1. Introduction

The inertial navigation system (INS), a fully self-contained system, could provide motion information including position, velocity, and orientation without external references, commonly used in the carriers such as vehicles, ships and aircraft [1–3]. Fiber optic gyroscope (FOG), owing to its small size, low cost, short start-up time, high reliability, small random walk coefficient and wide dynamic range, is often used for constructing the inertial navigation system with accelerometers. However, compared to other inertial sensors such as laser gyro and liquid gyro, the accuracy of FOG is limited owing to its drift. Rotation modulation technology could modulate the constant and slow changing drift to the sine/cosine form by rotating a strap-down INS about one or more axes periodically. Therefore, the gyro drift can be cancelled automatically after integrated to suppress the navigation errors effectively and improve the navigation accuracy greatly [4–8]. However, when the carrier is in angular motion, the angular velocity of the motion would be coupled with the rotational rate of the inertial measurement unit (IMU), so that the modulated equivalent gyro drift is no longer a strictly periodic sine/cosine form. After integrated, the drift could not be mitigated completely and then the navigation accuracy could be reduced [9–14].

Taking a single-axis rotation INS (SRINS) as an example, at present, the SRINS mainly uses the measurement of the encoder to control the IMU rotating about the z-axis [15–17]. However, the encoder measurement contains only the rotation velocity of the IMU with respect to the platform, but not the rotation velocity of the carrier. Therefore, by this control scheme, the carrier motion cannot be insulated. Therefore, the scheme, depending on encoder measurement to drive the IMU rotating, could reduce the gyro drift to a certain extent but could not insulate the azimuth motion of the carrier.

Herein, with the existed rotating mechanism already in the SRINS, a novel rotation control scheme is proposed. The control scheme employs the measurements of gyros and accelerometers, rather than the encoder, to generate control instructions to drive the IMU rotating periodically, which makes the INS not only own the function of rotation modulation, but also own the ability to isolate the azimuth motion. This INS can be called single-axis rotation/azimuth-insulation INS (SRAINS). Because the gyro measurement contains not only the rotation information of the IMU with respect to the platform, but also the angular motion information of the carrier, using the gyro measurement information to drive the IMU rotation could achieve the functions of rotation modulation and carrier azimuth motion insulation. Therefore, the proposed control scheme theoretically may further improve the INS navigation accuracy without increasing the complexity of the system structure.

In addition, in the rotation INS (RINS), the control error of the system can also reduce the navigation accuracy [15, 16]. In order to improve the control accuracy, optimizing the control strategy and shortening the delay time are adopted. The optimized control strategy is, regardless of whether the carrier has an angular movement or not, to control the IMU rotating about the z-axis of the platform frame with respect to the navigation frame, rather than the up-axis in the navigation frame [12, 18]. The method for shortening delay time is to apply interrupt mode instead of inquiry mode [9] to complete the data transfer between the navigation and control processors.

In brief, the proposed scheme is expected to improve the navigation accuracy of the INS by rotating the IMU about the z-axis with respect to the navigation frame with the measurements of the fiber optic gyros to insulate the azimuth motion, and optimizing the control strategy and shortening the delay time to improve the control accuracy.

2. Theoretical analysis

2.1 Configuration of the SRAINS

The SRAINS used in this study, as shown in Fig. 1, is composed of an IMU, an azimuth gimbal, and four related printed circuit boards (PCBs). The IMU mainly contains three FOGs and three accelerometers. The constant biases (1σ) of the FOG and accelerometer are 0.02°/h and 50μg, respectively.

 

Fig. 1 Configuration of the SRAINS.

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The azimuth gimbal contains a torque motor, an angle encoder and the azimuth frame. The torque motor is a permanent magnet DC torque motor, named RBE-01810 and produced by KOLLMORGEN Company. The encoder is an incremental circle-grating produced by Renishaw and its resolution is 0.83”. PCBs combined the software complete the navigation solution, motor control, sensor signals acquisition such as gyroscopes, accelerometers, encoders, and other functions such as fault detection, data transmission and system power supply. Particularly, the core PCB consists of two processors: the navigation processor and the control processor. The navigation processor is TMS320F67XX owing to its powerful and fast calculation ability. The motor control DSP is TMS320F28XX owing to its various functional modules on motor control such as QEP, PWM, etc. The former DSP mainly completes navigation solution, gyroscope and accelerometer information acquisition and other functions. The latter DSP mainly completes the motor control, the encoder information acquisition, etc..

As seen from Fig. 1, the hardware configuration of the SRAINS is the same with the traditional SRINS. Therefore, the realization of rotation and azimuth-insulation functions is based on the existed rotation mechanism of the traditional SRINS, without increasing the complexity of the system structure.

