Jacobian Formulation Research Articles

Optimized Reduced Jacobian Formulation for Simultaneous Solution of Control Systems in Electromagnetic Transient Simulations

This paper proposes a new optimized and reduced order Jacobian formulation for the simultaneous solution of control system block-diagrams with feedback loops in the simulation of electromagnetic transients. A reduced order Jacobian matrix formulation has been previously introduced for block-diagrams by solving a minimum Feedback Arc Set (FAS) problem. This paper enhances the latter method by proposing a faster formulation and exploring efficient heuristics for the FAS problem. This optimized reduced Jacobian formulation is implemented in a generic tool for the simulation of electromagnetic transients and demonstrated for large-scale and practical power systems.

A Microrobot With an Attached Microforce Sensor for Transurethral Access to the Bladder Interior Wall

AbstractWe present the conceptual design and limited functionality prototype and characterization of a system for application in transurethral palpation of any targeted area of the bladder interior wall tissue consisting of a robotic manipulator and a microforce sensor attached at its tip all less than 3.5 mm in diameter. A hyper-redundant ten-joint six degrees-of-freedom (6DOF) manipulator (5DOF rigid and five-joint continuum segments) is presented along with the forward and inverse kinematics analyses based on a Jacobian formulation to prevent configuration singularities. Simulated motion studies demonstrate the ability of the proposed manipulator to attain a desired pose (normal to the tissue) with any area in the bladder including the difficult to reach trigone area. A strain gauge-based microforce sensor is designed using finite element analysis (safety factor > 3), prototyped using additive manufacturing, and characterized. The characterized sensor was used to acquire in vivo measurements to evaluate human palm tissue viscoelastic properties. A single module of the continuum segment is designed and prototyped using additive manufacturing, and used to characterize its tension-bend angle behavior. Finite element analysis is used to improve structurally weak regions of the vertebra. A three-joint four-vertebrae prototype was successfully actuated to reach a bend state using tendons. The developed robot and sensor prototypes demonstrate capabilities of the proposed concept which could be a possible solution to quantitatively evaluate localized biomechanical properties of the bladder tissue to improve treatment and provide better patient care.

An improved stiff-ODE solving framework for reacting flow simulations with detailed chemistry in OpenFOAM

The integration of stiff ordinary differential equation (ODE) systems associated with detailed chemical kinetics is computationally demanding in practical combustion simulations. Despite the various approaches in expediting the computational efficiency, it is still necessary to optimise the cell-wise calculation in operator-splitting type simulations of reactive flow. In this work, we proposed an improved stiff-ODE solver framework targeting to speed up the simulation of reactive flow in OpenFOAM. This framework combines the Radau-IIA and backward differentiation formula (BDF) ODE-integration algorithms, the pyJac-based fully analytical Jacobian formulation, and dense-based LAPACK and sparse-based KLU sophisticated linear system solvers. We evaluate the performance of the efficient solver framework on various benchmark combustion problems across a wide range of chemical kinetic complexities. A comprehensive investigation of the key elements of stiff ODE solvers is conducted in the homogeneous reactor, focusing respectively on the influences of error tolerance, integration time interval, Jacobian evaluation methodology, and linear system solver on the accuracy and efficiency trade-off. More realistic simulation results are presented regarding the one-dimensional laminar flame and three-dimensional turbulent flame. The results indicate that the Radau-IIA is more preferable in both efficiency and accuracy compared with the widely used BDF and Seulex methods for large integration interval, whereas the differences between three methods diminish as the integration time interval decreases. In all cases, it is found that the full analytical Jacobian is more advantageous for small mechanisms of species number around 50–100 while the approximated formulation of Jacobian is recommended for larger ones. Furthermore, the more robust linear system solvers provide significant improvement on computational efficiency with the dense-based LAPACK solver being more suitable for small to moderate-scale mechanisms while sparse-based KLU being superior for large-scale mechanisms. The proposed efficient solver framework in its optimal configuration obtains more than 2.6 times speedup in realistic high-fidelity flame simulation with a 57 species combustion mechanism.

