The degree of freedom (DOF) refers to the number of axes utilized to represent pose changes. In the context of IO, 3-DOF IO estimates the relative position changes in a 2D plane, as shown in Figure 1. During unit time intervals, the collected IMU data are fed to a 3-DOF IO model, which outputs the distance and heading. These resultant values can be represented as 2D vectors and are referred to as relative position changes, which are added to the previous location, indicating the current position. [26] represented the pioneering learning-based IO approach. The method utilizes a support vector machine to classify motions captured by IMUs, including standing, walking, and turning, and subsequently regress pose vectors. The pose vectors are integrated to generate trajectories in a 2D plane. IO-Net introduced the Oxford inertial odometry dataset (OxIOD), which comprises various types of inertial data. It estimates trajectories in a 2D plane by feeding 2-second-long accelerations and angular velocities into two layers of long short-term memory (LSTM) networks [27][28]. This was the first approach to apply deep neural networks to inertial data and generate trajectories by utilizing raw data. Furthermore, [29] proposed an approach to alleviate the computational burden of recurrent neural networks by introducing a lightweight variant of LSTM, called WaveNet. [30] introduced a loss function for orientation, orientation change, and velocity in the reference coordinate system to enhance velocity estimation performance. However, these methods are limited in that, similar to PDR, they can only generate trajectories in a 2D plane.

Figure 1: 
Framework of a 3-DOF IO

Six-DOF IO represents the change from the initial pose of the platform in 3D space, as shown in Figure 2. The pose includes the position, which can be expressed by utilizing the x-axis, y-axis, and z-axis coordinates, and orientation, represented by the roll, pitch, and yaw angles. The position and orientation changes in the body frame represent the changes in pose over a unit of time. These pose changes in the body frame can be transformed into a 3D translation vector in the reference coordinate system by considering the previous pose. The current pose can be expressed as the sum of a 3D translation vector from the starting point. Compared with 3-DOF IO, 6-DOF IO utilizes three additional coordinate representations, resulting in a considerable increase in drift error owing to the additional orientation computations required. The initial 6-DOF IONet employed a neural network composed of convolutional layers and LSTM to estimate 3D pose changes by inputting linear acceleration and gyro data [31]. These 3D pose changes, when combined with the initial pose, can represent a 3D trajectory. However, when utilizing this method, the error in the estimated orientation diverged with an increase in the estimation time. Therefore, a method to stabilize the orientation change estimation performance was required. The extended IONet proposed incorporating additional inputs of gravitational acceleration and geomagnetic data, which are directly related to orientation, as well as introducing a network to extract multidimensional features from gravitational acceleration [32]. The approach significantly reduced trajectory and orientation errors. However, orientation still diverges as the estimation time increases. DO IONet overcomes these limitations by directly estimating the orientation in the reference coordinate system [33]. It does not require an initial orientation and avoids drift errors because only the orientation values are utilized. Consequently, it exhibits consistent 3D translation vector estimation performance, even with longer estimation periods.

Figure 2: 
Framework of a 6-DOF IO

As shown in Figure 3, the DO IONet framework comprised an encoder to extract local features and a decoder for sequential data processing. The encoder aimed to reduce the complexity of inertial data and prevent overfitting. The decoder extracted the complete pattern by incorporating temporal dependencies into the features. It served as a connector that linked each time-step feature to regress to a pose.

Figure 3: 
Overview of the DO IONet framework

The direct orientation IO framework overcame the drift error associated with orientation estimation and accurately estimated position change and orientation in the reference coordinate system. This framework, similar to the DO IONet, comprised an encoder to extract local features and a decoder for sequential data processing. The encoder decreased the complexity of the inertial data and prevented over-fitting, whereas the decoder captured temporal dependencies in the features, enabling them to effectively regress to a relative position for each time step.

The architecture of the proposed network, which inputs four types of inertial data in a time-series format and outputs position changes and orientation is shown in Figure 4. The four types of inertial data include linear acceleration (a), angular velocity (w), gravitational acceleration (g), and geomagnetic value (m). One set of input data contained 200 inertial data points measured over a duration of 2 s. The position change △p was the velocity between the 95th and 105th data points within the body frame. q was the orientation at the 95th frame in the reference coordinate system, and quaternions were utilized instead of Euler angles, which had limited angular representation ranges, for learning convenience. The proposed network comprised a convolutional block as an encoder, a transformer block, and fully connected layers as decoders.

