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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Mobile mapping systems have been widely applied for acquiring spatial information in applications such as spatial information systems and 3D city models. Nowadays the most common technologies used for positioning and orientation of a mobile mapping system include a Global Positioning System (GPS) as the major positioning sensor and an Inertial Navigation System (INS) as the major orientation sensor. In the classical approach, the limitations of the Kalman Filter (KF) method and the overall price of multi-sensor systems have limited the popularization of most land-based mobile mapping applications. Although intelligent sensor positioning and orientation schemes consisting of Multi-layer Feed-forward Neural Networks (MFNNs), one of the most famous Artificial Neural Networks (ANNs), and KF/smoothers, have been proposed in order to enhance the performance of low cost Micro Electro Mechanical System (MEMS) INS/GPS integrated systems, the automation of the MFNN applied has not proven as easy as initially expected. Therefore, this study not only addresses the problems of insufficient automation in the conventional methodology that has been applied in MFNN-KF/smoother algorithms for INS/GPS integrated systems proposed in previous studies, but also exploits and analyzes the idea of developing alternative intelligent sensor positioning and orientation schemes that integrate various sensors in more automatic ways. The proposed schemes are implemented using one of the most famous constructive neural networks—the Cascade Correlation Neural Network (CCNNs)—to overcome the limitations of conventional techniques based on KF/smoother algorithms as well as previously developed MFNN-smoother schemes. The CCNNs applied also have the advantage of a more flexible topology compared to MFNNs. Based on the experimental data utilized the preliminary results presented in this article illustrate the effectiveness of the proposed schemes compared to smoother algorithms as well as the MFNN-smoother schemes.

The development of land based mobile mapping systems was initiated by two research groups in North America, The Center for Mapping at Ohio State University, USA, and the Department of Geomatics Engineering at the University of Calgary, Canada [

Since the early nineties, advances in satellite and inertial technology made it possible to think about mobile mapping in a new way. Instead of using ground control points as references for orienting the images in space, the trajectory and orientation of the imager platform can now be determined directly [

An INS is a self-contained navigation technique in which measurements provided by accelerometers and gyroscopes are used to track the position and orientation of an object relative to a known starting point, orientation and velocity [

In the classical approach, the KF is applied in real-time applications to fuse different data from various sensors while optimal smoothing is applied in the post-mission mode. The basic idea of using the KF in INS/GPS integration is to fuse independent and redundant sources of navigation information with a reference navigation solution to obtain an optimal estimate of navigation states, such as position, velocity and orientation. However, the limitations of the KF have been reported by several researchers [

These limitations, in turn, may result in sub-optimal performance or even filter divergence if the assumption of local linearity is violated [

In addition to these limitations, the problem of poor observation of inertial error states becomes the most critical issue, especially when integrating a low cost MEMS IMU with GPS [

Optimal smoothing algorithms, also known as smoothers, have been applied for the purpose of accurate positioning and orientation parameter determination through post-processing for most of surveying and mobile mapping applications with integrated sensors [

As mentioned in the previous section, the accuracies of the KF solutions sometimes cannot fulfill the requirements of mobile mapping applications. An integrated system has to predict state parameters such as position, velocity and orientation using KF when GPS signal blockages appear [

In order to achieve high accuracy for positioning and orientation determination in mobile mapping applications, the data is processed in post-mission mode with an optimal smoothing algorithm. Most of the commercial mobile mapping systems use an optimal smoothing algorithm to provide accurate information on position and orientation for direct geo-referencing [

Another effective way to improve the accuracies of low cost MEMS INS/GPS integrated solutions is through the improvement of sensor fusion algorithms. Compared to the hardware perspective mentioned above, the software perspective can be considered as a cost effective solution to develop a low cost sensor fusion solution for certain mobile mapping applications.

