ACHIEVING MINIMUM PHASE TRANSFER FUNCTION FOR A NONCOLLOCATED SINGLE‐LINK FLEXIBLE MANIPULATOR

A noncollocated system has the potential of providing more precise tracking, improved disturbance rejection and increased bandwidth at the sensor location, but is considerably more difficult to stabilize than a collocated system due to its nonminimum phase nature. For a flexible manipulator, the problem becomes even more complicated because the system is inherently infinite dimensional. In this paper, a single‐link flexible manipulator having a tip payload and a noncollocated actuator and sensor is investigated using a linear distributed parameter model. With the joint torque as the input and the joint angle plus a weighted value of tip deflection as the measured output, an exact transfer function involving transcendental functions is derived. Using the methods of infinite product expansion, the root locus, and the asymptotic property of the roots of the transcendental equation, a necessary and sufficient condition in terms of the weighting factor of tip deflection is obtained such that the transfer function does not have any open right‐half plane zeros. This condition depends neither on the physical properties of the link nor on the mass properties of the tip payload. In order to correct the misinterpretation which has occurred in some closely‐related works concerning the stability of the infinite‐dimensional zero‐dynamics, the equivalence of the zeros of the transfer function and the eigenvalues of zero‐dynamics is also verified. Numerical results for the closed‐loop performance of a single‐link flexible arm using collocated and noncollocated PD controllers are given to show the efficacy of the proposed minimum phase transfer function approach.     

Inverse cartesian trajectory control and stabilization of a three-axis flexible manipulator

This article treats the question of end point trajectory control of a flexible manipulator based on the nonlinear inversion technique. The manipulator has two rigid links and the third link is elastic. A parameterization of the Cartesian coordinates of a point close to the end effector position is suggested. Using these coordinates as output variables, an inverse feedback control law is derived for tracking reference Cartesian trajectories. The stability of the zero dynamics associated with the end point motion control is examined. It is shown that inverse control of the end point causes divergent oscillatory flexible modes. In addition, for regulating the end point to a fixed position, a linear stabilizer is designed to damp the elastic vibration. Simulation results are presented to show that in the closed-loop system, reference end point trajectories can be accurately followed in spite of the parameter uncertainty in the arm dynamic model. © 1994 John Wiley & Sons, Inc.     

Robust trajectory control in the workspace of a class of flexible robots

This paper presents a robust indirect adaptive controller for a class of robots which have a flexible beam as last link. Since the relation between the joint space and the workspace depends on both the kinematics and the dynamics, a virtual joint space is defined so as to be related kinematically to the workspace. In fact, the transformation is defined from the virtual joint space to the joint and deformation spaces. Since the robot is a non‐minimum‐phase system in the virtual joint space, the transformation is obtained, in‐line by the iterative use of the causal–anticausal approach. Based on that transformation, a robust adaptive controller can be designed to ensure robust and fast convergence of the tracking error in the joint, deformation and virtual joint spaces. The controller thus obtained is simulated for a manipulator having one rigid and one flexible link. The simulation results demonstrate the good performances and the robustness of the system. © 2001 John Wiley & Sons, Inc.     

Hybrid Control Scheme for Robust Tracking of Two-Link Flexible Manipulator

The hybrid control scheme is proposed to stabilize the vibration of a two-link flexible manipulator while the robustness of Variable Structure Control (VSC) developed for rigid manipulators is maintained for controlling the joint angles. The VSC law alone, which is designed to accomplish only the asymptotic decoupled joint angle trajectory tracking, does not guarantee the stability of the flexible mode dynamics of the links. In order to actively suppress the flexible link vibrations, hybrid trajectories for the VSC are generated using the virtual control force concept, so that robust tracking control of the flexible-link manipulator can also be accomplished. Simulation results confirm that the proposed hybrid control scheme can achieve more robust tracking control of two-link flexible manipulator than the conventional control scheme in the presence of payload uncertainty.     

