A Short-baseline Dual-antenna BDS/MIMU Integrated Navigation System

. This paper puts forward a short-baseline dual-antenna BDS/MIMU integrated navigation, constructs the carrier phase double difference model of BDS (BeiDou Navigation Satellite System), and presents a 2-position initial orientation method on BDS. The Extended Kalman-filter has been applied for the integrated navigation system. The differences between MIMU and BDS position, velocity and carrier phase information are used as measurements. The experiment results indicate that the position error is less than 1m, the pitch angle error and roll angle error are less than 0.1°, and the heading angle error is about 1°. It shows that the new integrated navigation system has good performance and can be applied in various fields including USV and UAV.


Introduction
Integrated navigation is a kind of navigation technique, integrating two or more navigation systems and using a complementary algorith m which obviously improves the system's navigation precision. This paper develops an integrated system based on the BeiDou System, Ch ina's domestically designed navigation system, wh ich provides accurate positioning informat ion within transmission range. However, it can't continuously navigate when the satellites are shielded as is the case with GPS [1]. Inertial navigation system based on MIMU is a highly independent and invisible navigation technology [2]. Although MIMU has the advantage of being small, lightweight and low-cost, its precision is lo w; there is a significant error divergence limit ing the working hours of positioning. Therefore, it's necessary to integrate BDS and MIMU, wh ich helps achieve better navigation effects.
Concerning single-antenna GNSS/MIMU integrated system, the significant noises of MIMU make it difficult to define heading angle in init ial align ment [3], so some assistant sensors are needed. In conventional solutions, magnetic co mpass is usually mentioned, but it brings along another problem in that the compass is not stable since it's easily interfered by external magnetic fields. With the development of satellite orientation determination technology, positioning and attitude measuring using mult iple antennas is becoming more advanced [4][5], so it's beneficial to imp lement a rotary dual-antenna BDS/MIMU integrated navigation system. Scholars have made some advancements in the field, for example, Russian researchers integrate rotary GNSS dual antennas and MIMU. The system uses an alternative heading determination algorithm and results show that the orientation error is less than 1° [6][7]. Researchers fro m the Un iversity of New Orleans put forward an IMU/ GPS co mpass, with Kalman -filter fusing data and the compass outputs roll angle whose calculation accuracy is 2.12° [8].
Usually the dual-antenna GNSS integrated system is based on the double difference carrier phase model, which eliminates or diminishes satellite ephemeris error, clock error, ionosphere error, troposphere error to ach ieve millimeter-scale precision. Co mpared to the integration based on pseudo range model, the double difference carrier phase model is much more accurate [9][10]. The key problem of GNSS orientation is determining an amb iguity solution. The main algorithms include amb iguity mapping function, least ambiguity decorrelation adjust [11]. Researchers using these methods can achieve high accuracy, but usually the length of baseline is over 1m [12][13]. Currently the research focus is on a method to shorten the baseline length and simu ltaneously maintain accuracy. The longer baseline length can surely improve the precision, but also makes the system larger and more expensive.
In this paper, we design a novel method to determine the integer ambiguity double difference based on a rotary short-baseline. By this method, the dual-antenna BDS/MIMU integrated navigation system with 0.3m baseline is able to provide precise position and heading angle continuously, which has practical value in shortbaseline dual-antenna integrated navigation technology.