The process of the KF is divided into two groups, those for prediction and updating [

The smoothed estimates at any epoch k are computed as a linear combination of the filtered estimate at that epoch and the smoothed estimate at the heading epoch k + 1. Thus, these smoothed estimates can be considered as updating the forward filtered solution to obtain improved estimates. The computation of the smoothed estimates at each epoch requires the storage of the KF predicted and updated (filtered) estimates and their corresponding covariances at each epoch [

Three approaches concerning the development of alternative multi-sensor integration algorithms to improve the ability of analysis and prediction of complicated kinematic and nonlinear models to reduce the magnitude of accumulated positional and orientation errors during frequent GPS outages in land applications have been identified [

The advantage of the sampling based approach is that the computation of derivatives is not applied [

Bergman [

Aggarwal [

According to Kubo and Wang [

To overcome the limitations of EKF instabilities and Jacobian evaluations, Julier and Uhlmann [

Generally speaking, the results shown in [

The artificial intelligence approach distinguishes itself from other types of estimation approaches by using inexplicit models, known as black box, to approximate the nonlinear relationships between system dynamics and measurements [

Chiang

Wang

Sharaf and Noureldin [

El-Sheimy

Although artificial intelligence approaches are easier to design and implement, there are also limitations to these types of approaches [

In addition, if the dynamics experienced by the vehicle exceeds the ranges of training data significantly, the performance of these approaches tends to deteriorate accordingly. Therefore, a frequent re-training procedure is required to guarantee the performance of these approaches. Chiang

The navigation parameters provided by these artificial intelligent models are limited to positional parameters only because they rely on GPS solutions to provide training target and the use of a GPS receiver is unable to provide any reliable orientation parameters. Therefore, all of these artificial intelligent models are implemented to provide 2-D or 3-D positional parameters and bridge the gaps during GPS outages for land vehicular navigation applications [

The hybrid approach is implemented by combing conventional EKF or smoother based approaches with artificial intelligent models. These artificial intelligent models are applied to model the error behaviors of conventional EKF or smoother and compensate for the errors of the positional and orientation parameters estimated by KF and Smoother during GPS outages. Goodall

Generally speaking, a MFNN with an optimal topology is expected to provide the best approximation accuracy to the unknown model using the most appropriate number of hidden neurons and hidden layers [

To avoid these limitations, several methods have been proposed in the last two decades to construct a neural network successively and automatically during the learning process. These methods are often recognized as constructive networks. The common principle is to start from a small network and then add hidden neurons and hidden layers as needed during the learning procedure using special algorithms. In other words, the networks are able to decide the appropriate topology based on the task given without human intervention. An overview of current constructive algorithms can be found in [

Consequently, the objectives of this article were to: (1) develop a CCNN-smoother scheme for precise sensor positioning and orientation, (2) verify the performance of the proposed scheme using a low cost MEMS INS/GPS integrated system and (3) compare the performance with previously developed MFNN-smoother scheme in terms of the topology applied and estimated accuracy during GPS outages.

ANN methodologies have been motivated by the recognition that the human brain works in an entirely different way from a conventional digital computer [_{a}^{(}^{h}^{)} and ^{(o)}, respectively.

Therefore, the outputs of hidden and output layers, _{a}^{(h)}^{(o)}

_{a}

A simplest way to understand how to adjust weight is to use standard backpropagation learning algorithm whose error function E is given below [

Then the weight update formation is given as [

The primary goal of developing ANN-aided schemes is to improve the positioning and orientation accuracies during frequent GPS outages, which are crucial for land mobile mapping applications. As shown in

Therefore, the inputs applied in this study include positional and orientation states estimated by the EKF and smoother along with time information while the outputs include the errors of those estimated states during GPS outages. In other words, the proposed topology realizes the nonlinear mapping relationships between system dynamics, the length of GPS outage and the error behaviors of the EKF and smoother during GPS outages. Through using proper training strategy with sufficient training data, the proposed scheme is able to generalize the nonlinear mapping relationships between system dynamics, the length of GPS outage and the error behaviors of the EKF and smoother during outages.

