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In this paper, a nonlinear model predictive control (NMPC) is designed for redundant robotic manipulators in non-stationary environments. Using NMPC, the end-effector of the robot could track predefined desired path and reaches a moving target in the Cartesian space, while at the same time avoids collision with moving obstacles and singular configurations in the workspace. To avoid collisions with moving obstacles and capturing moving target, the future position of obstacles and the moving target in 3D space is predicted using artificial neural networks. Using online training of the neural network, no knowledge about obstacles and motion of the moving object is required. The nonlinear dynamic of the robot including actuators dynamic is also considered. Numerical simulations performed on a 4DOF redundant spatial manipulator actuated by DC servomotors, shows effectiveness of the proposed method.

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In redundant manipulation systems the end-effector path does not completely determine the trajectories of all the individual degrees of freedom and this freedom can be used to enhance the performance in some sense. The paper deals with utilizing the redundancy to minimize energy consumption. It extends previous results by considering more general cases of possible coupling between the axes, e.g. three axes for planar motion, and more general paths comprising of several primitive motions connected dynamically. The solution is based on projections into lower subspaces that separate the system and the input into two parts. One that is completely determined by the end-effector path and the other that is free for optimization. Simulation results show that redundancy, even with limited joint motion, can lead to a considerable reduction in energy consumption.

In this chapter, a control system approach that is based on artificial intelligence is adopted to analyze the inflation targeting strategy. The input/output model is constructed using a multi-layered perceptron network and a closed loop control strategy is adopted using a genetic algorithm to control inflation through the manipulation of interest rates. Given the historical inflation rate data, a control scheme is used to determine the interest rate that is required to attain the given inflation rate. The calculated interest rate is then compared to the historical inflation rate to evaluate the effectiveness of the control strategy.

This paper presents a nonlinear model predictive control (NMPC) method with adaptive neuro-modelling for redundant robotic manipulators. Using the NMPC, the end-effector of the robot tracks a predefined geometry path in the Cartesian space without colliding with obstacles in the workspace and at the same time avoiding singular configurations of the robot. Furthermore, using the neural network for the model prediction, no knowledge about system parameters is necessary; hence, yielding robustness against changes in parameters of the system. Numerical results for a 4DOF redundant spatial manipulator actuated by DC servomotors shows effectiveness of the proposed method.

This article presents new closed-loop schemes for solving the inverse kinematics of constrained redundant manipula tors. In order to exploit the space of redundancy, the end- effector task is suitably augmented by adding a constraint task. The success of the technique is guaranteed either by specifying the constraint task ad hoc or by resorting to a task priority strategy. Instead of previous inverse kinemat ics schemes that use the Jacobian pseudoinverse, the schemes in this work are shown to converge using the Jacobian transpose. A number of case studies illustrate different ways of solving redundancy in the context of the proposed schemes.

This paper describes an efficient approach for nonlinear model predictive control, applied to a 6-DOF arm robot. The model is first linearized and decoupled by feedback, secondly a model predictive control scheme, implemented with an optimized dynamic model and running within small sampling period, is exhibited. Major simulation results performed using numerical values of an industrial PUMA 560 robot prove the effectiveness of the proposed approach. The nonlinear model-based predictive control and the widely used computed torque control are compared. Tracking performance and robustness with respect to external disturbances or errors in the model are enlightened.

This paper presents a Nonlinear Model Predictive Control (NMPC) for redundant robotic arms. Using NMPC, the end-effector of robotic arm tracks a predefined geometry path in the Cartesian space in such a way that no collision with obstacles in the workspace and no singular configurations for robot occurs. Nonlinear dynamic of the robot including actuators dynamic is also considered. Moreover, the on-line tuning of the weights in NMPC is performed using the fuzzy logic. The proposed method automatically adjusts the weights in cost function in order to obtain good performance. Numerical simulations of a 4DOF redundant spatial manipulator actuated by DC servomotors shows effectiveness of the proposed method.

