In (9.8), the initial orientation $ \theta(0)$ was assumed to be known. In practice, this corresponds to a registration problem, which is the initial alignment between the real and virtual worlds. To understand the issue, suppose that $ \theta $ represents the yaw direction for a VR headset. One possibility is to assign $ \theta(0) = 0$, which corresponds to whichever direction the headset is facing when the tracking system is turned on. This might be when the system is booted. If the headset has an ``on head'' sensor, then it could start when the user attaches the headset to his head. Often times, the forward direction could be unintentionally set in a bad way. For example, if one person starts a VR demo and hands the headset to someone else, who is facing another direction, then in VR the user would not be facing in the intended forward direction. This could be fixed by a simple option that causes ``forward'' (and hence $ \theta(t)$) to be redefined as whichever direction the user is facing at present.

An alternative to this entire problem is to declare $ \theta(0) = 0$ to correspond to a direction that is fixed in the physical world. For example, if the user is sitting at a desk in front of a computer monitor, then the forward direction could be defined as the yaw angle for which the user and headset are facing the monitor. Implementing this solution requires a sensor that can measure the yaw orientation with respect to the surrounding physical world. For example, with the Oculus Rift, the user faces a stationary camera, which corresponds to the forward direction.

Steven M LaValle 2020-01-06