In this study, one of the constructive ANNs, the CCNN, is implemented to learn and compensate for the residual errors of the KF and smoother, respectively, to improve the accuracies of the positional and orientation parameters. The proposed scheme is capable of learning how the state vector (

Two key ideas in implementing the CCNN algorithm include a cascade architecture and unique learning algorithm for automatically training and installing new hidden neuron [

The CCNN architecture starts with a minimal topology, consisting of only the input and output neurons. The optimal values of the direct input-output weight links are computed during the training procedure, and the training continues with a minimal topology for the entire training data set until no further improvement can be achieved, as shown in

To recruit a new hidden neuron, a pool of candidate neurons that have different sets of randomly initialized weight values is applied. All these candidate neurons within the pool receive the trainable input connections from the external inputs. In addition, they receive the same residual error for each training pattern sent from the output neurons through the pseudo connections shown in

During the first step of recruitment, each candidate neuron is connected to each of these input neurons, but is not connected to the output neurons. The primary task of pseudo connections shown in

The second step of recruitment process initializes after the neuron with the highest correlation is installed in the topology as a new hidden layer. Only one layer of weights is trained during the second step of recruitment process shown in

The second hidden neuron is then recruited using the process shown in

The process of recruiting new neurons, training their weights and training all connections to the output neurons, is continued until the errors reach the training goal or the maximum number of iterations or epochs (as defined by the user). According to [

According to [

Based on the training data applied in this study, the topologies of the proposed schemes are shown in

As indicated in

The EKF applied in this study has 21 states, which are given as follows:

As shown in _{a}_{,1×3} and _{g}_{,1×3}) and scale factors (_{a}_{,1×3} and _{g}_{,1×3}) can be estimated and feed back to the INS mechanization to correct these raw measurements provided by an IMU. The scope of this study is to improve the accuracies of positional and orientation parameters during GPS outages, only the components concerning these parameters are shown in

The errors of positional and orientation parameters estimated by the KF and smoother during GPS outages are used as the desired outputs or target values during the training process of various proposed ANN architectures, including MFNNs and CCNNs. The positional and orientation parameters estimated by the KF and smoother along with the time information in each scenario are used as the inputs of the proposed architectures. The goal of the proposed schemes is to compensate for the errors of the positional and orientation parameters estimated by the KF and smoother during GPS outages [

An ANN with optimal topology is expected to provide the best approximation accuracy for the unknown model using the most appropriate number of hidden neurons and hidden layers [

After being well trained, the proposed ANN compensation schemes are added to a loosely coupled INS/GPS integration architecture (closed loop), as shown in

Three field tests were used to evaluate the performance of the proposed schemes. The tests were conducted in land vehicle environments using different integrated systems consisting of one tactical grade IMU, Litton LN200 (1 deg/hr), a low cost MEMS IMU, BEI MotionPak II and two NovATel OEM-4 receivers. In this study, those IMUs were applied to collect inertial measurements in the field and then these along with carrier phase DGPS solutions were fed into software that has an inertial navigation algorithm and EKF to estimate inertial states optimally. The integrated system with LN200 IMU was used as the reference. The measurements and navigation solutions provided by the integrated system with MotionPak II were used to verify the performance of proposed schemes.

The GPS measurements were processed using GrafNavTM software (Waypoint Consulting Inc.) in carrier phase DGPS to achieve ten centimeter level accuracy. The reference trajectories were generated by the integrated system with LN 200 IMU. They were determined using 21-state EKF and smoother implemented in the Aided Inertial Navigation Software (AINS) from the Department of Geomatics Engineering at of the University of Calgary. These parameters of EKF and the smoother applied in this study were well tuned so that they can represent the best achievable navigation accuracy for tactical grade IMUs.

