Design of warp knitting electronic shogging system based on mixed-velocity planning curve

Abstract

To realize high-speed running of a warp knitting machine, the shogging motion should not only meet the requirement of high dynamic response but should also satisfy high positioning accuracy. Due to the large location disturbance and the dynamic response delay in the interpolation method or the single velocity planning curve method, an electronic shogging system for a warp knitting machine based on the mixed-velocity planning curve is proposed in the present study. Through the analysis of the shogging motion combined with the knitted structure, the optimal resolution of the instruction signal is calculated, which is 725 pulses for one needle step, and the velocity loop bandwidth of the servo driver is optimized. In addition, the motor with a load inertia ratio close to 1 is also selected. Analysis of the shogging motion vibration curve confirms that the shogging motion has advantages of high positioning accuracy and high dynamic response under the mixed-velocity planning curve. The response performance with the mixed curve is 12.5% higher than that with the quintic polynomial, and the positioning accuracy of the mixed curve is 26% higher than that with uniform acceleration–deceleration curve.

Keywords

warp knitting electronic shogging motion mixed-velocity planning curve optimal resolution servo bandwidth

The mechanical cam is employed in a traditional guide-bar shogging system, which can realize the position correspondence between the slave axis and the master axis, and thus the master–slave position of the mechanical cam can change with a linear or nonlinear relationship.¹ The electronic shogging motion is a control method, evolved from the mechanical cam, which can detach from the mechanical structure as well as maintain mechanical cam characteristics. An algorithm for electronic cam curve programming based on a motion controller and motor drivers is used for planning of the servo motor motion in accordance with the knitting structure.^2,3 The improvement of the high-speed warp knitted machine is directly affected by the motion-planning performance. There is less stringent requirement for shogging motion position trajectory because the shogging is a point-to-point motion with rapidly intermittent start–stop. However, there remains a high requirement for the control of real-time planning velocity during the motion of the guide bars. The accurate control of start–stop for shogging velocity can directly reflect the dynamic response performance and positioning accuracy of the guide-bar shogging,⁴ which ultimately affects the knitting speed. Currently, the requirement for high dynamic and high positioning accuracy^5,6 of the warp knitting machine cannot be met using a control system with the interpolation method⁷ or single velocity curve method. Due to the speed of electronic shogging there has been no technological breakthrough regarding the warp knitting machine, and the warp knitting machine cannot be fully computerized.

Xia and Ge⁸ applied a linear servo to control warp knitting machine guide-bar shogging. By employing a mathematical fuzzy comprehensive evaluation method, the modified trapezoidal acceleration motion law was selected as the servo drive curve for the electronic shogging system. However, the price of the linear servo was too expensive to be useful in practical application. A speed control mode with a flexible curve in modified trapezoidal acceleration curves without stop was employed to improve the servo responsiveness by Zhang et al.⁹ and Qi et al.¹⁰ Although the shake of the guide-bar shogging was reduced, the high dynamic response of the whole control system was not achieved. The maximum speed of the warp knitting machine reached only 1000 r/min. Liao et al.¹¹ put forward a method of fitting a cam profile with Lagrange polynomial. The dynamic constraints of speed, acceleration and impulse are solved. Compared with the traditional master–slave control method, the electronic cam possesses higher tracking accuracy. In addition, an optimal tuning PID control algorithm is proposed by Kralev et al.¹² to realize fast transient with minimum overshoot and steady-state error. Zhang et al.¹³ proposed a velocity programming interpolation algorithm curve based on NURBS. The second-order Taylor expansion was applied to the molecular parameter curve in the NURBS curve representation. According to the simulation results, the interpolation algorithm can meet the requirements of high speed and high precision of the CNC system. To solve the problem of residual vibration, Tho et al.¹⁴ explicitly considered the maximum velocity and acceleration of the actuator, also proposing a simple motion-planning method based on S-curve instruction. By solving constrained (discrete) nonlinear programming, the shortest time solution was successfully obtained. Zhu et al.¹⁵ emphasized that bandwidth is a critical specification for motion positioning systems, as fast response to reference and broadband disturbance rejection are highly desirable in industrial applications. Fu et al.¹⁶ proposed to use a modified double-T network notch filter to eliminate mechanical resonance, thus servo stiffness and response performance were improved. Nevertheless, using this method only the dynamic response performance was improved, rather than the whole system. Zheng et al.¹⁷ modeled the electronic shogging system, and the simulation results using Matlab revealed that increasing speed proportional gain was instrumental in improving the dynamic response performance of the electronic shogging system. However, when a certain value was achieved when increasing the gain, the system would be out of action due to the shake. In addition, the method to eliminate the shake was not mentioned and no experimental verification was carried out in their research.

The fabric structure or the pattern effect are decided by the law of guide-bar shogging motion,¹⁸ and the knitting speed is decided by dynamic response performance and the positioning accuracy of the guide-bar transverse motion. To improve the dynamic response performance and the positioning accuracy as well as achieving a better knitted effect, in the present study an electronic shogging system for the warp knitting machine via the mixed-velocity planning curve is designed. Combining with the fabric structure, the law of shogging motion is analyzed and velocity planning curves are designed in several combinations. In addition, the optimal resolution of control command signal for motion is calculated and the servo bandwidth is analyzed. Through the above setting the performance of whole system will be optimized. The results demonstrate that this system has advantages of high dynamic response performance and high positioning accuracy through an experimental test platform, greatly improving the knitting speed.

