# EFFICIENT MODULE TRANSFERS AND MOVEMENTS

When there is a need for a module at a particular location in a matrix where there currently isn’t one, there is more that one way to get a module to that location. Figure 5a shows a matrix of modules. The top row has a module A at the left end on the diagram. The bottom row has an arbitrary sized series of modules connected together, including modules B, C, and D, along with some unlabeled modules. A module at the right end of the matrix on top of the bottom row is desired, so that there is a pattern arrangment that consists of having a module mounted on top of the module that happens to be labeled D. The labels identifying these modules are only abstract labels, for showing where the modules physically move to when they move around.

One techique of accomplishing this is to take module A and transfer it along the top of the bottom row matrix modules so that the matrix will have a module present at the desired location. Figure 5b shows the matrix with module A moved to the location where a module is desired in the matrix.

If the bottom row of modules is significantly long, there’s a faster and simpler way of getting the same pattern without the need of transferring module A from the left end of the top of the matrix all the way to the right end. This is achieved by placing module D on top of module C to get the matrix arranged in the desired pattern. This is simpler and quicker because it can be done in one move using a different module, instead of several moves based on the number of modules in the bottom row using module A. Figure 5c shows the matrix pattern when this technique is used. In this version, module A can be moved down to the left end of the bottom row if a specific pattern is required on this side.

This technique is beneficial for replacing failed or missing modules, or reshaping a matrix of modules, by significantly reducing the amount of time and energy that would have been consumed if this technique was not utilized. A matrix of modules can easily change from one shape or application to another one that is completely different with a minimal amount of effort by utilizing this technique for every module that needs to be moved into a position in the matrix.

# ENERGY RECOVERY METHOD

It is possible to recover some of the energy that is expended to vertically move a module. Figure 14a shows a matrix of modules with a single module in the left-most column, shaded blue, and the 2nd from left column of modules are shaded green; these green modules will be used to raise the blue module, and to do this they will actuate the legs that connect to the blue module by pivoting it with an electric motor as they relay it to each other from one module to the one above it.

A module that is shaded green expends energy by actuating to raise the blue module; to indicate that a green module has done this it will be changed to red. Figure 14b shows the 2nd from left module in the bottom row lifting the blue module, which is also connecting to the 2nd from left module in the 2nd from bottom row for repeating the cycle.

As the blue module is raised by the 2nd column of modules each one of these modules expended some energy. All of these modules in the 2nd from left column that were green are now red to indicate that they expended energy after the blue module reached the top row of the matrix, as shown in Figure 14c. Here kinetic energy was converted to potential energy.

Some of the energy that was expended to raise the blue module can be recovered when it gets lowered back down to the bottom of the matrix. Figure 14d shows the blue module beginning its trip back down to the bottom, and it also indicates that the modules in the 2nd column that were red turn back to green as they pass the blue module down to the module below it, in the 2ndform left column, to indicate that energy was retrieved from the process of converting potential energy to kinetic energy with the use of gravity.

This energy can be recovered by using the same electric motor in the actuator that was used to raise the blue module by each of the modules in the 2nd from left column of modules in the matrix as a generator. This process could be used to recharge the same power sources that provided the energy to raise the blue module. This is beneficial because it both saves energy and reduces an undesired build up of heat waste.

# EXTENDING/RETRACTING MATRIX EFFECT

There are 3 basic means that a matrix can use to produce an extending or retracting effect; these are pivoting the legs (Figure 11a, below), telescoping the legs (Figure 11b, below), or rotating the connecting plate surface (Figure 11c, below). They can be performed in any combination with each other, and using all of them can be used to maximize the ratio between highest and lowest matrix densities. In the figures below, the upper image shows the matrix retracted and the lower image shows the matrix extended. In Figure 11c, the legs with the connecting plates that rotate are shown in green between the module cores (red and yellow rectangles) in a 2-layer 3-dimensional representation; for the upper image, the red modules are in the upper rows and the yellow modules are in the lower rows.

# MATRIX PRODUCTION OF ROTATIONAL MOTION

A matrix consisting of a large quantity of modules is capable of producing rotational motion. Figure 10a, shown below, shows a matrix in a ring configuration to produce this effect. It consists of an outer ring in red, and an inner ring in blue. It can be more than one ring layer of modules thick. The box shows an enlarged “under the microscope” view of the individual modules that make up a portion of the ring matrix. The center column, in green, consists of modules that repeatedly connect and disconnect to the red and blue columns of modules and forms a ring layer in the matrix between the red and blue rings. For the red ring to produce a clockwise rotation motion in the blue ring, the legs on the modules in the green column and on the modules in the columns on either side of the green column that connect to the green column pivot clockwise when not connected together and counter-clockwise when connected together. They do the reverse to produce a counter-clockwise rotational motion in the red ring.

