2} + J_{max} T_{1} T_{2}^{2} + 2 J_{max} T_{1}^{3}$
$s_{cl 3}$	If $v_{t} > \frac{a_{max}^{2}}{J_{max}}$ , let $T_{1} = T_{3} = T_{5} = T_{7} = T_{j} = \frac{a_{max}}{J_{max}}$ , $T_{2} = T_{6} = 0$ , $s_{cl 3} = 2 (\frac{1}{6} J_{max} T_{1}^{3} + \frac{1}{2} J_{max} T_{1}^{2} T_{3} + \frac{1}{2} J_{max} T_{1} T_{3}^{2} - \frac{1}{6} J_{max} T_{3}^{3}) = 2 J_{max} T_{1}^{3}$

Table 2.

Calculation of feedrate scheduling.

Types	Conditions	Duration of each phase
Type I	$v_{t} \leq a_{max}^{2} / J_{max}$ , $S > s_{cl 2}$	$T_{1} = T_{3} = T_{5} = T_{7} = T_{j} = \frac{a_{max}}{J_{max}}$ , $T_{2} = T_{6} = \frac{v_{t}}{J_{max} \cdot T_{j}} - T_{j}$ , and $T_{4} = \frac{S - s_{cl 2}}{v_{t}}$
Type II	$v_{t} \leq a_{max}^{2} / J_{max}$ , $s_{cl 3} \leq S \leq s_{cl 2}$	$T_{1} = T_{3} = T_{5} = T_{7} = T_{j} = \frac{a_{max}}{J_{max}}$ and $T_{4} = 0$ . Solving. $\frac{3}{2} J_{max} T_{1}^{2} T_{2} + J_{max} T_{1} T_{2}^{2} + 2 J_{max} T_{1}^{3} = S$ . $T_{2} = T_{4} = \frac{- B + \sqrt{B^{2} - 4 AC}}{2 A}$ , where $A = J_{max} T_{1}$ , $B = \frac{3}{2} J_{max} T_{1}^{2}$ , and $C = 2 J_{max} T_{1}^{3} - S$
Type III	$v_{t} \leq a_{max}^{2} / J_{max}$ , $S > s_{cl 1}$	$T_{1} = T_{3} = T_{5} = T_{7} = \sqrt{\frac{v_{t}}{J_{max}}}$ , $T_{2} = T_{6} = 0$ , and $T_{4} = \frac{S - s_{cl 1}}{v_{t}}$
Type IV	$v_{t} \leq a_{max}^{2} / J_{max}$ , $S \leq s_{cl 1}$ Or $v_{t} \leq a_{max}^{2} / J_{max}$ , $S \leq s_{cl 3}$	Solving $2 J_{max} T_{1}^{3} = S$ . $T_{1} = T_{3} = T_{5} = T_{7} = \sqrt[3]{\frac{S}{2 J_{max}}}$ and $T_{2} = T_{4} = T_{6} = 0$

Feedrate blending algorithm for linear segments

Based on the S-type acceleration and deceleration algorithm, the feedrate profile of each segment can be generated. As shown in Figure 3(a), there are two consecutive segments with different feedrate profiles. To improve the machining efficiency, when the first segment is decelerating and its feedrate is not zero, then the acceleration of second segment starts. Thus, the feedrate of two segments is blended in the time period $[t_{s}, t_{e}]$ , which is defined as the corner time $(T_{b})$ . This method is called the feedrate blending algorithm. Figure 3(b)–(d) depicts different types of feedrate blending. To preferably realize feedrate blending for linear segments, it should be noted that the corner time is limited to no longer than the duration of deceleration phase of first segment or the duration of acceleration phase of second segment. More importantly, in the blending process, the corner error, axial dynamic constraints should be considered.

Figure 3.

Different cases of feedrate blending: (a) case I, (b) case II, (c) case III, and (d) case IV.

Geometrical error control

As shown in Figure 4(a), the corner motion starts from point $P_{a}$ and ends at point $P_{b}$ . The two axes move at the same time in the process, so that geometrical errors occur, where $P_{e} P_{2}$ is defined as the limited corner error $(ε_{max})$ , and $P_{a} P_{2} and P_{2} P_{b}$ are defined as the terminal errors $(P_{a} P_{2} = l_{a} and P_{2} P_{b} = l_{b})$ of segments $P_{1} P_{2} and P_{2} P_{3}$ , respectively. In addition, the terminal error should be within the half of the linear segments to avoid the overlapping, which means that the corner time $(T_{b})$ cannot exceed the half of feedrate scheduling time of the linear segment.

