The utility of a four-gimbal system for positioning two reference frames with respect to each other without gim bal lock has long been recognized. As early as 1954 such a system to isolate an inertial platform was proposed by Arnold and Schlesinger1 . However, the exact law for driving the fourth angle has been a question. The situa tion had not improved through 1962 where, in Reference 2, "gimbal flip" (an instantaneous rotation of 180 de grees in two axes, similar to the three-axis gimbal be havior at gimbal lock) appears as an inherent part of the fourth angle's driving law. Clearly this behavior is not desirable for platform isolation or visual display systems used for vehicle motion cues in simulation. This paper describes developments made at Langley (NASA) in the last several years. A driving law for the fourth angle is developed for two different four-gimbal systems. The first is similar to the gimbal systems de scribed in References 1 and 2 and has advantages in design and implementation for platform isolation. The second is specifically designed to minimize occlusion in visual display systems3. In each system, the fourth angle driving law is a direct consequence of maximizing the angle between the two axes causing the singularity. Thus, a necessary differ ential constraint is found to maintain a nonsingular solu tion. A sufficiency condition for a nonsingular solution is also found. The second four-gimbal system that mini mizes occlusion has the interesting result that the maxi mum angle between the singularity-forming axes is not always 90 degrees.
Study of the pedigree polytope and a sufficiency condition for nonadjacency in the tour polytope
