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This paper examines artificial gravity from the point of view of a person living and moving within a rotating habitat. First, it reviews the literature on comfort conditions for rotation. Next, it analyzes the relative motion of freefalling objects and the apparent slopes of surfaces to reveal the geometry of the effective gravitational field, its variation from Earthnormal gravity, and its variation within the supposed comfort zone for rotation. Finally, it examines the role of gravity in perception psychology and architectural design theory to explore the implications of artificial gravity for habitat design. An architectural grammar comprising wall, floor, and ceiling elements in particular orientations with respect to gravity is common sense. Nevertheless, its application to artificial gravity may be inadequate. Due to Coriolis accelerations and crosscoupled rotations, not only the updown (radial) but also the eastwest (tangential) directions emerge as gravitationally distinct. Inasmuch as architecture is powerless to mask these distinctions, it would do well to help the inhabitants adapt to them. It may do this by providing visual or other cues to assist orientation, as well as by arranging activities to minimize offaxis motion.
INTRODUCTION
There have been many proposals for orbital habitats that incorporate artificial gravity. Most of the analysis has focused on an external view of the artifact: structure, mass, deployment, axis orientation, dynamic stability, and so on. In contrast, relatively little has been written about the internal artificialgravity environment and its implications for the architecture of the habitat and the performance of the crew. This is rather ironic, since crew support is the most common justification for investing in artificial gravity at all. Concept studies have implied, and sometimes stated outright, that artificial gravity should permit the use of common terrestrial design elements. They have downplayed the artificiality of the gravity, and have not explored its ramifications for habitat design. Terrestrial gravity is a quintessentially common experience. An architectural grammar comprising wall, floor, and ceiling elements in particular orientations with respect to gravity is common sense. Nevertheless, this grammar may not be adequate for artificial gravity. Relative motion in a rotating environment involves Coriolis accelerations and crosscoupled rotations that essentially twist the apparent gravitational field.
MATHEMATICAL NOMENCLATURE
The essential formulae for artificial gravity can be found in any textbook on physics or mechanical dynamics. This paper adopts the following nomenclature: boldface indicates vectors; italics indicate scalar magnitudes; dots above indicate derivatives with respect to time; uppercase symbols refer to the inertial frame, while lowercase symbols refer to the rotating frame; measurement units are radians, meters, and seconds, except where stated otherwise.
COMFORT IN ARTIFICIAL GRAVITY
In the early days of space station design, it was widely assumed that orbital habitats would provide artificial gravity. Tsiolkovsky, Oberth, Noordung, von Braun and other visionaries performed detailed calculations and published various concepts for rotating space stations, several decades before the first Sputnik entered Earth orbit. With the beginning of manned space flight in the 1960s, there was concerted effort to determine the comfort criteria for rotating habitats. As experience with weightless space flight has accumulated, artificial gravity has assumed a lower priority. Since the beginning of the Salyut and Skylab missions, access to a microgravity environment has been one of the main motivations for space flight. Ironically, while extended stays in weightlessness have revealed its dangers, they have also shown that it is survivable. Hence, much of the research into the human factors of rotating habitats is twenty or thirty years old. Over the past four decades, several authors have published guidelines for comfort in artificial gravity, including graphs of the hypothetical "comfort zone" bounded by values of acceleration, headtofoot acceleration gradient, rotation rate, and tangential velocity. Individually, these graphs depict the boundaries as precise mathematical functions. Only when studied collectively do they reveal the uncertainties.^{16} Table 1 summarizes the comfort boundaries proposed by several authors. In this context, comfort does not imply luxury, but merely mitigation of symptoms of motion sickness. The parameters are:
THE GEOMETRY OF ARTIFICIAL GRAVITY
Tables and graphs of mathematical limits are handy references for broadbrush engineering analysis, but they do nothing to convey the look and feel of artificial gravity. In fact, by presenting only the averages, and not the underlying statistics, they give a false sense of certainty regarding the comfort boundaries. Comfort is a subjective, psychological quality of individuals that depends on many things, including intangibles such as stress, motivation, and general satisfaction. It may be influenced by aspects of environmental design beyond the parameters of rotation. It's not clear that either the experimenters or the subjects have used a consistent definition of comfort in establishing the boundaries shown in Table 1. Perhaps a more intuitive way to compare artificial gravity environments with each other as well as with Earth is to observe the relative motions of freefalling particles and the apparent slopes of surfaces.
