On Animating Whip-type MotionsDinkar N. BhatAddress450, Computer Science Building,Department of Computer Science,Columbia University,New York, NY. 10027.email: bhat@cs.columbia.eduJoseph K. KearneyAddressDepartment of Computer Science,The University of Iowa,Iowa City, IA. 52242.email: kearney@cs.uiowa.eduTo Appear in Journal of Visualization and Computer Animation

AbstractThis paper presents algorithmic methods to generate progressive, whip-type motions,characteristic of experienced humans in high speed athletic activities like throwing andstriking. In deriving these methods, we deduce boundary conditions for dynamic quanti-ties by analyzing the motion of a two-link system. A control algorithm based on cascadinggains is introduced and is shown to be e�cient when compared to �xed-gain proportionalcontrol. Principles of energy redistribution within links are used to explain the mechani-cal advantage gained by a whip-type motion, and the e�ciency of cascading gain control.The results can be applied in motion synthesis for animation which is demonstratedthrough simulation of a multi-link system. Our aim has been to obtain biomechanicallyplausible solutions which approximate natural motions.Keywords: motion control, dynamic simulation, mechanics-based animation, roboticmethods for animation

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IntroductionWe routinely perform physical activities that require a coordinated motion of a series ofbody segments. Performance in actions like throwing and striking is largely governed bythe e�ciency with which we impart energy to the object of interest: a ball, a board oreven a human opponent. Biomechanical studies have shown that in such actions there isa sequential action of the body segments progressing from the heavier proximal segments{ like the torso { to the relatively lighter more distal segments { like the hand. It is notedthat proximal segments consistently reach lower maximal velocities at an earlier momentthan distal segments[10]. It appears that this sequential motion is naturally preferredbecause of the mechanical advantage gained. This type of motion is called whip-typemotion1. In this paper, we present control algorithms for synthesizing whip-type motionsin simulated robots.Both sport research and biomechanics literature are replete with descriptions ofsequential segmental motion. The motion patterns of experienced athletes like javelinthrowers [19], handball throwers [10] and karate masters [4] have been analyzed. Us-ing computer analysis [19], it was observed that champion javelin throwers consistentlyaccomplished a proper sequence of acceleration and deceleration of body segments, toachieve maximum velocity of release. Putnam [16] and Jorgensen [9] describe speci�cwhip-type actions like kicking a ball and the swinging of golf clubs. They used two linkmodels for the appendages involved in the actions. Kreighbaum and Barthels [12] ex-plain the mechanical advantage gained with the whip motion through the use of the lawof conservation of angular momentum. Cochran and Stobbs [5] presented an empiricalstudy of the golf swing using a mechanical model of the involved links and muscles. Adetailed kinematic method was developed to analyze the racquet swing technique in [18].It showed that the racquet head speed depends on both the angular velocity of segmentsand the instantaneous position of the head of the racquet. Energy transfer in multi-linksystems has received much attention, and many theories have been put forward to ex-plain the transfer of energy fractions within and across links [1], [14]. Aleshinsky [1], [2]in a series of papers, discussed mechanical energy expenditure in multi-link systems. Heproposed quantitative measures for estimating mechanical advantage gained in di�erenttypes of motion where energy transformation from one form to another occurs. However,the most appropriate one for estimating mechanical e�ort expenditure is still hotly de-bated. While all the above schemes provide valuable insight into sequential segmentalmotion, they do not directly facilitate the development of related control algorithms foranimation.On the other hand, animators and researchers in robotics have worked in simu-lating motion of linked �gures. In this context, Isaacs and Cohen [8] created a systemcalled DYNAMO. It introduces control by means of specifying kinematic constraints onparts of a �gure. The dynamic forces created by the motion of the constrained parts1The name, coined in [1], is derived from the motion of a bull whip where a relatively slow motion ofthe handle causes the tip to exceed the speed of sound.1

