Nonlinear Position Control Using Only Position Feedback under Position Errors and Yaw Constraints for Air Bearing Planar Motors

Abstract

In this paper, we propose a nonlinear position control using only position feedback to guarantee the tolerances for position tracking errors and yaw. In the proposed method, both mechanical and electrical dynamics are considered. The proposed method consists of the nonlinear position controller and nonlinear observer. The nonlinear position controller is designed by a backstepping procedure using the barrier Lyapunov function to satisfy the constraints of position error and yaw. The nonlinear observer is developed to estimate full state using only position feedback. The stability of the closed-loop system is proven using Lyapunov and input-to-state stabilities. Consequently, the proposed method satisfies the constraints of position error and yaw using only position feedback for the planar motor.

1. Introduction

A planar motor is a dual-axis linear motion motor that plays a significant role not only in semiconductor manufacturing systems and precision machine tools but also in automated assemblies.

A planar motor consists of four forcers that are symmetrically mounted on a puck, as shown in Figure 1 [1]. The position sensors are mounted on the puck to measure the positions for X and Y directions. Two position sensors are inserted in planar motor to calculate yaw angle using position difference between two position sensors in X direction. Each forcer is a one-axis force generation device. Forcers $X_1$ and $X_2$ generate force in the X direction whereas forcers $Y_1$ and $Y_2$ generate force in the Y direction. Yaw ($\theta$) is generated by asymmetries in the forcers and leads to loss of synchronization between motor and platen teeth. It is undesirable and results in a severe force drop problem for electro-magnetic force. These forcers can be operated as stepper motors. The linear motion is achieved by the application of a proper sequence of phase currents. The planar motor operates on a steel waffle platen and is floated on the platen by an air bearing. This planar motor has three degrees of freedom that consists of two translational degrees of freedom and an unwanted rotational degree of freedom (yaw) due to the fact that not all forcers can be mounted at the center of mass. This forcer is capable of high position resolution (2 $\mathsf{\mu }$m) and high speed moves (1 m/s) using open-loop microstepping. However, when these forcers are operated by open-loop microstepping, they may miss steps, or have long settle times Furthermore, the undesired rotation (yaw) produced by the asynchronous mounted forcers at the center of mass cannot be regulated by open-loop microstepping.

Various feedback control methods have been proposed to control planar motors [2,3,4,5,6]. A robust adaptive control was designed to improve the position tracking performance [2]. In Reference [3], a proportional-integral-derivative (PID) controller with a velocity estimator was developed to reduce the settling time. An adaptive variable structure controller was proposed to guarantees global asymptotic tracking of a reference trajectory in [4]. A robust backstepping method was proposed to control planar motors without parameter information and current measurements [5]. The lead and proportional-integral (PI) compensators were proposed to satisfy phase margin for planar motors [6]. A learning adaptive robust control motion controller was proposed for a magnetically levitated planar motor to achieve good tracking performance [7]. In Reference [8], a predictive position control method using trajectory gradient soft constraint with attenuation coefficients in the weighting matrix to achieve high-precision, time-varying, and long-stroke positioning was proposed for planar motors.

These methods improved the position tracking performance; however, several drawbacks still exist. First, for planar motor control, satisfying the constraint of yaw regulation is important to avoid step-out. The previously reported methods have improved the position control performance; however, the constraining of the yaw cannot be guaranteed by these methods. Furthermore, the position tracking error constraints in both X and Y directions are also important. These constraints may be satisfied by the use of high gain in the controller. However, the use of high gain amplifies the ripple because of modeling errors, parameter uncertainties, and measurement noise [9]; thus, the system may become unstable. In addition, it may also result in peaking phenomenon; thus, the lateral control in the transient response becomes poor. Thus, the design of the control method is required to satisfy the constrains of the outputs [10,11,12,13,14]. Furthermore, in the previous methods, the currents are regarded as the inputs of the planar motor because electrical dynamics are faster than mechanical dynamics. That is, the electrical dynamics are neglected. However, when current control is used, the phase currents are reduced by back-back electromotive forces (EMFs) and have phase lags because of inductances [15]. Therefore, a feedback controller considering electrical dynamics is required for improving the control performance of planar motors [16]. A robust adaptive control with the consideration of electrical dynamics was proposed for planar motors [16]. However, this method required full state feedback. Generally, the position can be obtained by using a laser interferometer, and subsequently, the velocity is estimated. Normally, the currents are measured using sensing resistors. However, high-frequency noises affect current measurements [17]; thus, an estimation of currents is necessary. To the best of our knowledge, there is no method to estimate full state using only position feedback for planar motors.

In this paper, we propose a nonlinear position control using only position feedback under the position errors and yaw constraints for air-bearing planar motors. In the proposed method, both mechanical and electrical dynamics are considered. The proposed method consists of the nonlinear position controller and nonlinear observer. The nonlinear position controller is designed using a backstepping procedure using barrier Lyapunov function (BLF) to satisfy the constraints of position errors and yaw. The nonlinear observer is developed to estimate full state using only position feedback. The stability of the closed-loop system is proven using Lyapunov stability and input-to-state stability (ISS). Consequently, the proposed method satisfies the constraints of position error and yaw using only position feedback for planar motors. The proposed method is validated via simulations. The main contributions of the proposed method can be summarized as follows:

• The whole dynamics including both the mechanical and the electrical dynamics is considered in the controller design.
• The tolerance for position tracking errors and yaw are guaranteed using only position feedback in the proposed method.
• The estimated state variables are used instead of the measured signals so that the measurement noise cannot affect the control performance.

2. Mathematical Model of Planar Motor

The planar motor consists of four forcers ($X_1$, $X_2$, $Y_1$, and $Y_2$) symmetrically mounted on a puck. The principles of each forcer are similar to those of a permanent magnet stepper motor. The electrical dynamics of the forcer $X_1$ is expressed [5] as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{i}}_{x_{1a}}=& \frac{1}{L}[v_{x_{1a}}-R{i}_{x_{1a}}-\kappa {\dot{x}}_{1}cos\left(\gamma x_1\right)]\hfill \\ \hfill {\dot{i}}_{x_{1b}}=& \frac{1}{L}[v_{x_{1b}}-R{i}_{x_{1b}}-\kappa {\dot{x}}_{1}sin\left(\gamma x_1\right)],\hfill \end{array}\end{array}$$

(1)

where $x_1$ is the position of the forcer $X_1$, $i_{x_{1a}}$ and $i_{x_{1b}}$ are the currents in the forcer $X_1$, $v_{x_{1a}}$ and $v_{x_{1b}}$ are the voltage inputs in the forcer $X_1$, $\kappa$ is the force constant, R is the resistance of winding, L is the inductance, and $\gamma =\frac{2\pi }{p}$ where p is the toothpitch. The electrical dynamics of the other forcers $X_2$, $Y_1$, and $Y_2$ are the same as those of $X_1$. The forces, $F_{x_1}$, $F_{x_2}$, $F_{y_1}$, and $F_{y_2}$, are generated by the forcers $X_1$, $X_2$, $Y_1$ and $Y_2$, respectively, as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_{x_1}=& \kappa (i_{x_{1a}}cos\left(\gamma x_1\right)+i_{x_{1b}}sin\left(\gamma x_1\right))\hfill \\ \hfill F_{x_2}=& \kappa (i_{x_{2a}}cos\left(\gamma x_2\right)+i_{x_{2b}}sin\left(\gamma x_2\right))\hfill \\ \hfill F_{y_1}=& \kappa (i_{y_{1a}}cos\left(\gamma y_1\right)+i_{y_{1b}}sin\left(\gamma y_1\right))\hfill \\ \hfill F_{y_2}=& \kappa (i_{y_{2a}}cos\left(\gamma y_2\right)+i_{y_{2b}}sin\left(\gamma y_2\right)).\hfill \end{array}\end{array}$$

(2)

The total forces $F_x$ and $F_y$ in the X and Y directions, and the torque $\tau$ are calculated as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_x=& F_{x_1}+F_{x_2}\hfill \\ \hfill F_y=& F_{y_1}+F_{y_2}\hfill \\ \hfill \tau =& (F_{x_1}-F_{x_2})r+(F_{y_1}-F_{y_1})r,\hfill \end{array}\end{array}$$