In the next section, the influence of the azimuth motion, control errors on the navigation accuracy and the time-delay effect on the control precision will be analyzed theoretically.

2.2 Influence of the azimuth motion on the navigation accuracy

In the strap-down INS (SINS), attitude and velocity errors propagation equations [19] are expressed in Eqs. (1) and (2):

In RINS, attitude and velocity errors propagation equations [9] are shown in Eqs. (3) and (4):

In Eqs. (1)–(4), $\varphi$ is attitude error vector; $V$ and $\delta V$ are ground velocity vector and its error; $\omega$ and $\delta \omega$ are angular velocity vector and its error; $f$is a force vector; $\delta g^n$ is a gravity acceleration error vector; $\epsilon$ and $\nabla$ denote the gyro drifts and the accelerometer biases;$\delta K$ is scale factor error matrix and $\delta K=diag({\left[\begin{array}{ccc}\delta K_x& \delta K_y& \delta K_z\end{array}\right]}^T)$. $\delta G$ and $\delta A$ are installation error matrices of gyros and accelerometers. The subscripts $n$, $b_0$, $b$, $i$, $e$ and $p$ indicate the navigation frame ($n$-frame, defined as east-north-up), the body frame at the initial time ($b_0$-frame), the body frame ($b$-frame), the inertial frame ($i$-frame), the Earth frame ($e$-frame) and the platform frame(p-frame), respectively [9]; The p-frame is defined as follows: $z_p$ axis coincides with the rotation axis of the azimuth gimbal, $x_p$ axis is defined by the projection of the sensitive axis of the gyroscope $x$ in the normal plane of $z_p$, and $y_p$ axis is defined according to the right-hand rule [8]. Therefore, the transformation matrix $C_p^b$ from the p-frame to the b-frame can be described in Eq. (5):

Comparing the error propagation equations of the SINS and RINS, the difference between them is that in the latter equations, the transformation matrix $C_p^b$ is added. Since$C_{b0}^n$ is a constant matrix and exists in the error propagation equations of the SINS and RINS, the matrix $C_{b0}^n$ is independent of the rotation modulation effect. The following analysis assumes that the body frame at the initial time coincides with the n-frame, which means $C_{b0}^n=I$. For the SRINS, if the IMU rotates about the $z_p$-axis bi-directionally and continuously, and the rotation angle is ${\phi }_z$, we see that $C_p^b$ can be described in Eq. (5). When the carrier is stationary, $C_b^{b0}=I$. Then according to Eq. (3), the gyro drifts are modulated into a sine or cosine form, as shown in Eq. (6).

As shown in Eq. (6), theoretically, when the carrier is stationary, after the rotation modulation, the gyro drifts are modulated into a positive and negative symmetrical periodic form, and the mean is zero after the whole period integration. This means that the horizontal gyro drifts in b-frame are zeroes and they can be completely eliminated. Therefore, if the carrier is not maneuvering, the gyro drifts can be compensated automatically by rotation modulation, thus improving the navigation accuracy.

However, if the carrier is in azimuth motion, $C_b^{b0}$ is a time-varying matrix.

Comparing Eqs. (6) and (8), we can see that if the azimuth motion is not insulated, the carrier angular motion would be coupled with the IMU rotational motion so that the modulated drifts no longer have the strictly cosine form of the strict period, which eventually reduces the integral mitigation effect. In particular, when the carrier maneuvering direction is opposite to the motor rotation and the magnitude is equal, the gyro drifts are not modulated in the body frame. While in the SINS, the gyro drifts will be modulated due to the carrier maneuvering in the body frame. In this case, navigation accuracy of the RINS is worse than that of the SINS. Assuming that the two horizontal gyro drifts are both 0.01 °/h and the magnitude of the carrier maneuvering is equal to the motor rotation magnitude while the direction is opposite, according to Eqs. (1)-(4), after the integration, the maximum position error in 8h is about 3000 m in the east and about 8000 m in the north. While after the azimuth motion insulated, the maximum position errors are about 1 m in the east and in the north, respectively.