induce motion in the remaining unconstrained sections of the �gure. The highlight ofthis approach is that it combines kinematic control and dynamic integrity while produc-ing the desired motion. However, it does not provide a general control framework forsegmental motion. While it can produce beautiful animations of basketball playing andwhip lashing, it does not use or capture the common characteristics of these motions.Armstrong [3] formulated a recursive solution for the dynamic parameters of linked �g-ures. But controlling motion was not the primary consideration. Kinematic control usingspacetime constraints [6], [20], [22] has found wide acceptance for interactive animationpartly due to simplicity of speci�cation. We too will use this approach in generatingan animation sequence of a linked system. We specify spacetime constraints that givethe boundary conditions for throwing and then, using numerical optimization, computea trajectory that minimizes an objective function based on e�ort expended [15]. Themotion generated is smooth and consistent with human motion patterns. However, as amethod of control, optimization is not always attractive. Optimization is implementedas iterative search which makes it ine�cient. Moreover, the optimal solution gives littleinsight into the underlying dynamic principles governing the behavior of the linkage, theoptimal controller being a black box that produces a least cost solution satisfying initialand �nal conditions. We would like to understand the mechanical advantage gained withwhip-type motions and discover general principles that govern the e�cient control ofmotion.To the end of developing practical schemes for animation, we derive control strate-gies that generate high tip velocity in a link chain and produce a whip-type motion char-acteristic of human throwing and striking motions. By analyzing the dynamics of a two-link system, we derive boundary conditions for dynamic quantities, so that a whip-typemotion can be achieved. Based on the dynamic analysis, we demonstrate two controlstrategies; one using step controllers and the other using tuned proportional positioncontrollers (analogous to simple springs). A third method accentuates the progressivemotion by creating a dependency between the spring gains and the motion of the linkbelow. By using dynamic, cascading gains it appears that we can achieve high e�ciency.Finally, we discuss how these methods apply principles of energy generation and transferto gain mechanical advantage. We emphasize that the goal of this paper has been toderive biomechanically plausible solutions rather than present ad hoc methods.The organization of the paper is as follows. We formulate the problem and studythe dynamics of a two-link planar system. We then derive our control methods andprovide a qualitative analysis of the whip-type motion achieved, using energy transferanalysis. We discuss extensions of our methods to multi-link systems. The reader mayappreciate that this paper is a blend of quantitative explanations, experiments, anddiscussion, which is partly due to the cross-disciplinary nature of the problem.2

link 0

link 1

link 2

link 3link 4

link n

(base)tip

joint 2

joint 3

joint 4

joint n

joint 1Figure 1: The linkage model is illustrated. Our goal is to achieve high tip velocity.Problem FormulationOur goal is to �nd e�cient methods to generate high velocity at the tip of an openkinematic chain of links. High tip velocity is characteristic in athletic tasks that requiretransmission of energy to other objects. We also want to achieve high velocity withminimal e�ort so that we conserve resources, avoid potential damage from high strain,and maximize performance within the bounds of actuator output.In earlier work [11], we presented two of our control methods without analysisof linkage dynamics. In this paper, we will study the dynamics of a linkage to deriveproperties governing diverse whip-type motions, and applications in multi-link systems.The general linkage model used in this paper is shown in �gure 1. Links are numberedstarting from the immobile base labeled link 0. The moving components are labeled link1 through link n. Link i is connected to link i+ 1 through joint i+ 1. The base end ofthe moving chain is referred to as the proximal end, and the free end is termed as thedistal end. The links are assumed to be cylindrical, rigid, interconnected by frictionlessrevolute joints, and are constrained to move in one plane. We refer to this model as then-link model where n refers to the number of moving links. This model is applicable fordescribing many actions like kicking, dismounting from a parallel bar in gymnastics [16],and the golf swing [9] (see Fig. 2). For instance, in an overhead throw, the moving linkscorrespond to the legs, the torso, and the arms which act together as if they are a singlechain. To simplify analysis, we assume motion in a gravity free environment. Further,each joint has a limited range of motion which prevents full circular rotations of links.Torque controllers acting at joints form the basis of our control strategies. Eachtorque produces joint moments which in human motions corresponds to the net e�ectof all antagonistic muscles, ligaments, tissues, and so on, surrounding that joint. Thecontrollers work in tandem to produce the desired motion. For example, in animatingthe kicking of a ball using a two-link system (the links corresponding to the lower legand the thigh), the control torques at the two joints (corresponding to the knee and the3

