Odometer Velocity and Acceleration Estimation Based on Tracking Differentiator Filter for 3D-Reduced Inertial Sensor System

Velocity information from the odometer is the key information in a reduced inertial sensor system (RISS), and is prone to noise corruption. In order to improve the navigation accuracy and reliability of a 3D RISS, a method based on a tracking differentiator (TD) filter was proposed to track odometer velocity and acceleration. With the TD filter, an input signal and its differential signal are estimated fast and accurately to avoid the noise amplification that is brought by the conventional differential method. The TD filter does not depend on an object model, and has less computational complexity. Moreover, the filter phase lag is decreased by the prediction process with the differential signal of the TD filter. In this study, the numerical simulation experiments indicate that the TD filter can achieve a better performance on random noises and outliers than traditional numerical differentiation. The effectiveness of the TD filter on a 3D RISS is demonstrated using a group of offline data that were obtained from an actual vehicle experiment. We conclude that the TD filter can not only quickly and correctly filter velocity and estimate acceleration from the odometer velocity for a 3D RISS, but can also improve the reliability of the 3D RISS.


Introduction
Most current land vehicular navigation is highly dependent on the Global Positioning System (GPS). However, in urban canyons, tunnels, and other GPS-denied environments, GPS service may suffer from possible signal outages, jamming, and multi-path effects. To maintain positioning availability and accuracy in such cases, GPS is augmented with the inertial navigation system (INS). As a standalone approach, INS is inherently immune to external disturbances and is able to provide continuous navigation solutions with short-term accuracy [1]. Therefore, one of the common solutions for vehicular positioning during GPS outages is to augment GPS with INS [2,3].
For low-cost objectives, instead of integrating GPS with a full inertial measurement unit(IMU) containing three accelerometers and three gyroscopes, the reduced inertial sensor system (RISS) has gained more and more attention. Only one azimuth gyroscope and an odometer or wheel encoders is integrated with GPS, referred to as 2D RISS, to provide 2D positioning solutions in planar environments [1,[4][5][6][7]. An integrated RISS/GPS module using a particle filter (PF) was proposed in [8] to provide 2D navigation solutions. A 3D RISS [2,9] composed of a 2D RISS and two horizontal accelerometers is another navigation solution suitable for all wheeled moving platforms, and could obtain the pitch and roll angles of a land vehicle. An enhanced version of PF called the Mixture PF was utilized in [10] to perform the tightly coupled integration of a 3D RISS with GPS. Currently, most

3D RISS Mechanization
A 3D RISS is comprised of one azimuth gyroscope providing the azimuth angular rate change, two horizontal accelerometers for calculating the pitch and roll angles in a horizontal plan, and an odometer with a moving velocity in a near-horizontal plan [2,6,10,30]. The 3D RISS mechanization schematic diagram is shown in Figure 1.

3D RISS Mechanization
A 3D RISS is comprised of one azimuth gyroscope providing the azimuth angular rate change, two horizontal accelerometers for calculating the pitch and roll angles in a horizontal plan, and an odometer with a moving velocity in a near-horizontal plan [2,6,10,30]. The 3D RISS mechanization schematic diagram is shown in Figure 1.   During the 3D RISS mechanization process, the pitch angle of the vehicle is calculated by the forward accelerometer output information after error compensation. The pitch angle calculation formula is yielded by Equation (1): where aod is the acceleration of the vehicle in a near-horizontal plan, and g is the Earth's gravity. aod is not directly measurable, and is obtained from the derivative of the odometer velocity νod. Generally, aod at each time epoch can be calculated as where νod(k) refers to the νod at each time step, and dt is the sample epoch.
After that, the roll angle of the dynamic vehicle is calculated by the transversal accelerometer information fx, the azimuth gyroscope measurement wz, and the odometer velocity information. Therefore, the roll angle calculation formula is yielded by Equation (3): 1 sin cos Simultaneously, the azimuth can be derived by Equation (4). During the 3D RISS mechanization process, the pitch angle of the vehicle is calculated by the forward accelerometer output information after error compensation. The pitch angle calculation formula is yielded by Equation (1): where a od is the acceleration of the vehicle in a near-horizontal plan, and g is the Earth's gravity. a od is not directly measurable, and is obtained from the derivative of the odometer velocity ν od . Generally, a od at each time epoch can be calculated as where ν od (k) refers to the ν od at each time step, and dt is the sample epoch. After that, the roll angle of the dynamic vehicle is calculated by the transversal accelerometer information f x , the azimuth gyroscope measurement w z , and the odometer velocity information. Therefore, the roll angle calculation formula is yielded by Equation (3): Simultaneously, the azimuth can be derived by Equation (4).
After the attitude calculation, the 3D velocity in the local level frame (LLF) can be derived using Equation (5): Finally, the 3D position can be obtained by integration. The step-by-step computation of the 3D position is yield by

