Feedrate scheduling for interpolation of parametric tool path using the sine series representation of jerk profile

Abstract

This article proposes the sine series representation of jerk profile for acceleration/deceleration feedrate scheduling of parametric interpolator used in computer numerical control machining tools. By selecting the geometric sequence as the coefficients of the sine series, the closed-form expressions of the feedrate, acceleration and jerk profiles are obtained. The resulting feedrate profile is more concise compared with the polynomial profile and more efficient compared with the trigonometric profile. It can be well accepted by the conventional acceleration/deceleration feedrate scheduling algorithms. Also, an approach is proposed to determine the optimal geometric sequence that makes the tracking time as small as possible. Simulations and experiments of tracking complex contours expressed as free-form non-uniform rational B-spline curves are conducted. The results demonstrate the effectiveness of the proposed feedrate profile and the resulting feedrate scheduling strategy.

Keywords

Feedrate scheduling acceleration/deceleration velocity profile sine series non-uniform rational B-spline curve

Introduction

The feedrate scheduling, which determines the smoothness, accuracy and stability of the machining process, is one of the most important factors in the interpolation of parametric curves. In order to achieve high-speed and high-accuracy machining, various techniques for feedrate scheduling of industrial robots and computer numerical control (CNC) machining tools have been proposed for both the linear and parametric tool paths. The existing feedrate scheduling algorithm for parametric tool paths can be roughly divided into two broad families of classes: the acceleration/deceleration (AD) approach and the time-optimal (TO) approach.

The TO approach was first applied to solve the minimal time control problem for robotic actuators, and the shape of a feedrate profile is not known in advance but determined by some differential equations.¹ This approach can be further classified into two categories: the numerical methods^2–7 and the analytic methods.^8,9 Dong and colleagues^2,3 and Beudaert et al.⁴ proposed two similarly practicable methods to handle the feedrate scheduling problem numerically by solving a series of constrained optimization sub-problems. Zhang et al.^8,9 proposed an analytic way towards this problem by searching the intersections of the so-called velocity limiting curves/surfaces deduced from the acceleration, jerk and chord error constraints. Although optimal, these methods cannot guarantee a feasible solution even after an intensive computation.^1,10 Moreover, the numerical off-line optimization method generally needs a re-interpolation process to obtain the feedrate values at each interpolation period, which may cause undesirable errors. The analytic methods can only be applied to simple parametric curves due to the requirement of solving differential equations. Recently, Sun et al.^11–13 proposed a novel adaptive feedrate scheduling method that considers drive constraints. A B-spline curve was adopted to represent the feedrate profile, and its control points were adjusted iteratively, leading to a near-optimal feedrate profile under the prescribed constraints. The re-interpolation was still involved in this method.

The AD approach was first applied to line segments^14,15 which are the most widespread representation for tool path in CNC machining. Detailed and nice work on how to generate a universal AD velocity profile for line segments was given in Jeon and Ha¹⁶ and Luo et al.¹⁷ Later, this method was extended to deal with parametric tool path.^18–24 As mentioned in Fan et al.¹⁰ and Luo et al.,¹⁷ the most widely used profiles in AD approach are polynomial (e.g. linear-velocity, trapezoidal-velocity and trapezoidal-acceleration) and trigonometric profiles. With the polynomial profile, the process of the feedrate scheduling becomes complicated and the expression of the profile becomes cumbersome for a smoother profile. A piecewise velocity function with seven sub-functions²⁵ was needed to describe an acceleration phase with trapezoidal jerk as illustrated in Figure 1(a). Moreover, four kinds of acceleration profiles were developed for different conditions.^14,23,26 With the trigonometric profile, the machining becomes less efficient because only one point can reach the given maximum value of the acceleration or jerk profile,²² as illustrated in Figure 1(b).

In summary, the AD feedrate scheduling method is now most widely used in CNC systems due to its stability and low computational burden,¹⁵ although there are some pitfalls, especially when the dynamic characteristics of each axis are dramatically unmatched.^10,17 The study towards AD feedrate scheduling focuses on polynomial^14,21,23 and trigonometric^16,17,22 profiles, which usually concerns the constraints of tangential velocity, tangential/normal acceleration, tangential/normal jerk and chord error (only for parametric curves). Comparatively, the polynomial profile is more efficient than the trigonometric one because it can make full advantage of the axial capabilities, but the trigonometric representation is more succinct than the polynomial one and is more convenient to generate smoother profiles.

