Robin J. Popplestone

Planning for assembly from solid models

The authors describe how to use geometric boundary models of assembly components to find mating features, thereby permitting simpler task specifications to be used. To mitigate the combinatorics of searching for a feasible correspondence between features of different bodies, boundary models produced by PADL2 (the geometric modeler) are matched against a library of standard compound features. The feasibility of mating compound features is analyzed using the symmetry groups of features.<<ETX>>

A group theoretic approach to assembly planning

HIGH LEVEL ROBOTIC ASSEMBLY PLANNING IS CONCERNED WITH HOW BODIES FIT TOGETHER AND HOW SPATIAL RELATIONSHIPS AMONG BODIES ARE ESTABLISHED OVER TIME. IN ORDER TO GENERATE AN ASSEMBLY TASK SPECIFICATION FOR ROBOTS, IT IS NECESSARY TO REPRESENT THE GEOMETRIC SHAPES OF THE ASSEMBLY COMPONENTS IN A COMPUTATIONAL FORM. ONE OF THE PRINCIPAL ASPECTS OF SHAPE REPRESENTA- TION THAT IS RELEVANT FOR ASSEMBLY TASKS IS THE SYMMETRY OF THE SHAPE. GROUP THEORY IS THE STANDARD MATHEMATICAL TOOL FOR DESCRIBING SYMMETRY. THE INTERACTION BETWEEN ALGEBRA AND GEOMETRY WITHIN A GROUP THEORETIC FRAMEWORK HAS PROVIDED US WITH A UNIFIED COMPUTATIONAL TREATMENT OF REASON- ING ABOUT HOW PARTS WITH MULTIPLE CONTACTING FEATURES FIT TOGETHER.

The Edinburgh designer system as a framework for robotics

The Edinburgh Designer System provides a high level of integration of knowledge about function and form for Mechanical Engineering Design. In this paper I discuss the extensions required to make this system suitable for assembly planning, showing that interface modules (eg. a seated-bearing) give rise to the basic acts of assembly, and that, with the provision of a suitable temporal formalism, the detailing of these actions can be accomplished within the Designer System framework.

A Group Theoretic Formalization of Surface Contact

The surface contacts between solids are always associated with a set of symmetries of the contacting surfaces. These symmetries form a group known as the symmetry group of the surface. In this article we develop a group theoretic formalization for describing surface contact between solids. In particular we define (1) primitive and compound features of a solid, (2) a topological characterization of these features, and (3) the symmetry groups of primitive and compound features. The symmetry group of a feature is a descriptor of the fea ture that is at once abstract and quantitative. We show how to use group theory concepts to describe the exact relative motion (position) of solids under surface contacts, which can be either rigid or articulated. The central result of this article is to prove the following: 1. When primitive features of a solid are mutually dis tinct, 1-congruent or 2-congruent, the symmetry group of a compound feature can be expressed in terms of the intersection of the symmetry groups of its primitive features. 2. When two solids have surface contact, their rela tive positions can be expressed as a coset of their common symmetry group, which in turn can be ex pressed in terms of the intersections of the symmetry groups of the primitive features involved in this con tact. These results show that using group theory to formalize surface contacts is a general approach for specifying spatial relationships and forms a sound basis for the automation of robotic task planning. One advantage of this formulation is its ability to express continuous motions between two surface- contacting solids in a computational manner and to avoid combinatorics arising from multiple relationships, especially from discrete symmetries in the assembly parts and their fea tures. At the end of this article, a geometric representation for symmetry groups and an efficient group intersection algorithm using characteristic invariants are described.

Symmetry groups in analysis of assembly kinematics

The use of symmetry groups in analysis of the kinematics aspects of assembly is described. The goal is to find how to assemble a product from a characterization of its final assembly configuration. From such an assembly configuration, a contact state lattice expressed in terms of the contacting faces of mating assembly components is generated. This contact state lattice contains all the partially ordered assembly motion sets (POAMS) that describe the intermediate contact states of an assembly component and the translational motions leading from one contact state to another. POAMS form a detailed task specification for a robot task-level planner and can be used for generating compliant control strategies. The work described forms part a kinematic assembly planning system that reasons about how parts with multiple contacting features fit together and generates assembly task specifications automatically from solid models using symmetry groups.<<ETX>>

Using characteristic invariants to infer new spatial relationships from old

The paper is concerned with the use of symmetry groups as descriptors for surfaces-such symmetry groups can be used to characterise the constraints imposed between two bodies by areal contact between their surfaces. A method is presented for combining constraints imposed by spatial relationships between features (infinite geometric entities) of bodies. It consists of an algorithm for computing the intersections of infinite symmetry groups from their characteristic invariants. It is demonstrated that invariants of symmetry groups for these combined features can be computed in an efficient and practical way.<<ETX>>

Symmetries in World Geometry and Adaptive System Behaviour

We characterise aspects of our worlds (great and small) in formalisms that exhibit symmetry; indeed symmetry is seen as a fundamental aspect of any physical theory. These symmetries necessarily have an impact on the way systems exhibit reactive behaviour in a given world for a symmetry determines an equivalence between states making it appropriate for an reactive system to respond identically to equivalent states. We develop the concept of a General Transfer Function (GTF) considered as a building block for reactive systems, define the concept of full symmetry operator acting on a GTF, and show how such symmetries induce a quotient structure which simplifies the process of building an invertible domain model for control.

Symmetry inference in planning assembly

When planning or describing an assembly, statements about how features should (or might) relate spatially can be expressed in terms of the symmetry groups of the features. The same is true of the interpretation of sense data. The authors show how to associate symmetry groups with features of body models created by the PADL2 system of A.A.G. Requicha and Voelcker (1982), and how to perform reasoning about spatial relationships between features of bodies occurring in an assembly in this group-theoretic context. Techniques for implementing this approach in numerical algorithms are described, with a focus on its extension to include finite rotational symmetries.<<ETX>>

From characteristic invariants to stiffness matrices

A fitting relationship in an assembly implies that the relative location of the bodies belongs to a coset of the symmetry group of the mating feature pair. When a symmetry group is continuous, there are infinitesimal displacements which preserve the relationship. Assembly of two bodies normally involves the establishment of successively more constraining relations, many of which are fitting relations. The continuous topological structure of the associated group determines possible directions of assembly at any state in the assembly process. To accommodate to errors, it is necessary to choose a stiffness matrix appropriate to a given assembly state, which allows the robot to comply with wrenches normal to the possible assembly directions. The derivation of such matrices from a computational geometric representation of the mating feature symmetry group is considered.<<ETX>>

Planning for Assembly with Robot Hands 1

If an autonomous robot system is to make effective use of a dextrous hand for assembly, it must be able to (1) reason about how objects are intended to fit together and design mating trajectories, (2) derive uncertainty constraints which accommodate perturbations from the nominal trajectories, (3) and determine how objects should be grasped to exert the forces expected during execution. This paper discusses a planning system which exploits the knowledge of symmetry incorporated in a group theory based reasoner to arrive at a nominal assembly plan. The plan is then refined by establishing bounds on the permissible uncertainty and required forces. A final stage plans force mediated interaction by incorporating multiple agents which enforce the wrench closure requirements and uncertainty constraints over the space of possible grasps.