Abstract
This paper aims to provide navigation system designers with a detailed examination of the impact of inertial measurement unit (IMU) rotation on error reduction or induction in a rotary inertial navigation system (INS). The designer of a rotary INS can select the optimal rotation rate and direction by considering the dynamics of the carrier and the dominant errors based on the findings reported in three concise tables within this paper. This paper presents a comprehensive analytical derivation of error equations for the attitude, velocity, and position of a rotational INS, validated through numerical simulations. Experiments using a rotation platform and actual data from a micro-electromechanical system IMU sensor are also conducted to verify the effectiveness of mitigating errors through IMU rotation. Moreover, a tensor-based modeling technique is employed to facilitate exploring the impact of IMU rotation around an arbitrary axis on accumulated errors. This approach provides a modular platform for further research on rotational navigation systems.
- inertial measurement unit
- navigation error equations
- rotation modulation
- rotational inertial navigation system
- self-calibration
1 INTRODUCTION
The accumulation of errors in an inertial navigation system (INS) as an isolated navigation solution depends on the error of the inertial sensors. Although advanced inertial sensors such as ring lasers and fiber-optic gyroscopes (FOGs) have very low error rates (Du et al., 2010), these sensors are usually very heavy and expensive and are limited to specific applications. In the 1990s, new types of sensors based on micro-electromechanical systems (MEMS) technology were introduced to the navigation community. Because of their economical pricing, compact dimensions, lightweight design, and minimal energy usage, inertial MEMS sensors are used in applications such as pedestrian movement detection and robotic navigation inside buildings (Karumuri et al., 2011).
Inertial MEMS sensors have significant errors, such as high-frequency noise, bias, scale factors, and misalignment. Consequently, navigational calculations accumulate a considerable error in a short period of time (Du et al., 2016). Furthermore, sensor biases, including gyroscope and accelerometer biases, contribute to positional errors with cubic, quartic, and quadratic dependencies on time, respectively, particularly in the short term (Titterton & Weston, 2004). Because sensor biases are significantly larger in MEMS sensors, the position error in MEMS-based INSs would be notable. If another information system or receiver is used as an INS aid, the accumulated error rates of the INS can be significantly reduced. For instance, data regarding velocity and position acquired from global navigation satellite systems (GNSSs) may aid in estimating navigation solution errors through the use of Kalman-filter-based methods, thereby restricting the buildup of inaccuracies and error accumulation in INS solutions (Du et al., 2015).
Although the accumulation of navigation errors can be significantly reduced by using an aided navigation system, the error accumulates rapidly when aided sensors are unavailable. For example, GNSS signals are not available for underwater vehicles or vehicles traveling within a tunnel. Furthermore, vision sensors are ineffective in poor weather because they cannot capture clear images under such conditions. According to previous research, accurate decimeter positioning can be achieved using a typical INS equipped with a MEMS inertial measurement unit (IMU) with a continual GNSS update interval. However, in the case of GNSS connection loss, position errors can reach 30–50 m in 30 s (Du et al., 2010). As a result, methods are required to effectively reduce navigational errors without needing an external aid.
As early as 1968, Geller (1968) described an inertial gimbal system with a platform that constantly rotates with respect to the local frame and concluded that the position error of the system is strongly reduced when the rotation frequency is greater than twice the Schuler frequency. Levinson and Giovanni (1980) applied the same method to a hull-mounted marine navigation system. While these studies demonstrated the beneficial effect of IMU rotation on navigation solutions, they did not include an analysis of errors or a derivation of analytical equations for navigation errors.
In the early 20th century, propelled by FOG technology advancements, the majority of research concentrated on FOG-based rotary systems. Yang and Miao (2004), Zhang et al. (2009), and Ben et al. (2010) reviewed the principles of error adjustment through constant IMU rotation around the vertical axis. Yang and Miao (2004) found that sensor errors on the horizontal axis are modulated into periodic signals and are significantly reduced after a full cycle. Nevertheless, it is not possible to adjust the vertical axis error. Simulations have also confirmed this reduction in navigation error. Zhang et al. (2009) and Ben et al. (2010) derived differential equations for navigation errors and demonstrated them analytically using the Laplace transformation.
Sun et al. (2009) found that the rotation of a gyroscope embedded in an IMU causes a bias in the vertical axis, caused by the scale factor present in the horizontal axes. This bias significantly decreases navigational performance. Du (2015) developed position and velocity error equations for mapping applications in a rotational INS (RINS), with a focus on integrating a GNSS with a RINS that utilizes an inexpensive MEMS-based IMU.
Liu et al. (2019) introduced a rapid self-alignment technique designed for hybrid INSs. Their method is founded on a novel two-position analytic approach to enhance the accuracy and efficiency of aligning the system’s inertial sensors. Zhang et al. (2023) focused on addressing gravity-related disturbances in a dual-axis rotary modulation INS. The authors proposed a method for compensating these disturbances to improve navigation accuracy in marine environments.
Seo et al. (2022) proposed a calibration method for gyroscope bias in ring-laser-gyroscope-based RINS. This method analyzes the impact of IMU position on gyroscope bias and navigation performance. The proposed method improves RINS performance compared with conventional calibration methods, as verified by long-term navigation tests. Jie and Shi (2021) introduced a refined alignment technique for single-axis RINSs using a fuzzy adaptive filtering approach, with the aim of enhancing the accuracy of alignment processes within such navigation systems.
