Optimal Movement Selection Author(s): David A. Rosenbaum, James D. Slotta, Jonathan Vaughan and Réjean Plamondon Reviewed work(s): Source: Psychological Science, Vol. 2, No. 2 (Mar., 1991), pp. 86-91 Published by: Sage Publications, Inc. on behalf of the Association for Psychological Science Stable URL: http://www.jstor.org/stable/40062645 . Accessed: 05/09/2012 09:44 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

.

Sage Publications, Inc. and Association for Psychological Science are collaborating with JSTOR to digitize, preserve and extend access to Psychological Science.

http://www.jstor.org

SCIENCE PSYCHOLOGICAL

Research Article OPTIMAL

MOVEMENTSELECTION

David A. Rosenbaum, James D. Slotta, Jonathan Vaughan, and Rejean Plamondon3 Departmentof Psychology, Universityof Massachusetts, Departmentof Psychology, HamiltonCollege, and 3Departementde Genie filectrique,£cole Polytechniquede Montreal - Mostphysicaltaskscan be performedwithan infinite Abstract numberof movementpatterns. How then are particularpatternsselected? Wepropose that the contributionsof individual limbsegmentsdependon theirown independentlyassessedfits to task demands.An advantage of this system is that coordination among limb segments can be achieved withoutexplicit control of limb-segmentinteractions.In addition, the system allows segments that are still functioning to compensatefor segments that are disabled. To test the model, we first asked subjectsto oscillate thefingertipover varyingdistancesat varying rates, using only the finger, hand, or forearm. Based on theirperformance,we identifiedthe optimalamplitudeandfrequencyof movementfor each limb segment. Thenwe allowed the subjectsto use thefinger, hand, andforearmhoweverthey wished. We demonstratethat the relativecontributionof each limbsegmenttofingertipdisplacementis predictedby the similarityof the optimalamplitudeandfrequencyof that segment to the requiredamplitudeandfrequency of fingertip displacement. Because our model is similar to models proposedfor learningand perception, common computationalapproaches appearviablefor motorcontroland othermore widelystudied activitiesunderlyinginformationprocessing and behavior.

