Synchronization control strategy in multi-layer and multi-axis systems based on the combine cross coupling error

Abstract

Considering the variety of coupling errors and disturbances in multi-layer and multi-axis control systems, a novel control multi-layer and multi-axis strategy for structure is proposed in this article. First, the equivalent conversion of two models is proposed between master–slave mode and minimum adjacent coupling in multi-layer and multi-axis control model. Second, in view of existing synchronization control problem, this article presents master reference synchronous model and master–slave synchronous model in multi-layer and multi-axis synchronous control system. Moreover, an optimization control strategy based on combine cross coupling error control strategy is introduced. The stability and superiority of the proposed controller are theoretically analyzed. The combine cross coupling error experimental results demonstrate that interlayer starting synchronizing error maximum is 0.042 rad/s. Meanwhile, the startup and shutdown points of synchronizing error are 0.005 rad/s, the sync error tends to 0 between two points. The results show that the proposed strategy has a good tracking error and synchronizing error behavior.

Keywords

Synchronization control combine-cross-coupling error master–slave approach cross coupled control multi-layer and multi-axis adjacent coupling error

Introduction

Multi-axis centers play a key role in industrial fields to manufacture complex parts, such as impellers. In the machining of complex parts, synchronized multi-axis motions are required. Hence, the synchronous accuracy of translational and rotary axes is one of the important factors in the machines. Synchronous accuracy of rotary and translational axes has been investigated up to now.

Classical synchronous control theory is mainly proposed and developed by Y Koren¹ in the 1980s. The theory has three classes: the same ways (synchronized master command approach (SMCA)), master–slave mode (master–slave approach (MSA)), and the cross coupling control (CCC). Turl et al.² propose coupling compensation control strategy, which improves the synchronization performance, but for the motor coupling compensation control (n > 2), it always exists.³ The scholars proposed the partial CCC.^4,5 The existing synchronous control algorithm is mostly limited to two motors, which is difficult in expanding into the axis control.^6–9

Wei et al.^10,11 propose the problem of H∞ model approximation for a class of two-dimensional (2D) discrete-time Markovian jump linear systems with state-delays and imperfect mode information in 2015. The finite-frequency technique has the advantages of robust control, estimation, and fault detection. H Zhang and JM Wang^12–14 propose the observer design problem for linear-parameter-varying systems with uncertain measurements on scheduling variables.

The existing control strategies are synchronization coefficient is 1 or the proportion for each motor speed, which does not involve nonlinear in multi-layer and multi-axis. For this problem, this article proposes a new combine cross coupling error control strategy of synchronous control thought and synchronous control algorithm, which is incorporated into combine ring coupling control through nonlinear synchronization coefficient in the controller. Combine cross coupling errors are defined to be differential system errors in multi-layer and multi-axis control model. Finally, the experiments verify the stability and superiority of multi-layer and multi-axis synchronous control algorithm. This article provides a new idea for analyzing multi-axis synchronous control and provides a method for analyzing complex multi-axis synchronous structures.

Change of model

When the input signal is only one signal in the multi-layer and multi-axis control model, the master–slave mode is minimum adjacent coupling in its essence. In the case of multi-layer and multi-axis, the two modes can be as the equivalent conversion between two modes in Figure 1. For different input signals, the two modes cannot be replaced between the minimum adjacent coupling and master–slave mode. The input signal of the minimum adjacent coupling model arises from the same signal; for master–slave mode, slave axis receives a given input signal from master axis, the input signal of third layer from output signal of the second layer in multi-layer and multi-axis system.

Figure 1.

Multi-layer and multi-axis transformation diagrams.

Where M defines master axis, where S defines slave axis. Figure 1 shows three layers, each layer has three axes. $S_{11}$ , $S_{12}$ , and $S_{13}$ are slave axes of axis $M_{0}$ in first layer; $S_{31}$ , $S_{32}$ , and $S_{33}$ are slave axes of axis $S_{11}$ in second layer; $S_{31}$ , $S_{32}$ , and $S_{33}$ are slave axes of axis $S_{22}$ in third layer, $S_{11}$ and $S_{21}$ are both slave axis and master axis, respectively.