Figure 4.

The corner error and terminal errors: (a) the definition of corner error and terminal error, (b) the corner error when the feedrate profile is symmetric, (c) the corner error when the feedrate is asymmetric, and (d) the relation of corner error and terminal error.

As shown in Figure 4(a), when the feedrate profiles are identical, the smoothed corner is symmetric around the corner bisector; hence, the maximum corner error $(| | P_{2} P_{b} | | = ε_{max})$ occurs in the middle of the cornering trajectory at $t = T_{b} / 2$ .^2,6 However, owing to the asymmetry of corner trajectory by the proposed method, the point $(P_{0.5})$ at $t = T_{b} / 2$ may not coincide with point $P_{e}$ , as shown in Figure 4(b) and (c). Furthermore, it should be noted that if $P_{0.5} and P_{e}$ do not coincide, the distance (e) of $| | P_{2} P_{0.5} | |$ must be longer than $ε_{max}$ , thus the following expression can be obtained

e \geq ε_{max}

(6)

As a result, as long as the distance of $| | P_{2} P_{0.5} | |$ is controlled within the limited corner error, the actual corner error will be controlled.

Supposing that $P_{0.5 a} and P_{0.5 b}$ are the points of segments $P_{a} P_{2} and P_{2} P_{b}$ at $t = T_{b} / 2$ , respectively, and the distances of $| | P_{0.5 a} P_{2} | | and | | P_{2} P_{0.5 b} | |$ are $l_{0.5 a} and l_{0.5 b}$ , respectively, as shown in Figure 4(c), then the relation between $l_{0.5 a}, l_{0.5 b}, and e$ can be expressed as

l_{0.5 a}^{2} + l_{0.5 b}^{2} - 2 l_{0.5 a} l_{0.5 b} \cos (π - θ) = e^{2}

(7)

According to the S-type acceleration and deceleration algorithm, introduced in section “Feedrate scheduling based on S-type acceleration and deceleration algorithm,” $l_{0.5 a} and l_{0.5 b}$ can be expressed as

\begin{matrix} l_{0.5 a} = s_{k} + v_{k} (t - T_{k}) + \frac{1}{2} a_{k} {(t - T_{k})}^{2} \\ + \frac{1}{6} J_{k} {(t - T_{k})}^{3} v_{k} \\ {l_{0.5}}_{b} = s_{j} + v_{j} (t - T_{j}) + \frac{1}{2} a_{i} {(t - T_{j})}^{2} + \frac{1}{6} J_{j} {(t - T_{j})}^{3} v_{j} \end{matrix}

(8)

where $t$ is half of the corner time; $k and j$ represent the critical state of first and second segments, respectively.

The values of $k and j$ can be determined by the following equations

\begin{matrix} s_{k}^{2} + s_{j}^{2} - 2 s_{k} s_{j} \cos (π - θ) > ε_{max}^{2} \\ s_{k + 1}^{2} + s_{j + 1}^{2} - 2 s_{k + 1} s_{j + 1} \cos (π - θ) \leq ε_{max}^{2} \end{matrix}

(9)

Thus, the following expression can be calculated by combining equations (7) and (8)

Ұ - Y 1 \cdot t^{6} + Ұ - Y 2 \cdot t^{5} + Ұ - Y 3 \cdot t^{4} + Ұ - Y 4 \cdot t^{3} + Ұ - Y 5 \cdot t^{2} + Ұ - Y 6 \cdot t + Ұ - Y 7 = 0

(10)

where $Ұ - Y 1, \dots, Ұ - Y 7$ are the coefficients of $t^{6}, \dots, t, t^{0}$ , and the detailed expression could be found in Appendix 2.

The time $t$ can be computed by solving equation (10) with the Newton iteration method. Then, the corner time $T_{b}$ can be decided by

T_{b} = {\begin{matrix} 2 t, & if 2 t \leq T_{3, \min} \\ T_{3, \min}, & otherwise \end{matrix}

(11)

where $T_{3, \min}$ is the minimum value in critical times $(t_{3})$ of the first and second segments.

Feedrate control in the blending process

As shown in Figure 5(a), when the tool moves along the blending trajectory, the feedrate is defined as the norm of the addition of two velocity vectors, which can be expressed as equation (12). According to the algorithm of vector addition, it can be demonstrated that the feedrate reaches the maximum value when two velocity vectors are in the same direction, as depicted in Figure 5(b). Therefore, as long as the feedrate, generated by two velocity vectors with identical direction, could be controlled within the specified limits, the actual feedrate in the blending process would not exceed the allowable value

\begin{matrix} \vec{v} = {\vec{v}}_{1} + {\vec{v}}_{2} \\ ‖ \vec{v} ‖ = \sqrt{‖ {\vec{v}}_{1} ‖ + ‖ {\vec{v}}_{2} ‖ + 2 ‖ {\vec{v}}_{1} ‖ ‖ {\vec{v}}_{2} ‖ \cos (π - θ)} \end{matrix}

(12)

Figure 5.

