Figure 5
Four vertically stacked graphs share a horizontal axis labeled “Sample Index” ranging from 0 to 14000 with an interval of 200. Each graph contains two curves labeled in the legend as “True” (dashed blue) and “Predicted” (solid orange), which closely overlap across most regions. In the top graph, the vertical axis is labeled “x (meters)” and ranges from negative 1 to 1 with an interval of 0.5. The curves oscillate with varying amplitude, showing frequent peaks and troughs between approximately negative 1 and 0.9 from sample index 0 to around 5000. Between 5000 and 9000, the oscillations become smaller and more irregular. From about 9000 to 11000, the curve rises gradually toward positive values near 0.5. After 11000, the curve drops sharply to around negative 1 near 12000 and then rises slightly toward the end. In the second graph, the vertical axis is labeled “y (meters)” and ranges from negative 1 to 1 with an interval of 0.5. The curves show oscillatory behavior with moderate amplitude. From 0 to around 4000, there are repeated peaks approaching 0.9 and troughs near negative 0.5. Between 4000 and 8000, the oscillations reduce in amplitude and become smoother. From 8000 to 12000, the curve shows smaller undulations around zero. Toward the end, the values fluctuate between approximately negative 0.5 and 0.3. In the third graph, the vertical axis is labeled “z (meters)” and ranges from negative 1 to 0 with an interval of 0.5. The curves remain mostly near negative 0.8 to negative 1 from 0 to around 9000, with small fluctuations. Around 10000, there is a sharp upward spike reaching close to 0, followed by rapid drops and rises between approximately negative 1 and negative 0.3. After 11000, the curve continues with irregular fluctuations and several peaks and dips before settling near negative values toward the end. In the bottom graph, the vertical axis is labeled “phi (radian)” and ranges from negative 0.2 to 0.4 with an interval of 0.2. From 0 to about 4000, the curves show oscillations between approximately negative 0.1 and positive 0.3. Between 4000 and 9000, the fluctuations become more irregular and dense, with frequent small peaks and dips across the range. After 9000, the curve stabilizes with smaller variations, remaining mostly between 0 and 0.1, with occasional minor deviations. Across all four graphs, the dashed blue and solid orange curves follow nearly identical paths with minimal visible separation. Note: All numerical data values are approximated.Prediction performance of the trained DK-MPC model. The figure shows the comparison between true and predicted trajectories for position (x, y, z) and roll angle (ϕ) over time. The predicted signals (orange) closely follow the ground truth (blue dashed), demonstrating the model's ability to accurately capture the system dynamics in both translational and rotational dimensions