A Real-Time Method for Solving the Forward Kinematics of a Tripod With Fixed-Length Legs

This paper presents a real-time method for solving the forward kinematics of a tripod with fixed-length legs. The basic idea is to model the problem at hand based on a spatial four-bar linkage through which three sliding legs can be interrelated by choosing one link as a driving variable and other two links as driven variables. As a result, the original multivariable nonlinear problem with three variables can be reduced to one variable problem. A complete approach is provided to solve the unitary nonlinear programing problem. This includes a method for solving the implicit functions in terms of the driving and driven variables, and an approximation method for selecting an initial value leading to a fast solution. The simulation results show that (i) the method is effective, (ii) can reach very accurate results within five iterations for an error bound of $10−10$⁠, and (iii) numerically very stable. The experiment results show that the proposed forward kinematic method is fast enough to be implemented in real time to provide an accurate prediction of the tool pose from the joint encoder measurement.