Any motion of a rigid body in 3-D space is accompanied with six small deviations along and around its six DOF in infinitesimal motions. Sources of these errors are considered in detail as applied to geometrical errors [1]. For the 6-DOF manipulators, the total geometrical error vector depends on (6 n + 6) components, where n is the number of links, among them only 6 errors are induced by actuated links. This approach is formulated through the variational method of accuracy calculation [2], which models any small displacement of the rigid body as a result of total variation of the nominal function (the form-shaping function [1, 2] fulfils the role of the nominal function as applied to machines and robots).

To clear up an interrelationship between the nominal displacements and their deviations in the accuracy-related formulae, the links of the 2-DOF manipulator are renumbered as is shown in Figure 10.1. In these terms, the manipulation matrix, Eqs. (4.7), becomes

Kinematic redundancy can be obtained when a robot manipulator possesses more DOF than the minimum number required to execute a given end-effector task. If a robot manipulator is redundant, the inverse-kinematics problem may admit infinite solutions. How to obtain the required optimal solution, that is, the redundancy resolution problem, is an appealing and significant topic in robotics areas. In the past, many redundancy resolution schemes have been presented, and they can be divided into two main categories; that is, globally optimal control of redundancy and local/instantaneously optimal control of redundancy. The globally optimal concept and method were originally suggested by Whitney, and the early effective global schemes were presented by Uchiyama et al . and Nakamura et al . The global optimal schemes can generate the global optimal solutions, but they require a large amount of computation and are not suitable for real-time kinematic resolution. The detailed discussion can be referred to [143]. Thus, the local/instantaneously optimal control schemes are widely investigated and used in robotics areas. The classic local optimal solution is pseudoinverse-type formulation, which is investigated in Chapter 1; that is, one minimum-norm particular solution plus a homogeneous solution. Based on such a pseudoinverse-type solution, many optimization performance criteria have been exploited in terms of manipulator configurations and interaction with the environment, such as joint-limits avoidance, singularity avoidance, manipulability enhancement, and obstacle avoidance. Recent research shows that the solutions to redundancy resolution problems can be enhanced by using optimization techniques based on quadratic program (QP) methods. Compared with the classic pseudoinverse-based solutions, such QP-based methods do not need to compute the inverse of the Jacobian matrix, and are readily to deal with the inequality and/or bound constraints. Thus, new interests in QP-based methods have been generated. In [90], considering the physical limits, Cheng et al . presented a compact QP method to resolve the constrained kinematic redundancy problem. As a specific example in [90], a repetitive motion planning (RMP) scheme was presented to illustrate the effectiveness of the compact QP method. Note that such an RMP scheme is based on Euclidean norm (or two-norm), and used to remedy the non-repetitive problem of the closed end-effector path (which is different from the path tracked in this chapter). The two-norm was widely used because of its analytical tractability, but minimizing the two-norm solution was not an ideal choice. Thus, some researchers began to investigate the optimization schemes based on the infinity norm and minimizing the infinity norm implied the determination of a minimum-effort solution as opposed to the minimum-energy criterion associated with the Euclidean norm. The torque-level and acceleration-level optimization schemes have also been investigated. It is worth pointing out that various criteria are exploited to optimize different objectives according to the requirements in general.

A manipulator is said to be redundant when more DOF are available than necessary for a given end-effector path-tracking task. This implies that redundancy can be established simply with respect to some particular tasks. For non-redundant manipulators, the joint motion is uniquely determined by a prescribed end-effector primary task, and thus there are no redundant freedoms left for executing some secondary tasks, such as handling joint physical limits, environmental constraints, and configuration singularities. In contrast, redundant manipulators have wider operational space and meet more functional constraints, because an infinite number of feasible joint configurations can be available. They have thus been widely applied, for example, in cleanup and remediation of nuclear and hazardous materials, and in space or sea exploration. Therefore, a lot of research into redundant manipulators has been carried out, and much attention has been paid by the research community to motion planning and control of robot manipulators.

Kinematically redundant manipulators are those that have more DOF than required for (specific) end-effectors’ position-and-orientation tasks. This implies that redundancy can also be established with respect to some particular tasks; for example, positioning only. For non-redundant manipulators, the joint motion is usually determined uniquely by a prescribed end-effector trajectory and, thus, there is no freedom left to handle secondary tasks, such as the handling of joint physical limits and environmental constraints (e.g., obstacles). In comparison, redundant manipulators have wider operational space and extra DOF to meet a number of functional constraints and performance criteria because there exist an infinite number of joint configurations as feasible solutions (corresponding to a prescribed end-effector task).

A manipulator is said to be redundant when more DOF are available than the minimum number required to perform a specific end-effector primary task. For redundant manipulators, as discussed in previous chapters, the inverse kinematic resolution is a fundamental issue in operating the robot systems. The general description of this problem is that, given the desired Cartesian path r d ( t ) ∈ ℝ m for the manipulator’s end-effector to track, how can we generate the corresponding joint trajectory θ ( t ) ∈ ℝ n in real time t ? Note that n > m and, thus, there exist multiple or an infinite number of feasible solutions to the inverse-kinematic problem. In this sense, the redundancy of joint motion, together with kinematic and dynamic nonlinearities, complicate the manipulator control problem significantly.