Attitude Errors Analysis
Differentiating Equations (1), (3), and (4), respectively, the attitude errors can be yield From Equation (7), the pitch error is determined by the forward accelerometer measurement error δf y , the odometer's forward accelerate error δa od , and the Earth's gravity error δg. The roll error is determined by the transversal accelerometer measurement error δf x , the odometer measurement error δv od , the vertical gyroscope measurement error δw z , the pitch error δp, and also the Earth's gravity error δg. Assuming that the Earth's radius R is a relative large value, the azimuth error can be simplified as Therefore, the azimuth error is mainly determined by the vertical gyro measurement error δw z and the latitude error δϕ.

Velocity Errors Analysis:
Differentiating Equation (5), the velocity errors can be yielded: δv n = δv od cos A cos p − δAv od sin A cos p − δpv od cos A sin p, δv u = δv od sin p + δpv od cos p,

Position Errors Analysis
Differentiating Equation (6), the position errors can be obtained: In above analysis, the ν od derived from the odometer is at least as important as the data from the gyroscope or accelerometers. The derivative a od of ν od is used to calculate the pitch and roll angles. The ν od is used to calculate the 3D position together with inertial sensors. Hence, in actual applications, it is of great significant to improve the measurement accuracy and reliability of the odometer velocity.

TD Filter Principle
In classical control theory, the differentiator is built using a small time-constant inertial unit. The first-order derivate of the input signal U(s) can be obtained by following a linear, time-invariant, continuous-time dynamic system like Equation (17): where Y(s) and U(s) are the output and input, respectively; T is the time constant; and s is the Laplace operator. In fact, when T is small enough, the inertia unit becomes an approximate time-delay unit. That is to say, 1/(Ts + 1) ≈ e -Ts . The inertia unit in Equation (17) could be treated as a time-delay unit with a small time constant. The inverse Laplace transform of Equation (17) is When signals u(t) are corrupted by noises, the noises are also amplified 1/T times; so, the differentiator obtained by the classical method is not suitable for most engineering applications. An improved differentiator of Equation (19) was proposed in [25].
The difference of the two inertia units is used as the differential to depress the noise amplification. In order to obtain the differential by the fastest dynamic part, a resulting control law that drives any initial state point to the origin in the minimum time is introduced to construct the noise-tolerant time optimal control (TOC)-based TD [27].
The double-integral system is defined as .
where |u| ≤ r, and r is a constant constraint of the control input. It was proven in [24] that the resulting feedback control law that drives the state from any initial point to the origin in the shortest time is where ν is the desired value for x 1 .
where x 1 is the desired trajectory and x 2 is its derivative. Via Euler's method discretization, the discrete form of TD is given: where ν is the input signal, x 1 is the filter value of ν, x 2 is the derivative of x 1 , r is the tracking velocity of the TD filter, and h is the step size in simulation. The nonlinear switching function fhan (x 1 -ν, x 2 , r, h 0 ) is given by Equation (25): The state x 1 tracks the input signal ν in the maximum velocity r without oscillation due to the function of fhan. The error between x 1 and ν goes to zero. The larger the speed factor r is, the faster the signal tracks. However, the larger of the speed factor r is, the stronger the noise amplification is. Moreover, the noise will reduce by adjusting the filter parameter h 0 (5~10 times of h). The larger the filter factor h 0 is, the better the filtering effect is. However, the larger the filter factor h 0 is, the greater the phase loss of the tracking signal. Therefore, in order to obtain a better filtering effect, coordinated adjustment of r and h 0 is required.
Generally, there is some phase lag on the output results of filters. Because TD can give the derivative of the input signal, the phase lag of the TD filter could be compensated with the following equations: where ν 1 is the new input signal composed of the original input signal ν, h 1 is the forecast time of ν with x 2 , and x 1 is the filter value of ν 1 . The forecast time h 1 is usually 1~1.5 times of h 0 .