Zero Reaction Torque Trajectory Tracking of an Aerial Manipulator through Extended Generalized Jacobian

Aerial manipulators are used in industrial and service robotics tasks such as assembly, inspection, and maintenance. One of the main challenges in aerial manipulation is related to the motion of the UAV base caused by manipulator disturbance torques and forces, which jeopardize the precision of the robot manipulator. In this paper, we propose two novel inverse kinematic control methods used to track a trajectory with an aerial manipulator while also considering resultant UAV base motions. The first method is adapted from the generalized Jacobian formulation used in space robotics and includes the change in system momentum resulting from gravity and UAV control forces in the inverse kinematic control equation. This approach is simulated for a 2 and 3 degree-of-freedom aerial manipulator tracking trajectories with the end-effector. Although the end-effector position error is approximately zero throughout the simulated task, we see significant undesired UAV base motions of several centimeters in magnitude. To ameliorate this by exploiting the kinematic redundancy, we modify the generalized Jacobian by adding an additional task constraint which minimizes the reaction torques from the manipulator, to form the extended generalized Jacobian. While the second approach results in improved precision and reduced base motion by an order of magnitude as compared to the generalized Jacobian, a drawback is the reduction in the available workspace as the solution seeks to minimize the manipulator center of gravity translation. We also demonstrate and compare both approaches in a load picking task. All the algorithms are completed computationally faster than real time in the MATLAB simulations, illustrating their potential for application in real-world experiments.

Effects of geometry in the operation of coaxial electrosprays

Coaxial electrospray is a well established procedure to generate coaxial micro- and nano-structures (microcapsules and coaxial fibers). In this work we show the capabilities of a high-precision numerical code using dynamic mapping, analytic Jacobian formulation and the Taylor–Melcher leaky dielectric model (LDM) to deal with real coaxial electrospray configurations. Since the number of parameters involved in the problem is large, we focus on an experimentally tested liquid pair configuration where the thickness, relative position, and end sharpness of the coaxial feeding tubes are geometrical parameters of interest. The effects of those parameters and the voltage applied at the tubes on the coaxial liquid menisci, liquid streamlines, size of jet and core, and electric current are discussed.

Volumetric multiphase ventilation imaging based on four-dimensional computed tomography for functional lung avoidance radiotherapy.

Current computed tomography (CT)-based lung ventilation imaging (CTVI) techniques derive a static ventilation image without temporal information. This research aims to develop a four-dimensional CT (4DCT)-based multiphase dynamic ventilation imaging framework capable of recovering the entire ventilation process throughout the breathing cycle for functional lung avoidance radiotherapy (FLART). A total of 15 free-breathing thoracic 4DCT scans of lung or esophageal cancer patients were collected from the public datasets. The lung region of each phase image was first delineated, and then the mask-free isotropic total variation image registration algorithm was used to derive the deformation vector fields between the end-expiration (EE) phase and other phases. As a surrogate of ventilation, the voxel-wise local expansion ratio of each phase relative to the EE phase was estimated using the parameterized Integrated Jacobian Formulation method in the EE phase coordinate. Lastly, the dynamic ventilation images were generated by warping these phase-specific local expansion distributions with a same geometry into their respective breathing phases. Quantitative analysis, including interphase Spearman correlation coefficients, voxel-wise, and regional-wise expansion/contraction tracking, were performed to indirectly validate the proposed method. The proposed method maintains the physiological meaning of ventilation on each phase and enables to recover the dynamic lung ventilation process. The mean interphase Spearman correlations ranged between 0.23±0.20 and 0.93±0.04 and decreased near the EE phase. Only 26.2% (2.59E+6 out of 9.89E+6) of lung voxels exhibited the same expansion/contraction pattern as the global lung. Qualitative and quantitative evaluations of the interphase ventilation distribution difference show that ventilation spatiotemporal heterogeneities generally exist during respiration. In contrast to conventional CTVI metrics, our method enables to extract additional phase-resolved respiration-correlated information and reflects the generally existed ventilation spatiotemporal heterogeneities. Subsequent studies with quantitative phase-by-phase cross-modality evaluations will further explore its potential to deepen our understanding of lung function and respiration mechanics and also to facilitate more accurate implementation of FLART.