Similarly, if the carrier is in azimuth motion, ${\omega }_{nb}^b$ equals ${[\begin{array}{ccc}0& 0& {\omega }_{nbz}^b\end{array}]}^T$ and ${\omega }_{bp}^p$equals${[\begin{array}{ccc}0& 0& {\omega }_c\end{array}]}^T$. According to Eq. (3), ignoring the angular velocity of the Earth, the angular velocity error due to $\delta K_G$ and $\delta G$ can be expressed as follows:

As seen from Eq. (9), if the carrier is only in azimuth motion, just the errors $\delta G_{31}$, $\delta G_{32}$ and $\delta K_{Gz}$ can cause the angular velocity error. For the errors $\delta G_{31}$ and $\delta G_{32}$, no matter whether the carrier is in azimuth motion or not, the angular velocity errors due to $\delta G_{31}$ and $\delta G_{32}$ embody the characteristics of a single frequency sine and cosine correlated with ${\phi }_z$ or $-\psi +{\phi }_z$. After integration, angular velocity errors are nearly zeros. Therefore, the influence of the azimuth motion on the navigation accuracy is small due to the installation errors. For the error $\delta K_{Gz}$, if the carrier is in azimuth motion, the extra angular velocity error caused by $\delta K_{Gz}$ is ${\omega }_{nbz}^b\times \delta K_{Gz}$, which eventually reduces the modulation effect of the RINS. For example, when the carrier unidirectional rotates continuously at speed 1°/s, if the scale factor error $\delta K_{Gz}$ is 20 ppm, the angular velocity error due to $\delta K_{Gz}$ is approximately 0.072°/h, and the extra positioning error in 8 h is approximately 18000 m in the east and 35000 m in the north.

From the above analysis, we can see that when the carrier is stationary, in the traditional RINS, rotation modulation can automatically compensate for the biases of the inertial sensors, and improve the navigation accuracy. However, when the carrier is in azimuth motion, the carrier azimuth motion is coupled with the IMU rotational motion and then reduce the modulation effect of the RINS. Therefore, it is necessary to insulate the azimuth motion and realize the rotation modulation simultaneously to further improve the navigation accuracy of the INS.

2.3 Influence of control error on navigation accuracy

At present, the studies on RINS, such as initial alignment, error analysis, self-calibration, are assumed that the rotation control is error-free [20–22]. But actually, the error during the rotation control process is inevitable. Control errors could cause the gyro constant drift to be unequal to zero after a positive and negative rotation period integral, which is called non-modulatable drift and lead to a decrease in navigation accuracy. In this section, taking the SRINS with the bi-directional and continuous rotation scheme as an example, during the clockwise (CW) or counter clockwise (CCW) rotation process and the reversing process, the influences of the control error on navigation accuracy are analyzed. The control scheme is shown in Fig. 2.

 

Fig. 2 The control scheme diagram.

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In the control scheme, control errors during the CW and CCW rotation processes are similar, which called steady-state error. While the control error during the reversing process is called overshoot error.

2.3.1 Influence of the control error on navigation accuracy during the CW and CCW rotation processes

During the CW and CCW rotation processes, the steady-state control error could increase the angular random walk error of the azimuth gyro, and the random walk error is one of the main factors that restrict the long time navigation accuracy of INS [23].

The angle random walk (ARW) error can be expressed as:

Then the root-mean-square deviation (RMSD) of the attitude error due to the ARW error can be expressed by the standard deviation of $\theta (t)$, as shown in Eq. (12).

In the INS, the drifts of the inertial sensors could cause two kinds of periodical errors, i.e. the Schuler oscillation and the earth self-rotating oscillation. However, the influence of the Schuler oscillation is much smaller than that of the rotation of the Earth on the long navigation accuracy. Ignoring the Schuler effect the position errors in the east and north,$\Delta S_E$ and $\Delta S_N$, due to the z-gyro drift ${\epsilon }_z$ can be written as [19]:

Similarly, the position errors in the east and north due to the z- gyro ARW error (PSD = ${\sigma }_{{\epsilon }_z}^2$) can be written as:

In Eqs. (13) and (14), ${\omega }_e$ and $R$ are the rotational angular velocity and radius of the Earth. $L$ is the latitude. From Eq. (14) we can see that the gyro random walk error can cause the divergence position error over time, and the position error divergence term is also a random walk process. It is assumed that the random walk coefficient $K$ is increased by 0.001°/$\sqrt{h}$due to the control error, and$L$ = 40°, then according to 10 simulation results, the RMSD of the position errors due to the ARW in the east and north are 259.2m and 202.6m in 8h.

In addition, during the CW and CCW rotation processes, the control error will cause the fluctuation of the angular velocity over time, which results in the non-modulatable drifts of the horizontal gyros in the b-frame. In theory, according to Eq. (6), when the carrier is stationary, after the rotation modulation, horizontal gyro drifts in b-frame can be completely eliminated by integration. However, if the angular velocity of the IMU is fluctuant, assuming that the rotation angle interval [0,$2\pi$] is divided into N segments, each segment is $\Delta {\phi }_z$. If $\Delta {\phi }_z$ is small enough, the angular velocity in each segment can be considered as a constant, symbolized as ${\omega }_1$, ${\omega }_2$$\cdots$${\omega }_N$. According to Eq. (6), the integration of ${\epsilon }_X$ in one rotation cycle can be expressed as:

In practice, because${\omega }_1$ and ${\omega }_N$ are almost equal, assuming${\omega }_1$ = ${\omega }_N$, we get:

Similarly, the integration of ${\epsilon }_Y$ in one rotation cycle can be expressed as:

In Eqs. (16) and (17), if the sampling frequency of angle and the rotation velocity are fixed, $\Delta {\phi }_z$ is also fixed. As seen from Eqs. (16) and (17), the residual error of drift is proportional to the severity of fluctuation. The greater the value of $(1/{\omega }_{k+1})-(1/{\omega }_k)$, the more the residual error of drift, and the worse the rotation modulation effect. In rotation control process, if the fluctuation of the rotation velocity is 50ʺ/s, according to Eqs. (16) and (17), the equivalent non-modulatable drifts are: $E_X=$6.1e-007${\epsilon }_x^p$-0.0171${\epsilon }_y^p$ and $E_Y=$0.0171${\epsilon }_x^p$-6.1e-007${\epsilon }_y^p$.

2.3.2 Influence of control overshoot error on navigation accuracy during the reversing process

During the reversing process, the control overshoot error lead to the sensitive axis drifts in one period cannot be completely modulated, which can cause navigation errors in long time navigation. Assuming that the overshoot angle is $\alpha$ and $\beta$ in the CW and CCW rotation process, respectively, then after a CW and CCW rotation, the attitude error in b-frame is:

As seen from Eq. (19), the overshoot angles $\alpha$ and $\beta$ lead to the horizontal gyro drifts in b-frame are not zeroes after integration, which means $\alpha$ and $\beta$ will cause the attitude errors and then decrease the navigation accuracy.

As seen from the above analysis, in the CW and CCW rotation processes, the control error will cause the ARW error of the azimuth gyro and the fluctuation of the angular velocity which results in the non-modulatable drifts of the horizontal gyros in the b-frame; in the reversing process, the control overshoot error will lead to the attitude errors. All these errors will decrease the navigation accuracy. Therefore, it is necessary to reduce the control error.

2.4 Influence of time-delay on control accuracy

In the traditional RINS, the feedback signal of the rotation control system is provided by the encoder and obtained by the SPI module or QEP module of the control DSP [15–17]. Since the motor control and the feedback signal are both accomplished in the control DSP independently, data exchange is not required between the navigation DSP and control DSP, and there is no time-delay problem in the control system.

However, if the RINS has the azimuth insulation function, the feedback signal of the control system is the angular velocity information of the navigation solution. While the navigation solution is accomplished in the navigation DSP due to its powerful calculation ability and the motor control is accomplished in the control DSP due to its various functional modules on the motor control. Therefore, data exchange is necessary between the navigation DSP and control DSP, which can lead to a time delay problem in the control system. The time delay could reduce the rotation control accuracy and then reduce the rotation modulation effect. The influence of time delay on control accuracy is analyzed as follows:

From section 2.1 we know that the motor used in this study is DC torque motor, and its transfer function can be expressed as [12]:

Table 1. Parameters of the motor.

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The control algorithm exploited in the system is PII² [12], which proposed by Lei Wang in [12]. Reference [12] pointed out that PII² control algorithm in the motion-insulated control system is more effective to decrease steady-state angle error than the PID control algorithm. The open - loop transfer function of PII² control algorithm is shown in Eq. (21):

Further considering the delay time $\tau$ in the communication process between the two DSP, the control transfer function of the control system is:

If the time delay$\tau$is increased from 0.0005s to 0.01s, the root locus of the control system is shown in Fig. 3(a).

 

Fig. 3 (a) Root locus of the control system with the different delay-time. (b) Step response of the control system with the different delay-time.

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As seen from Fig. 3(a), with the time delay increasing, the root trajectory is closer to the imaginary axis, resulting in narrowing the stability of the closed-loop system, thereby reducing the stability of the system. In order to further verify the impact of time-delay on the dynamic performance of the control system, the step response is given in Fig. 3(b) with $\tau$ = 0.0005s, $\tau$ = 0.0025s and$\tau$ = 0.01s. From Fig. 3(b) we can see that when the delay time increases, the dynamic performance of the system is significantly reduced. The fluctuation amplitude becomes larger and the adjustment time becomes longer, which is consistent with the above root locus analysis results.

Hence, the time delay will reduce the control system performance, and even cause system instability. Therefore, if the designed controller does not take into account the impact of time delay, the actual control system performance will be greatly reduced, and may even become unstable. To sum up, in the design of the control system, in order to improve the control accuracy, to minimize the time delay of the control system is essential.

At present, the control requirements from engineering experience are: the steady-state control error is less than 20” in the CW and CCW rotation processes, and 50” in the reversing process, the effect of which on navigation can be negligible.