FreeFall and the Involute Curve
Abnormalities in freefall motion reveal abnormalities in the gravity itself. To facilitate a sidebyside comparison of various gravity environments, it's useful to specify a few standard tests:
The following discussion uses a Cartesian coordinate system x,y,z tied to the rotating habitat. The x,y,z axes initially coincide with inertial X,Y,Z axes, but rotate about the Z axis with angular velocity .
Drop
Figure 1 shows the motions of an observer and of a particle that drops from a height h. The particle's initial position and velocity in the rotating frame are: In the inertial frame: Before striking the floor, the particle travels a linear distance in the inertial frame: and subtends an angle of: If the observer had not dropped it, the particle would have traveled the same distance on a circular path subtending an angle of: This is the angle that the observer subtends while the particle is falling. In the observer's rotating frame of reference, the particle falls along an involute curve. It strikes the floor at an arc distance from the observer: where positive is east (prograde) and negative is west (retrograde). The particle always deflects to the west, because: The angles _{p} and _{o} depend only on the initial height h and the floor radius r_{f}. If the expressions are rewritten in terms of the ratio h/r_{f}, they become: Smaller ratios of h to r_{f} result in smaller angles, smaller angle differences, and a more vertical path as seen from the rotating frame. The angular velocity and centripetal acceleration A_{cent} influence the speed at which the particle falls, but not the path it follows. The elapsed time is:
Hop
Figure 2 shows the motions of an observer and of a particle that hops vertically from the floor with an initial relative velocity v. The particle's initial position and velocity in the rotating frame are: In the inertial frame: The particle traces a chord through a circle in the inertial frame. The slope angle of the chord is: The chord subtends an angle of: The Xaxis displacement, due entirely to the tangential velocity at the beginning of the hop, is: During this time, the observer travels this same distance along a circular path that subtends an angle of: Here, the angles _{p} and _{o} depend on the ratio of the relative velocity v to the environment's tangential velocity V_{t}: As in the drop, the particle strikes the floor at an arc distance from the observer: In contrast to the drop, the hop always deflects to the east (prograde), because: Though this is not immediately obvious, it can be seen by plotting the functions and also by comparing the derivatives: Equations 19 through 21 show that _{p} and _{o} are zero when v is zero, and positive when v is positive. However, _{p} increases faster than _{o}. In fact, if the ratio v/V_{t} exceeds 1, then _{o} decreases toward zero. The ratio v/V_{t} determines the proportions of the particle's path as seen in the rotating frame. Smaller ratios yield slimmer, more vertical, more Earthnormal paths. The overall size of the path depends on the floor radius and the apparent gravity. For a given tangential velocity V_{t} , a larger floor radius r_{f} yields weaker acceleration A_{cent} and increases the breadth and height of the path proportionately. From Figure 2 and Equation 17, the elapsed time is:
Variation within the Supposed Comfort Zone
In the analysis above, the height h and velocity v represent parameters of human size and speed. The designer should assume values adequate to support the activities of the inhabitants. To provide some semblance of normal life, the designer should select values of r_{f} and V_{t} that are large with respect to h and v. Unfortunately, that increases both the mass and kinetic energy of the rotating spacecraft. Economics pushes in the opposite direction, toward minimizing the radius and tangential velocity. As judged by the behavior of freefalling particles, this produces the least normal gravity environment. Table 2 summarizes, for each of several estimates of the comfort zone, the boundary point with the minimum acceptable radius and tangential velocity. In each row, the two limiting parameters are formatted in boldface; the other parameters are computed from them. Where radius is a limiting parameter, it's the inverse of the maximum acceptable gravity gradient per meter.
