Kinematics of rigid bodies can be analyzed in many different ways. The advantage of using Euler parameters is that the resulting equations are polynomials and hence computational algebra, in particular Gröbner bases, can be used to study them. The disadvantage of the Gröbner basis methods is that the computational complexity grows quite fast in the worst case in the number of variables and the degree of polynomials. In the present article we show how to simplify computations when the mechanism contains revolute joints. The idea is based on the fact that the ideal representing the constraints of the revolute joint is not prime. Choosing the appropriate prime component reduces significantly the computational cost. We illustrate the method by applying it to the well known Bennett's and Bricard's mechanisms, but it can be applied to any mechanism which has revolute joints.