3. Design and implementation scheme of the rotation/azimuth-insulation inertial navigation control system

From the analysis in section 2, we can see that the azimuth motion and poor control accuracy could reduce the rotation modulation effect and then reduce the navigation accuracy. Therefore, in the design of the control system not only need to achieve rotation/azimuth-insulation function, but also need to improve the control system control accuracy.

3.1 Control strategy of the SRAINS

In the platform INS, controlling the frame rotation with gyro measurements can make the platform coordinate always point to the geographical coordinate (coincident with the n-frame), and ultimately to insulate the carrier movement. It can be expressed in math as ${\omega }_{np}^n=0$, which means in the n-frame, the angular velocity vector of the n-frame relative to the p-frame is zero. On the basis of the insulation method of the platform INS, in the SRINS, we can also control the azimuth motor to meet ${\omega }_{npz}^n=0$ to isolate the carrier angular motion. Furthermore, in order to achieve both azimuth-insulation and rotation modulation functions, we can control the azimuth motor to meet ${\omega }_{npz}^n=\Omega$, where $\Omega$ is the angular velocity of the rotation modulation. Namely, regardless of whether the carrier is in azimuth movement, control the IMU rotating uniformly about the up axis of the n-frame with respect to the p-frame, to ensure that ${\omega }_{npz}^n=0$, thereafter the azimuth-insulation/rotation modulation function can be achieved. In this control strategy, the calculation principle of the motor control angular velocity is as follows:

In the INS, the relationship between the n-frame and the p-frame is:

Combining Eqs. (23)–(25) give:

Substituting Eqs. (24) and (28) into Eq. (27), ${\omega }_c$ can also be expressed as:

As seen from Eq. (29): 1) If the carrier is stationary, ${\omega }_c=\Omega /(\cos \theta \cos \gamma )$, which means the control angular velocity of the motor ${\omega }_c$ is related to the initial attitude. ${\omega }_c$ changes with the variety of the initial horizontal attitude $\theta$ or$\gamma$, which is detrimental to the motor control. 2) If the pitch or roll is 90°, ${\omega }_c$ should be infinite theoretically because $\tan \theta \to \infty$ and $1/\cos \gamma \to \infty$, which indicates that the control model has a singularity. 3) if the pitch or roll is close to 90°, because$\sin \gamma (\tan \theta -1)\dot{\theta }$,$\dot{\gamma }\tan \theta /\cos \gamma$ and $({\sin }^2\gamma \cos \gamma \sin \theta -{\sin }^2\theta )/(\cos \theta \cos \gamma )$ are close to infinite, the small changes in attitude can result in infinite rotation velocity of ${\omega }_c$, i.e., the motor is uncontrollable.

3.2 Improved control strategy of the SRAINS

In order to overcome the shortcomings described above, this paper presents an improved scheme: regardless of whether the carrier is in azimuth movement or not, control the IMU rotating uniformly about the z-axis of the p-frame with respect to the n-frame, to ensure that${\omega }_{npz}^p=\Omega$ to accomplish azimuth-insulation and rotation modulation. In this control strategy, the calculation principle of the motor control angular velocity is as follows:

In SRINS, combining Eqs. (23) and (24) we get:

In order to control the IMU rotating about the z-axis of the p-frame with respect to the n-frame, i.e. to meet ${\omega }_{npz}^p=\Omega$, ${\omega }_c$ should meet:

As seen from Eq. (31), 1) If the carrier is stationary, ${\omega }_c=\Omega$, which means the control angular velocity of the motor ${\omega }_c$ is independent of the initial attitude. ${\omega }_c$ would not change with the change of the initial horizontal attitude $\theta$ or$\gamma$. 2) If the pitch or roll is 90°, ${\omega }_c$ should not be infinite, which indicates that the control model has no singularities. 3) If the pitch or roll is close to 90°, the small changes in attitude cannot result in infinite rotation velocity of ${\omega }_c$. Therefore, all the problems mentioned in the original control strategy have been solved.

To sum up, for the SRINS, both control strategies can achieve the azimuth-insulation to a certain extent. However, the improved control strategy is more conducive to improving the control accuracy. Therefore, the improved control strategy will be used in this study for insulating the azimuth motion and modulating the inertial sensors drifts.

3.3 Implementation of the SRAINS

The control system of the SRAINS with the improved control strategy mainly consists of three parts: processors – the navigation processor, control processor and dual port RAM for data transferring, measuring devices – gyros, accelerometers, the encoder, and so on, rotating mechanism – the frame, the motor and its drive circuits. The configuration of the control system of the SRAINS is shown in Fig. 4.

