Spatial Operator Algebra Research Articles

Adaptive Fuzzy Integral Sliding Mode Cooperative Control Based on Time-Delay Estimation for Free-Floating Close-Chain Manipulators.

Space manipulators are expected to perform more challenging missions in on-orbit service (OOS) systems, but there are some unique characteristics that are not found on ground-based robots, such as dynamic coupling between space bases and manipulators, limited fuel supply, and working with unfixed bases. This paper focuses on trajectory-tracking control and internal force control for free-floating close-chain manipulators. First, the kinematics and dynamics of free-floating close-chain manipulators are given using the momentum conservation and spatial operator algebra (SOA) methodologies, respectively. Furthermore, an adaptive fuzzy integral sliding mode controller (AFISMC) based on time delay estimation (TDE) was designed for trajectory-tracking control, and a proportional-integral (PI) control strategy was adopted for internal force control. The global asymptotic stability of the proposed controller was proven by using the Lyapunov methodology. Three cases were conducted to verify the efficiency of the controller by using numerical simulations on two six-link manipulators with a free-floating base. The controller presents the desired tracking capability.

Efficiency assessment of SOA-based computed torque control: A comparative analysis with NE-based approach

This paper proposes the spatial operator algebra-based computed torque control scheme applied to an anthropomorphic 3-degree of freedom robotic manipulator, which aims to reduce the computational cost of the classical methods and integrate the advantages of low computational cost into advanced robotic control systems. The computational cost is increased due to the calculations of the inverse dynamic and the feedback control loop. The spatial operator algebra (SOA) algorithm provides a systematic derivation, evaluation, and subsequent conceptual interpretation of the manipulator dynamics model. This paper presents a powerful and efficient solution for controlling the dynamics and trajectory of the manipulator. In order to show the efficiency of the solution, the Newton Euler (NE) based control schemes are also applied to the manipulator. The SOA-based controller significantly reduced the computational cost and performed approximately 60%–70% faster than the NE-based controller. Furthermore, different initial states, disturbances, and uncertainty tests are implemented and the SOA-based controller demonstrated successful performance under various conditions while maintaining a lower computational cost. In this study, the advantages and limitations of each method (SOA-based, NE-based) are evaluated and the potential benefits of using SOA in the computed torque control scheme are highlighted. The SOA-based controller, which was verified by simulation, is then implemented in real-time and showed successful performance.

Minimum Base Attitude Disturbance Planning for a Space Robot During Target Capture

This paper presents a method to minimize the base attitude disturbance of a space robot during target capture. First, a general dynamic model of a free-floating space robot capturing a target is established using spatial operator Algebra, and a simple analytical formula for the base angular velocity change during the impact phase is obtained. Compared with the former models proposed in the literature, this model has a simpler form, a wider range of applications, and O(n) computation complexity. Second, based on the orthogonal projection matrix lemma, we propose the generalized mass Jacobian matrix (GMJM) and find that the base angular velocity change is a constant multiple of the component which the impact impulse projects to the column space of the GMJM. Third, a new concept, the base attitude disturbance ellipsoid (BADE), is proposed to express the relationship between the base attitude disturbance and the impact direction. The impact direction satisfying the minimum base attitude disturbance can be straightforwardly obtained from the BADE. In particular, for a planar space robot, we draw the useful conclusion that the impact direction unchanged base attitude must exist. Furthermore, the average axial length of the BADE is used as a measurement to illustrate the average base attitude disturbance under impact impulses from different directions. With this measurement, the desired pre-impact configuration with minimum average base attitude disturbance can be easily determined. The validity and the efficiency of this method are verified using a three-link planar space robot and a 7DOF space robot.

An analytical workbench for system level multibody dynamics

There has been considerable focus in the research community on developing accurate models, as well as on fast algorithms for solving the equations of motion of multibody systems required for simulating the dynamics of such systems. This paper focuses on the less explored complementary topic of evaluating system level dynamics properties of multibody systems. Examples of such dynamics properties are the system mass matrix, Jacobians and sensitivities of these quantities. These system level quantities manifest the dynamical properties of the system and are important for design, optimization and control. While such system level quantities are often used in theory to describe the underlying mathematical physics of the systems, due to their complexity there is a lack of a systematic methods for computing them when desired. In this paper we describe a computational workbench framework that provides a bridge between theory and the computation of such system level quantities. This workbench framework builds upon the Spatial Operator Algebra (SOA) that has been used for analysis and algorithm development for multibody dynamics. Mathematical operator expressions from the SOA can be transcribed literally to the workbench command line to allow the easy evaluation of complex dynamics quantities. We use a specific Python/C++ implementation of a workbench called PyCraft to illustrate the structure and use of such a workbench. Several examples illustrating operator-based analysis and corresponding PyCraft-based computation are included.

