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, ih 1, h 2, , m . (4) The advantage of determining the explicit equations of motion for constrained discrete dynamical system through We can differentiate standard constraint equations in this new theory is obvious. Compared to the Lagrange Lagrangian mechanics which are usually in pfaffian form. multiplier method, there is no need for this new approach to Under the assumption of sufficient smoothness, we take the determine the multiplier which is often very difficult to time derivative once on nonholonomic constraints and obtain for systems having a large number of degrees of freedom and many non-integrable constraints. Unlike the twice on holonomic constraints, to derive the constraint formulations offered by Gibbs, Volterra, Appell and equations of the matrix form:
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·841·
UDWADIA and KALABA have proved that the ideal Aqq(, , tt ) q bqq (, , ), (5) constraint force takes the form where Aq(, q , t )is referred to as constraint matrix and c 1/2 1 Qid M B( b AM Q ), (11) bq(, q , t ) is a m-vector. The final step is to form the explicit equations of motion and the non-ideal constraint force takes the form with constraints. Due to the presence of constraints, c 1/2 + 1/2 additional “generalized forces of constraints” should be Qnid M( I B BM ), c (12) imposed on the system. So, the actual explicit equation of motion of the constrained system could be assumed to take where B AM 1/2 and the superscript “” denotes the the form: Moore-Penrose generalized inverse [17–18]. From Eqs. (6), (7), (11), and (12), the explicit equation c Mq(, t ) q Qqq (, , tt ) Qqq (, , ), (6) of motion that governs the evolution of the constrained system (including both ideal and non-ideal constraints) is where Qq(, q , t ), arising by virtue of the holonomic and nonholonomic constraints, is the additional term of forces Mq Q+( M1/2 B + b AM1 Q )+(M1/2 I B + BM ) 1/2 c , imposed on the system. In Lagrangian mechanics, (13) c Qqq(, ,t ) is considered to be ideal and it is governed by D’ Alember’ s Principle which indicates that forces of where vector c is determined by the mechanician and could constraint do zero work under virtual displacements. be obtained by experimentation and/or observation. However, the constraints can also be non-ideal. Ideal When c is always zero, Eq. (13) reduces to D’ constraints generate ideal constraint forces subject to D’ Alember’ s Principle, which means the total work done Alember’ s Principle, while non-ideal constraints generate under virtual displacement is zero, the constraints are ideal, non-ideal constraint forces such as friction force, the constraint force is electro-magnetic force, etc. If there exist both ideal and non-ideal constraints in the system, Udwadia put c c 1/2 + 1 Q Qid M B( b AM Q ), (14) Qqqc (, , t ) in the form of
and the explicit equation of motion of the constrained Qqqccc(, , tt ) Q (, qq , ) Q (, qq , t ), (7) id nid system (only including ideal constraints) is
c where Qid (, qq ,t )is the ideal constraint force and 1/2 + 1 c Mq Q M B( b AM Q ). (15) Qnid (, qq , t ) is the non-ideal constraint force. Udwadia generalizes D’ Alember’ s Principle to include forces of constraint that may do positive, negative, or zero Thus, at each instant of time t, the constrained system is c work under virtual displacement at any instant time during subjected to an additional constraint force F ()t given by the motion of the constrained system. That is, he extends c 1/2 + 1 Lagrange’ s form of D’ Alember’ s Principle to include F(t ) M B ( b AM Q ). (16) non-ideal constraints. We denote constraint force as cq(, q , t ) Rn . The constraint force does the work When matrix M is a constant diagonal matrix so that W vcT (the same as the work done by Qqqc (, , t )in M mI, then Eq. (16) simplifies to any displacement v which is subject to Aq( , q , t ) v 0. So we write these equations in the form of Fc+(t ) MA ( b AM 1 Q ). (17)
Tc T W vQ vc, (8) When MQ1 is zero, Eq. (17) becomes which is the extended Lagrange’ s form of D’ Alembert’ s Fc+()t MA b . (18) Principle (We henceforth omit the arguments of functions where there is no confusion in our notations). The work For Eq. (13), if the mass matrix M is singular, we should c [11] done by the ideal constraint force Qid under virtual use the Udwadia-Phohomsiri equation instead of displacements is Udwadia-Kalaba equation[7–10] to get the equation of motion for the constrained system. Hence, the equation of Tc vQid 0, (9) motion is while the work done by non-ideal constraint force is + + Q I AAM q . (19) Tc vQnid 0. (10) A b
ZHAO Han, et al: Dynamic Modeling and Simulation of Multi-body Systems ·842· Using the Udwadia-Kalaba Theory
