Inertial Vector Research Articles

Angular and linear velocity observer design using vector and landmark measurements

AbstractThis paper presents an approach for designing angular and linear velocity observers for rigid bodies using two inertial vectors and one landmark measurements directly. Compared with the classical velocity estimation algorithms, the proposed observer does not need to reconstruct the pose information from measurements nor be coupled with velocity sensors. The linear‐time varying dynamics of the estimation error equation are analyzed, rigorously showing the uniform local exponential stability of the observer. Simulations are conducted to illustrate the significant performance of the proposed method.

Fault-tolerant attitude coordination control for multiple flexible spacecraft using vector measurements

This paper addresses the control problems of distributed attitude coordination for multiple flexible spacecraft based on the feedback of inertial vector measurements. It is assumed that the angular velocity information is not available, and the modal variable cannot be measured. The orientation error model with vector measurements is introduced. And then the distributed attitude synchronization control scheme is presented without actuator faults or disturbances. It is guaranteed that the desired equilibrium is locally asymptotically stable and the undesired equilibrium is unstable. Meanwhile, the domain of attraction is given by inequality constraints. In the case that there exist actuator faults and external disturbances, a robust fault-tolerant control strategy is developed to realize attitude synchronization. The hyperbolic tangent function in place of sign function is adopted to eliminate external disturbances. Meanwhile, the adaptive laws are proposed to estimate actuator faults. The salient feature of these control approaches is that no modal variable observers are utilized and the control schemes are robust to the inertia uncertainty of spacecraft. Finally, numerical simulations are implemented to manifest the validity of the presented control schemes.

Uniting Attitude Estimation With Global Asymptotic Stability

This paper approaches the problem of attitude estimation using biased gyro and body-referenced measurements of inertial vectors. First, a new observer for attitude estimation is proposed through introducing a family of synergistic potential functions with explicit expression of synergistic gap. Furthermore, it is convenient to adjust the design by adding additional state-dependent gains without adapting the switching rules. The proposed observer exhibits faster convergence for large attitude estimation errors than similar work in literature. Second, to achieve better steady-state performance, we unite this global observer with a traditional local observer, i.e., multiplicative extended Kalman filter. The new uniting observer inherits the desired transient and steady-state performance from the two sub-observers respectively. The performance of proposed methods has been evaluated via simulation and experiments.

Asymptotically stable optimal multi-rate rigid body attitude estimation based on lagrange-d'alembert principle

<abstract><p>The rigid body attitude estimation problem is treated using the discrete-time Lagrange-d'Alembert principle. Three different possibilities are considered for the multi-rate relation between angular velocity measurements and direction vector measurements for attitude: 1) integer relation between sampling rates, 2) time-varying sampling rates, 3) non-integer relation between sampling rates. In all cases, it is assumed that angular velocity measurements are sampled at a higher rate compared to the inertial vectors. The attitude determination problem from two or more vector measurements in the body-fixed frame is formulated as Wahba's problem. At instants when direction vector measurements are absent, a discrete-time model for attitude kinematics is used to propagate past measurements. A discrete-time Lagrangian is constructed as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential energy-like term obtained from Wahba's cost function. An additional dissipation term is introduced and the discrete-time Lagrange-d'Alembert principle is applied to the Lagrangian with this dissipation to obtain an optimal filtering scheme. A discrete-time Lyapunov analysis is carried out to show that the optimal filtering scheme is asymptotically stable in the absence of measurement noise and the domain of convergence is almost global. For a realistic evaluation of the scheme, numerical experiments are conducted with inputs corrupted by bounded measurement noise. These numerical simulations exhibit convergence of the estimated states to a bounded neighborhood of the actual states.</p></abstract>

Earth Velocity and Rigid-Body Attitude Estimation on SO(3) Using Biased Measurements

This article addresses the problem of simultaneously estimating: 1) the attitude of a rigid body on the special orthogonal group of order 3 ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {SO}(3)$</tex-math></inline-formula> ); 2) the Earth’s spin vector expressed in body-fixed coordinates; and 3) a pair of time-varying sensor measurement biases. The available sensor readings include biased angular velocity measurements collected from a set of high-grade rate gyros sensitive to the rotation of the planet and biased vector observations of some known inertial reference field. The estimation solution consists of a Lagrangian-based pseudo Kalman filtering routine, which is partially constrained to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {SO}(3)$</tex-math></inline-formula> vis-á-vis the attitude. Most noticeably, the geometric relationship between the Earth’s angular velocity and the inertial field is leveraged to relax common practical, as well as theoretical, demands closely linked to conventional attitude estimation solutions, namely, the need for explicit body-fixed measurements of two inertial vector fields. The proposed algorithm is validated through a set of realisticsimulations. Experimental results are also included to further demonstrate the performance of this strategy in a real-world application.

