CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 26, No. 5, 2013 ·839· DOI: 10.3901/CJME.2013.05.839, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn Dynamic Modeling and Simulation of Multi-body Systems Using the Udwadia-Kalaba Theory ZHAO Han1, ZHEN Shengchao1, 2, *, and CHEN Ye-Hwa2 1 School of Mechanical Engineering, Hefei University of Technology, Hefei 230009, China 2 School of Mechanical Engineering, Georgia Institute of Technology, Atlanta 30332, USA Received December 20, 2012; revised March 20, 2013; accepted March 27, 2013 Abstract: 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. Key words: Udwadia-Kalaba equation, multi-body systems, falling U-chain, fish robot developed equations of constrained motion when the 1 Introduction∗ constraints satisfy D’ Alember’ s principle. In the treatise [5] on the analytical dynamics, PARS refers to the The general problem of obtaining equations of motion Gibbs-Appell equations as “probably the most for constrained discrete mechanical systems has been an comprehensive equations of motion so far discovered”. But area of considerable interest among scientists and engineers. the Gibbs-Appell equations require a “lucky” choice of It is also one of the central issues in multi-body dynamics. problem-specific quasi-coordinates and they suffer from The problem has been aggressively and continuously similar problems when dealing with systems with a large pursued by many scientists, engineers and mathematicians number of degrees of freedom and many non- integrable [6] since constrained motion was initially described by constraints. DIRAC used Poisson brackets, a recursive LAGRANGE[1]. He invented the special Lagrange scheme for determining the Lagrange multipliers, for multiplier method to deal with constrained motion. singular Hamiltonian systems where the constraints do not However, the Lagrange multiplier method relies on exactly depend on time. [7–8] problem-specific approaches to determine the multipliers; it On the other hand, UDWADIA, et al , obtained a is often very difficult to find the multipliers to obtain the concise, explicit set of equations of motion for constrained explicit equations of motion for systems with large discrete dynamic systems which lead to a simple and new numbers of degrees of freedom and a mass of fundamental view of Lagrangian mechanics. They derived non-integrable constraints. GAUSS[2] introduced a new the fundamental equation of motion that describes the general principle of mechanics for handling constrained dynamics of constrained systems from Gauss’ s principle motion. Gauss’s Principle gives a clear description of the which seems somewhat less popular than the principles of general nature of constrained motion through minimization Lagrange, Hamilton, Gibbs and Appell. The equations can of a function of the accelerations of the particles of a deal with holonomic and also non-holonomic constraints. [9–10] system. GIBBS[3] and APPELL[4] have independently UDWADIA, et al , observed that all the research above has used D’ Alember’ s principle as their starting point. D’ Alember’ s principle assumes that the forces of constraints * Corresponding author. E-mail: [email protected] are considered to be ideal and the total work done by the © Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2013 ZHAO Han, et al: Dynamic Modeling and Simulation of Multi-body Systems ·840· Using the Udwadia-Kalaba Theory forces of constraints under virtual displacement is always Boltzmann, we are not required to develop a system of zero. This assumption works well in many situations and is proper quasi-coordinates. regarded as the core of classical analytical dynamics, but it The significant advantages of Udwadia and Kalaba’ s is not applicable when the constraints are nonideal. Thus, new theory are threefold. First, it allows descriptions of UDWADIA and KALABA generalized their previous mechanical systems that use more than the minimum equations to constrained mechanical systems that may not number of required coordinates which speeds and eases the satisfy D’ Alember’ s principle. Systems with singular setup of the equations of motion of complex systems. mass matrices are not common in classical dynamics when Second, it applies to systems whether or not their mass dealing with unconstrained motion. PARS[5] proposed that matrices are singular, is applicable to systems with when the minimum number of coordinates is employed for holonomic and/or nonholonomic constraints and to systems describing the unconstrained motion of mechanical systems, whose constraint forces may or may not be ideal. Third, the the corresponding Lagrange equations usually yield new theory opens up a new way of modeling complex non-singular, symmetric and positive definite mass multi-body systems. It permits decomposition of such matrices. However, singular mass matrices can arise when systems into sub-systems whose equations of motion are one wants greater flexibility in modeling complex known, and then combine the sub-system equations to get mechanical systems by using more than the minimum the composite system’ s equations of motion in a number of required generalized coordinates. Thus, straightforward and simple manner. UDWADIA, et al[11], developed general and explicit equations of motion to handle systems whether or not their 2 Explicit General Equations of Motion mass matrices are singular. We call these discoveries the for Multi-body System Udwadia-Kalaba theory. About 20 years has passed, we seldom realize its importance only because it is abstract. In Using Udwadia and Kalaba’ s approach, we first fact, the Udwadia-Kalaba theory can be applied to consider an unconstrained discrete dynamical system constrained discrete mechanical systems with unmatched whose configuration is described by the n generalized ease, clarity and elegance, especially to multi-body systems T coordinates q : [q12 ,q , ,qn ]. Its equation of motion such as mechanical manipulators, space vehicles, robots can be obtained, using Newtonian or Lagrangian mechanics, and similar systems. by the relation: As we know, the principles of mechanics[12–16] are so perfect that it is impossible to create a totally new Mq(, tt ) q Qq (, q , ), (1) fundamental principal for the theory of motion and nn equilibrium of discrete, dynamical systems. However, an where Mq( , t ) R is symmetric and positive definite n n additional perspective has been proposed by Udwadia and inertia matrix, q R is the velocity and q R is the n Kalaba which is useful to help us understand nature’ s law acceleration, and Qq(, q , t ) R is the force imposed on from new points of view[7–11]. the system whose constraints are released. The imposed Using Udwadia and Kalaba’ s marvelous theory, we can forces could include centrifugal force, gravitational force formulate the explicit, general equations of motion for and control input. The generalized acceleration of the constrained discrete dynamic systems in three steps：First, unconstrained system, which we denote by aq(, q , t ) is in terms of the generalized coordinates, we consider the thus given by unconstrained discrete dynamic system whose equations of 1 motion can be written using Newtonian or Lagrangian q M(, qtt ) Qqq (, , ) aqq (, , t ). (2) mechanics. Then we form the constraint equations which Second, constraints present in the system should be could include the usual holonomic, non-holonomic, considered. We shall assume that the system is subjected to scleronomic, rheonomic, catastatic and acataetatic variaties h holonomic constraints and mh nonholonomic of constraints or even combinations of such constraints. constraints of the form: Finally, we impose the additional generalized forces of constraint on the system. That is to say, we add the force ϕi (q , ti ) 0, 1, 2, , h , (3) resulting from the presence of constraints to the unconstrained system’ s force. ϕi (qq , , t ) 0,