To obtain a jerk-continuous feedrate profile with both simplicity and efficiency, this article proposes an approach to represent the jerk profile using the sine series. By selecting the geometric sequence as the coefficients of the sine series, the closed-form expressions of the feedrate, acceleration and jerk profiles are obtained. By selecting the optimal geometric sequence, a short AD time duration is obtained. The remainder of this article is organized as follows. The feedrate profile based on the sine series representation of jerk profile is presented in section ‘Feedrate profile based on the sine series representation of jerk profile’. The procedure for AD feedrate scheduling using this kind of feedrate profile is detailed in section ‘AD feedrate scheduling for parametric curves’. Simulation and experimental results are given in section ‘Simulation and experimental results’, and conclusions are summarized in section ‘Conclusion’.

Feedrate profile based on the sine series representation of jerk profile

Principles of AD feedrate profile

An AD feedrate scheduling is to use a certain feedrate profile to connect the feedrate from one value to another one under given constraints. Following the idea of Luo et al.,¹⁷ a profile with only an acceleration phase is adopted to illustrate the techniques of AD feedrate scheduling. Let the start velocity, the end velocity and the length of the path be $v_{s}$ , $v_{e}$ and s, respectively, then the feedrate function $f (t)$ should satisfy the following constraints

{\begin{matrix} f (0) = v_{s}, f' (0) = 0, f ″ (T) = 0 \\ f (T) = v_{e} \leq v_{\max}, f' (T) = 0, f ″ (T) = 0 \\ v_{s} \leq f (t) \leq v_{e}, | f' (t) | \leq a_{\max}, | f ″ (t) | \leq j_{\max}, t \in [0, T] \\ \int_{0}^{T} f (t) dt = s \end{matrix}

(1)

where t denotes the time; T is the time required to finish the path; $v_{\max}$ , $a_{\max}$ and $j_{\max}$ are the maximum allowable velocity, acceleration and jerk, respectively, which are defined as the minimal limits of all axes, that is, $v_{\max} = min {v_{x}, v_{y}, v_{z}}$ , $a_{\max} = \min {a_{x}, a_{y}, a_{z}}$ and $j_{\max} = min {j_{x}, j_{y}, j_{z}}$ .^10,17

For a general AD feedrate profile, three principles are normally accepted to guarantee the machining efficiency according to some studies.^{21–24,26,27}

Principle 1

There are three basic phases of an AD feedrate profile, that is, ACC (ACCeleration), CF (Constant Feed) and DEC (DECeleration), as illustrated in Figure 2. And there are seven practicable types of feedrate profiles, that is, ACC + CF + DEC, ACC + DEC, ACC + CF, CF + DEC, ACC, DEC and CF, as illustrated in Figure 3. If the symmetrical AD characteristics are applied, which is normally accepted^17,21,22,26 and will be accepted in this article, it is adequate to only study the ACC phase to determine a proper feedrate profile.

Principle 2

The main aim of the feedrate scheduling is to make the feedrate as large as possible, rather than maximize the jerk or acceleration.

Principle 3

If the feedrate profile consists of more than a CF phase, for example, an ACC + CF profile, the velocity at the CF phase must be the maximum velocity according to Principle 2. Thus, the CF + DEC and ACC + CF profiles are rare in practice because the start or end velocity of the segment is usual not the maximum velocity.

Representation of AD feedrate profiles

Following the works of Luo et al.¹⁷ and Lee et al.,²² the two conventional profiles, that is, the polynomial and the single sine feedrate profiles, are briefly revisited. Then, a novel feedrate profile derived from the sine series representation of jerk profile is presented with the aim of combining the advantages of the two conventional ones.

