Here is an Arduino robot that you can control via Bluetooth. The CurieBot Arduino 101 Mini Robot Rover has a 6-axis accelerometer/gyro and Bluetooth LE module. You will be able to interact with it from your smartphone or tablet.
The kit comes with battery, three-layer robot chassis, 1 x Adafruit Motor/Stepper/Servo Shield v2, and everything you need to complete this robot. More info is available here.