Figure 2: A golf swing, approximated from a stroboscopic photograph of a real golfer,is shown (reprinted with permission from Jorgensen-Jr., 1970). The two links representthe golfer's arm and the golf club.shank) act in combination to achieve the require velocity at the foot. The controller atjoint i is termed �i. To evaluate the total e�ort required to produce a given motion, weuse the sum of the squared torques applied to move the linkage [13], [15]. Thus, the totale�ort in moving an n-link system, evaluated over m equally spaced discrete time stepsof size �t, is given by: R = m mXj=1 nXi=1 � 2i (tj)�t (1)We call the measure R as the total applied torque (units of Nm2s). Other de�nitions ofe�ort have been used in animation systems [20], [22]. While measures based on mechan-ical energy expenditure may seem more natural, they have to be de�ned with much care.For example, if the measure is based on the classical de�nition of mechanical work, thenit cancels over di�erent phases of periodic motions, and hence the total e�ort expendedcan be vastly underestimated [1]. Thus, we prefer a simpler and more easily quanti�ableterm like the total applied torque. 4

DynamicsIn this section, we analyze the dynamics of a two-link system in order to derive generalproperties relating di�erent whip-type motions. The analysis of the two-link systemalso provides insight into whip-type motions for multi-link chains. Figure 3 illustratesthe two-link system along with the free body diagrams of the two links. A left handedcoordinate system is adopted such that its origin coincides with joint 1 denoted by h1, they-axis is directed upwards in the plane of the paper, and the z-axis points perpendicularlyinto the plane of the paper. O1 and O2 represent the centers of mass of the two links,respectively. The proximal and distal joint angles, identical to �1 and �1��2, respectively,and measured in the clockwise direction, are represented by �1 and �2, respectively.Tables 1 and 2 detail the symbols associated with static and kinematic parametersof the links, respectively. A particular parameter is denoted by its symbol with the linknumber as subscript. Additionally, let r1 be a vector directed along O1h1, r2 be a vectordirected along O2h2, l1 be a vector along h2h1, and rh2O1 be a vector along h2O1. TheParameter SymbolLength lMass mMoment of Inertia ITable 1: Symbols associated with static parameters of a link are tabled.Parameter SymbolOrientation angle �Angular vel. _�Angular acc. ��Linear acc.(of center of mass) aTable 2: Symbols associated with kinematic parameters of a link are tabled.Newtonian equations of motion [17] for the two links can now be formulated. For thedistal link, the equations are:F2 = m2a2 5

.

.

O1

O2

h1

h2

Tipφ

θ1

θ2

F2

1F

F2-

2T

link 2

link 1

2

1φ=

2τ

1τFigure 3: A two-link system is illustrated. (a) The schematic of the system; (b) The freebody diagrams of the two individual links. 6