Figure 4: 
Structure of the extended DO IONet

The convolutional block comprised two convolutional layers and one max-pooling layer. It compressed the local features of inertial data to reduce complexity. Each compressed feature was concatenated and combined with a positional vector for sequential data processing in the transformer block. The transformer block comprised two layers, each of which employed multi-head attention and a feedforward network, as shown in Figure 5. The transformer captured the relative importance and context of the compressed features and functioned as a decoder, connecting each time-step feature for regression. Finally, △p and q were output by utilizing a fully connected layer. Subsequently, q, transformed by the rotation matrix along with △p , was multiplied to generate the trajectory over a unit of time, which is the 3D translation vector.

Figure 5: 
Transformer structure

The current position p_t, previous position p_t-1, orientation R(q_t-1), and position change △p were utilized to represent the pose changes in the reference coordinate system. △p indicates the value in the body frame. The 3D translation vector was determined by multiplying the previous orientation with △p. Orientation calculations adopted a unit quaternion because the network could not learn by utilizing Euler angles. The current position can be calculated as follows:

where p_t represents the current position, which is the sum of the previous position and the 3D translation vector. It provides a concise pose representation and structurally reduces drift errors by utilizing the DO IONet framework, which directly estimates the orientation.

The 6-DOF IONet designed a loss function for independent position values along the three axes and dependent direction values for the four quaternion vectors. The consideration of the homoscedastic uncertainty for each task is necessary due to the different scales of position and orientation. Therefore, a multitask loss was adopted to learn various quantities at different scales [34].

where σ_i and L_i denote the variance and loss functions of the i^th, respectively. To prevent a division by zero within the total loss, the utilization of exponential mapping removed constraints on scalar values, and the incorporation of log variance enhanced network stability.

Two loss functions were utilized, as expressed in Equation (3).

where L_DPMAE and L_QME represent the delta position mean absolute error and quaternion multiplicative error, respectively.

where L_DPMAE and L_QME denote the loss functions associated with the displacement of the sensor unit within the body frame and the orientation within the reference coordinate system, respectively. △imag(q) resulted in a complex number being treated as a real number. Meanwhile, the ⊗ represents the Hamilton product operator, signifying the addition of two angles that indicate the complex conjugate of q.

Experiments were conducted by utilizing OxIOD, which provides four types of inertial data. Inertial data from OxIOD were captured by utilizing an iPhone 7 and the ground truth for pose was established by utilizing the Vicon motion capture system [27]. OxIOD offers four different form factors based on how the device is transported: handheld, pocket, handbag, and trolley. We utilized the handheld case in our experiments, which provided the longest data collection time and distance. The handheld case comprised 24 sequences. We discarded the initial 12 s and the final 3 s of each sequence to eliminate errors in the Vicon motion capture system and facilitate a clear comparison with existing models. Furthermore, during OxIOD training, we mitigated an error of approximately 1% that occurred when the standard deviation of the difference between consecutive frames and orientations reached 20°. In testing, data with errors were incorporated for comparison with other models. In total, 54,885 training samples were employed.

The core libraries utilized for training included Tensorflow 2.4.1, Keras 2.4.3, tfquaternion 0.1.6, and numpy-quaternion 2022.4.2. We employed an Adam optimizer with a learning rate of 0.0001 to converge the loss function. The batch size was set to 32 and training ran for 500 epochs. A validation data ratio of 10% was utilized to monitor the training progress. Our model was trained by utilizing an NVIDIA RTX 6000 GPU.

