Robust adaptive control for robot manipulators

A composite adaptive control law for robot manipulators in task space, which uses both the tracking error and the prediction error to drive parameter estimation, is developed in this paper. It is shown that global stability and convergence can be achieved for the adaptive control algorithm in the ideal case, and furthermore that the algorithm can be easily modified by using parameter projection to achieve robustness with respect to a class of unmodelled dynamics. In addition, the algorithm has the advantage that no requirement is needed for the inverse of the jacobian matrix or for the bounded inverse of the estimated inertia matrix. A simulation example is provided for performance demonstration.     

Global positioning of robot manipulators with mixed revolute and prismatic joints

The existing controllers for robot manipulators with uncertain gravitational force can globally stabilize only robot manipulators with revolute joints. The main obstacles to the global stabilization of robot manipulators with mixed revolute and prismatic joints are unboundedness of the inertia matrix and the Jacobian of the gravity vector. In this note, a class of globally stable controllers for robot manipulators with mixed revolute and prismatic joints is proposed. The global asymptotic stabilization is achieved by adding a nonlinear proportional and derivative term to the linear proportional-integral-derivative (PID) controller. By using Lyapunov's direct method, the explicit conditions on the controller parameters to ensure global asymptotic stability are obtained.     

Simulation of Industrial Manipulators Based on the UDU T Decomposition of Inertia Matrix

The UDU ^T – U and D are respectively the upper triangular and diagonal matrices – decomposition of the generalized inertia matrix of an n-link serial manipulator, introduced elsewhere, is used here for the simulation of industrial manipulators which are mainly of serial type. The decomposition is based on the application of the Gaussian elimination rules to the recursive expressions of the elements of the inertia matrix that are obtained using the Decoupled Natural Orthogonal Complement matrices. The decomposition resulted in an efficient order n, i.e., O(n), recursive forward dynamics algorithm that calculates the joint accelerations. These accelerations are then integrated numerically to perform simulation. Using this methodology, a computer algorithm for the simulation of any n degrees of freedom (DOF) industrial manipulator comprising of revolute and/or prismatic joints is developed. As illustrations, simulation results of three manipulators, namely, a three-DOF planar manipulator, and the six-DOF Stanford arm and PUMA robot, are reported in this paper.     

On the boundedness of the Hessian of the potential energy of robot manipulators

Several results on robot manipulator motion control require a uniform bound for the Hessian of the potential energy or equivalently the Jacobian of the gravity vector. Not all robot manipulators, however, ensure the existence of such a uniform bound. The first contribution of this article is the complete characterization of this class which is referred to as class ℬ︁𝒢𝒥 manipulators. The uniform bound of the Hessian is typically part of the control law expression and hence it plays an important role in controller gain synthesis. The second contribution of this article consists of deriving, for class ℬ︁𝒢𝒥 robot manipulators, an easy to compute explicit expression of the uniform bound in terms of kinematic and inertial link parameters. If for a particular robot manipulator the Hessian of potential energy is not uniformly bounded, a bound exists that is valid within the physical workspace of the manipulator. The third contribution of this article is the derivation of an explicit expression for the latter bound which is useful in the design and controller gain synthesis of control laws that are valid locally. ©1999 John Wiley & Sons, Inc.     

Digital high-gain PD control of robot manipulators

For all its simplicity, local proportional and derivative (PD) control remains an effective and popular tool in robot manipulation. The analysis of such a control system has usually been based on the assumption that the PD algorithm is implemented continuously. This article explores the issue of digital high-gain PD control on robot manipulators from the viewpoint of singular perturbation. It is shown that the fast subsystem becomes unstable when the gains are high enough, due to the effect of sample-and-hold. The stability is related to the feedback gains, the length of sampling period, the computation time, and the eigenvalues of the mass matrix. With reasonable approximations, simple and explicit stability criteria are derived. It is also shown that when the gains reach the critical values, chattering in the control signal occurs, similar to what happens in a variable structure system with sliding mode. Since the eigenvalues of the mass matrix vary with robot positions, the system stability and control chattering are also affected by the trajectory that the robot is planned to follow. © 1997 John Wiley & Sons, Inc.     

