Forward kinematics of three classes of 3-RRR spherical parallel mechanisms admitting closed-form solutions

3-RRR spherical parallel mechanisms (SPMs) have been extensively studied due to their numerous applications. Substantial effort has been devoted to their forward kinematics (FK), which is essential for their calibration and feedback control. However, despite their simple architecture, rather few 3-RRR SPMs with closed-form FK solutions (CFFKS) have been reported; iterative procedures are thus required in most cases. This paper presents three classes of 3-RRR SPMs with CFFKS, with the univariate polynomials for their FK being linear, quadratic, or quartic. These classes include a large set of designs, thereby enhancing the flexibility in selecting their architecture parameters. Moreover, they cover the majority of 3-RRR SPMs with special geometries that have been reported, while encompassing 3-RRR SPMs with certain special geometries yielding exceptional features such as unlimited rotation capacity about certain directions. Notably, these formulations are also applicable to many SPMs with alternative topologies and certain parallel mechanisms of other types. This work expands the family of SPMs with CFFKS, highly desirable in many practical applications.