Vector addition of the feedrate: (a) the feedrate along the toolpath and (b) different situations of the vector addition.

As mentioned above that the corner time is limited to no longer than the duration of deceleration phase of first segment or the duration of acceleration phase of second segment, therefore, the addition of the acceleration with the same direction will be negative first and then it becomes positive, as shown in Figure 6, which means that the feedrate, generated by two velocity vectors with identical direction, will decelerate first and then accelerate. As a result, the maximum feedrate in the blending process will occur at the start or the end. What’s more, since the corner time is limited to no longer than the durations of the first deceleration phase or the second acceleration phase, the feedrate will be controlled within the limited feedrate (constant feedrate phase), as shown in Figure 6. Therefore, the corner feedrate will be constrained within the programmed value by the proposed feedrate blending method (FBM).

Figure 6.

The addition of the acceleration with the same direction: (a) case I, (b) case II, (c) case III, and (d) case IV.

Variable acceleration and jerk in feedrate scheduling

For machining process, the machining precision is influenced not only by the feedrate but also by the limited drive ability of the machine tool, including the axial acceleration and jerk.¹ Therefore, to avoid violating the capacities of each individual drive, the drive constraints in feedrate scheduling should be limited.

As Figure 7 depicts, the axial feedrate of the addition of two velocity vectors can be expressed as

\begin{matrix} {\vec{v}}_{x} = {\vec{v}}_{x 1} + {\vec{v}}_{x 2} \\ {\vec{v}}_{y} = {\vec{v}}_{y 1} + {\vec{v}}_{y 2} \end{matrix}

(13)

Figure 7.

The addition of the axis feedrate.

Then, the following equation can be obtained by the algorithm of vector addition

\begin{matrix} ‖ {\vec{v}}_{x} ‖ = ‖ {\vec{v}}_{x 1} ‖ + ‖ {\vec{v}}_{x 2} ‖ \\ ‖ {\vec{v}}_{y} ‖ = ‖ {\vec{v}}_{y 1} ‖ + ‖ {\vec{v}}_{y 2} ‖ \end{matrix}

(14)

As a result, the acceleration and the jerk of axes x and y in the blending process can be expressed

\begin{matrix} a_{x} = a_{x 1} + a_{x 2} = a_{1} \cos (θ_{1}) + a_{2} \cos (θ_{2}) \\ a_{y} = a_{y 1} + a_{y 2} = a_{1} \sin (θ_{1}) + a_{2} \sin (θ_{2}) \\ J_{x} = J_{x 1} + J_{x 2} = J_{1} \cos (θ_{1}) + J_{2} \cos (θ_{2}) \\ J_{y} = J_{y 1} + J_{y 2} = J_{1} \sin (θ_{1}) + J_{2} \sin (θ_{2}) \end{matrix}

(15)

where $a_{1}, a_{2}, J_{1}, and J_{2}$ are the tangential acceleration and jerk of the first and second segments, respectively. $a_{x 1}, a_{x 2}, a_{y 1}, and a_{y 2}$ are the acceleration of axes x and y of the first and second segments, respectively. $J_{x 1}, J_{x 2}, J_{y 1}, and J_{y 2}$ are the jerk of axes x and y of the first and second segments, respectively.

Acceleration selection for feedrate scheduling and axial limitation

The acceleration and the jerk of axes x and y, as shown in Figure 8, correspond to the feedrate profile shown in Figure 6(a). The possible situations of feedrate blending are presented in Figure 8. In the feedrate blending process, $a_{1} \leq 0 and a_{2} \geq 0$ . Thus, $a_{x 1} \leq 0 and a_{x 2} \geq 0$ ; hence, the absolute value of $a_{x}$ will not exceed the maximum one between the absolute values of $a_{x 1} and a_{x 2}$ . As a result, the axial acceleration could be limited within the constraint, as long as the maximum tangential acceleration $(a_{1} and a_{2})$ follows equations (16) and (17)

\begin{matrix} a_{x_{max}} \leq a_{1} | \cos (θ_{1}) | \\ a_{y_{max}} \leq a_{1} | \sin (θ_{1}) | \end{matrix}

(16)

\begin{matrix} a_{x_{max}} \leq a_{2} | \cos (θ_{2}) | \\ a_{y_{max}} \leq a_{2} | \sin (θ_{2}) | \end{matrix}

(17)

Figure 8.