Phase Compensation for Signal Filtering
Let y(t) = sin(20πt) be an original input signal. A TD filter is set with the given design parameters: h = 0.005 s, h 0 = 5 h, r = 30,000, and h 1 = 1.2 h 0 . The original input signal y(t), the filtering result of TD without phase compensation and the filtering result of TD with phase compensation are shown in Figure 2a. As Figure 2a shows, the phase lag of the TD filter is obviously reduced by the phase compensation. The numerical differential result of y(t), the differential result of TD without phase compensation and the differential result of TD with phase compensation are shown in Figure 2b. As Figure 2b shows, after half a cycle of y(t), the differential result of TD with phase compensation can track the numerical differential result of y(t) very well. So, in the following part of this paper, TD refers in particular to TD with phase compensation.
where ν1 is the new input signal composed of the original input signal ν, h1 is the forecast time of ν with x2, and x1 is the filter value of ν1. The forecast time h1 is usually 1~1.5 times of h0.

Phase Compensation for Signal Filtering
Let y(t) = sin(20πt) be an original input signal. A TD filter is set with the given design parameters: h = 0.005 s, h0 = 5 h, r = 30000, and h1 = 1.2 h0. The original input signal y(t), the filtering result of TD without phase compensation and the filtering result of TD with phase compensation are shown in Figure 2a. As Figure 2a shows, the phase lag of the TD filter is obviously reduced by the phase compensation. The numerical differential result of y(t), the differential result of TD without phase compensation and the differential result of TD with phase compensation are shown in Figure 2b. As Figure 2b shows, after half a cycle of y(t), the differential result of TD with phase compensation can track the numerical differential result of y(t) very well. So, in the following part of this paper, TD refers in particular to TD with phase compensation.

Noise Reduction
A random noise with uniform distribution in [0,0.1] is added to y(t). The given design parameters of the above TD filter remain unchanged. The original input signal y(t), the noisy signal y(t) Figure 2. Filtering result and differential result of TD.

Noise Reduction
A random noise with uniform distribution in [0,0.1] is added to y(t). The given design parameters of the above TD filter remain unchanged. The original input signal y(t), the noisy signal and the filtering result of TD are shown in Figure 3a. TD can extract the original signal from the noisy signal with a small phase lag. The numerical differential result of original signal and noisy signal, and the differential result of TD are shown in Figure 3b. Compared to the numerical differential result of the original signal, the differential result of TD is obviously more accurate than the numerical differential result of the noisy signal. and the filtering result of TD are shown in Figure 3a. TD can extract the original signal from the noisy signal with a small phase lag. The numerical differential result of original signal and noisy signal, and the differential result of TD are shown in Figure 3b. Compared to the numerical differential result of the original signal, the differential result of TD is obviously more accurate than the numerical differential result of the noisy signal.

Outliers Exclusion
Here, a hypothetical outlier point is magnified 10x from the noisy signal referred to in Section 3.1.2. In addition, the given design parameters of the above TD filter remain unchanged. The original input signal y(t), the noisy signal and the filtering result of TD are shown in Figure 4a. TD cannot only extract the original signal from the noisy signal with a small phase lag, but is also hardly affected by the outlier point. The numerical differential result of the original signal and noisy signal, and the differential result of TD are shown in Figure 3b. The outlier point leads to the remarkable error of the numerical differential result of the noisy signal. However, there is almost no effect caused by the outlier point on the differential result of TD.

Outliers Exclusion
Here, a hypothetical outlier point is magnified 10x from the noisy signal referred to in Section 3.2.2. In addition, the given design parameters of the above TD filter remain unchanged. The original input signal y(t), the noisy signal and the filtering result of TD are shown in Figure 4a. TD cannot only extract the original signal from the noisy signal with a small phase lag, but is also hardly affected by the outlier point. The numerical differential result of the original signal and noisy signal, and the differential result of TD are shown in Figure 3b. The outlier point leads to the remarkable error of the numerical differential result of the noisy signal. However, there is almost no effect caused by the outlier point on the differential result of TD. and the filtering result of TD are shown in Figure 3a. TD can extract the original signal from the noisy signal with a small phase lag. The numerical differential result of original signal and noisy signal, and the differential result of TD are shown in Figure 3b. Compared to the numerical differential result of the original signal, the differential result of TD is obviously more accurate than the numerical differential result of the noisy signal.    As Section 3 demonstrated, TD achieves a better performance on random noises and outliers than traditional numerical differentiation. Using the odometer velocity data of a 3D RISS as the input signal of a TD filter, the filtered value of the odometer velocity data can be obtained from x1 of this TD with noise reduction and outlier exclusion, and the acceleration data can also be obtained from x2 of this TD filter rather than being calculated using the numerical differential method.

