Decoupled Natural Orthogonal Complement Research Articles

Impact modeling and reactionless control for post-capturing and maneuvering of orbiting objects using a multi-arm space robot

Autonomous on-orbit servicing, such as capture, refuel, repair and refurbishment of on-orbit satellites using a robotic arm mounted on servicing satellite is one of the important components of future’s space missions. Space robots increase reliability, safety, and ease of execution of space operations, but pose a novel challenge due to micro-gravity and space environments. While capturing high speed orbiting objects, robotic arms undergo impact and require appropriate modeling of the system. In this paper, a unified framework is provided for modeling impact dynamics, post-capture stabilization and target maneuvering of a multi-arm robotic system mounted on a servicing satellite while capturing orbiting objects. The dynamic model of multi-arm space robot is obtained using the Decoupled Natural Orthogonal Complement (DeNOC) based formulation and closed-loop constraint equations. All three phases of the capturing operation, namely, approach, impact, and post-impact are modeled using Impulse-momentum approach and conservation of momentum. In the approach phase, robot arms are planned to move from its initial configuration to the desired capture configuration. In the impact phase, a framework is developed to estimate the impulse forces and changes in the generalized velocities caused by the impact. In post-impact phase, these velocities are used as initial conditions for the post-impact dynamics simulations. The uncontrolled dynamics during post-impact will result in an undesirable motion, thus post-impact reactionless control (minimum base disturbance) strategy is used to maneuver the space robot’s arms and target object. As such, the robotic arms can be used to maneuver an astronaut for repair of satellite. Most of the times the parameters of target object are not known. Hence, an adaptive reactionless control strategy has been devised for capturing object with unknown parameters. The effectiveness of the framework is shown using a dual-arm robot mounted on a servicing satellite performing capturing operation for multiple objects. The effects of relative velocity and angle of approach on the impact forces are also investigated.

Dynamic simulation of serial robots under force control

The advantages of using force control in industrial robots are well known. Study of such systems in virtual environments in the form of simulation is of great help as most of the force controlled task works in close contact with the environment. In this paper, we show how to simulate different force control algorithms of a typical serial robot used in industries before deciding to choose a suitable one for real implementation. Hence, a proper dynamic model of the robot is essential which should be able to emulate the real robot, particularly if the robot moves at relatively higher speeds. This is done here using the concept of the decoupled natural orthogonal complement (DeNOC) matrices which is known to provide a recursive forward dynamic algorithm that is not only efficient but also numerically stable. Such simulation of robots under force control will allow users to tune the control gains without stopping the real robot on the production floor. Besides, such simulation can be used as an education tool as well to help beginners to explore various types of control algorithms and their performances. In addition, the framework for simulation proposed in this paper can work as a good test bench to test the performances of either a new control law or a different dynamic algorithm. As an illustration, the DeNOC based dynamics was substituted with MATLAB's SimMechanics which can also perform dynamic simulation. The comparison of the results validated the concept and correctness of the numerical simulations.

Forward dynamic analysis of parallel robots using modified decoupled natural orthogonal complement method

This paper aims at the forward dynamic analysis of parallel robots by adding an effective modification on the Decoupled Natural Orthogonal Complement (DeNOC) method incorporating Virtual Spring Method (VSM). Forward dynamic analysis of parallel manipulators have been studied by many researches during the last decades, however, due to the complexity of closed-loop structures, it is still an attractive research subject in robotics. Developing VSM in energy based dynamic methods i.e., Lagrange formulation, has been demonstrated in literature before. In the current work, first, the VSM is briefly reviewed and its liaison with the Augmented Lagrange Method (ALM) is discussed. Next, among the Newtonian methods, the DeNOC which is based on the screw theory is elaborated. Then, inspired by the relation between VSM and ALM, the modified DeNOC (MDeNOC) is introduced. For more clarification, a 2DOF-five bar planar mechanism is chosen as case study and the aforementioned dynamic approaches are applied. Finally, the advantages and disadvantages of each method are illustrated. The key advantage of MDeNOC method is the ability of dynamic response and constraint reactions calculations with less preparation effort compared to DeNOC, while its accuracy is similar to Lagrange-VSM.

