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

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.     

Comparative study of cable parallel manipulators with and without hybrid-driven planar five-bar mechanism

Cable parallel manipulators (CPMs) relay on cables instead of rigid links to support and manipulate the end-effector. The CPMs are required not only for operations with lower inertia and high payload, but also for output with greater flexibility. The paper is devoted to present and analyze a cable parallel manipulator with and without hybrid-driven planar five-bar mechanism (HDPM). The cable parallel manipulator with the HDPM combines positive features of both the cable parallel manipulator and the HDPM. Comparative study of kinematics and dynamics of the CPMs with and without HDPM are studied. Drive torques and drive powers are given to compare the load carrying capacity of the two manipulators, and workspace, stiffness performance, singularity analysis are also carried out. Simulation examples are presented to demonstrate the CPMs with and without HDPM and their mechanics performance.     

Mathematical modeling and trajectory planning of mobile manipulators with flexible links and joints

This paper is concerned with mathematical modeling and optimal motion designing of flexible mobile manipulators. The system is composed of a multiple flexible links and flexible revolute joints manipulator mounted on a mobile platform. First, analyzing on kinematics and dynamics of the model is carried out then; open-loop optimal control approach is presented for optimal motion designing of the system. The problem is known to be complex since combined motion of the base and manipulator, non-holonomic constraint of the base and highly non-linear and complicated dynamic equations as a result of the flexible nature of both links and joints are taken into account. In the proposed method, the generalized coordinates and additional kinematic constraints are selected in such a way that the base motion coordination along the predefined path is guaranteed while the optimal motion trajectory of the end-effector is generated. This method by using Pontryagin’s minimum principle and deriving the optimality conditions converts the optimal control problem into a two point boundary value problem. A comparative assessment of the dynamic model is validated through computer simulations, and then additional simulations are done for trajectory planning of a two-link flexible mobile manipulator to demonstrate effectiveness and capability of the proposed approach.     

Optimal structural design of a planar parallel platform for machining

Parallel manipulators have many advantages over traditional serial manipulators. These advantages include high accuracy, high stiffness and high load-to-weight ratio, which make parallel manipulators ideal for machining operations where high accuracy is required to meet the requirements that modern standards demand.Recently, the finite element method has been used by some workers to determine the stiffness of spatial manipulators. These models are mainly used to verify stiffness predicted using kinematic equations, and are restricted to relatively simple truss-like models. In this study, state-of-the-art finite elements are used to determine the out of plane stiffness for parallel manipulators. Euler–Bernoulli beam elements and flat shell elements with drilling degrees of freedom are used to model the platform assembly.The main objective of this study is to quantify the stiffness, particularly the out of plane stiffness, of a planar parallel platform to be used for machining operations. The aim is to obtain a design that is able to carry out machining operations to an accuracy of 10 μm for a given tool force.Reducing the weight of a parallel manipulator used in machining applications has many advantages, e.g. increased maneuverability, resulting in faster material removal rates. Therefore the resulting proposed design is optimized with respect to weight, subject to displacement and stress constraints to ensure feasible stiffness and structural integrity. The optimization is carried out by means of two gradient-based methods, namely LFOPC and Dynamic-Q.     

Full forward kinematics of redundant kinematic hybrid manipulator

A unified recursive mathematic model is established for full forward kinematics analysis of various redundant kinematic hybrid manipulators which are formed by several different parallel manipulators connected in series. In the established model, the relationships between the general input velocity/acceleration and the general output velocity/acceleration are discovered. The Jacobian matrices for mapping the general output velocities to the general input velocity and the Hessian matrices for mapping the general output velocities to the general input acceleration are derived. The unified iteration recursive formulae are derived for solving the general output velocity/acceleration of the end moving platform of the redundant kinematic hybrid manipulator by only giving the general input velocity/acceleration of every parallel manipulator. A 3D model of a novel (4SPS + SPR )+ (4SPS/SP) + 3SPR type hybrid manipulator is constructed and its full forward velocity/acceleration formulae are derived from the established unified recursive mathematic model and are proved to be correct by utilizing a simulation solution of the novel redundant kinematic hybrid manipulator.     

Kinematics of an offset 3-UPU translational parallel manipulator by the homotopy continuation method

For most parallel manipulators, the inverse kinematics is straightforward, while the direct kinematics is challenging. The latter requires the solution of a system of nonlinear equations. In this paper we use the homotopy continuation method to solve the forward and inverse kinematic problems of an offset 3-UPU translational parallel manipulator. The homotopy continuation method is a novel method which alleviates drawbacks of the traditional numerical techniques, namely; the acquirement of good initial guess values, the problem of convergence and computing time. The direct kinematics problem of the manipulator leads to 16 real solutions.     

