Figure 6
Four vertically stacked graphs share a horizontal axis labeled “Time (seconds)” ranging from 0 to 50 with an interval of 10. Each graph contains three curves identified in the legend as “Reference” (dashed blue), “Nonlinear M P C” (solid orange), and “D K M P C” (solid green). In the top graph, the vertical axis is labeled “x (meters)” and ranges from 0 to 3 with an interval of 1. The dashed blue reference and nonlinear M P C curves overlap and form step-like levels: near 0.4 from 0 to about 10 seconds, rising to around 0.7 from 10 to 20 seconds, then to about 0.8 from 20 to 30 seconds, and dropping back near 0.4 after 30 seconds. The orange curve shows a sharp peak above 2.3 near the start, then quickly settles and follows the step levels with small oscillations around each transition. The green curve follows the step pattern more closely with smaller overshoot at transitions. In the second graph, the vertical axis is labeled “y (meters)” and ranges from 0 to 1 with an interval of 0.5. The dashed blue reference and nonlinear M P C curves overlap and show step changes: around 0.4 from 0 to 10 seconds, rising to about 0.7 from 10 to 20 seconds, dropping to around 0.4 from 20 to 30 seconds, and increasing again to about 0.7 after 30 seconds. The orange curve exhibits oscillations near the start, including a peak above 1, then settles and tracks the step levels with mild overshoot at transitions. The green curve follows the same step pattern with smaller deviations and smoother transitions. In the third graph, the vertical axis is labeled “z (meters)” and ranges from 0 to 4 with an interval of 2. The dashed blue reference and nonlinear M P C curves overlap and remain near about 2.5 from 0 to 10 seconds, increase slightly to around 2.8 between 10 and 30 seconds, and then step up to about 3.2 after 30 seconds. The orange curve shows a large initial rise from near 0 to above 2, followed by oscillations before settling close to the reference levels. The green curve tracks the reference closely with minimal deviation and small adjustments at the step near 30 seconds. In the bottom graph, the vertical axis is labeled “phi (radian)” and ranges from 0 to 4 with an interval of 2. The dashed blue reference remains near 1 from 0 to 20 seconds, then drops to near 0 from 20 to 30 seconds, and stays close to 0 afterward. The orange curve shows initial oscillations near 0, then remains close to the reference with small fluctuations. The green curve shows sharp spikes at the beginning and around 30 seconds, reaching values above 4, and then quickly returns toward the reference level. Across all four graphs, the three curves are closely aligned after initial transients, with differences mainly visible near the beginning and at step transitions around 10, 20, and 30 seconds. Note: All numerical data values are approximated.Point stabilization performance of the DK-MPC (green) compared to nonlinear MPC (orange). The plots show the evolution of position (x, y and z) and roll angle (ϕ) as the quadrotor tracks stepwise changes (blue dashed) in the target position and orientation