Description of Hardware
The system is composed of a rotary module, a navigation computer, a M icro Inertial Measurement Unit (MIMU), dual-antenna BDS receiver and electronic circuit, as described in Fig. 1. The OEM BDS board provides the system with position, velocity, carrier phase and satellite ephemeris at a frequency of 5HZ. MIM U, model STIM 300 adopted specifically, includes a 3-axis gyroscope, a 3-axis accelero meter and a 3-axis inclinometer, provid ing data at a frequency of 125HZ. The rotary module drives the OEM BDS board and MIMU rotating. The navigation co mputer consists of a DSP and a FPGA, as Fig. 2 shows. The FPGA connects peripherals and DSP, and sends navigation results to PC; the DSP runs navigation algorith m. The two units are connected through EMIFA interface.    As Fig. 3 shows, represents BDS satellite i, a baseline vector A is composed of two antennas R1 and R2, 1 and 2 represent the actual distance between satellite and antennas, ∆ is the range difference of the two antennas and satellite, is regarded as a unit direction vector since the baseline is far shorter than the distance between satellites and antennas. ∆ can be written as (2) where is the carrier phase measurement of antenna k, is wavelength of BDS B1 carrier, is the actual distance between satellite and antenna k, c is speed of light, ( ) is the clock b ias of receiver, is the clock bias of satellite , is integer ambiguity, ∆ is tropospheric delay between satellite and receiver, ∆ is ionosphere delay between and receiver, is bias of carrier phase measurement, such as mu ltipath error and noise.
In a short-baseline dual-antenna system, the influence caused by tropospheric delay and ionosphere delay can be regarded as the same. So the difference between two antennas' carrier phase measurement is where ∆ = 2 − 1 is the measurement difference, ∆ = 1 − 2 is the range difference between and two antennas, ∆ = 2 ( ) − 1 ( ) is the clock error between two receivers, ∆ = 2 − 1 is the single difference between carrier phase integer amb iguity of two antennas, ∆ = 2 − 1 is the difference between measurement bias of two antennas. Fro m equation (1) and (3), the following equation can be written as = − ( ∆ − ∆ ) + ∆ + ∆ .
(4) Equation (4) shows that the single difference can eliminate the satellite clock error, but the BDS receiver clock error still exists. To eliminate the receiver clock error, the carrier phase's difference between different satellites is needed. Supposed R1 and R2 receives 4 or more satellites simu ltaneously, the satellite ( ≠ ) is the referenced satellite, then the double difference carrier phase model can be written as

Fast orientation method with short-baseline
Rotating the dual-antenna baseline by 180°, the double difference carrier phase of the two positions, 0° and 180°, can be calculated through Equation (5). Then the float amb iguity double difference can be calculated by adding the double difference at two positions, ∆ = ( ∆ 1 + ∆ 2 )/2.

Error model
There are 12 errors in the inertial navigation system, that is ].
There are 3 groups of observation in the filter, that is, velocity observation, position observation, satellite double difference carrier phase observation.
Velocity measurement equation is where , , ℎ are latitude, longitude, height output fro m INS respectively; , , ℎ are lat itude, longitude, height output from BDS board; , , are position measurement error of BDS board in eastward, northward, upward direction.
Satellite double difference carrier phase measurement equation is (20) In addition, the noise vector W can be written as where , , are white noise of gyroscope, and , , are white noise of accelerometer.

Experiment
To verify the result of the algorithm, an experiment has been undertaken. In the experiment, the navigation system was bound to a handcart, as was a Strapdown Inertial Navigat ion System (SINS) based on laser gyroscope with high accuracy. The two systems were fixed on an alu miniu m alloy sheet with screws, ensuring the heading directions were the same. The SINS is regarded as a reference. After the initial align ment of the SINS, the handcart was pushed along a specified route. Fig. 4 shows the trace plot of the system, where the blue solid line is the trace output fro m the system and red dotted line is the real trace. The plot shows that the position error of the system is less than 1m. The attitude output from the SINS over a short period of time is very accurate and stable. Therefore, the attitude measurement error of the system is the attitude difference between the SINS and the system. As Fig. 5 shows, the blue line is the pitch angle error, roll angle error and heading angle erro r of the system. Figure 5 and Tab le 1 indicate that the system's pitch angle error and roll angle error are less than 0.1°, and heading angle erro r is about 1°.  In Fig. 6, the blue line is eastward and northward velocity which is calcu lated by the integrated system. The red line is eastward and northward velocity, which is calculated by the SINS. The figure shows that the two lines are almost overlapping. Regarding the output from the SINS as a reference, the system's velocity error can be calculated. Fig. 7 and Table 2 indicate that the integrated system's velocity error in eastward and northward directions are both less than 0.07m/s.

Conclusions
The  that the position error is less than 1m, the pitch angle error and roll angle error are less than 0.1°, and the heading angle error is about 1°. The integrated system has an advantage of being compact, low-power, low-cost, and it can be applied in various fields including USV and UAV.