Tracking control of shape memory alloy (SMA) actuators is essential in many applications such as vibration controls. Due to the hysteresis, an inherent nonlinear phenomenon associated with SMAs, open-loop control design has proven inadequate for tracking control of these actuators. Aimed at to eliminate the position sensor to reduce cost of an SMA actuator system, in this paper, a neural network open loop controller is proposed for tracking control of an SMA actuator. A test stand, including a titanium-nickel (TiNi, or Nitinol) SMA wire actuator, a position sensor, bias springs, and a programmable current amplifier, is used to generate training data and to verify the neural networks open loop controller. A digital data acquisition and real-time control system was used to record experimental data and to implement the control strategy. Based on the training data obtained from the test stand, two neural networks are used to respectively model the forward and inverse hysteresis relations between the applied voltage and the displacement of the SMA wire actuator. To control the SMA actuator without using a position sensor, the neural network inverse model is used as a feedforward controller. The experimental results demonstrate the effectiveness of the neural network open loop controller for tracking control of the SMA wire actuator.