Biol Cybern (2005) 93: 288–306DOI 10.1007/s00422-005-0003-2

ORIGINAL PAPER

Emanuele Lindo Secco∗ · Luca Valandro∗Roberto Caimmi∗ · Giovanni MagenesBenedetto Salvato

Optimization of two-joint arm movements: a model techniqueor a result of natural selection?

Received: 2 July 2004 / Accepted: 21 June 2005 / Published online: 29 September 2005© Springer-Verlag 2005

Abstract The fossil record of early hominids suggests thattheir Arm length, and presumably stature and weight, hada tendency to increase. Using the minimum jerk principleand a related formulation of averaged specific power, ASP,with regard to selected two-jointArm movements, the currentpaper explores relationships between ASP, hand trajectorylength (or Arm length, or body mass) and mean movementspeed, deriving relationships which indicate that ASP is pro-portional to cubic mean movement speed, but inversely pro-portional to hand trajectory length (or Arm length, or 1/3power of body mass). Accordingly, an ‘ecological niche’ ismodeled in a three-parameter space. Either ASP maximiza-tion for fixed movement time, or ASP minimization for fixedmean movement speed, taken as selective optimization cri-terion, allows the increasing of human Arm length duringevolution, regardless of the arm-to-forearm length ratio.

Abbreviations ASP Averaged specific power

* These authors contributed equally to this work

Emanuele Lindo Secco (B) · Giovanni MagenesDipartimento di Informatica e Sistemistica, Universita di Pavia, ViaFerrata, 1, 27100 Pavia, ItalyTel: +39-0382-985352Fax: +39-0382-985373E-mail: Emanuele@bioing.unipv.it,E-mail: Giovanni@bioing.unipv.it

Luca Valandro · Benedetto SalvatoDipartimento di Biologia, Universita di Padova, Via U. Bassi,58/B, 35121 Padova, ItlayTel: +39-049-8276187,Fax: +39-049-8072213,E-mail: valandro@yahoo.com,Tel: +39-049-8276339Fax: +39-049-8276300E-mail: Salvato@civ.bio.unipd.it

Roberto CaimmiDipartimento di Astronomia, Universita di Padova,Vicolo dell‘Osservatorio, 5, 35122 Padova, ItalyTel: +39-049-8278238Fax: +39-049-8278212E-mail: Caimmi@pd.astro.it

1 Introduction

A general tendency for hominid species to become larger ormore complex over time, in sufficiently productive and stableenvironments, is usually associated with different selectionpressures that a posteriori confer greater fitness to those spe-cies that increase their probability of survival and that com-pensate for their reduced number of offspring by acquiringemergent properties (see also Valandro et al. 2003). On theother hand, no general consensus still exists on what kind ofoptimization criteria have driven the natural selection towardsincreasing size and weight (Mathers and Henneberg 1995).

In human evolution, being particularly marked by changesin the locomotor apparatus, a large role for biomechanicalfactors is expected (Preuschoft 1971; Rose 1991; Witte et al.1991; Wang and Crompton 2003). The hominids are thus anappropriate family for searching optimization criteria of thekind considered (see also Ruff et al. 1993). To this aim, afirst possibility consists in focusing on the locomotor appa-ratus i.e. lower limbs: the achievement of minimum powerrequired for motion is a valid optimization criterion towardsincreasing stature and weight (Wang and Crompton 2003).

The current paper is intended to exploit a different alter-native, which relies on an apparently unquestionable fact: theuse of the hand, and relatedArm movements, played an essen-tial role in human evolution. Although different authors haveshown this is not exclusive to Homo (e.g., Goodall 1986),the ability to manufacture tools and to use them with efficacythanks to a “perfect hand andArm” was long ago (since beforeDarwin) recognized as an important feature of Homo evolu-tion in relation to the encephalization process. An increas-ing hominid Arm length in time can be deduced from fossiland present-day data both directly from Arm bone lengths(Churchill and Smith 2000; Pheasant 1996) and indirectlyfrom estimated height (Aiello and Dean 1990,pp. 264, 267)in the absence of Arm bones.

Averaged arm lengths deduced from fossil (Churchill andSmith 2000) and British present-day (Pheasant 1996) data arelisted in Table 1.As forearm fossils are still lacking to this day,the related values have been calculated under the assumption

Optimization of two-joint arm movements: a model technique or a result of natural selection? 289

Table 1 Arm length and angular admissible excursions

Model angular excursions L1(arm)[mm] L2 (forearm) [mm] L1 + L2 [mm] No. of the fossils− 30.0◦ ≤ ϑ1 ≤ 90.0◦0.0◦ ≤ ϑ2 ≤ 90.0◦

1. Present-day man/British adult man 365.0 380.0 745.0 –2. Neanderthal females maximum length 298.6±20.6 (310.9) (609.5) 43. Neanderthal males maximum length 313.1±12.9 (326.0) (639.1) 84. Vogelherd man 338 (351.8) (689.8) 15. Upper paleo females 320.3±9.1 (333.5) (653.8) 36. Upper paleo males 346.3±24.2 (360.5) (706.8) 10

Limb lengths (Churchill and Smith 2000; Pheasant 1996) and typical angular excursions, ϑ1 and ϑ2, related to human articulations.Data in the brackets are extrapolated under the assumption of a constant length ratio, h = L1/L2, for fossil species, as in the case of Present-dayman, according to Eq. 1 - see the text for details.

that the arm-to-forearm length ratio, h = L1/L2, has remainedunchanged during evolution. The specific choice:

h = L1

L2= 365

380= 0.9605; (1)

has been made, with regard to the average proportion betweenarm and forearm in today’s age.1

It must be emphasized that the values listed in Table 1are intended to be illustrative only, to serve as data set for anexploration of size relationships, and do not purport to resultfrom a detailed survey. Nevertheless, it appears that humanArm length has been increasing during evolution. An incre-ment in hominid stature in time, which implies increasingArm length, has been pointed out by Wang and Crompton(2003). Then a spontaneous question arises, regarding theexistence of valid optimization criteria, related to Arm move-ments, towards increasing Arm length during evolution.