Adaptive control of robot manipulators in task space

Adaptive control of robotic manipulators in task space or Cartesian space is considered. A general Lyapunov-like concept is used to design an adaptive control law. It is shown that the global stability and convergence can be achieved for the adaptive control algorithm. The algorithm has the advantage that the inverse of Jacobian matrix is not required. The algorithm is further modified so that the requirement for the bounded inverse of estimated inertia matrix is also eliminated. In addition, two approaches are presented to achieve robustness to bounded disturbances     

Upper Bounding Estimation for Robustness to the Parameter Uncertainty with Trigonometric Function in Trajectory Control of Robot Arms

In this paper, a new robust control law is considered for controlling robot manipulators subjected to uncertainties. The control law is derived as a result of analytical solution from the Lyapunov function, thus stability of the uncertain system is guaranteed. Apart from previous studies, uncertainty bound and adaptation gain matrix are updated in time with the estimation law to control the system properly and uncertainty bound is determined using a trigonometric function of robot kinematics, inertia parameters and tracking error while adaptation gain matrix is determined using a trigonometric function of robot kinematics and tracking error. Application to a two-link robotic manipulator is presented and numerical simulations are included.     

Adaptive explicit force control of position-controlled manipulators

This article presents a new adaptive outer-loop approach for explicit force regulation of position-controlled robot manipulators. The strategy is computationally simple and does not require knowledge of the manipulator dynamic model, the inner-loop position controller parameters, or the environment. It is shown that the control strategy guarantees global uniform boundedness of all signals and convergence of the position/force regulation errors to zero when applied to the full nonlinear robot dynamic model. If bounded external disturbances are present, a slight modification to the control scheme ensures that global uniform boundedness of all signals is retained and that arbitrarily accurate stabilization of the regulation errors can be achieved. Additionally, it is shown that the adaptive controller is also applicable to robotic systems with PID inner-loop position controllers. Computer simulation results are given for a Robotics Research Corporation (RRC) Model K-1207 redundant arm and demonstrate that accurate and robust force control is achievable with the proposed controller. Experimental results are presented for the RRC Model K-1207 robot and confirm that the control scheme provides a simple and effective means of obtaining high-performance force control. © 1996 John Wiley & Sons, Inc.     

Linearized dynamic models of robot manipulators in cartesian space

This article presents development of the linearized dynamic models of robot manipulators in Cartesian space. A clever method is proposed to formulate the linearized dynamic models of robot manipulators in Cartesian space. Efficient methods are developed to compute various matrices involved in the models (or the manipulator sensitivity matrices in Cartesian space), such as the Jacobian matrix and the first and second derivatives of it with respect to time as well as the manipulator sensitivity matrices in joint space. The proposed methods are simple and systematic and have computational complexity of the order O(n²) only, where n is the number of degrees-of-freedom of the manipulator.     

On the robust control of robot manipulators

A simple robust nonlinear control law for n-link robot manipulators is derived using the Lyapunov-based theory of guaranteed stability of uncertain systems. The novelty of this result lies in the fact that the uncertainty bounds needed to derive the control law and to prove uniform ultimate boundedness of the tracking error depend only on the inertial parameters of the robot. In previous results of this type, the uncertainty bounds have depended not only on the inertia parameters but also on the reference trajectory and on the manipulator state vector. The presented result also removes previous assumptions regarding closeness in norm of the computed inertia matrix to the actual inertial matrix. The design used thus provides the simplest such robust design available to date     

Logarithmic based robust approach to parametric uncertainty for control of robot manipulators

In this paper, a new robust control law for controlling robot manipulators with parameter uncertainty is presented. A controller is designed based on the Lyapunov function and the control law that guarantees the system stability is derived as a result of analytical solution. Apart from previous studies, uncertainty bound and adaptation gain matrix are determined using the estimation law to control the system properly, and so this estimation law is developed as a logarithmic function depending on robot kinematics inertia parameters and tracking error. An application of the proposed control input to a two‐link robot manipulator is presented and numerical simulations are included. Copyright © 2005 John Wiley & Sons, Ltd.     

Output feedback tracking control of robot manipulators with model uncertainty via adaptive fuzzy logic

Many robot controllers require not only joint position measurements but also joint velocity measurements; however, most robotic systems are only equipped with joint position measurement devices. In this paper, a new output feedback tracking control approach is developed for the robot manipulators with model uncertainty. The approach suggested herein does not require velocity measurements and employs the adaptive fuzzy logic. The adaptive fuzzy logic allows us to approximate uncertain and nonlinear robot dynamics. Only one fuzzy system is used to implement the observer-controller structure of the output feedback robot system. It is shown in a rigorous manner that all the signals in a closed loop composed of a robot, an observer, and a controller are uniformly ultimately bounded. Finally, computer simulation results on three-link robot manipulators are presented to show the results which indicate good position tracking performance and robustness against payload uncertainty and external disturbances.     

