This paper presents a trajectory planning algo-rithm for person following that is more comprehensive thanexisting algorithms. This algorithm is tailored for a front-wheel-steered vehicle, is designed to follow a person while avoidingcollisions with both static and moving obstacles, simultaneouslyoptimizing speed and steering, and minimizing control effort.This algorithm uses nonlinear model predictive control, wherethe underling trajectory optimization problem is approximatedusing a simultaneous method. Results collected in an unknownenvironment show that the proposed planning algorithm workswell with a perception algorithm to follow a person in unevengrass near obstacles and over ditches and curbs, and on asphaltover train-tracks and near buildings and cars. Overall, theresults indicate that the proposed algorithm can safely followa person in unknown, dynamic environment