Simulation Experiments
In this section, a group of offline data prepared for a study of RISSs by the Navigation Instrumentation Research Group in Royal Military College of Canada is used to demonstrate the effect of velocity and acceleration estimation through an actual vehicular experiment based on TD. The 3D RISS mechanization is constructed from a low-grade Xbow IMU with the odometer output at 10 Hz update rates, and high-end Novatel SPAN IMU mechanization results are used as the reference. The odometer velocity (preprocessed by an offline wavelet filter) of this group is used as the original odometer velocity. The original velocity is filtered through a TD filter set with the given design parameters: h = 0.1 s, h0 = 5 h, r = 20,000, and h1 = 1.4 h0. In the 3D RISS with the original odometer velocity filtered by TD, the x1 takes the place of the original velocity νod and the x2 takes the place of aod calculated by Equation (2). The errors in attitude, velocity, position, and 2D trajectory will be plotted for comparison. Finally, error analysis will be discussed.

Validation of a 3D RISS with Velocity Filtered by TD
The three navigation simulation results of the 3D RISS with the original odometer velocity without a TD filter, the 3D RISS with the original odometer velocity filtered by TD, and the Novatel IMU are compared in Figures 5-8. In Figures 5-8, "3D-RISS without TD" refers to the simulation results of the 3D RISS with the original odometer velocity without a TD filter, "3D-RISS with TD" refers to the simulation results of the 3D RISS with the original odometer velocity filtered by TD, and "Novatel" refers to the simulation results of the Novatel IMU. The standard deviations of the main navigation errors between the two 3D RISSs and Novatel IMU are listed in Table 1. Figure 5a depicts a comparison of the tri-axial attitude curves. The two attitude errors between the two types of 3D RISSs and the Novatel IMU are plotted in Figure 5b. The pitch and roll data of the two types of 3D RISSs have a similar variation tendency with the Novatel attitude reference and no divergence trend. However, the azimuth error curves are divergent with time, which is mainly caused by the vertical gyro measurement error δωz. According to the 3D RISS mechanization, the azimuth error divergence cannot be corrected with the odometer velocity. The standard deviations of the pitch and roll errors between the 3D RISS with the original odometer velocity and the Novatel IMU are 0.88° and 0.32°, respectively. The standard deviations of the pitch and roll errors between the 3D RISS with the original odometer velocity filtered by TD and the Novatel IMU are 1.16° and 0.36°, respectively. Because of the filter phase lag, the horizontal attitude error with the TD filter is a little bigger. As Section 3 demonstrated, TD achieves a better performance on random noises and outliers than traditional numerical differentiation. Using the odometer velocity data of a 3D RISS as the input signal of a TD filter, the filtered value of the odometer velocity data can be obtained from x 1 of this TD with noise reduction and outlier exclusion, and the acceleration data can also be obtained from x 2 of this TD filter rather than being calculated using the numerical differential method.

Simulation Experiments
In this section, a group of offline data prepared for a study of RISSs by the Navigation Instrumentation Research Group in Royal Military College of Canada is used to demonstrate the effect of velocity and acceleration estimation through an actual vehicular experiment based on TD. The 3D RISS mechanization is constructed from a low-grade Xbow IMU with the odometer output at 10 Hz update rates, and high-end Novatel SPAN IMU mechanization results are used as the reference. The odometer velocity (preprocessed by an offline wavelet filter) of this group is used as the original odometer velocity. The original velocity is filtered through a TD filter set with the given design parameters: h = 0.1 s, h 0 = 5 h, r = 20,000, and h 1 = 1.4 h 0 . In the 3D RISS with the original odometer velocity filtered by TD, the x 1 takes the place of the original velocity ν od and the x 2 takes the place of a od calculated by Equation (2). The errors in attitude, velocity, position, and 2D trajectory will be plotted for comparison. Finally, error analysis will be discussed.