Using a source of opportunity for self-localization of mobile underwater vector sensor platforms

Transforming a network of few compact mobile underwater platforms, each equipped with a single acoustic sensor, into a distributed sensing array requires precise positioning of each mobile sensor. But conventional accurate underwater positioning tools rely on active acoustic sources (e.g., acoustic pingers), which imposes additional hardware and operational complexity. Hence self-localization (i.e., totally passive) methods using only acoustic sources of opportunity (such as surface vessels) for locating the mobile sensors of a distributed array appear as an attractive alternative. Existing underwater self-localization methods have mainly been developed for mobile platform equipped with time-synchronized hydrophones and rely only on the time-difference of arrivals between multiple pairwise combinations of the mobile hydrophones as inputs for a complex non-linear inversion procedure. Instead, we introduce a self-localization method, which uses a linear least square formulation, for two mobile time-synchronized vector sensor platforms based on their acoustic recordings of a distant surface vessel and their inertial navigation systems (INS) measurements. This method can be generalized to multiple vector sensor pairs to provide additional robustness towards input parameter errors (e.g., due to a faulty INS) as demonstrated experimentally using drifting buoys with inertial vector sensors deployed ∼100 m apart in shallow water.

Self-localization of mobile underwater vector sensor platforms using a source of opportunity.

Using a network of a few compact mobile underwater platforms, each equipped with a single acoustic sensor, as a distributed sensing array is attractive but requires precise positioning of each mobile sensor. However, traditional accurate underwater positioning tools rely on active acoustic sources (e.g., acoustic pingers), which implies additional hardware and operational complexity. Hence, self-localization (i.e., totally passive) methods using only acoustic sources of opportunity (such as surface vessels) for locating the mobile sensors of a distributed array appear as a simpler alternative. Existing underwater self-localization methods have mainly been developed for mobile platforms equipped with time-synchronized hydrophones and rely only on the time-differences of arrival between multiple pairwise combinations of the mobile hydrophones as inputs for a complex non-linear inversion procedure. Instead, this article introduces a self-localization method, which uses a linear least-square formulation, for two mobile time-synchronized vector sensor platforms based on their acoustic recordings of a distant surface vessel and their inertial navigation system (INS) measurements. This method can be generalized to multiple vector sensor pairs to provide additional robustness toward input parameter errors (e.g., due to a faulty INS) as demonstrated experimentally using drifting buoys with inertial vector sensors deployed ∼100 m apart in shallow water.

Analysis and control of a Stewart platform as base motion compensators - Part I: Kinematics using moving frames

Kinematics and its control application are presented for a Stewart platform whose base plate is installed on a floor in a moving ship or a vehicle. With a manipulator or a sensitive equipment mounted on the top plate, a Stewart platform is utilized to mitigate the undesirable motion of its base plate by controlling actuated translational joints on six legs. To reveal closed loops, a directed graph is utilized to express the joint connections. Then, kinematics begins by attaching an orthonormal coordinate system to each body at its center of mass and to each joint to define moving coordinate frames. Using the moving frames, each body in the configuration space is represented by an inertial position vector of its center of mass in the three-dimensional vector space ℝ3, and a rotation matrix of the body-attached coordinate axes. The set of differentiable rotation matrices forms a Lie group: the special orthogonal group, SO(3). The connections of body-attached moving frames are mathematically expressed by using frame connection matrices, which belong to another Lie group: the special Euclidean group, SE(3). The employment of SO(3) and SE(3) facilitates effective matrix computations of velocities of body-attached coordinate frames. Loop closure constrains are expressed in matrix form and solved analytically for inverse kinematics. Finally, experimental results of an inverse kinematics control are presented for a scale model of a base-moving Stewart platform. Dynamics and a control application of inverse dynamics are presented in the part II-paper.

Superfluidity of non-condensate bosons in Bose–Einstein condensed systems

We present a general theory for superfluidity of non-condensate bosons in Bose–Einstein condensed systems. Considering a system enclosed in a cylindrical vessel rotating about its axis, the superfluid density is determined by analyzing the response of the system to an inertial force field specified by a vector potential in the frame rotating together with the vessel. We derive a general formula for the superfluid density of non-condensate bosons expressed in terms of Green’s functions. It is shown that the superfluidity of non-condensate bosons is caused by the anomalous self-energy describing the pairing correlation between them. The total superfluid density is given by the sum of the condensate density and the superfluid density of non-condensate bosons, reducing to the Landau formula within mean-field approximations neglecting the damping of quasiparticles. We generalize the linear relation between the current and inertial vector potential to satisfy the gauge invariance, introducing the pairing wave function for the superfluid component of non-condensate bosons. We find that the phase of the pairing wave function is twice that of the condensate wave function and the circulation is still quantized in units of 2π∕m even in the presence of the pairing current of non-condensate bosons.