Polynomial feedrate profile

The simplest polynomial profile is the linear-velocity profile, of which the acceleration is confined. By selecting the order of the polynomial functions, one can get different feedrate profiles with desired order of continuity.^12,19,23,26 A polynomial feedrate profile with a trapezoidal jerk and confined jounce is depicted in Figure 1(a). The ACC profile given in Figure 1(a) should be divided into 7 regions, and an ACC + CF + DEC profile should be divided into up to 15 regions. The expression of the limited-jounce feedrate profile and procedure of the scheduling are presented in Fan et al.,²³ which are very cumbersome and will become more complicated as the profile gets smoother.

Single sine feedrate profile

Lee et al.²² proposed a feedrate scheduling method with a velocity profile expressed as a single sine function and got the acceleration and jerk by the time derivatives of velocity. Unlike the piecewise function for a polynomial profile, the single sine feedrate for an ACC profile can be expressed with only one formula. The concision of expression, however, cannot cover its inefficiency. As shown in Figure 1(b), a single sine function results in only one point that can reach the maximum acceleration or jerk for the acceleration or jerk profile, respectively, which surely cannot make full advantage of the dynamic characteristics of the servo system.

Sine series feedrate profile

To combine the advantages of the polynomial and the single sine feedrate profile, a novel feedrate profile is derived here from the sine series representation of jerk profile. Contrary to Lee’s work, the jerk profile is determined first, and then the acceleration and velocity profiles are obtained by the time integrations, which results in a feedrate profile with a continuous jerk rather than a confined jerk.

Note that the jerk profile in an ACC phase usually has the shape of an odd function with translation along the x-axis as shown in Figure 1; thus, the jerk profile in an ACC phase can be expressed as a sine series according to the theory of Fourier series. The involved sine functions have the following forms

{\tilde{j}}_{n} (t) = J_{t} \sin (\frac{2 n π}{T} t), n = 1, 2, 3, \dots

(2)

where $J_{t}$ is a coefficient to normalize the maximum velocity for the concise expression of the velocity function, and n is the order of the sine functions.

Considering the boundary condition ${\tilde{a}}_{n} (T) = 0$ , we get the acceleration and velocity through integration

\begin{matrix} {\tilde{a}}_{n} (t) = \int_{0}^{t} {\tilde{j}}_{n} (u) du = - \frac{J_{t} T}{2 n π} \cos (\frac{2 n π}{T} t) + \frac{J_{t} T}{2 n π} \\ {\tilde{v}}_{n} (t) = \int_{0}^{t} {\tilde{a}}_{n} (u) du = - J_{t} {(\frac{T}{2 n π})}^{2} \sin (\frac{2 n π}{T} t) + \frac{J_{t} T}{2 n π} t \end{matrix}

(3)

Normalizing the velocity yields

{\tilde{v}}_{n} (T) = \frac{J_{t} T^{2}}{2 n π} = 1 \Rightarrow J_{t} = \frac{2 n π}{T^{2}}

(4)

Substituting equation (4) into equation (3), we get the normalized jerk, acceleration and velocity as shown in equation (5)

{\begin{matrix} {\tilde{j}}_{n} (t) = \frac{2 n π}{T^{2}} \sin (\frac{2 n π}{T} t) \\ {\tilde{a}}_{n} (t) = - \frac{1}{T} \cos (\frac{2 n π}{T} t) + \frac{1}{T} \\ {\tilde{v}}_{n} (t) = - \frac{1}{2 n π} \sin (\frac{2 n π}{T} t) + \frac{t}{T} \end{matrix}, t \in [0, T]

(5)

In a general situation with start velocity $v_{s}$ and end velocity $v_{e}$ , the feedrate profile can be expressed as

{\begin{matrix} j_{n} (t) = {\tilde{j}}_{n} (t) Δ v \\ a_{n} (t) = {\tilde{a}}_{n} (t) Δ v \\ v_{n} (t) = {\tilde{v}}_{n} (t) Δ v + v_{s} \end{matrix}, t \in [0, T]

(6)

where $Δ v = v_{e} - v_{s}$ . Obviously, we have $v_{n} (0) = v_{s}$ and $v_{n} (T) = v_{e}$ .

