THREE-WHEEL DRIVE FOR SPHERICAL SURFACES

Abstract

This invention uses three fixed-position omniwheels and their associated motors to drive any possible rotation of a sphere. The mechanism is much simpler than mechanisms using multidirectional wheels (i.e., conventional wheels with a mechanism which changes the orientation of their axes). The invention can be applied for rotating surfaces which are approximately, but not perfectly, spherical.

Claims

1. A mechanism frictionally moving a spherical surface, said mechanism comprising three rotationally-driven omniwheels; the said omniwheels mounted in mutually fixed positions; the rotational axes of the planetary wheels of each omniwheel being orthogonal to the rotational axis of the omniwheel; each omniwheel touching the spherical surface at a point on one of the planetary wheels of said omniwheel, thereby providing the friction necessary to drive the motion of the spherical surface; with the three great circles of motion driven by the three omniwheels mutually orthogonal.

2. A mechanism as in claim 1 for which the three omniwheels are replaced by three sets of omniwheels, each set of omniwheels having one or more rotationally-driven omniwheels; with the great circle of motion for each omniwheel coinciding with the great circles of motion for all other omniwheels in its set; and the three great circles of motion for the three sets mutually orthogonal.

3. A mechanism as in claim 1 for which the axes of one or more omniwheels are not orthogonal to the axis of the corresponding omniwheel.

4. A mechanism as in claim 1 for which the driven surface is not a complete sphere.

5. A mechanism as in claim 1 for which the driven surface is approximately, but not perfectly, spherical.

Description

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

[0015] FIG. 1 is a schematic illustration of a sphere S driven in any rotation by three omniwheels. Part of the front surface is broken out to show the third wheel and great-circle wheel paths passing along the rear surface. Omniwheel positions are indicated by ellipses indicating the outer-edge paths of the planetary wheels.

[0016] FIG. 2 is an enlargement of part of FIG. 1 with an added path 8 showing that a motion produced by one wheel 3 crosses another wheel 2, thereby causing wheel binding if wheel 2 is a conventional wheel rather than an omniwheel, or if the great circles corresponding to the wheels are not orthogonal.

[0017] FIG. 3 is a schematic illustration of a sphere S similar to FIG. 1 except that the wheel orientations have been changed so that the associated great circles are mutually orthogonal.

DETAILED DESCRIPTION OF THE INVENTION

[0018] According to Euler's theorem on rotations about a point (see Whittaker, p.3; Goldstein, p. 118), any rotation of a rigid body about a fixed point can be represented by a single vector, the length of which is proportional to the angle of rotation. Furthermore, simple vector calculations show that if a rigid body undergoes a sequence of such rotations, the result is a rotation for which the vector is the algebraic sum of the vectors of the rotations in the sequence. This also applies if the actions to create the rotations in the sequence are applied simultaneously and continuously.

[0019] Consider a surface S such as the surface of a sphere. Let V.sub.1, V.sub.2, and V.sub.3 be the rotation vectors of wheels frictionally driving the motion of S. Then, assuming the motions of the wheels do not mutually interfere (i. e., there is no wheel binding), the resultant vector of rotation of the sphere S is V=(V.sub.1+V.sub.2+V.sub.3), the minus sign being necessary because each wheel drives a sphere rotation opposite to its own.

[0020] Let U.sub.1, U.sub.2, and U.sub.3 be unit vectors in the axial directions of the the wheels, so that the rotation vectors are V.sub.1=.sub.1U.sub.1, V.sub.2=.sub.2U.sub.2, and V.sub.3=.sub.3U.sub.3, where .sub.1, .sub.2, and .sub.3 are the respective angles of rotations of the wheels (which angles can be negative, of course). If the vectors U.sub.1, U.sub.2, and U.sub.3 are linearly independent, then any rotation vector V is a linear combination of those vectors, so any desired rotation V of the surface can be obtained by choosing the angles of rotation of the wheels so that V=(.sub.1U.sub.1+.sub.2U.sub.2+.sub.3U.sub.3).