The above equation is valid when the matrix dependence of the velocity and the acceleration of the chain T [11] MA| has full rank . This full rank condition can serve tip for several initial conformations of the chain and also as a check on whether we have obtained a correct model, give us a simple analytical model of the system. because it is also the condition required for the equation of As seen above, research on falling U-chain problem has motion of the constrained system to be unique. seemingly been conducted well for years. However, they all Remark. UDWADIA and KALABA have provided focus their attention on whether the system is energy explicit general equations of motion for constrained conservative or non-conservative, results of laboratory discrete dynamical systems with their newly initiated experiments and numerical simulations based on a simple approach which is applicable to all holonomical and non- analytical model and arguments with each other. None of holonomical constrained systems no matter whether they them attempt to present the explicit analytic form of the satisfy D’ Alember’ s Principle. They generalize Lagrangian mechanics to include both ideal and non-ideal general equations of motion for the falling U-chain system. constraint forces by using a new fundamental principle In this paper, we creatively apply the Udwadia-Kalaba governing the motion of constrained systems. The equation theory to solve the falling U-chain problem and to get its of motion obtained by using Udwadia and Kalaba’ s theory explicit analytic form of the general equations of motion. is general, simple and understandable. 4 Analytical-form Equations of Motion 3 Case 1: Falling U-chain of the Falling U-chain
The falling U-chain (Fig. 1) is a folded flexible heavy We apply Udwadia and Kalaba’ s theory to the falling chain with one end suspended from a rigid support and the U-chain problem to formulate the analytical-form equation other end lifted up to form a U-shaped fold at the bottom, of motion. We do not focus on the inclined angle of each then the lifted end is released to move down in the manner link. Therefore, to simplify, we assume the U-chain [19] of a bungee fall . The whole chain is in the gravitational consists of n particles which are connected with n–1 field while falling. This U-chain case is actually so old that massless links. The length of each link is 1, and the mass of it has been studied frequently by many dissertations[20–25]. each particle is m . Through this simplification, we can We are interested in it because it is an ideal constrained i describe the position of each particle with two degrees of dynamic system. Also, compared to the bottom-pile chain freedom, thus it will make the process of obtaining the and top-pile chain, the U-chain is conceptually easier to model because there are no actual collisions, no made or analytical-form equation of motion easier and speed up the broken contacts and thus no explicit collisions even in a simulation. The chain is constrained to move only in the discrete chain. This model provides ease and reliability for vertical plane with each particle denoted by (xi, yi), i1, 2, numerical simulation[19]. , n. All links are considered to be rigid and cannot be deformed. Consecutive particles are connected by massless links regardless of the friction, since the friction in the falling U-chain system is small and makes no difference to the performance of the dynamic system.
4.1 Falling U-chain: unconstrained equation of motion If there are no constraints placed on the links, connections between links are removed and the last link is Fig. 1. Falling U-chain not attached to the fixed support. The unconstrained U-chain’ s equation of motion can be obtained easily as WONG, et al[19], have given us a detailed and critical review of the history of falling U-chain problems which Mq Q( q , q , ti ), 1, 2, , n , (20) mainly include some erroneous approaches. They identify the source of error and propose a fool-proof Lagrangian m1 000 00 approach to the falling U-chain problem. They imply that 0m1 00 00 [23] the method of SOUSA, et al , is not reliable because its 00m2 0 00 solution is an erroneous for the falling U-chain and they also conclude that Lagrange’ s method gives definitive M 0 0 0 m2 0 0, answers which are better and more easily developed. 00 [24] TOMASZEWSKI, et al , analyze the dynamics of the tip 00 0 00mn 0 of the falling U-chain with laboratory experiments and numerical simulations. They have determined the time 00 0 000mn
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·843·
x 0 1 2xx−−x 22x +2yy −−y 22y 12 1 2 12 1 2 ymg 22 22 11 2xx−−x x +2yy −−y y 23 2 3 23 2 3 22 22 x2 0 b 2xx−−x x +2yy −−y y . (27) 34 3 4 34 3 4 qQym22, g , (21) 2xxx−2 −x 2+2yyy −2 −−−yxy 222 nn11 n n nn 1 n 1 n n n xn 0 ymnn g 4.3 Udwadia-Kalaba equation Considering Eq. (15), detailed Udwadia and Kalaba where m1, m2, , mn are the mass of each link, x1, x2, , eqution of the whole U-chain can be rewritten as xn, y1, y2, , yn are coordinates, Q is the external force.
