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Solving the direct kinematics of parallel spherical mechanisms with l legs is basically solving systems of l-1 second-order multinomials. This paper presents a recurrent expression for the control points of these multinomials when expressed in the Bernstein form. This result allows one to propose a technique for solving the direct kinematics of these mechanisms that takes advantage of the sub-division and convex hull properties of polynomials in the Bernstein form. Contrary to other numerical approaches, the one presented here is clearly less involved and, although it can be classified within the same category as interval-based techniques, it does not require any interval arithmetic computation.