Principle of the velocity planning curve

Generation of the velocity planning curve

During the knitting process, the guide bar needs to follow the spindle position of the warp knitting machine to start up quickly according to the structure, aiming to finish the work of overlapping and underlapping. The movement angle area of overlapping and underlapping is related to the design of the equipment mechanism, and the angle area does not need to be adjusted secondarily after the machine leaves the factory. The angle of each process of knitting stays constant, meaning that the guide bars are required to move a certain distance according to the technology structure. The yarns can be lapped efficiently following the requirement of pattern only when the speed law of the guide bars is controlled precisely.

The planning velocity curve is generated in the motion controller. In the interpolation method currently used, the motion controller obtains the number of pulses of each needle derived from the chain notation, the pitch of screw rod and the number of pulses in a motor cycle. Zhou et al.¹⁹ presented a velocity-adaptive control methodology to adjust the steps of interpolation and reduce error. Besides, the consecutive interpolation point was pre-calculated to guarantee the accuracy of the interpolation velocity. The calculated pulses are distributed into overlapping and underlapping movements, and they are planned by the tracing point method. Combining with the following motion controller model, the velocity planning curve is interpolated as the secondary conversion. A center velocity trajectory transformational method is proposed by Zhang et al.²⁰ in combination with the motion controller, and can be employed for path planning to optimize travel time. However, the obvious drawback of this method is that the actual command curve and the original curve are quite different. In the best case, all of the control commands are on the original curve. On the contrary in the worst case, none of the control command covers the original curve. This random state means that the theoretical performance cannot work in the user curve.²¹

The single velocity curve method aims to build a motion curve in the motion controller, such as S-shaped curve and quintic polynomial curve. The motion control command and the built-in curve are fitted in real time to ensure that the guide bars move strictly following the curve. In addition, this method is also called the formula method. The motion command of the formula method is calculated online in real time, which can guarantee that all the control commands are on the user curve and the effect of the user curve can be truly reflected.^22,23

Analysis of the velocity planning curve

For the velocity planning curve, most researchers have put their emphasis on positioning accuracy during the stop process of the guide bars, because the guide-bar shogging is a motion with high-speed intermittent start–stop. However, this focuses too much on the flexibility of bar movement. Quintic polynomial curve, S-shaped curve, modified trapezoid curve and others are also involved in their research. The velocity and acceleration of curves above all indeed have good continuity, which can eliminate acceleration steps and reduce the impact of mechanical movement effectively.²⁴ Nevertheless, the problem is that the speed is not able to reach a maximum value in an extremely brief period when the guide bars start and accelerate, meaning that with the increasing knitting speed, the time for guide-bar shogging is shortened. Lapping is challenging, as chain notation expresses in a short time so that the low dynamic response performance is shown and the knitting speed cannot be further improved.²⁵

Besides, to enhance the dynamic response performance of the system, the velocity planning curve is defined with uniform acceleration or deceleration. Through this method, the shogging speed can reach its maximum within the acceptable range of mechanical vibration. However, the overshoot at the stopping point appears at the same time, leading to low positioning accuracy.

Thus, a reasonable planning of velocity curve will affect the improvement of knitting speed directly.

Proposal of the mixed-velocity planning curve

In view of the existing deficiencies in these velocity curves, a mixed-velocity planning curve of uniform acceleration–deceleration and the quintic polynomial is proposed in the present study. The control principle is shown in Figure 1.

Figure 1.

Control principle of mixed-velocity planning curve.

The pulse volume of each process spindle motion is calculated by industrial computer (IPC) according to the spindle angle and technology structure, which is defined as the planning pulse. The planning pulse is sent to the motion controller in real time, and then the motion controller receives the planning pulse of each spindle.²⁶ The instruction signals sent by the upper computer need to go through a series of control processes, including three loops of cascade control, namely position loop, velocity loop and current loop, which correspond to position controller, velocity controller and current controller, separately. With the curve fitting of built-in functions, the deviation between the planning position and the feedback position of the motor encoder is adjusted by the position controller in the motion controller with the PID algorithm. Subsequently, the planning pulse is transformed into the voltage signal (the planning speed), which is sent to the driver in real time. The velocity loop and current loop are controlled by the built-in speed controller and current controller. Finally, the motor is controlled by a method of current (torque) so as to drive the guide bar to lap.

In addition, there is a time process for the command signal to arrive at the motor through the three-loop cascade control; this time process is called the response lag time in practical application. To compensate the retardation time, the lag compensation is designed innovatively, by which the parameter of the lag compensation can be adjusted according to the final pattern effect of yarn lapping. The planning pulse is sent to the motion controller in advance in order to achieve the effect of lag compensation.

Optimization of the electronic shogging system

Optimization of command signal resolution

Command signal resolution refers to the number of pulses sent to the servo driver from the motion controller within one rotation of the motor. Theoretically, the higher the command signal resolution, the better the shape-preserving property of the planning curve and the higher precision of the control system. However, too high a resolution may result in a larger pulse frequency, which further disturbs the transmission signal. Therefore, it is extremely important to select the optimal command signal resolution.²⁷ The maximum pulse frequency of the servo driver is less than 2 MHz, and is generally not more than 1 MHz (1 MHz =1,000,000 Hz). The formula is shown as follows:

R_{c} \times \frac{n_{m}}{60} \leq 1000000

(1)

R_{c} \leq 1000000 \times \frac{60}{n_{m}}

(2)

where, R_c denotes command signal resolution, n_m is rated speed of motor, and the rated speed of the motor used in practical application is n_m = 3000 rpm; rpm is the unit of speed, which is the abbreviation of rotation per minute. The following formula is obtained:

R_{c} \leq 20000

(3)

In practical application, the resolution should be as large as possible within the allowable range. Therefore, the spindle resolution R_c for one revolution is 20,000 pulses. Then, the number of pulses of each needle step can be calculated, which is defined as R_nc:

R_{n c} \leq d \times \frac{R_{c}}{P_{}}

(4)

where, d is needle step, the unit is millimeter and the gauge of the warp knitting machine is E28, so the needle step is defined as d = 25.4/28 = 0.907mm; P is the lead of ball screw, the unit is millimeter and the selected lead is P = 25mm.