We can conclude that $f (x)$ is as nonlinear function by the analysis above

{\begin{matrix} S_{11} = f_{11} (x) M_{0} \\ S_{12} = f_{12} (x) M_{0} \\ S_{13} = f_{13} (x) M_{0} \\ S_{21} = f_{14} (x) M_{0} \\ S_{22} = f_{15} (x) M_{0} \\ S_{23} = f_{16} (x) M_{0} \end{matrix}

(1)

{\begin{matrix} S_{21} = f_{21} (x) S_{11} \\ S_{22} = f_{22} (x) S_{11} \\ S_{23} = f_{23} (x) S_{11} \end{matrix}

(2)

{\begin{matrix} S_{11} = f_{11} (x) M_{0} \\ S_{12} = f_{12} (x) M_{0} \\ S_{13} = f_{13} (x) M_{0} \\ S_{21} = f_{21} (x) f_{11} (x) M_{0} \\ S_{22} = f_{22} (x) f_{11} (x) M_{0} \\ S_{23} = f_{23} (x) f_{11} (x) M_{0} \end{matrix}

(3)

In the relationship between the master axis $M_{0}$ and the slave axis of layer 1, layer 2, and the nonlinear function in equation (1), $f_{1 i} (x)$ defines the relation between the slave axis of layers 1 and 2, which is established directly with the master axis $M_{0}$ . Equation (2) shows the nonlinear functional relationship between the slave axis $S_{11}$ of layer 1 and the slave axis of layer 2, $f_{2 i} (x)$ defines the nonlinear function between the slave axis of the layer 2 and the slave axis $S_{11}$ of layer 1.

From equations (1)–(3), we obtain establish the relationship with master axis $M_{0}$ , equation (3) is complicated. It is complex relationships between $S_{14}$ , $S_{15}$ , $S_{16}$ and axis $M_{0}$ , $S_{21}$ , $S_{22}$ , $S_{23}$ are slave axis of axis $S_{11}$ in second layer, it is easy to build relationships between $S_{21}$ , $S_{22}$ , $S_{23}$ and axis $S_{11}$ , which decreases the relationship between master axis $M_{0}$ . Master–slave axis cannot replace ring cross coupling error by equation (1) in multi-layer and multi-axis system, which has superiority for synchronization and stability.

A limitation of minimum adjacent coupling is that it is unable to realize the master–slave control, which is given in multi-layer and multi-axis model. It is easy to establish mathematic relationship between master axis and layer 1, or the other relationships between the layer 2 and layer 1 in Figure 1(a). If the second layer 2 directly establish the mathematical relationship is very complicated with the master axis. So the system model of multi-layer and multi-axis is meaningful, the existing research method is based on the synchronization coefficient of 1, and the requirements of each axis running situation has significant limitations. There is no master–slave relationship between minimum coupling, which could not run the nonlinear between layers^5,15,16

ε_{i} (t) = y_{i} e_{i} (t) - y_{i + 1} e_{i + 1} (t)

(4)

$ε_{i} (t)$ defines synchronization error of the same layer axis, $e_{i} (t)$ defines the tracking error of the same layer axis, and $y_{i}$ is the nonlinear coefficient of the same layer axis.

For front layer of multi-layer multi-axis, the relationship between the position and the velocity is given as $v = ds / dt$ , and there is a nonlinear relationship between axes, so this article discusses the proportion of nonlinear multi-layer and multi-axis model.

For master–slave model, linear change is constant A, and if change relationship is nonlinear between master axis and slave axis, which defines $f (x)$ , and if the other layers build relationships with master axis, two problems exist. First, it is difficulty in building relations with master axis; second, which is nonlinear relationship between master axis and slave axis; second, the axes of each layer should establish relationship with master axis in the same layer. This article proposes a combine cross coupling error synchronous control strategy in multi-layer and multi-axis system.

Combine cross coupling error synchronization control thought

According to different applications,^10,22,23 combine cross coupling error control strategy is in multi-layer and multi-axis system, and due to the existence of multi-layer operates in master–slave synchronization control method, the same layer axes are given the same control and error compensation control idea of loop coupling control mode. Synchronizing error will cause any motor speed change between the motor and the adjacent motor in ring coupling error control strategy. Compared to the input mode of signal, it is clear that all the electrical signals arise from one signal, and output signals are not the same.

This control strategy achieves consistent follow and axis coupling synchronizing error compensation for the same master axis signal, so the system is guaranteed in the process of starting performance and anti-jamming performance. For multi-layer and multi-axis complex system, this control strategy is a kind of ideal synchronization control strategy.^8,16,17

$U (s)$ defines the input reference signal, $f (s)$ , $g (s)$ defines transfer function of the disturbance point, $T (s)$ is the load torque, $ω (s)$ defines the axis outputs angular velocity in Figure 2.

