柔性臂漂浮基空间机器人建模与轨迹跟踪控制

利用拉格朗日法和假设模态方法建立了末端柔性的两臂漂浮基空间机器人的非线性动力学方程．通过坐标变换,推导出一种新的以可测关节角为变量的全局动态模型，并在此基础上运用基于模型的非线性解耦反馈控制方法得到关节相对转角与柔性臂的弹性变形部分解耦形式控制方程．最后，讨论了柔性臂漂浮基空间机器人的轨迹跟踪问题，并通过仿真实例计算,表明该模型转换及控制方法对于柔性臂漂浮基空间机器人末端轨迹跟踪控制的有效性．     

欠驱动柔性机器人的振动可控性分析

欠驱动柔性机器人的可控性分析是对其进行有效控制的关键问题. 本文以具有柔性杆的3DOF平面欠驱动机器人为例, 分两步分析系统的可控性. 首先,忽略杆件的弹性变形, 研究欠驱动刚性系统在不同驱动电机位置的状态可控性；然后, 考虑柔性因素, 研究欠驱动柔性系统的结构振动可控性. 结果表明振动可控性是随机器人关节位形和驱动电机位置而变化的, 并且欠驱动刚性机器人的状态可控性对相应的柔性系统的振动可控性有很重要的影响. 最后, 将上述研究方法扩展到具有一个被动关节的N自由度平面欠驱动柔性机器人.     

Robust adaptive controller design for flexible joint manipulators

In this paper the control problem for robot manipulators with flexible joints is considered. A reduced-order flexible joint model is constructed based on a singular perturbation formulation of the manipulator equations of motion. The concept of an integral manifold is utilized to construct the dynamics of a slow subsystem. A fast subsystem is constructed to represent the fast dynamics of the elastic forces at the joints. A composite control scheme is developed based on on-line identification of the manipulator parameters which takes into account the effect of certain unmodeled dynamics and parameter variations. Stability analysis of the resulting closed-loop full-order system is presented. Simulation results for a single link flexible joint manipulator are given to illustrate the applicability of the proposed algorithm.     

Symbolic modeling and dynamic simulation of robotic manipulators with compliant links and joints

The explicit, non-recursive symbolic form of the dynamic model of robotic manipulators with compliant links and joints are developed based on a Lagrangian-assumed mode of formulation. This form of dynamic model is suitable for controller synthesis, as well as accurate simulations of robotic applications. The final form of the equations is organized in a form similar to rigid manipulator equations. This allows one to identify the differences between rigid and flexible manipulator dynamics explixitly. Therefore, current knowledge on control of rigid manipulators is likely to be utilized in a maximum way in developing new control algorithms for flexible manipulators.

Computer automated symbolic expansion of the dynamic model equations for any desired manipulator is accomplished with programs written based on commercial symbolic manipulation programs (SMP, MACSYMA, REDUCE). A two-link manipulator is used as an example. Computational complexity involved in real-time control, using the explicit, non-recursive form of equations, is studied on single CPU and multi-CPU parallel computation processors.     

Maximum allowable load of flexible manipulators for given dynamic trajectory

This paper presents the formulation and numerical solution of the dynamic load carrying capacity (DLCC) problem of flexible manipulators. For manipulators under the rigid body assumption, the major limiting factor in determining the maximum allowable load (load mass and load moment of inertia) for a prescribed dynamic trajectory (positions, velocities and accelerations) is the joint actuator capacity. But for a flexible robot, an additional constraint on allowable deformation at the end effector must be imposed because either lighter-weight links or operating at a higher speed could cause unacceptable fluctuations when moving along a trajectory. A Lagrangian assumed mode method was used to model the manipulator and load dynamics, including both joint and deflection motions. The deflection equations are then coupled with robot kinematics to solve for the generalized coordinates. A strategy to determine the DLCC subject to both constraints mentioned above is formulated where the end effector deflection is specified in terms of a series of spherical bounds with a radius equal to the allowable deformation. A general computational procedure for the multiple-link case given arbitrary trajectories is described in detail. Symbolic derivation and simulation by using a PC-based symbolic language MATHEMATICA® was carried out for a two-link planer robot. The results confirmed the necessity of the dual constraints and showed that which constraint is more critical for a given robot and trajectory depends on the required accuracy.     