Planning and executing reachingArm movements engagean incredible variety of control and biomechanical actions inhumans and primates: since the planning of the motor task,the central nervous system predicts desired skills by inter-nal representation of the limb and the environment dynamics(Shadmehr and Mussa-Ivaldi 1994). Then motor executionobeys to a match between moving points of the Arm equilib-rium (Mussa-Ivaldi and Bizzi 2000; Gomi and Kawato 1997),‘task-relevant’ feedback control (Todorov and Jordan 2002)and low-dimensional synergies of muscles (d’Avella et al.2003).

1 Data of the today man arm are from (Pheasant (1996)):British adults(age: 19–65), 50th percentile• shoulder to elbow length men: 365 mm; women: 330 mm;• elbow to finger tip men: 475 mm; women: 430 mm;• hand length (to subtract) men: 190 mm; women: 175 mm.We quoted the average values. To be complete, one should consider thereasonable variations (normally 5th to 95th percentile), and of course,whether the population of consideration is similar to that of the Britishone studied by Pheasant. Arm model comes from experiments wherethe wrist is normally bended and subjects held the handle of a two-link mechanical manipulandum (Flash and Hogan (1985); Secco et al.(2003)). Therefore, being the instrumented handle grasped in the middleof the palm and referring to male dimensions, it yields:

• Arm: 365 mm; forearm: 475−(190/2) = 380 mm.These are the L1 and L2 values reported in Table 1 for the ‘Present-dayMan’.

From an engineering viewpoint, the dynamic control of atwo degrees of freedom robot mimicking the Arm is compu-tationally onerous (Hollerbach and Flash 1982; Mussa Ivaldiet al. 1988) and complexity is even more growing whenskeleton and muscular redundancy of the natural system areintroduced in the model. Furthermore — because tendonssynergies are the final effectors of motion, involving highnon-linear contribution of muscle that depends on the ac-tual angular position of the activated joint — hand patternsof reaching movements exhibit surprisingly smooth, straightand bell shape speed profiles (Flash and Hogan 1985; Morasso1981; Shadmehr and Mussa-Ivaldi 1994).

It has been suggested that an internal kinematic modelis hierarchically predominant in planning movements, whiledynamics set-up follows as a secondary step (Morasso andSanguineti 1997); Arm stiffness (Mussa Ivaldi et al. 1985;Gomi and Kawato 1997) and even geometric stiffness (Burdet2000a; Burdet 2001; Burdet et al. 2000b) seem adapting to theenvironmental dynamics preserving kinematic features evenwhen disrupting the consistency of the sensorial information(Secco et al. 2003).

High invariance of trajectory and speed profile shapes(Morasso 1981) has yielded numerous optimization func-tionals in literature as principle driving motion planning andmovement execution (Nelson 1983). Common examples are:Arm movement driven by minimum potential-energy crite-rion on muscles (Massone and Bizzi 1989); a cost functionwith minimum in the middle of limb angular range (Bruwerand Cruse 1990); a minimum torque-change model (Uno et al.1989); a minimum jerk functional (Flash and Hogan 1985;Suzuki et al. 1996); the 2/3 Power Law (Lacquaniti et al.1983); the Delta Log-Normal Law (Plamondon 1995a,b); andvariations on the minimum jerk and the minimum torqueand/or miscellanea of both of them.

In the current attempt, the investigation of selective opti-mization criteria towards increasing Arm length during hu-man evolution, shall be related to the minimum jerk only.The reasons for this choice are owing to, among others, thesimplicity of its formulation, the large documented consis-tency of the model with experimental data, and finally theapplicability of this criterion in a broader context like sportbiomechanics and bio-mimetic robotics (Popescu 1999).

290 Emanuele Lindo Secco et al.

The minimum jerk model assumes that smooth naturalmovement results from the integral minimization of the thirdderivative (i.e. the jerk) computed on the free extremity po-sition of the limb (i.e. the hand). Kinematic features of Armmovement emerge directly from the functional minimiza-tion. Under realistic boundary conditions (zero initial andfinal speed and acceleration of the hand) the analytical re-sult gives a rectilinear trajectory of the hand with a bell-shape speed profile. Limb dynamics follows automaticallyafter imposing the kinematics.

The paper is organized as follows: in the first part wedefine a simple model of the human Arm. The workspace ofthe model is restricted by angular constraints as for the natu-ral limits of the shoulder and elbow articulations. The hand,arm and forearm motions are characterized through the min-imum jerk model (namely the trajectory, the speed and theacceleration). From the minimum jerk principle we deducean expression of the averaged specific power (ASP), relatingmovement extent of the hand with the execution time and themean velocity. The next step is a formulation of an ecologicalniche in terms of the power consumption. Then we simulatemovement sets performed with differentArm lengths, consid-ering two alternative optimization criteria, namely: (a) ASPmaximization for fixed movement time, and (b) ASP mini-mization for fixed mean movement speed. As a final point,human and hominid performances — in terms of movementcontrol and power — indicate that longer upper limbs anddifferent length ratios are consistent with the optimizationpresumed to occur as a result of natural selection.

For sake of clarity, through the current paper the termsarm and Arm shall be intended as the first segment i.e. fromshoulder to elbow, and the whole system i.e. arm, forearm,and hand, respectively.