Mq Q M1/2( AM−− 1/2 ) + ( b− AM1 Q ), (28) 4.2 Falling U-chain: constraint equation For i 1, 2,, n1, the constraint between i and i1 is where MqQ, , can be found in equation (21), A in equation 2 22 (xxyyi+11 i ) +( i+ i ) l . (22) (26) and b in Eq. (27). This is the explicit analytical-form equation of motion for the whole falling U-chain. Taking the time derivative twice, we get To summarize, we can derive the analytical-form equation of motion of the whole U-chain: form the unconstrained equation of motion at first, then write the (xi x i+i1 )+( x y i y i+i1 )+( y x i+ 1 xx ii )+( +1 y i+ 1 yy ii ) +1 constraint equation so that the additional imposed y2 y 2+2yy x 22x +2xx (23) i i+1 i+ 1 i i i+1 i+ 1 i constrained force can be derived, and finally add the constrained force to the equation of motion of the For in, there is constraint from the n–1 link and the other unconstrained system. constraint is
x222 +y l . (24) 5 Numerical Simulation of the Falling nn U-chain and analysis
Taking the time derivative twice, we get Through simulations, insights into the dynamics of the falling U-chain can be developed. To simplify, we assume 22 xnn x +y nn yx n y n. (25) li 1, g9.8, mi 1 for i1, 2, , n and n17. We choose n17 through trial and analysis. n17 is large enough to
simulate the dynamics and can get good results. Since we Considering all the constraints which have been know the whole U-chain’ s unconstrained equation of described above, the whole U-chain’s constraint equation motion and the constraint equation, we now know how to Aq b can be derived as follows: calculate the constrained force, and we can simulate the falling U-chain system in Matlab. We select the ode15i A algorithm to implement the numerical simulation. It is a variable order method that can solve fully implicit xxyyxxy121221 21 y 0000 differential equations. 00xxy23233232 yxxyy 00 0000xxyyxxyy We give five different initial separations xD between two 34344343 ends of the chain: (a) x012, (b) x013, (c) x014 (d) x015, (e) x016. 00000000 Table 1 provides the maximum vertical fall distance h max 00000000 of the falling chain tip and the time it takes of each case, also, it provides the vertical fall distance h of the freely falling 0000 body. From Table 1, we can determine if two ends of the 0000 chain get closer, the maximum vertical fall distance of the 0000 chain tip is reduced, and the time it takes gets less. In cases , (26) (a), (b), (c), (d), (e), the vertical fall distance of the chain tip, x−−−− xy yxx yy up to the time tmax at which the vertical fall distance of the n-1 n n-1 nnn-1 nn-1 chain tip reaches its maximum value hmax, is seen to always 00xynn be ahead of the vertical fall distance h of the freely falling
ZHAO Han, et al: Dynamic Modeling and Simulation of Multi-body Systems ·844· Using the Udwadia-Kalaba Theory body. This can be summarized by the general statement that the chain falls faster than a freely falling body.
Table 1. Chain tip’s five cases
Initial Maximum fall Corresponding Fall Case separation distance time distance
x0mm hmaxmm tmaxmm hmm a 12 15.943 1.635 13.099 b 13 15.990 1.685 13.912 c 14 16.070 1.749 14.989 d 15 16.323 1.802 15.911 Fig. 3. Vertical force Fy1 on the chain tip of case (b) e 16 16.671 1.842 16.625
In the simulation, we also find that the vertical distance of a freely falling ball is always less than that of the falling
U-chain tip before the time tmax in all the five cases which reconfirm the conclusion above. To further understand this behavior, we analyze the time dependence of the vertical acceleration ay1 of the chain tip (horizontal acceleration is not so important for the analysis).
Downward is the positive direction and Fy1 denotes the vertical force imposed on the falling chain tip. We know m1, so Fy1 is numerically equal to ay1. We performed Fig. 4. Vertical force Fy1 on the chain tip of case (c) simulations also with five initial separations between the ends of the chain (a) x012, (b) x013, (c) x014, (d) x0
15, (e) x016. Figs 2, 3, 4, 5, 6 show that as the initial separation between the chain ends is reduced, the time of the chain tip’s vertical acceleration larger than g (vertical acceleration of the freely falling body, equal to 9.8) gets more, and the value(about from 10 to 25) of chain tip’s vertical acceleration ay1 larger than g generally gets bigger. In other words, as the initial separation between chain ends is reduced, Fy1 and ay1 grow and the vertical velocity of the falling chain tip grows. Fig. 5. Vertical force F on the chain tip of case (d) Furthermore, two other characteristic features in Figs. 2 y1 –6 are the negative peak heights of the vertical acceleration and the time at which these maxima occur. As the initial separation lessens, the negative peak heights lessen and the time lessens. Next, we examine the values of the negative peak heights of the vertical acceleration of the falling chain tip as shown in Figs. 2–6. They are all large between –200 and –350 (far more than g). This means the tip is subjected to a great pulling force when hitting the lowest position.
