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The purpose of this paper is to propose an error model for serial robot kinematic calibration based on dual quaternions.

The dual quaternions are the combination of dual-number theory and quaternion algebra, which means that they can represent spatial transformation. The dual quaternions can represent the screw displacement in a compact and efficient way, so that they are used for the kinematic analysis of serial robot. The error model proposed in this paper is derived from the forward kinematic equations via using dual quaternion algebra. The full pose measurements are considered to apply the error model to the serial robot by using Leica Geosystems Absolute Tracker (AT960) and tracker machine control (T-MAC) probe.

Two kinematic-parameter identification algorithms are derived from the proposed error model based on dual quaternions, and they can be used for serial robot calibration. The error model uses Denavit–Hartenberg (DH) notation in the kinematic analysis, so that it gives the intuitive geometrical meaning of the kinematic parameters. The absolute tracker system can measure the position and orientation of the end-effector (EE) simultaneously via using T-MAC.

The error model formulated by dual quaternion algebra contains all the basic geometrical parameters of serial robot during the kinematic calibration process. The vector of dual quaternion error can be used as an indicator to represent the trend of error change of robot’s EE between the nominal value and the actual value. The accuracy of the EE is improved after nearly 20 measurements in the experiment conduct on robot SDA5F. The simulation and experiment verify the effectiveness of the error model and the calibration algorithms.

To realize the smooth interpolation of orientation on robot end-effector, this paper aims to propose a novel algorithm based on the unit quaternion spline curve.

This algorithm combines the spherical linear quaternion interpolation and the cubic B-spline quaternion curve. With this method, a C2-continuous smooth trajectory of multiple teaching orientations is obtained. To achieve the visualization of quaternion curves on a unit sphere, a mapping algorithm between a unit quaternion and a point on the spherical surface is given based on the physical meaning of the unit quaternion.

Finally, the curvature analysis of a practical case shows that the orientation trajectory (OT) constructed by this algorithm satisfied the C2-continuity.

This OT satisfies the requirement of smooth interpolation among multiple orientations on robots in industrial applications.

The purpose of this paper is to propose a smooth double-spline interpolation method for six-degree-of-freedom rotational robot manipulators, achieving the global C2 continuity of the robot trajectory.

This paper presents a smooth double-spline interpolation method, achieving the global C2 continuity of the robot trajectory. The tool center positions and quaternion orientations are first fitted by a cubic B-spline curve and a quartic-polynomial-based quaternion spline curve, respectively. Then, a parameter synchronization model is proposed to realize the synchronous and smooth movement of the robot along the double spline curves. Finally, an extra u-s function is used to record the relationship between the B-spline parameter and its arc length parameter, which may reduce the feed rate fluctuation in interpolation. The seven segments jerk-limited feed rate profile is used to generate motion commands for algorithm validation.

The simulation and experimental results demonstrate that the proposed method is effective and can generate the global C2-continuity robot trajectory.

The main contributions of this paper are as follows: guarantee the C2 continuity of the position path and quaternion orientation path simultaneously; provide a parameter synchronization model to realize the synchronous and smooth movement of the robot along the double spline curves; and add an extra u-s function to realize arc length parameterization of the B-spline path, which may reduce the feed rate fluctuation in interpolation.

The purpose of this paper is to propose a high-precision attitude solution to solve the attitude drift problem caused by the dispersion of low-cost micro-electro-mechanical system devices in strap-down inertial navigation attitude solution of micro-quadrotor.

In this study, a three-stage attitude estimation scheme that combines data preprocessing, gyro drifts prediction and enhanced unscented Kalman filtering (UKF) is proposed. By introducing a preprocessing model, the quaternion orientation is calculated as the composition of two algebraic quaternions, and the decoupling feature of the two quaternions makes the roll and pitch components independent of magnetic interference. A novel real-time based on differential value (DV) estimation algorithm is proposed for gyro drift. This novel solution prevents the impact of quartic characteristics and uses the iterative method to meet the requirement of real-time applications. A novel attitude determination algorithm, the pre-process DV-UKF algorithm, is proposed in combination with UKF based on the above solution and its characteristics.

Compared to UKF, both simulation and experimental results demonstrate that the pre-process DV-UKF algorithm has higher reliability in attitude determination. The dynamic errors in the three directions of the attitude are below 2.0°, the static errors are all less than 0.2° and the absolute attitude errors tailored by average are about 47.98% compared to the UKF.

This paper fulfils an identified need to achieve high-precision attitude estimation when using low-cost inertial devices in micro-quadrotor. The accuracy of the pre-process DV-UKF algorithm is superior to other products in the market.

The purpose of this paper is to employ an extended Kalman filter for implementing an AHRS (attitude and heading reference system) with acceleration compensation, thereby improving the reliability of such systems, since this removes the usual restrictive assumption that the vehicle is undergoing a non‐accelerated maneuver.

MARG (magnetic, acceleration and rate gyros) sensors constitute the basic hardware, which are integrated by the Kalman filter. The error dynamics for attitude and gyro biases is obtained in the navigation frame, providing a much simpler approach than usually taken in the literature, since it relies on direct quaternion differentiation. The state vector associated to the error dynamics possesses six components: three are associated to the quaternion error and three concern gyro bias estimates.

