Body Reference Frame Research Articles

Robocentric Model-Based Visual Servoing for Quadrotor Flights

This article proposes a robocentric formulation for quadrotor visual servoing. This formulation presents the task-specific state dynamics of the quadrotor in its body reference frame. Compared with other visual servoing methods, our method allows tightly and integrated state estimation and control on the same robocentric model, and allows a faster system response in aggressive quadrotor flights. On the theory level, we prove the controllability and observability of the proposed robocentric model. Then, we design an on-manifold Kalman filter for the estimation and an on-manifold iterative model predictive control for motion planning and control. We verify our proposed formulation and controller in two crucial quadrotor flight tasks: 1) hovering; and 2) dynamic obstacle avoidance. Experiment results show that the quadrotor is able to resist large external disturbances and recover its position and orientation from two reference visual features. Moreover, the quadrotor is able to avoid dynamic obstacles reliably at a relative speed up to 7.4 m/s, demonstrating the effectiveness of our visual servoing methods in agile quadrotor flights.

Application of Dynamic Neural Network Model for Magneto- Coulombic Attitude Control of Earth-Pointing Satellite

An active control of the satellite attitude using magneto-Coulombic actuators is proposed along the circular orbit. Equation for the required charges on the Coulomb shells is derived, which is used to find the available control torque along the three body axes. It is proved that the eigenvalues of the averaged control matrix of the magneto-Coulombic system in the body reference frame for different orientations and angular velocities components will converge to the same values as in the case of the eigenvalues of the averaged control matrix in the orbital reference frame. A dynamic neural network based control law is adapted and trained for the proposed system. The neural network based dynamic neural network controller has the advantages of handling the non-linearity, adaptation and on-board training, and robust to large changes in the parameters in the plant, such as the moments of inertia. It is shown that the stabilization accuracy of the dynamic neural controller is higher as compared to the proportional-derivative based controller.

Alignment scheme optimization when gyro accuracy inconsistency for inertial navigation system

Rotation modulation technology can average out constant errors in fiber optic gyro (FOG) rotational inertial navigation system (RINS). As a result, angle random walk (ARW) of FOGs plays a significant role in the precision of initial alignment. Using different gyros as horizontal gyros gradually transitioned from theory to practical application with the advancement of design philosophy. The theoretical formula between azimuth alignment accuracy and random walk coefficients (RWCs) of two horizontal gyros is deduced in this paper. A completely new multiple rotations and stationary alignment scheme is suggested. A recursive least-squares filter with azimuth variation in body reference frame and accelerometer outputs in navigation frame as measurements is constructed. The efficiency of this method is demonstrated through simulation and experimentation. The findings demonstrate that using this formula and alignment scheme, azimuth alignment accuracy could be greatly increased, which can be improved by 10.85% compared to common continuous rotation alignment scheme.

An INS-aided MASS autonomous navigation algorithm considering virtual motion constraints and the leeway and drift angle

Autonomous navigation is one of the critical enabling technologies in developing MASS or intelligent ships. This study proposes an autonomous navigation algorithm with INS-aided shipborne GNSS. The proposed algorithm constructs the ship's virtual motion attitude by considering the leeway and drift angle, and takes into account the one-dimensional speed constraints in the virtual ship body reference frame. This approach addresses the issue that the conventional INS and shipborne GNSS integrated navigation scheme can utilize the three-dimensional position information only, without considering the measurement update from the one-dimensional speed obtained by shipborne GNSS. The algorithm is verified by actual ship departure and cruising experiments in natural marine environments. The experimental results show that the proposed algorithm has a higher accuracy than the conventional algorithm when the GNSS speed and position information are available. In the scenario when the GNSS position information is temporarily lost, the proposed algorithm can continue to derive relatively reliable navigation parameters for the safe navigation of ships with only one-dimensional speed measurement.