A systematic method of dynamics for flexible robot manipulators

In this article, a systematic method to derive dynamic equations of motion for flexible robot manipulators is developed by using the Lagrangian assumed modes method. The proposed method can be applied to dynamic simulation and control system design for flexible robot manipulators. In the proposed method, the link deflection is described by a truncated modal expansion. The operations of only 3x3 matrices and/or 3 × 1 vectors exist in the method. All the dynamics computations are performed in the link coordinate systems, where the kinematics informations are computed with the forward recursion from the base to the hand tip and the dynamics informations are computed with the return recursion. As generally compared with other existing methods, the method proposed in this article is, computationally, more simple, systematic, and efficient. A computational simulation for a single-link flexible robot manipulator is presented to verify the proposed method. © 1992 John Wiley & Sons, Inc.     

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. The work has been carried out in the Dept. of Mechanical Engineering, Indian Institute of Technology Delhi, New Delhi 110016, India.     

A Perturbation Method for Dynamic Analysis and Simulation of Flexible Manipulators

This paper presents a perturbation method for the dynamicsimulation of flexible manipulators. In this method thevibrational motion of the manipulator is modeled as a first-orderperturbation of the nominal rigid link motion. For that purposethe flexible dynamic model is split into two parts. A rigidifiedsystem describes the nominal rigid link motion of themanipulator. A linear system linearized about the nominaltrajectory describes the vibrational motion. These equations arecomputationally more efficient than the non-linear dynamicequations and offer more insight in the dynamic phenomena of thesystem. The method is based on a full non-linear finite elementformulation which treats the general case of coupled largedisplacements motion and small elastic motion. A planar one linkmanipulator and a spatial two link manipulator with flexiblelinks are used for case studies. A comparison is made between thenon-linear and the perturbation analyzes. The perturbation methodappears to be a very efficient approach for dynamic analyzes offlexible manipulators and yields accurate results even forsystems with low frequency elastic modes.     

Underactuated Flexible Aerial Manipulators: a New Framework for Optimal Trajectory Planning Under Constraints Induced by Complex Dynamics

The aerial manipulators (AMs) are a new class of unmanned aerial systems (UASs) that are created in response to the ever-increasing demand for autonomous object transportation and manipulation. Because of power supply restrictions, the load carrying capacity is limited and therefore it is necessary to reduce the overall weight of these UASs. The past works in the field of AMs consider the multi-rotor unmanned aerial vehicles (UAVs) as the base and manipulators with rigid links as the interactive elements with the environment which are bulky and heavy. To overcome the issue, this paper introduces the AMs endowed with flexible manipulators, their dynamic modeling, a new method for trajectory planning and control algorithm such that the unfavorable effects of using flexible elements like vibrations are minimized. Due to lack of kinematic constraints and the presence of flexibility conditions, conventional methods of trajectory planning for ground wheeled-mobile manipulators (GWMMs) such as extended and augmented Jacobian matrix cannot be applied to AMs. The addition of flexibility to the manipulator increases underactuation degrees (UADs), the complexity of trajectory planning and control synthesis. Considering large deformation assumption for flexible links, the dynamic equations and their induced nonholonomic constraints are derived applying Lagrangian formulation. Then, these constraints with that part of equations of motion corresponding to the links flexibility are solved simultaneously in the context of an optimization algorithm resulting in optimized trajectories. Through simulation results, the proposed method of trajectory planning and vibration control of underactuated flexible AMs has been shown to be effective.     

Optimal Trajectory Planning for Flexible Link Manipulators with Large Deflection Using a New Displacements Approach

The main objective of the present paper is to determine the optimal trajectory of very flexible link manipulators in point-to-point motion using a new displacement approach. A new nonlinear finite element model for the dynamic analysis is employed to describe nonlinear modeling for three-dimensional flexible link manipulators, in which both the geometric elastic nonlinearity and the foreshortening effects are considered. In comparison to other large deformation formulations, the motion equations contain constant stiffness matrix because the terms arising from geometric elastic nonlinearity are moved from elastic forces to inertial, reactive and external forces, which are originally nonlinear. This makes the formulation particularly efficient in computational terms and numerically more stable than alternative geometrically nonlinear formulations based on lower-order terms. In this investigation, the computational method to solve the trajectory planning problem is based on the indirect solution of open-loop optimal control problem. The Pontryagin’s minimum principle is used to obtain the optimality conditions, which is lead to a standard form of a two-point boundary value problem. The proposed approach has been implemented and tested on a single-link very flexible arm and optimal paths with minimum effort and minimum vibration are obtained. The results illustrate the power and efficiency of the method to overcome the high nonlinearity nature of the problem.     