While linear model predictive control is popular since the 70s of the past century, only since the 90s there is a steadily increasing interest from control theoreticians as well as control practitioners in nonlinear model predictive control (NMPC). The practical interest is mainly driven by the fact that today's processes need to be operated under tight performance specifications. At the same time more and more constraints, stemming for example from environmental and safety considerations, need to be satisfied. Often, these demands can only be met when process nonlinearities and constraints are explicitly taken into account in the controller design. Nonlinear predictive control, the extension of the well established linear predictive control to the nonlinear world, is one possible candidate to meet these de- mands. This paper reviews the basic principle of NMPC, and outlines some of the theoreti- cal, computational, and implementational aspects of this control strategy.

One important issue in the motion planning and control of kinematically redundant manipulators is the obstacle avoidance. In this paper, a recurrent neural network is developed and applied for kinematic control of redundant manipulators with obstacle avoidance capability. An improved problem formulation is proposed in the sense that the collision-avoidance requirement is represented by dynamically-updated inequality constraints. In addition, physical constraints such as joint physical limits are also incorporated directly into the formulation. Based on the improved problem formulation, a dual neural network is developed for the online solution to collision-free inverse kinematics problem. The neural network is simulated for motion control of the PA10 robot arm in the presence of point and window-shaped obstacle.

We present an efficient obstacle avoidance control algorithm for redundant manipulators using a new measure called collidability measure. Considering moving directions of manipulator links, the collidability measure is defined as the sum of inverse of predicted collision distances between links and obstacles: This measure is suitable for obstacle avoidance since directions of moving links are as important as distances to obstacles. For kinematic or dynamic redundancy resolution, null space control is utilized to avoid obstacles by minimizing the collidability measure: We present a velocity-bounded kinematic control law which allows reasonably large gains to improve the system performance. Also, by clarifying decomposition in the joint acceleration level, we present a simple dynamic control law with bounded joint torques which guarantees tracking of a given end-effector trajectory and improves a kinematic cost function such as collidability measure. Simulation results are presented to illustrate the effectiveness of the proposed algorithm.

This paper presents a non-linear model predictive control (NMPC) for redundant robotic manipulators. Using NMPC, the end-effector of the robotic manipulator tracks a predefined geometry path in Cartesian space in such a way that no collision with obstacles in the workspace and no singular configurations for the robot occurs. Non-linear dynamic of the robot, including actuators dynamic, is also considered. Moreover, the online tuning of the weights in NMPC is performed using the fuzzy logic. The proposed method automatically adjusts the weights in the cost function in order to obtain good performance. Furthermore, using neural networks for model prediction, no prior knowledge about system parameters is necessary and system robustness against changes in its parameters is achieved. Numerical simulations of a 4DOF redundant spatial manipulator actuated by DC servomotors show effectiveness of the proposed method.

An algorithm for kinematic motion planning of redundant planar robots, having revolute joints, in an unknown dynamic environment is presented. Distance ranging sensors, mounted on the body of each manipulator link, are simulated here to estimate the proximity of an obstacle. The sensory data is analyzed through a fuzzy controller which estimates whether a collision is imminent, and if so, employs a geometric approach to compute the joint movements necessary to avoid the collision. Obstacles can sometimes move uncompromisingly in the environment attempting a deliberate collision. Strategies to deal with such cases are presented and recovery procedures to circumvent the obstacle from tight corners are suggested. Cases of link overlap have been avoided by considering each link as a body which is sensed as an obstacle by every other link of the same manipulator. Suitable examples are presented to demonstrate the algorithm.

A method for computing the optimal motions of robot manipulators in the presence of moving obstacles is presented. The algorithm considers the nonlinear manipulator dynamics, actuator constraints, joint limits and obstacle avoidance. The optimal traveling time and the minimum mechanical energy of the actuators are considered together to build a multicriterion function. Sequential unconstrained minimization techniques have been used for the optimization. Given the initial and final points the trajectories are defined using spline functions and are obtained through off-line computation for on-line operation. The obstacles are considered as objects sharing the same workspace performed by the robot. The obstacle avoidance is expressed in terms of the distances between potentially colliding parts and the motion is represented using translation and rotational matrices. Numerical applications involving a Stanford manipulator are presented.

