Local Coordinate Frame Research Articles

Three-dimensional deformation monitoring of internal nodes of large-span suspended dome structure using videogrammetry under camera instability

Large-span suspended domes often serve as landmarks for cities or countries. This paper proposes a videogrammetric method for monitoring the 3D deformation of internal key nodes in such structures, even under conditions of camera instability. Utilizing circular targets strategically placed on the measurement area, a dynamic parameter calibration method is developed to estimate the instantaneous position and orientation of cameras relative to the world coordinate system, minimizing camera motion-induced errors. Combined with surface targets on the rigid structure and local coordinate frame transformation, the 3D displacement response of structural internal nodes can be precisely measured using the proposed tracking method. The effectiveness and accuracy of the proposed methods were validated through exhaustive experiments, improving positioning accuracy for the structural internal nodes from 47.24 mm to 1.62 mm after dynamic parameter correction. This methodology offers a low-cost, easily implemented, and non-destructive vision-based solution for the long-term monitoring of large-span suspended domes.

The derivation of Jacobian matrices for the propagation of track parameter uncertainties in the presence of magnetic fields and detector material

In high-energy physics experiments, the trajectories of charged particles are reconstructed using track reconstruction algorithms. Such algorithms need to both identify the set of measurements from a single charged particle and to fit the parameters by propagating tracks along the measurements. The propagation of the track parameter uncertainties is an important component in the track fitting to get the optimal precision in the fitted parameters. The error propagation is performed at intersections between the track and local coordinate frames defined on detector components by calculating a Jacobian matrix corresponding to the local-to-local frame transport. This paper derives the Jacobian matrix in a general manner to harmonize with numerical integration methods developed for inhomogeneous magnetic fields and materials. The Jacobian and transported covariance matrices are validated by simulating the propagation of charged particles between two frames and comparing with the results of numerical methods.

Bearing-Based Formation Tracking Control of Nonholonomic Mobile Agents With a Persistently Exciting Leader

This paper investigates the leader-following formation control problem of nonholonomic agents with a persistently exciting leader based on bearing measurements. A follower needs to maintain a desired relative position to the leader in its local coordinate frame using bearing measurements. An integrated method consisting of a bearing-based estimator and a tracking controller is proposed to achieve the leader-following control objective. The estimator is designed to estimate the relative position to the leader in the follower's local coordinate frame. Based on the estimated relative position, the controller is proposed to drive the follower to achieve the desired formation. Different from existing integrated methods, our method does not require any conditions on the relative bearing between the leader and the follower in the global frame, which are difficult to guarantee. We characterize the conditions locally on the leader only, namely, conditions on the leader's linear and angular velocities with persistently exciting steering speed, such that the overall system is asymptotically stable with our integrated method. Besides, the proposed method is extended to the multi-agent formation control problem. A reasonable solution is proposed to the multi-agent case with a minimal leader follower structure. Simulations prove the effectiveness of the proposed methods in both cases.

Formation Control in Agents' Local Coordinate Frames for Arbitrary Initial Attitudes

Formation Control in Agents' Local Coordinate Frames for Arbitrary Initial Attitudes

Fixed-time Localization of Local Coordinate Frames: Interpretation and Applications to Formation Control Problems

Fixed-time Localization of Local Coordinate Frames: Interpretation and Applications to Formation Control Problems

Orbit-attitude-structure coupled modelling method in local translational coordinate frame for multibody systems

The absolute coordinate-based (ACB) method, including absolute nodal coordinate formulation and natural coordinate formulation, is a widely-used dynamic modelling method for multibody systems. However, the problem of accuracy loss would arise when multi-scale motions are described in the same global coordinate system, such as large-scale orbital motions, small-scale attitude motions and structural vibrations, and micro-scale contact/collision. To address this problem, an ACB method in local translational coordinate system is proposed such that the large-scale orbital motions can be separated from other smaller-scale motions. The dynamic equations are derived by the Hamilton's equations and the Kronecker product. The validity of the proposed method is studied using an orbit-attitude coupled model of an orbiting rigid plate. It is found that the attitude errors of the original ACB method increase as time step size decreases. The reason for this counterintuitive phenomenon is that more serious accuracy loss is encountered for smaller time step size. In addition, more iterations are required for the original ACB method to obtain the prescribed tolerable error in each time step due to accuracy loss. In comparison, the proposed method is more accurate and efficient in numerical simulations. Finally, the proposed method is applied to the orbit-attitude-structure coupled simulation of a space robotic system.