Coupling characteristics of rigid body motion and elastic deformation of a 3-PRR parallel manipulator with flexible links

Modeling of multibody dynamics with flexible links is a challenging task, which not only involves the effect of rigid body motion on elastic deformations, but also includes the influence of elastic deformations on rigid body motion. This paper presents coupling characteristics of rigid body motions and elastic motions of a 3-PRR parallel manipulator with three flexible intermediate links. The intermediate links are modeled as Euler–Bernoulli beams with pinned-pinned boundary conditions based on the assumed mode method (AMM). Using Lagrange multipliers, the fully coupled equations of motions of the flexible parallel manipulator are developed by incorporating the rigid body motions with elastic motions. The mutual dependence of elastic deformations and rigid body motions are investigated from the analysis of the derived equations of motion. Open-loop simulation without joint motion controls and closed-loop simulation with joint motion controls are performed to illustrate the effect of elastic motion on rigid body motions and the coupling effect amongst flexible links. These analyses and results provide valuable insight to the design and control of the parallel manipulator with flexible intermediate links.     

Adaptive control based on explicit model of robot manipulator

Adaptive and nonadaptive control algorithms, which make use of a fundamental mathematical property concerning positive definite matrices and Lyapunov stability theory, are proposed for the control of robot manipulators. Using the fact that the matrix dD(q)/dt-2C(q, dq/ dt) is skew symmetric, nonadaptive controllers which have a simplified structure with less computational burden are proposed. Using the dynamic equations for robot manipulators, parameter adaptation rules are developed for updating the controller's partially or totally unknown parameters, generalizing them to model reference adaptive controllers. To further take advantage of the simplified structure of the proposed adaptive controllers, a method for deriving the dynamic model of a robot manipulator which is linear in terms of its parameters is given. This dynamic model is also suitable for the pure identification of the parameters of links and payload of the manipulator     

多机器人协同吊运系统的协调动态载荷分配

针对多机器人共同吊运同一个物体形成闭运动链的协调系统,利用用牛顿-欧拉法建立了系统的广义动力学方程.研究了多机器人协调吊运系统动态载荷的一种实时性分配方法.根据各机器人的承载力,用线性加权方法使被吊运物质心处的内力有限并获得了系统的载荷分配原则.随后根据被吊运物的期望运动,给出了系统动态载荷分配的解析表达式,并进而可计算得到各机器人的期望工作载荷.最后,采用ABB工业机器人搭建吊运平台,实验验证所获得的载荷分配方法.研究结果为进一步研究多移动机器人协调吊运系统的载荷分配问题打下基础.     

Kinematics and the implementation of an elephant's trunk manipulator and other continuum style robots

Traditionally, robot manipulators have been a simple arrangement of a small number of serially connected links and actuated joints. Though these manipulators prove to be very effective for many tasks, they are not without their limitations, due mainly to their lack of maneuverability or total degrees of freedom. Continuum style (i.e., continuous "back-bone") robots, on the other hand, exhibit a wide range of maneuverability, and can have a large number of degrees of freedom. The motion of continuum style robots is generated through the bending of the robot over a given section; unlike traditional robots where the motion occurs in discrete locations, i.e., joints. The motion of continuum manipulators is often compared to that of biological manipulators such as trunks and tentacles. These continuum style robots can achieve motions that could only be obtainable by a conventionally designed robot with many more degrees of freedom. In this paper we present a detailed formulation and explanation of a novel kinematic model for continuum style robots. The design, construction, and implementation of our continuum style robot called the elephant trunk manipulator is presented. Experimental results are then provided to verify the legitimacy of our model when applied to our physical manipulator. We also provide a set of obstacle avoidance experiments that help to exhibit the practical implementation of both our manipulator and our kinematic model.     

