Mathematical Problems in Engineering

Research Article

Using the Bezier Curve and Particle Swarm Optimization in Trajectory Planning for Overhead Cranes to Suppress the Payloads’ Residual Swing

Table 1

The control points of the optimal Bezier curve.
(m) (rad)Control points of the optimal Bezier curve ()
0.6-0.087(0, 0, 0, 0.065, 0, 0.055, 0.109, 0.283, 0.600, 0.461, 0.600, 0.739, 1.155, 0.045, 0.177, 0.322, 0.467, 0.626, 0.6, 0.6, 0.6)
1-0.087(0, 0, 0.091, -0.223, -0.018, 0.278, 0.574, 0.96, 0.449, 0.66, 0.79, 0.920, 0.970, 0.754, 0.907, 0.808, 0.710, 0.361, 0.875, 1, 1)
1.4-0.087(0, 0.005, 0, 0, 0.281, 0.440, 0.599, 0.635, 1.400, 1.400, 1.400, 1.400, 1.400, 1.400, 1.400, 1.400, 1.400, 1.400, 1.4, 1.4, 1.4)
(0, 0.009, 0, 0, 0.096, 0.516, 0.935, 1.678, 0.924, 1.363, 1.589, 1.814, 1.826, 1.800, 1.799, 1.800, 1.801, 1.803, 1.8, 1.8, 1.8)
1.8-0.087
2.2-0.087(0, 0, 0, 0, 0.016, 0.424, 0.832, 1.634, 1.354, 1.588, 1.711, 1.835, 1.850, 2.200, 2.200, 2.194, 2.187, 2.175, 2.2, 2.2, 2.2)
2.6-0.087(0, 0, 0, 0, 0, 0.332, 0.663, 1.326, 1.313, 1.368, 1.578, 1.788, 2.153, 2.367, 2.104, 2.171, 2.237, 2.634, 2.6, 2.6, 2.6)
3-0.087(0, 0, 0, 0, 0.239, 0.746, 1.253, 2.028, 1.371, 1.824, 2.177, 2.530, 2.782, 2.926, 2.787, 2.809, 2.830, 3.014, 3, 3, 3)
3.4-0.087(0, 0.3, 0, 0.877, 1.004, 1.509, 2.013, 2.895, 0.661, 3.171, 3.400, 3.629, 1.578, 3.4, 3.4, 3.4, 3.4, 3.4, 3.4, 3.4, 3.4)
3.8-0.087(0, 0, 0.001, 0, 0.145, 0.515, 0.886, 1.483, 1.844, 1.764, 2.103, 2.441, 3.199, 3.304, 2.975, 3.114, 3.253, 3.861, 3.8, 3.8, 3.8)
4-0.087(0, 0, 0, 0.886, 0.927, 1.055, 1.183, 1.397, 1.713, 2.040, 2.468, 2.897, 3.428, 3.196, 2.839, 2.874, 2.909, 3.337, 4, 4, 4)
0.60(0, 0, 0.132, 0, 0.108, 0.162, 0.217, 0.217, 0.331, 0.381, 0.426, 0.471, 0.511, 0.581, 0.579, 0.600, 0.621, 0.664, 0.6, 0.6, 0.6)
10(0, 0.056, 0.113, 0.167, 0.227, 0.29, 0.353, 0.419, 0.518, 0.531, 0.6, 0.671, 0.799, 0.715, 0.828, 0.867, 0.907, 0.874, 1, 1, 1)
1.40(0, 0, 0, 0.026, 0, 0.048, 0.097, 0.220, 0.257, 0.413, 0.494, 0.574, 0.579, 0.983, 0.981, 1.040, 1.099, 1.219, 1.4, 1.4, 1.4)
1.80(0, 0.13, 0.271, 0.444, 0.541, 0.688, 0.835, 1.033, 1, 1.201, 1.314, 1.426, 1.47, 1.644, 1.647, 1.678, 1.7, 1.768, 1.8, 1.8, 1.8)
2.20(0, 0.24, 0.446, 0.687, 1.03, 1.152, 1.272, 1.165, 2.2, 1.743, 1.863, 1.983, 2.679, 1.74, 2.08, 2.187, 2.29, 2.168, 2.2, 2.2, 2.2)
2.60(0, 0.19, 0.38, 0.599, 0.756, 0.96, 1.168, 1.423, 1.523, 1.705, 1.855, 2, 2.123, 2.308, 2.354, 2.41, 2.469, 2.538, 2.6, 2.6, 2.6)
30(0, 0.411, 0.832, 1.149, 1.503, 1.787, 2.071, 2.286, 2.636, 2.637, 2.750, 2.862, 3.087, 2.998, 3, 3, 3, 2.998, 3, 3, 3)
3.40(0, 0.359, 0.717, 1.047, 1.457, 1.68, 1.905, 1.94, 3.185, 2.642, 2.83, 3.02, 3.942, 2.58, 3.235, 3.4, 3.565, 3.239, 3.4, 3.4, 3.4)
3.80(0, 0.466, 1.07, 0.688, 1.887, 2.15, 2.418, 1.749, 3.754, 3.04, 3.295, 3.546, 4.759, 2.714, 3.543, 3.78, 4, 3.667, 3.8, 3.8, 3.8)
40(0, 0.393, 0.718, 1.23, 1.498, 1.907, 2.316, 2.866, 2.647, 3.142, 3.399, 3.657, 3.677, 4, 3.930, 3.999, 4.069, 4.277, 4, 4, 4)