Hybrid Control Scheme for Robust Tracking of Two-Link Flexible Manipulator

A hybrid control scheme is proposed to stabilize the vibration of a two-link flexible manipulator while robustness of Variable Structure Control (VSC) developed for rigid manipulators is maintained for controlling the joint angles. The VSC law alone, which is designed to accomplish only the asymptotic decoupled joint angle trajectory tracking, does not guarantee the stability of the flexible mode dynamics of the links. In order to actively suppress the flexible link vibrations, hybrid trajectories for the VSC are generated using the virtual control force concept, so that robust tracking control of the flexible-link manipulator can also be accomplished. Simulation results confirm that the proposed hybrid control scheme can achieve more robust tracking control of two-link flexible manipulator than the conventional control scheme in the presence of payload uncertainty.     

Closed‐loop inverse kinematics algorithm for constrained flexible manipulators under gravity

This work deals with the inverse kinematics problem for a flexible robot manipulator under gravity in contact with a stiff surface. This problem consists of finding the joint and deflection variables for a given tip position and constraint force. The solution algorithm is based on the well‐known closed‐loop inverse kinematics (CLIK) scheme, using the Jacobian transpose, developed for rigid manipulators. The Jacobian employed in the algorithm is obtained by correcting the equivalent rigid manipulator Jacobian with two terms that account for the static deflections due to gravity and contact force, respectively. The algorithm is tested in a number of case studies on a planar two‐link arm. ©1999 John Wiley & Sons, Inc.     

On the force control problem for flexible joint manipulators

It is shown, using a singular perturbation model of the elastic joint manipulator dynamics and the concept of corrective control. how force control techniques developed for rigid manipulators can be extended to the flexible joint case. It is shown that the overall control law can be implemented in an inner loop/outer loop structure, where the inner loop is a nonlinear control that linearizes the system restricted to a suitable integral manifold in state space and the outer loop is a linear control that can be designed independently of the nonlinear inner loop, using any number of force control schemes designed for rigid manipulators to extend all of the standard techniques for force control of rigid manipulators to the flexible joint case, including hybrid position/force control, impedance control, or any other suitable design     

Inverse dynamics of spatial open-chain flexible manipulators with lumped and distributed actuators

This article addresses the problem of inverse dynamics for three-dimensional flexible manipulators with both lumped and distributed actuators. A recursive procedure is presented for computing the lumped inverse dynamic torques and the distributed piezoelectric actuator inputs for simultaneously tracking a prescribed end-point trajectory and reducing induced vibrations in the manipulator. The procedure sequentially solves for the non-causal inverse dynamic torques and piezoelectric voltages applied to each link in the manipulator, starting from the last element in the chain and proceeding to the base element. The method allows trajectory tracking wherein controllability of the structural vibrations is assured in all possible configurations through the use of only one motor at each intermediate joint and three motors at the ground. Numerical simulation shows that the elastic vibrations can be reduced significantly through the use of distributed actuators while at the same time satisfying the trajectory tracking requirement through the use of inverse dynamics. © 1994 John Wiley & Sons, Inc.     

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.     

Partial differential equation modeling and vibration control for a nonlinear 3D rigid‐flexible manipulator system with actuator faults

In this paper, modeling and controlling problem for a two‐link rigid‐flexible manipulator in three‐dimensional (3D) space is studied under actuator faults. For modeling, the dynamics of the 3D mechanical system is represented by nonlinear partial differential equations, which is first derived in infinite dimension form. Based on the nonlinear model, the controller is proposed, which can achieve joint angle control and vibration suppression control in the presence of actuator faults. The stability analysis of the closed‐loop system is given based on LaSalle invariance principle. Numerical simulations illustrate the effectiveness of the proposed controller. This study will promote the development of nonlinear flexible manipulator systems in 3D space.     

Parameterizing robust manipulator controllers under approximate inverse dynamics: A double‐Youla approach

We consider the goal of ensuring robust stability when a given manipulator feedback control law is modified online, for example, to safely improve the performance by a learning module. To this end, the factorization approach is applied to both the plant and controller models to characterize robustly stabilizing controllers for rigid‐body manipulators under approximate inverse dynamics control. Outer‐loop controllers to stabilize the nonlinear uncertain loop that results from approximate inverse dynamics are often derived by lumping uncertainty in a single term and subsequent analysis of the error system. Here, by contrast, the well‐known norm bounds of these uncertain dynamics are first recast into a generalized plant configuration that preserves the characteristic uncertainty structure. Then, the overall loop uncertainty is expressed with respect to the nominal outer‐loop feedback controller by means of an uncertain dual‐Youla operator. Therefore, using the dual‐Youla parameterization, we provide a novel way to rigorously quantify permissible perturbations of robot manipulator feedforward/feedback controllers. The method proposed in this paper does not constitute another robust control law for rigid‐body manipulators, but rather a characterization of a set of robustly stabilizing controllers. The resulting double‐Youla parameterization for the control of robot manipulators is amenable to numerous advanced design methods. The result is thoroughly discussed by a planar elbow manipulator and exemplified with a six‐degree‐of‐freedom robot scenario with varying payload.     