He et al. (2023) introduced a combination approach involving pure strapdown and dual-axis RINSs. The aim of their approach was likely to leverage the strengths of both systems to enhance navigation accuracy and reliability. Zhang et al. (2019) focused on determining the optimal modulation angular rate for MEMS-based rotary semi-strapdown INSs (semi-SINSs). The objective of this design is likely to find the best modulation angular rate that enhances the performance of a semi-SINS.
Xing et al. (2019) examined how changes in the rotation rates of a rotary MEMS IMU affect the accuracy of the navigation system. The authors investigated the relationship between rotation rate variations and navigation performance, which is valuable for understanding the limitations and sensitivities of a rotary system in real-world navigation scenarios. Wang et al. (2021) focused on developing an auto-calibration method for addressing axis misalignment in a RINS. The authors introduced a technique to automatically calibrate misalignment issues between different sensor axes, enhancing the accuracy and reliability of the integrated navigation system.
This paper aims to provide navigation system designers with a detailed examination of the impact of IMU rotation on error reduction or induction in a RINS. The designer of a RINS can select the optimal rotation rate and direction by considering the dynamics of the carrier and the dominant errors based on the results reported in three concise tables within this paper. This paper presents a comprehensive analytical derivation of attitude, velocity, and position error equations for a RINS, which are subsequently validated through numerical simulation. It is important to note that, to the best of the authors’ knowledge, no literature has reported a comprehensive analytical derivation and verification analysis of these three error equations for a MEMS-based RINS. Moreover, a tensor-based modeling technique is employed to facilitate future investigations, such as exploring IMU rotation around an arbitrary axis and its impact on accumulated errors. This approach will enable the development of a modular platform for further research on rotational navigation systems.
This paper is organized as follows: Section 2 introduces the notation, reference frames, and models employed to define the problem addressed in this paper. Section 3 presents a derivation of error equations for both conventional and rotational navigation systems. Section 4 investigates the verification of the derived error equations through numerical simulation and compares them with equations for a conventional INS (CINS). Corresponding experimental results based on a rotation platform and actual data for a consumer-grade MEMS IMU sensor are presented in Section 5, and finally, Section 6 concludes the paper by presenting three tables that summarize the results of this work.
2 MODELING
The RINS algorithm is generally similar to the CINS algorithm. However, these two algorithms differ in the sensor outputs and the coordinate transformation as a result of IMU rotation in the RINS algorithm. Equations (1) and (2), respectively, show the gyroscope and accelerometer outputs in a RINS:
In Equation (1), ωBI represents the angular velocity of the body frame (carrier) with respect to the inertial frame, ωBS represents the angular velocity of the body frame with respect to the sensor or IMU frame, and ωSI represents the angular velocity of the IMU frame with respect to the inertial frame. In a CINS, ωBS is zero. However, rotating the IMU at a constant rate around each of its axes gives this term a nonzero value. This distinction is the primary difference between conventional and rotary systems.
In Equation (2), fBI represents the linear acceleration of the body frame (carrier) with respect to the inertial frame, fBS represents the linear acceleration of the body frame with respect to the sensor frame, and fSI represents the linear acceleration of the IMU frame with respect to the inertial frame. It is evident that in a CINS, fBS is equal to zero. Because the rotation of the IMU at a constant rate around each of its axes does not introduce any linear acceleration to the sensor frame, this term is also equal to zero in a RINS. Thus, because the gyroscope and accelerometer outputs are in the sensor frame, the coordinates are transformed according to the following equations:
Here, [T]BS represents the transformation matrix from the sensor frame to the body frame. Obviously, it is necessary to know the rotation angle between the body frame and the sensor frame to perform a coordinate transformation. These data are typically delivered to the processor via an encoder sensor. With the specific force and angular velocity transferred from the sensor frame to the body frame, the CINS and RINS equations are no longer different. Figure 1 displays a schematic representation of the RINS structure installed on a moving carrier, including the associated frames.
The RINS mechanism
2.1 Inertial Sensor Modeling
The angular velocity and specific force, which are the gyroscope and accelerometer outputs, respectively, have multiple error sources, particularly in MEMS sensors, including noise, bias, scale factors, and misalignment. The gyroscope and accelerometer errors are modeled as follows:
In Equation (5), [δωSI]S is the gyroscope error, corresponding to the angular velocity error of the sensor frame with respect to the inertial frame, expressed in the sensor frame. [dg]S is the gyroscope drift, expressed in the sensor frame. [Sg]S and [Mg]S are the scale factor and misalignment matrices, respectively, in the gyroscope installation, and [ωSI]S is the true (error-free) angular velocity.