Validation of a 3D RISS with Velocity Filtered by TD
The three navigation simulation results of the 3D RISS with the original odometer velocity without a TD filter, the 3D RISS with the original odometer velocity filtered by TD, and the Novatel IMU are compared in Figures 5-8. In Figures 5-8, "3D-RISS without TD" refers to the simulation results of the 3D RISS with the original odometer velocity without a TD filter, "3D-RISS with TD" refers to the simulation results of the 3D RISS with the original odometer velocity filtered by TD, and "Novatel" refers to the simulation results of the Novatel IMU. The standard deviations of the main navigation errors between the two 3D RISSs and Novatel IMU are listed in Table 1. Figure 5a depicts a comparison of the tri-axial attitude curves. The two attitude errors between the two types of 3D RISSs and the Novatel IMU are plotted in Figure 5b. The pitch and roll data of the two types of 3D RISSs have a similar variation tendency with the Novatel attitude reference and no divergence trend. However, the azimuth error curves are divergent with time, which is mainly caused by the vertical gyro measurement error δω z . According to the 3D RISS mechanization, the azimuth error divergence cannot be corrected with the odometer velocity. The standard deviations of the pitch and roll errors between the 3D RISS with the original odometer velocity and the Novatel IMU are 0.88 • and 0.32 • , respectively. The standard deviations of the pitch and roll errors between the 3D RISS with the original odometer velocity filtered by TD and the Novatel IMU are 1.16 • and 0.36 • , respectively. Because of the filter phase lag, the horizontal attitude error with the TD filter is a little bigger.       The three 2D trajectory simulation results of the 3D RISS with the original odometer velocity, the 3D RISS with the original odometer velocity filtered by TD, and the Novatel IMU are compared in Figure 8. The two trajectory errors between the two types of 3D RISSs and the Novatel IMU are almost identical.
As Figures 5-8 and Table 1 show, the two navigation errors of the 3D RISS with the original odometer velocity and the 3D RISS with the original odometer velocity filtered by TD occur at the same level. Using the original odometer velocity, TD could quickly and correctly filter the velocity and estimate the acceleration for a 3D RISS.   The three 2D trajectory simulation results of the 3D RISS with the original odometer velocity, the 3D RISS with the original odometer velocity filtered by TD, and the Novatel IMU are compared in Figure 8. The two trajectory errors between the two types of 3D RISSs and the Novatel IMU are almost identical.

Anti-Interference Ability of a 3D RISS with Velocity Filtered by TD
As Figures 5-8 and Table 1 show, the two navigation errors of the 3D RISS with the original odometer velocity and the 3D RISS with the original odometer velocity filtered by TD occur at the same level. Using the original odometer velocity, TD could quickly and correctly filter the velocity and estimate the acceleration for a 3D RISS. The three 2D trajectory simulation results of the 3D RISS with the original odometer velocity, the 3D RISS with the original odometer velocity filtered by TD, and the Novatel IMU are compared in Figure 8. The two trajectory errors between the two types of 3D RISSs and the Novatel IMU are almost identical.
As Figures 5-8 and Table 1 show, the two navigation errors of the 3D RISS with the original odometer velocity and the 3D RISS with the original odometer velocity filtered by TD occur at the same level. Using the original odometer velocity, TD could quickly and correctly filter the velocity and estimate the acceleration for a 3D RISS.