R_{n c} = 0.907 \times \frac{20000}{25} = 725.6 \approx 725

(5)

The optimal resolution for each needle gauge is 725 pulses, derived from the optimum design of command signal resolution, which not only ensures the control accuracy of the system, but also avoids the distribution phenomenon in the process of signal transmission.

Optimization of velocity loop bandwidth

The bandwidth in the control system refers to the following ability of the system to the inputting command signal. The electronic shogging control system has three control loops including current loop, velocity loop and position loop from within. Additionally, their bandwidths are decreased in turn.²⁸

A simple PID system is established by Simulink using Matlab as shown in Figure 2. The PID parameters are adjusted automatically using Tune on PID adjustment interface, the results of which can be found in Figure 3(a).

Figure 2.

The PID system built by Simulink.

Figure 3.

Correspondence between time domain index and frequency domain index in PID system. (a) PID control interface; (b) The response speed in time domain; (c) The bandwidth in frequency domain.

Figure 3(b) and Figure 3(c) reveal correspondence between time domain index and frequency domain index directly. Response speed in the time domain corresponds to the bandwidth in the frequency domain. The larger the system bandwidth, the faster the response speed. Besides, transient characteristics in the time domain correspond to phase margin under the frequency domain. Moreover, the larger phase margin illustrates the better robustness.

The default bandwidth of the current loop is enough in servo control, and is also closely related to the internal hardware of the selected servo driver. The built-in hardware and algorithm cannot be changed once they are determined. In practical application, it is more directly significant to optimize the bandwidth of the velocity loop. The improvement of the velocity loop generates a higher gain of position loop, and the position command can be followed more quickly by the guide bars.

The factors affecting the bandwidth need to be analyzed in order to improve the velocity loop bandwidth. It can be concluded that the velocity loop bandwidth is related to the load inertia rate, encoder accuracy, speed controller gain, velocity estimation algorithm, and so on, based on theoretical and experimental analysis.²⁹

The load inertia rate should be controlled at about 1 when selecting the motor. Too high a load inertia rate limits physical bandwidth, causing significantly increased loss of the guide bar during acceleration and deceleration. As a result, acceleration and deceleration are unable to be realized quickly and the bandwidth is reduced drastically due to the mechanical resonance.

The load of the electronic shogging mechanism includes coupling, ball screw, guide device (guide shaft, guide seat and push rod) and guide bar, as shown in Figure 4. The motor rotates following the driver’s instruction signal. Meanwhile, the motor is connected with the ball screw through the coupling. The ball screw converts the rotary motion of the motor into linear reciprocating motion, and then drives the guide seat to reciprocate. The guide bar is connected with the guide seat through the push rod and steel wire rope, aiming to follow the guide seat for lapping. The load inertia is calculated.

Figure 4.

Connection mechanism of motor drive guide-bar shogging.

The groove part of the coupling is ignored, and the coupling is regarded as a constant cylinder. The formula of the rotary inertia for the coupling can be expressed as:

J_{a} = \frac{π}{32} ρ l (D_{c 1}^{4} - D_{c 2}^{4})

(6)

where coupling density is ρ = 2.7 × 10³ kg·m³, the length is l = 55 mm, the outside diameter is D_c1 =42 mm and the inside diameter is D_c2 = 16 mm. By substituting these values, we can obtain J_a = 0.44 ×10⁻⁴ kg·m².

The ball screw is composed of two parts. The screw thread is thicker, the diameter (D_s1) is 25 mm, the length (l_s1) is 218 mm, and the connection part with the motor is relatively thin. The diameter (D_s2) is 16 mm, and the length (l_s2) is 40 mm. In the calculation process, the factor of screw groove on the screw is ignored, and the screw can be regarded as a constant cylinder. The rotary inertia of the ball screw can be calculated as:

J_{b} = \frac{π}{32} ρ l_{s 1} D_{s 1}^{4} + \frac{π}{32} ρ l_{s 2} D_{s 2}^{4}

(7)

where, the density of screw material is ρ = 7.9 ×10³ kg·m³. The J_b = 0.68 × 10⁻⁴ kg·m² can be obtained by substituting these values.

As the guide bar and guide device (guide seat, guide shaft and push rod) are driven by the ball screw and then move horizontally, the equivalent moment of inertia of guide bar and guide device can be computed by the following formula:

J_{c} = M {(\frac{P}{2 π})}^{2}

(8)

In practical application, the helical pitch of the ball screw is P = 25 mm, the mass of the guide bar is M₁ =10.5 kg, the mass of the guide device is M₂ = 3.5 kg, and the total mass of guide bar and guide device is M = M₁+M₂ = 14 kg. Through substituting these values, J_c = 2.22 × 10⁻⁴ kg·m².

The total inertia (J) of system load is that:

J = J_{a} + J_{b} + J_{c}

(9)

Substituting the above calculation results, J =3.34 × 10⁻⁴ kg·m².