Ten GPS outages, marked with circles and each lasting 30 seconds, were simulated using the measurements collected in the third field test, as indicated in

In addition, a total sixty four GPS outages, each has 30 seconds in length, were simulated randomly in four scenarios using the measurements collected in the first and second field tests using the INS/GPS integrated with the MotionPak II (MEMS) IMU. Both field test data sets are applied to verify the performance of the proposed schemes.

As show in

Similarly, the 99-Percentile of errors of CCNN-KF/smoother and MFNN-KF/smoother based schemes are reduced by 99% on average compared to KF/smother solutions. However, the performance of the proposed schemes still needs to be verified using other independent data sets and this is presented in the next section.

Since the objective of this study was to verify the performance of proposed CCNN-smoother and compare to previously developed MFNN-smoother and commercially available smoother based schemes, those results concerning CCNN-KF, MFNN-KF, and commercially available KF are not provided to condense the discussions of this study. To verify the performance of well-trained ANN-smoother schemes, four sets of testing scenario are created. Each scenario consists of eight simulated GPS outages (30 seconds each) which are randomly selected from test trajectory 1 and trajectory 2, respectively. Therefore, the proposed algorithms are validated with 16 randomly selected GPS outages in each scenario.

As shown in

Since it is decided through empirical trials during the training stage, the number of hidden layers and neurons are determined by the designer. Insufficient or excess hidden layers and hidden also lead to the impact of overtraining or over-fitting, although this can be mitigated by reducing the number of training epoch, increasing the number of training goal or adjusting the number of hidden neurons. However, the ratios of improvement are not as good as with the CCNN-smoother scheme. The number of training epochs for MFNN applied in this study is reduced from 500 to 200. As shown in

As shown in

Generally speaking, the majority of positional and orientation errors produced by proposed CCNN-smoother and previously developed MFNN-smoother schemes concentrate around 0, which means that they are able to compensate various systematic errors automatically and increase the stability. On the other hand, the distributions of positional and orientation errors produced by CCNN-smoother and MFNN-smoother schemes are comparable.

On the other hand, the performances of proposed CCNN-smoother and previously developed MFNN-smoother schemes are similar in terms of RMS errors, medians and 99-Percentile of positional and orientation errors. The results presented in this study indicate that the proposed CCNN-smoother scheme is able to achieve comparable performance when undertaking the prediction task with fewer hidden neurons and less training efforts. In addition, based on the characteristics of the INS/GPS integration applications, the scheme applied is able to reflect the impact of new information and so catch the latest dynamic and sensor error variations. In a neural network, this can occur with a continuous learning process to adjust the weights and appropriate variation of topology if needed. Consequently, self-growing or construct networks such as CCNNs are better than fixed topology networks such as MFNNs.

The error behaviors of KF and smoother during GPS outage are coupled with vehicle dynamics, inertial sensor errors, the length of GPS outage and the quality of the system and measurement model applied. However, these error behaviors are too complicated to describe through proper mathematical models. The general idea of ANNs is to build up the nonlinear mapping relationship between inputs and outputs and learns for examples and generalizes for applications. Therefore, the proposed topology realizes the nonlinear mapping relationships between system dynamics, the length of GPS outage and the error behaviors of the EKF and smoother during GPS outages. Through using proper training strategy with sufficient training data, the proposed scheme is able to generalize the nonlinear mapping relationships between system dynamics, the length of GPS outage and the error behaviors of the EKF and smoother during outages

Therefore, the proposed CCNN-smoother scheme significantly improves the accuracies of all these positional and orientation parameters estimated by smoother applied in most of the commercial packages for mobile mapping applications for a low cost MEMS integrated system based on the field test data applied. Among these parameters compensated by proposed CCNN-smoother scheme, the improvement of the orientation parameters is more significant than that of positional ones. However, more training data sets are required in future studies to generalize the contributions of this study.

In addition, the use of CCNNs for developing an alternative INS/GPS fusion scheme has several advantages over MFNNs. First, the best topology can be decided automatically based on the complexity of the applications, and there is no need to perform extensive empirical trials to determine the size and depth of the network.