Anti-Interference Ability of a 3D RISS with Velocity Filtered by TD
In order to demonstrate the anti-interference effect of TD on a 3D RISS, a white Gaussian noise (with 0.002 variance and a 0 mean value) and a hypothetical outlier point (2 times amplitude and 100 s interval) are added to the original odometer velocity. The white Gaussian noise model is in accordance with our Kalman filter for the RISS/GPS integration, and the hypothetical outlier point is to simulate the odometer velocity error caused by vehicle sideslips or jumps off the ground. The given design parameters of the above TD filter remain unchanged. The results in Section 4.1 of the 3D RISS with the original odometer velocity are used as the reference. The three navigation simulation results of the 3D RISS with a noisy odometer velocity without a TD filter, the 3D RISS with a noisy odometer velocity filtered by TD, and the 3D RISS with the original odometer velocity are compared in Figures 9-12. In Figures 9-12, "3D RISS with noise" refers to the simulation results of the 3D RISS with a noisy odometer velocity without a TD filter, and "3D RISS with TD" refers to the simulation results of the 3D RISS with a noisy odometer velocity filtered by TD, and "Original 3D RISS" refers to the simulation results of the 3D RISS with the original odometer velocity. The standard deviations of the navigation errors between the two 3D RISSs with a noisy odometer velocity and the reference are listed in Table 2. Figure 9a depicts a comparison of the tri-axial attitude curves. The two attitude errors between the two 3D RISSs with noisy odometer velocities and the reference are plotted in Figure 9b. Because the a od is in the calculation of the pitch, the pitch errors are most obviously affected in the comparison of attitude curves. The max pitch error between the 3D RISS with the original odometer velocity and the 3D RISS with a noisy odometer velocity reaches 91 • . However, the max pitch error between the 3D RISS with the original odometer velocity and the 3D RISS with a noisy odometer velocity filtered by TD is 11 • . The standard deviations of the pitch and roll errors between the 3D RISS with the original odometer velocity and the 3D RISS with a noisy odometer velocity are 10.61 • and 0.065 • , respectively. The standard deviations of the pitch and roll errors between the 3D RISS with the original odometer velocity and the 3D RISS with a noisy odometer velocity filtered by TD are 1.75 • and 0.18 • , respectively. Figure 10a depicts a comparison of the tri-axial velocity curves. The two velocity errors between the two 3D RISSs with noisy odometer velocities and the reference are plotted in Figure 10b. The standard deviations of the V E , V N , and V U errors between the 3D RISS with the original odometer velocity and the 3D RISS with a noisy odometer velocity are 24.95 m/s, 24.19 m/s, and 34.79 m/s, respectively. The standard deviations of the V E , V N , and V U errors between the 3D RISS with the original odometer velocity and the 3D RISS with a noisy odometer velocity filtered by TD are 0.26 m/s, 0.25 m/s, and 0.64 m/s, respectively. Due to the excellent filtering performance of TD in velocity estimation, the velocity results of the 3D RISS with a noisy odometer velocity filtered by TD is hardly influenced by the hypothetical noises. Figure 11a depicts a comparison of the tri-axial position curves. The two position errors between the two 3D RISSs with noisy odometer velocities and the reference are plotted in Figure 11b. The standard deviations of latitude and longitude errors between the 3D RISS with the original odometer velocity and the 3D RISS with a noisy odometer velocity are 0.0021 • and 0.0029 • , respectively. The standard deviations of latitude and longitude errors between the 3D RISS with a noisy odometer velocity filtered by TD and the 3D RISS with the original odometer velocity are 0.000039 • and 0.000077 • , respectively. Similarly, the differences of the above two velocity errors are reflected in the two position errors. The three 2D trajectory simulation results of the three types of 3D RISS are compared in Figure 12.
As Figures 9-12 and Table 2 show, with the hypothetical significant noises introduced into the original odometer velocity, the divergence speed of navigation errors of the 3D RISS with odometer velocity filtered by a TD filter is much slower than the pure 3D RISS. Thus, the reliability of a 3D RISS is obviously improved by a TD filter used for estimating velocity and acceleration. Additionally, the velocity filtered by TD is favorable to reduce the error divergence risk of the integrated navigation Kalman filter.

Conclusions
Compared to a lot of studies on RISSs that have focused on filters of RISS/GPS integration or on the error correction of inertial sensors, there have seldom been studies on odometer velocity for a 3D RISS. However, in 3D RISS mechanization and error analysis, velocity information and its derivatives are at least as important as the information from the other inertial sensors of the RISS. Velocity information from an odometer is prone to noise corruption, which further leads to the noise amplification of acceleration information in a conventional differential method. This paper has presented a solution for odometer velocity and acceleration estimation using a 3D RISS based on a TD filter.
A TD filter does not depend on an object model and has less computation. With a TD filter, an input signal and its differential signal are estimated fast and accurately. Additionally, using the differential signal output by the TD filter, the filter phase lag can be decreased with the prediction method. As Section 3 demonstrated, TD achieves better performance on random noises and outliers than traditional numerical differentiation.
Using a group of offline data obtained from an actual vehicle experiment, the effectiveness of a TD filter on a 3D RISS was demonstrated through simulation experiments. With the odometer velocity data of a 3D RISS as the input signal of a TD filter, the filtered value with noise reduction and the exclusion of outlying odometer velocity data can be obtained from tracking the signal output from this TD, and the acceleration data can be obtained from the derivative output of this TD also, rather than calculated using the numerical differential method. As the results show, a TD filter could not only correctly and quickly filter the velocity and estimate the acceleration from the odometer velocity using a 3D RISS, but could also improve the reliability of a 3D RISS.
The future work will be to transform the TD filter algorithm into a navigational computer program of an actual 3D RISS.