 

Fig. 4 Configuration of the control system of the SRAINS.

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As seen from Fig. 4, the control system exploits the motion information ${\omega }_{ip}^p$ and$f_{}^p$ measured by gyros and accelerometers in the IMU, generates the control instruction ${\omega }_{cmd}$ by the navigation solution and then sends it to the control processor by dual port RAM. The control processor uses the control algorithm PII² to generate the PWM control quantity, which is converted into the current signal by means of the motor drive circuit. Finally, the current signal drives the motor rotating to realize the rotation modulation and azimuth insulation. In addition, the motor rotation angle ${\phi }_z$ is an indispensable parameter in the navigation solution. While ${\phi }_z$ is measured by the encoder and is obtained by the QEP function module in the control processor. Therefore, the control instruction ${\omega }_{cmd}$ and motor rotation angle ${\phi }_z$ must be exchanged between the navigation and control processors. At present, the data transmission are mainly with CAN bus [10], serial RS232, RS422 etc., which are appropriate for the case that a long distance between the navigation DSP and control DSP. However, these devices will result in indefinite delay time. The analysis of section 2.4 shows that the time-delay will decrease the control accuracy, and then reduce the navigation accuracy. The dual-port RAM is a distinctive data storage chip, which provides two independent ports with separate control, address, and I/O pins that permit two DSP access for reads or writes the same unit in memory. The dual-port RAM also provides two completely independent ‘busy’ logic to ensure the correctness of two CPU simultaneously read and write the same unit [24]. Dual-port RAM features a fast access time, and both ports are completely independent of each other. This means that the dual-port RAM can decrease the delay time of the control system due to its hardware, and then improve the control accuracy. Therefore, the dual-port RAM is well adapted for data transmission between two DSP, and it is used in this study for transmitting data between the navigation DSP and the control DSP to improve the control accuracy.

The detailed implementation of the control system of the SRAINS is as follows:

In RINS,

After the control instruction angular velocity ${\omega }_{np}^p$ is obtained by Eq. (32). ${\omega }_{np}^p$ is transmitted from the navigation DSP to the control DSP by the dual port RAM. In the control DSP, deducting the rotation control angular velocity ${\omega }_R$ = $\Omega$ from ${\omega }_{np}^p$, we get the angular deviation $\Delta \omega$,$\Delta \omega ={\omega }_{npz}^p-\Omega$. Then the deviation $\Delta \omega$ is used for calculating the control voltage $U_C$ with the PII² algorithm.

 

Fig. 5 Detailed implementation of the control system of the SRAINS.

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In Fig. 5, since ${\omega }_{npz}^p$ contains the angular velocity of the carrier motion and the motor rotation, it is possible to control the IMU rotating about the z-axis at an angular velocity $\Omega$ regardless of whether the carrier has an azimuth motion or not. Therefore, when drive the torque motor rotating in accordance with the control instruction angular velocity ${\omega }_{np}^p$,the biases of the inertial sensors can be modulated and the carrier azimuth motion can be insulated.

3.4 Implementation of the software

In combination with the system described above, the control software is programmed on the control DSP. In order to shorten the delay-time, software scheme exploits the interrupt mode instead of the inquiry mode to obtain the control instruction angular velocity ${\omega }_{np}^p$ from the navigation DSP. After the control instruction ${\omega }_{np}^p$ is solved in the navigation DSP, the interrupt signal is sent to the control computer immediately, and then the control computer completes the control instruction read, the angular deviation $\Delta \omega$ and control voltage $U_C$ calculation and other functions. The software implementation process is shown in Fig. 6.

 

Fig. 6 Software flowchart of the control DSP. (a) Inquiry mode. (b) Interrupt mode.

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The flowchart of the inquiry mode is shown in Fig. 6(a). In Fig. 6(a), 5ms is the inertial navigation computation period. Substitute the procedures in Fig. 6(b) for the procedures in the red box in Fig. 6(a), which is the flowchart of the interrupt mode.

As seen from Fig. 6, the maximum delay time for transmitting the control instruction is 5ms in the inquiry mode. If the flowchart apply the interrupt mode, data exchange time with dual-port RAM is only a few microseconds, therefore the delay time for transmitting the control instruction is ignorable in the interrupt mode. In brief, in the implementation of the control system, the interrupt mode replacing the inquiry mode in software flowchart and the dual port RAM replacing CAN bus etc. in hardware designing, shorten the delay time for transmitting the control instruction between the two DSP, to improve the control precision.

