This paper introduces a method to solve the crane motion trajectory control problem. A dynamic model is proposed based on the Udwadia–Kalaba equation, which can be solved without extra parameters, such as the Lagrange multiplier. The motion trajectory of a crane is used as a constraint (referred to as trajectory tracking constraint). To satisfy the system trajectory, a method to calculate the driving conditions on the basis of the above conditions is proposed. A 2D plane dynamic model of a crane is established. Five stages of crane movement are obtained. Simulation is performed with Matlab. Simulation results simulation show that the Udwadia–Kalaba equation can be well applied to trajectory tracking control of cranes.
MooreEH. On the reciprocal of the general algebraic matrix. Bull Am Math Soc1920; 26: 294–395.
2.
Abdel-RahmanEMNayfehAHMasoudZN. Dynamics and control of cranes: A review. J Vib Control2003; 9: 863–908.
3.
BartoliniGPisanoAUsaiE. Second-order sliding-mode control of container cranes. Automatica2002; 38: 1783–1790.
4.
ChangCChiangK. Fuzzy projection control law and its application to the overhead crane. Mechatronics2008; 18: 607–615.
5.
Rodriguez FO, Yu W and Moreno-Armendariz MA. Anti-swing control for overhead cranes with neural compensation. In: Proceedings of the international joint conference on neural networks, 2006, vol. 18, pp.4697–4703.
6.
UdwadiaFEKalabaRE. On motion. J Franklin Inst1993; 330: 571–577.
7.
UdwadiaFEKalabaRE. A new perspective on constrained motion. Math Phys Sci1992; 439: 407–410.
8.
UdwadiaFEKalabaRE. On the foundations of analytical dynamics. Int J Non Linear Mech2002; 37: 1079–1090.
9.
UdwadiaFEKalabaRE. Analytical dynamics: A new approach, Cambridge: Cambridge University Press, 1996.
UdwadiaFEKalabaRE. Explicit equations of motion for mechanical systems with non-ideal constraints. Appl Mech2001; 68: 462–467.
13.
UdwadiaFEKalabaRE. On constrained motion. Appl Math Comput2005; 164: 313–320.
14.
UdwadiaFEPhohomsiriP. Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proc Royal Soc2006; 462: 2097–2117.
15.
Ye-Hwa C. Mechanical systems under scrvo constraints:the Lagrange's approach. Mechatronics2005; 15: 317–337.
16.
ChenYH. Equations of motion of mechanical systems: Given force depends on constraint force. Mechatronics1999; 9: 411–428.
17.
ChenYH. Equations of motion of mechanical systems under scrvo constraints: the Maggi approach. Mechatronics2008; 18: 208–217.
18.
ShengchaoZHanZKangHet al.On Kepler's law: application of the Udwadia-Kalaba theory. Scientia Sinica2014; 44: 24–31–24–31.
19.
ZhaoHZhenSCChenYH. Dynamic modeling and simulation of multi-body systems using the Udwadia–Kalaba theory. Chin J Mech Eng2013; 26: 839–850.
ZhangBZZhenSCZhaoHet al.A novel study on Kepler’s law and inverse square law of gravitation. Eur J Phys2015; 36: 035018–035018.
22.
ZhenSCHuangKZhaoHet al.Why can a free falling cat always manage to land safely on its feet?Nonlinear Dyn2015; 79: 2237–2250.
23.
ZhaoHZhenSCChenYH. Dynamical modeling and simulation of multi-body systems by using Udwadia-Kalaba theory. Chin J Mech Eng2013; 26: 839–850.
24.
Peng-ChengWYong-ChunFZi-YaJ. A direct swing constraint-based trajectory planning method for underactuated overhead cranes. Acta Automat Sini2014; 40: 2414–2419.
25.
PengKCCSinghoseWFrakesDH. Hand-motion crane control using radio-frequency real-time location systems. IEEE/ASME Transac Mech2012; 17: 464–471.
26.
Burg T, Dawson D, Rahn C, et al. Nonlinear control of an overhead crane via the saturating control approach of Teel. In: Proceedings of the 1996 IEEE international conference on robotics and automation, 22–28 April 1996, pp.3155–3160.