Fast reactive flow simulations using analytical Jacobian and dynamic load balancing in OpenFOAM

Detailed chemistry-based computational fluid dynamics (CFD) simulations are computationally expensive due to the solution of the underlying chemical kinetics system of ordinary differential equations (ODEs). Here, we introduce a novel open-source library aiming at speeding up such reactive flow simulations using OpenFOAM, an open-source software for CFD. First, our dynamic load balancing model by Tekgül et al. [“DLBFoam: An open-source dynamic load balancing model for fast reacting flow simulations in OpenFOAM,” Comput. Phys. Commun. 267, 108073 (2021)] is utilized to mitigate the computational imbalance due to chemistry solution in multiprocessor reactive flow simulations. Then, the individual (cell-based) chemistry solutions are optimized by implementing an analytical Jacobian formulation using the open-source library pyJac, and by increasing the efficiency of the ODE solvers by utilizing the standard linear algebra package. We demonstrate the speed-up capabilities of this new library on various combustion problems. These test problems include a two-dimensional (2D) turbulent reacting shear layer and three-dimensional (3D) stratified combustion to highlight the favorable scaling aspects of the library on ignition and flame front initiation setups for dual-fuel combustion. Furthermore, two fundamental 3D demonstrations are provided on non-premixed and partially premixed flames, viz., the Engine Combustion Network Spray A and the Sandia flame D experimental configurations, which were previously considered unfeasible using OpenFOAM. The novel model offers up to two orders of magnitude speed-up for most of the investigated cases. The openly shared code along with the test case setups represent a radically new enabler for reactive flow simulations in the OpenFOAM framework.

An adaptive time-integration scheme for stiff chemistry based on computational singular perturbation and artificial neural networks

We leverage the computational singular perturbation (CSP) theory to develop an adaptive time-integration scheme for stiff chemistry based on a local, projection-based, reduced order model (ROM) freed of the fast time-scales. Its construction is such that artificial neural networks (ANN) can be plugged-in as cheap surrogates of the local projection basis, which is a state function, to alleviate the computational cost, without sacrificing the geometrical and physical foundation of the method. In fact, the solver relies on the synthetic basis in place of the more expensive on-the-fly calculated basis, i.e. the eigenvectors of the Jacobian matrix of the chemical source term, to define the local slow invariant manifold (SIM) and the projection matrix, then integrates explicitly the projected, i.e., non-stiff, chemical source term.We explore the feasibility of the ANN-accelerated CSP solver by training a set of ANNs to predict the projection basis vectors given the local chemical composition in a hydrogen/air homogeneous reactor problem. To enhance the smoothness of the basis vectors and reduce the reconstruction error, we introduce a constrained Jacobian formulation which removes the state heterogeneity due to the presence of temperature along with chemical species, and takes the derivatives enforcing the absolute enthalpy conservation. The test problem highlights the robustness of this ANN approach, arising from the relatively low requirements on the basis accuracy with respect to the requested integration accuracy.

A direct Jacobian total Lagrangian explicit dynamics finite element algorithm for real‐time simulation of hyperelastic materials

AbstractThis article presents a novel direct Jacobian total Lagrangian explicit dynamics (DJ‐TLED) finite element algorithm for real‐time nonlinear mechanics simulation. The nodal force contributions are expressed using only the Jacobian operator, instead of the deformation gradient tensor and finite deformation tensor, for fewer computational operations at run‐time. Owing to this proposed Jacobian formulation, novel expressions are developed for strain invariants and constant components, which are also based on the Jacobian operator. Results show that the proposed DJ‐TLED consumed between 0.70× and 0.88× CPU solution times compared to state‐of‐the‐art TLED and achieved up to 121.72× and 94.26× speed improvements in tetrahedral and hexahedral meshes, respectively, using GPU acceleration. Compared to TLED, the most notable difference is that the notions of stress and strain are not explicitly visible in the proposed DJ‐TLED but embedded implicitly in the formulation of nodal forces. Such a force formulation can be beneficial for fast deformation computation and can be particularly useful if the displacement field is of primary interest, which is demonstrated using a neurosurgical simulation of brain deformations for image‐guided neurosurgery. The present work contributes towards a comprehensive DJ‐TLED algorithm concerning isotropic and anisotropic hyperelastic constitutive models and GPU implementation.