Figure 3 shows, for each condition listed in Table 2, the paths followed by particles:
Though somewhat arbitrary, these actions represent a rough envelope for common human motion. A height of 2 meters is within reach of most adults, somewhat above the top of the head. On Earth, a 2meterpersecond hop raises one's center of mass by 0.204 meters (about 8 inches); in Figure 3, a short horizontal line marks this height. On Earth, both of these actions would trace vertical paths. Figure 3 shows that the various estimates of the comfort boundaries permit considerable deviations from Earthnormal gravity. Judging from Figure 3, it appears that Stone's criteria of 6.4 rpm and 10.2 m/s provide one of the more normal gravity environments. Hop #5 is the closest to normal breadth and height, and is proportionally more vertical than all except #3 (Gilruth's "optimum" comfort limit). Drop #5 is intermediate. Unfortunately, this figure doesn't tell the full story of gravitational distortion. Stone achieves that normallooking hop by imposing an exceptionally high angular velocity, to provide a higher tangential velocity and centripetal acceleration. Graybiel's findings suggest that the inhabitants may have a tough time overcoming dizziness.^{10} Dizziness is inherently counterintuitive  almost by definition  and difficult to represent on paper. One of the lessons of Figure 3 is that, in order to avoid the problems of dizziness associated with high angular velocities, while minimizing radius and tangential velocity, one may need to tolerate more deviant behavior of freefalling particles. If cost is no obstacle, then the best solution is to provide a maximal radius and tangential velocity and a minimal angular velocity. One can simulate an Earthnormal gravity environment within any nonzero tolerance, provided the radius is sufficiently large. In this respect, there is no substitute for a large radius.
Linear Motion and the Catenary Curve
Linear motion in the plane of rotation requires Coriolis acceleration in order to maintain constant velocity in the rotating frame. The Coriolis acceleration is perpendicular to the relative velocity. If the relative velocity is not tangential to the rotation, then the Coriolis acceleration will not be radial and will not align with the centripetal acceleration. The nonaligned Coriolis component skews the total apparent gravity and alters the apparent slope of surfaces. Two examples of such motion arise in many designs for artificialgravity habitats:
Both of these actions involve motion along a straight chord in the plane of rotation. This chord is tangent to the rotation only at its midpoint; at all other points, the chord is sloped with respect to the artificial gravity. Figure 4 shows an observer walking west to east (prograde) with relative velocity v along a straight chord in a rotating habitat. The chord is parallel to the x axis, at y = r_{c}, where r_{c} is the radius to the chord's midpoint. The total apparent gravity is the sum of the centripetal and Coriolis accelerations: where i and j are unit vectors parallel to the x and y axes. It's convenient at this point to define an intermediate constant term q: The dotted lines in Figure 4 show the apparent gravitational field. The acceleration vector A converges on a point, offset above the center of rotation by the Coriolis component. As the observer moves through his environment, he carries with him his own coordinate system that defines his sense of horizontal and vertical. Figure 4 designates these as and , respectively. The axis aligns with the acceleration vector A, which defines "up". In the , coordinate system, the apparent slope of the chord is a function of arc length x, where x seems to be measured on a curved path that initially slopes upward, levels off at midchord (x=0), and then slopes down. Figure 5 shows the apparent slope of the chord and strength of gravity in the observer's , coordinate system. At each point, the apparent slope of the chord is: Replacing x with an expression for arc length measured from midpath in the , coordinate system, this becomes: Thus, the apparent slope of the chord follows a catenary curve: The magnitude of the acceleration follows a similar catenary curve. From Equations 26, 28, and 29: As in Figure 4, the dotted lines in Figure 5 show the apparent gravitational field. Figures 6 and 7 show the situation for an observer walking in the opposite direction. Because the Coriolis acceleration is reversed, the apparent gravity is weaker but the apparent slope is steeper.