(3)

where r is the distance from the center of the motor to the forcer. Therefore, the dynamics of the planar motor are obtained as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{x}=& x_v\hfill \\ \hfill {\dot{x}}_v=& \frac{1}{M}(F_x-{\eta }_x{x}_v)\hfill \\ \hfill \dot{y}=& y_v\hfill \\ \hfill {\dot{y}}_v=& \frac{1}{M}(F_y-{\eta }_y{y}_v)\hfill \\ \hfill \dot{\theta }=& {\theta }_v\hfill \\ \hfill {\dot{\theta }}_v=& \frac{1}{I}(\tau -{\eta }_{\theta }{\theta }_v)\hfill \\ \hfill {\dot{i}}_{x_{1a}}=& \frac{1}{L}[v_{x_{1a}}-R{i}_{x_{1a}}-\kappa x_{1_v}cos\left(\gamma x_1\right)]\hfill \\ \hfill {\dot{i}}_{x_{1b}}=& \frac{1}{L}[v_{x_{1b}}-R{i}_{x_{1b}}-\kappa x_{1_v}sin\left(\gamma x_1\right)]\hfill \\ \hfill {\dot{i}}_{x_{2a}}=& \frac{1}{L}[v_{x_{2a}}-R{i}_{x_{2a}}-\kappa x_{2_v}cos\left(\gamma x_2\right)]\hfill \\ \hfill {\dot{i}}_{x_{2b}}=& \frac{1}{L}[v_{x_{2b}}-R{i}_{x_{2b}}-\kappa x_{2_v}sin\left(\gamma x_2\right)]\hfill \\ \hfill {\dot{i}}_{y_{1a}}=& \frac{1}{L}[v_{y_{1a}}-R{i}_{y_{1a}}-\kappa y_{1_v}cos\left(\gamma y_1\right)]\hfill \\ \hfill {\dot{i}}_{y_{1b}}=& \frac{1}{L}[v_{y_{1b}}-R{i}_{y_{1b}}-\kappa y_{1_v}sin\left(\gamma y_1\right)]\hfill \\ \hfill {\dot{i}}_{y_{2a}}=& \frac{1}{L}[v_{y_{2a}}-R{i}_{y_{2a}}-\kappa y_{2_v}cos\left(\gamma y_2\right)]\hfill \\ \hfill {\dot{i}}_{y_{2b}}=& \frac{1}{L}[v_{y_{2b}}-R{i}_{y_{2b}}-\kappa y_{2_v}sin\left(\gamma y_2\right)],\hfill \end{array}\end{array}$$

(4)

where x is the X axis position of the center of the puck, y is the Y axis position of the center of the puck, and $\theta$ is the yaw rotation. $x_1=x+rsin\left(\theta \right)$, $x_2=x-rsin\left(\theta \right)$, $y_1=y+rsin\left(\theta \right)$, and $y_2=y-rsin\left(\theta \right)$ are the positions of force $X_1$, $X_2$, $Y_1$, and $Y_2$, respectively. $x_{1_v}$, $x_{2_v}$, $y_{1_v}$, and $y_{2_v}$ are the linear velocity of the forcers, ${\theta }_v$ is the angular velocity of the yaw rotation. $i_i^{\prime }$s represent the currents in the forcers, $v_i^{\prime }$s represent the voltage inputs in the forcers, and, ${\eta }_x$, ${\eta }_y$, and ${\eta }_{\theta }$ are the coefficient of viscous friction.

3. Nonlinear Position Controller Design

3.1. Backstepping Control Using BLF for Mechanical Dynamics

In this subsection, the mechanical controllers are derived by a backstepping procedure using the BLF. In the context of mechanical dynamics, $F_x$, $F_y$, and $\tau$ can be regarded as virtual inputs, which are expressed as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{x}=& x_v\hfill \\ \hfill {\dot{x}}_v=& \frac{1}{M}(F_x-{\eta }_x{x}_v)\hfill \\ \hfill \dot{y}=& y_v\hfill \\ \hfill {\dot{y}}_v=& \frac{1}{M}(F_y-{\eta }_y{y}_v)\hfill \\ \hfill \dot{\theta }=& {\theta }_v\hfill \\ \hfill {\dot{\theta }}_v=& \frac{1}{I}(\tau -{\eta }_{\theta }{\theta }_v).\hfill \end{array}\end{array}$$

(5)

The mechanical errors are defined as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e_x=& x-x^d\hfill \\ \hfill e_{x_v}=& x_v-x_v^{*}\hfill \\ \hfill e_y=& y-y^d\hfill \\ \hfill e_{y_v}=& y_v-y_v^{*}\hfill \\ \hfill e_{\theta }=& \theta -{\theta }^d\hfill \\ \hfill e_{{\theta }_v}=& {\theta }_v-{\theta }_v^{*},\hfill \end{array}\end{array}$$

(6)

where $x^d$ and $y_d$ are the desired positions; and ${\theta }_d$ is the desired yaw, which is generally zero. Furthermore, $x_v^{*}$, $y_v^{*}$, and ${\theta }_v^{*}$ will be defined. The mechanical error dynamics are expressed as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{e}}_x=& x_v-{\dot{x}}_d\hfill \\ \hfill {\dot{e}}_{x_v}=& \frac{1}{M}(F_x-{\eta }_x{x}_v)-{\dot{x}}_v^{*}\hfill \\ \hfill {\dot{e}}_y=& y_v-{\dot{y}}_d\hfill \\ \hfill {\dot{e}}_{y_v}=& \frac{1}{M}(F_y-{\eta }_y{y}_v)-{\dot{y}}_v^{*}\hfill \\ \hfill {\dot{e}}_{\theta }=& {\theta }_v-{\dot{\theta }}_d\hfill \\ \hfill {\dot{e}}_{{\theta }_v}=& \frac{1}{I}(\tau -{\eta }_{\theta }{\theta }_v)-{\dot{\theta }}_v^{*}.\hfill \end{array}\end{array}$$

(7)

Theorem 1.

Consider the mechanical error dynamics (7) with $|e_x\left(0\right)|<b_x$, $|e_y\left(0\right)|<b_y$, and $|e_{\theta }\left(0\right)|<b_{\theta }$ where $b_x$, $b_y$, and $b_{\theta }$ are the positive constants and tolerances of the constraints for $e_x$, $e_y$, and $e_{\theta }$, respectively. If the virtual inputs are expressed as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill x_v^{*}=& -k_x{e}_x(b_x^2-e_x^2)+{\dot{x}}_d\hfill \\ \hfill y_v^{*}=& -k_y{e}_y(b_y^2-e_y^2)+{\dot{y}}_d\hfill \\ \hfill {\theta }_v^{*}=& -k_{\theta }{e}_{\theta }(b_{\theta }^2-e_{\theta }^2)+{\dot{\theta }}_d,\hfill \end{array}\end{array}$$

(8)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_x=& -k_{x_v}{e}_{x_v}+{\eta }_x{x}_v+M{\dot{x}}_v^{*}-\frac{e_x}{b_x^2-e_x^2}\hfill \\ \hfill F_y=& -k_{y_v}{e}_{y_v}+{\eta }_y{y}_v+M{\dot{y}}_v^{*}-\frac{e_x}{b_y^2-e_y^2}\hfill \\ \hfill \tau =& -k_{{\theta }_v}{e}_{{\theta }_v}+{\eta }_{\theta }{\theta }_v+I{\dot{\theta }}_v^{*}-\frac{e_{\theta }}{b_{\theta }^2-e_{\theta }^2},\hfill \end{array}\end{array}$$

(9)

where $k_x$, $k_{x_v}$, $k_y$, $k_{y_v}$, $k_{\theta }$, and $k_{{\theta }_v}$ are the positive controller gains that are applied to the mechanical tracking error dynamics (7), then $|e_x\left(t\right)|<b_x$, $|e_y\left(t\right)|<b_y$, and $|e_{\theta }\left(t\right)|<b_{\theta }$, $\forall t>0$ and $e_x\left(t\right)$, $e_y\left(t\right)$, and $e_{\theta }\left(t\right)$ asymptotically converge to zero.

