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This letter presents a new approach to solve the time-optimal path tracking under limited joint range and bounds on velocity, acceleration, and jerk. To obtain smooth and continuous accelerations, with beneficial effects for the load and wear on the actuators but a limited impact on performance, we state the minimum-time path tracking problem with the jerk as the control input. The main contribution of this letter is a formulation that includes the jerk constraints in the optimization problem and that, even if the resulting non-linear programming problem is non-convex, allows to perform an efficient and reliable convex relaxation using McCormick envelopes. Simulations and experimental tests on two 7-DoF manipulators have been carried out to show the benefits of the proposed approach and to compare it to state-of-the-art techniques.