The principle of selecting the motor is that the load inertia should be close to 1 based on the required parameters. The YASKAWA motor typed SGM7A-25A is selected in the current study. The parameters of this motor are shown in Table 1. The ratio of load inertia is 1.05:1.

Table 1.

The parameters of SGM7A-25

Parameters	YASKAWA SGM7A-25A
Working voltage/VAC	200
Continuous locked rotor torque/Nm	7.96
Rated torque/Nm	7.96
Rated speed/rpm	3000
Rated power/KW	2.5
Peak torque/Nm	23.9
Continuous locked rotor current/A	15.6
Peak current/A	51
Moment of inertia/kg·m²	3.19 × 10⁻⁴
Load inertia ratio	1.05：1

The speed feedback value is usually collected in the current position during the sampling period of the velocity loop after moving average filter. It can be assumed that the sampling periods of speed is 0.125 ms and depth of moving average filter is 4, indicating that the speed can be derived through average movement within 0.5 ms. An absolute encoder more than 17 bits (23 bits is the best) is selected to implement the high-precision analysis of motor position and speed. In addition, the higher the resolution ratio of the encoder, the less high the frequency burr within the system.

υ = \frac{S_{2} - S_{1}}{n T}

(10)

The v is feedback speed. S₁ and S₂ are the first sampling position and the fourth one. The n is the depth of moving average filter and T is the sampling periods of the speed.

Quantization error of velocity measurement can be decreased by the optimization of velocity measurement algorithm in the case of the same encoder. However, what needs to be assured is that the small lag of the velocity measurement algorithm is the precondition of the optimization. Otherwise, the system will be unstable.

The velocity loop bandwidth is increased by the improvement of gain of speed loop controller based on the support of software and hardware. If the system gain is increased blindly, more high-frequency burr signals will be pulled in the closed-loop system, which results in the increasing error of output torque caused by variation error of velocity measurement.

Design of mixed-velocity planning curve

Analysis of the movement track of guide bar

Guide bars perform a three-dimensional motion when knitting, and a three-dimensional motion coordinate system is established, as shown in Figure 5.

Figure 5.

The guide-bar three-dimensional trajectory coordinates: (a) three-dimensional trajectory coordinate; (b) an intuitive graphic of machine.

Taking the plane with y-axis and z-axis as a knitting needle plane, in the direction of the positive axis x is defined as front part of needle while the negative position is defined as the back part of needle. Forward and backward swing amplitudes of the guide bar are measured by the spindle angle, which is represented by θ. The z-axis denoted by z is the height difference between the upper and lower position when swinging.

The guide bar of GB1 with chain notation “0-1/2-1//” is analyzed as an example in the current work. The shogging angle is shown in Table 2.

Table 2.

The shogging angle region of GB1

GB1	Overlapping		Underlapping
Status	Start	Stop	Start	Stop
Angle(°)	135	205	260	75

Point A (θ₁, d₁, h₁) is at the back part of the needle bar while Point B (θ₂, d₂, h₂) is at the front part of the needle bar. The high-speed back-and-forth motion between A and B is carried out when the guide bar is lapping.

According to the transmission mechanism of the warp knitting machine, the guide bar moves as underlap and stops when swinging at the plane with y-axis and z-axis, with the spindle angle being 75°. After swinging through this plane, the guide bar moves as overlap to the x-axis positive direction and then accelerates at the spindle angle 135° (along the y-axis positive direction). The guide bar stops moving at the spindle angle 205° before swinging to the knitting needle plane. The guide bar continues to move back through the knitting needle plane to the underlapping part and starts to move as underlap when the spindle angle is 260°.

The swing distance of the guide bar and the height difference that results from swing is decided by the crankshaft and spindle of the transmission mechanism. The crankshaft track is determined when designing the machine, and cannot be altered. The shogging movement of the guide bar is controlled by the electronic shogging system. Therefore, to improve the performance of the system, it is necessary to study the shogging movement.

Design of the shogging velocity curve

Table 2 shows that the angle of the overlap shogging is θ_f = 205–135 = 70° and the angle of the underlap is θ_b = 360–260 + 75 = 175°. Thus, the shogging speed of the overlap is faster than that of the underlap during the constant-speed movement of the spindle. In case of the same lapping distance, the requirement for dynamic response performance and the positioning accuracy of the guide bar when overlapping is much higher than that of underlapping, and thus the movement of overlap is analyzed in the present study.

The velocity in x-axis direction is zero at the angle of 135° and the guide bar moves speedily. The guide bar should speed down to zero when swinging at the angle of 205° and move one needle gauge (d = 0.907 mm) along the x-axis. The shogging time (t_f) for overlapping in one cycle (360°) is calculated with the spindle speed w = 2000 rpm as follows:

t_{f} = \frac{60}{w} \times \frac{θ_{f}}{360} = \frac{60}{2000} \times \frac{70}{360} = 0 .0058 s =5 .8 ms

(11)

The guide bar should move quickly and accurately at the speed from zero to maximum and then to zero with one-gauge movement within this short time. As high dynamic response and high positioning accuracy is required, a more reasonable velocity curve is needed in this system.

According to kinematical theory, uniform acceleration and deceleration are the fastest velocity curves of the shogging movement. As shown in Figure 6(a), the guide-bar speed is accelerated to the maximum with uniform acceleration and then decelerated constantly to zero. However, a hard shock during the fast deceleration resulted in position overshoot. The knitting speed cannot be increased due to the inaccuracy of positioning.