Balancing Rotation Minimizing Frames with Additional Objectives

AbstractWhen moving along 3D curves, one may require local coordinate frames for visited points, such as for animating virtual cameras, controlling robotic motion, or constructing sweep surfaces. Often, consecutive coordinate frames should be similar, avoiding sharp twists. Previous work achieved this goal by using various methods to approximate rotation minimizing frames (RMFs) with respect to a curve's tangent. In this work, we use Householder transformations to construct preliminary tangent‐aligned coordinate frames and then optimize these initial frames under the constraint that they remain tangent‐aligned. This optimization minimizes the weighted sum of squared distances between selected vectors within the new frames and fixed vectors outside them (such as the axes of previous frames). By selecting different vectors for this objective function, we reproduce existing RMF approximation methods and modify them to consider additional objectives beyond rotation minimization. We also provide some example computer graphics use cases for this new frame tracking.

On transformations of ellipsoidal (triaxial) orthogonal curvilinear coordinates

Ellipsoidal orthogonal curvilinear coordinates referred to a triaxial reference ellipsoid are not used in geodesy perhaps because of their theoretical complexity. However, many surveying and geodetic problems could be resolved by simply transforming variables between local frames. Herein it is shown that the computations of local displacements and variance-covariance matrices referred to currently used local curvilinear geodetic coordinate frames gives approximately the same answers than when referred to local ellipsoidal (triaxial) curvilinear frames. The theoretical procedures to obtain the required transformations between local geodetic and local ellipsoidal (triaxial) frames are introduced and examples explained.

Commensurate and spiral magnetic order in the doped two-dimensional Hubbard model: Dynamical mean-field theory analysis

We develop a dynamical mean-field theory approach for the spiral magnetic order, changing to a local coordinate frame with preferable spin alignment along the $z$-axis, which can be considered with the impurity solvers treating the spin diagonal local Green's function. We furthermore solve the Bethe-Salpeter equations for nonuniform dynamic magnetic susceptibilities in the local coordinate frame. We apply this approach to describe the evolution of magnetic order with doping in the $t-t'$ Hubbard model with $t'=0.15$, which is appropriate for the description of the doped La$_2$CuO$_4$ high-temperature superconductor. We find that with doping the antiferromagnetic order changes to the $(Q,\pi)$ incommensurate one, and then to the paramagnetic phase. The spectral weight at the Fermi level is suppressed near half filling and continuously increases with doping. The dispersion of holes in the antiferromagnetic phase shows qualitative agreement with the results of the $t$-$J$ model consideration. In the incommensurate phase we find two branches of hole dispersions, one of which crosses the Fermi level. The resulting Fermi surface forms hole pockets. We also consider the dispersion of the magnetic excitations, obtained from the non-local dynamic magnetic susceptibilities. The transverse spin excitations are gapless, fulfilling the Goldstone theorem; in contrast to the mean-field approach the obtained magnetic state is found to be stable. The longitudinal excitations are characterized by a small gap, showing the rigidity of the spin excitations. For realistic hopping and interaction parameters we reproduce the experimentally measured spin-wave dispersion of La$_2$CuO$_4$.

Decentralized Circular Formation Control of Nonholonomic Mobile Robots Under a Directed Sensor Graph

This paper investigates the circular formation control problem of multiple unicycle-type mobile robots under a directed sensor graph. The topology of the sensor graph among all robots is described by a directed graph containing a spanning tree, and the node denoting the center is only required to be globally reachable in the sensor graph. A dynamic control law is developed such that the multi-robot systems with speed constraint are able to globally converge to a circular formation prescribed by a stationary center, a radius, and a spacing configuration. Remarkably, the proposed control law does not rely on any communication, and only requires each robot to use local measurements with respect to its neighbors based on its local coordinate frame, which makes it feasible to implement this control law in a decentralized manner. The global asymptotic stability of the closed-loop multi-robot systems is established by small gain theorem of which the small-gain condition is guaranteed by a low gain parameter in the designed observer.