For the sine feedrate function with order n, we obtain the tracking length in the ACC phase by integrating the velocity w.r.t. time

s_{n} = \int_{0}^{T} v_{n} (t) dt = v_{s} T + Δ v \int_{0}^{T} {\tilde{v}}_{n} (t) dt = \frac{(v_{s} + v_{e}) T}{2}

(7)

It can be deduced from equation (7) that the tracking length is independent of the order of the sine function and determined only by the start velocity, the end velocity and the time duration.

Therefore, we get the general feedrate profile as follows

{\begin{matrix} v (t) = \sum_{i = 1}^{n} w_{i} v_{i} (t) = - \frac{Δ v}{2 π} \sum_{i = 1}^{n} \frac{w_{i}}{i} \sin (\frac{2 i π}{T} t) + \frac{Δ vt}{T} + v_{s} \\ a (t) = Δ v \sum_{i = 1}^{n} w_{i} a_{i} (t) = - \frac{Δ v}{T} \sum_{i = 1}^{n} w_{i} \cos (\frac{2 i π}{T} t) + \frac{Δ v}{T} \\ j (t) = Δ v \sum_{i = 1}^{n} w_{i} j_{i} (t) = \frac{2 Δ v π}{T^{2}} \sum_{i = 1}^{n} i w_{i} \sin (\frac{2 i π}{T} t) \\ \sum_{i = 1}^{n} w_{i} = 1, w_{i} > 0 \end{matrix}, t \in [0, T]

(8)

where $w_{i}, i = 1, 2, \dots, n$ denote the coefficients of the sine series. It is seen that the tracking length is always $s_{n}$ .

According to equation (5) and $w_{i} > 0$ , it is obvious that $a (t) \geq 0$ and $v (t)$ is monotonically increasing w.r.t. t. Thus, the velocity constraint $v_{s} < v (t) < v_{e}, t \in [0, T]$ expressed in equation (1) is always satisfied. Only the acceleration and jerk constraints required by equation (1) need to be considered. They are re-formulated as

| a (t) | \leq a_{\max}, | j (t) | \leq j_{\max}

(9)

In order to reduce the total time for the contour tracking, the time for the acceleration phase should be minimized, that is, the time duration T for the ACC phase should be as small as possible. From equations (8) and (9), the time duration T can be determined by the following equation

T = \max (\frac{Δ v \cdot \max_{θ, w_{i}} f_{1} (w_{1}, \dots, w_{n}, θ)}{a_{\max}}, \sqrt{\frac{2 π Δ v \cdot \max_{θ, w_{i}} f_{2} (w_{1}, \dots, w_{n}, θ)}{j_{\max}}})

(10)

where $f_{1} (w_{1}, \dots, w_{n}, θ) = 1 - \sum_{i = 1}^{n} w_{i} \cos i θ$ and $f_{2} (w_{1}, \dots, w_{n}, θ) = \sum_{i = 1}^{n} i w_{i} \sin i θ, θ \in [0, 2 π]$ . If T is determined by the maximum value of function $f_{1}$ , at least one point on the acceleration profile reaches the maximum allowable acceleration. In this sense, the acceleration constraint is active. Otherwise, at least one point on the jerk profile reaches the maximum allowable jerk, and the jerk constraint is active.

Coefficients of the sine series

To make sure that the series in equation (8) have closed-form expressions and the maximum acceleration and jerk can be efficiently determined, we use the geometric sequence as the coefficients of the sine series. Other coefficients are feasible, but the geometric sequence is one of the most intuitive and convenient choices.

Given the coefficients as geometric sequence defined in equation (11)

{\begin{matrix} w_{i} = \frac{m - 1}{m^{i}}, m > 1 \\ \sum_{i = 1}^{\infty} w_{i} = 1 \end{matrix}

(11)

we obtain

{\begin{matrix} v (t) = - \frac{Δ v}{2 π} (m - 1) \\ (atan (\frac{m + 1}{m - 1} \tan (0.5 θ)) - 0.5 θ) + \frac{Δ vt}{T} + v_{s} \\ a (t) = - \frac{Δ v}{T} \frac{(m - 1) (m \cos θ - 1)}{m^{2} - 2 m \cos θ + 1} + \frac{Δ v}{T} \\ j (t) = \frac{2 Δ v π}{T^{2}} \frac{m (m - 1) (m^{2} - 1) \sin θ}{{(m^{2} - 2 m \cos θ + 1)}^{2}} \\ θ = \frac{2 π t}{T} \end{matrix}, t \in [0, T]

