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Quadruped robots possess significant mobility in complex and uneven terrains due to their outstanding stability and flexibility, making them highly suitable in rescue missions, environmental monitoring, and smart agriculture. With the increasing use of quadruped robots in more demanding scenarios, ensuring accurate and stable state estimation in complex environments has become particularly important. Existing state estimation algorithms relying on multi-sensor fusion, such as those using IMU, LiDAR, and visual data, often face challenges on non-stationary terrains due to issues like foot-end slippage or unstable contact, leading to significant state drift. To tackle this problem, this paper introduces a state estimation algorithm that integrates an invariant extended Kalman filter (InEKF) with a disturbance observer, aiming to estimate the motion state of quadruped robots on non-stationary terrains. Firstly, foot-end slippage is modeled as a deviation in body velocity and explicitly included in the state equations, allowing for a more precise representation of how slippage affects the state. Secondly, the state update process integrates both foot-end velocity and position observations to improve the overall accuracy and comprehensiveness of the estimation. Lastly, a foot-end contact probability model, coupled with an adaptive covariance adjustment strategy, is employed to dynamically modulate the influence of the observations. These enhancements significantly improve the filter’s robustness and the accuracy of state estimation in non-stationary terrain scenarios. Experiments conducted with the Jueying Mini quadruped robot on various non-stationary terrains show that the enhanced InEKF method offers notable advantages over traditional filters in compensating for foot-end slippage and adapting to different terrains.

This research used an invariant extended Kalman filter (IEKF) for the navigation of an unmanned aerial vehicle (UAV), and compared the properties and performance of this IEKF with those of an open-source navigation method based on an extended Kalman filter (EKF). The IEKF is a fairly new variant of the EKF, and its properties have been verified theoretically and through simulations and experiments. This study investigated its performance using a practical implementation and examined its distinctive features compared to the previous EKF-based approach. The test used two different types of UAVs: rotary wing and fixed wing. The method uses sensor measurements of the location and velocity from a GPS receiver; the acceleration, angular rate, and magnetic field from a microelectromechanical system-attitude heading reference system (MEMS-AHRS); and the altitude from a barometric sensor. Through flight tests, the estimated state variables and internal parameters such as the Kalman gain, state error covariance, and measurement innovation for the IEKF method and EKF-based method were compared. The estimated states and internal parameters showed that the IEKF method was more stable and convergent than the EKF-based method, although the estimated locations, velocities, and altitudes of the two methods were comparable.

This paper presents a learning-assisted approach for state estimation of quadruped robots using observations of proprioceptive sensors, including multiple inertial measurement units (IMUs). Specifically, one body IMU and four additional IMUs attached to each calf link of the robot are used for sensing the dynamics of the body and legs, in addition to joint encoders. The extended Kalman filter (KF) is employed to fuse sensor data to estimate the robot’s states in the world frame and enhance the convergence of the extended KF (EKF). To circumvent the requirements for the measurements from the motion capture (mocap) system or other vision systems, the right-invariant EKF (RI-EKF) is extended to employ the foot IMU measurements for enhanced state estimation, and a learning-based approach is presented to estimate the vision system measurements for the EKF. One-dimensional convolutional neural networks (CNN) are leveraged to estimate required measurements using only the available proprioception data. Experiments on real data from a quadruped robot demonstrate that proprioception can be sufficient for state estimation. The proposed learning-assisted approach, which does not rely on data from vision systems, achieves competitive accuracy compared to EKF using mocap measurements and lower estimation errors than RI-EKF using multi-IMU measurements.

