The double wishbone suspension is used commonly in high performance vehicles due to its superior kinematic response. However, its kinematics is very complicated, and to the best of the authors' knowledge, no reported analysis of the same for the full spatial model of the suspension exists in literature. This paper presents such a solution, building upon two key elements in the formulation and solution stages, respectively: the use of Rodrigue's parameters to develop an algebraic set of equations representing the kinematics of the mechanism, and the computation of Gröbner basis as a method of solving the resulting set of equations. It is found that the final univariate equation representing all the kinematic solutions for a given pair of steering and road profile inputs is of 64 degree - which explains the complexity observed in the kinematics of the mechanism. The real roots of this polynomial are extracted, and the solutions to the kinematic problem are computed for a particular set of inputs for the sake of illustration of the proposed formulation. The numerical accuracy of the solutions is verified by computing the residuals of the original set of kinematic constraints. The configurations of the mechanism for the real solutions are shown graphically.