Figure 2.

Block diagram of master reference synchronous control system.

In this kind of control mode, all units of the system share an input signal, the same as the ring cross coupling error, and the reference signal input signal is the same. Each partition by a single-axis drive does not share the same motor output power. Each unit obtains consistent input signal, and each unit of the input signal is not affected by any other factors besides the reference signal, so the disturbance in any unit will not affect other units.^8,18,19 The transfer function of first motor output angular velocity is derived as follows

G (s) = \frac{ω (s)}{U (s)}

(5)

The motor load torque transfer function is relative to the first motor (No. 1 axis)

H = \frac{ω (s)}{T (s)}

(6)

\begin{matrix} G_{1} (s) - G_{2} (s) & = \frac{f (s) g (s)}{1 + f (s) g (s)} - \frac{f' (s) g' (s)}{1 + f' (s) g' (s)} \\ = \frac{f (s) g (s) - f' (s) g' (s)}{(1 + f (s) g (s)) (1 + f' (s) g' (s))} \end{matrix}

(7)

\begin{matrix} G_{2} (s) - G_{3} (s) & = \frac{f' (s) g' (s)}{1 + f' (s) g' (s)} - \frac{f'' (s) g'' (s)}{1 + f'' (s) g'' (s)} \\ = \frac{f' (s) g' (s) - f'' (s) g'' (s)}{(1 + f' (s) g' (s)) (1 + f'' (s) g'' (s))} \end{matrix}

(8)

The selected parameter values are the same

\begin{matrix} f (s) = f' (s) = f'' (s) \\ g (s) = g' (s) = g'' (s) \end{matrix}

(9)

If the reference signal changes, the command reference control system can complete the synchronization

G_{1} (s) = G_{2} (s) = G_{3} (s)

(10)

One obtains

G_{1} (s) = G_{2} (s) = \dots = G_{n} (s)

(11)

Therefore, multi-axis master reference synchronous control system is the same tracking error with the same parameters

{\begin{matrix} H_{1 n} - H_{nn} = - \frac{g^{(n - 1)} (s)}{1 + f^{(n - 1)} (s) g^{(n - 1)} (s)} \\ H_{1 n} - H_{(n - 1) n} = 0 \\ ⋮ \\ H_{1 n} - H_{2 n} = 0 \end{matrix}

(12)

{\begin{matrix} H_{2 n} - H_{nn} = - \frac{g^{(n - 1)} (s)}{1 + f^{(n - 1)} (s) g^{(n - 1)} (s)} \\ ⋮ \\ H_{(n - 1) n} - H_{nn} = - \frac{g^{(n - 1)} (s)}{1 + f^{(n - 1)} (s) g^{(n - 1)} (s)} \end{matrix}

(13)

For the N-axis master reference control multi-axis synchronous control model, when a disturbance occurs on any ith axis, the synchronization between the other axes and the disturbance axis is the same, and there is no effect on the other axes in the control model.

In master reference control mode, Figure 3 shows that $U (s)$ is input signal, $ω (s)$ is output signal, $T (s)$ is interference signal, and $f (s)$ and $g (s)$ are function models.

Figure 3.

Block diagram of master–slave synchronous control system block diagram.

The following performance and synchronization performance of the N-layer multi-axis synchronization control model can be derived as follows

\begin{matrix} G_{1} (s) - G_{2} (s) & = \frac{f (s) g (s)}{1 + f (s) g (s)} (1 - \frac{f' (s) g' (s)}{(1 + f' (s) g' (s)}) \\ = \frac{f (s) g (s)}{1 + f (s) g (s)} \frac{1}{1 + f' (s) g' (s)} \end{matrix}

(14)

\begin{matrix} G_{1} (s) - G_{3} (s) = & \frac{f (s) g (s)}{1 + f (s) g (s)} \\ (1 - \frac{f' (s) g' (s) f'' (s) g'' (s)}{(1 + f' (s) g' (s)) (1 + f'' (s) g'' (s))}) \end{matrix}

(15)

The selected parameter values are the same

\begin{matrix} f (s) = f' (s) = f'' (s) \\ g (s) = g' (s) = g'' (s) \end{matrix}

(16)