perception and learning (Holland et al., 1986; Holyoak, Koh, & Nisbett, 1989; Rumelhart, McClelland, & the PDP Research Group, 1986), its underlying mechanisms appear general and powerful. The heart of our model is that limb segments are recruited according to their fits to task demands. To understand the model, consider the task of moving the fingertip back and forth at a frequency and amplitude that can be achieved with the finger alone, hand alone, or arm alone.2 The question we wish to answer is how the required frequency and amplitude of fingertip displacement determine which limb segments are used. We emphasize that we regard this as an example task and question. To approach the problem, consider the fact that when people are instructed to move individual limb segments at varying frequency and with minimal effort, the amplitude they produce decreases as the frequency increases (Kay et al., 1987). Thus minimal effort defines an optimal amplitude for each frequency. If subjects were asked to perform this task with increased effort- an experiment that has not been tried, as far as we know- a family of amplitude-frequency curves would presumably emerge, with each curve corresponding to a given effort level. Each limb segment would presumably have a characteristic set of curves. A fundamental question in psychology is how and whether Suppose now that there are control modules for the finger, people perform optimally in the face of multiple, sometimes hand, and arm, and that each module makes an independent conflicting, constraints. This issue has been pursued in many "bid" to a higher mechanism about the output it can provide. contexts, including decision-making under uncertainty (TverSuppose that when a particular frequency is required, each sky & Kahneman, 1974), perceptual judgment (Green & Swets, module bids the optimal amplitude that its segment(s) can pro1966), and memory organization (Anderson & Milsom, 1989). duce given that required frequency; the optimal amplitude corThe present article is concerned with a domain in which optito the location on the "least-effort" curve associated mization (and for that matter, psychology) has played a more responds with that For example, for a given required fingertip frequency. limited role- human motor performance. The aim of the article M, the finger module might bid an amplitude displacement, is to show that optimization provides a useful way of underequal to Vathe maximum amplitude, F, of finger rotation about standing how limb segments are recruited for physical tasks.1 the knuckle, the hand module might bid an amplitude equal to Vi We propose a model for limb-segment recruitment and describe the maximum amplitude, H, of rotation of the hand about the an experiment that supports the model's predictions. Because wrist, and the arm module might bid an amplitude equal to Va the model has features in common with others proposed for the maximum amplitude, A, of rotation of the forearm about the elbow. To express all these bids in common units, suppose that = Va(A)and H = Vi(A)\ these hypothetical values are for F should be sent to David A. Rosenbaum at the Correspondence Departmentof Psychology, Tobin Hall, University of Massachu- illustration only. The total bid would then be (Va)(Va)A+ setts, Amherst, MA 01003 (electronic mail address ROSENB@ (Vi)(V2)A+ VaA= Wi6(A).The contribution of the finger as a UMASS.BITNET). proportion of the total would be (3/i6)A/(!!/i6)A= 3/n, the con1. In the past, optimizationanalysisas appliedto motorcontrolhas tribution of the hand as a proportion of the total would be concernedspace-timerelationsof single moving points, such as the = Vn, and the contribution of the arm as a propor(Va)/(1Vi6)A hand duringaimed hand movements(Hogan & Flash, 1987;Meyer, tion of the total would be (VaVQV^A = Vn. If the produced Abrams,Kornblum,Wright,& Smith, 1988)or the eye duringsaccades (Bahill & Stark, 1979). Optimizationanalysis has also been used to understandchoices of gait patterns,such as the switchfromwalkingto 2. In this context, by "arm"we meanthe forearm,hand,andfinger, and by "hand"we meanthe handand finger. runningduringincreasesin locomotionspeed (Alexander,1984). 86

Copyright© 1991AmericanPsychologicalSociety

VOL. 2, NO. 2, MARCH1991

PSYCHOLOGICAL SCIENCE

David A. Rosenbaum et al. movement were distributed among the segments in proportion to the bids that were made, all three limb segments would be activated and the difference between the actual movements and the optima for the segments would be minimized. For this computational scheme to achieve desired amplitudes, the total bid must ensure the required amplitude. If the initial total bid is too small, the limb segments can be made to " try harder"- for example, through uniform upward scaling of their bids. If the initial net bid is too large, the individual bids can be lowered- for example, through uniform downward scaling of their bids. In this article we do not try to evaluate these possibilities. Our more modest aim is to demonstrate that the relative contributions of the limb segments depend on their fits to task demands; this is the cornerstone assumption of our model. Before turning to the experiment, we wish to point out that a major advantage of the model we are proposing is that if the optimal amplitude for a limb segment happens to change, other segments compensate. For example, suppose that an injury changes the optimal amplitude of the hand at some required frequency from H = Va(A)to H = Vfe(A).The net bid from the arm, hand, and finger modules would then be Vi(A). The relative contribution of the hand would drop from Vu to !/s, but the relative contribution of the finger would increase from Vu to 3/s and the relative contribution of the arm would increase from 4/n to Vi. Thus the intact limb segments would take over for the injured limb segment. The ability of the model to provide for compensation is appealing. Behavioral studies have demonstrated that such compensation occurs (Abbs, Gracco, & Cole, 1984; Cole, Gracco, & Abbs, 1984; Gallistel, 1980, chapter 5; Kelso et al., 1984). In the remainder of this article we present a two-part experiment that used the task described above - oscillating the fingertip over required amplitudes at required frequencies. In the first part of the experiment (the single-effector study) we identified optimal amplitudes and frequencies for the finger alone, hand alone, and arm alone. In the second part of the experiment (the multiple-effector study), we determined whether the optima for the individual segments predicted their observed contributions when all segments could be used. The results supported the predictions and so encourage further exploration of the model.