Dynamics and actuating torque optimization of planar robots

An optimization methodology is presented for design of serial-chain planar robots for minimizing torque at joints, when its endeffector is supposed to move on a prescribed path. In particular, the end-effector of the robot is allowed to move on a circular path. For the respective joint trajectories, the weighted sum of root mean square (RMS) of the actuating torques is minimized by the mass redistribution of the links. To achieve the goal, the DeNOC (Decoupled natural orthogonal complement) based dynamics was formulated by representing the rigid links as a set of rigidly connected point-masses known as equimomental system. The methodology is illustrated using a planar two-degree-of-freedom (DOF) robot with two revolute joints.

Dynamics of serial kinematic chains with large number of degrees-of-freedom

This paper investigates the dynamic behaviour of serial chains with degrees-of-freedom (DOF) as large as 100,000. A recursive solver called Recursive Dynamic Simulator (ReDySim), based on the Newton–Euler formulation and the Decoupled Natural Orthogonal Complement (DeNOC) matrices, was used to simulate the dynamics of these systems. Planar, as well as spatial motions of chains with moderate- (DOF≤1,000), large- (1,000 10,000) DOF were simulated. The results were validated by several means, such as comparisons with reported results wherever available, results obtained from commercial software, energy checks, etc. The study shows that ReDySim is capable of analysing serial chains of huge-DOF with acceptable numerical accuracy. The scheme is found to be numerically stable as well as computationally efficient, owing to the linear-time computation of the joint accelerations. Numerical studies were conducted to establish the theoretical basis for better performance of the proposed ReDySim solver. With the demonstrated capabilities of ReDySim, it may be found suitable for a large number of applications involving serial systems with huge-DOF.

Reduced-order forward dynamics of multiclosed-loop systems

In this work, a reduced-order forward dynamics of multiclosed-loop systems is proposed by exploiting the associated inherent kinematic constraints at acceleration level. First, a closed-loop system is divided into an equivalent open architecture consisting of several serial and tree-type subsystems by introducing cuts at appropriate joints. The resulting cut joints are replaced by appropriate constraint forces also referred to as Lagrange multipliers. Next, for each subsystem, the governing equations of motion are derived in terms of the Lagrange multipliers, which are based on the Newton–Euler formulation coupled with the concept of Decoupled Natural Orthogonal Complement (DeNOC) matrices, introduced elsewhere. In the proposed forward dynamics formulation, Lagrange multipliers are calculated sequentially at the subsystem level, and later treated as external forces to the resulting serial or tree-type systems of the original closed-loop system, for the recursive computation of joint accelerations. Note that such subsystem-level treatment allows one to use already existing algorithms for serial and tree-type systems. Hence, one can perform the dynamic analyses relatively quickly without rewriting the complete model of the closed-loop system at hand. The proposed methodology is in contrast to the conventional approaches, where the Lagrange multipliers are calculated together at the system level or simultaneously along with the joint accelerations, both of which incur higher order computational complexities, and thereby a greater number of arithmetic operations. Due to the smaller size of matrices involved in evaluating Lagrange multipliers in the proposed methodology, and the recursive computation of the joint accelerations, the overall numerical performances like computational efficiency, etc., are likely to improve. The proposed reduced-order forward dynamics formulation is illustrated with two multiclosed-loop systems, namely, a 7-bar carpet scrapping mechanism and a 3-RRR parallel manipulator.

Evolution of the DeNOC-based dynamic modelling for multibody systems

Abstract. Dynamic modelling of a multibody system plays very essential role in its analyses. As a result, several methods for dynamic modelling have evolved over the years that allow one to analyse multibody systems in a very efficient manner. One such method of dynamic modelling is based on the concept of the Decoupled Natural Orthogonal Complement (DeNOC) matrices. The DeNOC-based methodology for dynamics modelling, since its introduction in 1995, has been applied to a variety of multibody systems such as serial, parallel, general closed-loop, flexible, legged, cam-follower, and space robots. The methodology has also proven useful for modelling of proteins and hyper-degree-of-freedom systems like ropes, chains, etc. This paper captures the evolution of the DeNOC-based dynamic modelling applied to different type of systems, and its benefits over other existing methodologies. It is shown that the DeNOC-based modelling provides deeper understanding of the dynamics of a multibody system. The power of the DeNOC-based modelling has been illustrated using several numerical examples.