Modelling the dynamic characteristics of a robot arm joint

The dynamic behavior of a robot arm has been modelled, taking into account the elasticity and damping of the joints and the backlash introduced by the gear pairs of the transmission mechanism. The links of the robot arm are considered to be rigid and its joints rotational. A technique for automatic formulation of the differential equations of the lumped mass dynamic model of the gear train, incorporating the backlash nonlinearity, has been developed. Each link and the accompanying joint comprise a module. The dynamic equations of the manipulator are formulated through an iterative technique. The algorithm will provide automatic formulation of the dynamic equations of the model of the arm including the referenced nonlinearities of the joint drives. Finally, an application of the algorithm to a planar manipulator arm with two links and two rotational joints is presented.     

Adaptive Tracking Controller for Rigid-Link Elastic-Joint Robots with Link Acceleration Estimation

This paper presents a adaptive switching control scheme for elastic joint robot manipulators. The characteristics of both flexible and rigid subsystems are assumed unknown except the joint stiffness. An adaptive estimator compensates the uncertainty due to the unknown robot characteristics. The actuators and links position, velocity and an estimation of the link acceleration are used as feedback to the control law. A filter is designed to estimate the links acceleration and its accuracy is insured by the linear control theory. Lyapunov theory is used to verify the asymptotic stability of the control law. The performance of the proposed control method is tested using a simulated planar robot with two rotational degrees of freedom for high and low joint stiffness.     

A simple and efficient computational approach for the forward dynamics of elastic robots

A simple and efficient Lagrangian formulation is presented for the forward dynamic analysis of elastic robots. The proposed method formulates the equations of motion with respect to a floating frame that follows the rigid motion of the links. By virtue of the proposed formulation the constraint conditions are inserted in the Hamilton's principle by means of a penalty formulation rather than by the classical Lagrange's multiplier technique. As a consequence, the number of equations that define the behavior of the robot does not increase. The numerical implementation of the new method is very simple and always leads to the solution of positive definite matrices. A series of elastic robots are analyzed and the results demonstrate the capabilities of the new formulation for the forward dynamics.     

Integrated distributed sensing and active vibration suppression of flexible manipulators using distributed piezoelectrics

Structural oscillation of flexible robot manipulators would severely hamper their operation accuracy and precision. This article presents an integrated distributed sensor and active distributed vibration actuator design for elastic or flexible robot structures. The proposed distributed sensor and actuator is a layer, or multilayer of piezoelectric material directly attached on the flexible component needed to be monitored and controlled. The integrated piezoelectric sensor/actuator can monitor the oscillation as well as actively and directly constrain the undesirable oscillation of the flexible robot manipulators by direct/converse piezoelectric effects, respectively. A general theory on the distributed sensing and active vibration control using the piezoelectric elements is first proposed. An equivalent finite element formulation is also developed. A physical model with distributed sensor/actuator is tested in laboratory; and a finite element model with the piezoelectric actuator is simulated. The distributed sensing and control effectiveness are studied.     

Adaptive Control of a Two-Link Robot Using Batch Least-Square Identifier

We design a regulation-triggered adaptive controller for robot manipulators to efficiently estimate unknown parameters and to achieve asymptotic stability in the presence of coupled uncertainties. Robot manipulators are widely used in telemanipulation systems where they are subject to model and environmental uncertainties. Using conventional control algorithms on such systems can cause not only poor control performance, but also expensive computational costs and catastrophic instabilities. Therefore, system uncertainties need to be estimated through designing a computationally efficient adaptive control law. We focus on robot manipulators as an example of a highly nonlinear system. As a case study, a 2-DOF manipulator subject to four parametric uncertainties is investigated. First, the dynamic equations of the manipulator are derived, and the corresponding regressor matrix is constructed for the unknown parameters. For a general nonlinear system, a theorem is presented to guarantee the asymptotic stability of the system and the convergence of parameters’ estimations. Finally, simulation results are discussed for a two-link manipulator, and the performance of the proposed scheme is thoroughly evaluated.