An assessment of the correlation between robust CT-derived ventilation and pulmonary function test in a cohort with no respiratory symptoms.

To evaluate CT-ventilation imaging (CTVI) within a well-characterized, healthy cohort with no respiratory symptoms and examine the correlation between CTVI and concurrent pulmonary function test (PFT). CT scans and PFTs from 77 Caucasian participants in the NORM dataset (clinicaltrials.gov NCT00848406) were analyzed. CTVI was generated using the robust Integrated Jacobian Formulation (IJF) method. IJF estimated total lung capacity (TLC) was computed from CTVI. Bias-adjusted Pearson's correlation between PFT and IJF-based TLC was computed. IJF- and PFT-measured TLC showed a good correlation for both males and females [males: 0.657, 95% CI (0.438-0.797); females: 0.667, 95% CI (0.416-0.817)]. When adjusting for age, height, smoking, and abnormal CT scan, correlation moderated [males: 0.432, 95% CI (0.129-0.655); females: 0.540, 95% CI (0.207-0.753)]. Visual inspection of CTVI revealed participants who had functional defects, despite the fact that all participant had normal high-resolution CT scan. In this study, we demonstrate that IJF computed CTVI has good correlation with concurrent PFT in a well-validated patient cohort with no respiratory symptoms. IJF-computed CTVI's overall numerical robustness and consistency with PFT support its potential as a method for providing spatiotemporal assessment of high and low function areas on volumetric non-contrast CT scan.

Technical Note: On the spatial correlation between robust CT‐ventilation methods and SPECT ventilation

PurposeThe computed tomography (CT)‐derived ventilation imaging methodology employs deformable image registration (DIR) to recover respiratory motion‐induced volume changes from an inhale/exhale CT image pair, as a surrogate for ventilation. The Integrated Jacobian Formulation (IJF) and Mass Conserving Volume Change (MCVC) numerical methods for volume change estimation represent two classes of ventilation methods, namely transformation based and intensity (Hounsfield Unit) based, respectively. Both the IJF and MCVC methods utilize subregional volume change measurements that satisfy a specified uncertainty tolerance. In previous publications, the ventilation images resulting from this numerical strategy demonstrated robustness to DIR variations. However, the reduced measurement uncertainty comes at the expense of measurement resolution. The purpose of this study was to examine the spatial correlation between robust CT‐ventilation images and single photon emission CT‐ventilation (SPECT‐V).MethodsPreviously described implementations of IJF and MCVC require the solution of a large scale, constrained linear least squares problem defined by a series of robust subregional volume change measurements. We introduce a simpler parameterized implementation that reduces the number of unknowns while increasing the number of data points in the resulting least squares problem. A parameter sweep of the measurement uncertainty tolerance, τ, was conducted using the 4DCT and SPECT‐V images acquired for 15 non‐small cell lung cancer patients prior to radiotherapy. For each test case, MCVC and IJF CT‐ventilation images were created for 30 different uncertainty parameter values, uniformly sampled from the range 0.01,0.25. Voxel‐wise Spearman correlation between the SPECT‐V and the resulting CT‐ventilation images was computed.ResultsThe median correlations between MCVC and SPECT‐V ranged from 0.20 to 0.48 across the parameter sweep, while the median correlations for IJF and SPECT‐V ranged between 0.79 and 0.82. For the optimal IJF tolerance τ=0.07, the IJF and SPECT‐V correlations across all 15 test cases ranged between 0.12 and 0.90. For the optimal MCVC tolerance τ=0.03, the MCVC and SPECT‐V correlations across all 15 test cases ranged between −0.06 and 0.84.ConclusionThe reported correlations indicate that robust methods generate ventilation images that are spatially consistent with SPECT‐V, with the transformation‐based IJF method yielding higher correlations than those previously reported in the literature. For both methods, overall correlations were found to marginally vary for τ∈[0.03,0.15], indicating that the clinical utility of both methods is robust to both uncertainty tolerance and DIR solution.

