Figure 10
The composite layout is arranged in two rows. The top row contains four three-dimensional trajectory plots labeled “(a)”, “(b)”, “(c)”, and “(d)”. Each 3 D plot has the same axes: the bottom left axis is labeled “X (meters)” and ranges from negative 5 to 5 with an interval of 5; the bottom right axis is labeled “Y (meters)” and ranges from negative 5 to 0 with an interval of 5; and the vertical axis on the left is labeled “Z (meters)” and ranges from 0 to 12 with an interval of 2. In all 3 D plots, a dashed red curve forms a smooth upward helical spiral, looping around a central axis while rising vertically. A solid blue curve follows a similar path. In panel (a), the blue trajectory starts below the spiral and slightly offset outward, then curves inward and upward toward the lower turns of the spiral. The quadrotor is positioned near the lower section, below the main spiral loop. In panel (b), the blue curve bends smoothly and attaches to the spiral path along a mid-level segment, following the curvature of one loop. The quadrotor is positioned along this mid-height section on the left where the blue and red curves nearly coincide. In panel (c), both curves trace a higher portion of the spiral, following a curved arc that wraps around the central axis with minimal separation. The quadrotor is located along this upper-middle segment, aligned with the direction of the spiral. In panel (d), the blue curve closely overlaps the red spiral along the upper loops, continuing the circular upward pattern. The quadrotor is positioned near the top of the spiral, where the path flattens slightly before continuing upward. Across all four panels, the blue trajectory shows a progression from an initial offset path to full alignment with the red helical spiral, maintaining the same circular looping pattern around the vertical axis. The bottom row contains four two-dimensional plots aligned with the panels above and labeled collectively as “(e)”. Each plot has a horizontal axis labeled “Sample” ranging from 0 to 1000 with an interval of 500. Each plot includes two curves mentioned in a legend identifying the dashed red curve as “Reference” and the solid blue curve as “D K-M P C”. In the first plot, the vertical axis is labeled “x (meters)” and ranges from negative 5 to 10 with an interval of 5. The dashed red and solid blue curves follow an oscillatory pattern, starting near about 5 at sample 0, decreasing to around negative 4 near sample 400, rising to about 5 near sample 800, and ending near 0. The blue curve closely follows the red curve with small deviations near the start. In the second plot, the vertical axis is labeled “y (meters)” and ranges from negative 4 to 4 with an interval of 2. The curves form a sinusoidal pattern, starting near negative 2, rising to about 3 near sample 200, dropping to around negative 4 near sample 600, and increasing again to about 3 near sample 1000. The blue curve closely tracks the red dashed curve and lies slightly above at the peaks. In the third plot, the vertical axis is labeled “z (meters)” and ranges from 0 to 10 with an interval of 2. Both curves show a steady increasing trend, starting near about 1 at sample 0 and rising to around 10 at sample 1000, with the blue curve almost completely overlapping the red curve. In the fourth plot, the vertical axis is labeled “phi (radian)” and ranges from negative 0.2 to 0.2 with an interval of 0.1. The curves exhibit periodic oscillations, starting near 0, rising to about 0.25, dropping to around negative 0.2 near sample 500, and rising again toward about 0.25 by sample 1000. The blue curve closely follows the red dashed curve throughout. Note: All numerical data values are approximated.Helical trajectory following of the quadrotor using the proposed DK-MPC framework with prediction horizon H = 50. The red dashed line indicates the reference trajectory, while the blue line represents the actual quadrotor path. Snapshots (a)–(d) illustrate different stages of the helical maneuver. The controller achieves accurate real-time tracking with an average computation cost of 15 ms per control step. Plots in (e) shows the corresponding trajectory tracking in x, y, z and/phi