Path defined directed graph vector (Pgraph) method for multibody dynamics

Dynamical modeling of complex systems may include multiple combinations of serial and tree topologies, as well as closed kinematic chain systems. Simulating of such systems is a costly task not only for run time but also for construction time when a model description and constraints are introduced. This paper presents a systematic framework of a construction time efficient modeling methodology for complex systems by introducing a path defined directed graph vector (\(\mathit{Pgraph}\)) and the associated methodology based on Linear Graph Theory (LGT). Updating of body coordinate frame vectors which can be a challenge in complex topology systems is easily performed using the proposed method. This technique is especially useful for systems changing topology, such as walking mechanisms where bilateral and unilateral constraints are conditionally embedded. Although this is a general methodology, to be combined with many dynamical modeling algorithms available in the literature, here we demonstrate it using Spatial Operator Algebra (SOA), which is a recursive algorithm based on the Newton–Euler formalism.

Pre-Impact Configuration Designing of a Robot Manipulator for Impact Minimization

This paper studies the collision problem of a robot manipulator and presents a method to minimize the impact force by pre-impact configuration designing. First, a general dynamic model of a robot manipulator capturing a target is established by spatial operator algebra (SOA) and a simple analytical formula of the impact force is obtained. Compared with former models proposed in literatures, this model has simpler form, wider range of applications, O(n) computation complexity, and the system Jacobian matrix can be provided as a production of the configuration matrix and the joint matrix. Second, this work utilizes the impulse ellipsoid to analyze the influence of the pre-impact configuration and the impact direction on the impact force. To illustrate the inertia message of each body in the joint space, a new concept of inertia quasi-ellipsoid (IQE) is introduced. We find that the impulse ellipsoid is constituted of the inertia ellipsoids of the robot manipulator and the target, while each inertia ellipsoid is composed of a series of inertia quasi-ellipsoids. When all inertia quasi-ellipsoids exhibit maximum (minimum) coupling, the impulse ellipsoid should be the flattest (roundest). Finally, this paper provides the analytical expression of the impulse ellipsoid, and the eigenvalues and eigenvectors are used as measurements to illustrate the size and direction of the impulse ellipsoid. With this measurement, the desired pre-impact configuration and the impact direction with minimum impact force can be easily solved. The validity and efficiency of this method are verified by a PUMA robot and a spatial robot.

Dynamic Modeling and Analysis of a Lightweight Robotic Manipulator in Joint Space

Abstract The primary importance of the paper is the application of the efficient formulation for the simulation of open-loop lightweight robotic manipulator. The framework employed in the paper makes use of the spatial operator algebra and the associated equations are expressed in joint space. This compact representation of the manipulator dynamics makes it possible to solve the robot forward and inverse dynamics problems in a recursive and fast manner. In the current form, the presented algorithm can be applied for the dynamics simulation of an open-loop chain system possessing any number of joints. Specifically, the formulation has been successfully applied for the analysis of the 7DOF KUKA LWR robot. Results from a number of test cases for the robot demonstrate the verification of the calculations

Internal coordinate molecular dynamics: a foundation for multiscale dynamics.