Proof of Theorem 1.

The BLF, $V_{m_1}$, is defined as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_{m_1}=& \frac{1}{2}log\left(\frac{b_x^2}{b_x^2-e_x^2}\right)+\frac{1}{2}log\left(\frac{b_y^2}{b_y^2-e_y^2}\right)\hfill \\ \hfill & +\frac{1}{2}log\left(\frac{b_x^2}{b_x^2-e_x^2}\right).\hfill \end{array}\end{array}$$

(10)

The derivative of $V_{m_1}$ with respect to time is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_{m_1}=& \frac{e_x{\dot{e}}_x}{b_x^2-e_x^2}+\frac{e_y{\dot{e}}_y}{b_y^2-e_y^2}+\frac{e_{\theta }{\dot{e}}_{\theta }}{b_{\theta }^2-e_{\theta }^2}\hfill \\ \hfill =& \frac{e_x(x_v^{*}+e_{x_v}-{\dot{x}}_d)}{b_x^2-e_x^2}+\frac{e_y(x_y^{*}+e_{x_y}-{\dot{y}}_d)}{b_y^2-e_y^2}\hfill \\ & +\frac{e_{\theta }({\theta }_v^{*}+e_{{\theta }_v}-{\dot{\theta }}_d)}{b_{\theta }^2-e_{\theta }^2}.\hfill \end{array}\end{array}$$

(11)

With $x_v^{*}$, $y_v^{*}$, and ${\theta }_v^{*}$ (8), ${\dot{V}}_{m_1}$ becomes

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_{m_1}=& -k_x{e}_x^2+\frac{e_x{e}_{x_v}}{b_x^2-e_x^2}-k_y{e}_y^2+\frac{e_y{e}_{y_v}}{b_y^2-e_y^2}-k_{\theta }{e}_{\theta }^2+\frac{e_{\theta }{e}_{{\theta }_v}}{b_{\theta }^2-e_{\theta }^2}.\hfill \end{array}\end{array}$$

(12)

Let us define $V_{m_2}$ as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_{m_2}=& V_{m_1}+\frac{1}{2}{e}_{x_v}^2+\frac{1}{2}{e}_{x_y}^2+\frac{1}{2}{e}_{{\theta }_v}^2.\hfill \end{array}\end{array}$$

(13)

We obtain ${\dot{V}}_{m_2}$ as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_{m_2}=& -k_x{e}_x^2+\frac{e_x{e}_{x_v}}{b_x^2-e_x^2}+e_{x_v}{\dot{e}}_{x_v}-k_y{e}_y^2+\frac{e_y{e}_{y_v}}{b_y^2-e_y^2}+e_{y_v}{\dot{e}}_{y_v}\hfill \\ \hfill & -k_{\theta }{e}_{\theta }^2+\frac{e_{\theta }{e}_{{\theta }_v}}{b_{\theta }^2-e_{\theta }^2}+e_{{\theta }_v}{\dot{e}}_{{\theta }_v}\hfill \\ \hfill =& -k_x{e}_x^2+\frac{e_x{e}_{x_v}}{b_x^2-e_x^2}+e_{x_v}\left(\frac{1}{M}(F_x-{\eta }_x{x}_v)-{\dot{x}}_v^{*}\right)\hfill \\ \hfill & -k_y{e}_y^2+\frac{e_y{e}_{y_v}}{b_y^2-e_y^2}+e_{y_v}\left(\frac{1}{M}(F_y-{\eta }_y{y}_v)-{\dot{y}}_v^{*}\right)\hfill \\ \hfill & -k_{\theta }{e}_{\theta }^2+\frac{e_{\theta }{e}_{{\theta }_v}}{b_{\theta }^2-e_{\theta }^2}+e_{{\theta }_v}\left(\frac{1}{I}(\tau -{\eta }_{\theta }{\theta }_v)-{\dot{\theta }}_v^{*}\right).\hfill \end{array}\end{array}$$

(14)

With virtual inputs (9), ${\dot{V}}_{m_2}$ becomes

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_{m_2}=& -k_x{e}_x^2-k_y{e}_y^2-k_{\theta }{e}_{\theta }^2-k_{x_v}{e}_{x_v}^2-k_{y_v}{e}_{y_v}^2-k_{{\theta }_v}{e}_{{\theta }_v}^2.\hfill \end{array}\end{array}$$

(15)

Thus, $|e_x\left(t\right)|<b_x$, $|e_y\left(t\right)|<b_y$, and $|e_{\theta }\left(t\right)|<b_{\theta }$, $\forall t>0$ and $e_x\left(t\right)$, $e_y\left(t\right)$, and $e_{\theta }\left(t\right)$ converge to zero. □

Note that $F_x$, $F_y$, and $\tau$ are not actual inputs in the mechanical dynamics (5). Thus, the virtual inputs (9) are the desired X-axis and Y-axis forces and torque, which are expressed as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_x^d=& -k_{x_v}{e}_{x_v}+{\eta }_x{x}_v+M{\dot{x}}_v^{*}-\frac{e_x}{b_x^2-e_x^2}\hfill \\ \hfill F_y^d=& -k_{y_v}{e}_{y_v}+{\eta }_y{y}_v+M{\dot{y}}_v^{*}-\frac{e_x}{b_y^2-e_y^2}\hfill \\ \hfill {\tau }^d=& -k_{{\theta }_v}{e}_{{\theta }_v}+{\eta }_{\theta }{\theta }_v+I{\dot{\theta }}_v^{*}-\frac{e_{\theta }}{b_{\theta }^2-e_{\theta }^2}.\hfill \end{array}\end{array}$$

(16)

Note that the desired X-axis and Y-axis forces and torque can guarantee the constraints of $e_x$, $e_y$, and $e_{\theta }$.

3.2. Commutation Scheme

In the planar motor, the actual inputs are voltages and not forces and torque. To obtain the desired forces and torque, the desired currents are defined as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill i_{x_{1a}}^d=& \left(\frac{F_x^d}{2\kappa }+\frac{{\tau }^d}{4\kappa r}\right)cos\left(\gamma x_1\right)\hfill \\ \hfill i_{x_{1b}}^d=& \left(\frac{F_x^d}{2\kappa }+\frac{{\tau }^d}{4\kappa r}\right)sin\left(\gamma x_1\right)\hfill \\ \hfill i_{x_{2a}}^d=& \left(\frac{F_x^d}{2\kappa }-\frac{{\tau }^d}{4\kappa r}\right)cos\left(\gamma x_2\right)\hfill \\ \hfill i_{x_{2b}}^d=& \left(\frac{F_x^d}{2\kappa }-\frac{{\tau }^d}{4\kappa r}\right)sin\left(\gamma x_2\right)\hfill \\ \hfill i_{y_{1a}}^d=& \left(\frac{F_y^d}{2\kappa }+\frac{{\tau }^d}{4\kappa r}\right)cos\left(\gamma y_1\right)\hfill \\ \hfill i_{y_{1b}}^d=& \left(\frac{F_y^d}{2\kappa }+\frac{{\tau }^d}{4\kappa r}\right)sin\left(\gamma y_1\right)\hfill \\ \hfill i_{y_{2a}}^d=& \left(\frac{F_y^d}{2\kappa }-\frac{{\tau }^d}{4\kappa r}\right)cos\left(\gamma y_2\right)\hfill \\ \hfill i_{y_{2b}}^d=& \left(\frac{F_y^d}{2\kappa }-\frac{{\tau }^d}{4\kappa r}\right)sin\left(\gamma y_2\right)\hfill \end{array}\end{array}$$

(17)

where $i_i^d$, $i\in [x_{1a},x_{1b},x_{2a},x_{2b},y_{1a},y_{1b},y_{2a},y_{2b}]$, are the desired phase currents.