(12)

and

\begin{matrix} f_{1} = \frac{(1 - \cos θ) (m^{2} + m)}{m^{2} - 2 m \cos θ + 1} \\ f_{2} = \frac{m (m - 1) (m^{2} - 1) \sin θ}{{(m^{2} - 2 m \cos θ + 1)}^{2}} \end{matrix}

(13)

The derivations of equations (12) and (13) are given in Appendix 1.

Optimal coefficients of the sine series

In order to obtain the optimal coefficients of the sine series which minimize T, we need to minimize the maximum value of the functions $f_{1} (θ, m)$ and $f_{2} (θ, m)$ . For a given m, the maximum value of $f_{1} (θ, m)$ is determined by the following equation

f_{1}^{\max} = 2 - \frac{2}{m + 1} < 2

(14)

The maximum value of $f_{2} (θ, m)$ can be determined by solving for θ from the following equation, and then substituting the positive solution into function $f_{2} (θ, m)$

\cos θ = \frac{\sqrt{32 m^{2} + {(1 + m^{2})}^{2}} - (1 + m^{2})}{4 m} < 1, m > 1

(15)

The detailed derivations of equations (14) and (15) are given in Appendix 2. The maximum values of $f_{1} (m)$ and $f_{2} (m)$ w.r.t. m are shown in Figure 4.

For the conventional single sine function, we have $n = 1$ , $w_{1} = 1$ , $f_{1}^{\max} = 2$ and $f_{2}^{\max} = 1$ . For the sine series function, as shown in Figure 4, if m approaches 1, we have $f_{1}^{\max} \approx 1$ , which means that the resulting feedrate profile can reduce the time duration by up to 50% as compared to the trigonometric profile if the acceleration constraint is active; on the other hand, $f_{2}^{\max}$ quickly declines below 1 as m increases, which means that resulting feedrate profile can slightly reduce the time duration if the jerk constraint is active. Therefore, the proposed feedrate profile is more efficient than the single sine profile and can improve the efficiency significantly in the case that the acceleration constraint is active.

Let

T_{a m} = \frac{Δ v \cdot f_{1}^{\max} (m)}{a_{\max}}, T_{j m} = \sqrt{\frac{2 π Δ v \cdot f_{2}^{\max} (m)}{j_{\max}}}

(16)

Obviously, $Δ v$ has great influence on $T_{am}$ than $T_{jm}$ . If $Δ v$ is large enough, the acceleration constraint will be active. For any segment with start velocity $v_{s}$ , end velocity $v_{e}$ and length s, m should be determined by equation (16) to minimize the larger one of $T_{am}$ and $T_{jm}$ . If an ACC + DEC or ACC + CF + DEC profile is considered, we should determine the optimal values of m for the ACC and DEC phases, respectively.

As shown in Figure 4, $f_{1}^{\max}$ is a monotonically increasing function w.r.t. m, while $f_{2}^{\max}$ is a monotonically decreasing function when $m < 8$ and almost a constant when $m \geq 8$ . As a result, the optimal m is found within the interval $(1, 8]$ .

Let h be the search step, then a numerical search algorithm is proposed to determine the optimal m, which goes as follows:

Initialize

m = 1 + h

while

m \leq 8

Calculate

T_{am}

and

T_{jm}

T_{am} < T_{jm}

m = m + h

else

break

end while

Terminate Choose the current value $m^{*}$ as the optimal one.

This search algorithm has three resulting cases as shown in Figure 5. In case I, $T_{jm} > T_{am}$ for $1 < m \leq 8$ and $m^{*} = 8$ ; In case II, $T_{jm} > T_{am}$ for $1 < m < m^{*}$ , while $T_{jm} < T_{am}$ for $m^{*} < m \leq 8$ and $1 < m^{*} < 8$ ; In case III, $T_{jm} < T_{am}$ for $1 < m \leq 8$ and $m^{*} = 1 + h$ .

