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RSPM is an overconstrained parallel mechanism that has a variety of applications in medical procedures such as ankle rehabilitation because of its precision and accuracy. However, obtaining a complete explicit dynamic model of these mechanisms for tracking purposes has been a problematic challenge due to their inherent singularities, coupling effects of the limbs, and redundant constraints imposed by the intermediate joints. This paper presents a novel algorithm to obtain the analytical kinematic solutions of RSPMs based on the closed-loop vector method, which includes constraint analysis. By incorporating constrained kinematics into the dynamic model, a comprehensive explicit dynamic solution of the non-overconstrained version 3-RCC of RSPM is developed in task space, based on screw theory and the linear homogeneous property of algebraic equations on the manipulator twist. Based on the proposed computational framework, a robust self-tuning backstepping control (STBC) strategy is applied to the robot to overcome the effect of external disturbances and time-varying uncertainties. Furthermore, an observer-based compensation (OBC) method is presented for dealing with the nonlinear hysteresis loops of the ankle during trajectory tracking purposes. The closed-loop stability of the whole system including STBC and OBC is theoretically performed by Lyapunov methods. The proposed methodologies are validated by realistic co-simulations in different scenarios. For instant, in the presence of external disturbances, the maximum tracking error norm of STBC is 37.5% less than the sliding mode approach.<\/jats:p>","DOI":"10.1017\/s0263574724000390","type":"journal-article","created":{"date-parts":[[2024,4,12]],"date-time":"2024-04-12T07:07:27Z","timestamp":1712905647000},"page":"1568-1596","source":"Crossref","is-referenced-by-count":11,"title":["Adaptive backstepping controller based on a novel framework for dynamic solution of an ankle rehabilitation spherical parallel robot"],"prefix":"10.1017","volume":"42","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-8207-1042","authenticated-orcid":false,"given":"Ali","family":"Ahmadi N","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ali","family":"Kamali 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