�2 + r2xF2 = I2 ��2a2 = [ ��2xr2 + _�2x( _�2xr2)] + [ ��1xl1 + _�1x( _�1xl1)] (2)Similarly, the equations for the proximal link are:F1 � F2 = m1a1�1 � �2 + r1xF1 � rh2O1xF2 = I1 ��1a1 = [ ��1xr1 + _�1x( _�1xr1)] (3)By appropriately manipulating the above equations, we arrive at the following scalarequations for the angular accelerations of the two links:��1 = �1 � k1 _�12 sin�2 � ��2(k1 cos �2 + k2)� k1 _�22 sin�2k2 � k1 cos�2��2 = �2 � k1 ��1 cos �2 + k1 _�12 sin�2k3 (4)where k1 = m2r2l1; k2 = (I1+m2l21 +m1r21); k3 = (I2+m2r22). Finally, the tip velocity isgiven by: vtip = ql21 _�12 + l22 _�22 + 2l1l2 _�1 _�2 cos �2 (5)The above equations show the interdependency of the motions of the two links. Giventorque pro�les, dynamic simulation systems [7] solve the above equations iteratively in or-der to obtain link accelerations, velocities, and orientations. On the other hand, optimalcontrol approaches [13] formulate suitable optimization problems, and solve them nu-merically for optimal control pro�les. In any case, closed-form solutions are not possible.Here we derive boundary conditions relating di�erent whip-type motions by examiningthe above equations.First, the torque �2 should always be positive, since the distal link must acceleratecontinuously. Consequently, the rotational energy of the distal link increases over time.Next, to determine the conditions under which maximal tip velocity is attained, we dif-ferentiate vtip with respect to time and set the resulting equation to zero. It can be shownthat a solution to the equation is, ��1 = ��2 = �2 = 0. In other words, a su�cient conditionfor the tip velocity to be maximum2 is that the two links simultaneously attain extremalangular velocities, and are in phase, i.e the two links are oriented nearly identically whenmaximal tip velocity is attained[18]. This can be corroborated with cricket bowling wherean experienced bowler invariably releases the ball when the lower and upper arms are2The solution could also imply minimal tip velocity, but we are interested in motion in which tipvelocity is increasing after being initially zero. 7

oriented identically, never when the lower arm is lagging the upper arm (�2 > 0)3. Theextremal angular velocities of the distal and proximal link must correspond to maximumand minimum, respectively, since the distal link is accelerating over the forward rotationand the proximal link is decelerating. We show this formally in the next section when thestep control strategy is discussed. It may seem a little counter-intuitive that the proximallink decelerates during high speed movements, however, as we shall see later, it is thisdeceleration that actually promotes energy conversion from translational to rotationalenergy in the distal link. It is shown in [11] that a control strategy in which both linksare accelerating when maximal tip velocity is attained is much less e�cient than a controlmethod which achieves whip-type motion, for the same tip velocity.Control MethodsWe are interested in devising methods to synthesize whip-type motions for virtual linksof arbitrary length. Applying the results of previous section, we derive methods for atwo-link chain. We extend the approach to multi-link chains in a subsequent section.link 0

link 1

link 2

(base)

tip

joint 2

joint 1

(m=2.0kg, l=4.0m, I=2.667 kgm2)

(m=4.0kg, l=4.0m, I=5.334 kgm2)Figure 4: The initial stationary con�guration of the two-link model used in our experi-ments is shown.The two-link system used is shown in �gure 4 along with the mass, the length,and the moment of inertia of each link. Note that the distal link is lighter than theproximal link which is characteristic of body linkages involved in whip-type motions. Weare interested in achieving maximal tip velocity starting from the initial stationary con-�guration as shown. We assume that this initial con�guration has been reached througha backswing phase which prepares the system for forward rotation. The backswing phasecan be observed in many actions like the swinging of a golf club [5]. In the golf swing, apause normally occurs at the top of the swing, between downswing and backswing, indi-cating that the angular velocities of the club and body (representing a two-link system)are approximately zero at that instant. Thus, the backswing can be modelled indepen-dently from the rest of the stroke. In addition, there can be a dissipation phase after3The lower arm cannot lead the upper arm (�2 < 0) because the elbow joint prevents full circularmotion in that plane of rotation. 8