One obtains

G_{1} (s) > G_{2} (s) > G_{3} (s)

(17)

\begin{matrix} G_{n} (s) = \frac{ω^{(n - 1)} (s)}{U (s)} = \frac{f (s) g (s) f' (s) g' (s) f'' (s) g'' (s) \dots f^{(n - 1)} (s) g^{(n - 1)} (s)}{(1 + f (s) g (s)) (1 + f' (s) g' (s)) (1 + f'' (s) g'' (s)) \dots (1 + f^{(n - 1)} (s) g^{(n - 1)} (s))} (18) \end{matrix}

(18)

G_{1} (s) > G_{2} (s) > G_{3} (s) > \dots > G_{n} (s)

(19)

The performance of the system following reduced with the increase in the number of layers

{\begin{matrix} H_{1 i} - H_{ii} = 0 - \frac{g^{(i - 1)} (s)}{1 + f^{(i - 1)} (s) g^{(i - 1)} (s)} = - \frac{g^{(i - 1)} (s)}{1 + f^{(i - 1)} (s) g^{(i - 1)} (s)} \\ H_{2 i} - H_{ii} = 0 - \frac{g^{(i - 1)} (s)}{1 + f^{(i - 1)} (s) g^{(i - 1)} (s)} = - \frac{g^{(i - 1)} (s)}{1 + f^{(i - 1)} (s) g^{(i - 1)} (s)} \\ ⋮ \\ H_{ni} - H_{ii} = (\frac{f^{(n - 1)} (s) g^{(n - 1)} (s)}{1 + f^{(n - 1)} (s) g^{(n - 1)} (s)} - 1) \frac{g^{(i - 1)} (s)}{1 + f^{(i - 1)} (s) g^{(i - 1)} (s)} \end{matrix}

(20)

H_{1 i} - H_{ii} = H_{2 i} - H_{ii} \dots < H_{ni} - H_{ii}

(21)

The ith layer load disturbances $(0 < i \leq n)$ .

During ith axis disturbance, the synchronization performance of axis is same from No. 1 axis to No. i − 1 axis, and equation is same. The synchronization performance of the system reduced with the increase in the number of layers.

As it can be seen by the above equations, synchronization performance of axis is the same axis which is relative to the N axis during N layer load disturbances.

Ring CCC strategy is considering each motor speed with a given speed in Figure 4, where $ω_{d}$ is reference speed, $E_{i}$ is speed difference, $ω_{i}$ is output speed, and the tracking error is also considering between the motor speed and the speed of adjacent error at the same time.⁶ Synchronization error will be fed back to the motor itself, the adjacent synchronizing error compensates between the adjacent two motors, so that all coupling motor arises from a ring. Combine cross coupling error is for different application objects, such as the production of high precision, and linear coil winding of high precision.^17,19,20

Figure 4.

Block diagram of ring cross coupling structure.

Based on the tracking error and the definition of the synchronizing error, axis synchronous control problem can be described as follows: the design of controller and control of the torque make the speed tracking error $e_{i} (t)$ and synchronizing error $ε_{i} (t)$ converge to 0.

Combine cross coupling synchronizing control strategy: control axis torque of each layer should be able to stable convergence,^21,24 which make this layer axis tracking error with upper layer of the synchronizing error, and two axes and the adjacent tree synchronizing error at the same time. The ith layer of control torque should be able to make the interlayer of i, $i - 1$ sync, which makes synchronizing error $η_{i - 1} (t) \to 0$ and $η_{i} (t) \to 0$ converges to 0. The interlayer adjacent axis control torque of ith layer should be able to make $η_{i - 1} (t) \to 0$ and make the axis $i + 1$ , i, $i - 1$ sync, and make synchronizing error $ε_{i - 1} (t) \to 0$ , $ε_{i} (t) \to 0$ converges to 0.

The combine cross coupling error method is proposed above in several ways. The single-layer multi-axis synchronous control system has advantage of synchronicity and following performance by ring cross coupling error control strategy. The multi-layer and multi-axis synchronous control system decreases the complexity of relationships with master axis by combine cross coupling error control strategy.