The maximum rotational speed produced by this matrix structure is fairly slow; for example, suppose that it has a radius of 10 feet; it will hypothetically only be able to produce 1 cycle per minute of rotational motion (or 1 RPM). By connecting several matrices of rings together in series in a certain way the angular speed can be doubled for each additional layer. Figure 10b, shown below, shows a matrix in image (1) consisting of two cylinders in red and blue connected together by a truncated cone section in gray. This matrix structure by itself does not produce rotational motion; it is a rigid object. Image (2) shows how two matrix objects structured this way can combine to form another complex matrix structure that layers the blue section for the matrix on the right in the red section of the matrix on the left to produce rotational motion, which corresponds to the matrix in Figure 10a. The five matrices connected together in series in image (3) have 4 layers that can produce a relative rotational speed that is 8 times the rotational speed of a single layer of 2 matrices connected together between the matrix structures at the ends of this complex matrix structure. The calculation is 25-2 = 8, where the 5 is the number of matrix structures and the rest of the numbers are constants. If 1 layer is capable of producing a maximum of 1 rotation per minute, then 4 layers is capable of producing 8 rotations per minute and 12 layers is capable of producing somewhere in the vicinity of two thousand rotations per minute (hypothetically, 2,048 RPM).

In general, the hypothetical formula for the total maximum rotational speed output is

sn-1 = 2n-2∙s1

where sn-1 is the total maximum rotational speed, n is the number (greater than 1) of matrix structure objects such as the one in image (1) of Figure 10b, and s1 is the maximum rotational speed output of two of these matrix structure objects. This means a very large rotational speed can be produced by having many matrices that produce rotational motion in series, but this is limited by the tension produced by the centrifugal forces in the matrices that are rotating at faster speeds. Factors such as radius, density of the modules (mass), friction from wind resistance, and maximum tension that the modules can handle have to be taken into consideration to determine the actual upper limit of sn.

# MAKING ACCELEROMETERS USING MODULES IN A MATRIX

Although the modules can be produced so that each one has its own accelerometer, it is perhaps better that this option not be utilized, especially for matrices that would consist of large numbers of modules, for the following reasons:

1. It is more expensive to mount an accelerometer element in each module.
2. Generally it is unnecessary for a matrix to consist of modules that each contain an accelerometer.
3. A matrix can build accelerometers that are located throughout the structure positioned where desired.

In Figure 15a, the matrix is a simple example of a structure with a pendulum, the blue modules, for an accelerometer. The red modules symbolize a simplified cross section of an implemented structure. When the structure is subjected to horizontal acceleration, the pendulum, which is connected to the rest of the structure by the green module, will pivot the leg of the green module to which it is connected and in neutral (i.e., it is not locked from moving). The green module’s leg, which is connected to the pendulum, uses the sensors for measuring the angle of the leg’s position to measure acceleration of the structure in the horizontal plane.

Figure 15b shows how a matrix can build a gravimeter. As the structure, which is shown simplified as the red shaded modules and in 2 of 3 dimensions, is subjected to forces, the block of blue modules affect the green modules that connect it to the rest of the structure. In a similar way that the leg of the green module is driven by the pendulum in Figure 15a to act as a sensor, the green modules in Figure 15b also use the pivoting and telescoping position sensors in their legs that are chained together to measure forces placed on the structure. An actual gravimeter would consist of having a 3rd row of green modules perpendicular to the plane of the image and connected to a 3-dimensional block of the blue shaded modules. The forces that it measures can be both translational as well as rotational. Since the effects of external forces are conservative, it is necessary for the green modules to implement feedback systems that poll for the actual presence of forces by measuring the resisting effect of actuating the leg. For example, when the external forces are removed, the green modules have to re-center the block of blue modules for “synthesizing” the effect of equilibrium.

In practice, a matrix implemented into a structure can randomly use almost any module “on-the-fly” as an accelerometer. This basically means that any matrix implemented into a structure practically has the inherent built-in feature of having accelerometers, even though there are no elements present in the modules designed to function as accelerometers.

# MODULE CONNECTION ARRANGEMENT PATTERNS OF A CELL

A cell can be composed of a varying number modules, sometimes in various combinations for a given number of modules. When there can be various combinations for a given number of modules, there can be some arrangements that are not appropriate. Figure 19a shows 3 Core Base Assemblies, the objects consisting geometrically of two rectangular boxes and a square truncated pyramid with sides that slope at 45° angles that are part of the modules, connected together in such a way that they share a common corner. One of these modules has a gray shaded truncated pyramid for future reference. In this case the arrangement of modules is appropriate.