Different cases of additions of axis acceleration and jerk: (a) case I, (b) case II, (c) case III, and (d) case IV.

Therefore, the tangential acceleration amplitude used in feedrate scheduling can be determined by equation (18)

\begin{matrix} a_{1} = min (\frac{a_{x_{max}}}{| \cos (θ_{1}) |}, \frac{a_{y_{max}}}{| \sin (θ_{1}) |}) \\ a_{2} = min (\frac{a_{x_{max}}}{| \cos (θ_{2}) |}, \frac{a_{y_{max}}}{| \sin (θ_{2}) |}) \end{matrix}

(18)

where $a_{x_{max}} and a_{y_{max}}$ are the limited acceleration of axis x and axis y.

Optimal jerk determination to minimize the acceleration time

Figure 8(b) indicates the possible maximum axial jerk; therefore, if the tangential jerk $(J_{1} and J_{2})$ can satisfy equation (19), the axial jerk will not exceed the limits. It should be noted that if the current segment is not the first segment or the last segment, the calculation of the tangential jerk should consider both the former and the latter segments

\begin{matrix} min (\frac{J_{x_{max}}}{2}, \frac{J_{y_{max}}}{2}) \leq J_{1} \leq \frac{J_{x_{max}}}{| \cos (θ_{1}) |} \\ min (\frac{J_{x_{max}}}{2}, \frac{J_{y_{max}}}{2}) \leq J_{1} \leq \frac{J_{y_{max}}}{| \sin (θ_{1}) |} \\ min (\frac{J_{x_{max}}}{2}, \frac{J_{y_{max}}}{2}) \leq J_{2} \leq \frac{J_{x_{max}}}{| \cos (θ_{2}) |} \\ min (\frac{J_{x_{max}}}{2}, \frac{J_{y_{max}}}{2}) \leq J_{2} \leq \frac{J_{y_{max}}}{| \sin (θ_{2}) |} \\ J_{1} | \cos (θ_{1}) | + J_{2} | \cos (θ_{2}) | \leq J_{x_{max}} \\ J_{1} | \sin (θ_{1}) | + J_{2} | \sin (θ_{2}) | \leq J_{y_{max}} \end{matrix}

(19)

where $J_{x_{max}} and J_{y_{max}}$ are the limited jerk of axis x and axis y.

To optimize the machine performance, one optimization index is considered, which is to minimize jerk time as depicted in equation (5). For the problem of minimizing the sum of the jerk time $(T_{j min})$ , the objective function is

T_{j min} = min (T_{j 1} + T_{j 2})

(20)

where $T_{j 1} and T_{j 2}$ are the maximum jerk time of first and second segments, respectively.

Based on equations (5) and (18), the optimization function can be written as

T_{j min} = min (\frac{a_{1}}{J_{1}} + \frac{a_{2}}{J_{2}})

(21)

where $J_{1} and J_{2}$ are limited by the jerk constraints as equation (19).

As shown in Figure 9, the feasible search region of $J_{1} and J_{2}$ is surrounded by a polygon filled with green color. Then, the optimal values of $J_{1} and J_{2}$ can be determined on an edge of the polygon. As a consequence, the tangential acceleration and jerk used in feedrate scheduling can be calculated by equations (18), (19), and (21), which can keep not only the axial dynamics within the limits but also the machining efficiency.

Figure 9.

The optimization of jerks.

Simulation and experiment

In this section, the simulations and the experiments are conducted to evaluate the performance of the proposed FBM. Meanwhile, a TCM, which fits cubic B-spline around the corners,⁹ is also employed for comparison with the same S-shape feedrate profile. Besides, the parameters used in this section are listed in Table 3.

Table 3.

The parameters used in the experiments.

Methods	Parameters	Values	Units
FBM and TCM	Interpolation time	0.002	s
FBM and TCM	Programmed feedrate	50	mm/s
FBM	Jerk of axis x	15,000	mm/s³
	Jerk of axis y	20,000	mm/s³
	Acceleration of axis x	1200	mm/s²
	Acceleration of axis y	1500	mm/s²
TCM	Tangential jerk	20,000	mm/s³
TCM	Tangential acceleration	1500	mm/s²

FBM: feedrate blending method; TCM: two-step corner smoothing method.