This study developed a CCNN embedded sensor fusion algorithm to improve the accuracy of positional and orientation parameters during GPS outages using a novel procedure that combines CCNN architecture and the smoother applied in most of the software packages for mobile mapping applications for post-mission processing.

The preliminary results presented in this study indicate that the proposed CCNN-smoother scheme outperforms commercially available smoother schemes using a low cost MEMS INS/GPS integrated system in terms of RMS errors, medians and 99-Percentile of orientation errors by 80% in average based on the field test data applied in this study. Similarly, it outperforms smoothers in terms of RMS errors, medians and 99-Percentile of positional errors by 60% in average. Among these parameters compensated by proposed CCNN-smoother scheme, the improvement of orientation parameters is more significant than that of positional ones.

On the other hand, the performances of proposed CCNN-smoother and previously developed MFNN-smoother schemes are comparable in terms of RMS errors, medians and 99-Percentile of positional and orientation errors. The results presented in this study indicate that the proposed CCNN-smoother scheme is able to achieve comparable performance when undertaking the prediction task with a simpler and flexible topology decided automatically and with less training efforts. However, more training data sets are required to generalize the contributions and validate the applicability of propose algorithms for real life applications in future studies.

The authors would like to acknowledge the financial support provided by the National Science Council of the Executive Yuan, ROC (Taiwan) (NSC 98-2221-E-006 -223-MY2). Naser El-Sheimy and Xiaoji Niu from the MMSS group at the Department of Geomatics engineering, the University of Calgary, are also acknowledged for providing the field test data sets applied in this research. Eun-Hwan Shin is acknowledged for providing the INS mechanization and INS/GPS extended Kalman filter used in this article. In addition, Robert Sulej is acknowledged for providing the technical assistance and knowledge for developing CCNN tool used in this study.

An example of a land based mobile mapping system.

The impact of KF’s limiting factors on positional error during GPS signal blockage.

A loosely coupled INS/GPS integration architecture (closed loop).

The error behaviors of sampling filters and smoothers during GPS outages.

The error behavior of artificial intelligent models during GPS outages.

The error behaviors of hybrid model during GPS outages.

An example of a MFNN.

The nonlinear mapping capability of a MFNN.

Supervised learning for function approximation.

The initialization of a CCNN.

The recruitment and installation of the first hidden neuron.

The recruitment and installation of the second hidden neuron.

The comparison between finalized CCNN and MFNN topologies.

The comparison of the learning behaviors of CCNN-KF and MFNN-KF based schemes.

An ANN training architecture (adopted from [

The implementation of ANN embedded KF and smoother (adopted from [

The tested systems and experimental trajectories.

The samples of compensated positional and orientation errors (Training scenario).

The samples of compensated positional and orientation errors (Testing scenario-I with trojectory_1).

The samples of compensated positional and orientation errors (Testing scenario-I with trojectory_2).

The histograms of orientation errors (Testing scenario-I).

The histograms of positional errors (Testing scenario-I).

The topology comparison of CCNN and MFNN based schemes.

P: position. |
P | A | P | A | P | A | P | A |

Input neurons | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |

Hidden layer/neuron | 32/32 | 35/35 | 34/34 | 35/35 | 1/60 | 1/65 | 1/60 | 1/65 |

Output neurons | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |

Training time(s) | 355 | 378 | 359 | 366 | 583 | 645 | 586 | 650 |

The summary of the experimental conditions.

4 | 10 | 7 | 1.2 | 5.8 | 2.2 | 0 | 22 | 7.5 | 03.17.2005 | 2,400 | |

4 | 10 | 7 | 1.1 | 5.8 | 2.1 | 0 | 22 | 8.2 | 03.17.2005 | 1,850 | |

4 | 10 | 7.5 | 1.4 | 5.8 | 2.4 | 0 | 22 | 7.8 | 03.16.2005 | 1,700 |

NVS: Number of visible satellites, Min.: Minimum, Max.: Maximum, Avg.: Average, Speed: Horizontal velocity.