2 Methods and theoretical background

2.1 The minimum jerk principle

According to the minimum jerk model (Flash and Hogan1985) horizontal reaching Arm movements in humans re-sults from minimizing the acceleration derivative of the hand,namely a C cost function:

C = 1

2

tf∫

t0

j 2 (t)dt = 1

2

tf∫

to

[(d3x1

dt3

)2

+(

d3x2

dt3

)2]

dt; (2)

where (x1, x2) are the hand position coordinates on the planeof motion and tf − t0 the execution time – Fig. 1.

Given Eq. 2, kinematics can be deduced by solving theassociated Eulero-Poisson equations2 (Pontryagin et al.,1962). Morover Eq. 2 guarantees unimodal bell speed profile

2 The Eulero-Poisson equations — or the equivalent Hamiltonianapproach — have been reported on the International Society of Bio-mechanics (ISB) web site: http://isb.ri.ccf.org/biomch-l/files/MinJerk-English1.html. For direct and inverse kinematics equations see (Secco2001), App. C with L3 = 0. Sign conventions of ϑ comes from the DHrobotic notation (see Denavit-Hartenberg in Fu et al. 1987)

of the hand as observed in real Arm movements (Morasso1981; Flash and Hogan 1985). Thanks to the direct and theinverse kinematics — Eqs. 3 and 4 respectively — hand andlimb positions are related one to each other:{

x1 = L1 cos(ϑ1) + L2 cos(ϑ1 + ϑ2)

x2 = L1 sin(ϑ1) + L2 sin(ϑ1 + ϑ2)(3)

and, vice versa:

ϑ1 = arctan(

x2x1

)− arctan

(k2k1

);

ϑ2 = arctan(

s2c2

):

(4)

being L1, L2, the lengths and ϑ1, ϑ2 the angles of arm andforearm respectively, k1, k2, c2, s2, :

k1 = L1 + L2c2

k2 = L2s2

c2 = cos (ϑ2) = x21 +x2

2 −L21−L2

22L1L2

s2 = sin (ϑ2) = ±√

1 − c22

(5)

Inverse kinematics — Eq. 4 — yields two solutions for anygiven position of the extremity, being one not compatiblewith the anatomical angular constraints.

The formulation of Eqs. 3 is universal in dimensionlesscoordinates, ξi= xi /L, since it depends on the length ratio,h=L1/L2, but it is independent of the Arm length, L (for de-tails, see Appendix A). Starting from the Flash and Hogan(1985) formulation, we model the human Arm as a two de-gree of freedom planar system with two links and two joints(arm, forearm, and shoulder, elbow, respectively — Fig. 1).We assume:

1. The motion of the Arm model being without any frictioncontribution on the joints.

2. The Arm moving on horizontal plane (Ox1 x2), with theorigin fixed in the “shoulder”.

3. Null velocities and null accelerations of the free extrem-ity i.e. the “hand”, at the beginning and at the end of themovement (time t0 and tf respectively):

xi (t0) = xi

(tf) = 0; i = 1, 2; (6)

xi (t0) = xi

(tf) = 0; i = 1, 2; (7)

where t0 = 0, without loss of generality.

Neglecting 3-D movement and viscous/elastic propertiesof the cartilaginous joints3. lead to an essential model thesimplicity of which implies parsimony on one hand, and a

3 The ‘mechanical’ model adopted so far does not supply any vis-cous contribute on the two hinges of the model, considering shoulderand elbow articulations as two ideal revolute joints. A viscous contrib-ute as a linear function of the joint angular speed could be introduced inthe model, even if it is a simplification too: in fact viscosity of humanarticulations is quite nonlinear (see a review on Hasler et al. 1999; seealso Esteki and Mansour 1996). The simplicity of the proposed modelis highlighting the main role of the internal kinematic model (implic-itly introduced by the minimum jerk theory) assuming that is the maintask of the controller guaranteeing the typical invariant bell shape speedprofile always observed in the reaching movements

Optimization of two-joint arm movements: a model technique or a result of natural selection? 291

x2

x1

L1=arm

L2=forearm

(x1, x2)

ϑ1

ϑ2

Fig. 1 Model of the Arm: two rods (arm L1 and forearm L2) and two hinges (the shoulder and the elbow joints). The hand position (x1, x2) is‘uniquely’ defined when the two angular displacements (ϑ1, ϑ2) are fixed. Conventions are consistent with the common DH robotic notation (Fuet al. 1987). The model is simplifying the anatomical structure, constraining the motion to be planar, as it is in the majority of the experimentalset-up on upper limb reaching movement. According to the anatomical constraints, angular excursions of the model are limited as reported inTable 1

limited heuristic value on the other hand. Then the followingproposition holds:

Principle of minimum jerk Among an infinity ofallowed trajectories, the free extremity of the Arm fol-lows a special one, for which the cost function, C,expressed by Eq. 2,takes a minimum value.

Accordingly, the kinematic equations of hand motion are(for details see note 2 above):xi (τ ) = xi0 + (

xif − xi0)f (τ) ; i = 1, 2; (8)

where indices o and f denote beginning and ending time and:

f (τ) = τ 3(6τ 2 − 15τ + 10

) ; τ = t

tf; dτ

dt= 1

tf; (9)

for a dimensionless formulation of Eqs. 8, see Appendix A.After performing repeated derivatives on both sides of

Eqs. 8 and 9, velocity and acceleration take the form:

xi (τ ) = xif − xi0

tf

df

dτ; i = 1, 2; (10)

xi (τ ) = xif − xi0

t2f

d2f

dτ 2; i = 1, 2; (11)

df

dτ= 30τ 2 (1 − τ)2 ; (12)

d2f

dτ 2= 60τ (1 − τ) (1 − 2τ) ; (13)

in particular, Eqs. 10 and 12 show the velocity symmetry withrespect to τ =1/2 and a single point of maximum (i.e. the bellshape profile). In addition, the combination of Eqs. 8 yields:x1 − x10

x1f − x10= x2 − x20

x2f − x20; (14)

which shows that the hand path is a straight line between theinitial and final position.