Dynamic model of a flying manipulator with two highly flexible links

Nonlinear dynamic model of a flying manipulator with two revolute joints and two highly flexible links is obtained using Hamilton’s principle. Flying base of the manipulator is a rigid body. Stress is treated three dimensionally in the isotropic linearly-elastic links, but the in-plane and out-of-plane warpings of the links’ cross-sections are neglected. Although the links’ cross-sections undergo negligible elastic orientation, their models are more accurate than a nonlinear 3D Euler–Bernoulli beam. Tension, compression, twisting and spatial deflections of each link are coupled to each other by some nonlinear terms including two new ones. In the issue of flying flexible-link manipulators new terminologies, namely forward/inverse kinetics instead of forward/inverse kinematics are suggested, since determination of position and orientation of the end-effector is coupled to the partial differential motion equations.     

Dynamic modeling of a Stewart platform using the generalized momentum approach

Dynamic modeling of parallel manipulators presents an inherent complexity, mainly due to system closed-loop structure and kinematic constraints.In this paper, an approach based on the manipulator generalized momentum is explored and applied to the dynamic modeling of a Stewart platform. The generalized momentum is used to compute the kinetic component of the generalized force acting on each manipulator rigid body. Analytic expressions for the rigid bodies inertia and Coriolis and centripetal terms matrices are obtained, which can be added, as they are expressed in the same frame. Gravitational part of the generalized force is obtained using the manipulator potential energy. The computational load of the dynamic model is evaluated, measured by the number of arithmetic operations involved in the computation of the inertia and Coriolis and centripetal terms matrices. It is shown the model obtained using the proposed approach presents a low computational load. This could be an important advantage if fast simulation or model-based real-time control are envisaged.     

Dynamic Modeling and Tracking Control of 2-Joint Manipulators in Ocean Current Environment北大核心CSCD

The hydrodynamic performance of the underwater manipulator is greatly influenced by the current load. The underwater environment is assumed to be a still water environment and the current load is only considered as a simple random disturbance in the current control research, and the traditional control precision is usually rather low. Based on the Lagrange method and the Newton-Euler method, a dynamic model for 2-joint manipulators in the uniform ocean current environment was derived. In view of the relative motion of the ocean current and the manipulator, the Morison formula was introduced to calculate the water resistance and the inertia force of the ocean current on the manipulator. Based on this dynamic model, the sliding mode control strategy was used to achieve accurate tracking of the ideal trajectory of the manipulator. The simulation results show that, compared with the PD (proportional derivative) control, the sliding mode control strategy has better control effects. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Spatial impedance control of two cooperative industrial manipulators

Nowadays robotic manipulators are considered to perform a wide range of tasks. Since robotic systems consisting of multiple manipulators are capable of handling a variety of tasks that cannot be executed by single manipulators, cooperation of multiple manipulators is becoming increasingly interesting in particular for industrial applications. The interaction with other manipulators, respectively the environment, requires an extension of the conventional position control in order to achieve a desired compliance, and thus to limit the interaction wrench so to avoid damaging the involved objects. In this contribution the cooperation of two industrial manipulators, a Stäubli RX130L (6-DOF) and a Stäubli TX90L mounted on a linear axis (constituting a redundant 7-DOF manipulator), is addressed applying an impedance control scheme. The manipulation task is to grasp an object with both manipulators and to follow a prescribed trajectory by simultaneously limiting the contact forces between the manipulators and the object and ensuring a compliant behavior towards the environment. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Control for Static Stiffness and Thermal Sensitivity of a Hexapod Machine Tool

Beneath many conceptual advantages, parallel kinematic manipulators suffer some disadvantages which complicate their operation as machine tool. One of them is their inhomogeneous stiffness characteristic within the workspace, another is the high thermal sensitivity of their static accuracy due to the large strut lengths. Both properties, stiffness and thermal sensitivity, depend strongly on the configuration of the manipulator. As for machining purposes only five degrees of freedom are necessary, the 6th axis can be utilized to choose an optimal configuration with respect to stiffness or thermal sensitivity. In this contribution, an optimization method based on Jacobian matrix analysis is introduced and demonstrated for a model of the hexapod milling machine HEXACT. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Energy-Optimal Multi-Goal Motion Planning for Planar Robot Manipulators

In this work, the energy-optimal motion planning problem for planar robot manipulators with two revolute joints is studied, in which the end-effector of the robot manipulator is constrained to pass through a set of waypoints, whose sequence is not predefined. This multi-goal motion planning problem has been solved as a mixed-integer optimal control problem in which, given the dynamic model of the robot manipulator, the initial and final configurations of the robot, and a set of waypoints inside the workspace of the manipulator, one has to find the control inputs, the sequence of waypoints with the corresponding passage times, and the resulting trajectory of the robot that minimizes the energy consumption during the motion. The presence of the waypoint constraints makes this optimal control problem particularly difficult to solve. The mixed-integer optimal control problem has been converted into a mixed-integer nonlinear programming problem first making the unknown passage times through the waypoints part of the state, then introducing binary variables to enforce the constraint of passing once through each waypoint, and finally applying a fifth-degree Gauss–Lobatto direct collocation method to tackle the dynamic constraints. High-degree interpolation polynomials allow the number of variables of the problem to be reduced for a given numerical precision. The resulting mixed-integer nonlinear programming problem has been solved using a nonlinear programming-based branch-and-bound algorithm specifically tailored to the problem. The results of the numerical experiments have shown the effectiveness of the approach.     

