(covers information from several alternate timelines)
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A wheelchair was a device used to permit locomotion in a humanoid who was unable to use his or her legs due to accident, disease, or natural causes. It was also used by people who could walk, but had a limited ability to do so. It could be wheeled or be supported by analogous anti-gravity generators.
In Detroit of the year 2004, Lawrence Strode used a wheelchair. It was taken by Loomis – when he abducted Strode – to a makeshift infirmary in an abandoned factory formerly run by the American Wrecking Company, along with Strode himself, who Loomis rendered unconscious. The wheelchair's arrival was secretly watched by time travelers Jonathan Archer and T'Pol. The same wheelchair was subsequently used by Loomis to wheel an apparently unconscious Archer into the makeshift infirmary. (ENT: "Carpenter Street")
The USS Kelvin had at least one wheelchair. This was used in the alternate reality to transport the heavily pregnant Winona Kirk from sickbay to Medical shuttle 37 during evacuation from the Narada's attack. (Star Trek)
After he was injured in a training accident, Fleet Captain Christopher Pike became a wheelchair user, unable to walk or speak. The wheelchair provided movement as well as simple communication via a series of lights on the front of the device. Pike was given the illusion of leaving the wheelchair when his former colleague Spock brought him to Talos IV aboard the USS Enterprise in 2268. (TOS: "The Menagerie, Part I", "The Menagerie, Part II")
Admiral Mark Jameson had to use a wheelchair due to the debilitating effects of Iverson's Disease. He left the wheelchair after taking a rejuvenation treatment on the planet Cerberus II, which cured the disease and caused him to become appreciably younger. However, the treatment proved much too dangerous for Humans. (TNG: "Too Short a Season")
Melora Pazlar used what she referred to as a "trolley car" to move around in areas of high gravity due to her being an Elaysian, which prevented her from a full range of movement while in high gravity areas. She sent the specifications of the wheelchair to Julian Bashir before her arrival on Deep Space 9 in 2370. A low-technology version with wheels had to be used because Cardassian artificial gravity technology was not compatible with a standard Federation anti-grav unit. Upon viewing the device, Jadzia Dax exclaimed that she had not seen one in three hundred years. (DS9: "Melora")
The use of a wheelchair prop in the "The Menagerie" two-parter was meant to help camouflage the fact that a replacement actor was going to be playing Christopher Pike, instead of Jeffrey Hunter. (Star Trek: Aliens & Artifacts, p. 38) When Gene Roddenberry initially interviewed Sean Kenney about taking the part, Roddenberry stated, "All questions posed to you will be answered by you with a button, 'one beep' for yes and 'two beeps' for no, on a specially rigged light system, out of sight of the camera." (Captain Pike Found Alive!, "Chapter Two: Ground Zero") The prop consisted of an actual wheelchair which was motorized and outfitted with an outer plastic shell. Kenney himself maneuvered the wheelchair and operated the light. (Star Trek: Aliens & Artifacts, p. 40)
The wheelchair seen in "Melora" was initially to have utilized anti-grav technology. The chair from "Too Short a Season" was to be pulled out of storage and remodeled. However, the chair had originally been designed with the larger set of the USS Enterprise-D in mind and it was quickly realized that it would not be practical in the relatively small Deep Space Nine corridor sets. As a result, a simplified 21st century wheelchair was used instead. (The Making of Star Trek: Deep Space Nine, p. 108)
The design of Melora's chair was greatly influenced by writer Evan Carlos Somers' personal experiences of navigating the studio offices. As Somers explained, "So Bashir has to replicate a much simpler wheelchair for her and she encounters all the problems that I did whenever I went down to the DS9 set to snoop around." (Star Trek: Deep Space Nine Companion, p. 93)
Set Decorator Laura Richarz purchased the chair, which was revamped by Property Master Joe Longo. Longo later remembered how the production crew decided to keep the wheelchair as simple as possible, recalling that he "added a control panel and some wheel covers to block out the spokes in the wheels, and changed the joystick. Basically we tried to keep it as simple as possible, because of our experience on 'Too Short a Season'. We had made a big albatross of a moving chair for that, and it was bad. But this one worked great; the actress drove it everywhere." (Star Trek: Deep Space Nine Companion, p. 93)