3.3. Nonlinear Current Controllers for Electrical Dynamics

The phase currents of the planar motor decreased because of back-EMFs and have phase lags by inductances during operation. Now, we design the nonlinear controller to guarantee the desired currents (17) for the electrical dynamics controller. First, we design the controller for the forcer $X_1$. Let us define the current errors for the forcer $X_1$ as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill e_{x_{1a}}=& i_{x_{1a}}-i_{x_{1a}}^d\hfill \\ \hfill e_{x_{1b}}=& i_{x_{1b}}-i_{x_{1b}}^d.\hfill \end{array}\end{array}$$

(18)

The current error dynamics for the forcer $X_1$ are

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{e}}_{x_{1a}}=& \frac{1}{L}[v_{x_{1a}}-R{i}_{x_{1a}}-\kappa x_{1_v}cos\left(\gamma x_1\right)]-{\dot{i}}_{x_{1a}}^d\hfill \\ \hfill {\dot{e}}_{x_{1b}}=& \frac{1}{L}[v_{x_{1b}}-R{i}_{x_{1b}}-\kappa x_{1_v}sin\left(\gamma x_1\right)]-{\dot{i}}_{x_{1b}}^d.\hfill \end{array}\end{array}$$

(19)

Theorem 2.

Consider the current error dynamics for the forcer $X_1$ (19). If the voltage inputs for the forcer $X_1$ are given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill v_{x_{1a}}=& (R{i}_{x_{1a}}+\kappa x_{1_v}cos\left(\gamma x_1\right))+L({\dot{i}}_{x_{1a}}^d-k_e{e}_{x_{1a}})\hfill \\ \hfill v_{x_{1b}}=& (R{i}_{x_{1b}}+\kappa x_{1_v}sin\left(\gamma x_1\right))+L({\dot{i}}_{x_{1b}}^d-k_e{e}_{x_{1b}}),\hfill \end{array}\end{array}$$

(20)

where $k_e$ is a positive controller gain applied to the current error dynamics for the forcer $X_1$ (19), then $e_{x_{1a}}$ and $e_{x_{1b}}$ exponentially converge to zero.

Proof of Theorem 2.

The Lyapunov candidate function, $V_e$ is defined as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_e=& \frac{1}{2}{e}_{x_{1a}}^2+\frac{1}{2}{e}_{x_{1b}}^2.\hfill \end{array}\end{array}$$

(21)

Differentiating $V_e$ (21) yields

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{V}=& e_{x_{1a}}({\dot{i}}_{x_{1a}}-{\dot{i}}_{x_{1a}}^d)+e_{x_{1b}}({\dot{i}}_{x_{1b}}-{\dot{i}}_{x_{1b}}^d)\hfill \\ \hfill =& e_{x_{1a}}(\frac{1}{L}[v_{x_{1a}}-R{i}_{x_{1a}}-\kappa x_{1_v}cos\left(\gamma x_1\right)]-{\dot{i}}_{x_{1a}}^d)\hfill \\ \hfill & +e_{x_{1b}}(\frac{1}{L}[v_{x_{1b}}-R{i}_{x_{1b}}-\kappa x_{1_v}sin\left(\gamma x_1\right)]-{\dot{i}}_{x_{1b}}^d).\hfill \end{array}\end{array}$$

(22)

With the nonlinear controller (20), ${\dot{V}}_e$ becomes

$$\begin{array}{c}\hfill {\dot{V}}_e=-k_e{e}_{x_{1a}}^2-k_e{e}_{x_{1b}}^2.\end{array}$$

(23)

□

The electrical dynamics of the forcers $X_1$, $X_2$, $X_3$ and $X_4$ are all the same, thus the controllers for other forces are analogous to (20). From the controller for the forcers $X_1$, the controllers for all forcers $X_1$, $X_2$, $X_3$ and $X_4$ can be designed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill v_{x_{1a}}=& (R{i}_{x_{1a}}+\kappa x_{1_v}cos\left(\gamma x_1\right))+L({\dot{v}}_{x_{1a}}^d-k_e{e}_{x_{1a}})\hfill \\ \hfill v_{x_{1b}}=& (R{i}_{x_{1b}}+\kappa x_{1_v}sin\left(\gamma x_1\right))+L({\dot{v}}_{x_{1b}}^d-k_e{e}_{x_{1b}})\hfill \\ \hfill v_{x_{2a}}=& (R{i}_{x_{2a}}+\kappa x_{2_v}cos\left(\gamma x_1\right))+L({\dot{v}}_{x_{2a}}^d-k_e{e}_{x_{2a}})\hfill \\ \hfill v_{x_{2b}}=& (R{i}_{x_{2b}}+\kappa x_{2_v}sin\left(\gamma x_2\right))+L({\dot{v}}_{x_{2b}}^d-k_e{e}_{x_{2b}})\hfill \\ \hfill v_{y_{1a}}=& (R{i}_{y_{1a}}+\kappa y_{1_v}cos\left(\gamma y_1\right))+L({\dot{v}}_{y_{1a}}^d-k_e{e}_{y_{1a}})\hfill \\ \hfill v_{y_{1b}}=& (R{i}_{y_{1b}}+\kappa y_{1_v}sin\left(\gamma y_1\right))+L({\dot{v}}_{y_{1b}}^d-k_e{e}_{y_{1b}})\hfill \\ \hfill v_{y_{2a}}=& (R{i}_{y_{2a}}+\kappa y_{2_v}cos\left(\gamma y_1\right))+L({\dot{v}}_{y_{2a}}^d-k_e{e}_{y_{2a}})\hfill \\ \hfill v_{y_{2b}}=& (R{i}_{y_{2b}}+\kappa y_{2_v}sin\left(\gamma y_2\right))+L({\dot{v}}_{y_{2b}}^d-k_e{e}_{y_{2b}}).\hfill \end{array}\end{array}$$

(24)

4. Observer Design

In the previous section, for the proposed controller design, we assumed that the full states are measurable. In this paper, we assume that the positions for X and Y directions are measurable, that is, x, y, and $\theta$ are available. $x_{1_v}=x_v+r\dot{\theta }cos\left(\theta \right)$, $x_{2_v}=x_v-r\dot{\theta }cos\left(\theta \right)$, $y_{1_v}=y_v+r\dot{\theta }cos\left(\theta \right)$, and $y_{2_v}=y_v-r\dot{\theta }cos\left(\theta \right)$; thus, the dynamics of the planar motor (4) can be rewritten as

$$\begin{array}{cc}\dot{x}=\hfill & x_v\hfill \\ {\dot{x}}_v=\hfill & \frac{1}{M}(F_x-{\eta }_x{x}_v)\hfill \\ \dot{y}=\hfill & y_v\hfill \\ {\dot{y}}_v=\hfill & \frac{1}{M}(F_y-{\eta }_y{y}_v)\hfill \\ \dot{\theta }=\hfill & {\theta }_v\hfill \\ {\dot{\theta }}_v=\hfill & \frac{1}{I}(\tau -{\eta }_{\theta }{\theta }_v)\hfill \\ {\dot{i}}_{x1a}=\hfill & \frac{1}{L}[v_{x1a}-R{i}_{x1a}-\kappa x_{v}cos\left(\gamma x_1\right)-\kappa r{\theta }_{v}cos\left(\theta \right)cos\left(\gamma x_1\right)]\hfill \\ {\dot{i}}_{x1b}=\hfill & \frac{1}{L}[v_{x1b}-R{i}_{x1b}-\kappa x_{v}sin\left(\gamma x_1\right)-\kappa r{\theta }_{v}cos\left(\theta \right)sin\left(\gamma x_1\right)]\hfill \\ {\dot{i}}_{x2a}=\hfill & \frac{1}{L}[v_{x2a}-R{i}_{x2a}-\kappa x_{v}cos\left(\gamma x_2\right)+\kappa r{\theta }_{v}cos\left(\theta \right)cos\left(\gamma x_2\right)]\hfill \\ {\dot{i}}_{x2b}=\hfill & \frac{1}{L}[v_{x2b}-R{i}_{x2b}-\kappa x_{v}sin\left(\gamma x_2\right)+\kappa r{\theta }_{v}cos\left(\theta \right)sin\left(\gamma x_2\right)]\hfill \\ {\dot{i}}_{y1a}=\hfill & \frac{1}{L}[v_{y1a}-R{i}_{y1a}-\kappa y_{v}cos\left(\gamma y_1\right)-\kappa r{\theta }_{v}cos\left(\theta \right)cos\left(\gamma y_1\right)]\hfill \\ {\dot{i}}_{y1b}=\hfill & \frac{1}{L}[v_{y1b}-R{i}_{y1b}-\kappa y_{v}sin\left(\gamma y_1\right)-\kappa r{\theta }_{v}cos\left(\theta \right)sin\left(\gamma y_1\right)]\hfill \\ {\dot{i}}_{y2a}=\hfill & \frac{1}{L}[v_{y2a}-R{i}_{y2a}-\kappa y_{v}cos\left(\gamma y_2\right)+\kappa r{\theta }_{v}cos\left(\theta \right)cos\left(\gamma y_2\right)]\hfill \\ {\dot{i}}_{y2b}=\hfill & \frac{1}{L}[v_{y2b}-R{i}_{y2b}-\kappa y_{v}sin\left(\gamma y_2\right)+\kappa r{\theta }_{v}cos\left(\theta \right)sin\left(\gamma y_2\right)].\hfill \end{array}$$