The purpose of this paper is twofold. In the first part, we give a review on the current state of nonlinear model predictive control (NMPC). After a brief presentation of the basic principle of predictive control we outline some of the theoretical, computational, and implemen-tational aspects of this control strategy. Most of the theoretical developments in the area of NMPC are based on the assumption that the full state is available for measurement, an assumption that does not hold in the typical practical case. Thus, in the second part of this paper we focus on the output feedback problem in NMPC. After a brief overview on existing output feedback NMPC approaches we derive conditions that guarantee stability of the closed-loop if an NMPC state feedback controller is used together with a full state observer for the recovery of the system state.

The key features of a control system being developed for whole arm
collision avoidance for kinematically redundant manipulators are
described. The control system simultaneously deals with multiple
detections and provides controlled end-effector path deviation in the
event that the range of self-motion available is not sufficient. The
system has undergone extensive simulation using an animated model of a
manipulator in a variety of workcells. Examples are given

In this paper, we firstly propose a robust optimal predictive control (ROPC) scheme. In the ROPC scheme proposed, an optimal predictive control (OPC) is combined with a robust control that is constructed on the Lyapunov min-max approach. Since the control design of a real system may often be made on the basis of the imperfect knowledge about the model, it is an important trend to design the robust control law that guarantees the desired properties of the system under uncertain elements. By introducing the OPC technique in robust control here, we can find out much more deterministic controller for both the stability and the performance of uncertain nonlinear systems. Secondly, we propose the adaptive version of the ROPC scheme (AROPC). In the AROPC scheme proposed here, the adaptation law estimates the bounds of uncertainties of the system, and the robust control uses these estimates. This scheme is strongly useful in the case that the bounds of uncertainties are not known or known roughly. The analysis of the stability in the sense of Lyapunov is performed. We apply these algorithms to a simple type of 2-Iink robot manipulator and perform the simulations.

Several aspects of the path planning problem for highly redundant manipulators are dealt with in this paper. A new method is presented for the path planning. The basic idea is to find a smooth path consisting of points close enough to each other using harmonic potential fields, and then to keep the tip of each link on these path points until the manipulator reaches the goal. The concept of master link is introduced and applied to three path planning algorithms for the smooth motion of the manipulator. A reversing procedure is included to take the manipulator to its initial position. Besides, software developed in C++ for Windows platforms is introduced. The main features of the software are to draw obstacles and manipulators on the screen, to obtain two- and three-dimensional images of potential fields and implement path planning algorithms.

This paper focuses on autonomous motion planning of manipulators in known environments and with unknown dynamic obstacles. The navigation technique of robot control using artificial potential functions is based on fuzzy logic and stability is guaranteed by Lyapunov theory. A fuzzy logic system or fuzzy system is a universal approximator which provides a rule-based mapping between the input and the output space, while classical approaches make use of analytic harmonic functions to solve the navigation problem. In this particular application, the fuzzy system proposed is used to approximate the gradient of the harmonic functions.

The manipulator with a large degree of redundancy is useful for realizing multiple tasks such as maneuvering the robotic arms in the constrained workspace, e.g. the task of maneuvering the end-effector of the manipulator along a pre-specified path into a window. This paper presents an on-line technique based on a posture generation rule to compute a null-space joint velocity vector in a singularity-robust redundancy resolution method. This rule suggests that the end of each link has to track an implicit trajectory that is indirectly resulted from the constraint imposed on tracking motion of the end-effector. A proper posture can be determined by sequentially optimizing an objective function integrating multiple criteria of the orientation of each link from the end-effector toward the base link as the secondary task for redundancy resolution, by assuming one end of the link is clamped. The criteria flexibly incorporate obstacle avoidance, joint limits, preference of posture in tracking, and connection of posture to realize a compromise between the primary and secondary tasks. Furthermore, computational demanding of the posture is reduced due to the sequential link-by-link computation feature. Simulations show the effectiveness and flexibility of the proposed method in generating proper postures for the collision avoidance and the joint limits as a singularity-robust null-space projection vector in maneuvering redundant robots within constrained workspaces.