amplitude-driven and frequency-driven conditions, cross, or if the curves do not cross, where the mean amplitude and mean frequency are located. In order to find the optimal amplitude and frequency for each limb segment, we studied frequencydriven performance and amplitude-driven performance of the finger, hand, and arm. Seven University of Massachusetts students participated. All were seated and used the right hand only. The tip of the extended index finger was moved approximately parallel to the frontal plane of the body, with the center of the workspace directly in front of the right shoulder and at a distance slightly shorter than the distance between the elbow and the fingertip. The finger, hand, and forearm remained in the horizontal plane at all times; the upper arm was oriented vertically. The movement was performed with the palm facing sideways and with the thumb curled over the middle, ring, and little fingers. The session began with subjects producing preferred frequencies for each of four target amplitudes with the finger alone (rotating the finger about the knuckle), with the hand alone (rotating the hand about the wrist), and with the arm alone (rotating the arm about the elbow). The target amplitudes were indicated by two holes containing diodes sensitive to light from high-intensity bulbs. The holes were 2, 4, 6, or 8 cm apart when the task was performed with the finger alone, 4, 8, 12, or 16 cm apart when the task was performed with the hand alone, and 8, 16, 24, or 32 cm apart when the task was performed with the arm alone. For each combination of amplitude and limb segment, subjects were asked to move the fingertip back and forth at a comfortable rate just past the holes. A computer recorded when the light falling on each diode was blocked by the passing fingertip. Performance was videotaped.3 In the second part of the single-effector study, the frequency that each subject generated for each target amplitude and for each limb segment was fed back to that subject in the form of a computer-generated tone sequence. Subjects were told which limb segment to use (the one used to generate that frequency) and were asked to move at the frequency specified by the tones, covering an amplitude that they found comfortable. The amplitudes that subjects selected were later evaluated through a video scoring procedure designed for this purpose (Barnes, Vaughan, Jorgensen, & Rosenbaum, 1989). Analysis of the videotapes confirmed that subjects moved only the required effector and that the distance by which the fingertip passed the diodes (when particular amplitudes were THE SINGLE-EFFECTORSTUDY required) was small and not systematically related to the task As stated above, previous work has shown that amplitudes conditions. For each limb segment, amplitude and frequency of hand rotation about the wrist decrease with required frequen- were negatively related, both in the amplitude-driven and frecies of hand rotation (Kay et al., 1987). Varying amplitude when quency-driven cases (see Fig. 1). For each limb segment, the different frequencies are required can be referred to as fre- best-fitting straight line for the points obtained in the amplitudequency-driven performance. We also studied amplitude-driven driven condition crossed the best-fitting straight line for the performance, where subjects generate their own preferred frequencies given different required amplitudes. Our reason for 3. Each trial lasted 25 s. Ten seconds into the trial, the photodiodes studying amplitude-driven as well as frequency-driven perfor- were enabled, allowing data collection to begin. Two criteria had to be mance was to arrive at an unbiased estimate of each segment's met during the data collection period for the trial to be acceptable: The optimal amplitude-frequency combination; this estimate was photodiodes had to be passed in the correct order, and the coefficient of needed for the design of the multiple-effector study, as seen variation of the movement times (half a complete movement cycle) had below. We assumed that the optimal amplitude-frequency com- to be less than 0.5. For each limb, the same four amplitudes were used bination for a limb segment could be identified by finding the by all subjects and were presented in random order. Complete details point where the amplitude-frequency curves, obtained in the are presented in Slotta (1989). VOL. 2, NO. 2, MARCH 1991