In Equation (6),
2.2 Error Model for a RINS
Because the IMU rotation does not cause any linear displacement in the system, the CINS error model in the navigation frame is used as the error model for the RINS. This error model consists of several differential equations describing position, velocity, and attitude errors. The RINS error equations are based on the following equations (Du et al., 2015):
Here, [δrBI]N, [δvBI]N, and [δΦBI]N are position, velocity, and attitude errors expressed in the navigation frame. [δωBI]N and
3 ERROR INVESTIGATION
In this section, the accumulated error caused by the inertial sensor error is investigated for a CINS and a RINS with IMU rotation around the x-axis of the body frame at rate Ω. To model the sensor rotation with respect to the body frame, the rotation matrix expressed in Equation (12) is employed:
These investigations have considered constant bias, scale factor, and misalignment errors for the gyroscope and accelerometer sensors. For all analyses, it is assumed that the stationary state, body frame, and navigation frame are aligned. Accordingly, the rotation matrix from the body frame to the navigation frame, i.e., [T]NB, is equal to the identity matrix. This assumption is made to simplify our presentation and discussion of the equations. It should be noted that the [T]BS matrix is equal to the identity matrix in a CINS, because of the lack of IMU rotation with respect to the body frame. According to the inertial sensor error models in Equations (5) and (6), sensor error sources can be divided into three categories: bias, scale factor, and misalignment. For gyroscope and accelerometer sensors, the following equations apply:
where:
In the following, the accumulation of error caused by each sensor error source is discussed based on the differential equations for RINS error. Although the bias, scale factor, and misalignment of inertial sensors vary over time, to simplify the analysis, their values are assumed to be constant over a short period of time (during a complete cycle of the IMU) when temperature conditions are stable.
3.1 Effect of Sensor Bias
This section focuses on the accumulation of position, velocity, and attitude errors caused by the bias term of inertial sensors.
3.1.1 Attitude Error
The magnitude of measurement error caused by gyroscope drift is calculated in the navigation frame according to the following equation:
The attitude error caused by gyroscope bias in a CINS is calculated by integrating Equation (17) over the time of one rotation cycle (i.e., 360° rotation of the IMU, which takes T seconds):
Here, [I] represents the identity matrix. By substituting Equation (12) into Equation (17), we obtain the gyroscope error caused by the gyroscope bias in a RINS:
Integrating Equation (19) over one cycle yields the attitude error caused by the gyroscope bias:
A comparison of Equations (18) and (20) clearly shows that with the rotation of the IMU around its own x-axis, the errors caused by the bias of the y- and z-axis gyroscopes are removed.
3.1.2 Velocity Error
To calculate the cumulative velocity error caused by the constant bias of the accelerometers, the accelerometer bias must first be defined in the navigation frame:
The velocity error caused by the accelerometer bias in a CINS is calculated by integrating Equation (21) over one time cycle:
By substituting Equation (12) into Equation (21), we modulate the accelerometer bias in the navigation frame as follows:
By integrating Equation (23) over one cycle, we obtain the velocity error due to accelerometer bias for a RINS:
By comparing Equations (22) and (24), we can see that by rotating the IMU around the x-axis of the sensor, the errors caused by the bias of the y- and z-axes have been eliminated.
3.1.3 Position Error
The accumulated position error due to the constant bias of the accelerometers is determined by reintegrating Equation (21) over the period for a CINS, as shown in Equation (25):
Similarly, the cumulated position error due to the constant bias of the accelerometers is obtained by integrating Equation (23) twice:
In the above equation, it is clear that the increase in position error in the RINS in the north channel is the same as that of the CINS. However, the error in the east and down channels has increased by an order of 2 for the CINS and an order of 1 for the RINS.
3.2 Effect of the Sensor Scale Factor
This section investigates the cumulative position, velocity, and attitude errors due to the scale factor of the inertial sensor.
3.2.1 Attitude Error
To calculate the cumulative attitude error caused by the gyroscope scale factor, it is necessary to define the error of the gyroscope scale factor in the navigation frame:
In Equation (27), [ωSI]S denotes the angular velocity of the sensor frame with respect to the inertial frame, expressed in the sensor frame. This parameter is calculated as follows:
Here,
[T]SE is the transformation matrix from the Earth frame to the sensor frame, calculated as follows:
In Equation (30), [T]NE is the transformation matrix from the Earth frame to the navigation frame (Zipfel, 2000):
where λ is latitude and ℓ is longitude.
Because of the alignment of the body and navigation frames as well as the absence of IMU rotation in the CINS, the attitude error is calculated by integrating Equation (27) over one rotation cycle:
By substituting Equation (12) into Equation (27), we obtain the error of the gyroscope scale factor for the RINS:
In Equation (33), the appearance of the IMU rotation rate Ω in the north channel indicates an error increment: not only does the IMU rotation fail to reduce the attitude error compared with the CINS, the IMU rotation actually increases the error. Moreover, the increase or decrease in error in the down channel depends on the magnitude of the sum of the y- and z-axis scale factors.
3.2.2 Velocity Error
To calculate the cumulative velocity error caused by the accelerometer scale factor, the accelerometer scale factor error must first be defined in the navigation frame:
Here,
Assuming stationary conditions for the carrier’s position and attitude, the following equation applies:
Subsequently, the specific force sensed by the sensor with respect to the inertial frame, expressed in the navigation frame, is obtained:
Assuming that the body and navigation frames are aligned in the CINS, we obtain the error caused by the accelerometer scale factor in the navigation frame:
The velocity error caused by the accelerometer scale factor in the CINS is calculated by integrating Equation (38) over one rotation cycle as follows:
The error caused by the accelerometer scale factor in the RINS is obtained by substituting Equation (12) into Equation (34):
By integrating Equation (40) over a complete cycle, we obtain the velocity error caused by the accelerometer scale factor in the RINS:
According to Equation (41), for the RINS, the accumulation of the speed error caused by the accelerometer scale factor depends on the magnitude of the y- and z-axis scale factors and is approximately the same as that of the CINS.