Dynamic analysis of cable-driven parallel manipulators with time-varying cable lengths

In conventional researches, cables of cable-driven parallel manipulators are treated as simple linear elements that can only work in tension. This results in the fact that the effect of cable dynamics on the positioning precision of the end-effector is not adequately taken into account. To overcome this shortcoming, a dynamic model for cable-driven parallel manipulators with cables of slowly time-varying length is presented in this paper. The partial differential equation characterizing the dynamics of a cable with varying-length is deduced, and converted into ordinary differential equations through spatial discretization by finite difference approximation. Then, the dynamic model for cable-driven parallel manipulators is achieved considering the relationship between the motion of the end-effector and the cable end force, in which the degrees of freedom of cables and the end-effector are all involved. Two numerical examples are demonstrated to validate the dynamic model, and also show that it is necessary to take into consideration the cable dynamics for manipulators of long-span cables.     

Flatness‐Based Trajectory Tracking Control of an Underconstrained Cable Suspension Manipulator

Cable suspension manipulators support a payload platform in space by several spatially arranged cables with computercontrolled winches. The winches are mounted either fixed or on movable trolleys. Compared to conventional cranes, it is possible to control not only the translational motion of the payload but also its orientation in order to perform, for example, assembly tasks. By this, cable suspension manipulators combine the ability of cranes to support heavy payloads in a large workspace with the dexterity of robot manipulators. The present contribution treats nonlinear trajectory tracking control of cable suspension manipulators that are kinematically undetermined and statically determined. In particular, the cable suspension manipulator Cablev [4] is considered that has been developed at University of Rostock (Fig. 1). Its payload platform is supported by three cables with winches mounted on trolleys that move themselves on a gantry. Thus, the payload platform may perform sway motions with three degrees of freedom while the cable lengths are kept fixed. Applications are, for example, precise handling and assembling large and heavy components on construction sites or on shipyards. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Neural network-based nonlinear tracking control of kinematically redundant robot manipulators

In this paper, neural network-based nonlinear dynamical control of kinematically redundant robot manipulators is considered. The neural network-based controller achieves end-effector trajectory tracking as well as subtask tracking effectively. A feedforward neural network is employed to learn the parametric uncertainties, existing in the dynamical model of the robot manipulator. The whole system is shown to be stable in the sense of Lyapunov. Numerical simulation studies are carried out for a 3R planar robot manipulator to show the effectiveness of the control scheme.     

Measuring the closeness to singularities of a planar parallel manipulator using geometric algebra

A new index for measuring the closeness to the singularities of parallel manipulators using geometric algebra is proposed in this paper. Constraint wrenches acting on the moving platform of a parallel manipulator are derived using the outer product and dual operations. Removing the redundant constraint wrenches, a singularity polynomial is obtained when the coefficient of the outer product of all the non-redundant constraint wrenches equals zero. A singularity surface can be drawn using the singularity polynomial. Similarly, an approximate singularity polynomial and approximate singularity surface can be obtained by imposing a threshold to the singular polynomial. Then the singularity volume is calculated as the space between singularity surface and approximate singularity surface. The new index is derived by calculating the ratio of the non-singularity workspace volume (the workspace volume minus the singularity volume) to the workspace volume. The proposed index is coordinate-free and has a clear geometrical and physical interpretation. This index can be a basis for selecting structural parameters, path planning and mechanism design.     

A model of parameter adaptive law with time varying function for robot control

In this article, a model of adaptive control law for controlling robot manipulators using the Lyapunov based theory of guaranteed the stability of uncertain a system is derived. The novelty of obtained result is that the adaptive control algorithm is developed using a parameter estimation rule depending on manipulator kinematic, dynamic parameters and tracking error. This study is supported by a computer simulation and tracking performance has been improved.     

Modelling Flexible Manipulators With Motors at the Joints

A computationally efficient recursive model of a flexible manipulator with motors at the joints is described in this paper. The model adopts a mixed Eulerian and Lagrangian formulation of the equations of a flexible body and exploits the chained structure of the equations for a serial manipulator. The dynamic effects of the motors at the joints, including gyroscopic terms, are fully taken into account. Symbolic manipulation is used in a newly developed package, whose performance in detailed reproduction of the dynamic effects due to the interplay between the motors and the flexible links is assessed through simulation. A comparison between the complete model and a simplified one, where the motors are considered as simple inertias rotating around their own axis, has been carried out, using both a time domain analysis and a frequency domain analysis, in order to show the relevance of gyroscopic effects in modelling flexible robots.