Gibbs Appell Formulation Research Articles

Theoretical and experimental investigation of viscoelastic serial robotic manipulators with motors at the joints using Timoshenko beam theory and Gibbs–Appell formulation

In this article, the mathematical modeling of n-viscoelastic-link robotic manipulators based on the Gibbs–Appell formulation and the assumed mode method is developed. The elastic properties of the links are modeled according to the Timoshenko beam theory and its associated mode shapes. Also the dynamic effects of the motors at the joints are fully taken into consideration. In the mathematical modeling, the effects of torsion, extensional deformation, bending in two directions, structural damping, and viscous air as well as the gravity effects have also been considered. A recursive algorithm based on 4 × 4 transformation matrices has been developed to ease the derivation of motion equations. The mismatch between the high-speed vibrations of flexible robotic arms and the low-speed operation of the software and hardware interface in the computer and the experimental setup is a serious obstacle for the measuring and controlling of such systems. So, to verify the simulation results and the results obtained from the experimental test bed, high-speed digital-to-analog converter components on an electronic interface board are used in conjunction with the National Instrument’s LABVIEW software package.

A new approach for dynamic modeling of n-viscoelastic-link robotic manipulators mounted on a mobile base

In this article, the mathematical modeling of a robotic system composed of n flexible links and a mobile platform has been considered. Most of the mechanical systems including the nonholonomic constraints are analyzed by Lagrangian formulation and its associated “Lagrange multipliers.” Eliminating these variables from the obtained equations is a time-consuming and cumbersome task. So, the Gibbs–Appell formulation and the assumed mode method are used to make the derivation of motion equations easier. Also, to model the system thoroughly and accurately, the dynamic interactions between the manipulator and the mobile platform and the coupling effects arising simultaneously from large motions and small deflections are taken into consideration. The links (assumed as 3D Timoshenko beams) undergo tension-compression, torsion and spatial bending (in two directions), while the effects of internal and external damping (as dissipative forces) are also considered in the mathematical modeling. A systematic approach is developed based on the derived formulation to establish the dynamic equations of motion. In order to alleviate the computational complexity of the suggested method, all the mathematical operations are carried out through \(3\times 3\) and \(3\times 1\) matrices only. Finally, this method is applied to a mobile manipulator with two flexible links to demonstrate the ability of the proposed method in deriving the equations of motion of such complex systems.

Motion equation of nonholonomic wheeled mobile robotic manipulator with revolute–prismatic joints using recursive Gibbs–Appell formulation

The main intent of this paper is to represent a symbolic algorithm, capable of deriving the equations of motion of N-rigid link manipulators with revolute–prismatic (R–P) joints, which mounted on a mobile platform. The presence of prismatic joints besides the revolute ones, as well as the nonholonomic characteristics of the mobile platform makes the derivation of governing equations difficult. So, to derive the kinematic and dynamic equations of motion of such a complex system, and also to avoid computing the Lagrange multipliers associated with the nonholonomic constraints, the application of recursive Gibbs–Appell (G–A) formulation is applied. For modeling the system completely and precisely, the dynamic interactions between the manipulator and the mobile platform, the coupling effects due to the simultaneous rotating and sliding motion of the rigid arms, as well as both nonholonomic constraints associated with the no-slipping and the no-skidding conditions are included. Moreover, to improve the computational efficiency of the proposed systematic algorithm, all mathematical operations are done by only 3×3 and 3×1 matrices. Finally, a numerical simulation for a mobile manipulator with three R–P joints is performed, by using a developed computer program, to show the ability of this algorithm in deriving and solving the equations of motion of such systems.

Symbolic derivation of governing equations for dual-arm mobile manipulators used in fruit-picking and the pruning of tall trees

Mobile manipulators with only a single robotic arm have been successfully exploited in many agricultural tasks. The performance of these kinds of robotic systems has been improved by adding another robotic arm. However, for some agricultural applications such as pruning and fruit picking from tall trees, the number of links of each robotic arm should increase so that the arm can reach the target. The increase in the number of links and dynamic interactions between the manipulator and the mobile platform as well as the nonholonomic constraints of the mobile base make the manual derivation of motion equations almost impossible. So, this paper proposes a new solution to the problem of dynamic modeling of wheeled mobile manipulators with dual arms in an automatic and systematic approach. To avoid computing the “Lagrange multipliers” associated with the nonholonomic constraints, the equations of motion are derived according to the recursive Gibbs–Appell formulation. Also all the mathematical operations are performed by 3×3 and 3×1 matrices which are more efficient compared with the 4×4 and 4×1 ones. Finally, this method is applied to a mobile manipulator with two robotic arms to demonstrate the ability of the proposed method in deriving the equations of motion for such complex systems.