3.2.3 Position Error
The cumulative position error due to the accelerometer scale factor is calculated by reintegrating Equation (39) over a complete time period:
In the same way, the cumulative error of the position caused by the scale factor of the accelerometers in the RINS is obtained by double-integrating Equation (40):
A comparison of Equations (42) and (43) clearly shows that the error caused by the scale factors of the accelerometers in the CINS and RINS depend on the magnitude of the y- and z-axis scale factors and are approximately similar.
3.3 Effect of Sensor Misalignment
In this section, the accumulation of attitude, velocity, and position errors caused by the installation error (non-coaxial) of the inertial sensors is investigated.
3.3.1 Attitude Error
To calculate the cumulative attitude error caused by gyroscope misalignment, it is necessary to define the gyroscope alignment error in the navigation frame as follows:
Considering the alignment of the body and navigation frames and the absence of IMU rotation in the CINS, the attitude error for the CINS is calculated by integrating Equation (44) over a rotation cycle as follows:
The attitude error caused by the non-coaxiality of the gyroscopes in the RINS is obtained by substituting Equation (12) into Equation (44) and integrating over a rotation cycle as follows:
A comparison of Equations (45) and (46) clearly shows that the error caused by gyroscope misalignment in the north channel has been eliminated. Additionally, errors in both the east and down channels in the RINS are no longer influenced by latitude.
3.3.2 Velocity Error
To calculate the cumulative velocity error caused by accelerometer misalignment in the stationary condition, it is necessary to define the misalignment error of the accelerometers in the navigation frame:
The velocity error caused by accelerometer misalignment in the CINS is calculated by integrating Equation (47) over a rotation cycle:
By substituting Equation (12) into Equation (47), we obtain the accelerometer error due to misalignment in the RINS:
By integrating Equation (49) over a complete cycle, we can calculate the velocity error caused by the accelerometer misalignment:
A comparison of Equations (48) and (50) shows that in the RINS, the velocity error caused by the misalignment of accelerometers in the north axis has been eliminated. The east and down channel errors are approximately identical for the CINS and RINS. It should be noted that according to Equation (50), if the misalignment coefficients are considered symmetrical, there will be no misalignment in the east channel.
3.3.3 Position Error
The cumulative position error caused by accelerometer misalignment is obtained for the CINS by reintegrating Equation (48) over one time period as follows:
In the same way, the cumulative error of position caused by accelerometer misalignment is obtained by double-integrating Equation (49):
By comparing Equations (51) and (52), we can clearly see that the position error due to accelerometer misalignment in the RINS in the north and east channels experiences an order reduction with respect to time, corresponding to a significant reduction compared with the CINS. According to Equation (52), an error is induced for the RINS in the down channel.
3.4 Effect of Rotating Platform Encoder Error
For the RINS, based on Equations (3) and (4), the process of transferring inertial data from the sensor frame to the body frame requires a knowledge of the IMU rotation rate and the rotation angle provided by the rotating platform encoder, which is usually corrupted by errors. The effect of these errors is investigated here.
As mentioned previously, for an IMU rotating at an angular velocity of Ω around the sensor x-axis, the transformation matrix from the sensor frame to the body frame is given by Equation (12). Considering the rotating angle error, the coordinate transformation process is calculated as shown in Equation (53):
In Equation (53), I represents the identity matrix, and θ× is defined as follows:
3.4.1 Attitude Error
According to Equation (3), the angular velocity in the body frame is calculated from Equation (55), considering the encoder errors:
In the above equation, [δωSI]S is the sensor error discussed in the previous sections. The terms [θ×][T]BS[ωSI]S and [δωBS]B represent the effects of encoder rotation angle and rotation rate reading errors on the attitude error, respectively. Thus, Equation (56) calculates the attitude error in the navigation frame caused by inaccuracy of the encoder in reading the rotation angle:
As shown in Equation (56), the encoder’s accuracy in reading the rotation angle directly influences the attitude error in the east channel of the RINS. However, because the Earth’s angular velocity with respect to the inertial frame is small and is thus neglected in the simulation, the value of the attitude error due to the rotation angle error of the turntable in the simulation is zero. Therefore, no form of simulation output is presented in Section 4. Nevertheless, the attitude error in the navigation frame caused by inaccuracy of the encoder in reading the rotation rate is calculated as follows:
According to the above equation, the error in the rotation rate reading leads to an attitude error in the north channel of the RINS.
3.4.2 Velocity and Position Error
By using Equation (4) and considering the encoder errors, we can calculate the specific force vector in the body frame as follows:
In the above equation, [δfSI]S is the sensor error discussed in the previous sections. The term [θ×][T]BS [fSI]S represents the effect of the rotation angle error of the turntable on the velocity error and, consequently, position error. Therefore, the velocity error in the navigation frame caused by inaccuracy of the encoder in reading the rotation angle is calculated by the following equation:
As Equation (59) illustrates, the encoder’s accuracy in reading the rotation angle is pivotal, as it directly influences the velocity error in the east channel of the RINS. In addition, the position error in the navigation frame caused by inaccuracy of the encoder in reading the rotation angle is calculated by the following equation:
According to Equations (56), (57), and (59), the accuracy of the rotating platform’s angular position directly affects attitude and velocity errors, with these errors increasing linearly over time. Additionally, Equation (60) shows that the accuracy of the turntable’s angular position impacts the RINS position error, with a quadratic dependency on time. Hence, RINS designers can determine rotating platform requirements based on the maximum navigation time and the required accuracy of the RINS.