Figures 8 and 9 show an observer climbing a ladder aligned on a radius. In this case, the radius r_{c} to the midpoint of the chord is zero. To keep himself on top of the catenary curve, rather than hanging beneath it, the observer should ascend on the ladder's west side and descend on its east side. Therefore, the ladder should be accessible from both sides. Alternatively, the ladder could be laterally offset or inclined from the radius by an amount sufficient to assure that apparent gravity always presses the observer into the ladder and never pulls him away. In any case, the plane of the ladder should be perpendicular to the plane of rotation. As a rough guide to discerning the amount of slope that might be acceptable in artificial gravity, Table 3 lists some typical slopes in terrestrial architecture.^{1113,18}
ADAPTING TO ARTIFICIAL GRAVITY
Studies indicate that familiarity with gravity is not innate, but is learned in infancy. At 4 months, infants begin to realize that a rolling ball cannot pass through an obstacle, but are not yet aware that an unsupported ball will fall. At 5 months, they discriminate between upward and downward motion. At 7 months, they show sensitivity to gravity and the "appropriate" acceleration of a ball rolling upward or downward. By adulthood, falling objects are judged to move naturally only if they accelerate downward on a parabolic path. These judgments are based not on mathematical reasoning, but on visual experience; when asked to reason abstractly about such motion, many adults are prone to error.^{1416} Gravity underpins the sense of direction. Six directions on three axes are innately perceptible: updown (height), leftright (breadth), and frontback (depth). The anisotropic character of this space is judged by the effort required to move in any given direction. On Earth, up and down are distinct irreversible poles, while left, right, front, and back are interchangeable simply by turning around. Thus, in Earth gravity, there are three principal directions  up, down, and horizontal  and three basic architectural elements  ceiling (or roof), floor, and wall. These commonsense ideas, rooted in the experience of terrestrial gravity, permeate architectural theory. ThiisEvensen builds his entire grammar around the three elements of floor, wall, and roof.^{17} Habitat design for a gravitational environment distinctly different from Earth's requires a fundamental reexamination of terrestrial design principles. The goal is not to mimic Earth, but rather, to help the inhabitants adapt to the realities of their rotating habitat. In artificial gravity, the effects of Coriolis acceleration and crosscoupled rotation arise only during relative motion within the rotating habitat. While stationary, one may forget about these effects  only to be rudely reminded of them when rising out of a chair, lifting a piece of equipment, or turning to the side. It's possible to minimize these effects by planning activities to avoid offaxis motion. Where offaxis motion is unavoidable, one may arrange things to provide inhabitants with the best mechanical advantage with respect to the Coriolis acceleration. One may also strive to keep the inhabitants passively oriented to the rotation of their habitat, allowing them to prepare themselves for the consequences of their actions. Hesselgren constructs his architectural theory on the foundations of perception psychology.^{18,19} He describes "transformation tendencies" between various senses, whereby a perception in one modality may produce a mental image of a perception in another. For example, visual texture gives rise to a mental image or expectation of tactile grain. One modality that he never discusses, which is taken for granted on Earth but cannot be in space, is vestibular perception. It might be possible, through experience in a properly designed habitat, to acquire a transformation tendency to vestibular perception from visual, acoustic, haptic, or other perceptions. The goal is not to induce motion sickness by the mere sight of some visual cue. Rather, it is to provide visual or other reminders that motion relative to these cues will result in certain inescapable side effects, inherent in the artificial gravity. These perceptual cues would act as signals, triggering adaptive coordination in the inhabitants. From the designer's point of view, a consistent vocabulary of such signals would have to arise from convention. From the inhabitants' point of view, these conventions might to some extent be taught, but the unconscious transformation to a vestibular image would rely on association based on direct experience. As to the kinds of cues that might be provided, that's open to debate. They might be as banal as stenciled labels: "This end up"; "This end forward". They might be more refined and symbolic, engaging multiple facets of perception such as advancing and receding colors and basrelief shapes. In a oneoff design, it might be argued that no special cues are necessary. As long as the habitat is asymmetrical, the inhabitants will come to recognize, for example, that the galley is fore and the lab is aft. On the other hand, if the habitat comprises several chambers, it may still be beneficial to provide consistent cues to orientation that carry through from one chamber to the next.
CONCLUSIONS
Adherence to the comfort zone for artificial gravity does not guarantee an Earthnormal gravity environment. Consequently, habitat designs that function well in terrestrial gravity are not necessarily appropriate for artificial gravity. Artificial gravity can approach Earthnormalcy within any nonzero tolerance, provided that the radius and tangential velocity are large and the angular velocity is small. A habitat carefully designed to accommodate the peculiarities of artificial gravity may provide adequate comfort at a smaller radius than would otherwise be acceptable. This would reduce the mass and energy necessary to provide such a habitat. Adaptations for artificial gravity can not be contained in some addon package, to be installed after the habitat design is otherwise complete. On the contrary, artificial gravity is an overarching concept that should be forefront throughout the design process. Gravity permeates the environment. Artificial gravity is distinct from both Earthgravity and weightlessness and demands the same attention to detail. At a minimum, activities should be arranged to avoid offaxis motion, and to provide the best stability and mechanical advantage with respect to Coriolis forces and crosscoupled rotations. There may be some benefit in providing consistent, explicit visual cues to keep the inhabitants oriented with respect to the rotation; this remains to be tested.
REFERENCES