Figure 6.

The three velocity planning curves: (a) The velocity planning curve of uniform acceleration–deceleration; (b) The velocity planning curve of quintic polynomial; (c) The mixed-velocity planning curve of uniform acceleration–deceleration and quintic polynomial.

When the guide bar moves one needle for overlapping, the motor rotation angle is θ (rad):

θ = 2 π \frac{d}{P}

(12)

where P is the helical pitch of ball screw, P = 25 mm.

The acceleration (a_u) is as shown in equation (13):

a_{u} = \frac{4 θ}{t_{f}^{2}}

(13)

Additionally, the uniform acceleration–deceleration velocity curve formula can be expressed as equation (14) and equation (15). t is a variable of time.

v_{u 1} (t) = a_{u} t_{} = \frac{4 θ}{t_{f}^{2}} t_{} (0 < = t < = \frac{t_{f}}{2})

(14)

v_{u 2} (t) = a_{u} \frac{t_{f}}{2} - a_{u} (t_{} - \frac{t_{f}}{2}) = \frac{4 θ}{t_{f}^{2}} (t_{f} - t_{}) (\frac{t_{f}}{2} < t < = t_{f})

(15)

If a quintic polynomial curve is used in the shogging velocity curve as shown in Figure 6(b), there is a smooth curve when accelerating, showing that the movement process is flexible and there is no shock during the deceleration and a higher positioning accuracy performance. However, due to the flexibility of the traverse process, it is difficult to achieve high-response performance.

Equation (16) is quintic polynomial formula. θ_p(t) denotes the relationship between the rotation angle of motor and the time for overlapping.

θ_{p} (t) = c_{0} + c_{1} t^{} + c_{2} t^{2} + c_{3} t^{3} + c_{4} t^{4} + c_{5} t^{5} (0 < = t < = t_{f})

(16)

where, C₀, C₁, C₂, C₃, C₄, C₅ represent constants.

Taking a derivative of θ_p(t), velocity curve formula v_p(t) can be obtained.

v_{p} (t) = \frac{d_{θ}}{d_{t}}

(17)

v_{p} (t) = c_{1} + 2 c_{2} t^{} + 3 c_{3} t^{2} + 4 c_{4} t^{3} + 5 c_{5} t^{4} (0 < = t < = t_{f})

(18)

Deriving v_p(t), the acceleration curve formula a_p(t) is shown in equation (20):

a_{p} (t) = \frac{d_{v_{p}}}{d_{t}}

(19)

a_{p} (t) = 2 c_{2} + 6 c_{3} t^{} + 12 c_{4} t^{2} + 20 c_{5} t^{3}

(20)

According to the movement characteristics of guide bar lapping, $θ_{p} (0) = 0$ , $θ_{p} (t_{f}) = θ$ , $v_{p} (0) = 0$ , $v_{p} (t_{f}) = 0$ , $a_{p} (0) = 0$ , $a_{p} (t_{f}) = 0$ , these parameters are substituted to equation (16), equation (18), and equation (20) respectively. The constants $c_{0} = 0$ , $c_{1} = 0$ , $c_{2} = 0$ , $c_{3} = \frac{10 θ}{t_{f}^{3}}$ , $c_{4} = - \frac{15 θ}{t_{f}^{4}}$ , $c_{5} = \frac{6 θ}{t_{f}^{5}}$ can be obtained.

Substituting C₀, C₁, C₂, C₃, C₄, C₅ into equation (16) and equation (18), the formulas of position trajectory θ_p(t) and velocity curve v_p(t) for quintic polynomial can be calculated as follows:

θ_{p} (t) = \frac{10 θ}{t_{f}^{3}} t^{3} - \frac{15 θ}{t_{f}^{4}} t^{4} + \frac{6 θ}{t_{f}^{5}} t^{5} (0 < = t < = t_{f})

(21)

v_{p} (t) = \frac{30 θ}{t_{f}^{3}} t^{2} - \frac{60 θ}{t_{f}^{4}} t^{3} + \frac{30 θ}{t_{f}^{5}} t^{4} (0 < = t < = t_{f})

(22)

A mixed-velocity curve is designed innovatively with the aim to overcome the defects of the above two curves. The curve with uniform acceleration and deceleration is used in the acceleration section and partial deceleration section separately to satisfy the requirement of high dynamic response performance. A quintic polynomial curve is used when entering the plane of needle, by which flexible high positioning accuracy is realized. In practical application, to better guarantee the response performance of the system, the proportion of the uniform acceleration–deceleration curve is set to be slightly greater than the quintic polynomial, that is 1/2<r<1. The r is the proportion of the uniform acceleration–deceleration curve to the mixed-velocity curve. To show the characteristics of the mixed-velocity curve, take r = 3/4. The uniform acceleration–deceleration curve accounts for 3/4 of the whole curve, while the quintic polynomial curve occupies 1/4. The two curves are combined in proportion whose intersection can move horizontally as shown in Figure 6(c).

