## 5 Determination of Translation Magnitude from GNSS Measurements  
In an effort to resolve the scale ambiguity issue of a monocular camera, there are several available approaches, such as using the wheel tick and inertial sensors [6]. Before resorting to the inertial-aided approach, a decorrelation solution between sensor error and inertial mechanization is desired. An  
Figure 4: Feature matching and outlier removal between two image frames. analogous situation applies to using a wheel tick sensor if its drift arises quickly. Conversely, as one of the most widely applied outdoor positioning technologies, GNSS does not yield accumulated ranging or positioning errors, and can provide the capability to resolve the scale ambiguity problem of a monocular camera by leveraging GNSS differential technique. If the acquired GNSS and image data are synchronized, the baseline of position change between two GNSS epochs coincides with the desired camera translation. The GNSS observation equation for pseudorange measurements is given by:  
\[P^{i}=\rho^{i}+c\cdot t_{r}-c\cdot T^{i}-d_{t}^{i}+d_{r}^{i}+\varepsilon^{i} \qquad i=1,...,k \tag{14}\]  
where:  
\(P^{i}\) is the pseudorange measurement;  
\(\rho^{i}\) is the true geometric range between the designated GNSS satellite and the vehicle;  
\(t_{r}\) is the receiver clock error;  
\(T^{i}\) is the satellite clock error;  
\(c\) is the speed of light;  
\(d_{t}^{i}\) and \(d_{t}^{i}\) are ionospheric and tropospheric corrections, respectively, along the signal propagation path;  
\(\varepsilon^{i}\) denotes the unmodeled observation errors containing the thermal noise and so forth;  
The superscript \(i\) indicates the satellite index and \(k\) represents the total tracked number of satellites. Resorting to the concept of differential positioning technology, Equation (14) between two epochs yield differential expression as follows:  
\[\delta P^{i}=\delta\rho^{i}+c\cdot\delta t_{r}+\delta\varepsilon^{i}\qquad \qquad i=1,...,k \tag{15}\]  
where the notation \(\delta\) depicts the differential operation for each term. Under low dynamic conditions with short time intervals, the differential operation adequately eliminates the ionospheric and tropospheric effects. For the differential term of the satellite clock error, it can be either calculated using the decoded ephemeris or simply ignored as the satellite clock drift can be neglected in a short temporal scale. According to [PERSON] [31] and [PERSON] [6], the true range differential \(\delta\rho^{i}\) correlates the baseline magnitude of position change and the relationship satisfies:  
\[\delta\rho^{i} =\rho^{i}(t)-\rho^{i}(t-1) \tag{16}\] \[=\mathrm{dot}(\vec{\mathrm{R}}^{i}(t)-\vec{\mathrm{R}}_{r}(t), \vec{\mathrm{u}}(t))-\mathrm{dot}(\vec{\mathrm{R}}^{i}(t-1)-\vec{\mathrm{R}}_ {r}(t-1),\vec{\mathrm{u}}(t-1))\] \[=\mathrm{dot}(\vec{\mathrm{R}}^{i}(t),\vec{\mathrm{u}}(t))- \mathrm{dot}(\vec{\mathrm{R}}^{i}(t-1),\vec{\mathrm{u}}(t-1))-\mathrm{dot}( \delta\vec{\mathrm{R}}_{r}(t),\vec{\mathrm{u}}(t))-\mathrm{dot}(\vec{ \mathrm{R}}_{r}(t-1),\delta\vec{\mathrm{u}}(t))\] \[i=1,...,k\]  
where \(\mathrm{dot}(,)\) defines the vector dot product; \(\vec{\mathrm{R}}^{i}\) and \(\vec{\mathrm{R}}_{r}\) denote the position vectors of the satellite and the receiver in the \(e\) frame; \(\vec{\mathrm{u}}=\dfrac{\vec{\mathrm{R}}^{i}-\vec{\mathrm{R}}_{r}}{\left\| \vec{\mathrm{R}}^{i}-\vec{\mathrm{R}}_{r}\right\|}\) is the unit vector along the signal reception path. According to the commutative and distributive laws of the vector dot product, Equation (16) yields the baseline of position change \(\delta\vec{\mathrm{R}}_{r}\) as well as the change of the unit vector \(\delta\vec{\mathrm{u}}\). With the satellite broadcast ephemeris, the satellite position vector \(\vec{\mathrm{R}}^{i}(t-1)\) and \(\vec{\mathrm{R}}^{i}(t)\) can be easily estimated.