4. Simulation and experimental results

4.1 Simulation

In the implementation of the control system, in order to improve the control accuracy, first, control the IMU rotating about the z-axis of the p-frame with respect to the n-frame rather than the up-axis of the n-frame; second, use interrupt mode and dual port RAM to replace the inquiry mode and CAN bus etc. to shorten the delay-time. In order to verify the effectiveness of the improved implementation for improving control accuracy, the following simulation is carried out: In the same attitude motion, 1) with the same delay time and different control strategies, simulate the control error of the control system; 2) with the improved control strategy, with different delay time, simulate the control error of the control system; 3) with different control strategy and different delay time, simulate the control error of the control system. The attitude motion of the pitch $\theta$, roll $\gamma$ and azimuth $\psi$ are the superposition of n-component sine curves of different amplitude, different frequency, and different phase, where $\theta$, $\gamma$ and $\psi$ can be expressed as:

 

Fig. 7 (a) Attitude in the motion. (b) Motor control rotation velocities with the traditional and improved strategy.

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According to Eqs. (29) and (31), the control rotation velocities of the motor with the traditional strategy and the improved strategy can be denoted as ${\omega }_{c1}$ and ${\omega }_{c2}$, respectively. The traditional strategy is to control the IMU rotating about the up-axis of the n-frame. The improved strategy is to control the IMU rotating about z-axis of the p-frame. ${\omega }_{c1}$ and ${\omega }_{c2}$ are shown in Fig. 7(b). The outputs of the control system with $\tau =0.5ms$ and different control strategy are shown in Fig. 8(a). The outputs of the control system with the improved strategy and $\tau =5ms,0.5ms$ are shown in Fig. 8(b). The outputs of the control system with the traditional strategy, $\tau =5ms$and the improved strategy, $\tau =0.5ms$ are shown in Fig. 8(c). The ideal angular velocity of the control system output is 6°/s.

 

Fig. 8 (a) Control outputs with different control strategy. (b) Control outputs with different delay time. (c) Control outputs with different control strategy and different delay time.

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As seen from Fig. 8(a), with the same delay time, the control error is smaller with the improved control strategy. As seen from Fig. 8(b), in the same attitude motion, the fluctuation of the control rotation velocity with the traditional strategy derived from the platform INS is greater than that with the improved strategy. Furthermore, the shorter the delay time, the shorter the stabilizing time of the control system. As seen from Fig. 8(c), if the delay time is shortened and the optimized control strategy is adopted, the response time of the control system becomes shorter and the stability control accuracy is improved. Therefore, optimizing the control strategy and shortening the time delay in this study could improve the control accuracy theoretically.

4.2 Experimental results

In order to verify the effectiveness of the implementation of the SRAINS proposed in this study, several experiments were carried out in a real SRINS. The experimental equipment in the test, as shown in Fig. 9, are composed of a single-axis RINS, a DC power supply and a laptop that received the experimental data at a frequency of 200Hz.

 

Fig. 9 The experimental equipment.

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4.2.1 Control performance and rotation/azimuth-insulation function verification

In order to verify that the improved control system can reduce the control error and achieve the rotation/azimuth-insulation function, the experiments on the stationary base and moving base were carried out respectively.

First, the experiment is on the stationary base: Put the SRAINS on the stationary base, and exploit the traditional implementation as well as the improved implementation to control the IMU rotating respectively. Use the formula $\Delta {\psi }_p={\displaystyle \int ({\omega }_p^{}-\Omega )dt}$ to calculate the control precision, which is utilized for verifying the performance of the control system. ${\omega }_p^{}$ is the angular velocity of the p-frame with respect to the n-frame and $\Omega$ = 6°/s, which is the ideal angular velocity of the control system. The traditional implementation is to control the IMU rotating about the up-axis of the n-frame and to apply the inquiry mode in the software design. The improved implementation is to control the IMU rotating about the z-axis of the p-frame and to apply the interrupt mode in the software design. The control error with the traditional implementation and the improved implementation are shown in Fig. 10(a).

 

Fig. 10 Performance verification of the control system of the SRAINS Control error with different implementation (b) Rotation/azimuth-insulation function verification.

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As seen from Fig. 10(a), the steady-state control error during the CW and CCW rotation processes is less than 5” with the improved implementation while which is less than 50” with the traditional implementation; the overshoot error during the reversing process is less than 50” with the improved implementation while which is less than 200” with the traditional implementation; Therefore, the accuracy of the improved control system is obviously improved.

Second, the experiment is on the moving base: First, put the SRAINS on the turntable and perform the initial alignment. Second, rotate the turntable during the first CW and CCW rotation of the navigation process so that the SRAINS is in azimuth motion. Third, keep the turntable stationary during the second CW and CCW rotation of the navigation process so that the SRAINS is stationary. Finally, compare the azimuth angular velocity of the p-frame ${\omega }_p^{}$ in the two CW and CCW rotations; if ${\omega }_p^{}$ in the two CW and CCW rotations is consistent, it proves that regardless of whether the carrier is in azimuth movement or not, the IMU will rotate at 6°/s uniformly, which verifies that the azimuth motion has been insulated in the SRAINS. The azimuth $\psi$, the angular velocity ${\omega }_p^{}$ and the control error $\Delta {\psi }_p$ in the experiment are shown in Fig. 10(b).

