When the snake-like robot with passive wheels tracks the trajectory of the head, there are infinite solutions for the input torque. The minimum amplitude torque corresponds to the minimum rated torque of the motor which is the optimal torque mentioned in this paper. When the snake-like robot has no sideslip, the normal velocity of the snake-like robot is zero. So, a velocity constraint can be introduced into each module, and the snake-like robot is a nonholonomic system. The minimum amplitude torque corresponds to the torque with the minimum infinity norm. In this paper, the nonholonomic dynamics equations are developed, and the problem of solving the minimum amplitude torque is transformed into the problem of solving the minimum infinity norm under the constraints of dynamics equations. Using the numerical algorithm of the minimum infinity norm, the minimum infinity norm solution of the joint torque is derived when the snake-like robot tracks the head velocity. Using the minimum infinity norm solution, optimal torque control with minimum amplitude can be realized. The dynamics numerical simulation proves that the algorithm is valid.