Fig. 6. Vertical force Fy1 on the chain tip of case (e)
6 Case 2: Fish Robot
There are over 28 000 species of fish and a wide variety of propulsive systems used by fish for maneuvering in the aquatic environment. Fish have numerous fins which act to transfer momentum to the surrounding water and shed reverse Karman vortex street in the wake. The undulating Fig. 2. Vertical force Fy1 on the chain tip of case (a) motion of fish can generate forward thrust as well as
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·845· receive drag from the water[26]. They exhibit amazing waters, while anguilliform fish exhibit remarkable swimming capabilities: high propulsive efficiency, maneuverability. We focus on the robot model of extraordinary maneuverability, long duration and station carangiform fish which typically have large, high-aspect- keeping ability[27]. Fish’ s excellent locomotion ability ratio tails. They swim primarily using the rear and tail, under water has stimulated growing interests in biomimetic while the front body remains moving forward and robot fish. Many researchers have attempted to study the swinging. mechanism of fish swimming and have developed We can idealize the main body of the fish as a rigid underwater fish robots which can be prospectively applied body. The body is connected to the tail by a peduncle (a in military detection, undersea operation, reconnaissance slender region of generally negligible hydrodynamic de-mining and so on[28]. influence). Fish tails are usually flexible, but the caudal fins We review briefly related research work on the fish’ s of carangiform fish are very stiff. So, we treat the tail as a swimming mechanism. Most of the early research focused rigid lifting surface. For simplicity, we analyze the on building hydrodynamic models. TAYLOR[29] employed motion of the robotic fish only in two dimensions of the steady-state flow theory to calculate the fluid force. horizontal plane. The fish-like mechanism can be modeled LIGHTHILL[30] developed the elongated-body theory to as a three link system as shown in Fig. 7. study the swimming of slender fish. A two-dimensional waving plate theory was originally proposed by WU[31]. He advised to study the swimming fish as an elastic plate. Thereafter, a large-amplitude elongated-body theory was developed to analyze rapid acceleration and steady swimming. An overview of fish swimming modes for aquatic locomotion was presented by SFAKIOTAKIS, et al[32]. TRIANTAFYLLOU, et al[33], initiated significant work to develop an artificial robot fish. They developed an eight-link, foil-flapping robotic fish mechanism(RoboTuna) Fig. 7. Fish-like mechanism model and investigated drag forces experimentally. Subsequently, more fish robots have been developed, e.g., the well-known To conduct a hydrodynamic analysis in a relatively Mitsubishi robotic fish, the lamprey robot, the ISRobotics simple way, we model the three-link fish robot using Ariel Robot, the robotic Blackbass, etc. carangiform propulsion in which the fish oscillates the As seen above, research on swimming fish has caudal fin and peduncle. The drag acts on all the three seemingly been conducted well for years. But they all focus links, but mainly on the body. We assume the flow is on the whole fish’ s locomotion which does little to aid in inviscid and calm. The second link(peduncle) is usually understanding the detailed swimming behavior of fish. hydrodynamically negligible, but here we only consider the They do not present the explicit analytic form of the drag in the lateral direction. The third link(caudal fin) is general equations of motion for the swimming fish system. regarded as a thin flat plate generating lift and lateral drag In this paper, we show the detailed locomotion of each part according to quasi-static two-dimensional wing theory of the fish robot and the explicit analytic form of the which is often used to approximately calculate fluid forces general equations of motion and its derivation. acting on a high-aspect-ratio wing. Because z-component is The fish robot can be regarded as a multi-body system usually not considered in steady swimming, the robotic or a constrained discrete mechanical system. In order to fish’ s locomotion is analyzed only in two dimensions of obtain explicit equations of motion of the fish robot system the horizontal plane. Thus, we refer to the direction of and get the propulsive characteristics of each part, we intended locomotion longitudinal to the fish body as the x choose Udwadia-Kalaba theory to model the multi-body direction and the lateral direction as the y direction. Fig. 8 system. Compared with former simple model, the model illustrates an analytical model of the fish robot’s dynamics. built by applying Udwadia-Kalaba theory is more reasonable and can clearly describe the detailed locomotion of each part of the fish robot, not only the whole fish robot’s locomotion described in the former simple model.
7 Fish Robot Model
The fish’ s nervous system controls muscle contraction making the fish’ s body and fins undulate. These motions generate thrust by transferring momentum to the Fig. 8. Analytical model of fish robot dynamics surrounding water. Of all the fishes’ swimming modes, carangiform fish maintain high-speed swimming in calm We introduce the following coordinate system,
ZHAO Han, et al: Dynamic Modeling and Simulation of Multi-body Systems ·846· Using the Udwadia-Kalaba Theory coordinates, variables and symbols. We take the quasi-steady wing theory to approximately x axis: longitudinal to the fish body; y axis: lateral calculate the lift force because the caudal fin can be direction; x1, y1, x2, y2, x3, y3: coordinates of the mass center; regarded as a high-aspect-ratio thin plate. Although this
θθθ123,,: undulating angles of the three links; l2: length of model of calculating forces acting on the fish body is the second link; FD1x: longitudinal drag acting on the first substantially simplified, it does a very good job of link; FD1y, FD2y, FD3y: lateral drag acting on the three links; describing the qualitative behavior of the system. Fx, Fy: x, y component of the lift force; T2, T3: input torques exerted on the two joints; a: length from mass center of 9 Constrained Equation of Motion of Fish first link to first joint; b: length from mass center of third Robot System link to second joint. We begin to derive the constrained equation of motion 8 Drag and Lift Force from this point by applying Udwadia-Kalaba theory. The fish robot system has three links and two joints. It can be The drag acting on the fish is empirically measured by regarded as a common multi-body system. So, the towing the fish through the water and is found to be constrained equation of motion can be obtained through approximately quadratic with velocity[34]. It can be Udwadia-Kalaba equation which has been introduced in calculated by detail.
