Laboratory experiments were conducted for falling U-chain,but explicit analytic form of the general equations of motion was not presented.Several modeling methods were developed for fish robots,however they just focused on the whole fish’s locomotion which does little favor to understand the detailed swimming behavior of fish.Udwadia-Kalaba theory is used to model these two multi-body systems and obtain explicit analytic equations of motion.For falling U-chain,the mass matrix is non-singular.Second-order constraints are used to get the constraint force and equations of motion and the numerical simulation is conducted.Simulation results show that the chain tip falls faster than the freely falling body.For fish robot,two-joint Carangiform fish robot is focused on.Quasi-steady wing theory is used to approximately calculate fluid lift force acting on the caudal fin.Based on the obtained explicit analytic equations of motion(the mass matrix is singular),propulsive characteristics of each part of the fish robot are obtained.Through these two cases of U chain and fish robot,how to use Udwadia-Kalaba equation to obtain the dynamical model is shown and the modeling methodology for multi-body systems is presented.It is also shown that Udwadia-Kalaba theory is applicable to systems whether or not their mass matrices are singular.In the whole process of applying Udwadia-Kalaba equation,Lagrangian multipliers and quasi-coordinates are not used.Udwadia-Kalaba theory is creatively applied to dynamical modeling of falling U-chain and fish robot problems and explicit analytic equations of motion are obtained.
