The Fourth Dimension

1

TheFourth Dimension

  1. The concept of existence of other dimensions apart from the conventional known two dimensions easily physically proven by science has been an imaginary and belief issue for a very long time. A square represents the professional gentlemen and men in the flatland society that for the most part of their life have been confined within their geometrical boundaries. A square believed in the things they could see because they had not seen an irregular before in their lives and their imagination was limited to its angularity. After the discovery of another third dimension that the square had never imagined to have existed and cannot comprehend how it exists he begins to wonder whether there is a fourth dimension and the idea is not farfetched from what he has seen. The statement of the square is the revelation of belief of what is possible to what we are told or confined to think is true. It also shows us by giving geometric examples, which attribute to scientific discoveries and its relationship to religious beliefs. The possibility of existence of higher dimension than the third dimension is a prevalent belief among many scientists.

  2. The fourth dimension is a direction that is perpendicular to each of the three dimensional space.

Fora length, width and height in three dimensions it will form a cubeand when transformed into the fourth dimension by an extension of 1inch in all directions will result to formation of a hypercube.

Cube hypercube.

Vertexes

Edges

Faces

Solids

Point

1

Segment

2

1

Square

4

4

1

Cube

8

8

2

1

Hypercube

16

16

4

2

Tomeasure the objects longth one has to locate all eight cubes thatultimately form the hypercube and measure the edges length andsumming them.

  1. Spherical robotic manipulators, also regarded as polar robots, are stationary robot arms whose spherical work envelopes can be positioned in a polar coordinate system. It allows three degrees of rotational freedom about the center of the joint using several arrangements of its links. These robots have major use in industrial manufacturing processes. Their extensive use includes spot welding, fettling machines, die-casting, handling machine tools, gas and arc welding. A typical three axis spherical robotic structure consists of a linear (L) and two rotary (R) joints or three rotary (R-R-R) joints on a fixed base to cover the work envelope. These joints enable the device to attain three degrees of freedom. Reachable places can be calculated in a polar coordinate system because it is not a sphere.

Mathematicalway of proving that the spherical manipulator has only 3 Degree ofFreedom in Planar Mechanisms is illustrated below.

M=3(L-1)-2J

M– Mobility or degree of freedom

L- Number of links

J- Number of joints

Inthis case, the spherical manipulator has 4 links and 3 jointsapplying them in the formulae

M=3(4-1)-2*3=3 degrees of freedom

  1. A 10’x10’ square room does not cover a lot of space hence a spherical robotic manipulator is suitable for use in this situation. The robotic system will adopt electric drive since they offer flexibility and are readily available in most buildings. The robotic arm is designed with a fixed base and a fixed end effector (paint sprayer) connected to the paint which is mixed in a separate container and delivered under pressure through plastic tubes. The type of spherical robotic manipulator that will be used will consists of two rotary joints and one linear joint fixed with the end effector. Above the fixed base is a 360 rotary joint that will enable the robotic arm to rotate and access all the three dimensions of the room. This joint provides the robotic arm with a three coordinate orientation and enables it to access any object in space in our case any point on the wall. The first link is connected to the second link by a rotary joint considering the Z and Y coordinates with covering an angle of 180 degrees. This will enable the robotic arm gain the up and down movement it requires to paint the walls. The third link is connected to the second link and provides translational motion that will allow painting of a larger wall surface area with a single back and forth motion. The height and length of the links will be adjusted to suit the height of the room to be spray painted and hence very flexible for application in other rooms with different dimensional orientation. This will reduce cost incurred and the spherical robotic design is suitable for spray painting as it is one of its area of application and does not require complex programing.

Joint

Degrees of freedom

1

3

2

2

3

2

Sketch

b).Building a robot with one type of joint for this task will not onlybe time costly but also ineffective because one type of joint doesnot offer the required flexibility with minimal costs to cover alarger area while painting. Coordination between the joints from thebase will be complicated hence a more complex algorithm will have tobe employed.

c)No it is not possible to design a robot with less than three jointsto accomplish this task because the wall area to be painted in allthree polar coordinates with one robot without provision of all themobility it requires in other words the three or more degrees offreedom.