Combine cross coupling error control algorithm

Existing algorithms usually consider only the synchronization performance in the same layers, which do not analyze the synchronization of multi-layer and multi-axis. Combine cross coupling error synchronous control algorithm can be summarized as the following: defines the ith axis of the tracking error in multi-layer and multi-axis, each layer of the axis control considering the state of the adjacent two axis. For different applications, combine cross coupling error synchronous control algorithm has two advantages in Figure 5, one is ring cross coupling error in the same layer, which makes the synchronization between the axes, and the other is master–slave mode between layers, which maintains the synchronization with the master axis.

Figure 5.

Block diagram of combine-cross-coupling error structure.

Where $x_{i}^{d} (t)$ is reference velocity, $x_{i} (t)$ is actual output velocity of axis ith layer. The tracking error between axis i and reference input is described by

e_{i} = x_{i}^{d} (t) - x_{i} (t)

(22)

The synchronization error in interlayers^4,5

{\begin{matrix} ε_{1} = y_{1} e_{1} (t) - y_{2} e_{2} (t) \\ ε_{2} = y_{2} e_{2} (t) - y_{3} e_{3} (t) \\ ⋮ \\ ε_{n} = y_{n} e_{n} (t) - y_{1} e_{1} (t) \end{matrix}

(23)

Equation (23) shows the ring cross coupling error in the same layer, where $ε_{i} (t)$ denotes the synchronizing errors of same layer, $e_{i} (t)$ denotes the tracking error of interlayer, $y_{i}$ is synchronization coefficient $(i = 1, 2, \dots, n)$ , which is a nonlinear coefficient. The axes are ring cross coupling

y_{i} = GS (M + K_{M}) + K_{S}

(24)

where GS denotes transfer function, M is the master value, K_M is offset master value, and K_S is offset slave value. Equation (24) shows the nonlinear relationship between the slave axis and the master axis

Define $GS = 1$ , $K_{M} = 0$ , $K_{S} = 0$ , $y_{i} = M$ ;

Define $GS = a$ , $K_{M} = 0$ , $K_{S} = 0$ , $y_{i} = aM$ , where a denotes the synchronization coefficient;

Define GS is nonlinear coefficient, $K_{M} \neq 0$ , $K_{S} \neq 0$ , $y_{i} = GS (M + K_{M}) + K_{S}$ .

Equation (24) shows the tracking error without synchronizing error, which is master–slave mode in multi-layers

g_{i} = g_{i}^{d} (t) - g_{i} (t)

(25)

{\begin{matrix} η_{1} (t) = φ_{0} g_{0} (t) - φ_{1} g_{1} (t) \\ η_{2} (t) = φ_{1} g_{1} (t) - φ_{2} g_{2} (t) \\ ⋮ \\ η_{n} (t) = φ_{n - 1} g_{n - 1} (t) - φ_{n} g_{n} (t) \end{matrix}

(26)

where $g_{i} (t)$ denotes the axis synchronizing errors of interlayer; $n_{i} (t)$ denotes the tracking error of interlayer; $φ_{i}$ is synchronization coefficient of interlayer $(i = 1, 2, \dots, n)$ ; the $φ_{i}$ parameter settings are the same as $y_{i}$ ,which is a nonlinear coefficient. Dynamic characteristics of ith axis can be described as⁵

H_{i} (x_{i}) {\overset{\cdot}{x}}_{i} (t) + C_{i} (x_{i}, {\overset{\cdot}{x}}_{i}) x_{i} (t) + T_{i} (x_{i}, {\overset{\cdot}{x}}_{i}) = τ_{i}

(27)

where $x_{i} (t)$ denotes the coordinate of motion in the ith axis, $H_{i} (x_{i})$ represents movement inertia, $C_{i} (x_{i}, {\overset{\cdot}{x}}_{i})$ represents nonlinear characteristic parameter, $T_{i} (x_{i}, {\overset{\cdot}{x}}_{i})$ represents the effect caused by frictional force and other unexpected disturbances. $τ_{i}$ represents input torque. $e_{i}^{*} (t)$ denotes

{\begin{matrix} e_{1}^{*} = e_{1} (t) + a \int_{0}^{t} η_{1} (τ) d τ + β \int_{0}^{t} [ε_{1} (τ) - ε_{n} (τ)] d τ \\ e_{2}^{*} = e_{2} (t) + a \int_{0}^{t} η_{2} (τ) d τ + β \int_{0}^{t} [ε_{2} (τ) - ε_{1} (τ)] d τ \\ ⋮ \\ e_{n}^{*} = e_{n} (t) + a \int_{0}^{t} η_{n} (τ) d τ + β \int_{0}^{t} [ε_{n} (τ) - ε_{n - 1} (τ)] d τ \end{matrix}