Refined MagSail thrust model for preliminary mission design and trajectory optimization

A magnetic sail is an advanced propellantless propulsion system concept for deep space missions, proposed by Dana Andrews and Robert Zubrin about 30 years ago. The thrust is generated by the electromagnetic interaction between the solar wind charged particles and an artificial magnetic field obtained through a large current-carrying coil made of superconducting material. This paper discusses an up-to-date single-loop magnetic sail thrust model based on the numerical and experimental results obtained in the last decade. In particular, a particle model is initially used to evaluate the performance of a reference configuration in terms of thrust components in a body reference frame. Then, some scaling relationships are used to estimate the magnetic sail performance as a function of the design parameters. The proposed thrust model is used to analyze the minimum-time transfer problem between two circular and coplanar heliocentric orbits. Minimum flight times necessary to complete ephemeris-free Earth-Venus and Earth-Mars transfers are calculated as a function of the magnetic sail characteristic size in a parametric way.

Quantifying and Analyzing Mosquito Movement from Video Tracking Results.

After tracking the kinematics of a moving mosquito from video, one needs to convert the tracking output into the kinematics data needed to answer the research question. Here, we provide general guidelines for how to do this by discussing how to quantify body position and orientation in the world reference frame and wing and leg orientation in the mosquito body reference frame. These guidelines should be adapted based on the goal of your research. To answer your research question, the resulting kinematics data must then be further analyzed. Because the nature of this analysis depends strongly on your specific research question, we refer to literature for designing these postprocessing routines.

Rotation-Induced Geometrical Stiffening of a Tapered, Pretwisted Blade

A linear geometrical stiffening model is developed for a rotating beam with an asymmetric cross section undergoing coupled stretch–bending–torsion motion. The model is verified for a twisted and tapered Timoshenko beam mounted on a rotating hub with a presetting angle. The floating frame of reference formulation is adopted where the configuration of the deformable beam is described by using different coordinate systems. Three coordinate systems, viz., inertial reference frame attached to the center of the rigid hub, local reference frame attached to an arbitrary point on the elastic axis of the undeformed beam, and body reference frame attached to the point on the elastic axis of the deformed beam, are defined. The global position vector of an arbitrary point on the beam is specified using a coupled set of reference and elastic coordinates. Reference coordinates define the location and orientation of a body reference frame, and elastic coordinates describe the beam deformation about the body reference frame. For a rotating beam problem, reference and elastic coordinates give the angular displacements of the hub and elastic deformations of the beam, respectively. The expressions for the kinetic and strain energy of the deformable beam are derived. The inertial coupling between the reference and elastic coordinates in the kinetic energy expression is identified as a gyroscopic coupling term. Further, the inertial mass matrix, spin softening matrix, and centrifugal stiffening matrix are obtained from kinetic energy expression. The elastic stiffness matrices, which are independent of elastic coordinates, are derived from strain energy expression. According to the Rayleigh–Ritz method, the elastic coordinates are expressed using a series of admissible functions that satisfy the geometric boundary conditions of a fixed-free beam. The ordinary differential equations of motion are then derived using Lagrange’s equations. A set of dimensionless parameters is identified after the nondimensionalization of the equations of motion. Due to the skew-symmetric nature of the gyroscopic coupling matrix, the eigenvalue problem is not amenable to classical modal analysis. Therefore, a complex modal analysis method is used to determine the modal characteristics. The results from the present model are verified with those available in the literature. The results are also compared with those obtained from the three-dimensional finite element method using ANSYS. The effect of these dimensionless parameters on the modal characteristics of a rotating blade is studied.

