# Native Smartphone Single- and Dual-Frequency GNSS-PPP/IMU Solution in Real-World Driving Scenarios
## 4 Results Analysis
### IMU Dead Reckoning Performance  
An extreme, but realistically meaningful scenario is complete GNSS outages such as driving in tunnels. This study first attempts to produce IMU-only solutions over two separate 60 s GNSS outages where the positioning solution is generated solely from IMU inputs. To simulate such outages, outages A and B are extended by not including GNSS measurements during the 60 s. The solution states are first initialized using a reference solution produced by SPAN, with lever arm correction to the smartphone frame. The horizontal errors are compared against the SPAN reference solution in post-processed RTK/TC with full access to all available measurements in Figure 5.  
Figure 5: Position estimates based on dead-reckoning of Xsens and MI 8 IMUs only within 60 s. (**a**) IMU dead-reckoning horizontal errors within 60 s at Outage A. (**b**) Google Earth view of IMU mechanization trajectories at Outage A. (**c**) IMU dead-reckoning horizontal errors within 60 s at Outage B. (**d**) Google Earth view of IMU mechanization trajectories at Outage B.  
The results show diverging tracks from the reference solution in the Google satellite views, and quantitatively demonstrated by strictly increasing horizontal errors with time. The time series and table show higher horizontal errors from Mi 8 (blue) than from Xsens (red) for the complete duration of Outage A and after 20 s for Outage B. The Xsens has a slower diverging rate, suggesting a less noisy raw IMU performance. In general, the result suggests robust ultra-low-cost IMU performance for an outage less than 10 s comparable to that of automotive-grade IMUs such as the Xsens.  
Another key observation is that the IMU-only solution shows better accuracy through Outage B compared to during Outage A. Theoretically, IMU error propagation is complicated and influenced by a range of factors including initialization errors, IMU timing discrepancy, finite iteration rates, and noises. Measurement biases, mainly from gyroscope also play a critical role in impacting IMU dead-reckoning solutions, and the positioning errors \(\delta X\) in the respective axis can be represented as Equation (4) [42]:  
\[\delta X_{a} \approx\frac{1}{2}b_{a}t^{2} \tag{4}\] \[\delta X_{g} \approx\frac{1}{6}b_{g}gt^{3}\]  
where \(\delta X_{a}\) and \(\delta X_{g}\) are the position errors caused by accelerometer and gyroscope biases (\(b_{a}\) and \(b_{g}\)), respectively; \(g\) is gravity and \(t\) is the IMU mechanization time. In other words, a \(10^{-4}\) m/s\({}^{2}\) accelerometer bias and a \(10^{-3}\) rad/s gyro bias theoretically produce position errors of 0.5 cm and 1.6 m, respectively, in 10 s.  
Further analysis of these IMU biases is presented in Figure 6. The experiments were conducted in kinematic environments that started with a straight path. The initial biases are set as zero and it takes time for both of IMUs' biases to converge. The engine has run for 5 min prior to Outage A compared to 15 min prior to Outage B. From Figure 6, it can be observed that the bias states are better converged at the 15-min mark than the 5-min mark, potentially giving a better bias estimation. In addition, the Xsens IMU has more stable bias estimation which converges faster than for the smartphone IMU. For the accelerometers, the Mi 8 IMU does not show a clear converging pattern. For the gyroscopes, the Mi 8 IMU takes almost 10 min more to reach a stable state compared to the Xsens. This comparative performance is expected owing to lower-cost and performance of the smartphone IMU. As a consequence of these two reasons, the early unconverged estimated biases lead to the lower dead-reckoning performance at Outage A compared to Outage B.  
Figure 6: Times series of estimated IMU accelerometer and gyroscope biases with along three axes.