As seen from Fig. 10(b), during the first CW and CCW rotation, when the carrier is in the azimuth motion, the IMU still rotate at $\pm$6°/s, which proves that the azimuth motion has been insulated and the sensors drifts can be modulated due to the IMU rotating at $\pm$6°/s. From Fig. 10(b), we can also see that: during the CW and CCW rotation, when the carrier is in the azimuth motion, the dynamic control error is less than 10”; when the carrier is stationary, the control error is less than 5”.

Therefore, the results of the experiments on the stationary base and moving base show that: the control system proposed in the study can achieve the rotation/azimuth-insulation function; in addition, the steady-state control error is less than 10” and the overshoot error is less than 50”, which can meet the requirements of INS.

4.2.2 Azimuth motion insulation effect verification Azimuth motion insulation effect verification

In order to verify that the necessity of azimuth motion insulation, the following test is carried out. According to the practical data of a vehicle test and a ship test, rotating the turntable to simulate the carrier azimuth motion and using the encoder measurement as well as gyro measurement respectively to control the IMU rotation, compare the position error before and after the insulation of the azimuth motion to verify that the necessity of azimuth motion insulation.

Because the encoder measurement only contains the rotation velocity of the IMU but not the carrier, using the encoder measurement to control the IMU rotation cannot realize the azimuth motion insulation. While using the gyro measurements can realize the azimuth motion insulation, according to the analysis of section 3. The attitude of the vehicle test, ship test and this test are shown in Figs. 11(a)–11(c). The position errors $\Delta S_E$ and $\Delta S_N$ in the east and north before and after the azimuth motion insulated are shown in Fig. 11(d).

 

Fig. 11 Azimuth motion insulation effect verification. (a) Attitude of the vehicle test; (b) Attitude of the ship test; (c) Attitude of this test;(d) position errors in the east and north before and after the azimuth motion insulated.

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As seen from Fig. 11(d), before the azimuth motion shown in Fig. 11(c) insulated, the maximum position error $\Delta S_E^{}$ is about 4000m, $\Delta S_N^{}$ is about 6000m in the north. While after the azimuth insulated, the maximum position errors in the east and north are about 2000m. The navigation position error can also be represented by a radial position error$\Delta S$,$\Delta S=\sqrt{\Delta S_E^2+\Delta S_N^2}$, and after the azimuth motion insulated, the maximum radial position error reduced to 2800m from 7200m. Therefore, in the SRINS, insulating the azimuth motion is an effective way to improve the navigation accuracy.

5. Conclusion

In this paper, by analyzing the influence of carrier azimuth motion and rotation control precision on the navigation accuracy of the RINS and the influence of time delay on the control precision, some conclusions are drawn as follows: First, the carrier azimuth motion is coupled to the IMU rotational motion, resulting in a reduction of rotation modulation effect. Second, the steady-state control error during the CW and CCW rotation would cause the ARW error of the azimuth gyro and the non-modulatable drifts of the horizontal gyros in the b-frame; the overshoot error during the reversing process would cause attitude errors. Last, the delay-time of the control system will reduce control accuracy. All these errors would decrease the navigation accuracy, and should be reduced.

In order to improve the navigation accuracy of the INS, reduce the influence of carrier azimuth motion and improve the control accuracy, a novel control scheme is proposed in this study. The control scheme exploits the gyro measurements to control the IMU rotating to achieve both functions: rotation modulation and azimuth-insulation. In the implementation of the control system, this study improved the control accuracy by two ways: first, exploit the control strategy to control the IMU rotating about the z-axis of the p-frame with respect to the n-frame rather than the up-axis of the n-frame; second, use the interrupt mode to replace the inquiry mode to shorten the delay-time due to the data transfer between the navigation and the control processors. The simulation and experimental results show that: first, the control system proposed in this study can achieve the rotation/azimuth-insulation function; second, compared to the traditional implementation, the improved implementation improves the steady-state control precision from 50” to 10” during the CW and CCW rotation and the overshoot control precision from 200” to 50” during the reversing process. It is obviously that the control accuracy improved by about 4 times; lastly, in a practical test, after the azimuth motion insulated, the maximum position error reduce to 2800m from 7200m in the radial direction, which means the position error can be reduced more than 50% in some navigation movement. It is obvious that the navigation accuracy is improved in the SRAINS.