The motion is characterized by (1) increasing the velocityof the hand when 0≤ τ ≤ 1/2, and (2) decreasing it when1/2≤ τ ≤ 1, where the velocity is null at the beginning andat the end, as the imposed boundary conditions dictate.

2.2 The ASP

With regard to a selected Arm movement, let us define:

r2of = (

x1f − x1o

)2 + (x2f − x2o

)2 ; (15)

vof = rof

tf; (16)

where rof is the distance between the initial and final posi-tion of the hand (namely the movement extent), and vof , therelated mean velocity.

With regard to a fixed dimensionless time, τ = t/tf , theArm kinetic energy is:

TArm(τ ) = 1

2mArmv2

Arm(τ ); (17)

v2Arm(τ ) = v2

Arm(t) = 1

mArm

×∫ ∫ ∫

SArm

ρArm(x1, x2, x3, t)v2Arm(x1, x2, x3, t)d

3S;(18)

where ρArm(x1, x2, x3, t) is the local density within an infin-itesimal Arm volume element, d3 S, and vArm(x1, x2, x3,t) isthe related velocity.

Let the function, TArm (τ ), exhibit (k+1) extremum points(maxima and minima) within the domain, 0 = τ0 ≤ τ1 ≤ τ2 ≤. . . ≤ τk =1. The condition of null velocity at the beginning(τ = 0) and at the end (τ = 1) of the movement, necessar-ily implies the existence of three extremum points at least,two minima at τ = 0,1 and one maximum at τ = τmax, 0 <τmax < 1, or k ≥ 2.

In dealing with reaching or reaching-like Arm move-ments, energy changes may safely be assumed to occur alongthe following steps: (1) conversion from internal energy ofthe body into kinetic energy of the Arm and attached externalmasses, with the addition of external work performed throughthe Arm, and (2) conversion of kinetic energy of the Arm and

292 Emanuele Lindo Secco et al.

attached external masses into heat, conform to the second lawof thermodynamics. Accordingly, the energy released by thebody for a selected Arm movement, is:

E =k∑

i=1

|T (τi)−T (τi−1)| − W ; (19)

where T (τj ) is the Arm plus attached external mass kineticenergy at the extremal point, τj , and E = E(1), W = W(1),are the released energy and performed work, respectively,during the movement time tf .

The related ASP is:

χ = E

tf m; (20)

where m is the total mass involved in the motion i.e. Armmass plus attached external mass.

In general, the explicit expression of the energy, E, re-leased by the body for a selected Arm movement, Eq. 19, isnot an easy matter.

From this point on, our attention shall be restricted toa simple situation, where the temporal behavior of Arm ki-netic energy exhibits a bell-shaped curve which is, in general,asymmetric, with a single maximum at τ = τmax. It is the casefor reaching or reaching-like Arm movements, provided theminimum jerk principle holds. Accordingly, E = 2 T(τmax )via Eq. 19, and Eq. 20 reduces to:

χ = 2T (τmax)

tf m= v2

max

tf= v2

of

tfImax; (21)

Imax = I (τmax) = v2Arm(τmax)

v2of

= v2max

v2of

; (22)

for a formal derivation, see Appendix A. In particular, thesquare velocity ratio, Imax, is dimensionless.

The expression of the ASP remains unchanged even ifsome external work is performed; for further details, seeAppendix A. Alternative formulations of Eq. 21 may be ob-tained by use of Eq. 16, as:

χ = r2of

t3of

Imax = v3of

rof

Imax; (23)

s in terms of either movement time, tf , or mean (hand) move-ment speed, vof . The above result may be expressed by asingle statement.

For an assigned Arm class, defined by the arm-to-fore-arm length ratio, and an assigned class of movements,defined by the Arm path in dimensionless coordi-nates, the ASP depends only on the ratio of eithersquare hand movement extent to cube movement time,r2of /t3

f ,or cube mean movement speed to movement

extent, v3of /rof .

The last proposition is rigorous in the limiting situation ofmass-less two-link Arm with massive free extremity, whereImax = 225/64, and the ASP may be written explicitly(AppendixA). With regard to a two-(homogeneous) link arm,a viable approximation to the ASP reads:

χ = 225

64

r2of

t3of

I ′max = 225

64

v3of

rof

I ′max;

rof

L> 0.45 ; (24)

I ′max =

(vof

v∗of

)2

=(

ξof

ξ ∗of

)2

; (25)

where ξ ∗of = r•

0f /L is the dimensionless hand path, denot-ing a special Arm movement for which mean square veloc-ities related to different approximations are coincident. Theratio, ξof /ξ ∗

of , depends only on the Arm workspace in dimen-sionless coordinates. For further details, see Subsect. 3.3 andAppendix B. The restriction of hand path to lengths largerthan about one half the Arm length — rof /L > 0.45 —includes about 80% allowed Arm movements. The remain-ing movements are expected to be of little importance withregard to natural selection, owing to related short hand paths.Accordingly, Eq. 24 shall be used in the following.

