Decrease in Accuracy of a Rotational SINS Caused by its Rotary Table's Errors

We call a strapdown inertial navigation system (SINS) that uses the rotation auto-compensation technique (which is a common method to reduce the effect of the bias errors of inertial components) a ‘rotational SINS’. In a rotational SINS, the rotary table is an important component, rotating the inertial sensor assembly back and forth in azimuth to accomplish error modulation. As a consequence of the manufacturing process, errors may exist in rotary tables which decrease the navigation accuracy of rotational SINSs. In this study, the errors of rotary tables are considered in terms of installation error, wobble error and angular error, and the models of these errors are established for the rotational SINS. Next, the propagation characteristics of these errors in the rotational SINS are analysed and their effects on navigation results are discussed. Finally, the theoretical conclusions are tested by numerical simulation. This paper supplies a good reference for the development of low-cost rotational SINSs, which usually have low accuracy rotary tables and which may be used in robots, intelligent vehicles and unmanned aerial vehicles (UAVs).


Introduction
Navigation systems supply the necessary information for the control systems of robots, vehicles and UAVs [1]. SINS is one of the most commonly-used navigation systems, since it can output comprehensive navigation information, such as attitude, velocity and position [2]. Besides, SINS is an autonomous navigation system, which means it is not influenced by external environmental interference, including natural factors and human factors [3].
The inertial measurement unit (IMU), consisting of gyros and accelerometers, is the main component of a SINS [4]. There are several types of gyros and accelerometers, such as liquid-floated gyros, optical gyros, microelectromechanical system (MEMS) gyros, flexure accelerometers, MEMS accelerometers, and so on. In robots' and small UAVs' navigation systems, MEMS gyros and MEMS accelerometers are most often used [5], mainly due to size and cost factors. However, MEMS gyros usually have low accuracy [6], which leads to large navigation errors.
The rotation auto-compensation technique can efficiently reduce the bias errors of inertial components so as to improve navigation accuracy [7]. The technique has been widely used in SINSs with optical gyros, which are usually equipped for ships and can supply accurate navigation information for several days, such as AN/WSN-7B [8] and MK49 [9]. Nowadays, the rotational SINS with MEMS gyros has not been produced yet, but the rotation auto-compensation technique has been proposed for use in MEMS IMUs and has significantly improved the accuracy of inertial components [10]. In addition, rotational north-finding systems with MEMS gyros have been studied and good experimental results have been seen [11][12]. So, it can be deduced that the rotation auto-compensation technique can greatly improve the navigation accuracy of MEMS SINS and that rotational MEMS SINSs show significant potential for application. In this study, some pre-work was conducted for the development of rotational MEMS SINSs.
In a rotational SINS, the IMU is mounted on a rotary table [13]. The rotary table rotates the IMU back and forth to modulate the bias errors of the gyros and accelerometers. In rotational SINSs with optical gyros, the rotary tables are highly accurate but are large and costly. If these rotary tables are used in rotational MEMS SINSs, the MEMS-advantages of size and cost will be greatly diminished. Therefore, a kind of small-sized, low-cost rotary tables should be used in rotational MEMS SINSs. However, this kind of rotary tables have low accuracy.
The rotary table is an important component of the rotational SINS, and its errors affect navigation accuracy. Especially for low-accuracy rotary tables, a large decrease in navigation accuracy may be brought about by the rotary tables' errors. Currently, most of the researches on the rotational SINS are about modulation theory [14][15], rotation scheme [16] and initial alignment [17], but the research on the effect of the rotary table's errors has not been found.
In this study, the errors of the rotary table in a rotational SINS are classified and modelled while its error propagation characteristics are analysed. Detailed navigation error formulae are derived which supply a good reference for the selection of rotary tables for lowcost MEMS rotational SINSs. Firstly, in Section 2, the basic principle behind the rotational SINS is introduced and the calculation scheme is analysed. Next, the rotary table's errors are modelled in Section 3, including installation error, wobble error and angular error, and the effects of these errors on navigation results are explored in Section 4. In Section 5, some numerical simulations are carried out to verify the theoretical analysis.