Simulation

In this section, a simple toolpath with different angles is employed, as depicted in Figure 11. The corner error is set to $ε = 0.1 mm$ .

For the proposed FBM, the final acceleration and jerk of each segment used in feedrate scheduling process are listed in Table 4, which can limit the axis jerk and acceleration in the feedrate blending by the algorithms introduced in section “Feedrate blending algorithm for linear segments.” The corner times determined by the corner error are presented in Table 5. It should be noted that there is no need to schedule the feedrate again after the corner time is obtained. When the first segment is decelerating to the start of the corner time, meanwhile the acceleration of second segment will start according to the original scheduled feedrate profiles. Thus, the blending feedrate of the two segments is generated, as depicted in Figure 10.

Table 4.

The actual acceleration and jerk in feedrate planning of the proposed method.

Segment	Start point (mm)	End point (mm)	Acceleration (mm/s²)	Jerk (mm/s³)
1	(0, 0)	(−10, 0)	1200	15,000
2	(−10, 0)	(−10, 10)	1500	9935
3	(−10, 10)	(−6.894, 21.591)	1553	10,350
4	(−6.894, 21.591)	(−3.788, 10)	1553	10,355
5	(−3.788, 10)	(1.079, 10)	1200	8110
6	(1.079, 10)	(5.540, 17.725)	1732	11,545
7	(5.540, 17.725)	(10, 10)	1732	11,545
8	(10, 10)	(10, 0)	1500	9980
9	(10, 0)	(0, 0)	1200	15,000

Table 5.

The corner time of each corner.

Corner	1	2	3	4	5	6	7	8
Corner time (ms)	64.376	121.447	62.145	74.993	77.612	62.149	94.036	64.346

Figure 10.

The process of feedrate blending.

For the TCM, the smoothed cubic B-splines at the corners are first generated, and then the centripetal acceleration, the centripetal jerk, and the chord error should be limited in feedrate scheduling process.⁹ In this section, the centripetal acceleration, the centripetal jerk, and the chord error are set to $a_{n} = 2000 mm / s^{2}, J_{n} = 12, 000 mm / s^{3}, and δ = 0.001 mm$ , respectively. Besides, in real-time interpolation, the second-order Taylor expansion (STE) is implied to generate the target position for its satisfactory feedrate fluctuation and computational performance.²⁴

The detailed smoothed trajectories of corners and scheduled feedrate by two algorithms are depicted in Figures 11 and 12, respectively. The maximum corner errors and the minimum feedrate around the corners are presented in Table 6. As presented in Figure 12, the machining time by FBM is 2.350 s, which is reduced by 9.65% compared with TCM. And it can be seen from Figure 11 and Table 6 that the maximum corner errors generated by two methods are controlled within the limitation. In addition, the theoretical corner errors generated by FBM are smaller than those by TCM, which means that the FBM can smooth the tested linear segments with higher efficiency and higher accuracy than TCM.

Table 6.

The maximum corner error and the minimum feedrate of each corner.

Corner	The angle of corner (°)	Maximum corner error (mm)		Minimum feedrate (mm/s)
		TCM	FBM	TCM	FBM
1	90	0.1	0.0870	6.463	8.304
2	165	0.1	0.0965	50.000	36.117
3	30	0.1	0.0947	1.11658	2.494
4	105	0.1	0.0939	9.2007	9.894
5	120	0.1	0.0875	13.446	11.886
6	60	0.1	0.0948	3.10766	5.376
7	150	0.1	0.0934	37.418	22.374
8	90	0.1	0.0876	6.463	8.356

TCM: two-step corner smoothing method; FBM: feedrate blending method.

Figure 11.

The smoothed toolpath in simulation.

Figure 12.

The scheduled feedrate by two algorithms.

Experiment

In order to further evaluate its applicability and performance in the real-time CNC system, two practical toolpaths with obtuse and acute angles together with longer and shorter segments are applied on a three-axis milling CNC machine tool as depicted in Figure 13. The algorithms are carried out on the self-developed CNC system with position feedbacks from motor encoders, and the key processor of which is an ARM 926EJ-STM 400 MHz processor. The cornering tolerance is set to 0.02 mm.

Figure 13.

The experimental setup.

First, the leaf-shaped toolpath is tested with the TCM and FBM, as shown in Figure 14. The kinematic profiles by two methods are illustrated in Figures 15 and 16. As depicted in Figure 15, no matter at the longer linear segments or the shorter segments, the feedrate will be blended to smooth the corner within the specified tolerances by two methods. As a consequence, the tool motion will be never stopped to improve machining efficiency. However, the cycle time with FBM is reduced by 8.26% compared with TCM. What’s more, the two algorithms constrain axis feedrate and axis acceleration within the limits, but the axis jerk cannot be limited to the user-specified limits by TCM, as presented in Figure 16, which will affect vibratory and tracking performance of the feed drive system.²⁵

Figure 14.