(28)

$u_{i} (t)$ is control function

r_{i} (t) = u_{i} (t) - {\overset{\cdot}{x}}_{i} (t) = {\overset{\cdot}{e}}_{i} (t) + Λ \overset{*}{e_{i}} (t)

(29)

{\begin{matrix} u_{1} (t) = {\overset{\cdot}{ω}}_{1}^{d} (t) + a η_{1} (τ) + β [ε_{2} (t) - ε_{n} (t)] + Λ e_{1}^{*} (t) \\ u_{2} (t) = {\overset{\cdot}{ω}}_{2}^{d} (t) + a η_{2} (τ) + β [ε_{2} (t) - ε_{1} (t)] + Λ e_{2}^{*} (t) \\ ⋮ \\ u_{n} (t) = {\overset{\cdot}{ω}}_{n}^{d} (t) + a η_{n} (τ) + β [ε_{n} (t) - ε_{n - 1} (t)] + Λ e_{n}^{*} (t) \end{matrix}

(30)

where Λ is positive number, $α$ is positive number, $β$ is positive coupling coefficient, master–slave axis follows gradually, which can reach identity

\begin{matrix} u_{i} (t) = x_{i}^{d} (t) + (2 β + Λ) e_{i} (t) + 2 β Λ \\ \int_{0}^{t} e_{i} (t) d τ + a \int_{0}^{t} η_{i} (t) d τ \\ - β [\frac{y_{i}}{y_{i + 1}} e_{i + 1} (t) + \frac{y_{i}}{y_{i - 1}} e_{i - 1} (t)] \\ - β Λ \int_{0}^{t} [\frac{y_{i}}{y_{i + 1}} e_{i + 1} (t) + \frac{y_{i}}{y_{i - 1}} e_{i - 1} (t)] d τ \end{matrix}

(31)

Control function of arbitrary axis tracking error can be described in multi-layer and multi-axis

u_{trai} (t) = (2 β + Λ) e_{i} (t) + 2 β Λ \int_{0}^{t} e_{i} (τ) d τ + a \int_{0}^{t} η_{i} (τ) d τ

(32)

lim_{t \to \infty} e_{i} = lim_{t \to \infty} [x_{i}^{d} (t) - x_{i} (t)] = 0

(33)

Substituting equation (32) to equation (33) yields

\begin{matrix} lim_{t \to \infty} u_{trai} (t) = lim_{t \to \infty} [(2 β + Λ) e_{i} (t) + 2 β Λ \int_{0}^{t} e_{i} (τ) d τ + a \int_{0}^{t} η_{i} (τ) d τ] \\ = lim_{t \to \infty} a \int_{0}^{t} η_{i} (τ) d τ = η_{n} (t) = lim_{t \to \infty} [a \int_{0}^{t} φ_{i - 1} g_{i - 1} (t) - φ_{i} g_{i} (t) d τ] \end{matrix}

(34)

One obtains

lim_{t \to \infty} u_{trai} (t) = {\begin{matrix} a φ_{0} g_{0} & i = 0 \\ lim_{t \to \infty} a \int_{0}^{t} φ_{i - 1} g_{i - 1} (t) - φ_{i} g_{i} (t) d τ & i \neq 0 \end{matrix}

(35)

The tracking error of combine cross coupling error control system depends on the tracking error of multi-layer synchronization. $φ_{0} g_{0}$ is a constant, which is one of the main operation parameters. Synchronizing error control function is

\begin{matrix} u_{syni} (t) = β [\frac{y_{i}}{y_{i + 1}} e_{i + 1} (t) + \frac{y_{i}}{y_{i - 1}} e_{i - 1} (t)] \\ + β Λ \int_{0}^{t} [\frac{y_{i}}{y_{i + 1}} e_{i + 1} (t) + \frac{y_{i}}{y_{i - 1}} e_{i - 1} (t)] d τ \end{matrix}

(36)

\begin{matrix} lim_{t \to \infty} u_{syni} (t) = lim_{t \to \infty} {β [\frac{y_{i}}{y_{i + 1}} e_{i + 1} (t) + \frac{y_{i}}{y_{i - 1}} e_{i - 1} (t)] \\ + β Λ \int_{0}^{t} [\frac{y_{i}}{y_{i + 1}} e_{i + 1} (t) + \frac{y_{i}}{y_{i - 1}} e_{i - 1} (t)]} d τ = 0 \end{matrix}

(37)

When time tends to infinity, synchronizing error tends to 0. Define the control function of the ith axis as

u_{i} (t) = x_{i}^{d} (t) + u_{trai} (t) - u_{syni} (t)

(38)

where $u_{trai}$ is following error function, and $u_{syni}$ is synchronization error function. When time tends to infinity, the synchronizing error of combine cross coupling error control system depends on interlayer synchronizing error. The error of the system mainly arises from the same between the layers.

