In this paper, two gait transition models of a quadruped are derived based on gait kinematics. The learning and generalization capability of the cerebellar model articulation controller (CMAC) neural network in learning gait transitions is then studied. The two gait transition models are the transition between two general periodic gaits and the transition between a periodic gait and a continuous follow-the-leader (FTL) gait, while maintaining FTL mode during the transition. These models are nonlinear and require either heuristic rules or simultaneous solution of several nonlinear equations. Many transition gaits are then generated by these kinematic gait transition models under various gait conditions and evaluated in terms of stability and smoothness of leg movements. The CMAC neural network is then applied to learn the good transition gaits in four transition conditions: (1) from wave gait to wave gait; (2) from wave gait to FTL gait; (3) from walk to trot; and (4) from trot to transverse gallop. The learning and generalization capability of the trained CMAC neural network is evaluated and found to be satisfactory. This study has demonstrated the potential of applying neural networks to learn walking machine gaits and gait transitions.

The authors formulate a compliance model of walking systems and develop a stiffness matrix method for their foot force distribution by considering leg structure, actuators, and terrain as compliant, and incorporating the compatibility condition of deformation into equilibrium equations. To demonstrate the method in an actual machine, the stiffness matrix of a walking machine with pantograph legs is derived analytically, and numerical results for a quadruped walking with different gaits are presented and discussed. Compared with previous approaches, this method is considered to be more realistic, since it satisfies material deformation laws. Moreover, it is computationally very efficient and can be used in real-time control or simulation.>

The gaits used in turning motion of a quadruped are studied. Depending on the location of the turning center, turning gaits, are divided into two types: spinning gait and circling gait. The former is applied when the turning center is at or close to the gravity center, whereas the latter is suitable for a more distant turning center. The spinning gait is developed on the basis of the z-type creeping gait and is optimized for stability. The optimization procedure issues in two special gaits-the wave-spinning gait and the equal-phase-spinning gait. The circling gaits are developed based on the wave-crab gaits and the gaits are called wave-circling gaits. The stability margin curve of a wave-circling gait is found to be similar to a wave-crab gait when the turning radius is large. In order to assist the selection of a turning gait, the stable areas of turning center of both spinning and circling gaits are generated by numerical method.>

The condition under which closed-form solutions of forward kinematics of parallel platforms are obtainable is explored. It is found that forward position analysis has closed-form solutions if one rotational degree of freedom (DOF) of a parallel platform is decoupled from the other five DOFs. Geometrically, this condition is satisfied when five end points at the platform or at the base are on the same line. A general case in which these five points do not coincide with each other is studied first and is shown to have 16 possible closed-form solutions. The computation of these solutions is very efficient compared to that of Stewart platforms. The different geometries of parallel platforms with closed-form solutions are discussed.>

A general method, called the stiffness matrix method, is developed to determine the foot forces of walking vehicles, taking into account the elasticity of the leg structures, the actuating systems, and the terrain. Also derived are the analytical expressions of the stiffness matrix for the popular pantograph-type legs when only the leg elasticity is considered. The method is effective and well suited for real-time control by onboard microcomputers since the major computation involved is only the solution of a linear system of six equations. As an example of its application, a quadruped walking chair is analyzed in terms of this approach.>

The study of the optimally stable range of the wave gait is extended to the 2n-legged case (n>or=2). To accomplish this, an analytical method is applied to compare the longitudinal stability margin of the wave gait with all periodic and regular gaits. It is shown that the wave gait is optimally stable in the range of 3/44, the wave gait is optimally stable in the range of 1/2or=4. If the wave gait is compared with periodic and regular gaits, which are either symmetric or constant-phase-incremented, the wave gait is optimally stable in the range of 3/4or=3.>

The authors present a study on fully automated gaits which can be used to cross over four standard types of obstacles: grade, ditch, step, and isolated-wall. These gaits are simulated using computer graphics. The designer can arbitrarily select the dimensions of one obstacle, the computer program then generates a series of preprogrammed movements which enables a hexapod to cross over the obstacle in a fully automated mode. This study demonstrates the feasibility of fully automated obstacle-crossing gaits for the use of an all-terrain walking machine.>

The method presented can be applied to both closed-chain and open-chain manipulators. It is proved to be more computationally efficient that the conventional Newton-Euler method in inverse manipulator kinetics analysis. The improvement in computational efficiency is due to two factors: First, the method takes advantage of the special geometry of the positioning mechanism of most manipulators. That is, the majority of joint axes are parallel to one another and those few which are not, are perpendicular to the majority of axes. Second, the method uses longitudinal and transverse force components at certain joints. Two examples, one closed-chain and one open-chain, are used as illustrations.>

The mechanical efficiency of pantograph mechanisms is studied using the concept of redefined geometric work, and is compared with a conventional open-chain and a closed-chain type manipulator. The kinematics of two types of six-degree-of-freedom pantograph manipulators are studied. Special mechanisms which simplify the kinematics are discussed. The computational complexity of both Cartesian and cylindrical pantograph manipulators are evaluated and compared with a PUMA-type manipulator.>

Fully automated gaits are studied that can be used to cross four types of simplified obstacles: grade, ditch, step, and isolated-wall. These gaits are simulated using computer graphics. The designer can arbitrarily select the dimensions of one obstacle; the computer program then generates a series of preprogrammed movements that enables a hexapod to cross over the obstacle in a fully automated mode. The study demonstrates the feasibility of fully automated gaits for walking over these four major types of obstacles. With further research, fully automated gaits for crossing over obstacles of general geometry may be expected.>

The kinematics of pantograph type manipulators were studied and found to be very computationally efficient due to the decoupled kinematics of pantograph mechanisms [14]. In this paper, the dynamics of a wrist-partitioned, pantograph type manipulators are studied. Both Lagrange's and an extended D'Alembert's methods are used to derive the equations of motion, in applying the extended D'Alembert's method, a special treatment of the force and moment components in free-body diagrams allows the three dimensional motion of the manipulator to be treated as a two dimensional case. This special treatment also eliminates the need of a simultaneous solution of many equations. The extended D'Alembert's formulation is found to be more computationally efficient than the Lagrange's. Moreover, the inverse dynamics of both methods are found to be more computationally efficient than that of a conventional open-chain manipulator.