87

PSYCHOLOGICAL SCIENCE

Movement Selection finger(shortamplitude,high frequency),hand(mediumamplitude, mediumfrequency),and arm (large amplitude,low frequency). The proportionalcontributionof each limb segment was largerin its associatedoptimal-conditionpanelthanin any otherpanel. As tested in separateanalysesof variancefor each limb segment, there was a highly significantmaineffect of requiredamplitudeand requiredfrequencyfor each segment,but no significantinteractionsbetween requiredamplitudeand requiredfrequency.5The results therefore supportthe model's predictions. REMAINING ISSUES AND CONCLUSIONS Some final issues remainto be discussed. One concernsthe basis for the optimal amplitudesand frequencies of the limb Fig. 1. Relationbetween angulardisplacementand frequency when the finger, hand, or forearm("arm")performedthe sin- segments. From physical mechanics, it is known that, in gengle-effectortask. Angulardisplacementsare in degreesof rota- eral, the greaterthe mass and lengthof an object, the lower its tion for each limbsegmentaboutits own axis of rotation(about resonantfrequency(Kittel, Knight, & Ruderman,1973).The the knucklefor the finger, about the wrist for the hand, and orderingof optimalfrequenciesfor the threelimb segmentscan about the elbow for the forearm).Frequencypoints are verti- therefore be ascribed to differences in their masses and cally aligned within each limb because amplitude-driven lengths.6The orderingof optimalamplitudesis less straightforfrequencieswere recorded as being within or not within an ward,for in a simplephysicalsystem such as a pendulumwith acceptabletoleranceaboutthe frequenciesobtainedin the fre- no frictionor viscosity, amplitudeand frequencyare indepenquency-drivenconditions; only acceptable cases were ana- dent. However, as a result of frictionor viscosity, the greater lyzed. the lengthandmassof an object, the smallerthe amplitudeof its oscillationswhen driven at or near resonance (French, 1971). points obtainedin the frequency-drivencondition.4Assuming Modelingof amplitude-frequency relationshipsis discussed in that the intersection point defines the optimal amplitude- the Appendix. An alternativeto the above physicallybased explanationof frequencycombinationfor a limb segment, it can be inferred from Figure 1 that the optimalamplitude,measuredin degrees the orderingof optimalfrequenciesand amplitudesis that the of rotation about each segment's own axis of rotation, was optimareflecteddifferencesin the speed-accuracytradeoffsof smallestfor the arm, intermediatefor the hand, and largestfor each segment.It maybe, for example,that the spatialaccuracy the finger;optimalfrequencywas lowest for the arm, interme- of finger movementsis least degradedby increases in speed, diate for the hand, and highestfor the finger. whereas the spatial accuracy of arm movements is most degraded.Consistentwith this interpretation,in discrete,visually THE MULTIPLE-EFFECTORSTUDY guidedaimingmovements,the precisionof the fingeris higher than the precision of the arm (Langolf, Chaffin, & Foulke, The multiple-effectorstudywas conductedthe next day with when subjects tap as quickly as possible, the same subjects.Now subjectswere free to move the finger, 1976). However, rates are achievedwith the armthanwith the finger(see higher hand, and arm however they wished, as long as their moveThe latterresultsuggeststhat precisionis not the ments were made in the horizontalplane. The experimental Keele, 1986). determinantof repetitivemovementrate. Because accuonly methodwas essentiallythe sameas in the single-effectorstudy. demandsin our task were minimal,as in othercomparable Each subjectwas given his or her optimalamplitude-frequency racy studies (Kay et al., 1987;Kugler & Turvey, 1987), we doubt combinationfor the finger,hand,andarm,as well as all remainthat the observed amplitudeand frequency optima reflected ing combinationsof amplitudesandfrequenciesfromhis or her precisiondifferences. Instead, we believe that the optimarepreferredset. Thus there were three amplitudesx three fre- sulted from the mechanical propertiesof the limb segments. quencies, yieldingnine conditions. The results are shown in Figure2. Each limb segmentcon5. For the arm, the main effects of amplitude and frequency were tributedmore to displacementof the fingertipas its optimal F(2, 12) = 37.39, p < .001, MSe = 119.94, and F(2, 12) = 15.96, p < and came closer to the amplitude frequency requiredamplitude andfrequencyof fingertipdisplacement.For example,consider .001, MSe = 23.44, respectively. For the hand, the main effects of the panels along the main diagonalof Figure 2, which corre- amplitude and frequency were F(2, 12) = 11.49, p < .003, MSe = and F(2, 12) = 11.13, p < .003, MSe = 109.20, respectively. spond to the optimal amplitude-frequencyconditions for the 204.69, 4. The r2 values for the best-fitting straight lines exceeded .94 in each of the six cases shown in Figure 1. For each limb segment, the interaction between test condition (amplitude-driven or frequencydriven) and frequency was highly significant: F(3, 18) = 8.77, p < .001, MSe = 20.71, for the finger; F(3, 18) = 21.50, p < .001, MSe = 24.09, for the hand; and F(3, 18) = 16.36, p < .001, MSe = 12.73, for the arm. 88

