Android Design

The Special Problems of Stairways (p. 34)

Back in the sixties,when lunar excursions were still in the future and NSAS's budgets were still sizable, the public was often reated (through the media) to artist's rendering of some fantastic but feasible schemes for moon buggies. They walked, they crawled, they slithered and rolled, and more than anything they kept us as excited about the future s any of the Gernsback science fiction covers of the thirties and forties did our fathers. One of those renderings came to mind when the problem of climbing stairs came up. While the rednering itself is not available, memories of it may describe the scheme in enough detail to entice some of you into building and experimenting with it.

Three wheels are used, with their centers at the points of an eqiulateral triangle; the wheels do not touch each other. A point determined to be at the triple junction of lines bisecting the three angles of the triangle is the center of the fourth, smaller wheel. This fourth wheel is the drive wheel for the entire mechanism (see Fig. 6-6).

The hubs of the three outer wheels are sprocketed, as is the outer rim of the fourth, inner wheel. The inner wheel diameter is chosen such that it touches a chain connecting the three outer hubs at three pionts, each midway between two adjacent hubs. The three outer wheels are chain driven by this fourth inner wheel in unique pseudo-planetary drive.

With this drive, two wheels normally stay in contact with the ground for travel over relatively smooth surfaces. Once the leading wheel meets an impasse, though, the drive treats its hub as the fulcrum of a lever and the remaining wheels leapfrog it. On a stairway, successive leapfrogging easily brings the mechanism upstairs; control during descent, however, remains a ticklish problem.

The dimensions for this wheel drive show in Fig 6-6 are for standard 8-inch steps. Otherwise:

• Step rise ~= step tread
• Let x = step rise
• Wheel diameter = x
• Wheel center spacing = x * sqrt(2)
• Overall height = x * (1 + sqrt(3 / 2))
• Between wheel spacing = x * (sqrt(2) - 1)
• Minimum driving wheel diamter = x * (sqrt(2 / 3))
• Driving wheel diameter = x * sqrt(2 / 3) + hub diameter

Tight turns can be accomplished by driving one side forward and the other backward. A low center of gravity is possible. All in all, the configration shows promise, and you are encouraged to pursue it.

The one disadvantage of the scheme is that there has not been a great deal of experimentation and development done with it, so we are very much on our own in pursuing it. There is yet one more approach that has had nearly a century of commercial development; track drive.

Designing the main mechanical drive (p. 58)

Caption to Fig. 7-26.
Basic geometry of triangular wheel drive. Three wheels of radius R are located with their centers at the corners of an equilateral triangle, of dimensions S on each side. Unless S > 2R, the wheels will run into each other. The triangle center is located sitance 2A from each wheel center.

The three wheels are located with their centers at the three corners of an equilateral triangle and are locked in rotational step with each other and the drive shaft (see Figs. 7-26 and 7-27).

Since this is an experimental mechanism and the mathematics of the mechanics are an order of magnitude of two more complex that those associated with our track mechanism, experimentation (an empirical approach) is likely to prove preferable to calculation (a mathematical approach). As a compromise,the geometries can be modeled on paper.

The following are the results of my own investigations, which have not yet been tried in hardware.

Clearly, the stair climbing mode offers more critical, more crucial demands that a run along a flat, level surface (See Fig. 7-28). Here, the difference between and lipped and aquare step geometries is an important one, with the lipped geometry presenting the more difficult problem (see Fig. 7-29).

Consider that, as previously described, the wheels are forced into a leapfrog mode. Moutning the first step is not so much a problem as proceeding from the first to the second. As the leapfrogging occurs and the top wheel assumes the forward position, the rear wheel must clear the ip on its way to the top position. Complicating this action, we must consider that before the leapfrogging beings, the rear and forward wheels had been driven forward until progress was inhibited, the condition that first prompted the leapfrogging. This same difficulty occurs for the forward wheel on its way to the rearward position.