2 9.1 Equations of Motion of Unconstrained fish robot FDD0.5ρ CV S , (29) If there are no constraints exerted on the three links, in
other words, connections between links are removed. where ρ is the mass density of water, V is the velocity of Referring to the analytical model of the fish robot’ s the link of fish robot relative to the water flow, it can be in dynamics (Fig. 8), we can get equations of each link as the direction of x axis or y axis. CD is the drag coefficient, S is the projection area of the link of the fish robot on the plane perpendicular to the velocity. mx1 1 F D1xy, my 1 1 F D1 , I 1θ 1 0, (35) We assume that the caudal fin is in a quasi-steady mx0, my F, Iθ T , (36) uniform flow. Following the notations of MASON[35] and 2 2 2 2 D2y 2 2 2 [28] YU , lift force acting on the caudal fin is represented as mx3 3 Fx, my3 3 F yx + FD3 , I 3θ 3 T 3 , (37)
L2πρ lC (vl3e ), v 3 (30) where m1, m2, m3 are the mass of each link and I1, I2, I3 are
the moment of inertia. Because we will constrainθ andθ where l is the span and C is half of the chord length, ρ is 2 3 respectively to a definite function later in the constraint the water density. le is a unit vector pointing in the direction θ θ of the leading edge of the caudal fin which can be equation, here equation IT22 2 and IT33 3should be represented as removed. After mathematical manipulation, the unconstrained equation of motion for the robot system can be written in matrix form as lecosθθ 33 ij sin , (31)
Mq(, tt ) q Qq (, q , ), (38) where i is the unit vector along x axis and j is the unit vector along y axis. v3 is the relative velocity at the center of the caudal fin, which can be represented as M: Mq (, t ) m 000 000 00 1 v3xy 33 ij , (32) 00m1 0I1 00 where x is x component velocity and y is y component 3 3 velocity at the center of the caudal fin. Substituting Eqs. (31), 00m2 (32) into Eq. (30), we can derive x component lift force 0 m2 0, 00 I2 0 2 Fx 2πρ lC ( x33 y sin θθ 3 y 3 cos 3 ), (33) 00m 3 00m and y component lift force 3 0 000 000 0I3 2 Fy 2πρθ lC ( x3 sin 3 x 33 y cos θ 3 ). (34)
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·847·
x F Aq b, (50) 1 D1x l2 y1 FD1y 1 0a sinθθ 1 0 sin 0 0 0 122 θ1 0 l 2 0 1a cosθθ12 0 1 cos 0 0 0 x2 0 2 qyt2 , Q : Qq ( , q , ) FD2y . (39) l 2 θθ A 0 0 0 1 0 sin23 1 0b sin , θ 0 2 2 l2 x3 Fx θθ 0 0 0 0 1 cos23 0 1b cos 2 y F +F 3 yxD3 00 0 00 1 00 0 θ3 0 00 0 00 0 00 1
9.2 Constraint Equations alθθ22cos ( 2) θθ cos For fish robot system, we get three links with two joints 1 12 2 2 which generate four constraint equations. Also two more θθ22 θθ al1sin 12 ( 2) 2 sin 2 constraint equations are used to constrain θ and θ . 2 3 θ22 θθθ (lb2 2) 2 cos 23 cos 3 For the first joint, the constraint equations are b . (51) 22 (lb2 2)θ 2 sin θθ 23 sin θ 3 θθ 2 xa1+ cos 122 x ( l 2)cos 2 , (40) 0.3(2ππ ) sin(2t ) yasinθθ y ( l 2)sin . (41) 1 122 2 0.6(2ππ )2 sin(2t 0.25)
Taking the time derivative twice, we get
9.3 Constrained equation of motion ll xxa sinθθ 22 sin θθ a θ 22 cos θ θ cos θ , (42) Eq. (38) constitutes the unconstrained equation of 1 2 11 222 2 1 1 2 2 motion of the fish robot system, and Ep. (50) specifies the ll y ya+ cosθθ22 cos θθ a θ 22 sin θ θ sin θ . (43) constraints. Since matrix M is singular, we should use the 1 2 11 222 2 1 1 2 2 Udwadia-Phohomsiri equation to get the equation of For the second joint, the constraint equations are motion for the constrained system. We have checked the full rank condition and it is fulfilled: the rank of [MA | ]T is 9 (full rank). Hence, the acceleration of the fish x22( l 2)cosθθ 23 xb cos 3 , (44) robot system is y22( l 2)sinθθ 23 yb sin 3 . (45)
+ + Taking the time derivative twice, we get I AAM Q q . (52) A b ll22 22 xx2 3 sinθθ22 b sin θθ 33 θ 2cos θ 2 +b θ 3 cos θ 3 , 22 (46) For this fish robot system, we can explicitly find closed ll22 22 form expressions for M and Q in (39), A and b in (51). yy2 3 cosθθ22 b cos θθ 33 θ 2sin θ 2 b θ 3 sin θ 3 . 22 Having thus obtained the matrices M, Q, A and b, the (47) equations of motion of the system can be simply generated