For the finger, the main effects of amplitude and frequency were F(2, 12) = 8.922, p < .005, MSe = 184.77, and F(2, 12) = 8.82, p < .005, MSe = 46.84, respectively. The p values for all amplitude x frequency interactions exceeded .50. 6. Heglund, Taylor, and McMahon (1974) interpreted stride frequency data for animals of varying size in a similar way, as did Kugler and Turvey (1987) in their analysis of preferred frequencies in the swinging of hand-held pendulums.

VOL. 2, NO. 2, MARCH 1991

PSYCHOLOGICAL SCIENCE

David A. Rosenbaum et al.

Fig. 2. Proportional contribution to displacement of the fingertip attributableto displacement of the finger (F), hand (H), and arm (A) when the requiredoscillation frequency was optimal for the finger alone (High), hand alone (Medium), or arm alone (Low), and when the required fingertip displacement was optimal for the finger alone (Short), hand alone (Medium), or arm alone (Long). Optimal frequency-displacement combinations are shaded. The measures were arrived at trigonometrically, based on positions of marks on the fingertip, knuckle, wrist, forearm, and upper arm, as observed in video stills at the spatial reversal points of the movements. This explanation is simpler than one based on precision, for it does not require a further account of the source of the underlying effects (i.e., why the limb segments had different speedaccuracy tradeoffs). A second remaining issue concerns the degree of autonomy among the limb segments. The model of movement selection presented here does not include limb-segment interactions; interaction terms were unnecessary to account for our data. However, mechanical and neurophysiological interactions between adjacent limb segments are known to be significant (Hollerbach, 1980; Kots & Syrovegnin, 1966). Thus it remains an open question whether such interactions will have to be postulated in future studies. An especially interesting way to test for interVOL. 2, NO. 2, MARCH 1991

actions is to lock the knuckle, wrist, or elbow. If the segments are controlled as implied by our model, subjects should compensate for locked joints with joints that are still active.7 7. A strong prediction is that if the relative contributions of the still-active limb segments are scaled up when one or more segments becomes dysfunctional, the proportional increases in their relative contributions should be identical. For example, if F + H + A = M denotes the sum of the contributions of the finger, hand, and arm to movement M when the knuckle, wrist, and elbow are free, and F' + A' = M denotes the sum of the contributions of the finger and arm to movement M when only the knuckle and elbow are free (i.e., when the wrist is locked), the model predicts FIA = F'lA'. 89

