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The inverse kinematics is a challenging problem for controlling a humanoid robot finger with nonlinearly coupled joints. In this paper, a novel approach is proposed to derive the algebraic-elimination based solutions of inverse kinematics (IK) for the robot fingers. First, the given position of the fingertip is transformed from the base frame to the solution frames by means of D-H transformation. Therefore, two algebraic equations are obtained using the vector-loop and coordinate transformation techniques. The trigonometric terms can be converted into an eight-degree polynomial by making the substitution of two variables which are relative to the proximal (PIP) and distal interphalangeal (DIP) joint angles. To generate the smooth jerk-limited feedrate profile and interpolation points, a dynamics-based interpolator with real-time look-ahead (DBLA) algorithm is applied to plan the motion trajectory of a robot finger. The iterative Newton-Raphson method with a suitable initial guess is adopted for solving the roots of the polynomial. Finally, simulations are performed to demonstrate the exactness of the proposed inverse kinematics solutions. Experiments using an embedded motion controller are carried out for validating the feasibility of the proposed approach.