Modeling and analysis of a fuel-injection pump used in diesel engines

The Fuel-Injection Pump (FIP) used in diesel engine has a higher-pair cam-mechanism to pressurize the fuel for injection. This paper proposes a methodology to model FIP from a multibody Dynamics (MBD) perspective. The results from the model include the temporal behavior of driving torque, contact Hertz stress and reaction forces at various joints. The model helps the designer to assess the effect of various cam profiles, link parameters and other design variables. It is necessary that these parameters be optimized for future high pressure applications. For this purpose, a cam-mechanism with offset follower axis is analysed. Decoupled Natural Orthogonal Complement (DeNOC) matrices based algorithm is used to model FIP without and with offset cam-mechanism. The study shows that, the offset cam-mechanism allows reduction in the side-thrust, reaction forces, and the contact Hertz stress acting on the cam-follower interface. As a typical case, for an FIP working around a pressure value of 600 bar, an optimum offset value is found to be 9.5 mm and it shows a reduction of about 45% in side thrust values. To validate the modeling approach, experimental studies are performed on pump without and with offset cam-mechanism. Experimental results are in qualitative agreement with the theoretical model results.

60109 A UNIFIED APPROACH TO DYNAMICS OF A BIPED(Robotics and Mechatronics)

This paper presents a unified approach for dynamic modeling and analysis of a biped. A biped is modeled as a floating-base system where the foot-ground interactions are modeled as external forces and moments. Dynamic modeling is proposed using the concept of the Decoupled Natural Orthogonal Complement (DeNOC) matrices. The equations of motion thus obtained are independent of topology of the biped. The advantage of the proposed modeling is highlighted in the paper. A numerical example of a 7-link biped is provided using the methodology presented.

An improved dynamic modeling of a multibody system with spherical joints

A dynamic modeling of multibody systems having spherical joints is reported in this work. In general, three intersecting orthogonal revolute joints are substituted for a spherical joint with vanishing lengths of intermediate links between the revolute joints. This procedure increases sizes of associated matrices in the equations of motion, thus in- creasing computational burden of an algorithm used for dynamic simulation and control. In the proposed methodology, Euler parameters, which are typically used for representation of a rigid-body orientation in three-dimensional Cartesian space, are employed to repre- sent the orientation of a spherical joint that connects a link to its previous one providing three-degree-of-freedom motion capability. For the dynamic modeling, the concept of the Decoupled Natural Orthogonal Complement (DeNOC) matrices is utilized. It is shown in this work that the representation of spherical joints motion using Euler parameters avoids the unnecessary introduction of the intermediate links, thereby no increase in the sizes of the associated matrices with the dynamic equations of motion. To confirm the efficiency of the proposed representation, it is illustrated with the dynamic modeling of a spatial four- bar Revolute-Spherical-Spherical-Revolute (RSSR) mechanism, where the CPU time of the dynamic modeling based on proposed methodology is compared with that based on the revolute joints substitution. Finally, it is explained how a complex suspension and steer- ing linkage can be modeled using the proposed concept of Euler parameters to represent a spherical joint.

A recursive, numerically stable, and efficient simulation algorithm for serial robots with flexible links

A methodology for the formulation of dynamic equations of motion of a serial flexible-link manipulator using the decoupled natural orthogonal complement (DeNOC) matrices, introduced elsewhere for rigid bodies, is presented in this paper. First, the Euler Lagrange (EL) equations of motion of the system are written. Then using the equivalence of EL and Newton–Euler (NE) equations, and the DeNOC matrices associated with the velocity constraints of the connecting bodies, the analytical and recursive expressions for the matrices and vectors appearing in the independent dynamic equations of motion are obtained. The analytical expressions allow one to obtain a recursive forward dynamics algorithm not only for rigid body manipulators, as reported earlier, but also for the flexible body manipulators. The proposed simulation algorithm for the flexible link robots is shown to be computationally more efficient and numerically more stable than other algorithms present in the literature. Simulations, using the proposed algorithm, for a two link arm with each link flexible and a Space Shuttle Remote Manipulator System (SSRMS) are presented. Numerical stability aspects of the algorithms are investigated using various criteria, namely, the zero eigenvalue phenomenon, energy drift method, etc. Numerical example of a SSRMS is taken up to show the efficiency and stability of the proposed algorithm. Physical interpretations of many terms associated with dynamic equations of flexible links, namely, the mass matrix of a composite flexible body, inertia wrench of a flexible link, etc. are also presented.