The leaf-shaped toolpath.

Figure 15.

The scheduled feedrate of leaf-shaped toolpath.

Figure 16.

Kinematic profiles by two methods: (a) feedrate of axis x, (b) feedrate of axis y, (c) acceleration of axis x, (d) acceleration of axis y, (e) jerk of axis x, and (f) jerk of axis y.

The estimated contour errors are presented in Figure 17 and summarized in Table 7, in which the performance index “RMS” means the root mean square of contour errors and “Max” means the maximum absolute contour errors. The machining results with 4000 r/min spindle speed on the 6061 aluminum alloy workpiece and the tested areas of the leaf-shaped toolpath are shown in Figure 18. The results of surface roughness at the bottom of the tested areas are shown in Figure 19. It can be seen that the average surface roughness with FBM is slightly better than that with TCM, besides FBM delivers smaller RMS contour error and provides about 13.05% reduction in maximum contour error in the real-time CNC system. As mentioned above, the proposed method can provide a smoother and more accurate trajectory with limited axis dynamic constraints and faster machining time.

Table 7.

The comparison of machining time and contour error of leaf-shaped toolpath.

Algorithms	Cycle time (s)	Contour errors (μm)
		RMS	Max
FBM	12.022	1.974	15.603
TCM	13.106	2.409	17.945

RMS: root mean square of contour errors; Max: maximum absolute contour errors; FBM: feedrate blending method; TCM: two-step corner smoothing method.

Figure 17.

The estimated contour errors of leaf-shaped toolpath.

Figure 18.

The machining results and the tested areas of the leaf-shaped toolpath: (a) FBM and (b) TCM.

Figure 19.

The surface roughness of the tested areas.

Finally, a more complicated eagle-shaped toolpath with 183 linear segments is performed by TCM and FBM, as observed in Figure 20. The kinematic motion profiles of the two algorithms are depicted in Figures 21 and 22. As shown in Figure 21, the FBM can deliver a faster cycle time than TCM. Observed from the axial jerk profiles in Figure 22(e) and (f), the proposed FBM clearly respects the jerk limits of the drives. However, the axial jerks generated by TCM are out of bounds at most of the corners, which is reflected in contouring performance as shown in Figure 23 and Table 8. As illustrated, the proposed FBM has a better contouring performance with a Max contour error reduction of 3.66% and generates a faster motion with a cycle time reduction of 5.34% compared with the TCM.

Table 8.

The comparison of machining time and contour error of eagle-shaped toolpath.

Algorithms	Cycle time (s)	Contour errors (μm)
		RMS	Max
FBM	42.073	3.726	19.075
TCM	44.445	4.019	19.800

RMS: root mean square of contour errors; Max: maximum absolute contour errors; FBM: feedrate blending method; TCM: two-step corner smoothing method.

Figure 20.

Eagle-shaped toolpath.

Figure 21.

The scheduled feedrate of eagle-shaped toolpath.

Figure 22.

The kinematic profiles of eagle-shaped toolpath: (a) feedrate of axis x, (b) feedrate of axis y, (c) acceleration of axis x, (d) acceleration of axis y, (e) jerk of axis x, and (f) jerk of axis y.

Figure 23.

The estimated contour errors of eagle-shaped toolpath.

What’s more, in the blending process, FBM calculates the next interpolation point with an average consuming time of 5.061 μs. Meanwhile, the TCM takes about 277.143 μs to calculate each interpolation point with STE in the real-time ARM-based platform, as illustrated in Figure 24, which will degrade the performance of CNC systems. As a result, the proposed method is computationally efficient and convenient to apply in the real-time CNC system.

Figure 24.

The consuming time to calculate interpolation points.

Conclusion

This article presents a local corner smoothing method with the feedrate blending algorithm, which takes the corner error and axis kinematic constraints into consideration. The proposed method needs only one-step to blend the feedrate and thus to generate the smoothed trajectory around the sharp corner. The corner time can be calculated by the specified corner error, besides the axis acceleration and jerk during the corner time will be limited, since the acceleration and the jerk have been optimized and adjusted in the feedrate scheduling stage. Compared with the TCM with cubic B-spline, the proposed method shows a better contouring performance with faster machining time and has a better real-time performance. Simulations and experiments validate the effectiveness and the feasibility of the proposed method for smoothing linear segments. In future work, the FBM for three-axis and five-axis linear toolpath will be studied.

