## 2 Related Works  
Models of computation (MoCs) for cyber-physical components can be categorized into dataflow models, network models, synchronous-reactive models, finite state machines, discrete event models, time models, etc. Most of them are founded on the separation-of-concerns design principle and consequently focus on only one aspect of the modeling problem, but a few consider both temporal characteristics and geometric information in a synthetic manner.  
The time states in a cyber-physical system (CPS) have a dual nature, being both discrete and continuous, and a corresponding mixed temporal description can be realized in the form of hybrid automata [8; 9; 10]. Hybrid automata are defined by a directed multivariate graph. Each vertex in the graph is defined as a mode, i.e., a discrete form. The lines connecting the vertices represent modal transformations, which may include transitions from a continuous state to a discrete state. The flow condition corresponding to each discrete point defines a continuously variable feature of the discrete state, which is expressed by means of calculus. However, hybrid automata do not provide a suitable programming model for implementation. Related modeling paradigms, such as Petri nets [11; 12] and Modelica [13], are mainly used for analysis and verification.  
An actor-oriented programming model has been proposed by the Ptolemy project [14]. The PTIDES model includes a temporal concept of interaction with the environment. It is based on a discrete-time or discrete-event (DE) computing model, in which the software and hardware components, called actors, send messages associated with time-stamped events [15]. The DE paradigm specifies that each actor should process events in time stamp order; therefore, the order in which events are processed is independent of the physical times at which the events are delivered to the participants. Traditionally, the DE approach is used to build simulations; however, in PTIDES, DE models are executable specifications. Nevertheless, as mentioned before, this actor-based programming model lacks any consideration of spatial semantics related to physical geometry.  
In the IEC 61499 standard, FBs are not explicitly equivalent to execution semantics. An enhanced FB programming model [16] has been proposed for DE-based deterministic execution semantics with timestamps. This enhanced FB model introduces the timestamped event paradigm proposedby the Ptolemy project into the DE-based computational model of the original FB concept and considers execution following several semantic rules. However, like the actor-based model, it still lacks consideration of the semantics of the physical geometric space.  
Several design and implementation models for industrial automation systems are compared in [17], including VDSL, IEC 61499 and IEC 61131, Petri nets, and their extensions. They are the formal methods currently used for dependability in industrial automation systems, but none of them provide the geometric semantics.  
In summary, the programming models described above have contributed greatly to the design of CPPS and serve as the cornerstone for implementing computational CPS components. Although a temporal modeling concept has been developed to enable the handling of concurrent events to ensure deterministic system behaviors, the geometric semantics of physical dynamic processes are still neglected. To address this issue, this paper proposes an FB-based programming model called GePro that considers geometric semantics along with an event-triggered temporal mechanism of IEC 61499.