Smartphone-based vehicle navigation has become the primary choice in everyday life due to low cost and real-time traffic updates. However, for vehicle navigation applications in the Global Navigation Satellite System (GNSS)-denied scenario such as parking lot and tunnel, it is quite difficult to maintain robust and continuous positioning based on consumer-grade sensors. In this paper, a novel method is proposed for accurate vehicular dead-reckoning based only on a smartphone inertial measurement unit. Robust vehicle dead reckoning can improve positioning performance in GNSS-degraded areas or where high-precision positioning sources are available at low-frequencies. The key components of the method are a Kalman filter with data-driven parameters adapter and a deep neural network that provides data-driven measurement estimation. A combined convolutional neural network and gated recurrent unit deep learning network, termed AVNet, is proposed to estimate the attitude and velocity of the vehicle. The learned measurements are integrated into an invariant Kalman filter to estimate Three-Dimensional (3D) attitude, velocity and position. The method was tested on custom datasets collected in a parking lot, and a 0.4 % relative horizontal translation error was achieved on average.

Indoor localization for pedestrians, which relies solely on inertial odometry, has been a topic of great interest. Its significance lies in its ability to provide positioning solutions independently, without the need for external data. Although traditional strap-down inertial navigation shows rapid drift, the introduction of pedestrian dead reckoning (PDR), and artificial intelligence (AI) has enhanced the applicability of inertial odometry for indoor localization. However, inertial odometry continues to be affected by drift, inherent to the nature of dead reckoning. This implies that even a slight error at a given moment can lead to a significant decrease in accuracy after continuous integration operations. In this paper, we propose a novel approach aimed at enhancing the positioning accuracy of inertial odometry. Firstly, we derive a learning-based forward speed using inertial measurements from a smartphone. Unlike mainstream methods where the learned speed is directly used to determine the position, we use the forward speed combined with non-holonomic constraint (NHC) as a measurement to update the state predicted within a strap-down inertial navigation framework. Secondly, we employ an invariant extended Kalman filter (IEKF)-based state estimation to facilitate fusion to cope with the nonlinearity arising from the system and measurement model. Experimental tests are carried out in different scenarios using an iPhone 12, and traditional methods, including PDR, robust neural inertial navigation (RONIN), and the EKF-based method, are compared. The results suggest that the method we propose surpasses these traditional methods in performance.

Underwater vehicles heavily depend on the integration of inertial navigation with Doppler Velocity Log (DVL) for fusion-based localization. Given the constraints imposed by sensor costs, ensuring the optimization ability and robustness of fusion algorithms is of paramount importance. While filtering-based techniques such as Extended Kalman Filter (EKF) offer mature solutions to nonlinear problems, their reliance on linearization approximation may compromise final accuracy. Recently, Invariant EKF (IEKF) methods based on the concept of smooth manifolds have emerged to address this limitation. However, the optimization by matrix Lie groups must satisfy the “group affine” property to ensure state independence, which constrains the applicability of IEKF to high-precision positioning of underwater multi-sensor fusion. In this study, an alternative state-independent underwater fusion invariant filtering approach based on a two-frame group utilizing DVL, Inertial Measurement Unit (IMU), and Earth-Centered Earth-Fixed (ECEF) configuration is proposed. This methodology circumvents the necessity for group affine in the presence of biases. We account for inertial biases and DVL pole-arm effects, achieving convergence in an imperfect IEKF by either fixed observation or body observation information. Through simulations and real datasets that are time-synchronized, we demonstrate the effectiveness and robustness of the proposed algorithm.

Land vehicles need high-precision navigational systems in which multi-sensor integration may be provided. Moreover, land vehicles regularly use Global Navigation Satellite Systems (GNSS) to estimate their position. Unfortunately, several locations, such as tunnels and inside parking garages, where GNSS signals cannot be detected. Several types of research have been conducted to improve positioning information using multi-sensor integration. Then, the vehicle needs another system for finding its location in GNSS-denied conditions, such as Inertial Navigation System (INS). Despite the accuracy of INS in short-time period use, inertial navigation systems (INS) are liable to drifts of their positioning solution due to the inertial sensor errors that are inherent to them; therefore, this problem leads to errors accumulation over time then integration techniques are used to eliminate the resulting errors. Moreover, many filters are used in the process of integration, such as the Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF), Particular Filter (PF) and Invariant Extended Kalman Filter (IEKF). Moreover, this work introduces the left-invariant extended Kalman filter (LIEKF) as a navigation filter for a loosely coupled integration to eliminate positioning errors. Furthermore, the LIEKF is based on the symmetry-preserving observer theory, which claims that the estimation error depends on the theory of a Lie group matrix, and the proposed system INS/GPS-based LIEKF converges to constant values, unlike the traditional INS/GPS. Moreover, the proposed system INS/GPS-based LIEKF depends on State-estimate-independent Jacobians, and the LIEKF is more efficient and has better performance due to results such as the 2D position RMS error due to the INS/GPS-based EKF is 19.43 m . However, the 2D position RMS error due to the INS/GPS-based LIEKF is 3.32 m with 83% improvement. Moreover, the 2D position errors were enhanced using the INS/GPS-based LIEKF system compared to the INS/GPS-based EKF system.