Previously, we had calculated the linear distance between stair crests as 11.3 inches for stairs with an 8-inch rise and 8-inch tread. If we kept each side of the triangle to this 11.3-inch dimension, the leading edge of arising wheel would tend to rip the lip off the step during its arc of rise. Even given and indesctructible step, the contact with the lip would tend to force the rising rear wheel backwards. If successful, it could bring the lead wheel back to a point where the leapfrogging would not be initiated, and the whole thing would bounce back and forth like a dribbling ball. Even if this didn't happen, opposing stresses could be harmful to the linking drive.

A solution can be found by lengthening the sides of the triangle to 12 inches (see Fig. 7-30). Since the step rise is fixed at 8 inches (the tread is not a limiting factor), the horizontal separation of the wheel centers comes out as 8-15/16 inches.

As a result, as the top wheel rotates into the front position, its arc carries it into the vertical riser. Note that during the leapfrogging, the rotational motion of the wheels has been translated into the rotational motion of the assembly, so the wheel is turning at exactly the same speed as it is falling (none at all vis-a-vis the triangle). Also, once the contact with the vertical riser has occured, the mechanism is balanced so that gravity will irrevocably complete the leapfrogging.

Caption for Fig. 7-29
For lipped stairs, wheel may rest on underside of lip, unable to reach riser (dotted line). Arc of rise of rear wheel then clears lip. Note: This happens here for square lip and wheel radius scaled as shown. In general, this clearing can be assured by increasing pivot radius. Pivot radius can be increased by increasing horizontal separation of centers; vertical separation is fixed at step rise dimension.
Caption for Fig. 7-30
Solving for wheel radius required to ensure stair-crest clearance for simple triangle mounting frame capable of shrouding 1-in. radius pulley at each wheel hub. Note: Solution calls for 10-in. diameter (5-in. radius) wheels. These also provide an excellent transition from rotational to pivotal modes for torque-producing drive elements.
Since it must continue forward, it guides itself (in part due to slipping from imperfect traction) precisely into the front corner of the step. This forces the rear wheel (which, too, has had all of its rotational motion linked into the leapfrogging rotation) to slip backwards a fraction under an inch. This is adequate to extend the leapfrogging arc past contact with the lip of the stair.

Even though this gives a suitable solution to the geometry of the triangle of wheel centers, a simple triangle can still get us into trouble with the lip of a stair if the wheel radius is too small; in this case, the triangle can hit the crest of the stair. The problem is complicated by a requirement for a timing belt pulley somewhere along a line extending from each apex to a point halfway along each opposite side.

Caption for Fig. 7-32
Final geometry of drive belt and pulleys for triangular wheel mechanism. Note: Rotation of center (driver) pulley is opposite that of wheels, pivot, and other pulleys on wheels but the same as other two idler pulleys.
Caption for Fig 7-33.
Uniroyal Twin Power &tm; HTD belts have teeth on both surfaces or otherwise parallel construction of standard HTD belts. (Courtesy of Uniroyal)

In any case, we must also provide worst-case clearance for a pulley extending beyond the triangle. A measurement ofr 1-1/2 inches in each (horizontal, vertical) dimension permits a total pulley plus belt diameter of about 2 inches -- ample for our needs.

Once again, sparing you the rigors of the extensive math, this yields a minimum wheel radius of 4.95 inches. A 10-inch wheel diameter both suits this requirement and gives us a good chance to find a comercially available part.

By the way, if we mount the wheel shaft in a small arc with springs rather than in a simple hole, the slippage we required earlier might be accomplished without stress on the floor (see Fig. 7-3 on p. 63).

Details on the selected geometry are spelled out in Fig. 7-32 on p. 63. Once again, Uniroyal Twin Power&tm; HTD timing belt drives provide the necessary components. This time, though, we can use much smaller components.

The belt (see Fig. 7-33) is a Model TP450L050; 3/8-inch pitch, 1/2-inch wide, 45 inches long, teeth on both sides. All six pulleys are identical, Model 10L050; 1.194-inch pitch diameter, 1-7/16-inch flange diameter, 3/4-inch flange width, 1-1/8-inch overall width. These pulleys need no bushings and can be ordered with bores (to mate to shafts) from 3/8-inch to 9/16-inch. (See Fig. 7-34).

The belt weight is 0.22 punds, and the pulley weight is 0.28 punds. So we appear to have, at last, a lightweight drive mechanism capable of doing everything we've asked of it.

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