The AHRS is implemented in an ARM (Advanced RISC Machine) processor and tested with experimental data. The accelerated case is treated by two complementary approaches: by changing the noise variance in the Kalman filter, and by obtaining an acceleration information from GPS (global positioning system) velocity measurements. Experimental results are presented and the performance is compared with commercial ARHS systems.

The proposed AHRS can be implemented with low cost MARG sensors, and GPS aiding, with use for instance in UAV (unmanned aerial vehicle) and small aircrafts' attitude estimation, for navigation and control applications.

Usually the AHRS designs employ as states total gyro bias and Euler angles, or quaternion, and do not consider the accelerated case. Here the state is comprised by gyro bias and quaternion error variables, which attenuates the effect of nonlinearities, and two complementary procedures tackle the accelerated case: acceleration correction by using a GPS derived acceleration signal and change in the output noise covariance used by the Kalman filter.

The purpose of this article is to improve detection of common movement. Detecting if two or multiple devices are moved together is an interesting problem for different applications. However, these devices may be aligned arbitrarily with regards to each other, and the three dimensions sampled by their respective local accelerometers can therefore not be directly compared. The typical approach is to ignore all angular components and only compare overall acceleration magnitudes – with the obvious disadvantage of discarding potentially useful information.

This paper contributes a method to analytically determine relative spatial alignment of two devices based on their acceleration time series. The method uses quaternions to compute the optimal rotation with regards to minimizing the mean squared error.

Based on real-world experimental data from smartphones and smartwatches shaken together, the paper demonstrates the effectiveness of the method with a magnitude squared coherence metric, for which an improved equal error rate (EER) of 0.16 (when using derotation) over an EER of 0.18 (when not using derotation) is shown.

After derotation, the reference system of one device can be (locally and independently) aligned with the other, and thus all three dimensions can consequently be compared for more accurate classification.

Without derotating time series, angular information cannot be used for deciding if devices have been moved together. To the best of the authors ' knowledge, this is the first analytic approach to find the optimal derotation of the coordinate systems, given only the two 3D time acceleration series of devices (supposedly) moved together. It can be used as the basis for further research on improved classification toward acceleration-based device pairing.

Though the problem of resolving translational motion in particle methods is a relatively straightforward task, the complications of resolving rotational motion are non‐trivial. Many molecular dynamics and non‐deformable discrete element applications employ an explicit integration for resolving orientation, often involving products of matrices, which have well‐known drawbacks. The purpose of this paper is to investigate commonly used algorithms for resolving rotational motion and describe the application of quaternion‐based approaches to discrete element method simulations.

Existing algorithms are compared against a quaternion‐based reparameterization of both the central difference algorithm and the approach of Munjiza et al. for finite/discrete element modeling (FEM/DEM) applications for the case of torque‐free precession.

The resultant algorithms provide not only guaranteed orthonormality of the resulting rotation but also allow assumptions of small‐angle rotation to be relaxed and the use of a more accurate Taylor expansion instead.

The approaches described in this paper balance ease of implementation within existing explicit codes with computational efficiency and accuracy appropriate to the order of error in many discrete element method simulations.

An introduction to air data attitude heading reference systems (ADAHRS) is presented along with the developments and discussion of software algorithms used in such systems. Both Kalman filtering and fuzzy logic adaptive signal processing are presented and discussed. Quaternions are used in this paper as the basis for the discussion of both algorithms. Flight results from Archangel Systems AHR150 ADAHRS are also presented and discussed.

The purpose of this paper is to propose a decentralized output feedback controller for cooperative attitude regulation of spacecraft formation in absence of angular velocity feedback.

The nonlinear relative attitude dynamic and kinematic equations represented by relative quaternion and relative angular velocity, respectively, are considered in this paper. The lead filter is employed to synthesize virtual angular velocity signal so that the design of output feedback controller is achieved. Lyapunov method is adopted to prove the stability of closed‐loop system. Considering the external disturbance, the theory of L2‐gain disturbance attenuation is employed to improve the designed controller. Numerical simulations are carried out to verify the controllers proposed.

It is found that the closed‐loop system can be guaranteed asymptotically stable in absence of external disturbance. When disturbance is considered, as long as the sufficient condition proposed is satisfied, the improved controller can render system uniformly ultimately bounded stable.

The proposed output feedback control scheme can be considered as a fall‐back alternative for the case that the angular velocity sensors fail, or seen as another option for the system without angular velocity sensors at all.

Unlike most classical works in the field of output feedback which focus on centralized scheme and neglect the disturbance, the controller proposed in this paper is able to handle the output feedback control problem of multi‐agent formation in a decentralized fashion, so as to avoid the single failure point of a centralized scheme. Meanwhile, the capability of L2‐gain disturbance attenuation is also achieved simultaneously.

Considers asymptotic distributions of likelihood ratio criteria for two testing problems: a covariance matrix with complex structure has quaternion structure; and a covariance matrix with quaternion structure has complex structure. In each problem, the likelihood ratio criteria and their asymptotic distributions up to first order of approximation are obtained for the class of quaternion elliptical distribution with kurtosis parameters K under the alternatives close to the null hypothesis.