Footnotes

Appendix 1

Appendix 2

\begin{matrix} Ұ - Y 1 = (\frac{j_{k}^{2}}{36} - \frac{\cos (θ) \cdot j_{k} \cdot j_{j}}{18} + \frac{j_{j}^{2}}{36}) \\ Ұ - Y 2 = \frac{j_{i} \cdot (\frac{a_{k}}{2} - \frac{T_{k} \cdot j_{k}}{2})}{3} + \frac{j_{j} \cdot (\frac{a_{j}}{2} - \frac{T_{j} \cdot j_{j}}{2})}{3} - \frac{j_{j} \cdot \cos (θ) \cdot (a_{k} - T_{k} \cdot j_{k})}{6} - \frac{j_{i} \cdot \cos (θ) \cdot (\frac{a_{j}}{2} - \frac{T_{j} \cdot j_{j}}{2})}{3} \\ Ұ - Y 3 = \frac{j_{k} \cdot (\frac{j_{k} \cdot T_{k}^{2}}{2} - a_{k} \cdot T_{k} + v_{k})}{3} + \frac{j_{j} \cdot (\frac{j_{j} \cdot T_{j}^{2}}{2} - a_{j} \cdot T_{j} + v_{j})}{3} + {(\frac{a_{k}}{2} - \frac{T_{k} \cdot j_{k}}{2})}^{2} + {(\frac{a_{j}}{2} - \frac{T_{j} \cdot j_{j}}{2})}^{2} \\ - \frac{j_{k} \cdot \cos (θ) \cdot (\frac{j_{j} \cdot T_{j}^{2}}{2} - a_{j} \cdot T_{j} + v_{j})}{3} - \cos (θ) \cdot (\frac{a_{j}}{2} - \frac{T_{j} \cdot j_{j}}{2}) \cdot (a_{k} - T_{k} \cdot j_{k}) \\ - \frac{j_{j} \cdot \cos (θ) \cdot (j_{k} \cdot T_{k}^{2} - 2 \cdot a_{k} \cdot T_{k} + 2 \cdot v_{k})}{6} \\ Ұ - Y 4 = 2 \cdot (\frac{a_{k}}{2} - \frac{T_{k} \cdot j_{j}}{2}) \cdot (\frac{j_{k} \cdot T_{k}^{2}}{2} - a_{k} \cdot T_{k} + v_{k}) + 2 \cdot (\frac{a_{j}}{2} - \frac{T_{j} \cdot j_{j}}{2}) \cdot (\frac{j_{j} \cdot T_{j}^{2}}{2} - a_{j} \cdot T_{j} + v_{j}) \\ + \frac{j_{i} \cdot (\frac{a_{k} \cdot T_{k}^{2}}{2} - \frac{j_{k} \cdot T_{k}^{3}}{6} - v_{k} \cdot T_{k} + s_{k})}{3} + \frac{j_{j} \cdot (\frac{a_{j} \cdot T_{j}^{2}}{2} - \frac{j_{j} \cdot T_{j}^{3}}{6} - v_{j} \cdot T_{j} + s_{j})}{3} \\ - \cos (θ) \cdot (\frac{a_{j}}{2} - \frac{T_{j} \cdot j_{j}}{2}) \cdot (j_{k} \cdot T_{k}^{2} - 2 \cdot a_{k} \cdot T_{k} + 2 \cdot v_{k}) - \cos (θ) \cdot (a_{k} - T_{k} \cdot j_{k}) \cdot (\frac{j_{j} \cdot T_{j}^{2}}{2} - a_{j} \cdot T_{j} + v_{j}) \\ - \frac{j_{k} \cdot \cos (θ) \cdot (\frac{a_{j} \cdot T_{j}^{2}}{2} - \frac{j_{j} \cdot T_{j}^{3}}{6} - v_{j} \cdot T_{j} + s_{j})}{3} - \frac{j_{j} \cdot \cos (θ) \cdot (a_{k} \cdot T_{k}^{2} - \frac{j_{k} \cdot T_{k}^{3}}{3} - 2 \cdot v_{k} \cdot T_{k} + 2 \cdot s_{k})}{6} \\ Ұ - Y 5 = 2 \cdot (\frac{a_{k}}{2} - \frac{T_{k} \cdot j_{j}}{2}) \cdot (\frac{a_{k} \cdot T_{k}^{2}}{2} - \frac{j_{k} \cdot T_{k}^{3}}{6} - v_{k} \cdot T_{k} + s_{k}) + 2 \cdot (\frac{a_{j}}{2} - \frac{T_{j} \cdot j_{j}}{2}) \cdot (\frac{j_{j} \cdot T_{j}^{2}}{2} - \frac{j_{j} \cdot T_{j}^{3}}{6} - v_{j} \cdot T_{j} + s_{j}) \\ + {(\frac{j_{k} \cdot