The output of the controller design for each layer and each axis in a given input torque of interlayer. $(H_{i} (x_{i}), C_{i} (x_{i}, {\overset{\cdot}{x}}_{i}), T_{i} (x_{i}, {\overset{\cdot}{x}}_{i}))$ defines the input and output^4,5

{\begin{matrix} τ_{1} = H_{1} (x_{1}) {\overset{\cdot}{u}}_{1} (t) + C_{1} (x_{1}, {\overset{\cdot}{x}}_{1}) u_{1} (t) + F_{1} (x_{1}, {\overset{\cdot}{x}}_{1}) + k_{r} r_{1} (t) + k_{ξ} η_{1} (τ) + k_{ε} [ε_{1} (t) - ε_{n} (t)] \\ τ_{2} = H_{2} (x_{2}) {\overset{\cdot}{u}}_{2} (t) + C_{2} (x_{2}, {\overset{\cdot}{x}}_{2}) u_{2} (t) + F_{2} (x_{2}, {\overset{\cdot}{x}}_{2}) + k_{r} r_{2} (t) + k_{ξ} η_{2} (τ) + k_{ε} [ε_{2} (t) - ε_{1} (t)] \\ ⋮ \\ τ_{n} = H_{n} (x_{n}) {\overset{\cdot}{u}}_{n} (t) + C_{n} (x_{n}, {\overset{\cdot}{x}}_{n}) u_{n} (t) + F_{n} (x_{n}, {\overset{\cdot}{x}}_{n}) + k_{r} r_{n} (t) + k_{ξ} η_{n} (τ) + k_{ε} [ε_{n} (t) - ε_{n - 1} (t)] \end{matrix}

(39)

where $k_{r}$ , $k_{ξ}$ , and $k_{ε}$ represent positive control gain coefficients. We can obtain available system for closed-loop dynamic model

\begin{matrix} H_{i} (x_{i}) {\overset{\cdot}{r}}_{i} (t) + C_{i} (x_{i}, {\overset{\cdot}{x}}_{i}) r_{i} (t) + k_{r} r_{i} (t) \\ + k_{ξ} η_{i} (t) + k_{ε} [ε_{i} (t) - ε_{i - 1} (t)] = 0 \end{matrix}

(40)

In this article, combine cross coupling error control algorithm not only makes the system good dynamic performance and anti-interference but also makes the system faster response speed at the same time, which improves the synchronization precision of the system for requiring higher synchronous precision in multi-layer and multi-axis model.

The computation complexity of multi-layer and multi-axis synchronization control scheme can be in high precision and high-speed machining, which effectively implements synchronous precision.

The advantages of multi-layer and multi-axis synchronization control scheme provide methods for analyzing complex topological multi-axis structures. The control scheme will provide a meaningful analysis basis for the motion control system of multi-layer complex structure and quantify the digital relationship between the layer and the axis of the system, which broadens the multi-axis motion control analysis.

The simulation parameters synchronous comparative analysis

In the process of multi-layer and multi-axis system, resistance mutations are changing, and the motor output axis torque must also adjust accordingly, to maintain the stability of the movement. Therefore, load disturbance is an important factor that affects synchronization performance of the system.

Based on the above model in Figure 5, the article uses MATLAB/Simulink platform to build a system simulation model.^25–27 Here, torque constant K_t = 2.2 N m/A, back electromotive force coefficient K_e = 0.33, stator resistance R = 0.53Ω, stator inductance L = 8.7 mH, current sensor feedback coefficient K_f = 0.57, current controller gain K_pi = 300, moment of inertia J = 0.0045 kg m², peak current I_max = 46 A.