In the process of mixed-velocity curve combination, the sum of motor rotation angles corresponding to the two curves should always be kept unchanged. In formula (23), θ_u is the angle of motor rotation in uniform acceleration–deceleration section of the mixed-velocity curve:

θ_{u} = θ - (θ_{p} (t_{f}) - θ_{p} (r t_{f}))

(23)

The v_ue is the velocity at the junction of the mixed speed, where the velocity for uniform acceleration–deceleration and the quintic polynomial must be kept consistent:

v_{u e} = v_{p} (r t_{f})

(24)

The maximum velocity at uniform acceleration processing is v_u_max:

v_{u \max} = \frac{{a_{u m} t}_{f}}{2}

(25)

Equations (26) to (28) are used to calculate the angular displacement of acceleration section and deceleration section in the process of uniform acceleration–deceleration, which are, respectively, θ_u1 and θ_u2:

θ_{u 1} = \frac{1}{2} a_{u m} {(\frac{t_{f}}{2})}^{2} = \frac{a_{u m} t_{f}^{2}}{8}

(26)

θ_{u 2} = θ - (θ_{p} (t_{f}) - θ_{p} (r t_{f})) - \frac{a_{u m} t_{f}^{2}}{8}

(27)

θ_{u 2} = \frac{v_{u e} + v_{u \max}}{2} (r - \frac{1}{2}) t_{f}

(28)

According to these equations (24) to (28), the acceleration of a_um can be calculated as:

a_{u m} = \frac{4}{{r t}_{f}^{2}} (θ - θ_{p} (t_{f}) + θ_{p} (r t_{f}) - \frac{v_{p} (r t_{f})}{2} (r - \frac{1}{2}) t_{f})

(29)

The functions of the mixed-velocity planning curve are obtained by substituting equation (29) into these formulas, which can be expressed as:

v_{u m 1} (t) = a_{u m} t_{} = \frac{4}{{r t}_{f}^{2}} (θ - θ_{p} (t_{f}) + θ_{p} (r t_{f}) - \frac{v_{p} (r t_{f})}{2} (r - \frac{1}{2}) t_{f}) t_{} (0 < = t < = \frac{t_{f}}{2})

(30)

v_{u m 2} (t) = a_{u m} \frac{t_{f}}{2} - a_{u m} (t_{} - \frac{t_{f}}{2}) = \frac{4}{{r t}_{f}^{2}} (θ - θ_{p} (t_{f}) + θ_{p} (r t_{f}) - \frac{v_{p} (r t_{f})}{2} (r - \frac{1}{2}) t_{f}) ({t_{f} - t}_{}) (\frac{t_{f}}{2} < t < = r t_{f})

(31)

v_{p m} (t) = \frac{30 θ}{t_{f}^{3}} t^{2} - \frac{60 θ}{t_{f}^{4}} t^{3} + \frac{30 θ}{t_{f}^{5}} t^{4} (r t_{f} < t < = t_{f})

(32)

where v_um1(t), v_um2 (t) and v_pm(t) are the functions of the mixed-velocity planning curve in the uniform acceleration section, uniform deceleration section and quintic polynomial, respectively.

Experimental verification

The mixed-velocity curve of the guide-bar electronic shogging on the high-speed warp knitting machine is designed and the required high dynamic response and high positioning accuracy are analyzed in the present study. Lots of experiments are conducted to test and verify the superiority on the motion control of the system. The experimental types of warp knitting machine are HKS4 and RSE4, which are conventional high-speed warp knitting machines. In addition, the machine gauges include E28 and E32. The dynamic response and positioning accuracy can be obtained from the analysis of dynamic curves collected from the actual shogging movement of the guide bar.

Test platform and methods

The hardware composition and system functions of the test platform are shown in Table 3. Hardware is connected according to respective performance, and thus the test platform is established as presented in Figure 7 and Figure 8.

Table 3.

Hardware and functions of test platform

Hardware	Quantity	Function
HKS4 warp knitting machine	1	The main structure
IPC control system	1	Human-computer interactive
Motion control card	1	Motion controller
YASKAWA servo system	1	Control mechanism and actuators
LMS vibrometer	1	Vibration test of guide bars motion
Monitoring software	1	Collecting velocity curve

Figure 7.

Collection of the vibration signal of guide-bar shogging: (a) the guide bar on the machine; (b) the guide bar structure.

Figure 8.

The LMS device for data processing and analysis.

This paper focuses on the design of a control system based on a mixed-velocity planning curve. This test aims to show the optimization of the mixed-velocity curve for the control system. The test was carried out in contrast mode. Different velocity planning curves were selected under the same conditions of hardware. The status of each curve was monitored by LMS vibrometer, and they were also analyzed to investigate the effect of properties on the actual knitting process.

The lapping motion of the guide bar is a motion of high-speed periodic reciprocating with start and stop. Both vibration frequency and amplitude are different at the underlapping motion part, overlapping motion part and the needle plane part. Consequently, the vibration signal duration and amplitude can be analyzed from the collected vibration signal, and thus the dynamic response performance and positioning accuracy are acquired.

Our experimental procedure is as follows:

Prepare the hardware and set parameters of the motion control card and servo driver. The software is set on the established test platform. Under the same lapping notation, the influence of GB1 on the machine speed is greater than that of other guide bars (GB2, GB3, GB4), which is mainly caused by the minimum overlap angle of GB1. The overlap of GB1 has the highest requirement on the response performance of the system. Therefore, the performance of the system can be better reflected by the analysis and test of the overlap of GB1. The chain notation of GB1 is 0-1/2-1//.

Install the LMS vibrometer sensor to the side of the guide bar, as shown in Figure 7. The sensor can shield signals in the other direction when collecting the signals in a certain direction. The sensor installed here collects the vibration signals in the direction of the guide shogging motion rather than that in the forward–backward swing direction.

Select different velocity planning curves at the IPC interface and then load them in the motion controller. After starting the warp knitting machine, the shogging vibration signals produced by different velocity planning curves are collected at the same speed of 2000 rpm.

Compare and analyze the collected vibration signals and then draw the conclusions.