T_{k}^{2}}{2} - a_{k} \cdot T_{k} + v_{k})}^{2} + {(\frac{j_{j} \cdot T_{j}^{2}}{2} - a_{j} \cdot T_{j} + v_{j})}^{2} - \cos (θ) \cdot (a_{k} - T_{k} \cdot j_{k}) \cdot (\frac{a_{j} \cdot T_{j}^{2}}{2} - \frac{j_{j} \cdot T_{j}^{3}}{6} - v_{j} \cdot T_{j} + s_{j}) \\ - \cos (θ) \cdot (\frac{j_{j} \cdot T_{j}^{2}}{2} - a_{j} \cdot T_{j} + v_{j}) \cdot (j_{k} \cdot T_{k}^{2} - 2 \cdot a_{k} \cdot T_{k} + 2 \cdot v_{k}) \\ - \cos (θ) \cdot (\frac{a_{j}}{2} - \frac{T_{j} \cdot j_{j}}{2}) \cdot (a_{k} \cdot T_{k}^{2} - \frac{j_{k} \cdot T_{k}^{3}}{3} - 2 \cdot v_{k} \cdot T_{k} + 2 \cdot s_{k}) \\ Ұ - Y 6 = (\begin{matrix} 2 \cdot (\frac{j_{k} \cdot T_{k}^{2}}{2} - a_{k} \cdot T_{k} + v_{k}) \cdot (\frac{a_{k} \cdot T_{k}^{2}}{2} - \frac{j_{k} \cdot T_{k}^{3}}{6} - v_{k} \cdot T_{k} + s_{k}) + 2 \cdot (\frac{j_{j} \cdot T_{j}^{2}}{2} - a_{j} \cdot T_{j} + v_{j}) \\ \cdot (\frac{a_{j} \cdot T_{j}^{2}}{2} - \frac{j_{j} \cdot T_{j}^{3}}{6} - v_{j} \cdot T_{j} + s_{j}) - \cos (θ) \cdot (\frac{j_{j} \cdot T_{j}^{2}}{2} - a_{j} \cdot Tj + v_{j}) \cdot (a_{k} \cdot T_{k}^{2} - \frac{j_{k} \cdot T_{k}^{3}}{3} - 2 \cdot v_{k} \cdot T_{k} + 2 \cdot s_{k}) \\ - \cos (θ) \cdot (j_{k} \cdot T_{k}^{2} - 2 \cdot a_{k} \cdot T_{k} + 2 \cdot v_{k}) \cdot (\frac{a_{j} \cdot T_{j}^{2}}{2} - \frac{j_{j} \cdot T j^{3}}{6} - vj \cdot T_{j} + s_{j}) \end{matrix}) \\ Ұ - Y 7 = {(\frac{a_{k} \cdot T_{k}^{2}}{2} - \frac{j_{k} \cdot {T_{k}}^{3}}{6} - v_{k} \cdot T_{k} + s_{k})}^{2} + {(\frac{a_{j} \cdot T_{j}^{2}}{2} - \frac{j_{j} \cdot T_{j}^{3}}{6} - v_{j} \cdot T_{j} + s_{j})}^{2} - ε_{max}^{2} \\ - \cos (θ) \cdot (a_{k} \cdot T_{k}^{2} - \frac{j_{k} \cdot T_{k}^{3}}{3} - 2 \cdot v_{k} \cdot T_{k} + 2 \cdot s_{k}) \cdot (\frac{a_{j} \cdot T_{j}^{2}}{2} - \frac{j_{j} \cdot T_{j}^{3}}{6} - v_{j} \cdot T_{j} + s_{j}) \end{matrix}

Acknowledgements

The authors express their deepest gratitude to their family for the support.

Author contributions

Y.-B.Z. and T.-Y.W. initiated and developed the ideas related to this research work. Y.-B.Z. derived relevant formulations and carried out the performance analyses of experimental results with the help of Y.-F.L. and R.-J.K. Besides, Y.-B.Z. wrote this article under the guidance of J.-C.D. and T.-Y.W.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported, in part, by the National Natural Science Foundation of China (grant nos 51475324 and 51605328).

ORCID iD

Yong-Bin Zhang

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