In this paper, we present a strategy for recovery from joint failures in a robot arm mounted on a mobile platform. We use the ability to reposition the base of the arm by relocating the platform and to change the position of the wrist by regrasping the scientific-tool package to provide design degrees of freedom (DOFs). The velocity and acceleration specifications of the task combine with the constraints imposed by the joint failure to yield values for these DOFs that ensure that the task can be achieved despite the joint failure.

Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked and handling singular solutions have made probability-one homotopy methods even more practical. POLSYS_GLP consists of Fortran 95 modules for finding all isolated solutions of a complex coefficient polynomial system of equations. The package is intended to be used on a distributed memory multiprocessor in conjunction with HOMPACK90 (Algorithm 777), and makes extensive use of Fortran 95-derived data types and MPI to support a general linear product (GLP) polynomial system structure. GLP structure is intermediate between the partitioned linear product structure used by POLSYS_PLP (Algorithm 801) and the BKK-based structure used by PHCPACK. The code requires a GLP structure as input, and although finding the optimal GLP structure is a difficult combinatorial problem, generally physical or engineering intuition about a problem yields a very good GLP structure. POLSYS_GLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem definition both at a high level and via hand-crafted code. Different GLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding.

In this paper we formulate the “generalized inverse kinematics problem” for a spatial serial chain, where the goal is to determine values for structural parameters as well as for the joint parameters. The kinematics equations of the chain are formulated first using matrix exponentials and then cast into a form based on exponentials in a Clifford algebra. These equations contain the coordinates of the joint axes explicitly and have a systematic structure that can be exploited in their solution. As an example we fit a seven degree-of-freedom CCS chain to a 12 position task trajectory. In this problem, we can also specify desired values for the first two joint angles, and compute the structural parameters and the remaining joint angles.

The paper considers the problem of setting the dimensions of a system of constrained parallel robots to ensure that together they can position an end-effector in a specified set of positions. The focus is on five degree-of-freedom serial chains that provide point contact, modelled as a spherical joint, with the end-effector. We identify seven such chains which can be used to form 756 parallel robots ranging from one to three degrees of freedom. The algebraic equations of the surfaces that contain the center of the spherical joint of these seven chains can be used to form the design equations for the parallel system. These equations are solved to determine some or all of the structural parameters depending upon the required number of task positions. We demonstrate this by computing the dimensions of a symmetric 3CS parallel robot such that it guides the end-effector through three arbitrarily defined task positions.

The synthesis theory for spatial linkage systems can be formulated in a way that is compatible with the geometric design of serial and parallel chain robotic systems. Mechanism design generally seeks exact solutions for a finite approximation, to the trajectory of a moving frame. In contrast, robots must have a six-degree-of-freedom workspace with specific characteristics such as shape and size. The paper describe an inventor's environment that fits these workspaces to a designers specification. Current results yield the five-degree-of-freedom TS robot and four-degree-of-freedom CC robot that fit a desired continuous task.

In this paper we use the fact that 4/spl times/4 homogeneous transforms can be viewed as limiting cases of rotations in four dimensional Euclidean space, E/sup 4/, to construct a metric for spatial displacements. For each spatial displacement, we compute an associated four-dimensional rotation and determine the associated biquaternion representation. We then use the standard Euclidean metric for these eight-dimensional vectors, in order to obtain a bi-invariant metric on SO(4). The result is an induced metric on SE(3) that is bi-invariant to a specific degree of approximation. As examples we determine the distance between two specified displacements, and find the reference frame "equidistant" between the two given frames for various positions of the global reference frame.