The number of DOF is very vital for a manipulator. A manipulator is kinematically and/or functionally redundant when more DOF are available than the minimum number required to perform the primary task of a specific manipulator. Compared with a non-redundant manipulator, a redundant manipulator has wider operational space and extra degrees to configure the manipulator to meet additional functional constraints without affecting the position (and/or orientation) of the end-effector. Therefore, much attention and effort in the robotic research community have been paid to studies on and applications of redundant robot manipulators.

Studies on and applications of a redundant robot manipulator are the topical research issues in the robotic community. In particular, the inverse-kinematics problem, which finds the joint motion for a given manipulator’s end-effector task is one of the important and challenging issues in operating the redundant manipulators. Note that there are an infinite number of joint configurations of a manipulator with respect to a specific task, due to the redundancy; and redundancy resolution is to make a suitable selection from the configurations. This kind of selection is usually used to accomplish some secondary tasks for robot manipulators or generate an optimal solution to achieve some performance criteria, besides finishing the main end-effector path-tracking task. The redundancy-resolution problem can be readily solved by optimization techniques, especially QP. Note that the quadratic program (QP) can be converted to a linear variational inequality (LVI), and the resultant LVI is equivalent to a piecewise-linear equation (PLE) that can be solved by many algorithms and techniques efficiently, such as numerical algorithm E47 and numerical algorithm 94LVI presented in previous chapters as well as recurrent neural networks.

Given they have more DOF than required to perform a given end-effector primary task, redundant robot manipulators can achieve subtasks such as obstacle avoidance, fault tolerance, repetitive motion planning, joint limits, and singularity avoidance. Existing optimal control methods for path planning and control of robot manipulators can be roughly categorized into two types: First, optimal control methods that can handle each robot link separately without regard to robot dynamics by computing set points for low-level (e.g., position- or velocity-level) single-input single-output controllers; Second, optimal control methods that can handle the robot directly by considering robot dynamics and computing motor torques for high-level controllers, specifically, torque-level controllers. The former methods pertain to the description of motion (position, velocity, acceleration, etc.). These methods can perform well if the desired motion is not too fast and does not require large acceleration. The latter methods are related to the explanation of motion in terms of forces and torques and are applicable to numerous practical manipulators. The level at which path planning and control is performed may depend on the type of robot controller. For example, if joints are driven by positionor velocity-controlled servo motors, then a path planning and control scheme at the position or velocity level will generally suffice. However, if controllers can be driven by joint torque or actuator force, then the path planning and control scheme will be designed at the torque level. The computer simulations and experiments in this chapter are based on a six-DOF planar redundant robot manipulator.The manipulator is operated and controlled by pulse signals transmitted from the host computer. Manipulator joints are operated by position- and velocity-controlled servo or stepper motors, and the resultant joint variables (i.e., joint angle and joint velocity) should be converted into pulse per second (PPS) for manipulator control. In consideration of these points, the former optimal control method, which is better suited and much simpler than the latter methods, is exploited to control the six-DOF planar redundant robot manipulator.

In this paper, we presents a manipulator with elastic elements for energy saving. With this manipulator the energy needed for driving the manipulator to lift a certain load decreases, so that the smaller actuators can be chosen. This structure is much more suitable for manipulators used in industry and wearable robot for the aged and disabled people. At the same time, we can find the effection of the stiffness of the elastic elements and load mass on torque minimization. The concept we used here is try to store energy get from the actuators and gravity into the springs, and then this saved energy can be part of the driving power to lift load.

This paper presents the control simulation of the 6-DOF parallel manipulator. The 6-DOF parallel manipulator consist of fix base and moveable platform coupled together by the six electrical actuators, each of which has two universal joints. This paper explains the modeling and simulation of the 6-DOF parallel manipulator consist of inverse kinematics, dynamic equations of motion of the 6-DOF parallel manipulator is expressed by the Lagrange formulation and computer model of control scheme using Simulink. The simulation results are presented.

The technique of reduction cell monitor with manipulator was proposed, the dynamic model of manipulator was established and fuzzy compensate control strategy was applied to this repeat monitor control system. The adaptive law was designed and the simulation shows adaptive fuzzy compensate control have good trajectory tracking ability, also the friction, disturbance, load variation was compensated and then the monitor automation realized.

Electrostatic discharge (ESD) of MOSFET has become one of the major concerns in electronic product design, so the grasping with manipulator was proposed. The dynamic model of manipulator was established and adaptive neural network control with RBF NN strategy was applied to this repeat motion system. The adaptive law was designed and the simulation shows adaptive RBFNN control have good trajectory tracking ability and can applied into electronic design techniques for grasping control.

Model of Powerbot mobile manipulator aiming at standard DH method and modified DH method were constructed in the paper, its kinematics, inverse kinematics and dynamics of mobile manipulator were simulated by use of robotics toolbox for MATLAB, and the results was verified with actual operations. In addition a three-dimensional mobile manipulator model was added to Microsoft Robotic Studio, a modeling and simulation platform of general mobile manipulator was established, it provides a framework and basis for further research planning and control.

The screw theory is adopted for the kinematic performance analysis of a fully-symmetrical 5-5R parallel mechanism firstly. Then the limb singularity analysis by using reciprocal screw theory is presented. Two cases of limb singular configurations are found for this parallel mechanism. When three parallel axes of rotating joints are coplanar or the plane determined by two intersecting axes is perpendicular to the base plane, the limb singular configurations will occur. The determination of singular configurations is critical to understanding a parallel manipulator's kinematics and can be important in the implementation of controller for manipulator.