Introduction  to  simulation

The hexapod robot is analysed in this chapter by three different methods:

Kinematic Analysis, Finite element Analysis and Dynamic Analysis. First a brief introduction of the theory of simulation is included.

9.1. Introduction  to  simulation    

Simulation is the process to design a real system model and perform experiments with it, with the idea to understand its behaviour or evaluate new strategies, with certain limits or a list of them for the performance of the system.

These experiments consist of certain types of mathematical and logical relations in order to evaluate the behaviour at long periods of time. Simulations are found in all sciences, biology and chemistry for example, allowing them to study the behaviour of the growth of populations, adapting to economy and social sciences to predict models to obtain optimal solutions.

There are different mathematical models:

Empirical: They use formulas, linear or not with one or more independent variables in order to perform the simulation

Conceptual: They reproduce with formulas and mathematical algorithms the physical process in nature

Stochastic: They use relations or statistical correlations within the independent and dependent variables. The optimization models are the one used to solve problems that are indeterminate; they present more than one solution possible. In general, the mathematical models of statistical systems that do not vary with time consist of algebraic equations, while the mathematical

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representation of dynamic systems and physic laws are integrated by differential equations.

Analogic: Are based in the analogies that can be observed by the point of view of physical systems that are ruled by identical mathematical formulations.

In occasions the phenomena that is needed to study is so complex, that is not enough to study them from the mathematical point of view, there is experimental techniques needed, to obtain practical solutions: the physical model.

Experimental models applied to physical models of reduced scale or of analogic type. Prototypes constructed give optimal results in terms of functionality, stability and economy taking into account all the variables that intervene in the problem.

The application of any of these models has its limitation, depending in the complexity of the model. Due to the fast development of computers, the mathematical simulation has grown over the physical and analogical simulation thanks to the efficiency of simulate complex systems in reduced time at low cost.

The design of a model and its simulation has different stages, being the first the definition of the system; it consists of studying the objectives of the problem, its modelization objectives and defines the system to modelate. Once the objectives are defined, the model its built. In the formulation of the model it is necessary to define all the variables that form part of them, its logical relations and flux diagrams that describes the complete model. Also for the adequate representation of the logic relationships and flow charts to fully describe the model. Besides, for the proper representation of the considered problem, it is important to define clear and exactly the inputs that the model is going to require to provide the desired results.

With the defined model, the next step is to decide the programme language or software package to run in the computer to provide the expected results. Once the corresponding results are obtained, the verification process is executed.

This task consists in verifying that the simulated model is compliant with the

stated design requirements. That is to evaluate that the model behaves according to its design.

Expert’s opinion on the results of the simulation.

The accuracy in the prediction of historical data.

The accuracy in future predictions.

The check of simulation model failure with data that drives the real model to fail.

The acceptance and confidence in the model by the person who is going to use the results of the simulated experiment.

The experimentation with the model is performed once this has been validated.

It implies the generation of the desired data and to perform a sensitivity analysis of the required indexes with the aim of it interpretation. In this stage of the study, the results of the simulation are evaluated and on this basis a decision is taken.

The computer simulation is the reproduction in the computer of the actual behaviour of a system or process to be studied. It takes advantage of the computing capabilities of the actual computers able to solve mathematical systems at a affordable cost.

Computer simulation is today one of the most polyfacetic field that a scientist or engineer can find in the professional world. It may also be one of the most relevant for a company independently of the sector involved. It is a fundamental tool to maintain and increase competitivity and profits of the production systems.

The ratio cost/efficiency and the improvement in the graphics display, provides a big impulse to the computer simulation companies and then to the simulation itself. Low cost and user-friendly simulation software packages are available today, unthinkable a few decades ago. Computer simulation has increased his value in other related industries.

Computer simulation may be classified according to several criteria:

Stochastic or deterministic. In the stochastic simulation random numbers generators are used to simulate all possible events.

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Dynamic or static. Static models use equations defining relation among elements of the modelated system and they try to find a balanced status. This kind of system is frequently used in physical systems before performing dynamic simulation. In dynamic simulation the model is changing in response to the input variables.

Continuous or discrete. In discrete simulation, is taken into account the time assigned to each event, keeping a queue of events to be simulated, sorted by his simulation time. The simulation reads the events queue and execute s the corresponding ones. In this type of simulation is not relevant the simulation in real time. In a continuous simulation a set of algebraic differential equations are solved. The simulation periodically solves the equations and uses the results to update the system status. These types of simulations were implemented in analog computers solving the differential equations by means of electronics components.

Local or distributed. This criteria refers to a simulation with a single computer or trough several interconnected computers.

To be confident with the computer simulation is necessary the validation of the simulation model. Verification and validation are fundamental aspects to keep in mind during the program development. Another aspect very important is to obtain the same results each time the simulation is executed. Although this may seem obvious, it is not in stochastic simulations where the random numbers will not be such, but semi-random. Obviously where a person is part of the simulation, results will not be the same.

The rest of this chapter deals with the ground reactions of the hexapod robot designed and all the forces and movement acting.

9.2. Ground  Reactions  of  the  Walking  Robot    

The first step is to find out what is the reactions (R) in each of the individual legs taking into consideration all of the weights of the structure and actuators and all