Internal coordinates such as bond lengths, bond angles, and torsion angles (BAT) are natural coordinates for describing a bonded molecular system. However, the molecular dynamics (MD) simulation methods that are widely used for proteins, DNA, and polymers are based on Cartesian coordinates owing to the mathematical simplicity of the equations of motion. However, constraints are often needed with Cartesian MD simulations to enhance the conformational sampling. This makes the equations of motion in the Cartesian coordinates differential-algebraic, which adversely impacts the complexity and the robustness of the simulations. On the other hand, constraints can be easily placed in BAT coordinates by removing the degrees of freedom that need to be constrained. Thus, the internal coordinate MD (ICMD) offers an attractive alternative to Cartesian coordinate MD for developing multiscale MD method. The torsional MD method is a special adaptation of the ICMD method, where all the bond lengths and bond angles are kept rigid. The advantages of ICMD simulation methods are the longer time step size afforded by freezing high frequency degrees of freedom and performing a conformational search in the more important low frequency torsional degrees of freedom. However, the advancements in the ICMD simulations have been slow and stifled by long-standing mathematical bottlenecks. In this review, we summarize the recent mathematical advancements we have made based on spatial operator algebra, in developing a robust long time scale ICMD simulation toolkit useful for various applications. We also present the applications of ICMD simulations to study conformational changes in proteins and protein structure refinement. We review the advantages of the ICMD simulations over the Cartesian simulations when used with enhanced sampling methods and project the future use of ICMD simulations in protein dynamics.

Multi-Objective Trajectory Planning of FFSM Carrying a Heavy Payload

Aiming at carrying a heavy payload to a desired pose (including position and orientation), a multi-objective optimization-based approach for maximum-payload trajectory planning of free-floating space manipulators (FFSM) is proposed in this paper. The presented approach corresponds to two typical applications: (i) the manipulator joints attain the desired states; (ii) the inertial pose of the end-effector (pose with respect to the inertial frame) attains the desired values, for which a novel two-stage method is presented. Firstly, for the purpose of reducing computational complexity, dynamics equations are derived using a spatial operator algebra (SOA) method. Secondly, objective functions are defined according to the improvement of load-carrying capacity and pose requirements of the end-effector. Then, the joint trajectories are specified using sinusoidal polynomial functions. Finally, a multi-objective particle optimization (MOPSO) algorithm is employed to obtain a non-dominated solution set, during which process particles that do not satisfy the constraints are eliminated. Simulations are performed for a 7-DOF FFSM, considering three and five objectives for optimization in the two applications, respectively. The results demonstrate that the proposed approach can provide satisfactory joint trajectories and improve load-carrying capacity effectively.

GNEIMO-Fixman: An Accurate Torsional Molecular Dynamics Simulation Method for Studying Biomolecular Dynamics

Torsion angle Molecular Dynamics simulations provide a powerful alternative to all-atom MD since they place constraints on the bond lengths and angles, and thereby efficiently simulate large scale conformational changes in torsional space. The previous uses of torsional MD have been severely limited by the lack of efficient algorithms to solve the mathematical complexities of torsional MD. Our development of an advanced Generalized Newton-Euler Inverse Mass Operator (GNEIMO) torsional MD method overcomes these limitations with several theoretical and algorithmic advancements. We will outline these advancements and the effectiveness of the GNEIMO method in critical applications such as protein homology model refinement and protein conformational dynamics.Due to the rigidity in the model the torsional MD simulations show a bias in the partition function, which results in inaccuracies in the calculated thermodynamic and kinetic properties. Fixman developed a compensating potential that rigorously annuls the bias in the partition function, but the calculation of the Fixman potential has remained mathematically intractable even for moderate size molecules. Recently, we have developed a low-cost algorithm, based on Spatial Operator Algebra for calculating Fixman potential and torque for any molecule. We will present results on the validation and the effect of the Fixman correction potential in recovering the correct population distribution function, transition rates, and free energy surface for large branched systems.We have designed a modular software platform for the GNEIMO-Fixman method that can be readily combined with any type of forcefield engine. We have combined GNEIMO algorithm with Rosetta forcefield and performed homology model refinement with torsional MD. Thus our GNEIMO-Fixman torsional MD method allows for the use of larger integration time steps than for all-atom MD simulations, and enables enhanced conformational sampling in the low frequency torsional degrees of freedom.

Fixman compensating potential for general branched molecules.

