Figure 8
Four vertically stacked graphs are shown, each with a horizontal axis labeled “Time (seconds)” ranging from 0 to 50 with an interval of 10. A legend in the top graph identifies three curves: “Reference” shown as a dashed blue line, “Nonlinear M P C” shown as a solid orange line, and “D K M P C” shown as a solid green line. 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 curve begins near 0.5, increases gradually toward about 0.9 around the middle, and then slightly declines toward about 0.4 by the end. The green curve rises smoothly and closely follows the dashed blue curve. The orange curve shows a sharp spike above 2.3 near the start, then settles and follows the same general path with small oscillations. In the second graph, the vertical axis is labeled “y (meters)” and ranges from 0 to 1.5 with an interval of 0.5. The dashed blue curve rises from near 0 to about 0.8 around 20 seconds, then drops toward about 0.3 near 30 seconds, and rises again toward about 0.7. The green curve follows the same overall pattern as the dashed blue curve with smoother transitions. The orange curve shows an early peak above 1.1 and then fluctuates around the dashed blue curve with visible oscillations. In the third graph, the vertical axis is labeled “z (meters)” and ranges from 0 to 4 with an interval of 1. The dashed blue curve increases gradually from about 2.1 to around 3.2 over time. The green curve closely follows the dashed blue curve with minimal variation. The orange curve rises sharply from near 0 to above 2.7 early on, then aligns with the gradual upward trend. In the bottom graph, the vertical axis is labeled “phi (radian)” and ranges from negative 2 to 4 with an interval of 2. The dashed blue curve forms a repeating oscillatory pattern with multiple peaks above 2 and dips below 0. The orange curve remains near 0 with small fluctuations in the early stage. The green curve follows the oscillatory pattern closely, matching the peaks and troughs in position and shape. All four graphs show three overlapping curves with distinct behaviors at the beginning and smoother alignment over the remaining time range. Note: All numerical data values are approximated.Comparison of control performance between the proposed DK-MPC (green) and a standard nonlinear MPC (orange) in a trajectory tracking task (blue dashed)