The kinesin family of motor proteins are involved in a variety of cellular processes that transport materials and generate force. With recent advances in experimental techniques, such as optical tweezers can probe individual molecules, there has been an increasing interest in understanding the mechanisms by which motor proteins convert chemical energy into mechanical work. Here we present a mathematical model for the chemistry and three dimensional mechanics of the kinesin motor protein which captures many of the force dependent features of the motor. For the elasticity of the tether that attaches cargo to the motor we develop a method for deriving the non-linear force-extension relationship from optical trap data. For the kinesin heads, cargo, and microscope stage we formulate a three dimensional Brownian Dynamics model that takes into account excluded volume interactions. To efficiently compute statistics from the model, an algorithm is proposed which uses a two step protocol that separates the simulation of the mechanical features of the model from the chemical kinetics of the model. Using this approach for a bead transported by the motor, the force dependent average velocity and randomness parameter are computed and compared with the experimental data.