Conclusiones sobre modelos para Predicción de Precios

The fundamental forces of nature are the electrostatic force, the gravitational force, the strong nuclear force and the weak nuclear force. These forces give rise to the complex interactions present in nature. If these forces could be utilised for the control of a robotic swarm then it could be possible to have complex behaviour emerge from a few simple rules.

The RVF framework utilises virtual analogues of the fundamental forces of nature to guide robots in exploration, mapping and collision avoidance. This not only enables complex behaviours from basic laws, it also allows heterogeneity to be defined intrinsically within the system. This is achieved through varying the virtual physical properties of robots, such as charge or mass.

This section aims to outline the virtual forces that are used in the RVF framework and explain how these forces are converted into motor velocities to actuate a differential wheel drive robot. This will explore: the electrostatic force; the gravitational force; the strong nuclear force; and finally actuation. It should be noted that the weak nuclear force is responsible for radioactive decay, and though no analogue is drawn in this paper, its utility could lie in distributing a swarm at a given rate, among other potential uses.

5.2.1 Electrostatic Force

The electrostatic force governs how two charged particles will interact. Like charges will repel, whilst opposing charges will attract; both in proportion to the total charge between the parti- cles. The RVF framework implements the electrostatic force for collision avoidance. As a robot approaches an obstacle, it is repelled by a force proportional to the distance from the obstacle. In contrast to the infinite range of this force present in nature, the RVF framework limits the range of the electrostatic force to a robot’s sensor range. To prevent the robot accelerating infinitely, a viscous friction coupling term was added, proportional to the speed of the robot. The equation used to calculate the magnitude of the virtual electrostatic force is given in equation 5.1.

(5.1) Fe=

Q qke rn −µv

Where ke is 8.99x109, q is the robot charge, Q is the obstacle charge and was determined through trial and error to be 2.5x10−7,µ is the coefficient of friction, r is the distance to the obstacle, v is the velocity of the robot, and n is the order to which r is raised to (in the case of the electrostatic force in physics this is 2). These parameters were determined experimentally, this is detailed in the subsequent section. The electrostatic force is calculated independently for each wheel of a differential wheel drive robot; this is because the viscous friction term is dependent on the individual velocity of each wheel.

In order to decide the direction of the electrostatic force, the robot increments its bearing 45 degrees away from the detected obstacle. The value of 45 degrees was chosen through trial and error and was determined to give the best collision avoidance response.

5.2.2 Gravitational Force

In nature the gravitational force attracts bodies of mass. The reactive virtual forces framework seeks to use this attractive quality to promote discovery of unexplored regions of the map. In this case the gravitational force is used to attract the robot towards the centre of mass of the unknown cells. The centre of mass is calculated by averaging the position of all unknown cells in the map, then finding the closest unknown cell to this average position. The magnitude of the gravitational force may then be calculated using equation 5.2.

(5.2) Fg=

G Mm r2

Where M is the mass of the centre of mass of the unknown cells, m is the mass of the robot, G is the gravitational constant and r is the distance to the centre of mass. M was determined experimentally and is a fixed value, this is detailed in section 5.3.2.

The direction of the gravitational force is determined to be the bearing given between the robot’s position and the position of the centre of mass of the unknown cells.

5.2.3 Strong Nuclear Force

The strong nuclear force bonds most matter together and acts over minute distances. Within the RVF framework it is used to attract robots to nearby goals, in the form of frontier cells. If a frontier cell is found to be in sensor range of a robot, then it becomes the robot’s goal. In the case where there is no frontier cell in range, the robot reverts to using the centre of mass of the unknown cells as a goal, in conjunction with the gravitational force.

This means that the robot is often moving towards frontier cells and thus areas that are yet to be explored. However, when the robot enters in an entirely explored region the gravitational force takes over and attracts the robot to a further unexplored area. Due to this the robot is nearly always gathering new data.

Currently there exists no simple analytical method for calculating the strong nuclear force. However, its relative strength is known when compared to the electrostatic force. This is a value of 137 times stronger than the electrostatic force. This leads to equation 5.3 for calculating the strong nuclear force in the reactive virtual forces framework.

(5.3) Fs= 137 ∗ Fe

Where Fe is the magnitude of the electrostatic force.

5.2.4 Actuation

Having calculated the forces acting on a robot, these forces must then be converted into motor velocities. This involves combining the two parts of the acting forces: the magnitude and the

direction. It should be noted that only two forces are ever acting on a robot: either a combination of the electrostatic force and the gravitational force, if there are no frontier cells within sensor range; or a combination of the electrostatic and the strong nuclear force, if frontier cells are found within sensor range. If there are no obstacles in sensor range, then the electrostatic force is determined to be zero.

The magnitude of the resultant force is calculated as the sum of the magnitudes of the forces acting on the robot. This force is then converted into a change in motor velocity by rearranging Newton’s second law, the resulting equation is given in equation 5.4.

(5.4) ∆v =F∆t

m

Where∆v is the change in motor velocity, F is the total force, ∆t is the time the force is acting over and m is the mass of the robot.

The direction of the resultant force is found by averaging the direction of the two acting forces. In order to actuate towards the appropriate heading, robots calculate the smallest angle between their current heading and their desired heading. This allows the robot to determine whether it should turn clockwise or anti-clockwise. After this decision is made the wheels are actuated as follows:

• Clockwise turn –∆v is subtracted from the right wheel and added to the left wheel. • Anti-clockwise turn –∆v is added to the right wheel and subtracted from the left wheel.

The robot continues turning until it reaches its desired heading ±10 degrees. This value prevented the robot from continuously turning due to missing the desired heading repeatedly, whilst also enabling a suitable level of accuracy.