Kinematic and dynamic modeling of viscoelastic robotic manipulators using Timoshenko beam theory: theory and experiment

This paper presents an investigation into the development of modeling of n-viscoelastic robotic manipulators. The dynamic model of the system is derived using Gibbs-Appell formulation and assumed mode method. When the beam is short in length direction, shear deformation is a factor that may have significant effects on system dynamic. So, in modeling, the assumption of Timoshenko beam theory and associated mode shapes has been considered. Although including the effect of damping in continuous systems makes the formulations more complicated, two important damping mechanisms, namely, Kelvin-Voigt damping as internal damping and the viscous air damping as external damping have been considered. Based on derived formulation, a non-linear recursive algorithm is developed for deriving the inverse dynamic equation of motion, systematically. The performance of the proposed algorithm was assessed in terms of the required mathematical operations for deriving the kinematic and dynamic equations of the mechanical system. Finally, to validate the proposed formulation, a comparative assessment between the results achieved from experiment and simulation is presented in time and frequency domains.

Systematic modeling of a chain of N-flexible link manipulators connected by revolute–prismatic joints using recursive Gibbs-Appell formulation

The main intent of this paper is to represent a systematic algorithm capable of deriving the equations of motion of N-flexible link manipulators with revolute–prismatic joints. The existence of the prismatic joints together with the revolute ones makes the derivation of governing equations difficult. Also, the variations of the flexible parts of the links, with respect to time cause the associated mode shapes of the links to vary instantaneously. So, to derive the kinematic and dynamic equations of motion for such a complex system, the recursive Gibbs-Appell formulation is applied. For a comprehensive and accurate modeling of the system, the coupling effects due to the simultaneous rotating and reciprocating motions of the flexible arms as well as the dynamic interactions between large movements and small deflections are also included. In this study, the links are modeled based on the Euler–Bernoulli beam theory and the assumed mode method. Also, the effects of gravity as well as the longitudinal, transversal and torsional vibrations have been considered in the formulations. Moreover, a recursive algorithm based on 3 × 3 rotational matrices has been applied in order to derive the system’s dynamic equations of motion, symbolically and systematically. Finally, a numerical simulation has been performed by means of a developed computer program in order to demonstrate the ability of this algorithm in deriving and solving the equations of motion related to such systems.

Application of recursive Gibbs–Appell formulation in deriving the equations of motion of N-viscoelastic robotic manipulators in 3D space using Timoshenko Beam Theory

The goal of this paper is to describe the application of Gibbs–Appell (G–A) formulation and the assumed modes method to the mathematical modeling of N-viscoelastic link manipulators. The paper’s focus is on obtaining accurate and complete equations of motion which encompass the most related structural properties of lightweight elastic manipulators. In this study, two important damping mechanisms, namely, the structural viscoelasticity (Kelvin–Voigt) effect (as internal damping) and the viscous air effect (as external damping) have been considered. To include the effects of shear and rotational inertia, the assumption of Timoshenko beam (TB) theory (TBT) has been applied. Gravity, torsion, and longitudinal elongation effects have also been included in the formulations. To systematically derive the equations of motion and improve the computational efficiency, a recursive algorithm has been used in the modeling of the system. In this algorithm, all the mathematical operations are carried out by only 3×3 and 3×1 matrices. Finally, a computational simulation for a manipulator with two elastic links is performed in order to verify the proposed method.

Dynamic modeling of nonholonomic wheeled mobile manipulators with elastic joints using recursive Gibbs–Appell formulation

This paper focuses on the study of dynamic modeling of nonholonomic wheeled mobile robotic manipulators, which consist of a serial manipulator with elastic joints and an autonomous wheeled mobile platform. To avoid computing the Lagrange multipliers associated with the nonholonomic constraints, the approach of Gibbs-Appell (G-A) formulation in recursive form is adopted. For modeling the system completely and precisely, dynamic interactions between the manipulator and the mobile platform, as well as both nonholonomic constraints associated with the no-slipping and the no-skidding conditions, are included. Based on developed formulation, an algorithm is proposed that recursively and systematically derives the equation of motion. In this algorithm, in order to improve the computational complexity, all mathematical operations are done by only 3×3 and 3×1 matrices. Also, all dynamic expressions of a link are expressed in the same link local coordinate system. Finally, two computational simulations for mobile manipulators with rigid and elastic joints are presented to indicate the capability of this algorithm in generating the equation of motion of mobile robotic manipulators with high degree of freedom.

4542869 Flap mechanism : Gerald T Brine assigned to McDonnell Douglas Corporation

In this paper, the dynamic equations of flexible cooperative manipulators that are kinematically and dynamically constrained to an object and are placed on a non-holonomic mobile platform are explored. The main objective is to develop the dynamics of closed kinematic cooperative manipulators by defining constraints in mobile form while considering the geometric dimensions of the common object instead of assuming a concentrated mass in the gripper as well as the application of manipulator with flexible links and revolute-prismatic joints in the constrained cooperative mobile form. Unlike robots with independent manipulators, the dynamic constraint relates the dynamic equations of object and arms, initiating the motion and introducing interaction forces from the object to arms. In addition, the kinematic constraint keeps the distance between the two arms constant by constraining the relative motion velocity of the arms’ gripper. The dynamic equations of manipulators are derived using the recursive Gibbs–Appell formulation while the object equations are obtained using the Newton-Euler approach. Finally, the motion equations for a cooperative mobile robot with two 2-flexible-link arms are simulated. The results are evaluated at two stages by incorporating dynamic constraints as well as concurrent consideration of kinematic and dynamic constraints. Also, each manipulator's dynamic model weighted by using the experimental test setup with single link and revolute-prismatic joint.