The technique of constraining high frequency modes of molecular motion is an effective way to increase simulation time scale and improve conformational sampling in molecular dynamics simulations. However, it has been shown that constraints on higher frequency modes such as bond lengths and bond angles stiffen the molecular model, thereby introducing systematic biases in the statistical behavior of the simulations. Fixman proposed a compensating potential to remove such biases in the thermodynamic and kinetic properties calculated from dynamics simulations. Previous implementations of the Fixman potential have been limited to only short serial chain systems. In this paper, we present a spatial operator algebra based algorithm to calculate the Fixman potential and its gradient within constrained dynamics simulations for branched topology molecules of any size. Our numerical studies on molecules of increasing complexity validate our algorithm by demonstrating recovery of the dihedral angle probability distribution function for systems that range in complexity from serial chains to protein molecules. We observe that the Fixman compensating potential recovers the free energy surface of a serial chain polymer, thus annulling the biases caused by constraining the bond lengths and bond angles. The inclusion of Fixman potential entails only a modest increase in the computational cost in these simulations. We believe that this work represents the first instance where the Fixman potential has been used for general branched systems, and establishes the viability for its use in constrained dynamics simulations of proteins and other macromolecules.

GNEIMO Constrained Dynamics Method: A Tool for Protein Structure Refinement and Conformational Changes

Constrained molecular dynamics methods, wherein the high frequency degrees of freedom are placed as hard holonomic constraints in the dynamic model of the protein have been developed several decades ago, but these methods are not used widely. There are several bottlenecks in using these methods, the most formidable being the computational time taken for solving the coupled equations of motion. Spatial operator algebra(SOA) techniques reduces the computational time for solving the equations of motion by two orders of magnitude. We have developed a computational framework called GNEIMO, that uses the SOA techniques combined with all-atom force field and appropriate integrators to solve the constrained equations of motion. The generalized constrained molecular dynamics method GNEIMO, allows the user to “freeze and thaw” torsional degrees of freedom as fit for the problem studied.We will demonstrate the use of GNEIMO method in protein structure refinement of low resolution homology models. Starting from low resolution homology models we observed that the all-torsion GNEIMO method leads to a 2A improvement in RMSD to the crystal structure, while the all-atom molecular dynamics method disrupts the starting model further. The GNEIMO method also showed enrichment in the population density of native-like conformations.We have also tested out the GNEIMO method for studying conformational transitions between two well characterized (crystal structures) conformations of a protein. Long time scale dynamics with GNEIMO on calmodulin and fasciculin shows that the transitions from one conformation to another happen with more facility than with all-atom MD simulations.

Spatial Operator Algebra for Free-floating Space Robot Modeling and Simulation

As the dynamic equations of space robots are highly nonlinear,strongly coupled and nonholonomic constrained,the efficiency of current dynamic modeling algorithms is difficult to meet the requirements of real-time simulation.This paper combines an efficient spatial operator algebra(SOA) algorithm for base fixed robots with the conservation of linear and angular momentum theory to establish dynamic equations for the free-floating space robot,and analyzes the influence to the base body's position and posture when the manipulator is capturing a target.The recursive Newton-Euler kinematic equations on screw form for the space robot are derived,and the techniques of the sequential filtering and smoothing methods in optimal estimation theory are used to derive an innovation factorization and inverse of the generalized mass matrix which immediately achieve high computational efficiency.The high efficient SOA algorithm is spatially recursive and has a simple math expression and a clear physical understanding,and its computational complexity grows only linearly with the number of degrees of freedom.Finally,a space robot with three degrees of freedom manipulator is simulated in Matematica 6.0.Compared with ADAMS,the simulation reveals that the SOA algorithm is much more efficient to solve the forward and inverse dynamic problems.As a result,the requirements of real-time simulation for dynamics of free-floating space robot are solved and a new analytic modeling system is established for free-floating space robot.

GNEIMO: Constrained Molecular Dynamics Methods For Long Time Scale Simulation of Macromolecules

The two leading causes for the limitation of the all-atom molecular dynamics (MD) simulation timescales are: 1. calculation of forces that scale as the square of the number of atoms and, 2. the integration time step is limited to 1fs due to the high frequency modes in the protein. High performance technologies and better force calculation algorithms have addressed the former, and we address the latter issue in this work.Here we report a constrained MD method, GNEIMO (Generalized Newton-Euler inverse mass operator method), that is capable of achieving stable dynamics with integration time steps as large as 10 to 20fs. The GNEIMO method provides a platform to perform long time scale hierarchical simulations ranging from all-atom simulations, coarse-grained dynamics of clusters of few atoms, to dynamics with larger motifs constrained, at lesser computational expense compared to all-atom MD. GNEIMO method uses spatial operator algebra to solve for the internal coordinate dynamics with computational cost scaling linearly as the number of degrees of freedom.The current implementation of GNEIMO is capable of performing constant temperature Nose-Hoover dynamics, with continuum Generalized Born solvation. We use adaptive step size integration to provide stable dynamics with larger time steps. We have carried out tens of nano-seconds of stable dynamics for proteins with time steps as large as 10 to 20fs for different integrators. We report results from conformational changes of domains in proteins from long time scale dynamics. This implementation integrates the force-field module from the MD program LAMMPS, with constrained dynamics module from NASA-Jet Propulsion laboratory.