There are two additional constraint equations: by substitution in the right-hand side of Eq. (52). We get nine coupled equations which describe the motion of the constrained fish robot system. Using the explicit θ2 0.3sin(2πt ), (48) expressions for M, Q, A and b, the nine coupled equations θ 0.6sin(2πt 0.25). 3 can be obtained directly in Maple. The results cover several pages. The complexity of the results precludes explanation Taking the time derivative twice, we get in this paper. However, through simulation, an insight into the dynamics of the fish robot system can be obtained. 2 θ2 0.3(2ππ ) sin(2t ), (49) 10 Numerical Simulation θ 0.6(2ππ )2 sin(2t 0.25). 3 It is necessary to first verify the model we have After mathematical manipulation, the constraint equations constructed. Therefore, numerical simulations have been can be written in matrix form as made and the results detailed. Through a series of
ZHAO Han, et al: Dynamic Modeling and Simulation of Multi-body Systems ·848· Using the Udwadia-Kalaba Theory manipulations, Eq. (52) is obtained which include nine fish’ s swimming behavior. The fish’ s main propulsive coupled equations. The solution of the equations can be characteristics include forward velocity, acceleration and obtained by solving the initial value problems for ODE swing angle of the body. We change the amplitude, through ode15i function in Matlab. frequency and phase difference to get the different simulation results and then compare them to draw some 10.1 Parameters and initial condition conclusions. Of course, if we change the amplitude, We assume that the fish robot is one kind of the frequency and phase difference, initial condition should carangiform fishes and that it is a fish of small size in a also be changed accordingly. calm water. Table 2 shows the parameter values used in the The influence of the amplitude of caudal fin's oscillating simulation. The initial conditions including initial on forward velocity, acceleration and swing angle of fish’s coordinate, velocity and acceleration of each link are ω π φ o presented in Table 3. main body: We keep A2 0.3, 2 , 15 , but A varies with the number of 0.1, 0.2, 0.4, 0.6. Figs. 9–12 3 show that the forward velocity increases very quickly in the Table 2. Parameters in simulation beginning, but as the fish speeds up, the drag force Object Variable Value increases. The velocity tends to become fixed velocity in 2 × –3 Moment of inertia I1 (kg m ) 1.23 10 the end. The forward velocity fluctuates slightly around a Mass of 1st link m1kg 0.409 1st link Length of link 2am 0.1875 mean value. This phenomenon is caused by the unsteady Projected area to vertical plane Sm2 7.069×10–3 thrust force coming from inherent oscillations in fish itself. 2 Projected area to horizontal plane S1 m 0.19 When the oscillating amplitude of caudal fin increases, the 2 × –5 Moment of inertia I2 (kg m ) 3.7 10 forward velocity of fish robot increases accordingly which 2nd Mass of 2nd link m kg 0.104 2 are consistent with the swimming performance of this type link Length of link l2m 0.062 5 2 Projected area to horizontal plane S2m 0.04 of fish. Figs. 13–16 show that the starting acceleration (i.e. 2 –7 Moment of inertia I3 (kg m ) 6.75×10 propulsive force) increases along with oscillating amplitude × –3 3rd Mass of 3rd link m3kg 9.0 10 and tends to zero because of the increasing drag force. The link Length of link 2bm 0.03 2 fluctuation around zero can be explained by the unsteady Projected area to horizontal plane S3m 0.008 3 thrust force. The frequency of the oscillating swing angle of Density of water ρ (kg m ) 998
Fluid Drag coefficient CD 0.5 the fish's main body is the same as that of caudal fin and force Fin span lm 0.075 the amplitude of the oscillating swing angle increases Fin chord length 2Cm 0.03 slightly as the caudal fin's oscillating amplitude increases.