The research on state estimation for quadruped robots is critical. Its result passed to motion controller makes the robot navigate autonomously and adjust the gait to a more stable motion. The current research depends on a multi-sensor fusion of cameras, lidars or other proprioceptive sensors, such as Inertial Measurement Unit (IMU) and encoders. The high-frequency data are generally derived from body sensors, which is to be fused with data from external sensors directly, or preprocessed with EKF first. Due to its unguaranteed convergence and robustness of tracking state mutations, EKF is insufficient. Therefore, we study state estimation for hydraulic quadruped robot based on the fusion of IMU measurement and leg odometry in this paper, and Invariant Extended Kalman Filter (IEKF) is successfully applied to quadruped robots by using this method. Besides, neural networks are utilized to train the weight functions of foot force and the state of leg odometry, and our trained functions improve the accuracy of observation compared with common weight average methods. Finally, our experiments of accuracy show that the root mean square error of our method is significantly reduced and the absolute trajectory error is reduced by 30% compared to traditional IEKF. The algorithm achieves the drift per distance travelled below 4 cm/m. Moreover, it has good robustness.

An application of moving-horizon strategy for nonlinear systems with possible outliers in measurements is addressed. With the increased success of movinghorizon strategy in the state estimation for linear systems with outliers acting on the measurement, investigating the nonlinear approach is highly required. In this paper we applied the nonlinear version which has been presented in the literature in term of discrete-time linear time-invariant systems, where the applied strategy considers minimizing a least-squares functions in which each measure possibly contaminated by outlier is left out in turn and the lowest cost is propagated. The moving horizon filter effectiveness as compared with the extended Kalman filter is shown by means of simulation example and estimation error comparison. The moving horizon filter shows the feature of resisting outliers with robust estimation.

In this paper, a novel navigation method with the assistance of a vehicle dynamic model (VDM), known as the VDM-aided navigation method, is introduced. This method is specifically designed for a subset of fixed-wing aerial robots within the broader category of unmanned aerial vehicles. Vision-based navigation (VBN) is employed to increase accuracy while maintaining reliability in Global Navigation Satellite System (GNSS) outages. In addition, an unscented Kalman filter (UKF) is used to estimate navigation parameters, including speed, position and attitude. This method uses the dynamic system as a process model and employs VBN, barometric altitude and vertical gyro as measurement inputs. In VBN, the method of scale-invariant feature transform is used as a method for image matching. To ensure the real-time capability of this method with the existing microprocessor, a hardware-in-the-loop (HIL) laboratory has been utilized. According to nonlinear observability methods, one can show the proposed integrated nonlinear navigation is observable under all conditions. Finally, the results of the HIL laboratory demonstrate that the proposed approach can estimate the robot navigation parameters with an acceptable level of precision even in the absence of an Inertial Navigation System (INS) and GNSS. It was validated even when there was an error of up to 20% in VDM parameters. Furthermore, an investigation was carried out regarding the use of Extended Kalman Filter instead of the UKF for the integrated navigation output. In GNSS outage conditions, considering both accuracy and cost, this method can serve as a valuable alternative for aerial robots. In addition, this approach can be recommended for INS fault detection with or without GNSS. Additionally, the integrated navigation provided can substitute the GNSS/INS system during fault conditions.