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a)sin algorithm ASME International basic test maneuver Block 2 Block Block 6 Gimbal commanded rates compute the gimbal discrete equations extended test maneuver figure 1(a flight simulations Four-Axis Gimbal Robot four-gimbal robot wrist Four-Gimbal Systems four-gimbal wrist fourth gimbal angle gimbal angle rates gimbal lock gimbal robot wrist gimbal wrist gles i i i I i i i Integration of quaternion integration step k+1 k+1 k+l k+l k+l Keith Barker Langley Research Center maneuver q maximum gimbal angle NASA NASA TN operational limit operator inputs overview Performing Organization probe-tip coordinates quaternion initialization quaternion rate equations rate limit rates are computed rates are scaled Research Center Hampton Robot Wrist Figure rotational rates scale factor scaling procedure sgn(sin shown in figure Simula singularity condition Theoretical Three three-gimbal robot wrist three-gimbal wrist transformation matrix Cfc+i updated gimbal angles updated quaternions vertical position worst-type wrist singularity