J. M. McCarthy

On grasping planar objects with two articulated fingers

The grasps attainable by mechanical hands with two opposing articulated fingers are examined. Such grasps are called planar, since all forces lie in plane defined by the contact points and the center of mass of the object. Assuming that the contact interaction can be modeled by point contact with Coulomb friction, the equilibrium equations for the grasped object are obtained. Satisfaction of force and moment equilibrium leads to the development of a compatability condition that relates object shape, contact locations and surface roughness as characterized by the coefficient of static friction μ. The set of all possible equilibrium grasps is determined for some examples and the results are presented as curves in a friction angle space. This representation permits choosing a grasp that is optimum in accordance with an independently developed criterion such as minimum dependence on friction forces.

The quartic singularity surfaces of planar platforms in the Clifford algebra of the projective plane

Abstract In this paper, we study the workspace and singular configurations of a planar platform supported by three linearly actuated legs, the 3-RPR parallel manipulator. The constraint equations of the platform are formulated in the Clifford algebra of the projective plane, C + (P 2 ) , which yields a manifold defining its set of reachable positions and orientations. We compute the Jacobian of these equations and derive the algebraic equation of the surface of points in C + (P 2 ) for which this Jacobian is singular, called the singularity surface of the manipulator. For the general planar platform manipulator this surface is a quartic surface with a double line. For the special case of the “proportional” planar platform, the surface factors into two planes and a circular hyperboloid. For the special case of the “in-line” planar platform, this surface reduces to a quartic ruled surface. Further special cases of this surface are examined and found to consist of pairs of hyperbolic paraboloids.

Dual orthogonal matrices in manipulator kinematics

The angular or linear displacements required at each of the joints of a robot manipulator to attain a specified position and orientation of its hand are obtained by solving the closure equations of the manipulator. These equations express the fact that the sequence of coordinate transformations between each link of the manipulator from its hand to its base must equal the transformation from the hand to the base directly. It is well known that the 4 X 4 homogeneous form of the point coordinate transformation matrix together with special coor dinate frames adapted to the structure of a manipulator yield a standard form of these coordinate transformation matrices known as the Denavit-Hartenberg matrix (Paul 1981). It is not so well known that the transformation equations of the coordinates of lines in these special coordinate frames may be used as well (Pennock and Yang 1985). Furthermore, by introducing dual numbers, the 6 X 6 matrices that arise col lapse into 3 X 3 orthogonal matrices with dual number ele ments, yielding a dual form of the Denavit-Hartenberg ma trix. Presented here is an elementary development of these results. Also discussed is a dual form of the Jacobian of a manipulator.

Dimensional Synthesis of Bennett Linkages

This paper presents a synthesis procedure for a spatial 4R linkage, known as Bennett’s linkage. It is known that the two solutions of the RR chain synthesis equations form a Bennett linkage. While analytical solutions to these equations have been developed previously, this paper uses the cylindroid that is known to exist for a Bennett linkage to simplify the solution process. It is interesting that geometric constraint associated with the spatial 4R chain simplifies the solution of the RR chain design equations. An example design is presented.

The number of saturated actuators and constraint forces during time-optimal movement of a general robotic system

The authors formulate the time-optimal control problem for general robotic systems and show that the required maximum (or minimum) value of the path acceleration is the solution of a linear programming problem. The fact that such a solution is an extreme point of the set of feasible solutions makes it possible to determine the minimum number of actuators and internal forces that must be saturated during the time-optimal movement. Specifically, it is proved that, if the dynamics of a general robot system are defined by n coordinates, m differential constraint equations, and p actuators, then some combination of at least L=m+p+1-n of the actuators and internal constraint forces is saturated during a time-optimal movement of the system along a prescribed path. The result applies to general class of dynamic systems with both holonomic and non-holonomic constraints. >

Spatial rigid body dynamics using dual quaternion components

The equations of motion of cooperating robot systems are obtained by connecting the individual equations of motion for each arm and the workpiece using the constraint equations of the closed chain. Dual quaternions have been shown to provide a convenient algebraic representation for these constraints. The equations of motion for a rigid body whose position is defined by the eight dual quaternion coordinates are derived. Because a rigid body has six degrees of freedom, the use of dual quaternion coordinates requires two additional differential constraint equations. The result is a set of ten differential equations prescribing the movement of the body. Use of these equations is demonstrated through a planar example of a double pendulum.<<ETX>>

Minimum-time trajectories for two robots holding the same workpiece

The authors present an algorithm which minimizes the time for two robots holding the same workpiece to move along a given path. The method can be applied to any constrained robot system, including the case where one robot arm moves in contact with a surface. The unique feature of these systems is that they are redundantly actuated because they have more actuators then degrees of freedom. In addition to finding the optimum torque histories, the algorithm determines the contact force between each robot and the workpiece throughout the motion. Constraints on these internal forces are easily introduced into the algorithm. Examples are given for two planar arms holding a common workpiece. It is shown that arbitrary bounds on the internal forces of interaction on the workpiece may easily be imposed; this causes a reduction in the path traversal time. The motions found using this algorithm establish a performance limit on cooperating robot systems, and can be used to guide the analysis and design of such systems.<<ETX>>

Planar Motion Synthesis Using an Approximate Bi-Invariant Metric

In this paper we present a technique for using a bi-invariant metric in the image space of spherical displacements for designing planar mechanisms for n (> 5) position rigid body guidance. The goal is to perform the dimensional synthesis of the mechanism such that the distance between the position and orientation of the guided body to each of the n goal positions is minimized. Rather than measure these distances in the plane, we introduce an approximating sphere and identify rotations which are equivalent to the planar displacements to a specified tolerance. We then measure distances between the rigid body and the goal positions using a bi-invariant metric on the image space of spherical displacements. The optimal linkage is obtained by minimizing this distance for each of the n goal positions.

An algebraic formulation of configuration-space obstacles for spatial robots

An algebraic formulation of the boundaries of configuration-space obstacles for wrist-partitioned spatial robots is presented. When the end-effector of the robot moves in contact with an obstacle, it is constrained not only by the robot reachability constraints but also by the link-obstacle contact constraints. The reachability constraint is modeled by a chain of two spherical joints, and the contact constraint is a combination of spherical joint and planar joint. Each of these constraints defines a manifold in the 6D space of rigid displacements. Parameterized and algebraic expressions defining these manifolds are obtained using dual quaternions. The obstacle boundary is obtained from the intersection of the manifolds associated with two types of constraints. An example is provided to show how this formulation leads to equations for the boundary of a joint obstacle for a PUMA robot.<<ETX>>

Bennett's linkage and the cylindroid

Bennetts linkage is a spatial 4R closed chain that can move with one degree of freedom. The set of relative displacement screws that form the one-dimensional workspace of this device defines a ruled surface known as a cylindroid. The cylindroid is generally obtained as a result of a real linear combination of two screws. Thus, the workspace of Bennetts linkage is directly related to a one-dimensional linear subspace of screws. In this paper, we examine in detail Bennetts linkage and its associated cylindroid, and introduce a reference pyramid which provides a convenient way to relate the two. These results are fundamental to efficient techniques for solving the synthesis equations for spatial RR chains.