4 SIMULATION
This section investigates the direct effect of sensor and rotating platform encoder errors on the RINS output through numerical simulation. The effects of sensor bias, scale factor, misalignment, encoder angular position error, and encoder angular rate error are considered for the x-axis RINS and compared with those of the CINS.
It is worth mentioning that the Earth’s rotation rate (ωEI) is assumed to be zero throughout the simulation. This assumption is made because the equations in this paper are derived for a MEMS-based RINS and, because of the lower accuracy of MEMS IMUs and the noise level of these sensors, it is typically not possible to measure the Earth’s rotation rate (Yongjian et al., 2015). This assumption will not have a substantial effect on the simulation results because the Earth’s rotation rate is much lower than the IMU rotation rate.
Numerical simulations were performed on a personal computer using MATLAB/Simulink software. “ODE4 Runge-Kutta” was selected as the equation solver, with a rate of 100 Hz. Figure 2 shows a schematic of the simulation procedure.
RINS simulation procedure
As shown in Figure 2, the simulation is divided into two main parts: RINS simulation and CINS simulation. In the RINS simulation, the vehicle’s true accelerations and angular velocities in the stationary state are first calculated, based on the initial conditions. Next, with the addition of the effect of IMU rotation and the effect of IMU errors on the accelerations and angular velocities and with a change in the data coordinate system from sensor to body, the outputs are sent to an INS solver to calculate the vehicle position, velocity, and attitude. The CINS simulation is similar to the RINS simulation, except that the effect of IMU rotation on the accelerations and angular velocities of the vehicle is not considered.
4.1 Errors Caused by Sensor Bias
The bias values applied to the inertial sensors are given in Table 1. Figure 3 shows the magnitude of position error due to gyroscope bias in the RINS compared with the CINS.
Comparison of cumulated CINS and RINS attitude errors due to gyroscope bias
Bias Applied to the IMU in the Simulation
According to Figure 3, the attitude errors in the roll, pitch, and yaw channels in the CINS increase linearly over time according to Equation (18). For the RINS, according to Equation (20), the error in the pitch and yaw channels has been removed. Figure 4 shows the velocity error due to the bias of the accelerometers in the RINS versus the CINS.
Comparison of CINS and RINS velocity errors due to accelerometer bias
Figure 4 shows that the velocity error in the north, east, and down channels in the CINS increases over time according to Equation (22). For the RINS, according to Equation (24), the error in the east and down channels has been eliminated. Figure 5 compares the position error caused by accelerometer bias in the RINS versus the CINS.
Comparison of CINS and RINS position errors due to accelerometer bias
As shown in Figure 5, the position error in the north, east, and down channels in the CINS increases over time by an order of 2 according to Equation (25). For the RINS, according to Equation (26), the position error in the east and down channels increases linearly, with a much smaller increase than that observed for the CINS.
4.2 Errors Caused by the Sensor Scale Factor
The scale factors applied to the inertial sensors are indicated in Table 2. Figure 6 shows the magnitude of position error due to the gyroscope scale factor in the RINS compared with that for the CINS.
Comparison of attitude errors caused by the accelerometer scale factor for the CINS and RINS
Scale Factor Applied to the IMU in the Simulation
As shown in Figure 6, with the assumption that the Earth’s rotation rate is negligible, the error in the roll, pitch, and yaw channels for the CINS remains zero over time according to Equation (32). For the RINS, according to Equation (33), the error in roll channel increases over time. In other words, an error is induced in the roll channel. Figure 7 shows the velocity error caused by the accelerometer scale factor in the RINS compared with that for the CINS.
Comparison of velocity errors caused by the accelerometer scale factor for the CINS and RINS
Figure 7 shows that for the CINS, the velocity error is approximately zero in the north and east channels and increases over time in the down channel according to Equation (39). Because the scale factors in the y- and z-axes are assumed to be unequal in the simulation, the error behavior of the RINS in the down axis is not completely aligned with that of the CINS, according to Equation (41). In Figure 8, the position error caused by the accelerometer scale factor is shown for the RINS in comparison with the CINS. It is shown that the position error behaviors in the CINS and RINS are similar, according to Equations (42)–(43). The difference in the CINS and RINS results is due to the inequality of the scale factor matrix elements.
Comparison of cumulative position errors caused by the accelerometer scale factor for the CINS and RINS
4.3 Errors Caused By Sensor Misalignment
The misalignments applied to the inertial sensors are given in Table 3. Figure 9 shows the attitude error due to gyroscope misalignment in the RINS and CINS.
Comparison of attitude errors caused by gyroscope misalignment for the CINS and RINS
Misalignment Applied to the IMU in the Simulation
According to Figure 9, the attitude errors in the roll, pitch, and yaw channels for the CINS equal zero over time, as given in Equation (45), as the Earth’s rotation rate is assumed to be zero in the simulation. For the RINS, the attitude error is bounded, and according to Equation (46), the pitch and yaw errors are zero at the end of each periodic cycle. Figure 10 displays the velocity error caused by accelerometer misalignment in the RINS and CINS.
