Kenneth J. Waldron

Direct kinematic solution of a Stewart platform

Abstmct-The Stewart platform is a fully parallel, six-degree-offreedom manipulator mechanism. Although its inverse kinematics have been extensively studied, no solutions to the direct position kinematics problem have been previously presented in the literature. Here, a solution of the direct kinematics proMem of the case in which the six limbs form three concurrent pairs at either the base or the “hand” member is presented. Even though it is not the most general possible configuration, this case does include many arrangements that have been used in practical robot mechanisms.

Force distribution in closed kinematic chains

The problem of force distribution in systems involving multiple frictional contacts between actively coordinated mechanisms and passive objects is examined. The special case in which the contact interaction can be modeled by three components of forces (zero moments) is particularly interesting. The Moore-Penrose generalized inverse solution for such a model (point contact) is shown to yield a solution vector such that the difference between the forces at any two contact points projected along the line joining the two points vanishes. Such a system of contact forces is described by a helicoidal vector field which is geometrically similar to the velocity field in a rigid body twisting about an instantaneous screw axis. A method to determine this force system is presented. The possibility of superposing another force field which constitutes the null system is also investigated. >

Kinematics of a Hybrid Series-Parallel Manipulation System

In this paper we first derive the coordinate transformations associated with a three-degree-of-freedom in-parallel-actuated micro-manipulator. Then we combine these results with the transformations associated with an in-series three-axis wrist on which the in-parallel micro-manipulator is mounted. The results are the basic transformation equations between joint-space position variables and end-effector (or task space) position variables for a hybrid series/parallel six-degree-of-freedom manipulator system. This structural combination results in a manipulator which exhibits desirable fine and gross motion characteristics as both a stand-alone device or as a sub-system of a more complex system with redundant degrees of freedom. The forward and inverse position kinematics and rate and force decomposition for this hybrid six-degree-of-freedom linkage are presented.

Series-parallel dualities in actively coordinated mechanisms

A deep symmetry between serial chain manipulators and fully parallel systems such as the Stewart platform is dem onstrated. This symmetry is shown to be a result of the well-known duality of motion screw axes and wrenches. The appearance of the inverse of the Jacobian matrix in force decomposition in the same role as the Jacobian in rate decomposition is also a consequence of this same duality and of the reciprocity relationship between the motion screw system and the wrench system of a kine matic joint. A geometric meaning of the columns of the Jacobian is demonstrated. A simple example of the appli cation of the ideas presented here to the understanding of the complex combinations of serial and parallel chains found in vehicle and multifingered hand problems is also presented.

An analytical approach for gait study and its applications on wave gaits

In the past, the determination of the gait stability margins of legged locomotion systems depended mainly on numerical computation assisted by graphical methods. Although some of these results were expressed as empirically derived equa tions, analytical derivations were lacking. The only exception was the equation of the longitudinal stability margin for quadruped wave gaits derived by McGhee and Frank (1968). They applied a complicated, nonlinear programming ap proach to the derivation. In this paper, we describe an ana lytical approach that has proved to be more efficient than previous gait study techniques. This analytical approach de fines the foot positions by means of the concept of local phase, which is the fraction of a cycle period by which the current foot position follows the placement of that foot. Based on this concept, basic theorems that simplify the study of periodic gaits are developed. This analytical approach is then applied to derive a general equation for the longitudinal stability margin for the 2n-legged wave gait. Also, this ap proach is applied to study the effects on the stability margin of the wave gait by varying the stroke and the pitch of the leg.

Geometric optimization of serial chain manipulator structures for working volume

Broadly speaking, the regional structure of a manipulator, which consists of the three inboard joints and their associated members, determines the workspace shape and volume. The orientation structure, which for a six-degrees-of-free dom manipulator consists of the three outboardjoints and members, determines the geometric dexterity or orientation potential of the manipulator. It is possible to determine the optimal dimensions of the regional structure for a given total length, using straightforward geometric arguments. By the use of the spherical counterpart of Grashofs theorem formu lated by Freudenstein (1964-65), it is also possible to show that there is an optimum geometry of the orientation structure. Two methods of characterizing geometric dexterity are utilized in this paper. The first is the concept of a dexterous workspace, which is a portion of the workspace within which the hand may assume any orientation. Although the dex terous workspace is a very useful concept for theoretical purposes, it is of limited practical utility because mechanical joint motion limits usually preclude its existence in real industrial robot structures. The second method of character izing geometric dexterity is to trace the portion of the work space within which the hand can assume a specified orienta tion. In this paper, the geometric conditions for the existence of a dexterous workspace are formulated for geometrically optimum, six-revolute manipulator structures. The optimiza tion criteria used include,freedom from geometric singular ities. We show that for an optimal geometry, singular posi tions can be completely excluded with small reductions of the joint motion ranges. These reductions have a negligible effect on the geometric performance of the system.

Technical description of the adaptive suspension vehicle

This paper contains descriptions and specifications of the major mechanical systems of the Adaptive Suspension Vehi cle. It also contains an overview of the computer software and hardware architectures. Experimental response curves for the principal servo systems are presented.

The adaptive suspension vehicle

This paper provides a description of the Adaptive Suspension Vehicle. The vehicle uses a legged, rather than a wheeled or tracked, locomotion principle, and is intended to demonstrate the feasibility of systems of this type for transportation in very rough terrain conditions. The vehicle is presently under test, with installation and validation of software modules for different operational conditions scheduled for completion by the end of 1986.

Sub-optimal algorithms for force distribution in multifingered grippers

The work described in this paper addresses the problem of determination of the appropriate distribution of forces between the fingers of a multifingered gripper grasping an object. The system is statically indeterminate and an optimal solution for this problem is desired for force control. A fast and efficient sub-optimal method for computing the grasping forces is presented. This method is based on the superposition of finger-interaction forces on equilibrating forces. An interaction force is defined as the component of the vector difference of the finger contact forces at any two fingers along the line joining the two contact points. They are computed based on the assumption that the normals at the point of contact pass through the centroid of the contact points and are therefore independent of the actual geometry of the object. The contact interaction is modelled as a point contact. The problems associated with making the algorithm independent of the object geometry are explored.

System Design of a Quadrupedal Galloping Machine

In this paper we present the system design of a machine that we have constructed to study a quadrupedal gallop gait. The gallop gait is the preferred high-speed gait of most cursorial quadrupeds. To gallop, an animal must generate ballistic trajectories with characteristic strong impacts, coordinate leg movements with asymmetric footfall phasing, and effectively use compliant members, all the while maintaining dynamic stability. In this paper we seek to further understand the primary biological features necessary for galloping by building and testing a robotic quadruped similar in size to a large goat or antelope. These features include high-speed actuation, energy storage, on-line learning control, and high-performance attitude sensing. Because body dynamics are primarily influenced by the impulses delivered by the legs, the successful design and control of single leg energetics is a major focus of this work. The leg stores energy during flight by adding tension to a spring acting across an articulated knee. During stance, the spring energy is quickly released using a novel capstan design. As a precursor to quadruped control, two intelligent strategies have been developed for verification on a one-legged system. The Levenberg-Marquardt on-line learning method is applied to a simple heuristic controller and provides good control over height and forward velocity. Direct adaptive fuzzy control, which requires no system modeling but is more computationally expensive, exhibits better response. Using these techniques we have been successful in operating one leg at speeds necessary for a dynamic gallop of a machine of this scale. Another necessary component of quadruped locomotion is high-resolution and high-bandwidth attitude sensing. The large ground impact accelerations, which cause problems for any single traditional sensor, are overcome through the use of an inertial sensing approach using updates from optical sensors and vehicle kinematics.