maximal velocity is reached, to bring the linkage back to rest. In e�ect, the dissipationphase drains energy from the system. The dissipation phase can be easily modelled usingderivative controllers at the joints which damp out oscillations in link motion. Therefore,the entire stroke can consist of three di�erent phases. We will focus on the middle phasewherein maximal tip velocity is reached.Step ControlOne way of achieving a whip-type motion is to use step torques at the joints. The twotorque functions, �1 and �2, are de�ned as follows:�1 = T1; 0 < t � ta= T2; ta < t < tf�2 = T3; 0 < t < tf (6)where T1 > 0; T2 < 0; T3 > 0, tf is the instant when maximal tip velocity is attained,and ta is some intermediate time. The positive torque T1 acting up to time ta drives theproximal link forward. Thereafter, the sudden application of a negative torque aids inenergy transfer between fractions within the distal link which consequently accelerates.This negative torque is also called a stop. In accordance with dynamic analysis in theearlier section, the torque on joint 2 is kept non-negative. The motivation of using astop at joint 1 comes from the observation that champion throwers achieve maximumvelocity of release by stopping the bottom part of the body (the proximal segments)almost violently and subsequently accelerate the parts (the distal segments) closer to theballistic object [19]. In Appendix A, we illustrate the e�ect of the stop on link kinematicparameters at the instant maximal tip velocity is achieved.The step control strategy is simple to implement and throws further light on whip-type motions. However, it has de�ciencies: (a) Applying violent torques is not so commonin humans4, since it can result in severed muscle �bres. The next two strategies tend toapply the negative torque more gently. (b) The free parameters in the torque equationsare di�cult to estimate. For instance, the time instant at which the torque on joint 1must be reversed is not obvious. We will not pursue this control strategy in more detailin this paper since we aim to generate feasible solutions.Proportional Position ControlWe demonstrate how proportional controllers (analogous to undamped springs) at jointscan be coordinated to produce whip-type motions. The torque �i applied at joint i is4Only very experienced athletes like javelin throwers use this method to good advantage. Even theysu�er from frequent muscle fatigue! 9

proportional to the di�erence between the current joint angle �i and the set point �di . IfKi is the gain of the controller at joint i, then the applied torque is given by:�i = Ki(�di � �i) (7)At the joint set point, the corresponding control torque drops to zero. Interior jointsreach their set point when the joint angle is 0, that is, when the two links connected bythe joint have the same orientation. The set point for the base joint is placed midwaythrough the joint range.The gain constants in the controller equations are implicit functions of the staticparameters of the chain. The advantages of using such lumped constants are that, (a)the control strategy becomes relatively independent of the structure of the chain, and(b) fewer independent parameters need to be speci�ed. The gain K1 for the base jointcontroller is arbitrarily set to 1000.0. The gain K2 for the joint 2 controller is set to50.0. This value was chosen experimentally to achieve greatest maximal tip velocity fora motion that starts from the con�guration shown in �gure 4. The variation in maximaltip velocity for a range of values for K2 is shown in �gure 5. It can be seen that themaximal tip velocity increases with K2 until a value of about 50.0. As the gain increasesfurther, the maximal tip velocity decreases. The reason is that if the gain is increasedtoo much, the distal link catches up too early with the proximal link, even before thebase joint reaches its set point. On the other hand, if the gain is too small, then thepeak angular velocity attained by the proximal link is too small, and hence maximal tipvelocity attained is not optimal.M

axim

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ip V

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5030200 4010 60 70 80 90 100

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Spring Constant of Distal LinkFigure 5: The variation in maximal tip velocity with gain of the controller acting at joint2, is plotted. The gain on the base joint controller is �xed at 1000.0.Using the above described control strategy, the motion of the two-link chain wassimulated using the model-based dynamic simulator Newton [7]. Pro�les of joint angles,angular velocities, joint torques, and the tip velocity are presented in �gure 6. The10

