Recursive Gibbs Appell Formulation Research Articles

Dynamic analysis of multiple inclined and frictional impact-contacts in multi-branch robotic systems

In this paper, the dynamic behavior of a multi-branch robotic mechanism made up of n rigid links and suspended inside an enclosure with curved walls has been investigated. The system's motion occurs in two stages: the suspension stage and the impact stage. The recursive Gibbs–Appell formulation and the regularized method (which is able to compute the impact forces occurring during a short collision time) have been used to extract the system's motion equations. Furthermore, to increase the accuracy of modeling, a friction force has also been considered at the locations where robot joints collide with enclosure walls. The stiffening of differential equations (which originates from the adoption of the regularized method in modeling the impact-contact phenomenon), the inclined impacts of the robotic system, the precise detection of impact time, and the existence of multiple impacts are some of the fundamental challenges faced during simulation; and the structured algorithm proposed in this article has been able to deal with these challenges quite successfully. Lastly, to show the precision and efficacy of the suggested algorithm, the motion of a 10-link, 4-branch robotic mechanism has been simulated and analyzed using several contact and friction force models.

A recursive algorithm for dynamics of multiple frictionless impact-contacts in open-loop robotic mechanisms

In this paper, the phenomenon of multiple impact-contacts has been dynamically modeled for an open kinematic chain with rigid links and revolute joints. The dynamic equations of the mentioned system have been extracted based on the recursive Gibbs–Appell formulation. The impact-contact phenomenon has been formulated by the regularized method in which the force of impact is a continuous function of the relative penetration and relative velocity of two colliding surfaces with respect to each other. The geometrical specifications and the mechanical properties of colliding surfaces are the two main parameters used in the modeling of viscoelastic contact force models. In this work, a recursive algorithm, which has been developed based on 3 × 3 rotation matrices to reduce the computational load, symbolically derives the motion equations of a multibody system that collides with surrounding surfaces at several points. The system under study includes the non-impact (flight) phase and the impact phase. Going from the flight phase to the impact phase and back, detecting the exact moment of impact, and also solving the differential equations of motion during a very short impact time have their particular challenges and complexities, which are dealt with in this work. In the next step, nine famous contact force models have been compared in order to select the most suitable model for simulation work. Finally, to show the accuracy and the capability of the presented algorithm, the dynamic behavior of an open-chain robotic mechanism consisting of 4 rigid links connected by revolute joints has been simulated and analyzed.

Dynamic modeling and extended bifurcation analysis of flexible-link manipulator

In this article, the nonlinear dynamic analysis of a flexible-link manipulator is presented. Especially, the possibility of chaos occurrence in the system dynamic model is investigated. Upon the occurrence of chaos, the system dynamical behavior becomes unpredictable which in turn brings about uncertainty and irregularity in the system motion. The importance of this investigation is pronounced in similar systems such as double pendulum and single-link flexible manipulator. What makes this study distinct from previous ones is the increase in the number of links as well as the changing the bifurcation parameters from system mechanical parameters to force and torque inputs. To this aim, the motion equations of the N-link robot, which are derived with the aid of the recursive Gibbs-Appell formulation and the assumed modes method, are used. In the end, the equations of motion are developed for a two-link flexible manipulator, and its nonlinear dynamical behavior is analyzed via numerical integration of discrete equations. The results are presented in the form of bifurcation diagrams (for variation of torque amplitude), time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms. The outcomes indicate that when there is no offset, the decrease in damping results in chaotic generalized modal coordinates. In addition, as the excitation frequency decreases from 2π to π, a limiting amplitude is created at 0.35 before which the behavior of generalized rigid and modal coordinates is different, while this behavior has more similarity after this point. An experimental setup is also used to check the torques as the system input.

Analytical and experimental investigation of the dynamic behavior of a revolute-prismatic manipulator with N flexible links and hubs

In this paper, the dynamic model and vibration analysis of a flexible manipulator composed of N elastic links and robot actuators for which structural vibration is considered are investigated. In view of the concurrent linear and rotary motions of the link caused by revolute-prismatic joints, the interaction of joint’s structural vibration and link fluctuation is taken as an effective model parameter. Utilization of prismatic joint with hub in the manipulator’s structure in question, its significant length and low weight result in the operation of hub akin to a flexible link attached to revolute joint. To model the expressed hub oscillation with respect to the link, the assumed modes method and mode shapes of Euler-Bernoulli beam with independent generalized modal coordinate with respect to the link are employed. Noting the complexity of the present model relative to studied flexible manipulators, the recursive Gibbs-Appell formulation is used to derive the motion equations. Therefore, the dynamic equations of hub are of time-independent form, while these equations are obtained as time-variable for links. Although the obtained equations are simultaneously solved in the coupled form, the derived equations for a single-link flexible manipulator are simulated in three cases of (1) rigid hub, (2) elastic hub, and (3) elastic hub and flexible joints. The simulation results are compared with a similar experimental setup, indicating that the flexible manipulator model with elastic link, hub, and joints yields satisfactory results with an error of less than 1 mm and 1° in the longitudinal and rotational motions, respectively. Moreover, the results of lateral vibration show good accuracy, with the endpoint of the robot having perfect precision with just 1% tolerance.

Dynamic modeling of flexible cooperative mobile manipulator with revolute-prismatic joints for the purpose of moving common object with closed kinematic chain using the recursive Gibbs–Appell formulation

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.