The principle behind the rotational SINS
Distinct from conventional SINSs, the IMU of a rotational SINS is fixed on a rotary table instead of directly on the vehicle (Figure 1 is a rotational SINS with fibre-optical gyros (FOGs) and flexure accelerometers, developed by the authors' department). When the rotary table rotates, the bias errors of the horizontal gyros and the horizontal accelerometers are modulated, yielding improved heading, attitude and position accuracy. The computation scheme of the rotational SINS is shown as Figure 2 [18].  In this paper, the coordinate systems are defined as follows: the inertial coordinate system is the reference coordinate for the inertial components, denoted by i; the navigation coordinate system is chosen as the local eastnorth-up (ENU) coordinate, denoted by n; the body coordinate system is vehicle-carried and denoted by b; the rotary table coordinate system is fixed with the rotary table and denoted by r. Take the gyros' errors for example, the principle behind the rotation auto-compensation technique can be simply explained as follows: the errors of the three gyros are denoted by r ε ; the rotation rate of the rotary table is denoted by c ω ; and then the projection of r ε on the n-frame can be expressed by: Similarly for the gyros, the two horizontal accelerometers are also modulated.

Error modelling of the rotary table
For the rotary table in a rotational SINS, there are three main types of errors: installation error, wobble error and angular error. In this section, the physical causes of these errors are analysed and then models are established.

Modelling of the installation error
The vertical axis of the rotary table should be parallel to the vertical axis of the body, which means that the z-axis of the r-frame should coincide with the z-axis of the bframe. However, there exists an error angle due to the installation technology, which is called the 'installation error'. Assuming that the r0-frame is the coordinate system coinciding with the initial r-frame, then the transition matrix from the r0-frame to the r-frame can be written as: The installation error can be expressed through the transition matrix from the b-frame to the r0-frame, which consists of two error-angle parameters, denoted by α β , : Assuming α β , are both small quantities, then Eq. (3) can be approximated as: Considering the installation error of the rotary table, the gyros' outputs can be expressed by: The accelerometers' outputs can be expressed by: (6) Usually, before using a rotational SINS, some calibration operations will be performed to compensate for the errors in the system. Eq. (5) and Eq. (6) supply the reference for designing the calibration experiment for the installation error of the rotary table.

Modelling of the wobble error
The vertical axis of the The wobble error changes the rotation rate between the rframe and the b-frame, which can be decomposed into two types of motion [19]: the r-frame does the coning motion relative to the b-frame caused by the wobble error, whereby the coning rate is c ω and the half-cone angle is θ ; the r-frame rotates at the speed of c ω around z r , caused by the rotation of the rotary table. Using the quaternion 1 ( ) Q t to express the coning motion, it can be written as: where η is a coning parameter describing the relation between the r-frame and the b-frame at the initial time.
Using the quaternion 2 ( ) Q t to express the rotation, then it can be written as: Then, the r-frame to the b-frame quaternion is: where ⊗ is the quaternion multiplication symbol, ( )  t Q is the differentiation of ( ) t Q and r br ω is the angular rate of the r-frame relative to the b-frame projected on the rframe. It can be got as: , which is equal to the rotation rate.
According to the relation between the quaternion and the transition matrix, b r C can be written as: 11 12 where: θ B = -cos(3ω t + 2η)sin ( ) + cos(ω t)cos ( ) Considering the wobble error of the rotary table, the gyros' outputs can be expressed by:  (15) The accelerometers' outputs can be expressed by: Eq. (15) and Eq. (16) supply the reference for designing the calibration experiment for the wobble error of the rotary table.