Results and discussion

The shogging vibration signals are collected by the signal pickup assembly on the LMS vibration tester and the vibration curves are shown on the signal analyzer by sensors. Figures 9, 10 and 11 present the shogging vibration curves controlled by the uniform acceleration–deceleration velocity planning curve, the quintic polynomial velocity planning curve and the mixed-velocity planning curve, respectively.

Figure 9.

The vibration curves under uniform acceleration–deceleration velocity planning curve at the speed of 2000 rpm: (a) The original curve graphic; (b) The graphic of adding cursors on original curve.

Figure 10.

The vibration curves under quintic polynomial velocity planning curve at the speed of 2000 rpm: (a) The original curve graphic; (b) The graphic of adding cursors on original curve.

Figure 11.

The vibration curves under mixed (uniform acceleration–deceleration and quintic polynomial) velocity planning curve at the speed of 2000 rpm: (a) The original curve graphic; (b) The graphic of adding cursors on original curve.

According to the yarn lapping rule analysis, the guide bars vibrate more when overlapping than when underlapping because the time of overlapping is shorter than underlapping in the same needle gauge. Therefore, it remains effective to analyze the data of overlapping. In order to be more convincing, multiple continuous values over a certain period of time are taken in Figures 9 –11. The average values of the collected continuous data are calculated to eliminate experimental deviation, and then the average values are compared and analyzed. The analyzed data are presented in Tables 4, 5 and 6, derived from the vibration curves under the uniform acceleration–deceleration velocity planning curve, the quintic polynomial velocity planning curve and the mixed-velocity planning curve, respectively.

Table 4.

Time and acceleration of overlapping on uniform acceleration–deceleration velocity planning curve

Uniform acceleration–deceleration	Period 1	Period 2	Period 3	Period 4
Time of overlapping (ms)	6.17	6.32	6.40	6.18
Average time (ms)	6.27
Acceleration(m/s²)	176.03	181.43	178.34	174.75
Average acceleration(m/s²)	177.64

Table 5.

Time and acceleration of overlapping on quintic polynomial velocity planning curve

Quintic polynomial	Period 1	Period 2	Period 3	Period 4
Time of overlapping (ms)	8.05	7.96	7.89	7.97
Average time (ms)	7.97
Acceleration(m/s²)	91.32	94.53	89.99	89.49
Average acceleration(m/s²)	91.33

Table 6.

Time and acceleration of overlapping on mixed-velocity planning curve

Mixed curve	Period 1	Period 2	Period 3	Period 4
Time of overlapping (ms)	6.95	7.04	6.87	7.03
Average time (ms)	6.97
Acceleration(m/s²)	133.43	130.30	132.30	129.46
Average acceleration(m/s²)	131.37

Through analyzing the data in Tables 4 –6, Figure 12 and Figure 13 are obtained.

Figure 12.

Time error bars and vibration acceleration error bars corresponding to average values on different velocity planning curves: (a) time error bars; (b) vibration acceleration error bars.

Figure 13.

Comparison of performance parameters for uniform acceleration–deceleration, quintic polynomial and mixed-velocity planning curves: (a) Average time of overlapping; (b) Average vibration acceleration of overlapping.

Figure 12(a) and Figure 12(b) present time error bars and vibration acceleration error bars corresponding to average time and average vibration acceleration on different velocity planning curves, respectively. It can be concluded from Figure 12 that the deviation values are extremely small compared with the corresponding average values, thus the deviation ranges of these experimental data are reasonable on different velocity planning curves. It is effective to analyze the average values of these experimental data.

It can be revealed from Figure 13(a) that the time of overlapping with the mixed-velocity planning curve is 1.0 ms less than that with quintic polynomial velocity planning curve. The less the overlapping time is, the higher performance the dynamic response is. The performance with the mixed-velocity planning curve is 12.5%, higher than that with the quintic polynomial velocity planning curve.

The vibration acceleration reflects the vibration degree of the guide-bar shogging. The greater the acceleration is, the greater the vibration is, causing the lower positioning accuracy of guide bars. From Figure 13(b), the vibration acceleration with uniform acceleration and deceleration curves is 46.27 m/s² faster than that with the mixed-velocity curve. In addition, it can also be revealed that the positioning accuracy of the mixed curve is also 26% higher than that with uniform acceleration and deceleration curves.

From the above analysis, the mixed-speed planning curve not only shows high dynamic response performance, but also satisfies the requirements of high positioning accuracy of guide-bar shogging motion.

Conclusion

To conclude, the warp knitting guide-bar shogging system with mixed-velocity curve has been applied successfully, which improves the performance during the yarn lapping on the high-speed warp knitting machine. The production speed of the warp knitting machine has been increased by approximately 300 rpm with a more stable condition. We summarize our contributions as follows:

The performance of the control system is improved by about 12% based on the optimization of the command signal pulse number and velocity loop bandwidth.

Guide bars move along the mixed-velocity planning curve to improve the dynamic response performance. In addition, the flexible shogging is achieved to reduce rigid impact on the guide bars.

To provide the control system with a mixed-velocity planning curve of a much higher control performance, the system parameters are optimized and adjusted during the shogging process. The positioning accuracy of the mixed-velocity planning curve is better than that of the velocity planning curve with acceleration and deceleration. Besides, the dynamic response also performs better than that of the quintic polynomial velocity planning curve.

Two curves covering acceleration–deceleration as well as the quintic polynomial velocity planning curves are screened out skillfully and combined into the mixed-velocity planning curve, and thus the knitting speed of the warp knitting machine can be significantly improved.