3 Results

With regard to a selected Arm movement, Eq. 24 impliesthat either an increase in ASP for fixed movement time, or adecrease in ASP for fixed mean movement speed, takes placefrom longer movement extents, and vice versa. Accordingly,two possible optimization criteria may be defined:

(a) ASP maximization for fixed movement time;(b) ASP minimization for fixed movement speed;

which offer an interpretation to an increasingArm length dur-ing human evolution, deduced from fossil and modern armdata listed in Table 1, and inferred from increasing hominidstature over the last four million years (Wang and Crompton2003, Table 1 and Fig. 1).

3.1 ASP related to two special optimization criteria

A comparison between the above optimization criteria, canbe performed in the following, idealized situation. Let a fisher(man) and his son (child) draw a rope, linked to a float, towardthemselves on the seashore. Both the rope and the float beingon the water surface, they may safely be considered as mass-less. Let the Arm movements occur along a horizontal plane,and be repeated identically for each individual. Finally, let Rbe the length of the rope, TF , TS , the time needed for drawing,(rof )F , (rof )S, the path of the hand, or “armful”, related toeach movement, and (tf )F , (tf )S , the respective time, of thefisher (F) and his son (S) to realize each movement.

The total number of armfuls (both forwards and back-wards) for drawing the entire rope and the float up to theseashore, is:

NI = R(rof

)I

; I = F, S; (26)

and the time needed for each armful is:(tf)I

= TI

(rof

)I

R; I = F, S; (27)

where the path of the hand per armful, rof , is related to thelength of the Arm.

Optimization of two-joint arm movements: a model technique or a result of natural selection? 293

For fixed movement time, (tf )F = (tf )S = tf , Eq. 27reduces to:

vI = (vof )I ; vI = R

TI

;

(vof )I =(rof

)I

tf; I = F, S; (28)

where v, vof , represent the mean velocity of the rope andthe armful, respectively. The combination of Eqs. 24 and 28allows an explicit expression of the ASP released to get therope, as:

χI = 225

64I ′

max(vof )3

I(rof

)I

; I = F, S; (29)

which may be written under the equivalent form:(rof

)−2I

χI = 225

64I ′

max1

t3f

; I = F, S; (30)

the longer is the armful, the larger is the ASP. In other words,with regard to a selected species, more massive, larger indi-viduals need more ASP and vice versa, for fixed movementtime.

Vice versa, for fixed mean movement speed, (vof )F =(vof )S = vof , Eq. 27 may be cast under the equivalent form:

v = vof ; v = R

T; vof =

(rof

)I(

tf)I

; I = F, S; (31)

where TF = TS = T . The combination of Eqs. 24 and 31allows an explicit expression of the ASP released to get therope, as:

χI = 225

64I ′

max

v3of(

rof

)I

; I = F, S; (32)

which may be written under the equivalent form:(rof

)IχI = 225

64I ′

maxv3of ; I = F, S; (33)

In this case the shorter is the armful, the larger is the ASP:therefore more massive, larger individuals need less ASPwhen movement speed is fixed.

3.2 Modeling an ecological niche in a 3-parameter space

Assuming either above mentioned optimization criterion, anecological niche can be modeled in a 3-parameter space, us-ing a similar procedure. For this reason, our attention shallbe limited to the latter criterion i.e. fixed mean movementspeed. Following the same line of thought, mutatis mutandis,the former criterion i.e. fixed movement time can be takeninto consideration.

The product on the left-hand side of Eq. 33 describes abranch of hyperbola, where the axes are equal, the centercoincides with the origin, and the asymptotes coincide withthe coordinate axes, rof and χ , as shown in Fig. 2. Differ-ent mean velocities of the armful, vof , correspond to similarhyperbolas with different axes.

With regard to the same kind of armful, the movementextent is proportional to the Arm length,

(rof

)I

∝ LI , which,in turn, is expected to be proportional to a typical body dimen-sion, LI ∝ bI . A recent investigation on size and powerrequired for motion of early hominids, relies on the assump-tion that any given biological material, such as tissue andbone, have the same density regardless from age or specimen(Wang and Crompton 2003).

If the body mean density does not appreciably changeamong individuals of the same species, or even among differ-ent species, then the above proportionality is equivalent toLI ∝ (mI )

1/3, being m the mass of the body, i.e. the individ-ual.

In dealing with adaptive and phylogenetic influences onmusculoskeletal design in cercopithecine primates, relativelimb lengths for adult monkeys are calculated as the lengthof each segment divided by the body mass to one third (Polk2002) which is expected to remain unchanged wheneverLI ∝m

1/3I i.e. LImI ∝ ρ−1

I = const, where ρI is the mean densityof an individual belonging to the I species. The above men-tioned power-law fits the related data (Polk 2002, Table 2Atherein) with a relative scatter of about 10%. Accordingly, aconstant body mean density may safely be assumed, and Eq.33 reads:

LI χI = 225

64I ′

max

v3of

kL

; I = F, S; (34)

where kL is a proportionality constant, or:

m1/3I χI = 225

64I ′

max

v3of

km

; I = F, S; (35)

where km is a proportionality constant. The curves repre-sented by Eqs. (34, 35) make families of hyperbolas withsame features as their analogue plotted in Fig. 2, providedrof is replaced by L and m1/3, respectively.

Since a long time, it has been established that the basalmetabolic rate, �, of mammalians and birds scales as the indi-vidual mass following a power-law, � ∝ m3/4, the so-called3/4 law, with evidence of a deviation to a smaller value forthe smallest mammals (see e.g., West and Brown 2004). Let� be the specific metabolic rate, � = � /m. The dependence:

� = C�m−β; 1

4� β <

1

3; (36)

where C� is a proportionality constant, is acceptable to agood extent. Then the combination of Eqs. 35 and 36 yields:

χI = 225

64I ′

max

v3of

km

�I

C�

m−(1/3−β)

I ; 0 <1

3− β � 1

12; (37)

in the case under discussion, the ASP is proportional to thespecific metabolic rate, owing to the extremely low depen-dence on the individual mass.