(25)

We proposed a nonlinear observer to estimate the full states as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{\widehat{x}}=& {\widehat{x}}_v+l_x(x-\widehat{x})\hfill \\ \hfill {\dot{\widehat{x}}}_v=& \frac{1}{M}[{\widehat{F}}_x-{\eta }_x{\widehat{x}}_v]+l_{xv}(x-\widehat{x})\hfill \\ \hfill \dot{\widehat{y}}=& {\widehat{y}}_v+l_y(y-\widehat{y})\hfill \\ \hfill {\dot{\widehat{y}}}_v=& \frac{1}{M}[{\widehat{F}}_y-{\eta }_y{\widehat{y}}_v]+l_{yv}(y-\widehat{y})\hfill \\ \hfill \dot{\widehat{\theta }}=& {\widehat{\theta }}_v+l_{\theta }(\theta -\widehat{\theta })\hfill \\ \hfill {\dot{\widehat{\theta }}}_v=& \frac{1}{I}[\widehat{\tau }-{\eta }_{\theta }{\widehat{\theta }}_v]+l_{\theta v}(\theta -\widehat{\theta })\hfill \\ \hfill {\dot{\widehat{i}}}_{x1a}=& \frac{1}{L}[v_{x1a}-R{\widehat{i}}_{x1a}-\kappa {\widehat{x}}_{v}cos\left(\gamma x_1\right)-\kappa r{\widehat{\theta }}_{v}cos\left(\theta \right)cos\left(\gamma x_1\right)]\hfill \\ & +l_{x1a}(x-\widehat{x})\hfill \\ \hfill {\dot{\widehat{i}}}_{x1b}=& \frac{1}{L}[v_{x1b}-R{\widehat{i}}_{x1b}-\kappa {\widehat{x}}_{v}sin\left(\gamma x_1\right)-\kappa r{\widehat{\theta }}_{v}cos\left(\theta \right)sin\left(\gamma x_1\right)]\hfill \\ & +l_{x1b}(x-\widehat{x})\hfill \\ \hfill {\dot{\widehat{i}}}_{x2a}=& \frac{1}{L}[v_{x2a}-R{\widehat{i}}_{x2a}-\kappa {\widehat{x}}_{v}cos\left(\gamma x_2\right)+\kappa r{\widehat{\theta }}_{v}cos\left(\theta \right)cos\left(\gamma x_2\right)]\hfill \\ & +l_{x2a}(x-\widehat{x})\hfill \\ \hfill {\dot{\widehat{i}}}_{x2b}=& \frac{1}{L}[v_{x2b}-R{\widehat{i}}_{x2b}-\kappa {\widehat{x}}_{v}sin\left(\gamma x_2\right)+\kappa r{\widehat{\theta }}_{v}cos\left(\theta \right)sin\left(\gamma x_2\right)]\hfill \\ & +l_{x2b}(x-\widehat{x})\hfill \\ \hfill {\dot{\widehat{i}}}_{y1a}=& \frac{1}{L}[v_{y1a}-R{\widehat{i}}_{y1a}-\kappa {\widehat{y}}_{v}cos\left(\gamma y_1\right)-\kappa r{\widehat{\theta }}_{v}cos\left(\theta \right)cos\left(\gamma y_1\right)]\hfill \\ & +l_{y1a}(y-\widehat{y})\hfill \\ \hfill {\dot{\widehat{i}}}_{y1b}=& \frac{1}{L}[v_{y1b}-R{\widehat{i}}_{y1b}-\kappa {\widehat{y}}_{v}sin\left(\gamma y_1\right)-\kappa r{\widehat{\theta }}_{v}cos\left(\theta \right)sin\left(\gamma y_1\right)]\hfill \\ & +l_{y1b}(y-\widehat{y})\hfill \\ \hfill {\dot{\widehat{i}}}_{y2a}=& \frac{1}{L}[v_{y2a}-R{\widehat{i}}_{y2a}-\kappa {\widehat{y}}_{v}cos\left(\gamma y_2\right)+\kappa r{\widehat{\theta }}_{v}cos\left(\theta \right)cos\left(\gamma y_2\right)]\hfill \\ & +l_{y2a}(y-\widehat{y})\hfill \\ \hfill {\dot{\widehat{i}}}_{y2b}=& \frac{1}{L}[v_{y2b}-R{\widehat{i}}_{y2b}-\kappa {\widehat{y}}_{v}sin\left(\gamma y_2\right)+\kappa r{\widehat{\theta }}_{v}cos\left(\theta \right)sin\left(\gamma y_2\right)]\hfill \\ & +l_{y2b}(y-\widehat{y}),\hfill \end{array}\end{array}$$

(26)

where $\widehat{•}$ denotes the estimation of •, ${\widehat{F}}_x=\kappa ({\widehat{i}}_{x1a}cos\left(\gamma x_1\right)+{\widehat{i}}_{x1b}sin\left(\gamma x_1\right))+\kappa ({\widehat{i}}_{x2a}cos\left(\gamma x_2\right)+{\widehat{i}}_{x2b}sin\left(\gamma x_2\right))$, ${\widehat{F}}_y=\kappa ({\widehat{i}}_{y1a}cos\left(\gamma y_1\right)+{\widehat{i}}_{y1b}sin\left(\gamma y_1\right))+\kappa ({\widehat{i}}_{y2a}cos\left(\gamma y_2\right)+{\widehat{i}}_{y2b}sin\left(\gamma y_2\right))$, $\widehat{\tau }=\left[\kappa r\right({\widehat{i}}_{x1a}cos\left(\gamma x_1\right)$$+{\widehat{i}}_{x1b}sin\left(\gamma x_1\right))-\kappa r({\widehat{i}}_{x2a}cos\left(\gamma x_2\right)+{\widehat{i}}_{x1b}sin\left(\gamma x_2\right))]+[\kappa r({\widehat{i}}_{y1a}cos\left(\gamma y_1\right)+{\widehat{i}}_{y1b}sin\left(\gamma y_1\right))-\kappa r({\widehat{i}}_{y2a}cos\left(\gamma y_2\right)+{\widehat{i}}_{y2b}sin\left(\gamma y_2\right))]$. The estimation errors are defined as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \tilde{x}=& x-\widehat{x}\hfill \\ \hfill \tilde{y}=& y-\widehat{y}\hfill \\ \hfill \tilde{\theta }=& \theta -\widehat{\theta }\hfill \\ \hfill {\tilde{i}}_{x1a}=& i_{x1a}-{\widehat{i}}_{x1a}\hfill \\ \hfill {\tilde{i}}_{x1b}=& i_{x1b}-{\widehat{i}}_{x1b}\hfill \\ \hfill {\tilde{i}}_{x2a}=& i_{x2a}-{\widehat{i}}_{x2a}\hfill \\ \hfill {\tilde{i}}_{x2b}=& i_{x2b}-{\widehat{i}}_{x2a}\hfill \\ \hfill {\tilde{i}}_{y1a}=& i_{y1a}-{\widehat{i}}_{y1a}\hfill \\ \hfill {\tilde{i}}_{y1b}=& i_{y1b}-{\widehat{i}}_{y1b}\hfill \\ \hfill {\tilde{i}}_{y2a}=& i_{y2a}-{\widehat{i}}_{y2a}\hfill \\ \hfill {\tilde{i}}_{y2b}=& i_{y2b}-{\widehat{i}}_{y2a}.\hfill \end{array}\end{array}$$