Modeling of centrifugal deployment of three-section minisatellite boom

The aim of the article is to model the processes of centrifugal deployment of a three-section boom and preliminary analyze the feasibility of this deployment method for an Earth remote sensing (ERS) minisatellite (MS). During the research, methods of theoretical mechanics, multibody dynamics, control theory, and computer modeling were used. Centrifugal deployment of multi-section booms have been successfully used on spin stabilized satellites, but not on ERS satellites, which have other features of operation and require additional studies. The main part of the MS is a platform to which a transformable antenna is attached by means of a transformable boom. Before deployment, the stowed boom and antenna are attached to the MS platform. The boom sections are connected by joints with one rotational degree of freedom and deployed sequentially due to centrifugal forces when the MS rotates in the required direction. Each of the boom joints has a locking mechanism that latches when a predetermined deploy angle is reached. To model the processes of the boom deployment, the MS is presented as a system of connected bodies, where the platform and the stowed antenna are absolutely rigid bodies, and the boom consists of three flexible beams of a tubular cross-section. The differential equations of the MS dynamics during the deployment are obtained using the Lagrangian formalism, which are supplemented by algebraic equations describing the constraints from the joints. The scenarios of the boom deployment with a constant control torque and a constant angular velocity of the MS are considered. These scenarios are simulated, and estimates of the control actions needed to ensure full deployment of the boom and the stabilization of the MS after latching of the joints are calculated. Dependences of variations of the loads on the boom structure during deployment are obtained. The simulation results allow us to conclude that it is feasible to implement the method of the boom centrifugal deployment for the MS, which can perform fast rotations about the three axes of the body reference frame. Implementation of this method allows designers to reduce mass of the MS because it does not require any servo drives in the boom deployment system.

A Joint Extrinsic Calibration Tool for Radar, Camera and Lidar

We address joint extrinsic calibration of lidar, camera and radar sensors. To simplify calibration, we propose a single calibration target design for all three modalities, and implement our approach in an open-source tool with bindings to Robot Operating System (ROS). Our tool features three optimization configurations, namely using error terms for a minimal number of sensor pairs, or using terms for all sensor pairs in combination with loop closure constraints, or by adding terms for structure estimation in a probabilistic model. Apart from relative calibration where relative transformations between sensors are computed, our work also addresses absolute calibration that includes calibration with respect to the mobile robot's body. Two methods are compared to estimate the body reference frame using an external laser scanner, one based on markers and the other based on manual annotation of the laser scan. In the experiments, we evaluate the three configurations for relative calibration. Our results show that using terms for all sensor pairs is most robust, especially for lidar to radar, when minimum five board locations are used. For absolute calibration the median rotation error around the vertical axis reduces from 1 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> before calibration, to 0.33 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> using the markers and 0.02 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">°</sup> with manual annotations.

Equivalence of Euler equations and torque-angular momentum relation

Euler derived equations for rigid body rotations in the body reference frame and in the stationary reference frame by considering an infinitesimal part of the rigid body.Another derivation is possible, and it is widely used: transforming torque-angular momentum relation to the body reference frame.However, their equivalence is not shown explicitly.In this work, for a rigid body with different moments of inertia, we calculated Euler equations explicitly in the body reference frame and in the stationary reference frame and torque-angular momentum relation.We also calculated equations of motion from Lagrangian.These calculations show that all four of them are equivalent.

Implementation of Translational Motion Dynamics for INS Data Fusion in DVL Outage in Underwater Navigation

Underwater navigation is generally accomplished through the data fusion of INS (Inertial Navigation System) and auxiliary sensors such as DVL (Doppler Velocity Logger) sensor. However, because of the possibility of DVL outage, alternative low-cost solutions are attractive. Among these, one is using vehicle kinetic model information extracted by the Newton-Euler equation to improve INS performance, which is called model-aided navigation. In this paper, only the vehicle translational motion dynamics are used to replace DVL in underwater navigation in DVL outage. The vehicle 3D translational dynamics has been obtained by using general Newton-Euler equations. Integrating these dynamics leads to the calculation of velocities in the body reference frame. Similar to DVL measurements, the calculated velocities (called “Pseudo-DVL” in this paper) are fused with INS. In this paper, a Kalman filter is used for INS/Pseudo-DVL data fusion. Field test data collected by an equipped research AUV called SUT-III is used to evaluate the proposed approach experimentally. Also, a comparison between the integrated navigation of INS-DVL and INS-Pseudo-DVL is performed and the results are provided. The results reveal that despite having low-cost MEMS IMUs, INS/Pseudo-DVL has effectively corrected standalone INS estimation. Finally it is concluded that despite slight degradation of navigation performance, an expensive DVL sensor can be replaced by using the proposed method in DVL outage.