The paper presents a software implementation of the nonlinear model predictive control (NMPC) algorithm of Wen and Jung (2002, 2003) to a 6-degree-of-freedom robot manipulator. The implemented algorithm attempts to reduce the error at the end of the prediction horizon rather than to find the optimal solution. This leads to reduction of computational burden. Simulations have been carried out for the mathematical model of the 6DOF robot manipulator (Puma 560) and some exemplifying results are presented.

Robot manipulators present restrictions on their performance, such as the maximum torque the motors can apply, limitations in their position, speed, acceleration, etc. This paper studies the application of a multivariable constrained predictive controller for robotic control, that considers these restrictions when calculating the control signal. The model used to calculate predictions is evaluated at every sample-time by linearization of the Denavit-Hartenberg model, and discretization using a bilinear transform. By using simulation, this paper presents the successful application of this technique to a direct-drive manipulator.

Three nonlinear continuous-time predictive control schemes are proposed to address the trajectory tracking control problem of rigid link robot manipulators. The control laws using state variable feedback minimize a quadratic performance index of the state predicted tracking error. Without online optimization, an asymptotic tracking of smooth reference trajectories is guaranteed. The proposed controllers achieve the positions and speed tracking objectives via link position measurements. Lyapunov theory is used to prove the boundedness and stability convergence of the state tracking. Robustness with respect to payload uncertainties and viscous friction is shown. Simulations for a two-link rigid robot are performed to validate the proposed controller.

An efficient approach for nonlinear model predictive control is proposed. Basically, the model is first linearized by feedback, secondly a model predictive control scheme, implemented with an optimized dynamic model and running within a small sampling period, is exhibited. Major simulation results performed using numerical values of an industrial SCARA type robot prove the effectiveness of the proposed approach. The nonlinear model-based predictive control and the commonly used computed torque control are compared. The tracking performances and the robustness with respect to external disturbances or model/robot mismatch are described.

We present a measure called collidability measure for obstacle
avoidance control of redundant manipulators. Considering moving
directions of manipulator links, the collidability measure is defined as
the inverse of sum of predicted collision distances between links and
obstacles. This measure is suitable for obstacle avoidance control since
directions of moving links are as important as distances to obstacles.
For dynamic redundancy resolution, null space control is utilized to
avoid obstacles by minimizing the collidability measure. Also, by
clarifying decomposition in the joint acceleration level, we present a
simple dynamic control law with bounded joint torques which guarantees
tracking of a given end-effector trajectory and improves a kinematic
cost function such as collidability measure. Simulation results are
presented to illustrate the effectiveness of the proposed
algorithm

The time-varying obstacle avoidance problem is considered mathematically. The manipulator motion is described in terms of constrained motions that are governed by the environment and the manipulator itself. Various constraints are identified and derived, which are classified into two categories: the environment constraints and the manipulator constraints. These constraints are converted into the reachable path segment at each servo time instant to verify the existence of a collision-free trajectory. Discussions with regard to time-varying obstacle avoidance are also presented.

Advanced Robotics Redundancy and Optimization

- Y Nakamuro

Nakamuro, Y. (1991). Advanced Robotics Redundancy and Optimization.
Addison-Wesley, Reading, MA, USA.

Finite horizon non linear predictive control with integral action of rigid link manipulators

- R Hedjar
- R Toumi
- P Boucher
- D Dumur
- S Tebbani

Hedjar, R., Toumi, R., Boucher, P., Dumur, D., and Tebbani, S. (2005).
Finite horizon non linear predictive control with integral action of rigid
link manipulators. IEEE Conference on Control Applications, Août,
Canada.