(27)

The estimation error dynamics are obtained by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \dot{\tilde{x}}=& {\tilde{x}}_v-l_x\tilde{x}\hfill \\ \hfill {\dot{\tilde{x}}}_v=& \frac{1}{M}[{\tilde{F}}_x-{\eta }_x{\tilde{x}}_v]-l_{xv}\tilde{x}\hfill \\ \hfill \dot{\tilde{y}}=& {\tilde{y}}_v-l_y\tilde{y}\hfill \\ \hfill {\dot{\tilde{y}}}_v=& \frac{1}{M}[{\tilde{F}}_y-{\eta }_y{\tilde{y}}_v]-l_{yv}\tilde{y}\hfill \\ \hfill \dot{\tilde{\theta }}=& {\tilde{\theta }}_v-l_{\theta }\tilde{\theta }\hfill \\ \hfill {\dot{\tilde{\theta }}}_v=& \frac{1}{I}[\tilde{\tau }-{\eta }_{\theta }\tilde{\theta }]-l_{\theta v}\tilde{\theta }\hfill \\ \hfill {\dot{\tilde{i}}}_{x1a}=& \frac{1}{L}[-R{\tilde{i}}_{x1a}-\kappa {\tilde{x}}_{v}cos\left(\gamma x_1\right)-\kappa r{\tilde{\theta }}_{v}cos\left(\theta \right)cos\left(\gamma x_1\right)]-l_{x1a}\tilde{x}\hfill \\ \hfill {\dot{\tilde{i}}}_{x1b}=& \frac{1}{L}[-R{\tilde{i}}_{x1b}-\kappa {\tilde{x}}_{v}sin\left(\gamma x_1\right)-\kappa r{\tilde{\theta }}_{v}cos\left(\theta \right)sin\left(\gamma x_1\right)]-l_{x1b}\tilde{x}\hfill \\ \hfill {\dot{\tilde{i}}}_{x2a}=& \frac{1}{L}[-R{\tilde{i}}_{x2a}-\kappa {\tilde{x}}_{v}cos\left(\gamma x_2\right)+\kappa r{\tilde{\theta }}_{v}cos\left(\theta \right)cos\left(\gamma x_2\right)]-l_{x2a}\tilde{x}\hfill \\ \hfill {\dot{\tilde{i}}}_{x2b}=& \frac{1}{L}[-R{\tilde{i}}_{x2b}-\kappa {\tilde{x}}_{v}sin\left(\gamma x_2\right)+\kappa r{\tilde{\theta }}_{v}cos\left(\theta \right)sin\left(\gamma x_2\right)]-l_{x2b}\tilde{x}\hfill \\ \hfill {\dot{\tilde{i}}}_{y1a}=& \frac{1}{L}[-R{\tilde{i}}_{y1a}-\kappa {\tilde{y}}_{v}cos\left(\gamma y_1\right)-\kappa r{\tilde{\theta }}_{v}cos\left(\theta \right)cos\left(\gamma y_1\right)]-l_{y1a}\tilde{y}\hfill \\ \hfill {\dot{\tilde{i}}}_{y1b}=& \frac{1}{L}[-R{\tilde{i}}_{y1b}-\kappa {\tilde{y}}_{v}sin\left(\gamma y_1\right)-\kappa r{\tilde{\theta }}_{v}cos\left(\theta \right)sin\left(\gamma y_1\right)]-l_{y1b}\tilde{y}\hfill \\ \hfill {\dot{\tilde{i}}}_{y2a}=& \frac{1}{L}[-R{\tilde{i}}_{y2a}-\kappa {\tilde{y}}_{v}cos\left(\gamma y_2\right)+\kappa r{\tilde{\theta }}_{v}cos\left(\theta \right)cos\left(\gamma y_2\right)]-l_{y2a}\tilde{y}\hfill \\ \hfill {\dot{\tilde{i}}}_{y2b}=& \frac{1}{L}[-R{\tilde{i}}_{y2b}-\kappa {\tilde{y}}_{v}sin\left(\gamma y_2\right)+\kappa r{\tilde{\theta }}_{v}cos\left(\theta \right)sin\left(\gamma y_2\right)]-l_{y2b}\tilde{y},\hfill \end{array}\end{array}$$

(28)

where ${\tilde{F}}_x=\kappa ({\tilde{i}}_{x1a}cos\left(\gamma x_1\right)+{\tilde{i}}_{x1b}sin\left(\gamma x_1\right))+\kappa ({\tilde{i}}_{x2a}cos\left(\gamma x_2\right)+{\tilde{i}}_{x2b}sin\left(\gamma x_2\right))$, ${\tilde{F}}_y=\kappa ({\tilde{i}}_{y1a}cos\left(\gamma y_1\right)+{\tilde{i}}_{y1b}sin\left(\gamma y_1\right))+\kappa ({\tilde{i}}_{y2a}cos\left(\gamma y_2\right)+{\tilde{i}}_{y2b}sin\left(\gamma y_2\right))$, $\tilde{\tau }=[\kappa r({\tilde{i}}_{x1a}cos\left(\gamma x_1\right)+{\tilde{i}}_{x1b}sin\left(\gamma x_1\right))-\kappa r({\tilde{i}}_{x2a}cos\left(\gamma x_2\right)$$+{\tilde{i}}_{x1b}sin\left(\gamma x_2\right)\left)\right]+[\kappa r({\tilde{i}}_{y1a}cos\left(\gamma y_1\right)+{\tilde{i}}_{y1b}sin\left(\gamma y_1\right))-\kappa r({\tilde{i}}_{y2a}cos\left(\gamma y_2\right)+{\tilde{i}}_{y2b}sin\left(\gamma y_2\right))].$

Assumption 1.

During the operation of the planar motor, $cos\left(\theta \right)\ge 0$, $\forall t>0$. ◊ The planar motor steps out when the yaw is relatively large. Thus, this assumption is reasonable.

Theorem 3.

Suppose the estimation error dynamics (28) with Assumption 1. If $l_x$, $l_y$, and $l_{\theta }$ are positive, $l_{xv}$ and $l_{yv}$ are $\frac{L}{M}$, and $l_{\theta v}$ is $\frac{L}{I}$, then the estimation errors exponentially converge to zero.

Proof of Theorem 3.