The master axis speed is set to 1 rad/s. The method was further tested by adding a force disturbance to axis M₀ and axis S₁₁ from 25 to 35 s during the motion, which makes it deviate from the reference speed.^5,8Figure 6 shows the combine cross coupling error structure in multi-layer and multi-axis synchronized control model. System is the action of uniform motion with unit step signal. The simulation results are shown in Figures 6 –8. The tracking error and synchronizing error curves without disturbance are shown in Figure 6. The tracking error and synchronizing error curves with axis M₀ disturbance are shown in Figure 7. The tracking error and synchronizing error curves with axis S₁₁ disturbance are shown in Figure 8.

Figure 6.

(a) Axis M₀, axis S₁₁, axis S₂₁, and axis S₃₁ tracking error curve without disturbance, (b) synchronizing error curve between axis M₀ and axis S₁₁, (c) synchronizing error curve between axis M₀ and axis S₂₁, and (d) synchronizing error curve between axis M₀ and axis S₃₁.

Figure 7.

(a) Axis M₀, axis S₁₁, axis S₂₁, and axis S₃₁ tracking error curve with axis M₀ disturbance, (b) synchronizing error curves between axis M₀ and axis S₁₁, (c) synchronizing error curves between axis M₀ and axis S₂₁, and (d) synchronizing error curves between axis M₀ and axis S₃₁.

Figure 8.

(a) Axis M₀, axis S₁₁, axis S₂₁, and axis S₃₁ tracking error curve with axis S₁₁ disturbance, (b) tracking error local curve, (c) synchronizing error curves between axis M₀ and axis S₁₁, (d) synchronizing error curves between axis M₀ and axis S₂₁, and (e) synchronizing error curves between axis M₀ and axis S₃₁.

Experimental results

In the process of start, the system of the tracking error converges to 0 quickly, interlayer starting maximum synchronizing error is 0.042 rad/s in Figure 6, the maximum starting synchronous control accuracy is 4.2%, the maximum synchronous error is 0.001 rad/s in steady state, synchronous control accuracy is close to 0, which can achieve high-speed synchronization control system of the performance index requirements.

Figures 7 and 8 show that the system tracking error and synchronizing error response in load mutation (25–35 s). When axis M₀ disturbed, interlayer tracking error and starting maximum synchronizing error are 0.05 rad/s and the starting maximum synchronous control accuracy is 5% in Figure 7. In the case of a disturbance, the startup and shutdown points of maximum synchronizing error are 0.12 rad/s, the sync error converges to 0 between two points.

Under the first-layer axis perturbation, the startup and shutdown points of synchronizing error is 0.005 rad/s, the sync error converges to 0 in Figure 8.

For multi-layer and multi-axis synchronous control model, it can be seen that the synchronizing error increases with the number of layers, the synchronizing error become main problems interlayer in interlayer synchronizing error, which proves the rationality of the type (30) and (31).

The results show that the method is less tracking error and synchronizing error, which effectively restrains the load mutation and the influence of parameter perturbation. The synchronicity of combine cross coupling error effectively achieves the requirements of high-precision synchronous control system performance index.

Conclusion

A novel control multi-layer and multi-axis strategy for structure is proposed in this article. Synchronization control strategy of combine cross coupling error is produced. Combine cross coupling error control strategy is proposed in multi-layer and multi-axis system, and the motor synchronous error is low. The axis load disturbance occurs, system synchronization error is effectively suppressed, axis disturbance has little impact on synchronization performance, and the dynamic synchronization performance is better than using minimum adjacent coupling control strategy. The simulation experimental results demonstrate that interlayer starting maximum synchronizing error is 0.042 rad/s; meanwhile, the startup and shutdown points of synchronizing error is 0.005 rad/s, and the sync error tends to 0 between two points. It is inherently capable of maintaining synchronization between the axes during startup and shutdown and even during extreme or abnormal load conditions. Simulation experimental results verify the effectiveness of the proposed approaches.

The research of complex multi-layer and multi-axis model will be in the direction of development.^28,29 For multi-layer multi-axis system, the local stability characteristics of the analysis will be of meaningful research. In the future research, we will employ convex optimization algorithms to improve multi-layer and multi-axis model.^30,31

Footnotes

Academic Editor: Hamid Reza Karimi

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 by Anhui Provincial Natural Science Foundation of China (grant no. 1608085ME106) and Natural Science Foundation of the Anhui Higher Education Institutions of China (grant no. KJ2015A063).

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