A unified robotic kinematic representation including position and orientation is presented. Based on a special Lie algebra which collects all real 3*3 skew-symmetric matrices with its exponential mapping, a 3-D orientation vector is defined for a possible unification with 3-D position vectors. All computational formulas are developed and illustrated for the PUMA 560 robot. Utilizing the 3-D orientation vector, and integral-force matrix that transforms the conventional kinematic Jacobian to a mathematical Jacobian matrix is derived. The unified method, which has applications to robotic control systems, was simulated by computer for the PUMA 560 robot with all six joints for tracking both the positional and orientational paths, and the simulation results are given.>

A number of conventional methods for orientation representation are reviewed and discussed. An analysis based on Lie algebra for exploring possible definitions of three-dimensional (3D) orientation vectors and for unifying representations of position and orientation is presented. The associated computation methods are developed and verified by application to the six-axis Puma 560 robot.>

The authors explore the use of neural networks in conjunction with a learning control scheme to simplify the mathematical complexity in nonlinear robotic controls. A sufficient-input requirement is proposed for neural network applications to dynamic systems based on the system invertibility theory. A lower bound is found which determines the least number of inputs required to excite a neural network designed to learn the inverse of a robotic system. In the proposed combination method of robotic learning control, a nonlinear learning law is first applied on a robotic system to find a desired basin of the output error. Then, using a back-propagation neural network (BPN) to learn the system input-output relation from previously recorded data, an expected control input with respect to the desired task for the robotic system can iteratively be resolved by a learning process. A simulation study has been carried out to verify the convergence of the combination learning method associated with a BPN. From the simulation results, convergence performance has been achieved for the Stanford-like robot arm.>

A unified formulation of kinematic equation and its general solution for both conventional and redundant robots is presented. It provides the dynamics and control analysis of redundant robots. Through this formulation, the dynamic modeling problem in robotic system with redundancy as well as the control strategies by utilizing several advanced control schemes are developed. The theoretical results show that a redundant robot has many advantages, but it may be more difficult to deal with than a nonredundant conventional counterpart. Explicit formulas in dynamics and control for redundant robots are outlined.>

A number of conventional orientation representation methods that are often used in kinematics and control system design are reviewed. Unlike position, orientation is relatively hard to represent by a three-dimensional vector to reflect its three degrees of freedom. A detailed analysis based on the original conceptions of kinematics and Lie algebra is presented and used to explore a unified representation method for both position and orientation. A set of formulas and relations between the old and new orientation representations are developed.>

A dynamic model that represents an exact linearization scheme with a simplified nonlinear feedback is presented. To realize this model for robotic systems, the output functions should be chosen so that a special decomposition of the total inertial matrix is satisfied. The concept of an imaginary robot is utilized to achieve the formulation and to solve the realization problem. Two illustrative examples are given in the paper, one for the Stanford arm and the other for a PUMA type of robot. An optimal robotic physical design and a control system design based on the new model are also discussed.>

A nonlinear transformation for simplifying the robot dynamic formulation and control system modeling is presented. This is a canonical transformation in the Hamiltonian formulation of robot dynamics, and it can be embodied in a special decomposition of the total inertia matrix. In simplification by using this canonical transformation, the dynamic system is not only transformed to a linear model with the Brunovsky canonical form, the input formulation is also reduced to a simple recovery form. By taking advantage of the linearity of the robot dynamic equation with respect to the total inertia matrix, a detailed procedure of the realization is then described. A 3-D robot example illustrates the feasibility and computational efficiency of this procedure. And it also shows that the realization procedure for the canonical transformation can be summed up in two major steps: seek an imaginary robot and solve a reduced equation.

In robotics, to deal with coordinate transformation in three-dimensional (3D) Cartesian space, the homogeneous transformation is usually applied. It is defined in the four-dimensional space, and its matrix multiplication performs the simultaneous rotation and translation. The homogeneous transformation, however, is a point transformation. In contrast, a line transformation can also naturally be defined in 3D Cartesian space, in which the transformed element is a line in 3D space instead of a point. In robotic kinematics and dynamics, the velocity and acceleration vectors are often the direct targets of analysis. The line transformation will have advantages over the ordinary point transformation, since the combination of the linear and angular quantities can be represented by lines in 3D space. Since a line in 3D space is determined by four independent parameters, finding an appropriate type of "number representation" which combines two real variables is the first key prerequisite. The dual number is chosen for the line representation, and lemmas and theorems indicating relavent properties of the dual number, dual vector, and dual matrix are proposed. This is followed by the transformation and manipulation for the robotic applications. The presented procedure offers an algorithm which deals with the symbolic analysis for both rotation and translation. In particular, it can effectively be used for direct determination of Jacobian matrices and their derivatives. It is shown that the proposed procedure contributes a simplified approach to the formulation of the robotic kinematics, dynamics, and control system modeling.