With regard to an assigned species, an ecological nichemay be represented as a window, defined by three ranges,namely the movement extent,

(rof

)min <

(rof

)<(rof

)max,

or the Arm length, Lmin < L < Lmax, or the body mass,m

1/3min < m1/3 < m

1/3max; the mean velocity of the armful, 0 <

vof <(vof

)max; and the ASP of the armful, (χ)min < (χ) <

(χ)max.

294 Emanuele Lindo Secco et al.

0.2 1 1.82

10

18

=χ

kgW

s

m3

2

maxχ

minχ

( )minofr ( )maxofr [ ]mrof

= ecological niche( )maxofv

( ) [ ]s/m0.3vI of3/1'

max =[ ]s/m5.2=[ ]s/m0.2=

[ ]s/m5.1=

Fig. 2 The ASP, χ , as a function of the length of the hand path, rof , and the mean velocity, vof , related to the armful. For fixed vof , the curve is abranch of hyperbola where the axes are equal, the center coincides with the origin, and the asymptotes coincide with the coordinate axes. Withregard to an assigned species, an ecological niche may be represented as a window, defined by three ranges, namely the length of the path of thearmful, (rof )min < (rof ) < (rof )max; the mean velocity of the armful, 0 < vof < (vof )max; and the ASP of the armful, (χ)min < (χ) < (χ)max.Instead of the length of the hand path, the Arm length, Lmin < L < Lmax, or the body mass, m

1/3min < m1/3 < m

1/3max, may also be considered. See

the text for further details

If, in absence of drastic external changes, survival isoptimized by minimizing the ASP for fixed mean move-ment speed, then evolution occurs towards larger dimensions,which implies lager masses. In fact, more massive individ-uals (or species) get less tired in competing with their lessmassive counterparts, and then are plausibly favored.

If, on the other hand, survival is optimized by maximizingthe ASP for fixed movement time, the above conclusion stillholds. In fact, more massive individuals (or species), mov-ing the hand cover a larger distance in competing with theirless massive counterparts, and then can be favorite in spiteof higher absolute energy dissipation (extending this idea,also throwing spears or bifacial could be better performed bylarger hominids).

3.3 Arm optimization

An interpretation of fossil data towards increasingArm lengthduring human evolution, as discussed in the last subsec-tion, implies a constant arm-to-forearm length ratio, h=L1/L2,regardless from specimen and age. In general, h is an addi-tional variable which, in turn, can be differentially optimizedover time depending on internal or external constraints. Tothis aim, an investigation on the dependence of Arm work-space and power performance on h, and related simulations,seem to be appropriate. Though h values cannot still be de-duced from most hominid fossil data, different speciescharacterized by different h may safely be considered, asit is the case for primates (Polk 2002).

As the dimensionless Arm workspace (in coordinates,ξi = xi/L) is constrained by the length ratio, h = L1/L2,different cases involving different morphological values arealso to be dealt with. Considering fossil records we expect tofind that an improvement of the efficiency of the upper limb,as a better reaching system, has occurred in the human Armsystem.

To this aim, a planar model of the human Arm consistingof two ideal links (two joints: the arm and the forearm) andtwo hinges (two articulations: the shoulder and the elbow),is adopted. The Arm configuration is defined by two angles,ϑ1 and ϑ2 (see Fig. 1).

The distance, d, from the fixed to the free extremity — i.e.from shoulder to hand — is not exceeding the Arm length, L,whereas the lower limit, dmin, maintains above the absolutedifference of arm and forearm length, L0. Therefore it holds:

L0 = |L1 − L2| � dmin

� d = (x2

1 + x22

)1/2 � L = L1 + L2; (38)

where L0 = 0 in the special case L1 = L2 = L/ 2.With regard to a mechanical two-segment Arm, affected

by no further constraint on the motion, the number, N, ofadmissible final positions, defined as absolute usability, isproportional to the surface of a circular corona, whose innerand outer radius equal L0 and L, respectively. Then the fol-lowing relation holds:

N = k · π · (L2 − L20

) = k · 4π · L1L2; (39)

Optimization of two-joint arm movements: a model technique or a result of natural selection? 295

where k is a proportionality constant. The special case, L1 =L2 = L/2, reads:

Nmax = k · 4π · L2

4; (40)

which corresponds to a maximum (for fixedArm length, L). Infact, among the infinite set of rectangles where L1 +L2 = L,the larger area is attained by the square,4 L/2 + L/2 = L.

The combination of Eqs. 38, 39, and 40, yields the explicitexpression of the ratio, N/Nmax, defined as relative usability:

N

Nmax= 4

L1L2

L2= 4

h

(1 + h)2 ; (41)

interestingly, the right-hand side of Eq. 41 remains unchangedby replacing h with the inverse value, h−1 .As an example, therelative usability, expressed by Eq. 41, is reduced by about25% passing from h = 1 to h = 3 or 1/3, or just by about4% when passing from h = 1 to h = 3/2 or 2/3.

With regard to a biological two-segmentArm, some move-ments are forbidden by structural constraints, but it exhibitsa similar dependence of the usability on the length ratio, h,as in its mechanical counterpart, where no such constraintsexist. In doing simulations for building up the effective Armworkspace, the region of analysis is an actual 14 × 14 gridover the L × L space (Fig. 3). The arm shoulder is fixed in(0,0) while the hand is located in (1,0) — star symbol in Fig.3, panel B. Simulated movements start from this configura-tion — arm completely stretched — and terminate in eachpoint of the grid. Taking into account the Arm constraintsand the admissible angular excursions (Table 1), the allowedconfigurations are selected among all the movements, andthe effective workspace of the Arm is built up (Fig. 3 b–d).