Figure 19b also shows 3 Core Base Assemblies connected together in such a way that they share a common corner, but they are a mirror image pattern of the 3 Core Base Assemblies of Figure 19a. This is also an appropriate arrangement of modules. The Core Base Assemblies of Figures 19a and 19b are opposite halves of a cell that can be joined together to form a whole cell.

Figure 19c shows two modules connected together. This pattern is the only way any two modules can be joined together when there are only two; the bottom one can, for example, be assigned identification number 1 (ID #1) and the other would be assigned ID #2. There are four sides and two types of sides; the type of side is used as a reference to determine which has ID #1 or ID #2. If a third module is added to the pattern, using the Cube Face Numbering Convention article as a convention reference, then to Figure 19c the module with the gray pyramid in Figure 19b would be assigned ID #3; and in Figure 19a it would be assigned ID #4 since the first two modules are in reverse positions. Three of the arrangements of modules in Figure 19c can be joined together to form a whole cell.

Figure 19d shows the same two modules of Figure 19c in a 2-dimensional side view.

Figure 19e shows a third module added to the arrangement of modules from Figure 19d. This is not an appropriate arrangement of modules, and can be determined because there are more than two modules connected together and none of them share a common corner. One reason it is not appropriate is because if the pins-and-hole system that joins two modules together are not designed to latch the modules together (e.g., the pin is a simple cylinder), then this arrangement will not allow the modules to latch together to bond.

Figure 19f shows the arrangement of Figure 19e with a fourth module added to the back, shown partially out of view, in green, and with a gray pyramid. This arrangement of modules is appropriate, since each module shares a common corner with two of its adjacent modules. The arrangement of modules in Figure 19c can be joined to the arrangement of modules in Figure 19f to form a whole cell.

Figure 19g shows three modules arranged in the complementary arrangement of the ones in Figure 19e, and is also not an appropriate arrangement of modules for the same reason, and also because the two opposite modules won’t have the direct electronic connections described in the Pattern for Element Core Electronic Connections article. The two arrangements of modules in Figures 19e and 19g can join together to form a whole cell.

Figure 19h shows four modules in a ring pattern, and it is also not an appropriate arrangement of modules for the same reasons given for the three of Figure 19g.

Although a fourth module with a gray pyramid is added, shown partially out of view and in green and yellow, to the arrangement of Figure 19g to form the arrangement of Figure 19i, there is no difference in the pattern arrangements of Figures 19i and 19f other than the gray one being in a different position in the pattern.

Figure 19j shows five modules connected together, made by connecting a fifth module to the top of the pattern of modules shown in figure 19i. This is the only way five modules can be connected together to form part of a cell.

In Summary, the following table shows the possible number of arrangements for a given number of modules in a cell:

number of modules in a cell 2 3 4 5 6
possible number of arrangements 1 4 2 1 1

There are only two number-of-modules-in-a-cell situations where there are multiple possible number-of-arrangements of modules, for 3 and 4 modules. For 3 modules, 2 are appropriate and 2 are inappropriate; for 4 modules, 1 is appropriate and 1 is inappropriate.

# CONNECTING PLATES ALIGNMENT REQUIREMENT

When the connecting plates of two different cells engage each other to connect to bond together correctly, the axis going though the center of each telescoping shaft and connecting plate have to be both parallel and aligned with each other; in other words, they both have to share a common axis. The reason this is necessary is to be able to utilize the entire area of the connecting plate of each module against moment forces, and large tension or compression forces along the telescoping shaft (module leg) axes. Figure 17a shows part of two telescoping shafts and side views of their connecting plates for two different modules; one of the connecting plates is blue and the other is red. The dashed green line through each telescoping shaft and connecting plate indicates the axis of the leg and connecting plate, and at the same time shows that they both share a common axis.

In figure 17b, the two legs and connecting plates of figure 17a are shown as parallel but not aligned to each other to share a common axis.

If the two connecting plates from figure 17b try to engage to bond together along their axes, they will come in contact as shown in figure 17c; since they aren’t aligned they won’t be able to bond, at least not properly if even possible at all. This type of contact is acceptable if it is not necessary for the two modules to bond together and if there isn’t going to be an excessive amount of force or pressure between them.

Figure 17d shows the two modules of figure 17a in contact for correctly connecting together; they might or might not be bonded together.