progressive nature of the movement is evident from the joint angle and angular velocitypro�les. Note that the tip velocity becomes maximum when three events occur; (a)the joint angle at joint 2 drops to zero, (b) the control torques becomes zero, and (3)the angular velocities of the two links attain local extrema. These events are consistentwith the observations made in analyzing the dynamics of a whip-type motion. As instep control, a negative torque is applied on the proximal link, only less violently. Theanimation frames generated by Newton were supplied as input to VORT[21], an easy touse software tool-kit for rendering. The output sequence of frames are overlayed in �gure7. To measure the e�ciency of the method, we calculated the total applied torque us-ing equation (1). This quantity measures the e�ort required to move from the stationarycon�guration to the con�guration at which maximal tip velocity is attained. The valuewas found to be 57; 530units. This value provides a basis for comparing proportionalcontrol to the next two methods.Cascading Gain ControlThe progressive nature of the motion obtained using proportional controllers can beaccentuated by de�ning a time-varying gain at the second joint. The gain depends onthe motion of links below it, and hence the control strategy is called cascading gaincontrol. In this strategy, the equation for the controller at joint i is given by:�i = �i(�di � �i); i = 2 : : : n (8)where �i is a variable gain that depends on the angle �i�1, the set point �di�1, and thestarting angle �si�1 at joint i� 1. At any instant, �i is de�ned as:�i = Ki (�i�1 � �si�1)(�di�1 � �si�1) (9)In the case of the two-link chain we are using, the gain of the controller acting at the joint2 is varied continuously. The base joint controller is a simple proportional controller whichdrives the system. The cascading gain value is initially small which causes the forwardrotation of the second link to be delayed with respect to the movement of the �rst link.This delay models the hinderance phenomenon - a lag in uncocking the distal joint -observed in the golf swing [9]. As the gain increases, we are e�ectively increasing thesti�ness of the spring. Finally, the gain drops back to zero as joint 2 reaches its set point.The cascading gain controller was simulated with the two-link model previouslydescribed. Gain parameters K1 and K2 were chosen experimentally so that the maxi-mal tip velocity was approximately the same as in the previous experiment with simpleproportional controllers5. With K1 = 600:0, we found that the greatest maximum tip5Using the same gains as in proportional control, we found tha the tip velocity attained was muchhigher using this control scheme. 11

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0.00 200.00 400.00 600.00Figure 6: Simulation results of the two-link chain using proportional controllers areillustrated. The graphs show the joint angles, the angular velocities of the links, the jointtorques, and the velocity of the tip of the distal link.12

Figure 7: A sequence of frames, obtained from the simulation of the two-link chain usingproportional controllers are overlayed.velocity was obtained with K2 = 100:0. Pro�les of the joint angles, angular velocities,joint torques, and tip velocity are shown in �gure 8. The rendered animation frames areoverlayed in �gure 9. As with the proportional controller, the cascading gain strategyleads to a progressive whip-type motion. The overall e�ort expended as measured byequation (1) is 28; 848units. This is a substantial improvement over the proportionalcontroller. We will discuss the reason for this e�ciency later.Optimal ControlIn the previous section, we demonstrated three control strategies. While each of themproduced realistic whip-type motion sequences, we found that the cascading gain controlmethod was more e�cient. In this section, we will compare their performance quantita-tively. To evaluate the performance of the proportional and cascading gain controllers,we determined the locally optimal trajectory satisfying initial and �nal conditions of themotion determined with the cascading gain controller. The tool employed for optimalmotion analysis is a software system called Optimizer [15]. Given a set of spacetime con-straints and an initial sequence, Optimizer formulates a nonlinear least-squares problemwhich is solved with a version of the Levenberg-Marquardt algorithm. The constraintson our optimization problem were derived from the simulation results of the cascadinggain controlled linkage. The optimizing function is the total applied torque. As in thesimulation, the initial joint velocities were zero. The �nal joint angles and velocities wereconstrained to match the corresponding quantities in the simulation when maximal tipvelocity was achieved.The pro�les of the variables monitored are shown in �gure 10. A whip-type motionis again evident from the angular velocity pro�les. The total applied torque was measured13

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0.00 200.00 400.00 600.00Figure 8: Simulation results of the two-link chain using cascading gain controllers areillustrated. The graphs show the joint angles, the angular velocities of the links, the jointtorques, and the velocity of the tip of the distal link.14