Table 3. Initial condition of each link Object Variable Value
(,xy111 , θ ) (0, 0, 0) 1st (,xy111 , θ ) (0, 0, 0) link (,xy111 , θ ) (0, 0, 0) θ (,xy222 , ) (a l2/2 , 0, 0) 2nd (lbθ22 sin θθ b θ cos θ , bcos θθ 22 33 3 3 33 (,xy222 ,θ ) 22 link bAθθ33sin , 3 ωsin( φ )) θ (l 2θ , 0,0) (,xy222 , ) 22 (,xy333 , θ ) (al +2 + b cosθθ 3 , bsin 33 , A sin( φ )) Fig. 9. Forward velocity of amplitude 0.1 3rd (,xy333 , θ ) (bθ3 sin θ 3 , lb 22 θ+ cos θθ 33 , A 3 ωcos( φ )) link (lbθ22 sin θθ b θ cos θ , bcos θθ 22 33 3 3 33 (,xy333 ,θ ) 22 bAθθ33sin , 3 ωsin( φ ))
10.2 Results and discussions In this simulation, we assume the peduncle and caudal fin’ s oscillating functions are θω22 Atsin( ), θ3
A3sin(ωφ t ). The peduncle and caudal fin oscillate with the same frequency but different amplitudes and there exists a phase difference between the two oscillations. We investigate the influence of caudal fin’ s oscillating on the Fig. 10. Forward velocity of amplitude 0.2
CHINESE JOURNAL OF MECHANICAL ENGINEERING ·849·
Fig. 11. Forward velocity of amplitude 0.4 Fig. 16. Acceleration of amplitude 0.6
The influence of the frequency of caudal fin’s oscillating on forward velocity, acceleration and swing angle of fish’s o main body: in this simulation, AA230.3, 0.6, φ 15 , but the oscillating frequency of the peduncle and caudal fin ω changes with 2π , 2.5 ππ , 3 , 3.5 π. We find that the final forward velocity and the oscillating frequency of the final forward velocity increase with the frequency of the caudal fin's oscillating increasing (figures omitted). Starting acceleration increases significantly when ω increases and Fig. 12. Forward velocity of amplitude 0.6 the frequency of final acceleration’ s oscillation around
zero also increases (figures omitted). ω has little influence on the oscillating amplitude of swing angle. The oscillating frequencies of peduncle, caudal fin and the main body are the same. The influence of phase difference of caudal fin’ s oscillating on forward velocity, acceleration and swing angle of fish’ s main body: the influence of phase difference φ on the swimming behavior is not very obvious as is the influence of oscillating amplitude and frequency of caudal fin. The final forward velocity (figures Fig. 13. Acceleration of amplitude 0.1 omitted), starting acceleration and oscillating amplitude of swing angle of the fish’ s main body decreases as the phase difference increasing.
11 Conclusions
Udwadia-Kalaba theory is applied to analyze the dynamics of the falling U-chain tip and 2-joint Carangiform fish robot. (1) The explicit analytical form of general equations of motion of the whole U-chain system is obtained. Fig. 14. Acceleration of amplitude 0.2 (2) Through numerical simulation, ideal and interesting
insights are obtained: as the chain ends are moved closer together, the maximum vertical fall distance of the chain tip and the falling time decrease. The chain will fall faster than freely falling body and the vertical acceleration will always be higher than g9.8. As the separation becomes less, the chain tip will fall faster. The chain tip is subjected to a greater pulling force which may be up to 40 times larger than one link's gravitational force when it hits the lowest position. (3) The explicit analytical form of general equations of Fig. 15. Acceleration of amplitude 0.4 motion of the 2-joint Carangiform fish robot is obtained.
ZHAO Han, et al: Dynamic Modeling and Simulation of Multi-body Systems ·850· Using the Udwadia-Kalaba Theory
The dynamic model including the forward velocity, [24] TOMASZEWSKI W, PIERANSKI P, GEMINARD J C. The motion of a freely falling chain tip[J]. American Association of Physics acceleration and also the swing angle of fish body is Teachers, 2006, 74(5): 776. derived which is a characteristic difference between this [25] GREWAL A, JOHNSON P. A chain that speeds up, rather than dynamic analysis and other conventional analysis of the slows, due to collisions: How compression can cause tension[J]. American Association of Physics Teachers, 2011, 79(7): 723. dynamics of Carangiform fish robot. [26] CHEN H, ZHU C A. Modeling the dynamics of biomimetic (4) By solving Udwadia-Kalaba equation, the swimming underwater robot fish [C]//IEEE International Conference on performance of the fish robot is obtained. The phenomenon Robotics and Biomimetics, Hong Kong, July 5–9, 2005: 478–483. [27] LIU L Z, YU J Z, WANG L. Dynamic modeling of that fish gain burst propulsive force, acceleration and a high three-dimensional swimming for biomimetic robotic fish[C]// velocity through raising oscillating amplitude and International Conference on Intelligent Robots and Systems, Beijing, frequency of caudal fin agrees with the simulation. Raising October 9–15, 2006: 3 916–3 921. the oscillating amplitude and frequency of the caudal fin [28] YU J Z, LIU L Z, WAN G L. Dynamic modeling and experimental validation of biomimetic robotic fish[C]//Proceedings of the 2006 will result in an increasing of the main body’s swing angle. American Control Conference, Minnesota, June 14–16, 2006: 4 129–4 134. References [29] TAYLOR G. Analysis of the swimming of long narrow animals[J]. [1] LAGRANGE J L. Mechanique analytique[M]. Paris: Mme ve Proceedings of the Loyal Society A: Mathematical, Physical & Courcier, 1787. Engineering Science, 1952, 214(1 117): 158–183. [2] GAUSS C F. Uber ein neues allgemeines Grundgsetz der [30] LIGHTHILL M J. Note on the swimming of slender fish[J]. The Mechanik[J]. Journal of die reine und angewandte Mathematik, Journal of Fluid Mechanics, 1960, 9(2): 305–317. 1829, 4: 232–235. [31] WU T Y. Swimming of a waving plate[J]. The Journal of Fluid [3] GIBBS J W. On the fundamental formulae of dynamics[J]. The Mechanics, 1961, 10(3): 321–344. American Journal of Mathematics, 1879, 2(1): 49–64. [32] SFAKIOTAKIS M, LANE D M, DAV IE S J B C. Review of fish [4] APPELL P. Sur une Forme Generale des Equations de la swimming modes for aquatic locomotion[J]. IEEE Journal of Dynamique[J]. Comptes rendus de l'Academie des sciences, 1899, Oceanic Engineering, 1999, 24(2): 237–252. 129(1): 459–460. [33] TRIANTAFYLLOU M S, TRIANTAFYLLOU G S. An efficient [5] PARS L A. A treatise on analytical dynamics[M]. Connecticut: Ox swimming machine[J]. Scientific American, 1995, 272(3): 64–70. Bow Press, 1965. [34] CHAN W L, KANG T. Simultaneous determination of drag [6] DIRAC P A M. Lectures in quantum mechanics[M]. New York: coefficient and added mass[J]. IEEE Journal of Oceanic Yeshiva University, 1964. Engineering, 2011, 36(3): 422–429. [7] UDWADIA F E, KALABA R E. A new perspective on constrained [35] MASON R. Fluid locomotion and trajectory planning for motion[J]. Proceedings of the Royal Society, 1992, 439(1 906): shape-changing robots[D]. California Institute of Technology, 2003. 407–410.