An efficient algorithm for generating manipulator inertia matrix using the minimum set of dynamics parameters

This article presents an efficient algorithm for computing the inertia matrix of rigid serial manipulators. The derivation of the algorithm is based on the closed-form formulation of the force and moment exerted on a link using a minimum set of dynamic parameters of the manipulator model. The minimum set of dynamic parameters can be derived completely from the original dynamic parameters using the recursive re-grouping method before starting the simulation and the control. The proposed computation method is suitable for the control and the simulation based on parameter estimates because the minimum set of dynamic parameters is an identifiable parameter set. The computational efficiency of the proposed methods is compared with other published methods. It is shown that the proposed algorithm is the most efficient approach for serial manipulators. As an example, the number of computations for the inertia matrix of a manipulator with n rotational joints is 11n²+9 n − 35 multiplications and 7n²+ 23 n − 57 additions by reformulating the dynamic model using the minimum set of dynamic parameters. © 1996 John Wiley &, Sons, Inc.     

On the adaptive control of robot manipulators with velocity observers

To achieve accurate tracking control of robot manipulators many schemes have been proposed. A common approach is based on adaptive control techniques, which guarantee trajectory tracking under the assumption that the robot model structure is perfectly known and linear in the unknown parameters, while joint velocities are available. Despite tracking errors tend to zero, parameter errors do not unless some persistent excitation condition is fulfilled. There are few works dealing with velocity observation in conjunction with adaptive laws. In this note, an adaptive control/observer scheme is proposed for tracking position of robot manipulators. It is shown that tracking and observation errors are ultimately bounded, with the characteristic that when a persistent excitation condition is matched then they, as well as the parameter errors, tend to zero. Simulation results are in good agreement with the developed theory.     

A new class of adaptive controllers for robot trajectory tracking

This article presents a new class of adaptive schemes for the motion control of robot manipulators. The proposed controllers are very general and computationally efficient because they do not require knowledge of either the mathematical model or the parameter values of the manipulator dynamics, and are implemented without calculation of the robot inverse dynamics or inverse kinematic transformations. It is shown that the control strategies are globally uniformly bounded in the presence of bounded disturbances, and that in the absence of disturbances the ultimate bound on the size of the tracking errors can be made arbitrarily small. Computer simulation results are given for a PUMA 560 manipulator, and demonstrate that accurate and robust trajectory tracking can be achieved by using the proposed controllers. © 1994 John Wiley & Sons, Inc.     

Adaptive tracking control of flexible‐joint manipulators without overparametrization

In this paper, an adaptive controller is designed for rigid‐link flexible‐joint robot manipulators based on link and actuator position measurements only. It is based on the adaptive integrator backstepping method and the link and actuator velocity filters are used to estimate the unknown velocity terms. Moreover, the proposed controller exploits the estimate of the joint stiffness matrix inverse to overcome the overparametrization problem, which has been a significant drawback in adaptive partial state feedback controllers. It achieves asymptotic tracking of link positions while keeping all states and signals bounded. The tracking capability of the presented method is shown through simulation results of one‐ and two‐link flexible joint manipulators. © 2004 Wiley Periodicals, Inc.     

一种基于操作空间的操作器分散自适应控制方法

本文提出了一种操作器分散自适应阻力控制方法．这种方法不要求知道操作器动态模型的结构和参数，采用分散控制的形式，可以对各自由度单独进行控制，因此计算简单有效，具有一定的容错能力，控制系统有较好的暂态性能，由于控制律以操作空间坐标形式描述，适合于具有冗余自由度的操作器的控制．计算机仿真表明了该方法的良好的控制效果     

Comments on `On adaptive inverse dynamics control of rigid robots'[with reply]

The commenters point out an error in the adaptive control approach for robotic manipulators given in the above-named work by M.W. Spong and R. Ortega (see ibid., vol.35, p.92-5, Jan. 1990). It is noted that the authors assert that their adaptive control scheme eliminates the restriction given in J.J. Craig (1986) on the estimated inertia matrix; however, the commenters point out that a form of the inverse of the estimated inertia matrix must be bounded for the acceleration to be bounded. In a reply, the authors acknowledge the error in their paper     

A class of nonlinear PD-type controllers for robot manipulators

In this article we present the stability analysis of a class of PD-type controllers for position and motion control of robot manipulators. The main feature of this class of controllers is that the proportional and derivative gains can be nonlinear functions of the robot states. These controllers can be obtained from control strategies that adjust the controller gains depending on the robot states. It is shown that global asymptotic stability of the control system is achieved provided that the P and D gains have suitable structure. As an outcome, we propose a global regulator constrained to deliver torques within prescribed limits of the actuator's capabilities. Experimental results on a two degrees of freedom direct drive arm show the usefulness of the proposed scheme. © 1996 John Wiley & Sons, Inc.