Figure 9: A sequence of frames, obtained from the simulation of the two-link chain usingcascading gain controllers, are overlayed.to be 29; 450units. This torque value is quite close to that obtained using cascading gaincontrol. Given the di�erent numerical algorithms used by Newton and Optimizer, theslight di�erence in e�ciency should not be considered signi�cant. Also, it can be seen thatthe �nal joint angle value also does reach zero though supplied as input value. Theseinconsistencies are due to numerical error. We believe that varying the discretizationsteps of the overall simulation can a�ect the accuracy of results in Optimizer. It can beobserved that the torque pro�le obtained is qualitatively more similar to that of cascadinggain control than in simple proportional control. Unlike the other two methods, inproportional control both torque pro�les decreased monotonically. The animation framesare overlayed in �gure 11.Qualitative AnalysisIn this section, we qualitatively describe the mechanical advantage gained by a whip-typemotion, and explain why cascading gain control is e�cient. This description is based onenergy-transfer analysis. We then describe how our control methods can be extended tomulti-link chains.Energy TransferThe total energy in a link chain is distributed across its component links. The action ofjoint forces, due to control moments at the articulations, promotes energy transfer withinlinks [1]. Aleshinsky [2] substantiated this observation by deriving the energy balanceequations for a linked system and showing that energy transformation from translationalto rotational form occurs in a whip-type motion due to joint forces.15

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0.00 200.00 400.00 600.00Figure 10: Optimization results based on the total torque object function. The initial and�nal angles and angular velocities are constrained to match the results of the cascadinggain simulation shown in �gure 8. The graphs show the joint angles, the angular velocitiesof the links, the joint torques, and the velocity of the tip of the distal link.16

Figure 11: A sequence of frames, obtained using optimization, are overlayed. Spacetimeconstraints on the joint positions and velocities were obtained from the cascading gaincontrolled linkage.The rotational energy (RE) of any link, given by RE = 12I _�2, arises due to twosources: (a) energy transformation from translational form of energy due to joint forces,and (b) energy due to joint control moments acting at articulations (corresponding to�i) in the direction of rotation. The second source corresponds to energy input. Forexample, the total rotational energy of the distal link at time t is given by,RE = FE + Z tt1 M2 _�2dtwhere FE is the energy transformation component over time due to the joint forces actingat joint 2 , M2 is the control moment due to torque �2, and t1 is the initial time instant.A whip-type motion is advantageous because of the e�ective conversion of translationalto rotational form of energy in the distal link. Speci�cally, the sharp deceleration ofthe proximal link in the latter part of motion causes a constraint force to be applied atjoint 2 in such a way that e�cient energy transfer occurs between the translational androtational forms of energy. This causes the distal link angular velocity to increase, andconsequently the the tip velocity increases.Now consider the two cases: proportional control and cascading gain control, asused in our two link example. We believe the e�ciency of cascading gain control comesfrom the improvement in conversion of energy from translational to rotational formsof energy. This is observed from the ratio of the tip velocity to angular velocity ofthe distal link over time. This ratio at each instant is proportional to the ratio of thekinetic energy to rotational energy of the distal link at the same instant. In the case ofcascading gain control the ratio falls over a much larger range than proportional controlwhich indicates better conversion between energy fractions. While our explanation isqualitative in nature, we believe quantitative experiments can be performed by measuringthe energy transformation due to individual joint force components. In fact, it may be17

very useful to measure quantitatively, using an index like the compensation coe�cient asde�ned in [2], the advantage gained by whip-type motions. This can be incorporated indynamic simulation systems which permit the measurement of joint forces.Multi-link SystemsOur methods can be used for simulating whip-type motions in multi-link chains too.The dynamics of multi-link chains is decidedly more complex. However, the progressivenature of the motion suggests that the overall motion can be treated as successive twolink systems, starting from the most proximal moving link, i.e �rst, link 1 and link 2rotate till the tip of link 2 attains peak velocity, other links remaining nearly stationary;then link 2 and link 3 move, and so on till the tip of the most distal link achieves maximaltip velocity. Figures 12 and 13 correspond to the forward rotation phase of a �ve linksystem controlled using tuned proportional controllers. The controller gains, startingfrom the most proximal joint, are: 1000:0; 600:0; 150:0; 80:0; and 25:0, respectively. Thelink masses (in kg), starting from the most proximal link, are: 4:0; 2:0; 1:0; 0:5 and 0:25,respectively. All links are identical in length. The progressive motion is evident in thiscase. It may be seen that the tip attains peak velocity at approximately the same instantwhen the most distal link attains peak angular velocity. This is similar to two linkmotion. In the same way, the cascading gain controller can also be used. Figure 14shows the corresponding animation and for clarity, the images in the sequence are notoverlayed. Recall that a swing typically consists of three phases, namely the backswing,the middle phase wherein peak tip velocity is reached, and a dissipation phase, as seenin the animation sequence.ConclusionIn this paper, we examined control algorithms for simulating whip-type motions in link-ages that correspond to natural motions like throwing and striking. The main resultsand contributions of this paper are:� Three control methods for generating whip-type motions in simulated robots weredevised. They can used in motion synthesis for animation. The methods are taskindependent since they depend only local state variable information.� The dynamics of a two link system was analyzed in order to derive boundaryconditions on dynamic variables for a whip-type motion to occur.� Energy transfer principles were used to explain the mechanical advantage gainedin whip-type motions. 18