The Lyapunov candidate function, $V_o$ is defined as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_o=& \frac{1}{2}({\tilde{x}}^2+\frac{M}{L}{\tilde{x}}_v^2+{\tilde{y}}^2+\frac{M}{L}{\tilde{y}}_v^2+{\tilde{\theta }}^2+\frac{Icos\left(\theta \right)}{L}{\tilde{\theta }}_v^2\hfill \\ \hfill & +{\tilde{i}}_{x1a}^2+{\tilde{i}}_{x1b}^2+{\tilde{i}}_{x2a}^2+{\tilde{i}}_{x2b}^2+{\tilde{i}}_{y1a}^2+{\tilde{i}}_{y1b}^2+{\tilde{i}}_{y2a}^2+{\tilde{i}}_{y2b}^2).\hfill \end{array}\end{array}$$

(29)

with Assumption 1, (29) can be the Lyapunov candidate function. The derivative of $V_o$ is as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_0=& -l_x{\tilde{x}}^2-\frac{{\eta }_x}{L}{\tilde{x}}_v^2-l_y{\tilde{y}}^2-\frac{{\eta }_y}{L}{\tilde{y}}_v^2-l_{\theta }{\tilde{\theta }}^2+\frac{{\eta }_{\theta }}{L}{\tilde{\theta }}_v^2\hfill \\ \hfill & -\frac{R}{L}{\tilde{i}}_{x1a}^2-\frac{R}{L}{\tilde{i}}_{x1b}^2-\frac{R}{L}{\tilde{i}}_{x2a}^2-\frac{R}{L}{\tilde{i}}_{x2b}^2\hfill \\ \hfill & -\frac{R}{L}{\tilde{i}}_{y1a}^2-\frac{R}{L}{\tilde{i}}_{y1b}^2-\frac{R}{L}{\tilde{i}}_{y2a}^2-\frac{R}{L}{\tilde{i}}_{y2b}^2.\hfill \end{array}\end{array}$$

(30)

If $l_x$, $l_y$, and $l_{\theta }$ are positive, $l_{xv}$ and $l_{yv}$ are $\frac{L}{M}$, and $l_{\theta v}$ is $\frac{L}{I}$, then ${\dot{V}}_o$ is negative definite. Thus, the estimation errors exponentially converge to zero. □

5. Analysis of Closed-Loop System

The backstepping controller (16), nonlinear controller (24), and nonlinear observer (26) were designed separately. Thus, the stability of the closed-loop system should be studied. The closed-loop system is expressed as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{e}}_m=& A_m{e}_m+B_{m_1}\left(X\right)e_e+B_{m_2}\tilde{X}\hfill \\ \hfill {\dot{e}}_e=& A_e{e}_e+B_e\left(X\right)\tilde{X}\hfill \\ \hfill \dot{\tilde{X}}=& A_{ob}\left(X\right)\tilde{X},\hfill \end{array}\end{array}$$

(31)

where $e_m={\left[e_x{e}_{x_v}{e}_y{e}_{y_v}{e}_{\theta }{e}_{{\theta }_v}\right]}^T$, $e_e={\left[e_{x_{1a}}{e}_{x_{1b}}{e}_{x_{2a}}{e}_{x_{2b}}{e}_{y_{1a}}{e}_{y_{1b}}{e}_{y_{2a}}{e}_{y_{2b}}\right]}^T$, $X={[x{x}_{v}y{y}_v\theta {\theta }_v{i}_{x_{1a}}{i}_{x_{1b}}{i}_{x_{2a}}{i}_{x_{2b}}]}^T$, $\tilde{X}={[\tilde{x}{\tilde{x}}_v\tilde{y}{\tilde{y}}_v\tilde{\theta }{\tilde{\theta }}_v{\tilde{i}}_{x_{1a}}{\tilde{i}}_{x_{1b}}{\tilde{i}}_{x_{2a}}{\tilde{i}}_{x_{2b}}{\tilde{i}}_{y_{1a}}{\tilde{i}}_{y_{1b}}{\tilde{i}}_{y_{2a}}{\tilde{i}}_{y_{2b}}]}^T$, $c_{i_i}=cos\left(\gamma i_j\right)$, $s_{i_i}=sin\left(\gamma i_j\right)$ for $i\in [x,y]$ and $i\in [1,2]$, $c_{\theta }=cos\left(\theta \right)$, $b_{m_2}=-\frac{R}{L}+k_e$

$$A_m=\left[\begin{array}{cccccc}-k_x& 0& 0& 0& 0& 0\\ 0& -k_{x_v}& 0& 0& 0& 0\\ 0& 0& -k_y& 0& 0& 0\\ 0& 0& 0& -k_{y_v}& 0& 0\\ 0& 0& 0& 0& -k_{\theta }& 0\\ 0& 0& 0& 0& 0& -k_{{\theta }_v}\end{array}\right]$$

$$A_e=\left[\begin{array}{cccccccc}-k_e& 0& 0& 0& 0& 0& 0& 0\\ 0& -k_e& 0& 0& 0& 0& 0& 0\\ 0& 0& -k_e& 0& 0& 0& 0& 0\\ 0& 0& 0& -k_e& 0& 0& 0& 0\\ 0& 0& 0& 0& -k_e& 0& 0& 0\\ 0& 0& 0& 0& 0& -k_e& 0& 0\\ 0& 0& 0& 0& 0& 0& -k_e& 0\\ 0& 0& 0& 0& 0& 0& 0& -k_e\end{array}\right]$$

$$B_{m_1}\left(X\right)=\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ \frac{\kappa }{M}{c}_{x_1}& \frac{\kappa }{M}{s}_{x_1}& \frac{\kappa }{M}{c}_{x_2}& \frac{\kappa }{M}{s}_{x_2}& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& \frac{\kappa }{M}{c}_{y_1}& \frac{\kappa }{M}{s}_{y_1}& \frac{\kappa }{M}{c}_{y_2}& \frac{\kappa }{M}{s}_{y_2}\\ 0& 0& 0& 0& 0& 0& 0& 0\\ \frac{r\kappa }{M}{c}_{x_1}& \frac{r\kappa }{M}{s}_{x_1}& -\frac{r\kappa }{M}{c}_{x_2}& -\frac{r\kappa }{M}{s}_{x_2}& \frac{r\kappa }{M}{c}_{y_1}& \frac{r\kappa }{M}{s}_{y_1}& -\frac{r\kappa }{M}{c}_{y_2}& -\frac{r\kappa }{M}{s}_{y_2}\end{array}\right]$$

$$B_{m_2}=\left[\begin{array}{cccccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& \frac{{\eta }_x}{M}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& \frac{{\eta }_y}{M}& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& \frac{{\eta }_{\theta }}{M}& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$$

$$B_e\left(X\right)=\left[\begin{array}{cccccccccccccc}0& -\frac{\kappa }{L}{c}_{x_1}& 0& 0& 0& -\frac{r\kappa }{L}{c}_{\theta }{c}_{x_1}& b_{m_2}& 0& 0& 0& 0& 0& 0& 0\\ 0& -\frac{\kappa }{L}{s}_{x_1}& 0& 0& 0& -\frac{r\kappa }{L}{c}_{\theta }{s}_{x_1}& 0& b_{m_2}& 0& 0& 0& 0& 0& 0\\ 0& -\frac{\kappa }{L}{c}_{x_2}& 0& 0& 0& -\frac{r\kappa }{L}{c}_{\theta }{c}_{x_2}& 0& 0& b_{m_2}& 0& 0& 0& 0& 0\\ 0& -\frac{\kappa }{L}{s}_{x_2}& 0& 0& 0& -\frac{r\kappa }{L}{c}_{\theta }{s}_{x_2}& 0& 0& 0& b_{m_2}& 0& 0& 0& 0\\ 0& 0& 0& -\frac{\kappa }{L}{c}_{y_1}& 0& -\frac{r\kappa }{L}{c}_{\theta }{c}_{y_1}& 0& 0& 0& 0& b_{m_2}& 0& 0& 0\\ 0& 0& 0& -\frac{\kappa }{L}{s}_{y_1}& 0& -\frac{r\kappa }{L}{c}_{\theta }{s}_{y_1}& 0& 0& 0& 0& 0& b_{m_2}& 0& 0\\ 0& 0& 0& -\frac{\kappa }{L}{c}_{y_2}& 0& -\frac{r\kappa }{L}{c}_{\theta }{c}_{y_2}& 0& 0& 0& 0& 0& 0& b_{m_2}& 0\\ 0& 0& 0& -\frac{\kappa }{L}{s}_{y_2}& 0& -\frac{r\kappa }{L}{c}_{\theta }{s}_{y_2}& 0& 0& 0& 0& 0& 0& 0& b_{m_2}\end{array}\right]$$