[0021] FIG. 1 schematically illustrates concepts related to the instant invention. Sphere S is supported on three omniwheels 1, 2, and 3, which are fixed in position so their axial vectors U.sub.1, U.sub.2, and U.sub.3, respectively, are fixed in space but are not coplaner (so they are linearly independent). The rotational axis of each planetary wheel is orthogonal to the rotational axis of its base disk. For simplicity, in the following description each omniwheel is treated as a circular disk lying in a plane passing through the center of the sphere S. A breakout in the front surface of S reveals omniwheel 3 contacting the rear surface of S, that surface being treated as transparent for purposes of illustration. The sphere is shown with lines of latitude and longitude (with equator 7) which move with the sphere.

[0022] If there is no friction at omniwheels 1 and 3, as omniwheel 2 rotates it frictionally turns sphere S, and the sphere's point of contact moves on great circle 5. The same applies mutatis mutandis for omniwheels 1 and 3 and their respective great circles 4 and 6. The great circle corresponding to one of the omniwheels will be referred to as the great circle of motion for that omniwheel. The three great circles of motion are fixed in space relative to the positions of the omniwheels: they do not move as the sphere rotates.

[0023] In FIG. 1 the omniwheels are positioned so that the great circles touch the latitude lines 30 degrees on each side of equator 7.

[0024] FIG. 2 shows what happens near omniwheel 2. Assuming that there is no friction at omniwheels 1 and 2, as omniwheel 3 turns it causes points on the line 8 parallel to great circle of motion 6 to move at an angle across wheel 2. At the point of contact of the omniwheel with the sphere, the motion can be resolved into two components, one tangent to the great circle of motion 5 and the other orthogonal to that one, with both components lying in the sphere's tangent plane at the point of contact. Now considering that there is friction at omniwheel 2, since the rotational axis of that sphere-contacting planetary wheel is tangent to great circle of motion 5, it offers zero (or very little) frictional resistance to the component orthogonal to the great circle of motion. That is not the case for the other component, for there must be considerable friction in that direction in order that the wheel drive the sphere. This is still the case if there is friction at all omniwheels and the omniwheels rotate. Therefore, the arrangement of FIGS. 1 and 2 has a large amount of wheel binding.

[0025] It is apparent from FIG. 2 that there will always be binding if conventional wheels are used.

[0026] On the other hand, omniwheels will have the desired effect of isolating the actions of each wheel from those of the other wheels if the great circles are orthogonal to each other. This is the same as saying that the omniwheel vectors U.sub.1, U.sub.2, and U.sub.3 are to be mutually orthogonal.

[0027] Such an arrangement is possible. Indeed, for such an arrangement, the angle between the normal vector of each great circle and a fixed central vector will satisfy cos=1/{square root over (3)}(so54.7 . FIG. 3 illustrates this. In that figure the omniwheel axes (and therefore the great circles of motion) are orthogonal. Except for great circle 6. the parts of the great circles on the back side of the sphere have been omitted for clarity.

[0028] The contact point of the driving omniwheel of a great circle of motion can be placed anywhere on the great circle of motion if the plane of the omniwheel coincides with the plane of the great circle. It is easy to see that there can be more than one omniwheel on a great circle.

[0029] It is not necessary that the arrangement of the driving omniwheels be symmetrical, and this provides some flexibility in designing a mechanism of this invention. However, since the great circles must be orthogonal to each other, there can be only three of them.

[0030] The description given above relates to omniwheels for which the axes of the planetary wheels lie in the plane of the corresponding base disk. As has already been mentioned, omniwheels with planetary-wheel axes not paallel to the plane of the base disk are known. Using such an omniwheel changes the angle between the axis of the omniwheel and that of the associated great circle; the tangent to the great circle is parallel to the rotational axis of the planetary wheel at the point of contact. This can be used to change the mechanical configuration (e.g., arrange the axes of the omniwheels to be parallel), but it does not eliminate the need for orthogonality of the great circles.

[0031] It is apparent that the invention can be applied to surfaces which are not complete spheres provided the rotation of the omniwheels is sufficiently restricted.

[0032] Persons knowledgeable of the appropriate art will see that the invention can be used to drive motions of surfaces that are approximately, but not perfectly, spherical. For such surfaces it is, in general, not possible to make the lines of motion (corresponding to great circles on spherical surfaces) orthogonal at every point, so there will be some frictional losses.

[0033] This description has focused on arrangements with omniwheel axes fixed in space, but it applies to any arrangement for which the relative positions of the driven great circles are fixed; i.e., for which the driven great circles are mutually orthogonal. Thus, for example, the invention could be used for a vehicle supported by a single spherical wheel, the omniwheels being mounted on the supported chassis.