Componentes del sistema

9.5.2. Results  by  Dynamic  Analysis    

There exist the need to verify that the system of equations here presented is working correctly. The easiest way is to check the resulting position of the coordinates shown in Figure   73 for one leg. In Table     7 are shown the solution of these equations for segments 1,2,3 and for a time interval of 0.1s, g=9.8 m/s², mass 1,2,3,4,5,6=1kg .

t 0 0.05 0.1

x1 0.2165 0.2146 0.2147

y1 0.1250 0.1283 0.1280

Ǿ1 60 59.1340 59.1962

x2 0.1083 0.1148 0.1143

y2 0.0625 0.0686 0.0879

Ǿ2 60 59.1340 56.6641

x3 0.2165 0.2193 0.2272

y3 -0.1250 -0.1201 -0.1042

Ǿ3 60 61.3003 65.3680

Table    7  Verification  of  equations

This position and angles of our leg segments are needed to verify that our relations between equations were performed in the correct way. This can be checked by trigonometry relations with the table shown above or it is easy to check by representing this values in a plot.

To show a more realistic robot, the real dimensions of the box that carries the instrument has been added and a second leg is shown. The actuators are represented as bars also to show their position. Both legs have been resolved with the same resolution scheme. The angular velocity is shown for the right leg

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and gravity acting for 0.1 seconds. No damping or external force was added to the actuators.

Figure  74  Movement due to mass and gravity

9.5.2.1. Dynamic  Simulations  adding  Forces  to  the  Actuators.  

The equations described in this chapter lets easily input and simulate forces in the actuators, with this can be obtained the position,velocity and acceleration of the leg and actuator due to the introduced force . In Figure 75 a force of 50N is making the legs fall freely, The same type of graph as the one used for the verification of the equations is shown. The angular velocity in point 1 changes compared to the previous graph which was free fall due to mass and gravity.

Figure  75  Movement due to 50N load in F1.

To check the equations, the instrument box has been drawn as a rigid element that does not move in space and only the legs were performing the real motion, that is not the reality, when one leg is lowered the box will follow , when working with the six legs of the real hexapod, these movements shall be studied carefully as can lead to high forces on the structure that can result in deformations or malfunction of one of the actuators. For a faster simulation time in Figure 76 the actuators have not been represented. A 50N force is been forced to the top actuator in the left leg. One of our conditions is that the box will have to remain parallel to the ground, we can study the behaviour of the system equations forcing the box movement.

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Figure  76  Real movement of the instrument box

Now that the equations have been verified, we will start working with the equations of the six legs. This will be a high time consuming simulation as each leg will have its independent set of equations. When the six legs are combined together care has to be taken in were you take the common origin of coordinates for all the legs. This chapter deals with two degrees of freedom in each leg but care will have to be taken with the rowing motion in further work.

Figure  77  Movement due to 50N load in F1 for six legs.

Once the system of equations is built , the ground reaction forces equations can be included by the dynamic method code. These are important values that will be both validated by Kinematic and dynamic models.

The reactions are found in the worst case assuming that the hexapod is standing in three legs. The other three legs do not have a ground load reaction but contribute with their weight and displacement configuration.

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Figure  78  Maximum  movement  of  the  legs  

The Figure 79 shows the ground reaction of Leg 6 for some of the worst configuration that the six legs will achieve.

Figure  79 X6 distance of Leg6 vs. Reaction

Figure  80  Comparison of Reaction Results

Figure 79 shows how the reaction with the ground value on leg1 changes with the distance. To validate our methods these values obtained in Figure 79 can be compared to Figure 80 which are the values obtained in the kinematic model in chapter 9.3

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9.6. Steps  Followed  for  the  leg  design    

The steps followed are clear through the chapter:

1.- Setting a system of equations for each leg: Care has to be taken in to have a unique origin of coordinates and the same sign reference throughout all of the different simulations

2.-Validation the results of one leg: geometrically and mathematically (Dynamic and Kinematic), these results can be verified by trigonometry rules, this equations involve very large matrices that can lead to common mistakes, care has been taken in this step to verify the performance of the equations

3.-Building the system of equations: Each leg has its own degrees of freedom and does not depend on the movement of its leg close by, that is not the case of the instrument box that is dependent of all the movements. In this chapter two leg system and the complete case was analysed.

A third method by finite element analysis has also been included, that also validates the design configuration.

4.- Calculation of the reactions for the worst case configuration: The reactions are key to the design, they provide the information to dimension the actuators which will be the heart of the system. It will all give important information about how the structure will suffer and how robust it will have to be.

5.-Building the complete 3D model: These system of equations provides very important information about each point evaluated, obtaining position, velocity and acceleration of the center of gravity of the different parts that form each of the hexapod legs. The system of equations can be solved in a time domain frame to observe the evolution of the movement this will help to predict the movement the robot will have before actually sending the real command, the real forces can be simulated in the actuators which is an extra benefit that these equation provides compared to other methods.