AD feedrate scheduling for parametric curves

Procedure for AD feedrate scheduling

The AD feedrate scheduling method was first developed for linear tool paths, and then applied to parametric tool paths. For a linear tool path, the nominal velocity at each corner point can be determined by the angle between two adjacent lines.^17,22 For a parametric tool path, the procedure for AD feedrate scheduling normally consists of the pre-processing, the feedrate profile generation and the interpolating stages. It has been well studied in Lin et al.,²¹ Lee et al.²² and Fan et al.²³ Here, our work focuses on the feedrate profile generation stage.

In the stage of pre-processing, the critical points of the tool path, where the curvature reaches local maximum values and exceeds the threshold curvature $κ_{cr}$ determined by equation (17), are picked out into a set which also contains the start and end points of the tool path^21,22

κ_{cr} = min (\frac{8 δ}{{(v_{\max} T_{s})}^{2} + 4 δ^{2}}, \frac{a_{\max}}{v_{\max}^{2}}, \sqrt{\frac{j_{\max}}{v_{\max}^{3}}})

(17)

With these critical points, the whole tool path is splitted into several segments, of which the length can be calculated with adaptive Simpson quadrature formula.²⁸ After that the nominal velocity at each critical point is calculated by the following equation^21,22

v^{i} = \min (\frac{2}{T_{s}} \sqrt{\frac{1}{κ_{i}^{2}} - {(\frac{1}{κ_{i}} - δ)}^{2}}, \sqrt{\frac{a_{\max}}{κ_{i}}}, \sqrt[3]{\frac{j_{\max}}{κ_{i}^{2}}}, v_{\max})

(18)

where $δ$ is the allowable chord error, $κ_{i}$ is the curvature of the critical point and $T_{s}$ is the interpolation period. It should be noted that the velocities at the start and end points must be zero. Unlike the linear tool path, the chord error, normal acceleration and normal jerk constraints are used to determine the critical points and the velocities at the critical points for a parametric tool path. However, it is difficult to consider these constraints in the feedrate profile generation stage if the curvature of the tool path is variable. Normally, a simple way which ignores these constraints is adopted to generate the feedrate profile between two adjacent critical points.^{20–24,26,27}

In the stage of feedrate profile generation, the proposed AD feedrate profile is applied to connect the nominal velocities at the critical points for each segment. The flowchart to determine the whole feedrate profile for the complete curved tool path is illustrated in Figure 6, which is similar to that in Lee et al.²² The only difference between them is that in our method, the optimal m employed in the ACC and DEC phases should be determined using the method presented in section ‘Optimal coefficients of the sine series’. So, the efficiency of the conventional AD feedrate scheduling algorithms is completely preserved.

The details to determine the complete feedrate profile for each segments with start velocity $v_{s}$ , end velocity $v_{e}$ and length s are described schematically as follows:

Step I

In the first step, the maximum tracking length $S_{1}$ for the segment which could be planned as a simple ACC or DEC profile with the velocity changing from $v_{s}$ to $v_{e}$ is calculated using equations (7) and (10). If $s \leq S_{1}$ , this segment is planned as a simple ACC or DEC profile, and a dichotomy technique discussed in section ‘Dichotomy technique for velocity modification’ is employed to modify the larger velocity of the two nominal velocities, that is, to ensure the tracking length is exactly the segment length. If $s > S_{1}$ , this segment is planned as an ACC + DEC or an ACC + CF + DEC profile and the process goes to Step II.

Step II

In the second step, the maximum tracking length $S_{2}$ for the segment which could be planned as an ACC + DEC profile is calculated by adding the tracking lengths of an ACC profile with the velocity increasing from $v_{s}$ to $v_{\max}$ and a DEC profile with the velocity decreasing from $v_{\max}$ to $v_{e}$ . If $s \leq S_{2}$ , this segment is planned as an ACC + DEC profile and again the dichotomy technique is employed to modify the actually maximum velocity $v_{fs}$ . If $s > S_{2}$ , this segment is planned as an ACC + CF + DEC profile and the process goes to Step III.