$$A_{ob}=\left[\begin{array}{cccccccccccccc}-l_x& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ -l_{xv}& -\frac{{\eta }_x}{M}& 0& 0& 0& 0& \frac{\kappa }{M}{c}_{x_1}& \frac{\kappa }{M}{s}_{x_1}& \frac{\kappa }{M}{c}_{x_2}& \frac{\kappa }{M}{s}_{x_2}& 0& 0& 0& 0\\ 0& 0& -l_y& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -l_{yv}& -\frac{{\eta }_y}{M}& 0& 0& 0& 0& 0& \frac{\kappa }{M}{c}_{y_1}& \frac{\kappa }{M}{s}_{y_1}& \frac{\kappa }{M}{c}_{y_2}& \frac{\kappa }{M}{s}_{y_2}\\ 0& 0& 0& 0& -l_{\theta }& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -l_{\theta v}& -\frac{{\eta }_{\theta }}{I}& \frac{r\kappa }{I}{c}_{x_1}& \frac{r\kappa }{I}{s}_{x_1}& -\frac{r\kappa }{I}{c}_{x_2}& -\frac{r\kappa }{I}{s}_{x_2}& \frac{r\kappa }{I}{c}_{y_1}& \frac{r\kappa }{I}{s}_{y_1}& -\frac{r\kappa }{I}{c}_{y_2}& -\frac{r\kappa }{I}{s}_{y_2}\\ -l_{x1a}& -\frac{\kappa }{L}{c}_{x_1}& 0& 0& 0& -\frac{r\kappa }{L}{c}_{\theta }{c}_{x_1}& -\frac{R}{L}& 0& 0& 0& 0& 0& 0& 0\\ -l_{x1b}& -\frac{\kappa }{L}{s}_{x_1}& 0& 0& 0& -\frac{r\kappa }{L}{c}_{\theta }{s}_{x_1}& 0& -\frac{R}{L}& 0& 0& 0& 0& 0& 0\\ -l_{x2a}& -\frac{\kappa }{L}{c}_{x_2}& 0& 0& 0& +\frac{r\kappa }{L}{c}_{\theta }{c}_{x_2}& 0& 0& -\frac{R}{L}& 0& 0& 0& 0& 0\\ -l_{x2b}& -\frac{\kappa }{L}{s}_{x_2}& 0& 0& 0& +\frac{r\kappa }{L}{c}_{\theta }{s}_{x_2}& 0& 0& 0& -\frac{R}{L}& 0& 0& 0& 0\\ 0& 0& -l_{y1a}& -\frac{\kappa }{L}{c}_{y_1}& 0& -\frac{r\kappa }{L}{c}_{\theta }{c}_{y_1}& 0& 0& 0& 0& -\frac{R}{L}& 0& 0& 0\\ 0& 0& -l_{y1b}& -\frac{\kappa }{L}{c}_{y_2}& 0& -\frac{r\kappa }{L}{c}_{\theta }{s}_{y_1}& 0& 0& 0& 0& 0& -\frac{R}{L}& 0& 0\\ 0& 0& -l_{y2a}& -\frac{\kappa }{L}{s}_{y_1}& 0& +\frac{r\kappa }{L}{c}_{\theta }{c}_{y_2}& 0& 0& 0& 0& 0& 0& -\frac{R}{L}& 0\\ 0& 0& -l_{y2b}& -\frac{\kappa }{L}{s}_{y_2}& 0& +\frac{r\kappa }{L}{c}_{\theta }{s}_{y_2}& 0& 0& 0& 0& 0& 0& 0& -\frac{R}{L}\end{array}\right].$$

In the closed-loop system (31), $A_m$ and $A_e$ are Hurwitz with the positive control gains. $B_{m_1}$, $B_{m_2}$, and $B_e$ are bouneded. Thus, $e_m$ and $e_e$ are ISS stable. In the previous section, it was proven that $\tilde{X}$ exponentially converge to zero. Thus, $e_e$ converges to zero. Consequently, $e_m$ converges to zero.

6. Simulation Results

We performed simulations to evaluate the performance of the proposed method. The simulations were performed using MATLAB/Simulink. The parameters of the planar motor, Normag XY1304, were used. The parameters, control gains, and observer gains used are listed in

Table 1. Plant parameters, control gains, and observer gains.

Plant Parameters and Gains
Plant parameters	M	$1.35$ kg
	I	$4\times {10}^{-3}$ kg/m2
	r	$0.0485$ m
	$\gamma$	$2\ast \pi /p$
	p	$1.0168$ mm
	$\kappa$	17
	L	$7\times {10}^{-4}$ H
	R	2 $\mathsf{\Omega }$
Control gains	$k_x,k_y$	$1\times {10}^{10}$
	$k_{\theta }$	$3\times {10}^{17}$
	$k_{x_v},k_{y_v}$	$1\times {10}^4$
	$k_{{\theta }_v}$	$6\times {10}^3$
	$\rho$	$1\times {10}^5$
Observer gains	$l_x,l_y$	1000
	$l_{\theta }$	$\mathrm{20,000}$
	$l_{xv},l_{yv}$	$5.18\times {10}^{-4}$
	$l_{\theta v}$	$0.175$
	$l_{x1a},l_{x1b}$	0
	$l_{x2a},l_{x2b}$	0
	$l_{y1a},l_{y1b}$	0
	$l_{y2a},l_{y2b}$	0

. The sampling frequency was set to be 1 MHz. To demonstrate the robustness of the proposed controller, the following disturbances of both forces and torque are included in the simulations.

• $F_{dx}=14(1+0.5cos\left(3t\right))\dot{x}+2sin\left(4\gamma x\right)$,
• $F_{dy}=14(1+0.5cos\left(3t\right))\dot{y}+2sin\left(4\gamma y\right)$,
• ${\tau }_d=5(1+0.5cos\left(2t\right))\dot{\theta }$.

In these simulations, the proposed method was compared to the conventional PID controller (32) as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill F_x=& k_{px}(x^d-x)+k_{ix}{\int }_0^t(x^d-x)d\tau +k_{dx}({\dot{x}}^d-\dot{x})\hfill \\ \hfill F_y=& k_{px}(y^d-y)+k_{iy}{\int }_0^t(y^d-y)d\tau +k_{dy}({\dot{y}}^d-\dot{y})\hfill \\ \hfill \tau =& k_{p\theta }({\theta }^d-\theta )+k_{i\theta }{\int }_0^t({\theta }^d-\theta )d\tau +k_{dy}({\dot{\theta }}^d-\dot{\theta }),\hfill \end{array}\end{array}$$

(32)

where $k_{px}=k_{py}=\mathrm{50,000}$, $k_{dx}=k_{dy}=50$, $k_{ix}=k_{iy}=500$, $k_{p\theta }=1000$, $k_{d\theta }=5$, and $k_{i\theta }=2000$. Two cases were tested to evaluate the effect of the proposed method, which are as follows:

• Case 1: The conventional PID controller (32) and the nonlinear controller (24).
• Case 2: The proposed controller (16) and (24) with the constraints ($b_x,b_y,b_{\theta }=1\times {10}^{-5}$).

For two cases, the commutation scheme (17) and the nonlinear observer (26) were used.

The desired positions for X and Y directions as shown in Figure 2 are used. The desired yaw was set to be zero.

The estimation performances of x, y, $\theta$, and i for Case 2 are shown in Figure 3 and Figure 4. It is observed that the estimated states tracked well the actual states by using the proposed nonlinear observer.

Figure 5, Figure 6 and Figure 7 show the simulation results for cases 1 and 2. Large acceleration and deceleration appear in the desired positions for the X and Y directions. Thus, large overshoots of the positions and yaw appear in case 1. The offset error in the positions exist during periods of constant velocity. Moreover, in case 2, the tolerances $1\times {10}^{-5}$ for the position tracking errors and yaw are guaranteed by the proposed method during both transient and constant periods. Furthermore, the offset error in the positions disappeared during the constant velocity periods.

7. Conclusions

In this paper, we proposed nonlinear position control using only position feedback to guarantee the tolerances for position tracking errors and yaw in air-bearing planar motors. The proposed method consisted of the nonlinear position controller and nonlinear observer. The nonlinear position controller was designed by a backstepping procedure using BLF to satisfy the constraints of position errors and yaw. The nonlinear observer was developed to estimate full state using only position feedback. In the simulations, we observed that the tolerances for the position tracking errors and yaw were guaranteed by the proposed method during both transient and constant periods. In the future work, we will design the controller to guarantee the constraints of both the control inputs and the outputs.