2-D distributed pose estimation of multi-agent systems using bearing measurements

This article studies distributed pose (orientation and position) estimation of leader–follower multi-agent systems over κ-layer graphs in 2-D plane. Only the leaders have access to their orientations and positions, while the followers can measure the relative bearings or (angular and linear) velocities in their unknown local coordinate frames. For the orientation estimation, the local relative bearings are used to obtain the relative orientations among the agents, based on which a distributed orientation estimation algorithm is proposed for each follower to estimate its orientation. For the position estimation, the local relative bearings are used to obtain the position constraints among the agents, and a distributed position estimation algorithm is proposed for each follower to estimate its position by solving its position constraints. Both the orientation and position estimation errors converge to zero asymptotically. A simulation example is given to verify the theoretical results.

Bearing-Only Formation Control With Bounded Disturbances in Agents’ Local Coordinate Frames

Bearing-Only Formation Control With Bounded Disturbances in Agents’ Local Coordinate Frames

Range-Only Bearing Estimator for Localization and Mapping

Navigation and exploration within unknown environments are typical examples in which simultaneous localization and mapping (SLAM) algorithms are applied. When mobile agents deploy only range sensors without bearing information, the agents must estimate the bearing using the online distance measurement for the localization and mapping purposes. In this paper, we propose a scalable dynamic bearing estimator to obtain the relative bearing of the static landmarks in the local coordinate frame of a moving agent in real-time. Using contraction theory, we provide convergence analysis of the proposed range-only bearing estimator and present upper and lower-bound for the estimator gain. Numerical simulations demonstrate the effectiveness of the proposed method.

Leader-Following Entrapping Control of AUVs Using Bearing Measurements in Local Coordinate Frames

In this paper, the entrapping control problem of discrete-time AUVs with local coordinate frames is studied. To achieve entrapment in arbitrarily shaped orbits and formations, we design a bearing-based leader-following framework fully in the discrete-time domain with four parts: the orientation estimation unit, estimator unit, controller unit and parameters tuning unit. With bearing measurements and communication information, the orientation estimation unit can estimate orientations of local coordinate frames in finite time, and the estimator unit can achieve local localization. Based on estimation, the controller unit can drive each AUV to track the desired orbit or formation with an arbitrary shape. We present sufficient conditions under which stability of the overall system is proved using the theorem of finite-time stability and LaSalle’s theorem for the discrete-time system. Moreover, the parameters tuning unit can calculate optimal parameters to improve overall performance. Additionally, we extend our schemes to nonholonomic AUVs with unicycle models. Finally, simulation results demonstrate the effectiveness of the proposed scheme.

Industrial Robot Contouring Control Based on Non-Uniform Rational B-Spline Curve

This paper presents a novel algorithm about the industrial robot contouring control based on the NURBS (non-uniform rational B-spline) curve. First, aiming at the error between the industrial robot’s actual trajectory and the desired trajectory, the contour error is proposed as the trajectory evaluation index, and the estimation algorithm of contour error based on the tangent approximation is proposed. Based on the tangent approximation algorithm, the estimation algorithm of contour error in the local task coordinate frame is proposed to realize the transformation from the Cartesian coordinate frame to the local task coordinate frame. Second, according to the configuration of the industrial robot, a modified cross-coupling control scheme based on the local task coordinate frame is designed. Finally, the Bernoulli’s lemniscate curves are constructed by NURBS curve and five-order polynomial curve, respectively, and they are symmetrical. The contrast experiment is designed using the two types of constructed Bernoulli’s lemniscate curves as the incentive trajectory. Through the analysis and comparison between the obtained uniaxial tracking error and the contour error curve of the two incentive trajectories, it is concluded that the incentive trajectory constructed by the NURBS curve has better contour control performance than that constructed by the five-order polynomial curve. The results drawn from this paper lay a certain foundation for the future high-precision contouring control of industrial robots.

Cooperative Adaptive Model-Free Control With Model-Free Estimation and Online Gain Tuning.

In this article, a distributed adaptive model-free control algorithm is proposed for consensus and formation-tracking problems in a network of agents with completely unknown nonlinear dynamic systems. The specification of the communication graph in the network is incorporated in the adaptive laws for estimation of the unknown linear and nonlinear terms, and in the online updating of the elements in the main controller gain matrix. The decentralized control signal at each agent in the network requires information about the states of the leader agent, as well as the desired formation variables of the agents in a local coordinate frame. These two sets of variables are provided at each agent by utilizing two recently proposed distributed observers. It is shown that only a spanning-tree rooted at the leader agent is enough for the convergence and stability of the proposed cooperative control and observer algorithms. Two simulation studies are provided to evaluate the performance of the proposed algorithm in comparison with two state-of-the-art distributed model-free control algorithms. With lower control effort as well as fewer offline gain tuning, the same level of consensus errors is achieved. Finally, the application of the proposed solution is studied in the formation-tracking control of a team of autonomous aerial mobile robots via simulation results.