Trajectory control of a two DOF rigid–flexible space robot by a virtual space vehicle

Model based control schemes use inverse dynamics of the robot arm to produce the main torque component necessary for trajectory tracking. For a model-based controller one is required to know the model parameters accurately. This is a very difficult job especially if the manipulator is flexible. This paper presents a control scheme for trajectory control of the tip of a two arm rigid–flexible space robot, with the help of a virtual space vehicle. The flexible link is modeled as an Euler–Bernoulli beam. The developed controller uses the inertial parameters of the base of the space robot only. Bond graph modeling is used to model the dynamics of the system and to devise the control strategy. The efficacy of the controller is shown through simulated and animation results.     

Neural network based dynamic modeling of flexible‐link manipulators with application to the SSRMS

This paper presents an approach for dynamic modeling of flexible‐link manipulators using artificial neural networks. A state‐space representation is considered for a neural identifier. A recurrent network configuration is obtained by a combination of feedforward network architectures with dynamical elements in the form of stable filters. To guarantee the boundedness of the states, a joint PD control is introduced in the system. The method can be considered both as an online identifier that can be used as a basis for designing neural network controllers as well as an offline learning scheme to compute deflections due to link flexibility for evaluating forward dynamics. Unlike many other methods, the proposed approach does not assume knowledge of the nonlinearities of the system nor that the nonlinear system is linear in parameters. The performance of the proposed neural identifier is evaluated by identifying the dynamics of different flexible‐link manipulators. To demonstrate the effectiveness of the algorithm, simulation results for a single‐link manipulator, a two‐link planar manipulator, and the Space Station Remote Manipulator System (SSRMS) are presented. ©2000 John Wiley & Sons, Inc.     

Modeling and robust adaptive iterative learning control of a vehicle‐based flexible manipulator with uncertainties

In this brief, this paper deals with a robust adaptive iterative learning control (ILC) problem for a flexible manipulator attached to a moving vehicle with uncertainties. To begin with, considering the infinite dimensionality of the flexible distributed parameter system, a coupled ordinary differential equation and partial differential equation model is established without any discretization. Then, it is followed by a presentation of an adaptive ILC strategy, which can drive the vehicle and joint to the desired positions based on a proportional‐derivative feedback structure with unmodeled dynamics and external disturbances. The deformation of the flexible manipulator can also be suppressed simultaneously under the proposed control laws. By using Lyapunov's direct method, the stability of the closed‐loop system is demonstrated. The simulation results are provided to illustrate the effectiveness of the proposed control laws.     

Dynamic modelling, simulation and control of a manipulator with flexible links and joints

The paper presents a dynamic modelling technique for a manipulator with multiple flexible links and flexible joints, based on a combined Euler–Lagrange formulation and assumed modes method. The resulting generalised model is validated through computer simulations by considering a simplified case study of a two-link flexible manipulator with joint elasticity. Controlling such a manipulator is more complex than controlling one with rigid joints because only a single actuation signal can be applied at each joint and this has to control the flexure of both the joint itself and the link attached to it. To resolve the control complexities associated with such an under-actuated flexible link/flexible joint manipulator, a singularly perturbed model has been formulated and used to design a reduced-order controller. This is shown to stabilise the link and joint vibrations effectively while maintaining good tracking performance.     