Stability of perception of the hand's aperture in a grasp.

How we judge the location of our body parts can be affected by a range of factors that change how our brain interprets proprioceptive signals. We examined the effect of several such factors on how we perceive an object's width and the spacing between our thumb and fingers when grasping. Grasp-related perceptions were slightly wider when using all digits, in line with our tendency to grasp larger objects with the entire hand. Surprisingly, these perceptions were not affected by the frames of reference for judgements (object width versus grasp aperture), whether the object was grasped actively or passively, or the strength of the grasp. These results show that the brain maintains a largely stable representation of the hand when grasping stationary objects. This stability may underpin our dexterity when grasping a vast array of objects. Various factors can alter how the brain interprets proprioceptive signals, leading to errors in how we perceive our body and execute motor tasks. This study determined the effect of critical factors on hand-based perceptions. In Experiment1, 20 participants grasped without lifting an unseen 6.5cm-wide object with two grasp configurations: thumb and all fingers, and thumb and index finger. Participants reported perceived grasp aperture (body reference frame) or perceived object width (external reference frame) using visual charts. In Experiment2, 20 participants grasped the object with three grasp intensities (1, 5 and 15% maximal grasp force) actively or passively and reported perceived grasp aperture. A follow-up experiment addressed whether results from Experiment2 were influenced by the external force applied during passive grasp. Overall, there was a mean difference of 0.38cm (95% confidence interval (CI), 0.12 to 0.63) between the two grasp configurations (all digits compared to thumb and index finger). Perceived object width compared to perceived grasp aperture differed by only -0.04cm (95% CI, -0.30 to 0.21). There was no real effect of grasp intensity on perceived grasp aperture (-0.01cm; 95% CI, -0.03 to 0.01) or grasp type (active versus passive; 0.18cm; 95% CI, -0.19 to 0.55). Overall, grasp-related perceptions are slightly wider when using all digits, in line with our tendency to grasp larger objects with the entire hand. The other factors - frame of reference, grasp intensity and grasp type - had no meaningful effect on these perceptions. These results provide evidence that the brain maintains a largely stable representation of the hand.

A 3-D Trailer Approach to Leader-Following Formation Control

A real-time 3-D trajectory planner for the leader-following formation control of autonomous vehicles is proposed. The planner relies on the definition of a trailer body whose hinge point is rigidly attached to the leader and assigns each follower to a distinct point of such a trailer. Convergence results for the trailer body reference frame are presented based on Lyapunov analysis, which guarantee that the planning can be independently implemented by $n$ followers, with a common leader, as they asymptotically behave as $n$ points of a unique trailer body. As such, the need for communication among followers is dispensable. The proposed strategy is divided into trajectory planning and trajectory tracking, the first problem being vehicle-independent. Experimental results obtained with quadrotor vehicles are presented, which demonstrate the richness and suitability of the planned trajectories.

An efficient formulation for flexible multibody dynamics using a condensation of deformation coordinates

This paper introduces a more efficient formulation for flexible multibody dynamics based on the floating frame of reference formulation (FFRF). A two-step condensation method is proposed. The first condensation involves coordinate reduction using a modal transformation, from nodal coordinates to modal coordinates, in the context of model order reduction (MOR). The second condensation involves re-fabrication of the equations of motion using a dynamic condensation of the deformation modal coordinates obtained from the first condensation. In the first condensation, constraint modes are corrected by condensation of the truncated modes that represent the effects of residual flexibility. As a result, the first condensation offers a more accurate deformation field in the reduced order model, which consequently leads to a more accurate flexible multibody simulation. In the second condensation, the equations of motion of a flexible multibody are once more reduced to the equations of motion in terms of absolute coordinates for the body reference frames. In this way, the proposed method deals with smaller sized equations of motion than in the conventional FFRF, where they are represented in terms of both absolute coordinates and deformation modal coordinates. The effect of the first condensation was verified using a benchmark example of a slider–crank mechanism with a flexible connecting rod. The efficiency of the proposed condensation method was also investigated using a computational complexity analysis with an $n$ -flexible body pendulum.