Gradient-based bearing-only formation control: An elevation angle approach

This paper proposes an elevation angle rigidity theory in both 2D and 3D spaces, and applies it to solve multi-agent formation control with only inter-agent bearing/direction measurements in agents’ local coordinate frames. Motivated by the sensor technology in measuring elevation angle and angular diameter, we develop elevation angle rigidity by attaching each agent in a multi-agent framework with a rod in 2D and a ball in 3D, respectively. By defining the elevation angle rigidity matrix, conditions for infinitesimal elevation angle rigidity are derived. Compared to previously developed angle rigidity-based and bearing rigidity-based formation control laws, the proposed elevation angle rigidity-based control law can maintain the gradient-based control property. Compared to distance-based formation control laws, less sensor measurements are required. The formation maneuvering with desired translation and rotation is also realized by using only local bearing measurements. Simulation examples illustrate the advantages and effectiveness of the proposed approach.

On invariance of spatial isogeometric Timoshenko–Ehrenfest beam formulations for static analysis

This study elaborates the invariance of spatial Timoshenko–Ehrenfest​ beam formulations in the context of isogeometric analysis. Such invariance confirms that zero strain measures are always generated by rigid transformations, i.e., rigid translations and rotations. The violation of this property can degrade the performance of the formulations in predicting structural responses. In the setting of linear analysis, the invariance of planar beam formulations has already been studied, but a similar investigation for spatial beam formulations is not yet carried out. Most of the spatial beam formulations are developed in the local coordinate frame, and components of unknown kinematics in this frame are discretized by using rational B-spline basis functions. Unfortunately, those formulations are found to be non-invariant under such a discretization scheme, and the degradation in their performance is demonstrated. On the other hand, the local coordinate frame is widely defined by the so-called natural Frenet–Serret frame. The sole utilization of this frame does not allow the consideration of beams having twisting configuration. In this study, these shortcomings are resolved.

Collision avoidance control for formation flying of multiple spacecraft using artificial potential field

• Trajectory design/control for spacecraft formation flying with obstacle avoidance is presented. • Rotational potential function for obstacle avoidance in 3-dimensional space is newly proposed. • The controller is proven to be asymptotically stable in the sense of Lyapunov and successfully tested. This study presents trajectory design/control for spacecraft formation flying with obstacle avoidance. Based on the artificial potential field (APF), a formation potential is first defined to derive a formation control law for virtual structure, which enables multiple spacecraft to maintain polygonal or tetrahedral formation. As an efficient method to circumvent local minima which often occur in the APF-based approach, a newly proposed rotational potential is derived in a local coordinate frame to add in the APF framework. The synthesized formation and rotational potential function is used to develop a gradient-based control law to design/control the formation flying trajectory while avoiding collision with obstacles. Proven to be asymptotically stable in the sense of Lyapunov, the proposed continuous feedback control law is demonstrated via formation keeping/reconfiguration examples. The proposed approach successfully maintains the trajectory in the desired formation without colliding with obstacles and without falling into local minimum. These results are comparatively analysed with those of other APF-based approaches. The overall analysis shows that the proposed rotational potential, which has been newly derived in this research, enables a group of spacecraft in formation to efficiently avoid collision with obstacles without convergence to a local minimum.

2-D Directed Formation Control Based on Bipolar Coordinates

This work proposes a novel 2-D formation control scheme for acyclic triangulated directed graphs (a class of minimally acyclic persistent graphs) based on bipolar coordinates with (almost) global convergence to the desired shape. Prescribed performance control is employed to devise a decentralized control law that avoids singularities and introduces robustness against external disturbances while ensuring predefined transient and steady-state performance for the closed-loop system. Furthermore, it is shown that the proposed formation control scheme can handle formation maneuvering, scaling, and orientation specifications simultaneously. Additionally, the proposed control law is implementable in agents' arbitrarily oriented local coordinate frames using only low-cost onboard vision sensors, which are favorable for practical applications. Finally, a formation maneuvering simulation study verifies the proposed approach.