Velocity error for the CINS and RINS due to accelerometer misalignment
As shown in Figure 10, the velocity error for the CINS increases in the north and east channels and equals zero in the down channel, according to Equation (48). For the RINS, according to Equation (50), the velocity error in the north channels is eliminated, the error growth rate in the east channel is reduced by the difference in misalignment coefficients, and the error in the down channel is equal to the error for the CINS. In Figure 11, the magnitude of position error caused by accelerometer misalignment is compared for the RINS and CINS. As can be seen, the increase in position error for the RINS in the north and east channels is lower than that of the CINS, following Equations (51)–(52). However, an error is induced in the down channel.
Position error caused by accelerometer misalignment for the CINS and RINS
4.4 Errors Caused by Rotation Platform Encoder Errors
This subsection investigates the validity of the analytical relationships describing the effect of rotation platform encoder errors on RINS navigation solutions through numerical simulation. To closely mirror practical applications, the simulation assumes the use of a commercial optoelectronic encoder (DS58-20, Netzer) as the angular position sensor for the rotation platform. This encoder has been implemented in a practical MEMS-based RINS (Wang, 2013). Table 4 presents the technical parameters of the DS58-20 encoder based on data provided by themanufacturer. According to Equation (57), the error in the rotation rate reading leads to an attitude error in the north channel of the RINS. Figure 12 displays the error caused by a rotation rate error of 0.008°/s.
Attitude error due to error in the turntable rotation rate for the RINS (x-axis rotation)
Technical Specifications of the DS58-20
Figure 12 shows that the error in the north channel of the RINS attitude solution increases. Simulation results for attitude errors indicate that the error magnitude in the north channel reaches 2° by the end of the simulation, consistent with Equation (57). According to Equation (59), the error in the turntable angular position leads to an attitude error in the east channel of the RINS. Figure 13 demonstrates the error caused by an angular position error of 0.008°.
Velocity error due to error in the turntable angular position for the RINS (x-axis rotation)
Figure 13 shows that the error in the east channel of the RINS velocity solution increases linearly over time. Simulation results for velocity errors indicate that the error magnitude in the east channel reaches 3.4 m/s by the end of the simulation, consistent with Equation (59).
According to Equation (60), the error in the turntable angular position leads to a position error in the east channel of the RINS. Figure 14 displays the error caused by an angular position error of 0.008°.
Position error due to error in the rotation angle reading for the RINS (x-axis)
As shown in Figure 14, the error in the east channel of the RINS position solution exhibits a quadratic increase over time. The simulation results for position errors indicate that the error magnitude in the east channel reaches 43 m by the end of the simulation, consistent with Equation (60).
5 EXPERIMENTAL VALIDATION
Experiments utilizing a rotation platform and actual data from a low-cost MEMS IMU sensor were conducted in the Hardware-in-the-Loop lab at the Amirkabir University of Technology. These experiments aimed to verify the effectiveness of mitigating navigation errors through IMU rotation.
The rotation platform comprises a single-axis turntable and a computer unit that controls the turntable’s angular position and rate. According to the findings of this paper, rotating the IMU around the z-axis can significantly reduce navigation errors in the horizontal plane. In contrast, rotations around the x- or y-axis primarily decrease errors in the vertical direction. Consequently, z-axis rotation is more advantageous for navigation applications than x- or y-axis rotation. Hence, in this experiment, the MEMS IMU is securely mounted to the rotating plate of the turntable, ensuring that the z-axis of the IMU is perpendicular to the rotating plate.
After initialization of the turntable, the MEMS IMU body frame is aligned with the local navigation frame. The computer unit logs the rotation angle data, the angle between the body frame and the sensor frame, and the MEMS IMU data at a frequency of 100 Hz. Table 5 presents the technical specifications of the rotation platform, whereas Table 6 and Table 7 provide the characteristics of the MEMS IMU tested.
Technical Specifications of the Rotation Platform
Characteristics of the Gyroscope Triad
Characteristics of the Accelerometer Triad
Two types of laboratory tests were conducted under static conditions using the described hardware setup: a CINS test and a RINS test. During the CINS test, inertial data are gathered while the system remains stationary on the turntable for 4 min. In the rotary test, the IMU rotates around the z-axis at a rate of 30°/s for the same duration. Figure 15 and Figure 16 illustrate the effect of IMU rotation on the reduction or induction of errors in navigation solutions.
Position error for the CINS and RINS
Velocity error for the CINS and RINS
According to Equations (78)–(80) in Appendix B, the RINS navigation solution errors are expected to be less than those of the CINS in the north and east channels. This claim is substantiated by Figure 15 and Figure 16, where the position and velocity errors for the RINS are significantly lower than those in the corresponding CINS channels. However, the position and velocity errors for the RINS and CINS behave similarly in the vertical channel. Additionally, Figure 17 shows that, similar to the position and velocity errors, the RINS exhibits superior attitude error reduction compared with the CINS in the north (roll) and east (pitch) channels. Despite this improvement, unlike the position and velocity errors, the RINS suffers from attitude error induction in the vertical channel.