An Estimator for UAV Attitude Determination based on Accelerometers, GNSS Sensors, and Aerodynamic Coefficients

Accurate navigation and control of Unmanned Air Vehicles (UAV) are strongly linked to the estimation of position and attitude. Rotation of an aircraft can be determined measuring two vectors, for example, velocity and gravity vectors, in two independent reference frameworks, for example, inertial and the body reference frames. The velocity vector can be determined in inertial reference frame from a GNSS-based sensor and in the body reference frame by integrating accelerometer measurements. A novel approach is presented for gravity vector estimation in the body reference frame, which is based on UAV aerodynamics and mass properties. Combining both vectors enables determination of body rotation. Nonlinear simulations show that, utilizing GNSS sensors, strap-down accelerometers and the proposed approach, aircraft attitude determination is accurate. The developed calculations are implemented on a fixed-wing UAV, permitting the substitution of specific attitude determination gadgets, which are normally costly for several applications. © 2018 Institute of Navigation.

A Jacobian formulation for efficient simulation of multibody chain dynamics

This research proposes an efficient analysis method with an implicit integration method for multibody chain dynamics. Absolute Cartesian coordinates which are related to the body reference frame are used as generalized coordinates. Contact between a sprocket and chain links is formulated as a circle to circle contact. Chain links are connected by bushing elements. Newton-Raphson method is used to solve the combined equations of motion, constraints, and implicit integration formulas. Jacobian matrix must be calculated and LU decomposed with the implicit integration method in Newton-Raphson method. Since Cartesian coordinates with respect to the body reference frame are used as the generalized coordinates and the contact and bushing elements exist between two bodies, the Jacobian matrix for these elements depended on only the relative motion of two adjacent bodies. Therefore, the Jacobian matrix hardly changes unless the relative motion changes significantly. This research investigates the Jacobian matrix to find small-changing and large-changing submatrices. Investigation showed that 65.4 % and 34.6 % of the Jacobian matrix belong to the small changing and the large changing submatrices, respectively. The small changing submatrices are calculated and LU decomposed only once at the initial time and reused at every time step afterwards. The large changing submatrices are updated and LU decomposed at every time step. As the sprocket rotates, the generalized coordinate sequence of the Jacobian matrix for the links is shifted one by one to keep the Jacobian matrix change of the small change submatrices small until the end of simulation time. The proposed method is implemented and its results are compared to those obtained from the original method which updates and LU decomposes the whole Jacobian matrix at every time step. Numerical experiments showed that the proposed method is about 11 times faster and yields almost identical results to the original method.

A passive camera based determination of a non-cooperative and unknown satellite's pose and shape

The relevance of autonomy in space systems during rendezvous and docking operations has been lately increasing. At the scope, a robust GNC architecture is required, which strictly relies on the navigation system's performance and must assure both high efficiency and safety, i.e. low errors and no collisions with the target satellite. One of the most explored fields is the optical navigation one. Using passive optical sensors such as cameras can give high benefit in terms of characterization of the observed scene, thus enlarging the consciousness of what is going on in the mission scenario. The present research investigates the development of a filter which can estimate the shape and relative attitude, position and velocity of a non-cooperative, possibly unknown satellite orbiting around Earth, observed by a camera and a distance sensor mounted on a chaser satellite, whose objective is to successfully complete a docking maneuver. The image taken at a certain time is processed, features are extracted from it and matched with the ones extracted from the image at the previous time step. The matched features along with the relative distance measured by the distance sensor are merged inside an unscented Kalman filter, which predicts, updates and improves the state's estimate throughout the iterations. The expedient used in the filter is to give a 3D characterization to the 2D features used as measurements. The filter estimates the 3D coordinates of these points, i.e. the target's shape, in the camera reference frame, which depend on the target's attitude dynamics and the chaser's relative orbital dynamics. Thus, the target's attitude parameters, i.e. the quaternions, and angular velocity vector, the relative position and velocity vectors and the tracked 3D points are all included in the state vector and estimated by the filter. Subsequently, the 3D point coordinates are determined in the body reference frame. By doing this for all the tracked points, a 3D map of the target can be built.