Step III

In this step, an ACC + CF + DEC profile is assigned to the segment, which means that the velocity first accelerates from $v_{s}$ to $v_{\max}$ , then maintains the maximum velocity $v_{\max}$ and finally decelerates from $v_{\max}$ to $v_{e}$ . The tracking length for the CF phase is $s - S_{2}$ .

In the stage of interpolating, the parametric interpolator based on the second-order Taylor expansion²⁹ is adopted to generate the reference position commands.

Dichotomy technique for velocity modification

As mentioned in section ‘Principles of AD feedrate profile’, there are seven practical types of an AD feedrate profile depending on the start velocity, the end velocity and the length of the segment. And in three cases, that is, ACC + DEC, ACC and DEC, a dichotomy technique is adopted to modify the actually maximum velocity, to ensure that the tracking length matches the segment length.

An ACC + DEC profile is taken as an example to illustrate the dichotomy technique. The tracking length of this profile is a monotonically increasing function of the actual maximum velocity $v_{fs}$ . Denoting the larger velocity of $v_{s}$ and $v_{e}$ as $v_{m} = \max (v_{s}, v_{e})$ , then the initial dichotomy search interval for the actually maximum velocity $v_{fs}$ is $[v_{m}, v_{\max}]$ , and the termination condition as

| s_{e} - s | \leq eps

(19)

where $s_{e}$ is the tracking length and eps is the threshold specifying the desired accuracy of the search algorithm. This method can be analogously applied to modify the end velocity $v_{e}$ of an ACC profile and the start velocity $v_{s}$ of a DEC profile.

Backward velocity modification for short blocks

According to Figure 6, if $s < S_{1}$ , the boundary conditions will be changed after feedrate scheduling, and this kind of block is termed as ‘short’ block. Other kinds of blocks are termed as ‘long’ block. If not carefully handled, the short blocks may cause discontinuity at the junction of two adjacent feedrate profiles. Following the idea of Lee et al.,²² a backward adjusting method is adopted to deal with the short blocks. To avoid the discontinuity at the junction of two feedrate profiles, it is necessary to check whether the (i + 1)th block is ‘short’ or not before scheduling the feedrate profile for the ith block. If so, first, all of the successive short blocks, for example, (i + 1)th, (i + 2)th, …, (i + k)th blocks, are found. Then, federate scheduling is applied to these short blocks and the ith long block backward, for example, from the (i + k)th block to the ith block. Finally, the feedrate scheduling goes to the (i + k + 1)th long block. The procedure to handle short blocks is illustrated in Figure 7.

Simulation and experimental results

Experimental setup

To illustrate the effectiveness of the proposed method, simulations and experiments of tracking complex contour expressed as non-uniform rational B-spline (NURBS) curve are conducted on a triaxial table driven by the YASKAWA SGDV series servo drivers and SGMJV motors, as shown in Figure 8. The lead of the ball screws used is 10 mm/rev, and the incremental encoder for each motor has a resolution of 10,000 pulses/rev. The off-line feedrate scheduling algorithm and the on-line control task are implemented on MATLAB environment and a dSPACE DS1103 system, respectively.

The parameters used in the feedrate scheduling and interpolation are listed in Table 1. The reference position commands generated by the interpolator are sent to the dSPACE system for motion control. The effectiveness of the sine series feedrate profile is demonstrated via the machining times in simulations and the contour errors in experiments.

Table 1.

Parameters of kinematic constraints and feedrate scheduling method.

Parameters	Symbols	Values
Interpolation period	$T_{s}$	$0.002 s$
Chord error	$δ$	$0.01 mm$
Maximum velocity	$v_{\max}$	$200 mm / s$
Maximum acceleration	$a_{\max}$	$1000 mm / s^{2}$
Maximum jerk	$j_{\max}$	$50, 000 mm / s^{3}$

Tracking of trident NURBS curve

The first NURBS curve is a trident curve shown in Figure 9(a), and the associated control points, weight vectors and knot vectors are listed in Table 2 of Appendix 3. The seven critical points determined by the given constraints are marked with star as shown in Figure 9, with which the curve is splitted into six segments. In Figure 9(a), the solid blue, the solid red and the dashed blue curves represent the shape, the control polygon and the lines connecting the critical points of the trident curve, respectively.