Attitude error for the CINS and RINS
6 CONCLUSION
With rotation of the IMU unit and the modulation of inertial sensor errors, the RINS leads to a reduction in navigation errors and thus improves navigation performance without using an external source. However, in addition to reducing errors caused by the sensors, the IMU unit rotation induces additional error in the RINS output. RINS designers must consider the induced errors and perform appropriate calibration before initializing the RINS. Alternatively, they can mitigate these errors by integrating the RINS with additional aiding sensors such as GNSS, compass, or vision systems. Table 8 shows the results of rotation around the x-axis of the sensor. In addition, Table 9 and Table 10 present the results of rotation around the y- and z-axes of the IMU. The relationships for the errors caused by rotation around the y- and z-axes are presented in Appendices A and B.
Effect of IMU Rotation Around the X-Axis on INS Error
Effect of IMU Rotation Around the Y-Axis on INS Error
Effect of IMU Rotation Around the Z-Axis on INS Error
HOW TO CITE THIS ARTICLE
Mohammadkarimi, H., Mozafari, S., Ghasemi, M., Parhizkar, M. A., & Mobtaker, M. (2025). Analysis of IMU rotation effects on inertial navigation system errors. NAVIGATION, 72(1). https://doi.org/10.33012/navi.680
APPENDIX A IMU ROTATION AROUND THE SENSOR Y-AXIS
This appendix presents the relationships expressing the error for IMU rotation around the y-axis. The process of extracting these relationships for this case is the same as that of extracting relationships for the state of rotation around the x-axis, except that the rotation matrix from the sensor frame to the body frame is changed. To model the sensor rotation with respect to the body frame, the rotation matrix expressed in Equation (61) is employed:
By replacing the rotation matrix of Equation (61) in the modeling relationships of the RINS, the error relationships for rotation around the lateral axis (y-axis) can be obtained as shown below.
A.1 Errors Caused by Sensor Bias
Attitude Error
A comparison of Equations (18) and (62) clearly shows that with IMU rotation around its own y-axis, the errors caused by the bias of the x- and z-axis gyroscopes are removed.
Velocity Error
By comparing Equations (22) and (63), one can clearly see that by rotating the IMU around the y-axis of the sensor, the errors caused by the bias of the x- and z-axes have been eliminated.
Position Error
By comparing Equations (25) and (64), we can easily see that by rotating the IMU around the y-axis of the sensor, the errors caused by the bias of the x- and z-axes have been eliminated.
A.2 Effect of the Sensor Scale Factor
Attitude Error
In Equation (65), the increase or decrease in error in the north and down channels depends on the magnitude of the sum of the x- and z-axis scale factors. Additionally, unlike the CINS case, the error in the RINS east channel is non-zero, which indicates an increase in error due to the IMU rotation.
Velocity Error:
According to Equation (66), for the RINS, the accumulation of the speed error caused by the accelerometer scale factor depends on the magnitude of the x- and z-axis scale factors and is approximately the same as that of the CINS.
Position Error
A comparison of Equations (42) and (67) clearly shows that the error increases in the north and down channels. The error caused by the accelerometer scale factor for the CINS and RINS depends on the magnitude of the x- and z-axis scale factors.
A.3 Effect of Sensor Misalignment
Attitude Error
By comparing Equations (45) and (68), one can easily see that the error caused by gyroscope misalignment in the east channel has been eliminated. Additionally, errors in both the north and down channels in the RINS depend on the misalignment of the x- and z-axes.
Velocity Error
A comparison of Equations (48) and (69) shows that, for the RINS, the velocity error caused by accelerometer misalignment in the east axis has been eliminated. The north and down channel errors are approximately identical in the CINS and RINS. It should be noted that according to Equation (69), if the misalignment coefficients are considered symmetrical, there will be no misalignment in the north channel.
Position Error
By comparing Equations (51) and (70), we see that the position error due to accelerometer misalignment increases in the down channel. In the east channel, the error is eliminated, and in the north channel, the error is smaller than the CINS error. However, if the misalignment coefficients are considered symmetrical, the error will also be zero in the north channel. Table 7 provides a summary of the results for IMU y-axis rotation.
A.4 Effect of Rotation Platform Encoder Errors
As previously mentioned, for an IMU rotating at an angular velocity of Ω around the sensor’s y-axis, Equation (61) gives the transformation matrix from the sensor frame to the body frame. Considering the angular position error of the turntable (δθ), Equation (71) describes the coordinate transformation process:
In Equation (71), I represents the identity matrix, and θ× is defined as follows:
Attitude Error
Similar to the previous steps and using Equation (55), the attitude error in the navigation frame due to the encoder angular position error of the turntable can be calculated by the following equation:
As Equation (73) illustrates, the encoder angular position error directly influences the attitude error in the north and east channels of the RINS. Additionally, the attitude error in the navigation frame, resulting from the encoder’s inaccuracy in reading the rotation rate, is calculated using the following equation:
According to Equation (75), the encoder rotation rate error leads to an attitude error in the east channel of the RINS, with a linear dependency on time.
Velocity and Position Error
Similar to the previous calculation and using Equation (58), the velocity error in the navigation frame due to the encoder angular position error of the turntable can be calculated by the following equation:
According to Equation (75), the encoder angular position error causes a velocity error in the north channel of the RINS, with a linear dependency on time. Additionally, the position error in the navigation frame, resulting from the encoder angular position error, is calculated using the following equation:
According to Equation (76), the encoder angular position error results in a position error in the north channel of the RINS, with a quadratic dependency on time.
