Cartons are a common means for packaging a variety of items。 These are usually ‘‘erected’’ by dedicated machines which fold the carton into shape from a flat net。 When there is a need to use new forms of cartons, dedicated machinery may not be appropriate and reconfigurable systems may be used instead。 There is a need to simulate the erection process itself to ensure that this behaves as expected。 This paper investigates the use of constraint-based techniques to produce such a simulation。 The necessary com- mands are generated from the geometry of the carton net。 Geometric constraints are imposed to ensure that the net remains intact and so resolve cases where loops of faces are present。84081

1。Introduction

Cartons represent a popular way of packaging a variety of consumer products including food items, electronic components and stationary supplies [1]。 They are versatile and they are likely to increase in popularity as they have good environmental properties in regard to recycling and degradability。

Cartons are formed or erected from flat nets of carton board which have been cut and printed。 Additionally creases are normally pre-formed by pressing thin metal rules into the flat board。 Typi- cally the erection process is carried out by dedicated packaging ma- chines which fold along the appropriate creases [2]。 The carton is usually held in its erected state by gluing or by tucking and locking of flaps。

For some ranges of products, there is a desire to adapt the carton to accommodate different sizes and quantities [3]。 There is also interest in ‘‘innovative’’ designs of cartons [4–6] so as to provide greater appeal for the customer。 These ideas can be ac- commodated by designing new dedicated machines or introducing reconfigurable processes [7]。 Reconfigurability can be obtained by the use of robots and these have been used in various ways to fold cartons [8–10]。 Additionally, there has been related interest in pa- per folding [11–14] and in the use of robots to form origami models from paper [15]。

When designing a packaging system, there is naturally a need to simulate the machinery and its interaction with the  packaging

material。 There are difficulties when the material is carton board as its properties are non-linear [16,17] and can vary with the properties of the environment in which processing takes place [18]。 Often numerical techniques such as finite element analysis, with a suitable representation of the material properties, are needed [19–21]。

When dealing with reconfigurable systems, it is also important to simulate the erection process of the carton itself。 This is to ensure that the process is feasible and no unwanted interferences occur between different parts of the carton。 It also helps in checking that parts of the net which are to be ‘‘captured’’ by other parts are moved to appropriate positions。 The simulation also provides data for the machine in terms of settings [22] and control parameters。

This paper looks at how to provide such a simulation of the motion of the carton during erection。 It aims to do this based on data from the original flat net for the carton。 Folding takes places along creases and these pide the net into faces。 The next section discusses the face graph [23] which shows the adjacency between faces。 When this has no loops (as can occur with simple cartons [9–11,24]), the erection process is one of driving all the creases through the required angles, assuming that there are no  cases of faces interfering with each other。 The interference problem is certainly an issue。 For simple cartons it is usually straightforward to see how interferences can be avoided, but this is not the case for more general folding operations [25]。

When loops are present, ‘‘gussets’’ exist in which panels move because of the motion of neighbours。 Section 3 discusses how transforms can be applied to simulate the motion of the faces of cartons and it shows how commands can be created to enable the angles for gusset folds to be found using a constraint-based approach。 An example of a tray carton is discussed in the   early