The special case of present-dayArm is represented in Fig.3, where panel a shows the region of analysis and panel b theeffective workspace (N = 86); panels c and d correspond toa change in the length ratio used in panel b, by a factor 3/2(N = 88) and 2/3 (N = 79), respectively. Final positionswithin the dashed circle are outside the validity limits of Eq.24, for calculating ASP. With regard to movement extents,r0f /L > 0.45 i.e. outside the dashed circle in Figs. 3b–d, theabsolute usability reduces to N = 65, 63, 63, respectively.

The above mentioned range in length ratio has no specialmeaning, and it is only intended to include most, if not all,biological Arms related to different species. Further analysisshows that values of h sufficiently close to unity (by a factorless than about two) allow a larger usability. On the contrary,values of h sufficiently distant from unity (by a factor largerthan about two), reduce the usability.

4 More specifically, the above results may be expressed by the fol-lowing proposition.

Theorem The surface of a circular corona, where the inner and theouter radius equal Lo = |L1− L2 | and L = L1 + L2, respectively, is4π times the surface of a rectangle, whose dimensions equal L1 andL2, respectively. The surface ratio of circular corona to circle, whoseradius equals L, coincides with the surface ratio of rectangle to square,whose side equals L/2.

In building up the effectiveArm workspace related to fos-sil species (Table 1), the forearm length, L2, or the arm-to-forearm length ratio, h, cannot be deduced from current data,and must necessarily be assigned. To this aim, two extremeArm classes shall be exploited, namely: (1) different arm-to-forearm length ratios assumed for different species, focusingon extreme situations, and (2) equal arm-to-forearm lengthratios assumed for different species, according to the datalisted in Table 1.

Thanks to Eqs. 8 and 24 describing the minimum jerkmovement in the workspace and the ASP related to eachmovement, now we can virtually reproduce Flash and Hogan(1985) protocol for different Arms, replicating the experi-ment with the Neanderthal and the Upper paleo females andmales, the Vogelherd man and the so called ‘Present-day’man.

The ASP, χ , defined by Eq. 24, may be expressed as theinverse of a dimensionless movement extent, r0f /L:

κ = L

rof

={(

xf − xo

)2

L2+(yf − yo

)2

L2

}−1/2

= 64

225

1

I ′max

L

v3of

χ; (42)

rof

L> 0.45; (43)

where Po(xo, yo) and Pf (xf , yf ) represent the initial andfinal hand position, respectively.

For fixed Arm length, let us take into consideration threedifferent values of the arm-to-forearm length ratio, namely:h1= 0.9605 (related to the Present-day man), h2 = 3/2 h1and h3 = 2/3 h1. The workspace of each Arm is representedin panels B, C, D of Fig. 3. In general, different Arm work-spaces are related to different values of arm-to forearm lengthratio, h. On the other hand, the ASP is independent of h fora selected Arm movement, according to Eq. 24, providedthe final position of the hand is allowed in all cases. Thistrend is apparent in Fig. 4, where the dimensionless ASP,κ = L/r0f , is plotted as a function of the dimensionlesscoordinates, x1/L = x/L and x2/L = y/L, for each admis-sible Arm movement. Final positions above the semi-trans-parent plane are not representative, lying outside the validitylimits of Eq. 42.

For fixed arm-to-forearm length ratio, h = 365/380, armand forearm length are deduced from fossil lengths — lastcolumn of Table 1 — via Eq. 1. The resulting ASP is plot-ted in Fig. 5 as a function of the hand movement extent, forfixed movement time (panel a) and mean movement speed(panel b). Different symbols in figure are related to the differ-ent species listed in Table 1, according to the figure legend.Final positions within the semi-transparent cylinder are notrepresentative, lying outside the validity limits of Eq. 24.

296 Emanuele Lindo Secco et al.

0 1

0

1

x

y

x/L

y/L

0 1

0

1

x/L

x

y

y/L

startingpoint

rof

/L=0.45

0 1

0

1

x

y

y/L

x/L

0 1

0

1

x

y

y/L

x/L

a c

db

Fig. 3 Region of analysis (a) and effective workspace of the Arm (b–d) due to the joint limitations of Table 1, related to different classes wherethe total Arm length, L = L1 + L2, remains unchanged, and (1) L1 = 365 mm, L2 = 380 mm, as in the modern man, see Table 1 (b, crosses);(2) L1/L2 = (3/2)(365/380) (c, triangles-right); (3), L1/L2 = (2/3)(365/380) (d, triangles-down). Distances are normalized to the total Armlength, L = L1 +L2 . The initial position of the hand, (1,0), is marked as a black star. The crude area (a) is an actual 14×14 square of unit side; acurved band (b–d) represents the allowed final positions when analyzing with a plotting grid of 14×14 (grid step of ∼ 50 mm with L = 745 mm).The ASP related to movement extents inside the dashed circle, where rof /L ≤ 0.45, cannot safely be expressed by Eq. 24. The value of theabsolute usability, with regard to the whole workspace and the region rof /L > 0.45, is N = 86, 88, 79, and N = 65, 63, 63, respectively, for b,c, d

As it was expectable, an increase of the mean Arm lengthwith time, which is inferred from the data reported in Table 1,is consistent with both the optimization criteria discussedabove, see Subsects. 3.1 and 3.2.