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.>

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 are saturated during a time-optimal movement of the system along a prescribed path. The result applies to a general class of dynamic systems with both holonomic and nonholonomic constraints.>

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.>

An evaluation is made of the performance of a cooperating robot system using movements planned for minimum time. Minimum-time movements characteristically require that a set of motors in the robot be driven at the maximum torque through the motion. Thus, these movements are limited by the combination of motor performance, mechanical advantage of the kinematic chain, and the location of the start and goal positions. Increasing the payload until a minimum-time solution is no longer feasible makes it possible to obtain the performance limit of the system for the associated path. The cases of two robots working individually and together are examined in order to find the performance enhancement obtained by having two robots cooperate.>

The characterization of the relative position of parts in an assembly can be viewed as a generalized robot kinematics problem in which the set of acceptable assemblies is the workspace of the system. The authors show how the Clifford algebra of projective space provides algebraic manifolds characterizing this allowable movement in an assembly. They examine six basic constraints that represent the mating of cylindrical, spherical, and polyhedral features: pure rotation about a line, pure rotation about a point, pure translation along a line, contact of a point with a plane, contact of a plane with a point, and contact of a line with a line. The authors derive parametric and algebraic formulas for the six primitive functional constraints. To illustrate how this theory gives a geometric form to functional constraints, the hyperboloid that defines the planar peg-in-hole constraint is derived. It is shown that the constraint that two pegs fit into two holes is the intersection of two hyperboloids.>

An examination is made of the equations for the boundaries of the joint-space obstacles in the workspace of a 5R closed-chain robot. This two-degree-of-freedom chain appears in many robotics applications. The equations describing the joint-space obstacle boundaries are derived, using three different sets of parameters, and the obstacles are graphically displayed. The boundaries of polygonal obstacles in contact with the polygonal links of a robot arm are studied using the 4D image space of planar displacements. The obstacles are mapped into the image space as the surface representing the position and orientation that a body assumes while maintaining contact with the obstacle. The intersection of the available robot positions and the obstacle contact positions is a representation of the obstacles in the robot's workspace. Three parameterizations of joint-space obstacles are determined from the image-space representation: the joint angles of one of the 2R robots, the end effector position, and the base joint angles of the 5R chain. The first and third parameter sets define two possible configurations for the 5R chain, and the second defines four possible configurations.>

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.>

Parameterized equations for the boundaries of joint space obstacles for planar robots are derived. The boundary of a joint obstacle prescribes the link motion of the closed chain formed by the robot in contact with a given obstacle. The theory of kinematic mappings is used to represent the link motion as a set of points in a projective three space. The link positions that are feasible to the arm define a manifold in this space, termed its reachable manifold. The link positions satisfying contact conditions define another manifold in the same space called the contact manifold. Their intersection is mapped to joint space to obtain equations defining the obstacle boundary. While the theory applies to general planar robots, the authors focus on the 3R robot arm. An example is also provided in which the boundaries of joint obstacles are computed.>

Simple kinematic mappings have been used in robotics for the specification of robot motion in obstacle avoidance problems. Different kinematic mappings have been highly developed by kinematics researchers for the design of mechanisms, focussed primarily on single degree of freedom motion. It is possible that a more general theory of kinematic mapping will unite this effort. Presented here is a survey of the theory of the kinematic mappings of planar, spherical, and spatial displacements used in mechanical design, together with examples of loci of planar and spherical displacements which satisfy a general constraint.

This paper examines the grasps attainable by mechanical hands with two opposing articulated fingers. 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 the contact interaction can be modeled by point contact with friction we obtain the equilibrium equations parametrized by the magnitude of one of the applied normal forces. Thus a curve of possible grasps in friction angle space is obtained from which an optimum may be chosen. Three examples are discussed in detail.

This paper formulates the static equilibrium equations for grasping a general solid object using a three fingered mechanical hand. The contact between the fingers and the object is modeled as point contact with friction. The primary result of this work is a procedure which allows the computation of the friction forces required to satisfy equilibrium for a known set of contact normal forces. By comparing these friction forces to the maximum sustainable between the contacting surfaces, we can determine if the object will slip from the grasp and further we have the twist which characterizes the slipping mode. Two examples are solved in detail to illustrate the procedure.

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.

A procedure is presented for computing the friction forces required to satisfy static equilibrium, given a set of normal contact forces exerted by a three-fingered mechanical hand. The contact between the fingers and the object is modeled as "point contact with friction. "This means the fingertip is free to rotate about the point of contact, but sliding along its surface is resisted by a friction force. Comparing these friction forces to the maximum friction force, which can be sustained between the contacting surfaces, it must be determined whether the object slips from the grasp and further the instantaneous motion of the impending movement. Two examples are solved in detail to illustrate the procedure.