Statistical Summary of the compensated positional and orientation solutions (Training scenario).

States | RMS | Median | 99-Percentile | |||||||

K | CCNN-K | MFNN-K | K | CCNN-K | MFNN-K | K | CCNN-K | MFNN-K | ||

Tj-3 | Roll(deg) | 0.919 | 0.049 | 0.038 | −0.932 | 0.002 | 0.003 | 0.611 | 0.039 | 0.048 |

Pitch(deg) | 0.573 | 0.04 | 0.034 | 0.623 | 0.003 | 0.004 | 0.521 | 0.028 | 0.034 | |

Heading(deg) | 6.434 | 0.921 | 0.496 | 1.823 | 0.002 | 0.002 | 3.410 | 0.750 | 0.886 | |

North (m) | 1.081 | 0.319 | 0.273 | −0.250 | −0.003 | 0.005 | 1.079 | 0.328 | 0.383 | |

East (m) | 1.781 | 0.342 | 0.215 | −0.364 | 0.004 | −0.004 | 1.780 | 0.342 | 0.415 | |

Height (m) | 0.365 | 0.077 | 0.068 | 0.636 | 0.006 | −0.001 | 0.363 | 0.067 | 0.078 | |

K: Kalman filter; CCNN-K: CCNN-Kalman filter; MFNN-K: MFNN-Kalman filter | ||||||||||

States | RMS | Median | 99-Percentile | |||||||

S | CCNN-S | MFNN-S | S | CCNN-S | MFNN-S | S | CCNN-S | MFNN-S | ||

Tj-3 | Roll(deg) | 0.929 | 0.031 | 0.023 | −0.932 | −0.002 | 0.001 | −0.246 | 0.028 | 0.050 |

Pitch(deg) | 0.586 | 0.020 | 0.019 | 0.595 | 0.001 | 0.003 | 0.302 | 0.046 | 0.041 | |

Heading(deg) | 2.776 | 0.567 | 0.483 | −0.345 | −0.002 | 0.002 | 4.630 | 0.302 | 0.426 | |

North (m) | 0.233 | 0.042 | 0.040 | −0.08 | 0.002 | 0.003 | 0.757 | 0.292 | 0.348 | |

East (m) | 0.217 | 0.039 | 0.047 | 0.045 | 0.003 | −0.003 | 0.823 | 0.109 | 0.172 | |

Height (m) | 0.103 | 0.036 | 0.034 | 0.021 | 0.002 | −0.002 | 2.277 | 0.125 | 0.139 | |

S: Smoother; CCNN-S: CCNN-Smoother; MFNN-S: MFNN-Smoother |

Statistical summary of the compensated positional and orientation solutions.

States | RMS | Median | 99-Percentile | |||||||

S | CCNN-S | MFNN-S | S | CCNN-S | MFNN-S | S | CCNN-S | MFNN-S | ||

Tj-1 | Roll(deg) | 0.935 | 0.043 | 0.052 | −0.933 | −0.003 | 0.002 | −0.476 | 0.088 | 0.050 |

Pitch(deg) | 0.524 | 0.020 | 0.040 | 0.537 | −0.002 | 0.001 | 0.702 | 0.046 | 0.041 | |

Heading(deg) | 3.577 | 0.428 | 0.646 | −2.198 | −0.002 | −0.003 | 8.630 | 1.302 | 1.426 | |

North (m) | 0.359 | 0.104 | 0.108 | 0.187 | 0.001 | −0.003 | 0.857 | 0.492 | 0.348 | |

East (m) | 0.224 | 0.068 | 0.076 | 0.042 | 0.000 | −0.001 | 0.623 | 0.309 | 0.172 | |

Height (m) | 0.374 | 0.080 | 0.083 | 0.025 | 0.001 | 0.004 | 1.977 | 0.295 | 0.139 | |