Ang

ular

Vel

coity

(rad

/s)

Time(s)

.

.

..

.. .

.Figure 12: The angular velocities of a �ve link system controlled using tuned proportionalactuators is shown. Notice how the peak values, corresponding to successive links, moveover time.T

ip V

eloc

ity (

m/s

)

Time(s)Figure 13: The tip velocity pro�le of the �ve link system controlled using tuned propor-tional actuators is shown. Notice that the tip velocity attains peak value at approximatelythe same instant when the most distal link attains peak angular velocity.19

(1) (2) (3) (4)(5) (6) (7) (8)(9) (10) (11) (12)(13) (14) (15) (16)Figure 14: A complete animation sequence of the �ve link system performing a whip likemotion. The sequence proceeds from top to bottom and left to right. It includes all thethree phases observed in typical whip type motions. Maximal tip velocity is reached atapproximately the 13th frame (the left most on the bottom row).20

Future work will be in two directions: (a) measuring mechanical e�ort expenditure usingmore sophisticated quantitative indices, and (b) developing more complex and completemodels for the muscles which will thereby constitute torque controllers. We aim tounderstand the e�ect of varying static link parameters in whip-type motions. Also,automating the selection of gain constants in the control schemes will ease their use ininteractive systems. AppendixA E�ect of Using a StopWith a step controller, the e�ect of using a stop { a negative torque { at the base jointof a two-link chain (see equation (6)) is examined.� Statement: When maximal tip velocity is attained, the proximal and distal linksattain minimum and maximum angular velocities, respectively.Proof: In the analysis of the dynamics of a two-link chain, it was shown that asmaximal tip velocity is attained, �2 ! 0. We will examine the angular accelerationsof the links just before maximal tip velocity is attained. Taking limits as �2 ! 0,the angular accelerations of the two links, from equation 4, are given by:��1 = �1 � ��2(k1 + k3)k2 � k1��2 = �2 � k1 ��1k3 (10)where k1; k2; k3 > 0.Eliminating ��2 and manipulating the resulting equation, we obtain,(k2 � k1 � k1(k1 + k3)k3 ) ��1 = �1k2 � k1 � k1(k1 + k3)�2k3(k2 � k1) (11)If we assume that the lengths of the links are identical, and m2 < m1 as requiredin whip-type motions, then we can show, k2 > k1 and (k2 � k1 � k1(k1+k3)k3 ) > 0.Since �1 < 0 and �2 > 0 in step control, it follows from equation 11 that ��1 < 0 and��2 > 0. We know from the discussion on dynamics of a two-link system that at theinstant maximal tip velocity is reached, the angular accelerations of the links are0. Therefore, it follows that the angular velocities of the proximal and distal linksreach minimum and maximum, respectively, when maximal tip velocity is attained.AcknowledgementsThis work was supported, in part, by National Science Foundation Grant IRI-8808896and O�ce of Naval Research Grant ONR 00014-88K-0632. We thank Bevra Prasad andSamuel Yuan for their assistance in this work.21

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