Footnotes

Declaration of conflicting interests

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

Funding

The authors disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was supported by the National Nature Science Foundation of China (61772238) and the Fundamental Research Funds for the Central Universities (JUSRP52013B).

ORCID iDs

Baoping Zheng

Gaoming Jiang

Haisang Liu

References

Amir

Saleh

Mohammadreza

, et al. On the running-in behavior of cam-follower mechanism. J Tribol Int 2018; 118: 301–313.

Gao

Zhang

SL.

Design of electronic cam control system based on PLC.

J Mech Elec Eng 2014; 31(11): 1365–1368.

Fan

Huang

, et al. Reliability design of an electronic cam curve for flying shear machine in short materials cutting. J Shanghai Jiaotong Uni Sci 2020; 25(2): 246–252.

Zhang

Xia

Liu

, et al. Analysis of shogging motion vibration of guide bars on warp knitting machines. J Text Res 2013; 34(7): 141–145.

Liu

YN.

S-model speed planning of NURBS curve based on uniaxial performance limitation. IEEE Access 2019; 7: 60837–60849.

Chen

Yang

ECH.

Near time-optimal S-curve velocity planning for multiple line segments under axis constraints. IEEE Trans Ind Electron 2018; 65(12): 9582–9592.

Rastegar

, et al. A real-time and look-ahead interpolation algorithm with axial jerk-smooth transition scheme for computer numerical control machining of micro-line segments. J Eng Manufact 2019; 233(9): 2007–2019.

Xia

MQ.

Motion rule of electronically pattern system on a high speed warp knitting machine.

Fibers Textil East Eur 2009; 17(4): 64–67.

Zhang

Jiang

PB.

Design of flexible electronic shogging system for high-speed warp knitting machine. J Donghua Uni Eng Ed 2013; 30(3): 202–206.

10.

Qin

Xia

Zhang

, et al. Control algorithm of electronic shogging system based on speed control mode in warp knitting. J Textil Res 2011; 32(6): 141–145.

11.

Liao

Jeng

Chieng

WH.

Electronic cam motion generation with special reference to constrained velocity, acceleration, and jerk. ISA Trans 2004; 43(3): 427–443.

12.

Kralev

Mitov

Slavov

, et al. Optimal three-loop cascade PI-P-PI controller for electro-hydraulic power steering system. Mater Sci Eng 2019; 664(1): 1–11.

13.

Zhang

Gao

Cheng

XY.

Analysis of velocity planning interpolation algorithm based on NURBS curve. AIP Conf Proc 2017, 1834: 040012(1–8).

14.

Tho

Akihiro

Kazuhiko

Minimum-time S-curve commands for vibration-free transportation of an overhead crane with actuator limits. Contr Eng Pract 2020; 98: 104390 (1-8).

15.

Zhu

Tat

Chee

KP.

Flexure-based magnetically levitated dual-stage system for high-bandwidth positioning. IEEE Trans Ind Inform 2019; 15(8): 4665–4675.

16.

Meng

JQ.

Improvement of servo stiffness of electronic shogging system on warp knitting machine.

J DongHua Uni Nat Sci 2019; 45(2): 270–274.

17.

Zheng

Xia

Liu

Simulation of electronic shogging system on warp knitting machine based on Simulink.

J Textil Res 2018; 39(2): 150–156.

18.

Zhao

Hasan

Development of auxetic warp knitted fabrics based on reentrant geometry. Textil Res J 2020; 90(3): 344–356.

19.

Zou

Guo

Hamimid

FY.

A novel robot trajectory planning algorithm based on NURBS velocity adaptive interpolation. Mech Mach Sci 2018; 55: 1191–1208.

20.

Zhang

Gao

Cheng

XY.

Smooth trajectory planning along Bezier curve for mobile robots with velocity constraints. Int J Contr Autom 2013; 6(2): 225–234.

21.

Shi

Dai

Liu

Linear interpolation control of numerical control machine tools and its realization based on multi-axis motion control card. Adv Mater Res 2011; 201: 2423–2426.

22.

Perumaal

Jawahar

Automated trajectory planner of industrial robot for pick-and-place task. Int J Adv Robot Syst 2013; 10(100): 1–17.

23.

Timar

Farouki

Smith

, et al. Algorithms for time–optimal control of CNC machines along curved tool paths. Robot Comput Integr Manufact 2005; 21(1): 37–53.

24.

Reiter

Müller

Gattringer

On higher-order inverse kinematics methods in time-optimal trajectory planning for kinematically redundant manipulators. IEEE Trans Industr Inform 2018; 14(4): 1681–1690.

25.

Liu

Xia

Zhang

Dynamic analysis of shogging motion mechanism of guide bar on warp knitting machine. J Textil Res 2012; 33(11): 121–126.

26.

Xie

Liu

JH.

The design of control system for multi joint robot based on PC+DSP. Mod Electron Technol 2018; 1: 1–5.

27.

Lorenz

Van Patten

KW.

High-resolution velocity estimation for all-digital, AC servo drives. IEEE Trans Ind Appl 1999; 27(4): 701–705.

28.

Dumanli

Sencer

Optimal high-bandwidth control of ball-screw drives with acceleration and jerk feedback. Precis Eng 2018; 54: 254–268.

29.

Gao

Tian

Wang

YT.

Fuzzy self-tuning tracking differentiator for motion measurement sensors and application in wide-bandwidth high-accuracy servo control. Sensors 2020; 20(3): 948–961.