Attitude Determination Algorithms through Accelerometers, GNSS Sensors, and Gravity Vector Estimator

Aircraft and spacecraft navigation precision is dependent on the measurement system for position and attitude determination. Rotation of an aircraft can be determined measuring two vectors in two different reference systems. Velocity vector can be determined in the inertial reference frame from a GNSS-based sensor and by integrating the acceleration measurements in the body reference frame. Estimating gravity vector in both reference frames, and combining with velocity vector, determines rotation of the body. A new approach for gravity vector estimations is presented and employed in an attitude determination algorithm. Nonlinear simulations demonstrate that using directly the positioning and velocity outputs of GNSS sensors and strap-down accelerometers, aircraft attitude determination is precise, especially in ballistic projectiles, to substitute precise attitude determination devices, usually expensive and forced to bear high solicitations as for instance G forces.

Modelling of flexible bodies with minimal coordinates by means of the corotational formulation

This paper is concerned with the extension of the minimal coordinates approach to flexible bodies. When using minimal coordinates, the number of configuration parameters corresponds exactly to the number of degrees of freedom and they can be chosen arbitrarily as far as there is a one-to-one relationship between the configuration of the system and the configuration parameters. In the rigid case, the equations of motion are obtained from the description of the translational and rotational motion of a frame attached to each body in terms of the chosen configuration parameters, and from the forces acting on each body. The extension to the simulation of flexible bodies naturally leads to a description of the motion of a flexible body from the one of its nodes. However, the relationship between the latter and the full internal motion of the body is not unique and is the subject of various developments. It was then proposed for the sake of generality to systematically treat flexible bodies as superelements, implemented according to the corotational approach, with a floating corotational frame. This allows to model any flexible body from its mass and stiffness matrices obtained from any available finite element code. Moreover, it doesn’t impose any restriction on the kinematics of the nodes which can then be expressed indifferently from absolute or relative coordinates as usually encountered with minimal coordinates. After a description of the adopted framework, the paper develops the equations of motion. Some test examples are presented, where the proposed approach will be compared to the ones obtained with the classical body reference frame approach and results from the literature. In some cases, the influence of the chosen corotational frame is analysed. The examples confirm that the corotational formulation should be restricted to flexible bodies involving only small deformations and rotational velocity. It is also shown that modelling can be adapted to improve the quality of the results.

Dynamic Inversion Control for Performing Herbst Manoeuver with Lateral Center-of-Gravity Offset

&lt;!--?xml version="1.0" encoding="UTF-8"?--&gt;&lt;div class="abstract"&gt;&lt;div class="abstract_label"&gt;The present study addresses the effects of lateral center-of-gravity (CG) movement, resulting from asymmetric firing of some of the onboard stores, on the dynamics and control of a combat aircraft while attempting the highly demanding Herbst manoeuver. The complete six degree-of-freedom equations of motion of the aircraft for such lateral CG offset are derived in two different body reference frames attached either to the symmetric nominal CG location or to the shifted asymmetric CG location. The Herbst manoeuver is first simulated using nonlinear dynamic inversion based control to handle the highly nonlinear post stall flight dynamics considering the standard equation of motion without considering any lateral CG variation. Thereafter, it is observed that if the same controller is retained, the manoeuver performance deteriorates significantly even when the CG undergoes a moderate lateral shift. To overcome this shortfall, closed loop controllers are next designed incorporating both the models of asymmetric dynamics as derived in this paper. It is validated through MATLAB simulations that both the controls, thus designed, can recover the original manoeuver performance almost completely; however, the first one requires more complex computations and hence increased computation time while the second one requires that all the measurements be transformed to the new body reference frame at every time step.&lt;/div&gt;&lt;/div&gt;