Passive and Explicit Attitude and Gyro-Bias Observers Using Inertial Measurements

This article addresses the problem of estimating the attitude and gyro bias using biased gyro and inertial vector measurements. Two nonlinear observers are presented. One of the two is the passive observer evolving on the semisphere of quaternion, which decouples the gyro measurements from the reconstructed attitude. The other is the explicit observer evolving on the SO(3), which uses directly body-frame measurements of known inertial vectors. The proposed observers achieve faster convergence for large attitude estimation errors than similar observers in literature, guarantee asymptotical stability for almost arbitrary initial conditions, achieve similar steady-state performance with lower computational cost than multiplicative extended Kalman filter, which is the workhorse of attitude estimation, and are suitable for implementation using a low-cost inertial measurement unit. Simulation and experimental results prove the fast convergence and steady-state performance of the proposed observers.

Kalman Filter Cascade for Attitude Estimation on Rotating Earth

This article presents a discrete-time attitude estimation solution based on a cascade of two linear time-varying Kalman filters (KFs). Under mild assumptions, the cascade's first KF resorts to body-fixed measurements of angular velocity and of a constant inertial vector to yield an estimate of Earth's angular velocity. The latter, in addition to all previous measurements, is fed to the second KF to obtain an estimate of the rotation matrix. Although topological constructions are lifted, a last-step projection operator is employed that maps the final rotation matrix estimate onto $SO(3)$ . Briefly, two linear time-varying systems are designed, with no linearizations whatsoever, that are shown to be uniformly completely observable, thus rendering the overall solution globally exponentially stable. Simulation results are presented that allow to assess the performance of the cascaded KF duo. A set of experimental results is also presented that validates the efficiency of the proposed solution and deems it a suitable attitude estimation choice for applications where only one body-vector measurement is available.

Hybrid Pose and Velocity-Bias Estimation on $SE(3)$ Using Inertial and Landmark Measurements

This paper deals with the design of globally exponentially stable invariant observers on the Special Euclidian group SE(3). First, we propose a generic hybrid observer scheme (depending on a generic potential function) evolving on $SE(3)\times \mathbb{R}^6$ for pose (orientation and position) and velocity-bias estimation. Thereafter, the proposed observer is formulated explicitly in terms of inertial vectors and landmark measurements. Interestingly, the proposed observer leads to a decoupled rotational error dynamics from the translational dynamics, which is an interesting feature in practical applications with noisy measurements and disturbances.

Inertial vector measurements based attitude synchronization control for multiple spacecraft formation

for multiple spacecraft with actuator and angular velocity constraints using vector measurements directly. First, the novel vector measurements-based cooperative attitude synchronization control scheme is presented by using barrier Lyapunov function. Especially, the proposed control algorithm is proven to be robust to time delays in the directed communication links, and the actuator and angular velocity constraints are satisfied simultaneously. Second, further analysis of the initial condition is performed to preclude the undesired equilibrium. It indicates that at least 2 orthogonal inertial vectors are required. Third, the fully distributed attitude synchronization control strategy with adaptive gains is developed. Moreover, the control algorithm is implemented without any global information of the graph. This affords a new approach to optimize the control performance. Finally, simulation results demonstrate the effectiveness and performance of the proposed control schemes.

A solution for the attitude determination of three-vehicle heterogeneous formations

This paper proposes a solution to a new attitude determination problem for a three-vehicle formation, where there are restrictions on the detection of other vehicles. Vision-based sensors are considered, which measure line-of-sight (LOS) vectors between different vehicles and inertial vectors that can vary according to the vehicle. There is a constraint on the LOS vectors which are only measured in relation to a chief vehicle. Moreover, each vehicle can measure only one inertial vector. The solution for the different attitude relations is devised geometrically, making use of a multi-stage process. First, two candidates for the relative attitude of each branch are determined. Then, these relative attitude candidates are used to compute inertial attitude candidates for the chief vehicle. The comparison between these results disambiguates the problem, which in general has a unique solution. There are some degenerate solutions, which are also determined. Finally, simulations are carried out considering noise in the sensors. The results are coherent with state-of-the-art approaches to similar problems.