The simulation results shown in Figure 10 indicate that the machining time reduces from 3.126 s obtained with the single sine method to 2.577 s obtained with the sine series method, decreasing by 17.6%. The experimental results are given in Figure 11. The maximum and mean contouring errors resulting from the sine series method are 15.7 and 3.6 µm, respectively, while those resulting from the single sine method are 13.8 and 2.2 µm, respectively. The performances of the two methods in terms of accuracy are very similar.

Tracking of butterfly NURBS curve

The second NURBS curve is a butterfly curve shown in Figure 12(a), and the associated control points, weight vectors and knot vectors are listed in Table 3 of Appendix 3. A total of 37 critical points are determined and marked with star in Figure 12(a). As a result, the butterfly curve is splitted into 36 segments.

The simulation results shown in Figure 13 indicate that the machining time decreases from 5.427 s obtained with the single sine method to 4.802 s obtained with the sine series method, decreasing by 11.52%. The experimental results are illustrated in Figure 14. The maximum and mean contouring errors resulting from the sine series method are 20.9 and 3.9 µm, respectively, which are also very close to the results 19.9 and 3.6 µm obtained with the single sine method.

As mentioned before, with the single sine method, only one point on the acceleration profile can reach the given maximum acceleration $a_{\max}$ during the ACC/DEC phase, while with the sine series method, the acceleration can maintain the maximum acceleration $a_{\max}$ for a while, which can be observed from the acceleration profiles in Figures 10, 11, 13 and 14. The contour errors of the two experiments are very small due to the continuous jerk of the feedrate profiles. As for the significantly larger mean contouring error in the first experiment for the sine series method, we attribute it to the longer time duration in contouring with the maximum velocity. For a given tool path, the higher the velocity, the larger the contouring error.^3,21 As shown in Figures 11 and 14, the contouring error is quite large when the feed reaches the given maximum value. The time durations with the maximum contouring velocity for both the single sine and sine series methods are nearly zero in the second experiment as shown in Figure 14. However, the time duration with the maximum contouring velocity for the single sine method is much shorter than that for the sine series method in the first experiment as shown in Figure 11. Hence, the mean contouring error for the proposed sine series method is notably large in the first experiment. In fact, the maximum contouring error usually plays a more important role than the mean contouring error in evaluating the contour following performance.^19–23 The proposed sine series method can generate a more efficient feedrate profile while keeping a similar maximum contouring error as compared with the conventional single sine method, which can be observed from Figures 11 and 14. The proposed method is more effective for the trident curve due to the larger difference between the velocities at the adjacent critical points, which can be seen from the velocity profiles in Figures 10 and 13. In this case, the acceleration constraint is the dominant factor that determines the AD time duration, which leads to a more significant improvement in efficiency for the proposed method.

Conclusion

This article presents the sine series representation of jerk profile for AD feedrate scheduling of parametric interpolator used in CNC machine tools. To the best knowledge of the authors, this work is the first attempt in introducing the sine series representation in AD feedrate scheduling under the velocity, acceleration, jerk and chord error constraints. The distinctive characteristics of the proposed method are summarized as follows:

It can generate a jerk-continuous feedrate profile and thus provide smooth feed motions for the machine tools, which benefits greatly to the contour following performance.

It is both efficient and concise. Using the sine series representation of jerk profile, various flexible and efficient jerk profiles can be obtained. By selecting the optimally geometric sequence as the coefficients of the sine series, the closed-form expressions of the feedrate profile that leads to a short AD time duration is obtained. The resulting feedrate scheduling method can generate a more efficient feedrate profile while keeping a similar maximum contouring error as compared with the conventional feedrate scheduling method that employs the trigonometric profile.

Footnotes

Appendix 1

Appendix 2

Appendix 3 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 partially supported by the National Natural Science Foundation of China under grant no. 51325502 and the Science & Technology Commission of Shanghai Municipality under grant nos 13JC1408400 and 14111104801.

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