# LATERAL LATTICE RECONFIGURATION USING LEG EXTENDING CAPABILITIES

This type of modular self-reconfigurable system is capable of lattice based transformations to change its shape by using the leg extending and retracting feature. Figure 6a shows several modules initially in a state where there is an empty space above module E and between modules B and C. The following example shows how the modules in this matrix can translate a module to fill in this empty space by moving another module to that location.

Figure 6b shows the first transition state of the matrix given in Figure 6a. It involves disconnecting and shortening the legs between modules B and D, and beginning the process of extending the legs between modules B and C to join them together.

Figure 6c shows module B translating horizontally by extending the leg connections between modules A and B until the extending legs between modules B and C join together.

Figure 6d shows modules A and B disconnecting from each other and retracting, and the use of module C to align module B above module E by shortening the legs connecting modules B and C together.

Figure 6e shows the legs for modules B and E extending and joining together and the legs that disconnected between modules A and B completely retracted; this completes the transition.

This matrix could also have accomplished the same goal, also with lattice transitions, by first extending the legs between modules B and C to connect to each other, then disconnecting and shortening the legs that join modules B and D together, then shortening the legs that join B and C together, then finally extending the legs between modules B and E to connect these modules to each other. This version describes how it is possible to accomplish this task without the first column of modules in the matrix.

# DIAGONAL LATTICE RECONFIGURATION USING LEG EXTENDING CAPABILITIES

This type of modular self-reconfigurable system is capable of lattice based transformations to change its shape by using the leg extending and retracting feature. Figure 7a shows a matrix consisting of a column on the left with 2 modules which is connected to another module in the middle column on the bottom row, and it is connected to the bottom of a column of 3 modules on the right. The following example shows how the modules in this matrix can translate a module to move it both horizontally and vertically.

Figure 7b shows the first transition state of the matrix given in Figure 7a. It involves extending the legs between modules A and B to translate module A vertically.

Figure 7c shows modules A and C extending legs between them and joining them together.

Figure 7d shows modules A and B disconnecting from each other and retracting the legs between them.

Figure 7e shows the legs between modules A and C retracting to translate module A horizontally.

Module A has now been translated out of the first column into the second column and out of the second row into the first row, and this was accomplished using only lattice transitions. In a similar matrix arrangement, the same type of goal could have been accomplished by first extending the legs of the two middle row modules to join, then disconnecting the legs between modules A and B, then shortening the legs that join module A to the module in the column of modules on the right to translate module A to the second column, then connecting the leg from module A to the module in the middle column and bottom row, then disconnecting the legs that join module A to the module in the column on the right, then extending the legs that join module A to the module in the bottom row to translate module A to the top row, then connecting the legs between modules A and C, then finally disconnecting and retracting the legs between module A and the module in the bottom row.

# ALIGNING LEGS FOR JOINING CONNECTING PLATES

To join together two modules that are not aligned, in positions where all that is necessary is to extend the appropriate legs, both of these legs have to pivot to the correct angle. Figure 9a, shown below, is an example of two modules A and B that are not connected directly to each other in a matrix.

Suppose the right leg of module A and top leg of module B are selected for connecting these modules together. Figure 9b, shown below, shows the common axis along which these legs must align for connection as a green line.

Figure 9c, shown below, shows the legs for modules A and B in new positions after they have pivoted to the angle to align them to their common axis, shown as a green line, for joining together.

These legs are now aligned so that all that is is necessary for modules A and D to join together is to extend the legs. Figure 9d, shown below, shows modules A and B joined together after the legs have been extended.

If modules A and B are in the same matrix system and approximal, then it is not necessary to use any sensors or external input data to determine that the legs are aligned for joining. If modules A and B are in different matrix systems or a significant enough distance apart to be able to geometrically introduce an alignment discrepancy in the same matrix system, then it is necessary to use sensors or external input data to determine that the legs are aligned for joining; this can be accomplished by placing several sensors on the surface of the connecting plate and a significant distance from the axis of the leg and connecting plate, near the perimeter of the connecting plate circle. Once they are in close enough proximity to detect the presence of each other by emitting signature signals, the differences in strength of the signals determines whether or not the legs need to pivot until the signal strengths are equal to each other. This is usually a basic “reflex” action; an exception would be when the legs reach their pivoting limit. This pivoting limit is a minimum of 45° in any direction from the center axis perpendicular to the face of the module core assembly, or core “cube,” and will probably not be more than a few degrees beyond this because of the limitations imposed by the design of the module’s mechanical structures. In a situation where the legs cannot pivot to align to their common axis, the modules can still join together by carrying out more complex movements involving having the matrices change the positions of these modules and other modules that are adjacent to them in such a way that they can join together.