A recursive, numerically stable, and efficient simulation algorithm for serial robots

Traditionally, the dynamic model, i.e., the equations of motion, of a robotic system is derived from Euler–Lagrange (EL) or Newton–Euler (NE) equations. The EL equations begin with a set of generally independent generalized coordinates, whereas the NE equations are based on the Cartesian coordinates. The NE equations consider various forces and moments on the free body diagram of each link of the robotic system at hand, and, hence, require the calculation of the constrained forces and moments that eventually do not participate in the motion of the coupled system. Hence, the principle of elimination of constraint forces has been proposed in the literature. One such methodology is based on the Decoupled Natural Orthogonal Complement (DeNOC) matrices, reported elsewhere. It is shown in this paper that one can also begin with the EL equations of motion based on the kinetic and potential energies of the system, and use the DeNOC matrices to obtain the independent equations of motion. The advantage of the proposed approach is that a computationally more efficient forward dynamics algorithm for the serial robots having slender rods is obtained, which is numerically stable. The typical six-degree-of-freedom PUMA robot is considered here to illustrate the advantages of the proposed algorithm.

Dynamics Modelling of Hexaslides using the Decoupled Natural Orthogonal Complement Matrices

In this paper, dynamic model of a class of parallel systems, namely, the hexaslides, is proposed. The model developed is based on the concept the decoupled natural orthogonal complement (DeNOC) matrices, introduced elsewhere. The dynamic model of hexslides, though complex due to the existence of multi-loop kinematic chains, is required for actuator power estimation, computed-torque control, optimum tool trajectory generation, etc. The use of DeNOC offers many physical interpretations, recursive algorithms, and parallel computations. Using the proposed dynamic model, a parallel inverse dynamics algorithm has been presented to compute the actuator forces. This is useful to choose suitable motors for an application. An illustration is provided using an existing machine tool based on hexaslides, namely, the HexaM, while it is carrying out a circular contouring. Secondly, the effect of leg and slider inertias is also studied, which clearly suggests that neither of these can be neglected while finding the actuator forces.

Modular and Recursive Kinematics and Dynamics for Parallel Manipulators

Constrained multibody systems typically feature multiple closed kinematic loops that con- strain the relative motions and forces within the system. Typically, such systems possess far more ar- ticulated degrees-of-freedom (within the chains) than overall end-effector degrees-of-freedom. Thus, actuation of a subset of the articulations creates mixture of active and passive joints within the chain. The presence of such passive joints interferes with the effective modular formulation of the dynamic equations-of-motion in terms of a minimal set of actuator coordinates as well the subsequent recursive solution for both forward and inverse dynamics applications. Thus, in this paper, we examine the development of modular and recursive formulations of equations-of-motion in terms of a minimal set of actuated-joint-coordinates for an exactly-actuated parallel manipulators. The 3 RRR planar parallel manipulator, selected to serve as a case-study, is an illustrative example of a multi-loop, multi-degree-of-freedom system with mixtures of ac- tive/passive joints. The concept of decoupled natural orthogonal complement (DeNOC) is combined with the spatial parallelism inherent in parallel mechanisms to develop a dynamics formulation that is both recursive and modular. An algorithmic approach to the development of both forward and inverse dynamics is highlighted. The presented simulation studies highlight the overall good nu- merical behavior of the developed formulation, both in terms of accuracy and lack of formulation stiffness.

Recursive Kinematics and Inverse Dynamics for a Planar 3R Parallel Manipulator

Abstract We focus on the development of modular and recursive formulations for the inverse dynamics of parallel architecture manipulators in this paper. The modular formulation of mathematical models is attractive especially when existing sub-models may be assembled to create different topologies, e.g., cooperative robotic systems. Recursive algorithms are desirable from the viewpoint of simplicity and uniformity of computation. However, the prominent features of parallel architecture manipulators-the multiple closed kinematic loops, varying locations of actuation together with mixtures of active and passive joints-have traditionally hindered the formulation of modular and recursive algorithms. In this paper, the concept of the decoupled natural orthogonal complement (DeNOC) is combined with the spatial parallelism of the robots of interest to develop an inverse dynamics algorithm which is both recursive and modular. The various formulation stages in this process are highlighted using the illustrative example of a 3R Planar Parallel Manipulator.

Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices

Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n—n being the degrees-of-freedom of the system at hand—inverse dynamics and order n3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the system’s accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.