SCIENCE PSYCHOLOGICAL

Movement Selection A final issue concerns planning versus feedback. Based on the data reported here, we cannot say whether the effects we obtained were planned or emerged only in response to feedback, acquired after several movement cycles. A preliminary analysis suggests that the limb-segment contributions made late in the trials were also present early (Slotta, 1989). This outcome suggests that the patterns were planned. Because planning is essential for some tasks (e.g., deciding whether to jump across a ravine), a planning interpretation of our results is as plausible, a priori, as a feedback interpretation. Consequently, the model of limb-segment recruitment that we have suggested may describe what happens before movements have begun as well as what happens once they are under way. Insofar as this conjecture is correct, our model- or at least our emphasis on optimization in limb recruitment- may provide a new way of understanding how we are able to select particular movements when, as is almost always the case, a given task can be performed with an infinite number of movement patterns (Bernstein, 1967; Jordan & Rosenbaum, 1989; Rosenbaum, 1991). - Based on a master'sthesis by the second auAcknowledgments thor.Aspectsof the resultswerepresentedat the AnnualMeetingof the NorthAmericanSociety for the Psychologyof Sportand Physical Activity, Kent, OH, June 1989. We thank B. Collett, H.J. Barnes,W. Hulstijn,M.J.Jorgensen,J. Kroll,D.E. Meyer,J. Summers, A. Thomassen,and P. van Wieringenfor helpfulcomments. Supportedin partby NSF grantsBNS-8710933and BNS 90-08665 and an NIH Research Career DevelopmentAward (to the first author),and NSF ResearchOpportunityAwardsto the first and thirdauthors.The article was preparedwhile the first and fourth authorswere Fellows at the NetherlandsInstitutefor Advanced Study, Wassenaar.The third author'scurrentaddress is Department of Psychology, CarnegieMellon University,Pittsburgh,PA 15213.

namics and a comparative study of dynamics formulation complexity. IEEE Transactions on Systems, Man, and Cybernetics, SMC-10, 730-736. Holyoak, K.J., Koh, K., & Nisbett, R.E. (1989). A theory of conditioning: Inductive learning within rule-based default hierarchies. Psychological Review, 96, 315-340. Houk, J.C., & Rymer, J.C. (1981). Neural control of muscle length and tension. In V.B. Brooks (Ed.), Handbook of physiology: Motor control (Section 1, Vol. 2). Bethesda, MD: American Physiological Society. Jordan, M.I., & Rosenbaum, D.A. (1989). Action. In M.I. Posner (Ed.), Foundations of cognitive science (pp. 727-767). Cambridge, MA: MIT Press. Kay, B.A., Kelso, J.A.S., Saltzman, E.L., & Schdner, G. (1987). Space-time behavior of single and bimanual rhythmical movements: Data and limit cycle model. Journal of Experimental Psychology: Human Perception and Performance, 13, 178-192. Keele, S.W. (1986). Motor control. In J.K. Boff, L. Kaufman, & J.P. Thomas (Eds.), Handbook of human perception and performance, Vol. II. New York: Wiley. Kelso, J.A.S., Tuller, B., Vatikiotis-Bateson, E., & Fowler, C.A. (1984). Functionally specific articulatory cooperation following jaw perturbations during speech: Evidence for coordinative structures. Journal of Experimental Psychology: Human Perception and Performance, 10, 812-832. Kittel, C, Knight, W.D., & Ruderman, M.A. (1973). Mechanics (Berkeley Physics Course, Vol. 1). New York: McGraw-Hill. Kots, Y.M., & Syrovegnin, A.V. (1966). Fixed set of variants of interactions of the muscles of two joints in the execution of simple voluntary movements. Biophysics, 11, 1212-1219. Kugler, P.N., & Turvey, M.T. (1987). Information, natural law and self-assembly of rhythmic movements: A study in the similitude of natural law. Hillsdale, NJ: Lawrence Erlbaum Associates. Langolf, G.D., Chaffin, D.B., & Foulke, J.A. (1976). An investigation of Fitts' Law using a wide range of movement amplitudes. Journal of Motor Behavior, 8, 113-128. Meyer, D.E., Abrams, R.A., Kornblum, S., Wright, C.E., & Smith, J.E.K. (1988). Optimality in human motor performance: Ideal control of rapid aimed movements. Psychological Review, 95, 340-370. Rosenbaum, D.A. (1991). Human motor control. San Diego: Academic Press. Rumelhart, D.E., McClelland, J.L., & the PDP Research Group. (1986). Parallel distributed processing: Explorations in the microstructure of cognition. Vol. 1: Foundations. Cambridge, MA: MIT Press. Slotta, J.D. (1989). The use of energetics in movement planning. Unpublished master's thesis. University of Massachusetts, Amherst, MA. Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185, 1124-1131.