Robust HU-based CT ventilation from an integrated mass conservation formulation.

Computed tomography (CT) ventilation algorithms estimate volume changes induced by respiratory motion. Existing Hounsfield Unit (HU) methods approximate volume change from the measured HU variations between spatially corresponding voxel locations within a temporally resolved CT image pair, assuming that volume changes are caused solely by changes in air content. Numerical implementations require a deformable image registration to determine the inhale/exhale spatial correspondence, a preprocessing lung volume segmentation, a preprocessing high‐intensity vessel segmentation, and a post‐processing smoothing applied to the raw volume change estimates obtained for each lung tissue voxel.PurposeWe introduce the novel mass‐conserving volume change (MCVC) method for estimating voxel volume changes from the HU values within an inhale/exhale CT image pair. MCVC is based on subregional volume change estimates that possess quantitatively characterized and controllable levels of uncertainty. MCVC is therefore robust to small variations in DIR solutions and the resulting ventilation images are overall more reproducible. In contrast to existing HU methods, MCVC does not require a preprocessing lung vessel segmentation or pre/post‐processing Gaussian smoothing.MethodsSubregional volume change estimates are defined in terms of mean density ratios. As such, the corresponding uncertainty is characterized using Gaussian statistics and standard error analysis of the sample density means. A numerical solution is obtained from the MCVC formulation by solving a constrained linear least squares problem defined by a series of subregional volume change estimates. Reproducibility of the MCVC method with respect to DIR solution was assessed using expert‐determined landmark point pairs and inhale/exhale phases from 10 four‐dimensional CTs (4DCTs) available on http://www.dir-lab.com. MCVC was also compared to the robust Integrated Jacobian Formulation (IJF), a transformation‐based ventilation method.ResultsThe ten Dir‐Lab 4DCT cases were registered twice with the same DIR algorithm, but using different degrees of freedom (DIR 1 and DIR 2). Standard HU ventilation (HUV) and MCVC ventilation images were computed for both solutions. The average spatial errors (300 landmarks per case) for DIR 1 ranged between 0.74 and 1.50 mm, whereas for DIR 2 they ranged between 0.68 and 1.18 mm. Despite the differences in spatial errors between the two DIR solutions, the average Pearson correlation between the corresponding HUV images was 0.94 (0.03), while for the MCVC images it was 1.00 (0.00). The average correlation between MCVC and IJF ventilation over the ten test cases was 0.81 (0.14), whereas for HUV and IJF it was 0.56 (1.11).ConclusionWhile HUV is robust to DIR solution, its implementation depends on heuristic Gaussian smoothing and vessel segmentation. MCVC is based on subregional volume change measurements with quantifiable and controllable levels of uncertainty. The subregional approach eliminates the need for Gaussian smoothing and lung vasculature segmentation. Numerical experiments are consistent with the underlying mathematical theory and indicate that MCVC ventilation is more reproducible with respect to DIR algorithm than standard HU methods. MCVC results are also more consistent with the robust IJF method, which suggests that incorporating robustness leads to more consistent results across both DIRs and ventilation algorithms.

Robust CT ventilation from the integral formulation of the Jacobian.