Data were divided into 17 training and 7 testing datasets. The model was evaluated from various perspectives by utilizing short- and long-term data. The evaluation criteria included the trajectory, orientation, and 3D translation vectors. For short-term data evaluation, time intervals from 0 s to 20 s and from 100 s to 120 s were utilized for each test data sequence. For long-term data evaluation, the time intervals ranged from 0 s to 100 s and from 100 s to 200 s for each sequence. Although the trajectory provided insights into the performance across various instances, it contained cumulative errors from continuous integration, making it less suitable to evaluate the performance of the model over a specific unit of time. Therefore, orientation and 3D translation vectors were utilized as evaluation metrics for the performance assessment per unit of time. Two existing approaches to estimate position and orientation changes were evaluated and compared with the DO IONet framework. For the DO IONet experiments, eight evaluations were performed based on different encoder and decoder configurations. Three evaluations were performed on the models designed by utilizing stacking transformers. Overall, the evaluations encompassed a comprehensive analysis of various approaches to estimate the position and orientation changes, including two existing methods and multiple configurations of the DO IONet framework.

Short-term evaluations were performed by utilizing data from 0–20 s and 100–120 s. The average trajectory, orientation, and 3D translation root-mean-square errors (RMSEs) at these two time points are listed in Table 1. The models that estimated the trajectory and 3D translation vector exhibited superior performance in the estimation of position and orientation change. However, for the orientation, the extended DO IONet with one convolutional block and two transformers outperformed the others.

Experiments with different numbers of convolutional blocks were performed to reduce the feature size input to the decoder that processed the time-series data. Based on the results, the model performed better when only one convolutional block was utilized, regardless of the decoder type. This is because the pooling layer within the convolutional block could omit essential details in the feature representation. Furthermore, experiments were performed with IONet models comprising only stacked transformers without a separate encoder. The performance improved as the number of stacked transformers increased. A model with three layers of transformers demonstrated a performance close to that of the extended DO IONet.

The long-term evaluation results for each model are listed in Table 2. The RMSE of the trajectory included errors from previous time points as the estimation time increased, resulting in divergence for all IONet models. The orientation RMSE of existing frameworks diverged because they relied on the integration of the initial values to output the final pose. However, the models that utilized the DO IONet framework exhibited values similar to those of the short-term results. Consequently, based on the 3D translation vector, calculated as the product of the position change and orientation, models that utilized the novel framework with accurate orientation values generally performed better.

In the case of the encoder, as demonstrated by the previous results, missing features and higher errors resulted from the stacking of additional convolutional blocks. Some models that only comprised transformers experienced increased errors despite being more stacked. In the long-term evaluation, the 9-axis IONet exhibited the lowest trajectory RMSE. However, the RMSE of the extended DO IONet orientation was approximately 73% and 11% lower than that of the 9-axis IONet and DO IONet, respectively. Furthermore, the RMSE of the extended DO IONet 3D translation vector was approximately 53% and 5% lower than that of the 9-axis IONet and DO IONet, respectively.

The existing IONet and proposed DO IONet frameworks exhibited different characteristics. The existing frameworks continuously integrated changes from the initial position and orientation, resulting in relatively small trajectory estimation errors in the short term. However, the 3D translation vector diverged owing to uncorrectable orientation errors after several tens of seconds. Therefore, although existing frameworks were utilized for short-term trajectory estimation with a single IMU, they did not fully exploit the advantages of IMUs in small-scale odometry for orientation estimation. Conversely, the DO IONet framework also estimated position changes; however, the orientation was output in the reference coordinate system. This enabled relative position estimation regardless of the current orientation, provided the initial position was known. Consequently, it can be utilized as part of a sensor-fusion odometry system or as a complementary metric for the current pose. Therefore, the DO IONet framework offered high scalability and flexibility, making it effective for various practical applications.

We conducted experiments to determine the optimal design for the encoder and decoder within the DO IONet framework. For the encoder, we investigated performance based on the number of convolutional blocks, with each block comprising two convolutional layers. Models with only one convolutional block outperformed those with two blocks, indicating that excessive convolutional layers could diminish sequential features. We examined the performance of different decoders, including LSTM, GRU, one transformer, and two transformers. Models with transformer blocks, commonly utilized for feature extraction in time-series data, demonstrated improved performance regardless of the estimation time. This suggests that transformers can be stacked to enhance performance and serve as a viable alternative to recurrent networks. Finally, we examined performance based on the number of transformers to determine whether the performance improved by increasing the number of stacked transformers. Models with two transformer blocks outperformed those with only one, whereas the performance difference between models with three stacked transformers and those with two stacked transformers was minimal.