(Received 3/16/90;Revision accepted 10/12/90)

REFERENCES Abbs, J.H., Gracco, V.L., & Cole, KJ. (1984). Control of multimovement coordination: Sensorimotor mechanisms in speech motor programming. Journal of Motor Behavior, 16, 195-231. Alexander, R.M. (1984). Walking and running. American Scientist, 72, 348-354. Anderson, J.R., & Milsom, R. (1989). Human memory: An adaptive perspective. Psychological Review, 96, 703-719. Bahill, A.T., & Stark, L. (1979). The trajectories of saccadic eye movements. Scientific American, 1, 108-117. Barnes, H.J., Vaughan, J., Jorgensen, M.J., & Rosenbaum, D.A. (1989). A lowcost method for recording videotaped continuous movements with the Macintosh. Behavior Research Methods, Instruments, and Computers, 21, 255-258. Bernstein, N. (1967). The coordination and regulation of movements. London: Pergamon. Cole, K.J., Gracco, V.L., & Abbs, J.H. (1984). Autogenic and nonautogenic sensorimotor actions in the control of multiarticulate hand movements. Experimental Brain Research, 56, 582-585. French, A.P. (1971). Vibrations and waves. New York: W.W. Norton & Co. Gallistel, C.R. (1980). The organization of action. Hillsdale, NJ: Erlbaum. Green, D.M., & Swets, J.A. (1966). Signal detection theory and psychophysics. New York: Wiley. (Reprinted, Huntingdon, NY: Kruger, 1974.) Heglund, N.C., Taylor, C.R., & McMahon, T.A. (1974). Scaling stride frequency and gait to animal size: Mice to horses. Science, 186, 1112-1113. Hogan, N., & Flash, T. (1987). Moving gracefully: Quantitative theories of motor coordination. Trends in the Neurosciences, 10, 170-174. Holland, J.H., Holyoak, K.J., Nisbett, R.E., & Thagard, P.R. (1986). Induction: Processes of interference, learning, and discovery. Cambridge, MA: MIT Press. Hollerbach, J.M. (1980). A recursive Lagrangian formulation of manipulator dy-

90

APPENDIX A more detailedaccountof the patternsof optimalamplitudesand frequenciescan be derivedby assumingthateach limbsegmentbehaves like a linear dampedoscillator. Each oscillatingsegment (the finger, hand,or arm)can thenbe representedas a systemwithconstantinertia, I, anda characteristicstiffnessandviscosity, movingundera sinusoidal drivingtorque,T0coso)t.ApplyingNewton's second law to this system, one obtainsan equationof a resonantsystem whose solution,the angulardisplacement,0, can be describedas a functionof the angular drivingfrequency,to. If ®(o>)is plotted againstu>,the curve ascends, reaches a maximum,and then descends. The steepness of the curve dependson the qualityfactor, Q, of the system; as Q decreases (because dampingincreasesor stiffnessdecreases),the maximumof ®(o>) decreases.The datain Figure1 can be viewedas pointslyingon or near line segmentsfroma set of such curves. The line segmentscorrespond to cases in whichthe drivingfrequencyequalsor exceeds the frequency that gives rise to maximumangulardisplacement.Furthermore,the arrangementof the theoreticalcurves for the finger, hand, and armis similarto what we observed. Maximumangulardisplacementand frequency should be inversely related, and segments with larger mass (largerinertia)shouldwork at lower amplitudesfor a given frequency. Thus as the inertia of a segment increases, its optimal amplitudefrequencypointshouldoccupya locationlowerandfartherto the left in the amplitude-frequency plane.The modelalso providesa possibleway of explainingthe differencesbetween the frequency-drivenand amplitude-drivenconditions.If Q were higherin the amplitude-driven conVOL. 2, NO. 2, MARCH1991