Nonlinear Observer on SO(3) for Attitude Estimation on Rotating Earth Using Single Vector Measurements

This letter presents a novel attitude estimation solution, built on $SO(3)$ , that resorts to single measurements of a constant inertial vector, in addition to angular velocity readings provided by a set of three high-grade fiber optic rate gyros, which are assumed to be sensitive to the angular motion of the Earth. This approach contrasts with typical attitude solutions that require either a single but time-varying inertial vector, or measurements of two non-collinear inertial vectors. The designed nonlinear observer features only one tuning scalar parameter that, in view of the region of convergence of the rotation matrix error, is shown to render the proposed solution almost globally asymptotically stable. Extensive simulation results with realistic noise, including Monte Carlo, are presented that allow to assess the achievable performance.

Globally exponentially stable attitude observer with Earth velocity estimation

AbstractThis paper presents a novel observer for attitude estimation based on a triad of high‐grade rate gyros aided by a body‐fixed vector measurement of a constant inertial vector, departing from the majority of solutions that consider at least two of these vectors. A cascade approach is proposed, where the first block estimates a vector that is related to the angular velocity of the Earth and a second block estimates the attitude. While the topological characteristics are relaxed in the attitude observer so that global exponential stability is achieved, an additional stage is also devised that yields estimates directly on the special orthogonal group, preserving global convergence of the estimation error. Simulation results with realistic sensor noise were performed, including extensive Monte Carlo runs. These results illustrate the convergence of the proposed solution, as well as the achievable performance and robustness to sensor noise.

Attitude observer on the special orthogonal group with Earth velocity estimation

This paper proposes a new nonlinear attitude observer based on high-grade rate gyros and single body-fixed vector measurements of a constant inertial vector, in contrast with typical solutions that require two of these vectors. The structure is cascaded, where in the first block a vector that is related to the angular velocity of the Earth is estimated and in the second block the attitude itself is obtained. The attitude is directly estimated on the special orthogonal group and the estimation error is shown to converge to zero, with a region of convergence that is best described as semi-global, with local exponential convergence. Simulation results illustrate the achievable performance of the proposed solution and the robustness to sensor noise.

Adaptive Attitude Tracking Control of Rigid Body Systems With Unknown Inertia and Gyro-Bias

This paper presents a new adaptive attitude tracking controller for rigid body systems, with unknown inertia and unknown gyro-bias, using inertial vector measurements. The proposed control scheme guarantees almost global asymptotic convergence of the attitude and angular velocity to their desired values. Simulation results are provided to illustrate the effectiveness of the proposed approach.

Einstein Equation for Nondual Field Matter Modifies Naiver-Stokes Dynamics

Cartesian extended matter has its own nondual analog of the 1915 Einstein Equation for pure field physics in nonempty moving space. This tensor balance of energy densities and local stresses leads to Maxwell-type equalities for inertial currents and vector geodesic equations for relativistic accelerations of the Ricci scalar for inertial and gravitational mass densities. Field inertia of slow energy flows reestablishes the living force feedback which is missed in Newton-Euler fluid dynamics and in the Navier-Stokes equation.

A methodological approach to forecasting spatial distribution of workplaces in an industrial metropolis

Many world cities retain their unique industrial status. Such a feature of the economy of an industrial metropolis imposes additional requirements on the development of the forecast of spatial distribution of workplaces. The article highlights the contradictions of the long-term development of an industrial megalopolis, which become scenic forks, when forecasted. These include optimization of the industrial and trade-service sectors of the economy, the ratio of inertial and innovative development vectors, variability of migration flows and the choice of the agglomeration model type. The article is devoted to the problem of forecasting the development of a large metropolis, where the industrial sector plays a significant role in the economy. At the methodological level, the article justifies principles of spatial development of an industrial metropolis. The article describes forecasting tools for spatial location of workplaces, based on a combination of several models. The study was performed through the example of Ekaterinburg – the industrial capital of Russia; the metropolis scenarios were justified until 2035; the forecast of spatial distribution was calculated through the example of the two sectors competing for investments – industrial and trade-service. The authors substantiate spatial distribution of workplaces taking into account the projected number of people employed, the number of population of working age and distinguishing features of transport behavior of citizens. The paper demonstrates that the number of large industrial enterprises in a historically industrial center and its first zone decreases, and the modern industry in the form of small and medium-sized businesses located in industrial parks commence gradually forming a circuit with nodes on transport routes towards the largest consumer territories.