Motion equations of cooperative multi flexible mobile manipulator via recursive Gibbs–Appell formulation

• The model of time-varying multi flexible mobile manipulators is proposed. • The cooperative mobile manipulator with the closed kinematic chain is illustrated. • The Gibbs–Appell formulation has been upgraded by using Lagrange multiplayer. • The maximum dynamic load caring capacity has been evaluates for a common object. In this paper, the developed model of an N-flexible-link mobile manipulator with revolute-prismatic joints is presented for the cooperative flexible multi mobile manipulator. In this model, the deformation of flexible links is calculated by using the assumed modes method. In additions, non-holonomic constraints of the robots’ mobile platforms that bound its locomotion are considered. This limitation is alleviated through the concurrent motion of revolute and prismatic joints, although it results in computational complexity and changes the final motion equations to time-varying form. Not only is the proposed dynamic model implemented for the multi-mobile manipulators with arms having independent motion, but also for multi-mobile manipulators in cooperation after defining gripper's kinematic constraints. These constraints are imported to the dynamic equations by defining Lagrange multipliers. The recursive Gibbs–Appell formulation is preferred over other similar approaches owing to the capability of solving the equations without the need to use Lagrange multipliers for eliminating non-holonomic constraints in addition to the novel optimized process of obtaining system equations. Hence, cumbersome simultaneous computations for eliminating the constraints of platform and arms are circumvented. Therefore, this formulation is improved for the first time by importing Lagrange multipliers for solving kinematic constrained systems. In the simulation section, the results of forward dynamics solution for two flexible single-arm manipulators with revolute-prismatic joints while carrying a rigid object are presented. Inverse dynamics equations of the system are also presented to obtain the maximum dynamic load-carrying capacity of the two-rigid-link mobile manipulators on a predefined path. Two constraints, namely the capacity of joint motors torque and robot motion stability are considered as the limitation criteria. The concluded motion equations are used to accurately control the movement of sensitive bodies, which is not achievable through the use of one platform.

Derivation of dynamic equation of viscoelastic manipulator with revolute–prismatic joint using recursive Gibbs–Appell formulation

In this paper, the motion analysis of a viscoelastic manipulator with N-flexible revolute–prismatic joints is being studied with the help of a systematic algorithm. The presence of prismatic joints, along with revolute ones, makes the derivation of the equations complicated. The link’s axial motions cause variation of its flexible parts with respect to time. In order to modify the associated mode shapes concerning an instant link length, dynamic interaction between the rotary reciprocating motion and transverse vibration of the flexible arm is evaluated. The links are modeled on the assumed mode method using the Timoshenko beam theory (TBT). Dynamic equations are derived from the recursive Gibbs–Appell formulation. The formulation involves fewer mathematical calculations but shows efficient computational performance when compared to other formulations. The dynamic model of each joint shows flexibility, damping, backlash and frictions resulting in accuracy of the formulations. Applying recursive algorithm based on the $$3\times 3$$ rotational matrix instead of the $$4\times 4$$ one causes the computational performance to fall by separating the rotating matrix. Furthermore, motion equations are obtained symbolically and systematically. Links linear motion causes TBT mode shapes changes with respect to time. This is implemented in a non-dimensional form to avoid computing for each step. Finally, the following dynamic equations are solved numerically by MATLAB software for a spatial two-armed manipulator. The outcome of the simulations represents the ability of the proposed algorithm to derive and solve the equations of motion. Moreover, the data are compared with the rigid and elastic links, modeled by the Euler–Bernoulli beam theory.

Dynamic Behavior of Flexible Multiple Links Captured Inside a Closed Space

This work presents a systematic method for the dynamic modeling of flexible multiple links that are confined within a closed environment. The behavior of such a system can be completely formulated by two different mathematical models. Highly coupled differential equations are employed to model the confined multilink system when it has no impact with the surrounding walls; and algebraic equations are exploited whenever this open kinematic chain system collides with the confining surfaces. Here, to avoid using the 4 × 4 transformation matrices, which suffers from high computational complexities for deriving the governing equations of flexible multiple links, 3 × 3 rotational matrices based on the recursive Gibbs-Appell formulation has been utilized. In fact, the main aspect of this paper is the automatic approach, which is used to switch from the differential equations to the algebraic equations when this multilink chain collides with the surrounding walls. In this study, the flexible links are modeled according to the Euler–Bernoulli beam theory (EBBT) and the assumed mode method. Moreover, in deriving the motion equations, the manipulators are not limited to have only planar motions. In fact, for systematic modeling of the motion of a multiflexible-link system in 3D space, two imaginary links are added to the n real links of a manipulator in order to model the spatial rotations of the system. Finally, two case studies are simulated to demonstrate the correctness of the proposed approach.

A systematic method for the hybrid dynamic modeling of open kinematic chains confined in a closed environment

This work presents a systematic method for the dynamic modeling of multi-rigid links confined within a closed environment. The behavior of the system can be completely characterized by two different mathematical models: a set of highly coupled differential equations for modeling the confined multi-link system when it has no impact with surrounding walls; and a set of algebraic equations for expressing the collision of this open kinematic chain system with the confining surfaces. In order to avoid the Lagrangian formulation (which uses an excessive number of total and partial derivatives in deriving the governing equations of multi-rigid links), the motion equations of such a complex system are obtained according to the recursive Gibbs–Appell formulation. The main feature of this paper is the recursive approach, which is used to automatically derive the governing equations of motion. Moreover, in deriving the motion equations, the manipulators are not limited to planar motions only. In fact, for systematic modeling of the motion of a multi-rigid-link system in 3D space, two imaginary links are added to the \(n\)-real links of a manipulator in order to model the spatial rotations of the system. Finally, a 2D and a 3D case studies are simulated to demonstrate the effectiveness of the proposed approach.

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.

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.