The paper describes the use of unit-quaternions for the kinematic modeling and control of mobile robots subject to the rolling-without-slipping constraint. The kinematic differential equation that describes the robot’s motion is unique in that it contains no trigonometric functions, and the most computationally expensive mathematical operation is a squaring operation. The paper shows that this feature leads to a drastic decrease in the time required to perform an Euler step when integrating the kinematic differential equation. Empirical tests across multiple PCs, single-board computers, and microcontrollers are presented to support the results. The paper also describes how position control can be accomplished with unit-quaternions. The control law contains no trigonometric functions and its most computationally expensive mathematical operation is a square-root operation. The control law will drive a robot to a waypoint with forward or reverse motion based upon the path that requires the least amount of rotation, and this can be determined in terms of unit-quaternions without any additional arithmetic operations. Both numerical and empirical results show the effectiveness of the control method applied to a differential drive mobile robot.