SCIENCE PSYCHOLOGICAL

David A. Rosenbaum et al. ditions, the slope of the resultingamplitude-frequencycurve would have been steeperin the amplitude-driven conditions,as we observed. It is plausiblethat Q was higher in the amplitude-drivenconditions because the amplitudesthat subjects spontaneouslyproducedin the frequency-drivenconditions were generally smaller than those they were requiredto producein the amplitude-driven conditions;the latter amplitudeswere selected somewhatarbitrarilyby us when we set up the experiment.Largeamplitudesdemandhigherstiffnessand/orlower

dampingbecause of the length-tensionpropertiesof muscle (Houk & Rymer, 1981), and these stiffness or dampingchanges could have boosted Q. The fact that subjectsspontaneouslyproducedsmallerangulardisplacementswhenthey could select theirown displacements(in the frequency-driven conditions)accordswith the assumptionthatthey soughtto minimizeeffort, for producingsmallerangulardisplacements can be achievedby loweringmusclestiffness,whichpresumablylowers effort.

MODERN CLASSICS BETWEENTHERAPISTAND CLIENT

INTRODUCTION TO MODERN BEHAVIORISM

The New Relationship Michael Kahn, Director, California Institute of Integral Studies, and Professor Emeritus, UniversityJ ofJ California J

Third Edition Howard Rachlin, SUNY, Stony Brook .... ,,_,. The writing is clear, the presentation excellent .... the best text available at this level."William M. Baum, University of New Hampshire

A compelling study of the therapist-client relationship that shows how classical transference analysis can be integrated

with the contemporary humanistic emphasis on warmth and empathy. January 1991, 192 pages, cloth: 2178-3, $19. 95;

paper:2194 5, $10 95

... 199Q ,Q2 paper 2176-7 $1595

2101 c ,26 95

SCHIZOPHRENIAGENESIS The Origins of Madness

INTERPERSONALPERCEPTION

Irving,. Gottesman, university of Virginia

Edward E.Jones, Princeton University

"Jones has done a masterful and elegant job of delineating one of the core issues in human psychology- how people come to make sense of each other. We are all the richer for

thissplendidbook.'' -Jerome Bruner,NewYorkUniversity 1990, 313 pages,cloth:2138-4, $2995; paper:2143-0, $18.95

"Gottesman combines intellectual passion and rigor with a deep compassion and concern for schizophrenic persons and their families .... This work is a real tour de force." -Dante Cicchetti, Director, Mt. Hope Family Center,

Universityof Rochester 1990, 296 pages,cloth: 2145-7, $24.95; paper:2147-3, $14.95

MIND SIGHTS

THE DEVELOPMENTOF CHILDREN

Original Visual Illusions, Ambiguities, and Other Anomalies, with a Commentary on the Play of Mind in Perception and Art Roger N. Shepard, Stanford University

Michac, Co| University of California at San Diego, and Sheila R Cole

Mind Sights is an elegant introduction to visual perception and the psychology of art that makes a unique and entertaining supplement for a broad range of psychology courses. "This book is an absolute gem.'-Stephen

M. Kosslyn,

Harvard

University 1990, 228 pages, 100 illustrations, paper: 2133-3, $14.95

"To do such a nice job of integrating theoretical perspectives which are normally opposed without distorting them, and to give readers such a good intuitive feel' for young children... is a real tour de force."- Robbie Case, Stanford University 1989, 702 pages, cloth: 1864-2, $39. 95

cloth: 2134-1,

$24.95;

Available at fine bookstores or order directly from W H. Freeman and Company, 4419 West 1980 South, Salt Lake City, UT 84104. To examine any of these texts for course adoption, please write to us on your school letterhead stating: course name, number, length, present text in use, enrollment, and adoption decision date. Write to: Marketing Department, W. H. Freeman and Company, 4 1 Madison Avenue, New York, NY 10010.

IB W. H. FREEMAN AND COMPANY TheBook Publishing Arm of Scientific American

VOL. 2, NO. 2, MARCH 1991

91