Computed tomography (CT) derived ventilation algorithms estimate the apparent voxel volume changes within an inhale/exhale CT image pair. Transformation‐based methods compute these estimates solely from the spatial transformation acquired by applying a deformable image registration (DIR) algorithm to the image pair. However, approaches based on finite difference approximations of the transformation's Jacobian have been shown to be numerically unstable. As a result, transformation‐based CT ventilation is poorly reproducible with respect to both DIR algorithm and CT acquisition method.PurposeWe introduce a novel Integrated Jacobian Formulation (IJF) method for estimating voxel volume changes under a DIR‐recovered spatial transformation. The method is based on computing volume estimates of DIR mapped subregions using the hit‐or‐miss sampling algorithm for integral approximation. The novel approach allows for regional volume change estimates that (a) respect the resolution of the digital grid and (b) are based on approximations with quantitatively characterized and controllable levels of uncertainty. As such, the IJF method is designed to be robust to variations in DIR solutions and thus overall more reproducible.MethodsNumerically, Jacobian estimates are recovered by solving a simple constrained linear least squares problem that guarantees the recovered global volume change is equal to the global volume change obtained from the inhale and exhale lung segmentation masks. Reproducibility of the IJF method with respect to DIR solution was assessed using the expert‐determined landmark point pairs and inhale/exhale phases from 10 four‐dimensional computed tomographies (4DCTs) available on http://www.dir-lab.com. Reproducibility with respect to CT acquisition was assessed on the 4DCT and 4D cone beam CT (4DCBCT) images acquired for five lung cancer patients prior to radiotherapy.ResultsThe ten Dir‐Lab 4DCT cases were registered twice with the same DIR algorithm, but with different smoothing parameter. Finite difference Jacobian (FDJ) and IFJ images were computed for both solutions. The average spatial errors (300 landmarks per case) for the two DIR solution methods were 0.98 (1.10) and 1.02 (1.11). The average Pearson correlation between the FDJ images computed from the two DIR solutions was 0.83 (0.03), while for the IJF images it was 1.00 (0.00). For intermodality assessment, the IJF and FDJ images were computed from the 4DCT and 4DCBCT of five patients. The average Pearson correlation of the spatially aligned FDJ images was 0.27 (0.11), while it was 0.77 (0.13) for the IFJ method.ConclusionThe mathematical theory underpinning the IJF method allows for the generation of ventilation images that are (a) computed with respect to DIR spatial accuracy on the digital voxel grid and (b) based on DIR‐measured subregional volume change estimates acquired with quantifiable and controllable levels of uncertainty. Analyses of the experiments are consistent with the mathematical theory and indicate that IJF ventilation imaging has a higher reproducibility with respect to both DIR algorithm and CT acquisition method, in comparison to the standard finite difference approach.

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.

Stable posture control for planar hyper-redundant arms using selective control points

The high degree of freedom in hyper-redundant arms makes resolving their redundancy using Jacobian techniques impractical. Furthermore, Jacobian techniques are adapted for the specification of tasks focused around the end-effector, they are not well equipped to capture tasks where arm posture control is preponderant like locomotion or manipulation. For these type of arms, parameterized modal curve based approaches are often deemed superior. However, these curves restrict the arms to a predefined type of postures and adding extra mode functions to account for more complex postures makes solving the inverse kinematics more challenging. This paper explores a method to reliably achieve posture tracking for hyper-redundant planar arms using the more simple Jacobian formulation. The introduced method attempts to track arbitrary target postures. This method is characterized by the ability for the arm to depart from, traverse or land on singular postures.

Dual-arm cooperative manipulation under joint limit constraints

Abstract Cooperative manipulation of a rigid object is challenging and represents an interesting and active research area, especially when these robots are subject to joint and task prioritization constraints. In cooperative manipulation, a primary task is to maintain the coordination of motions, to avoid severe damage caused by the violation of kinematic constraints imposed by the closed chain mechanism. This paper proposes a kinematic controller for dual-arm cooperative manipulation that ensures safety by providing relative coordinated motion as highest priority task and joint limit avoidance and world-space trajectory following at a lower priority. The coordination of motions is based on modular relative Jacobian formulation. The approach is applicable to systems composed of redundant or non-redundant manipulators. Experiments in simulation demonstrate the behavior of the approach under different redundancy configurations. Experiments on two robots with different number of redundant motions show the applicability of the proposed approach to cooperative manipulation under joint limit constraints.