[8] UDWADIA F E, KALABA R E. Analytical dynamics: A new approach[M]. Cambridge, UK: Cambridge University Press, 1996. Biographical notes [9] UDWADIA F E, KALABA R E. Explicit equations of motion for ZHAO Han, born in 1957, is currently a vice-president and a mechanical systems with nonideal constraints[J]. Journal of Applied doctoral supervisor at Hefei University of Technology, China. He Mechanics, 2001, 68(3): 462–467. is a committee member of International IFToMM Education [10] UDWADIA F E, KALABA R E. On constrained motion[J]. Applied Commission and also a member of Editorial Board of Chinese Mathematics and Computation, 2005, 164(2): 313–320. Journal of Mechanical Engineering. His research interests include [11] UDWADIA F E, PHOHOMSIRI P A. Explicit equations of motion for constrained mechanical systems with singular mass matrices and mechanical transmission, magnetic machine, vehicles, digital applications to multi-body dynamics[J]. Proceedings of the Royal design and manufacturing, information system, dynamics and Society, 2006, 462(2 071): 2 097–2 117. control. [12] ROUSEBALL W W. A short account of the history of Tel: 86-13805697995; E-mail: [email protected] mechanics[M]. New York, NY: Dover Publications, 1960. [13] SHAMES I H. Engineering mechanics: dynamics[M]. New Jersey, ZHEN Shengchao, born in 1988, is currently a PhD candidate at NJ: Englewood Cliffs, 1960. [14] SALETAN E J, CROMER A H. Theoretical mechanics[M]. New Hefei University of Technology, China and a visiting scholar at York, NY: Wiley Press, 1971. Georgia Institute of Technology, USA. His research interests [15] ROSENBERG R M. Analytic Mechanics of discrete systems[M]. include analytic mechanics, dynamics of multi-body systems, New York, NY: Plenum Press, 1977. optimal control, robust control, adaptive control, fuzzy [16] IRSCHIK H, HOLL H J. The equations of Lagrange written for a engineering, uncertainty management. non-material volume[J]. Acta Mechanica, 2002, 153(3–4): 231–248. [17] MOORE E H. On the reciprocal of the general algebraic matrix[J]. Tel: 1-404-9531295; E-mail: [email protected] Bulletin of American Mathematical Society, 1920, 26(5): 394–395. [18] PENROSE R. A generalized inverse for matrices[J]. Proceedings Chen Ye-Hwa, got his PhD degree at the University of California, of the Cambridge Philosophical Society, 1955, 51(3): 406–413. Berkely in 1985, is currently a professor at Georgia Institute of [19] WONG C W, YASUI K. Falling chains[J]. American Association of Technology, USA. He is a member of IEEE, ASME and Sigma Xi. Physics Teachers, 2006, 74(7): 490. [20] CALKIN M G. The dynamics of a falling chain:II[J]. American He is also a regional editor(North America) of Nonlinear Journal of Physics, 1989, 57(2): 157–159. Dynamics and System Theory, and an associate editor of [21] CALKIN M G, MARCH R H. The dynamics of a falling chain: I[J]. International Journal of Intelligent Automation and Soft American Journal of Physics, 1989, 57(2): 154–157. Computing. His research interests include advanced control [22] SCHAGERL M, STEINDL A, STEINER W, et al. On the paradox methods for mechanical manipulators, neural networks and fuzzy of the free falling folded chain[J]. Acta Mechanica, 1997, 125(1-4): 155–168. engineering, adaptive robust control of uncertain systems, [23] SOUSA C A, RODRIGUES V H. Mass redistribution in variable uncertainty management. mass systems[J]. European Journal Physics, 2004, 25(1): 41–49. Tel: 1-404-894-3210; E-mail: [email protected]