Tj-2 | Roll (deg) | 0.877 | 0.029 | 0.021 | −0.894 | −0.005 | 0.004 | −0.442 | 0.059 | 0.069 |

Pitch (deg) | 0.664 | 0.026 | 0.030 | 0.678 | 0.001 | 0.000 | 0.901 | 0.045 | 0.095 | |

Heading(deg) | 2.707 | 0.395 | 0.334 | −2.266 | −0.012 | 0.003 | 8.314 | 1.061 | 0.451 | |

North (m) | 0.478 | 0.370 | 0.432 | 0.177 | 0.000 | −0.002 | 0.651 | 0.536 | 1.853 | |

East (m) | 0.361 | 0.257 | 0.325 | −0.072 | −0.001 | −0.002 | 0.276 | 0.118 | 0.421 | |

Height (m) | 0.261 | 0.101 | 0.090 | 0.030 | −0.005 | −0.002 | 1.393 | 0.449 | 0.086 | |

S: Smoother; CCNN-S: CCNN-Smoother; MFNN-S: MFNN-Smoother |

States | RMS | Median | 99-Percentile | |||||||

S | CCNN-S | MFNN-S | S | CCNN-S | MFNN-S | S | CCNN-S | MFNN-S | ||

Tj-1 | Roll(deg) | 0.923 | 0.021 | 0.033 | −0.930 | 0.003 | −0.004 | −0.388 | 0.122 | 0.154 |

Pitch(deg) | 0.530 | 0.037 | 0.041 | 0.547 | −0.002 | 0.000 | 0.711 | 0.043 | 0.133 | |

Heading(deg) | 0.358 | 0.135 | 0.145 | −0.222 | 0.005 | 0.002 | 0.754 | 0.679 | 0.741 | |

North (m) | 0.307 | 0.218 | 0.277 | 0.019 | −0.002 | 0.003 | 0.619 | 0.412 | 0.301 | |

East (m) | 0.235 | 0.107 | 0.099 | 0.017 | 0.000 | 0.004 | 0.677 | 0.309 | 0.292 | |

Height (m) | 0.445 | 0.140 | 0.150 | 0.027 | −0.002 | −0.002 | 2.298 | 0.203 | 0.574 | |

Tj-2 | Roll (deg) | 0.843 | 0.130 | 0.160 | −0.840 | −0.006 | 0.004 | −0.232 | 0.071 | 0.224 |

Pitch (deg) | 0.671 | 0.102 | 0.110 | 0.672 | −0.001 | 0.001 | 0.971 | 0.136 | 0.142 | |

Heading(deg) | 0.302 | 0.192 | 0.204 | −0.224 | 0.000 | 0.000 | 0.889 | 0.143 | 1.265 | |

North (m) | 0.622 | 0.424 | 0.440 | 0.009 | 0.000 | 0.005 | 1.677 | 0.898 | 1.445 | |

East (m) | 0.844 | 0.687 | 0.766 | −0.083 | 0.000 | 0.000 | 0.438 | 0.395 | 0.500 | |

Height (m) | 0.327 | 0.124 | 0.145 | 0.033 | 0.001 | 0.001 | 1.697 | 0.217 | 0.598 | |

S: Smoother; CCNN-S: CCNN-Smoother; MFNN-S: MFNN-Smoother |

States | RMS | Median | 99-Percentile | |||||||

S | CCNN-S | MFNN-S | S | CCNN-S | MFNN-S | S | CCNN-S | MFNN-S | ||

Tj-1 | Roll(deg) | 0.925 | 0.023 | 0.033 | −0.931 | 0.003 | −0.003 | −0.384 | 0.127 | 0.173 |

Pitch(deg) | 0.531 | 0.038 | 0.038 | 0.547 | −0.002 | 0.001 | 0.711 | 0.044 | 0.149 | |