Computational Dynamics of Multi-body System Modeling Based on Spatial Operator Algebra Theory

In order to improve the modeling efficiency for the forward and inverse computational dynamics of the multi-body system, the spatial operator algebra theory is adopted.The realization of the forward and inverse computational dynamics of the multi-body system procedure is built in the environment of Mathematica 6.0 software.The generalized mass is an important factor to link rotational force vectors and rotational acceleration vectors,and the high effective iterative method can be used to compute forward and inverse dynamics.The equation for the generalized mass,forward and inverse dynamics based on the spatial operator algebra theory,has a simple math expression and a clear physical meaning.The software structure of the forward and inverse dynamics is built,the modeling for the IDEF0 method is presented and the multi-body system computational dynamics parameter simulation model is realized on the basic of integrated procedure VB.NET and Mathematica 6.0.According to the analysis,the multi-body system of PUMA560 robot forward and inverse dynamics is studied,and correctness and validity is verified by the computed examples.

Study on the Dynamics Control of Industrial Robot Modeling Based on Spatial Operator Algebra Theory

In order to improve the modeling efficiency of the industrial robot dynamics control, the recursive dynamics is studied based on the Spatial Operator Algebra (SOA) theory, and the procedure realization is built in the environment of Matahematica6.0 software. The high effective recursive forward and reward dynamics for the SOA theory, has a simple math expression and a clear physical meaning. The software structure of the forward and inverse dynamics is built and the industrial robot dynamics simulation model is realized based on the integrated procedure VB.NET and Mathematica 6.0. According to the analysis, the example of PUMA560 robot forward and inverse dynamics is studied, and correctness and validity is verified by computed examples.

Control of Manipulators with some Unactuated Joints

A mobile-based robot, a robot with an actuator failure, a manipulator with structural flexibility, and robots grasping an articulated object are systems with unactuated or "passive" joints. The dynamics and control of these robot systems is more complex than fully-actuated systems because unactuated joints may lead to uncontrollability. Based on a recursive formulation of the inverse dynamics for manipulators with some unactuated joints, control laws are developed for joint space control and task space control. Important in the task space control law for underactuated systems is the inversion of the generalized Jacobian. Spatial operator algebra techniques are used to develop computationally efficient algorithms for the inverse generalized Jacobian and the control laws.

Linearization of manipulator dynamics using spatial operators

Techniques from the spatial operator algebra are used to obtain closed-form operator expressions for two types of linearized dynamics models: the linearized inverse and forward dynamics models. Spatially recursive algorithms of O(n) and O(n/sup 2/) complexity for the computation of the perturbation vector and coefficient matrices for the linearized inverse dynamics model (LIDM) are developed. Operator factorization and inversion identities are used to develop corresponding closed-form expressions for the linearized forward dynamics model (LFDM). Once again, these are used to develop algorithms of O(n) and O(n/sup 2/) complexity for the computation of the perturbation vector and the coefficient matrices. The algorithms for the LFDM do not require the explicit computation of the mass matrix nor its numerical inversion and are also of lower complexity than the conventional O(n/sup 3/) algorithms. >

Recursive Formulation of Operational Space Control

A recently developed spatial operator algebra approach to modeling and analysis of multibody robotic systems is used to develop O(n) recursive algorithms that compute the op erational space mass matrix and the operational space coriolis/centrifugal and gravity terms of an n-link serial manipulator. These algorithms enable an O(n) recursive im plementation of operational space control.

A Spatial Operator Algebra for Manipulator Modeling and Control

A recently developed spatial operator algebra for manipu lator modeling, control, and trajectory design is dis cussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high- level framework for describing the dynamic and kinematic behavior of a manipulator and for control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced concep tual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus the transition from an abstract problem formulation and solution to the detailed mechanization of specific algorithms is greatly simplified.