Dynamic isotropy in 6-DOF kinematically constrained platforms by three elastic nodal joints

The principle of kinematic design has a wide range of applications e.g. from optical mirror mounts to parallel robots. Despite the importance of dynamic isotropy in the optimization of dynamic performance, a thorough analysis of dynamic isotropy in different kinematic arrangements has not yet been addressed in the literature. Dynamic isotropy, leading to equal eigenfrequencies, is a powerful optimization measure. In this paper, we present fully-parametric solutions for obtaining dynamic isotropy in general 3D platforms kinematically constrained by three elastic nodal joints in 6 DOFs. It is analytically shown that there exist two possible kinematic arrangements which are described by 3-2-1 and 2-2-2 kinematic node spaces. Both kinematic arrangements are studied with respect to their Jacobian formulation, Jacobian singularity and stiffness decoupling. It is proven that decoupling of stiffness matrices and accordingly dynamic isotropy for both kinematic arrangements are possible. Subsequently, conditions concerning geometry, stiffness and inertia in order to obtain dynamic isotropy are parametrically established. Finally, it is numerically demonstrated that the presented formulation is general enough even for being directly used, as a novel and efficient approach, in order to design dynamically isotropic 6-6 Gough–Stewart platforms (6-6 hexapods).

Investigating Task Prioritization and Holistic Coordination Using Relative Jacobian for Combined 3-Arm Cooperating Parallel Manipulators

A new modular relative Jacobian formulation for single end-effector control of combined 3-arm cooperating parallel manipulators is derived. It is based on a previous method of derivation for dual-arm robots, with the same approach of modularity and single end-effector control for combined manipulators. This paper will present this new formulation, as well as investigate task prioritization scheme to verify the claim that a single end-effector controller of combined manipulators will indeed implement a strict task prioritization, by intentionally adding more tasks. In addition, this paper will investigate a claim that the holistic approach to control of combined manipulators affords easier control coordination of each of the stand-alone components. Switching control from an individual manipulator control in the null space to relative control in the tasks space is shown to investigate the smoothness of task execution during switching. Simulation results using Gazebo 2.2.5 running in Ubuntu 14.04 is shown.

Solving the dynamic equations of a 3-PRS Parallel Manipulator for efficient model-based designs

Abstract. Introduction of parallel manipulator systems for different applications areas has influenced many researchers to develop techniques for obtaining accurate and computational efficient inverse dynamic models. Some subject areas make use of these models, such as, optimal design, parameter identification, model based control and even actuation redundancy approaches. In this context, by revisiting some of the current computationally-efficient solutions for obtaining the inverse dynamic model of parallel manipulators, this paper compares three different methods for inverse dynamic modelling of a general, lower mobility, 3-PRS parallel manipulator. The first method obtains the inverse dynamic model by describing the manipulator as three open kinematic chains. Then, vector-loop closure constraints are introduced for obtaining the relationship between the dynamics of the open kinematic chains (such as a serial robot) and the closed chains (such as a parallel robot). The second method exploits certain characteristics of parallel manipulators such that the platform and the links are considered as independent subsystems. The proposed third method is similar to the second method but it uses a different Jacobian matrix formulation in order to reduce computational complexity. Analysis of these numerical formulations will provide fundamental software support for efficient model-based designs. In addition, computational cost reduction presented in this paper can also be an effective guideline for optimal design of this type of manipulator and for real-time embedded control.

Distribution System State Estimation Model Using a Reduced Quasi-Symmetric Impedance Matrix

This paper presents a novel formulation of the Distribution System State Estimation (DSSE) optimization model. For a given electric three-phase circuit feeder, network models are built using a quasi-symmetric impedance matrix TRX representing the entire structure and topology of the radial network. As a key contribution, the state variables of demands and generators connected to large-scale distribution grids are obtained by using a convenient matrix reduction technique. As a result, the size of the optimization problem is considerably reduced with respect to the jacobian formulation by considering radial and weakly meshed exploitation and elimination of interconnecting nodes. Results and comparative analysis are presented using the IEEE 4-, 13-, 37-, 123-, and 8500-node test systems.