Heading(deg) | 0.372 | 0.179 | 0.184 | −0.222 | 0.005 | 0.002 | 0.755 | 0.573 | 0.639 | |

North (m) | 0.318 | 0.256 | 0.263 | 0.019 | −0.001 | 0.002 | 0.630 | 0.385 | 0.309 | |

East (m) | 0.244 | 0.122 | 0.114 | 0.017 | 0.000 | 0.004 | 0.694 | 0.292 | 0.294 | |

Height (m) | 0.455 | 0.134 | 0.150 | 0.027 | −0.002 | −0.001 | 2.209 | 0.220 | 0.656 | |

Tj-2 | Roll (deg) | 0.852 | 0.122 | 0.123 | −0.856 | −0.005 | 0.004 | −0.299 | 0.075 | 0.665 |

Pitch (deg) | 0.667 | 0.102 | 0.105 | 0.667 | −0.001 | 0.001 | 0.928 | 0.128 | 0.105 | |

Heading(deg) | 0.309 | 0.201 | 0.205 | −0.224 | 0.000 | 0.000 | 0.995 | 0.133 | 1.097 | |

North (m) | 0.481 | 0.347 | 0.428 | 0.013 | 0.000 | 0.005 | 1.702 | 0.833 | 1.543 | |

East (m) | 1.043 | 0.823 | 1.021 | −0.084 | 0.000 | 0.001 | 0.485 | 0.323 | 0.535 | |

Height (m) | 0.319 | 0.114 | 0.138 | 0.033 | 0.001 | 0.002 | 1.696 | 0.233 | 0.579 | |

S: Smoother; CCNN-S: CCNN-Smoother; MFNN-S: MFNN-Smoother |

States | RMS | Median | 99-Percentile | |||||||

S | CCNN-S | MFNN-S | S | CCNN-S | MFNN-S | S | CCNN-S | MFNN-S | ||

Tj-1 | Roll(deg) | 0.932 | 0.080 | 0.394 | −0.931 | 0.002 | −0.003 | −0.369 | 0.129 | 0.123 |

Pitch(deg) | 0.534 | 0.045 | 0.047 | 0.547 | −0.002 | 0.001 | 0.711 | 0.048 | 0.207 | |

Heading(deg) | 0.413 | 0.103 | 0.180 | −0.222 | 0.005 | 0.001 | 0.753 | 0.519 | 0.606 | |

North (m) | 0.367 | 0.256 | 0.271 | 0.019 | 0.000 | 0.000 | 0.618 | 0.338 | 0.264 | |

East (m) | 0.273 | 0.173 | 0.165 | 0.016 | 0.000 | 0.002 | 0.673 | 0.445 | 0.466 | |

Height (m) | 0.470 | 0.175 | 0.164 | 0.027 | −0.001 | 0.001 | 2.250 | 0.227 | 0.764 | |

Tj-2 | Roll (deg) | 0.864 | 0.220 | 0.271 | −0.869 | −0.006 | 0.000 | −0.082 | 0.068 | 0.073 |

Pitch (deg) | 0.731 | 0.163 | 0.176 | 0.685 | −0.001 | 0.003 | 1.784 | 0.129 | 0.970 | |

Heading(deg) | 0.315 | 0.228 | 0.289 | −0.226 | 0.000 | −0.001 | 0.937 | 0.127 | 0.427 | |

North (m) | 0.937 | 0.526 | 0.662 | 0.015 | 0.000 | 0.004 | 1.765 | 0.779 | 0.960 | |

East (m) | 1.550 | 1.322 | 1.345 | −0.089 | 0.000 | 0.001 | 0.450 | 0.350 | 0.424 | |

Height (m) | 0.326 | 0.140 | 0.187 | 0.033 | 0.000 | 0.002 | 1.693 | 0.289 | 0.714 | |

S: Smoother; CCNN-S: CCNN-Smoother; MFNN-S: MFNN-Smoother |

The comparison between MFNN and CCNN methodologies.