A Virtual Work Based Algorithm for Solving Direct Dynamics Problem of a 3-RRP Spherical Parallel Manipulator

In this paper, we use principle of virtual work to obtain the direct dynamics analysis of a 3-RRP spherical parallel manipulator, also called spherical star-triangle (SST) manipulator (Enferadi et al., Robotica 27, 2009). This manipulator has good accuracy and relatively a large workspace which is free of singularities (Enferadi et al., Robotica, 2009). The direct kinematics problem of this manipulator has eight solution (Enferadi et al., Robotica 27, 2009). Given a desired actuated joint trajectories, we first present an algorithm for selecting the admissible solution. Next, direct velocity and direct acceleration analysis are obtained in invariant form. The concept of direct link Jacobian matrices is introduced. The direct link Jacobian matrix relates motion of any link to vector velocity of actuated joints. Finally, dynamical equations of the manipulator are obtained using the principle of virtual work and the concept of direct link Jacobian matrices. This method allows elimination of constraint forces and moments at the passive joints from motion equations. Two examples are presented and trajectory of moving platform are obtained. Results are verified using a commercial dynamics modeling package as well as inverse dynamics analysis (Enferadi et al., Nonlinear Dyn 63, 2010).     

失重环境下可控柔性臂的模态特性

在非重力场中,考虑控制器动态反馈的影响,对存在控制器定位约束的柔性臂系统进行动力分析,研 究其在相对平衡位置的模态特性．以具有柔性关节和弹性臂杆的可控柔性臂为研究对象,分析了控制器作用下的反 馈约束特性,将控制器位置和速度增益引入力边界条件,得到了耦合控制器参量的模态特征方程,证明了反馈约束 的存在使得系统特征频率为复频率,且模态主振型是复变函数．通过数值仿真,明确了可控柔性臂的模态特性与控 制器增益之间的关系,得到了不同于经典振动理论的结论．设计了可控柔性臂的仿失重实验平台,试验模态结果证 明了理论分析的有效性．     

基于轨迹规划的平面三连杆欠驱动机械臂位置控制

针对中间关节为欠驱动的二阶非完整平面三连杆机械臂,提出一种基于轨迹规划的末端点位置控制策略.首先,建立系统的动力学模型,并根据几何关系利用差分进化算法求取所有连杆与目标位置相对应的目标角度;然后,根据驱动关节与欠驱动关节的耦合关系,采用时间缩放法和双向法分别规划两根驱动连杆的两条轨迹,并利用遗传算法优化合适的第1连杆中间位置,将两条轨迹拼接成一条完整可达轨迹;最后,设计滑模变结构控制器以跟踪完整可达轨迹,实现系统从初始位置到目标位置的控制目标.数值仿真结果表明了所提出控制策略的有效性.     

A Robust Controller for A 3‐DOF Flexible Robot with a Time Variant Payload

This article describes a new control scheme designed for a three degree of freedom (3‐DOF) flexible robot. The control scheme consists of two multi variable control loops. The inner loop is the motor's position control system, while the outer loop controls the robot tip's position, thus canceling vibrations which are originated by the structural flexibility of the manipulator during movement. As it will be shown, the outer control loop is robust to payload variations. The outer loop performance is based on a perfect cancelation of the inner loop dynamics. The effects of not achieving such perfect cancelation are also studied, and rules for designing a robust controller in this case are developed. Simulations assuming different payloads have been carried out with successful results for trajectory tracking. Trajectory tracking with a variable payload is also achieved.     

Feedback linearization control of flexible joint robots

In this paper, the control of a two link, flexible joint manipulator is examined. Among external forces, an exogenous constraint force acting on the end-effector is included. The manipulator dynamics are described by:

. On the assumption that f(x), g(x) and J^T f_e(x) are smooth vector fields, it is shown that the inner loop control u is of the form:

u=1/dT_n,g(v−dT_n,(ƒ(x)+J^Tƒ_e(x)))

where u is an outer loop control signal and y = T(x) is a diffleomorphism that transform (a) into linear system. As the position control scheme is adopted, the value of the contact force is not controlled.

The results for the inner loop control are substantiated by simulation of a two-link robot model. The robustness of the control method is examined and a Lyapunov-based control correction, similar to that of the free motion case, is implemented. Results are obtained for parametric errors of up to 50%. In the simulation, the manipulator is required to follow a specified joint trajectory such that the end-effector traces a sinusoidal path along a constraint surface. The results obtained illustrate the tracking of the link reference trajectory and indicate that the inner loop corrections are valid.