Modelling of the angular error
From Figure 2, it can be seen that after solving the attitude of the IMU, the rotation angle of the rotary table is needed to get the attitude of the vehicle. However, there exists an error in the rotation angle, which is called the 'angular error' and mainly comes from two aspects: one concerns the errors from the angular measuring device of the rotary Considering the angular error of the rotary table, the gyros' outputs can be expressed by: Eq. (18)  In this section, the effects of the rotary table's errors on position accuracy and attitude accuracy will be analysed and formulae will be derived.

The effect of the installation error
The installation error changes the relative attitude between the IMU and the vehicle and has no effect on the IMU position-solving. From Figure 2, it can be seen that the rotational SINS's position is equal to the IMU's position, so the installation error does not affect position accuracy.
From Figure 2, it can be seen that during the attitude solution process, the transition matrix from the b-frame to the r-frame is needed to get the vehicle attitude from the IMU attitude. But, according to the analysis of Section 3.1, the transition matrix is affected by the installation error, and so the attitude accuracy is also affected by the installation error.
The attitude of the vehicle is denoted as follows: the pitch angle is τ , the roll angle is γ , and the heading angle is ψ . Then, the transition matrix from the b-frame to the nframe can be expressed by [21]: where: where the r'-frame is the theoretical r-frame with no installation error. Let Δτ , Δγ and Δψ stand for the pitch angle error, the roll angle error and the heading angle error respectively. Then, according to the relation between n b C and the attitude, we have: can be calculated that Δτ = β, Δγ = α, Δψ = 0 , which is equal to the installation error. However, when the pitch angle approaches 90 degrees, Eq. (24) is no longer applicable. This is because the attitude of SINS degenerates to two degrees of freedom and it is the attitude which needs to be redefined rather than Eq. (20). The same problem occurs in the following analysis (and the explanation will not be repeated).

The effect of the angular error
It can be seen from Figure 2 that the rotation angle has nothing to do with the position solution, so its error does not affect the position accuracy. However, the rotation angle is needed in the attitude calculation process, so its error affects the attitude accuracy. Similarly with the derivations in Section 4.1 and 4.2, the calculated According to the relation between

Simulation results and discussion
From the above analysis, it can be concluded that the rotary table's errors have no effect on position accuracy and that only the installation error, the wobble error and the angular error affect attitude accuracy. In this section, the effects of the rotary table's errors to attitude will be simulated and the simulation results will be compared with the theoretical ones. In order to verify the derived formulas, the simulation conditions are set as follows: 1 According to the analysis in Section 3.1, the installation error can be expressed by two parameters α β 、 . In the simulation, the two error angles are both set at 5 minutes of angle, which corresponds to low installation accuracy. The attitude errors of the rotational SINS are shown in Figure 5 and the specific results of some special time points are referred to in Table 1, Table 2 and Table 3. It is clear that the calculated values agree well with the simulated values and that the attitude errors reach 0.1 degrees under the simulation conditions.    The diameter of the table board is assumed to be 100 mm and the maximum vibration amplitude is set as 1 mm, which represents low machining accuracy. Next, from Section 3.2, it can be seen that the half-cone angle is θ = 1 / 100rad = 0.01rad . Without loss of generality, the initial installation angle η is set as 30 degrees. The attitude errors are shown in Figure 6. It is clear that the calculated values agree well with the simulated ones, and the attitude errors almost reach 1 degree under the simulation condition, which greatly decreases the attitude accuracy. There are several spikes in the curves of Figure 6 corresponding to the rotation process which can be seen in detail in Figure 7. This is because the attitude errors are related to the rotation angle, as shown in Eq. (22). The specific results of some special time points are referred to in Table 4, Table 5 and Table 6.  The angular error is set as 1 minute of angle, which corresponds to a low-accuracy rotary table. The attitude errors are shown as in Figure 8. The specific results of some special time points are referred to in Table 7, Table  8 and Table 9. It is clear that the calculated values agree well with the simulated values and the attitude errors are less than 0.02 degrees under the simulation conditions.