APPENDIX B IMU ROTATION AROUND THE SENSOR Z-AXIS
In this section, the relationships expressing the error for IMU rotation around the z-axis are presented. The process of extracting relationships for this case is the same as that of extracting relationships for the state of rotation around the x-axis, except that the rotation matrix from the sensor frame to the body frame is changed.
To model sensor rotation with respect to the body frame, the rotation matrix expressed in Equation (77) is employed:
By replacing the rotation matrix of Equation (77) in the modeling relationships of the RINS, the error relationships for rotation around the directional axis (z-axis) will be obtained as described below.
B.1 Error Caused by Sensor Bias
Attitude Error
By comparing Equations (18) and (78), we can clearly see that with IMU rotation around its own z-axis, the errors caused by the bias of the x- and y-axis gyroscopes are removed.
Velocity Error
A comparison of Equations (22) and (79) shows that by rotating the IMU around the z-axis of the sensor, the errors caused by the bias of the x- and y-axes have been eliminated.
Position Error
By comparing Equations (25) and (80), one can see that by rotating the IMU around the z-axis of the sensor, the errors caused by the bias of the x- and y-axes have been eliminated.
B.2 Error Caused by the Sensor Scale Factor
Attitude Error
In Equation (81), the increase or decrease in error in the north and down channels depends on the magnitude of the sum of the x- and y-axis scale factors. In addition, the error in the RINS east channel is similar to that of the CINS and is equal to zero.
Velocity Error
According to Equation (82), IMU rotation around its z-axis does not affect the cumulation of velocity error caused by the sensor scale factor, and the velocity error for the RINS (due to the accelerometer scale factor) is similar to that of the CINS.
Position Error
By comparing Equations (42) and (83), we can clearly see that the RINS position error is similar to that of the CINS (as observed for the velocity error), and the rotation of the IMU in this direction has no effect on error accumulation.
B.3 Error Caused by Sensor Misalignment
Attitude Error
A comparison of Equations (45) and (84) clearly shows that the errors caused by gyroscope misalignment in the north and down channels have been eliminated. Additionally, errors in the east channel in the RINS depend on the misalignment of the x- and y-axes.
Velocity Error
By comparing Equations (48) and (85), one can see that in the RINS, the velocity error caused by accelerometer misalignment has been eliminated in all directions.
Position Error
Because the velocity errors have been eliminated in all directions, the position and velocity errors have also been eliminated. Therefore, the rotation of the IMU around its z-axis significantly affects error accumulation. Table 8 shows a summary of the results for IMU z-axis rotation.
B.4 Effect of Rotation Platform Encoder Errors
As previously mentioned, for an IMU rotating at an angular velocity of Ω around the sensor’s z-axis, Equation (77) gives the transformation matrix from the sensor frame to the body frame. Considering the angular position error (δθ) of the turntable, Equation (87) describes the coordinate transformation process:
In Equation (87), I represents the identity matrix, and θ× is defined as follows:
Attitude Error
Similar to the previous calculations and using Equation (55), the attitude error in the navigation frame due to the encoder angular position error of the turntable can be calculated by the following equation:
As Equation (89) illustrates, the encoder angular position error directly influences the attitude error in the east channel of the RINS. Additionally, the attitude error in the navigation frame, resulting from the encoder’s inaccuracy in reading the rotation rate, is calculated using the following equation:
According to Equation (85), the encoder’s rotation rate error leads to an attitude error in the down channel of the RINS, with a linear dependency on time.
Velocity and Position Error
Similar to the previous step and using Equation (58), the velocity error in the navigation frame due to the encoder angular position error of the turntable can be calculated by the following equation:
According to Equation (91), the encoder angular position error has no effect on the velocity error of the RINS. Additionally, the position error in the navigation frame, resulting from the encoder angular position error, is calculated using the following equation:
According to Equation (92), the encoder angular position error does not result in a position error in the RINS.
NOMENCLATURE
- N
- Navigation frame
- I
- Inertial frame
- B
- Body frame
- E
- Earth frame
- S
- Inertial sensor (IMU) frame
- t
- Time
- T
- IMU rotation period
- g
- Gravitational acceleration
- ωAB
- Angular velocity of frame A with respect to frame B
- fAB
- Linear acceleration of frame A with respect to frame B
- [T]AB
- Transformation matrix from frame B to frame A
- Ω
- IMU rotation rate
- λ, ℓ, h
- Latitude, longitude, altitude
- δω
- Gyroscope error
- δf
- Accelerometer error
- δr
- Position error
- δv
- Velocity error
- δΦ
- Attitude error
- ba / dg
- First-order tensor of the accelerometer/gyroscope bias (drift)
- Sa /Sg
- Second-order tensor of the accelerometer/gyroscope scale factor
- Ma /Mg
- Second-order tensor of the accelerometer/gyroscope misalignment
- bi / di
- Accelerometer/gyroscope bias error along the i-axis
- kai /kgi
- Accelerometer/gyroscope scale factor error along the i-axis
- kgij /kaij
- Accelerometer/gyroscope misalignment error along the i and j axes
- RN, RM
- Prime vertical and meridian radius of the Earth’s curvature
- vN, vE, vD
- Terrestrial velocity in the navigation frame
- fN, fE, fD
- Non-gravitational specific forces in the navigation frame
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
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