4 Discussion

In this work, we addressed both theoretical and practicalimplications of the minimum jerk hypothesis applied to the

Optimization of two-joint arm movements: a model technique or a result of natural selection? 297

Fig. 4 Distribution of the dimensionless ASP, expressed by Eq. 43, with regard to the effective workspace of the Arm, represented in Fig. 3,b–d. Symbol captions as in Fig. 3. The dimensionless ASP related to movement extents above the semi-transparent plane, where rof /L ≤ 0.45,cannot safely be expressed by Eq. 42

human Arm. The use of optimization techniques to modelnatural behavior is appealing because of the analogy it bearsto the optimization presumed to occur as a result of naturalselection (Flash and Hogan 1985). In fact, it is usually as-sumed that the continuous feedback and fine-tuning implicitin natural selection lead to near-optimized systems (e.g.,Westand Brown 2004). In other words the high invariability ofsome motion features — as, for example, the bell unimodalspeed profiles that we have guaranteed through the minimumjerk hypothesis — are clearly symptomatic of a high degreemodel of control in the central nervous system. In the end, themotor learning and planning optimality is an issue of controloptimality.

In terms of power consumption, a comparison withphysiological results strengths our prediction that the ASPis proportional to m−1/3 — Eq. 35 —, providing a selectiveoptimization criterion as the ASP minimization at constantmean movement speed: in fact metabolic energy consump-tion has been found proportional to m−0.316 (Taylor et al.1970, 1980; Taylor 1985) both in animals with a large spec-trum of masses (3×10−3/3×103 kg) and with different loco-motor design and behavior.

On the other hand, the model is also suggesting an inter-pretation of the documented increase of the humanArm lengthduring historical ages, as owing to either (a) ASP maximi-zation for fixed movement time, or (b) ASP minimizationfor fixed mean movement speed. With regard to the secondcriteria (b), in hunter populations, weapons projectile likespears or chipped stones could be thrown by young and more

expert adults in the same time interval but with better effi-cacy from the longer arm (rapid effort). The condition ofconstant time could possibly be extended to the compari-son of different populations in which the one with a betterneuro-motor reactivity or higher brain capability of move-ment control could reach the target more successfully (i.e.longer range and power). On the contrary, the impositionof constant average velocity — criteria (a) — could reflectagain the advantage of fishermen as well as gatherers withlonger arms in the reduction of power dissipation for the samemechanical work (endurance effort) eliminating the apparentparadox of inverse optimization in the two simple hypothesesanalyzed here.

Optimization seems to occur even on the arm-to-fore-arm length ratio (h), since the ‘modern Arm’ ratio (h = 0.96;Table 1) is strangely proximal to the maximum usability con-dition h = 1 in Eq. 40.

Human and hominid performances in terms of movementcontrol and power — calculated applying the minimum jerkmodel to a great variability of Arm lengths and upper fore-limb and hind limb ratios — indicate that longer upper limbsand different ratios are consistent with the optimization pre-sumed to occur as a result of natural selection.

For our species (except sportsmen) due to the wide useof technology, some selective pressures could be diminishedif not eliminated at all. The fitting of the minimum-jerk prin-ciple could be interpreted as modern humans are more finelyoptimized for “soft-functions” in order to improve fine andeffective control and the shift toward longer radius bone could

298 Emanuele Lindo Secco et al.

0

0.5

1

00.5

10

2

4

x/Ly/L

PresentNean.FemalesNean.MalesVogelherdUp.P.FemalesUp.P.Males

=χ

kgW

s

m

I 3

2

'max

0

0.5

1

0

0.5

10

4

x 10-8

x/L

y/L

PresentNean.FemalesNean.MalesVogelherdUp.P.FemalesUp.P.Males

=χ

kg

W

s

m

I 3

2

'max

a

b

Fig. 5 The fractional ASP, χ/I ′max, as a function of the length of the hand movement, rof , for fixed movement time, tf = 1 s (a) and constant

mean movement speed, vof = 1 m/s (b) with regard to the effective workspace of the Arm represented in Fig. 3b, but in connection with differentspecies listed in Table 1: Present-day man/British adult man (crosses); Neanderthal females (triangles-right); Neanderthal males (triangles-down);Vogelherd (squares); Upper Paleo females (stars); and Upper Paleo males (points). The fractional ASP related to movement extents inside thesemi-transparent cylinder, where rof /L ≤ 0.45, cannot safely be expressed by Eq. 24

Optimization of two-joint arm movements: a model technique or a result of natural selection? 299

be interpreted also as more advantageous from a controlstandpoint.

According to the model, equations also elicit some keyfeatures on the Arm mobility and on the tendency of increas-ing Arm length.

With regard to size and power required for lower limbmotion, it has recently been found that mobility is a factorwhich may limit increase of stature (Wang and Crompton2003).

Following the same line of thought, it can be shown thatthe above conclusion holds for upper limb motions under thevalidity of the minimum jerk principle. To this aim, let usconsider two individuals of different size, their characteristiclengths (i.e. stature or Arm length), who complete a similarArm movement (i.e. coinciding trajectories in dimensionlessArm workspace, see Fig. 3a) within the movement times,(tf )1 and (tf )2, respectively.A special case has been discussedin Subsect. 3.1: a fisher and his son draw a rope, linked to afloat, towards themselves on the seashore.

Using Eqs. 29 and 31, the ratio of related ASPs takes theexpression:

χ1

χ2=[(rof )1

(rof )2

]2 [(tf )1

(tf )2

]−3

=[(L)1

(L)2

]2 [(tf )1

(tf )2

]−3

; (44)

where the movement extent, in the case under discussion, hassafely been assumed proportional to theArm length, rof ∝ L.

The 3-D relationship among the ratios of ASPs, Armlengths, and movement times, is represented in Fig. 6. Aninspection of Fig. 6, taken due account of Eq. 44, disclosesthat the ASP ratio increases with increase in the Arm lengthratio, but decreases with increase in the movement time ra-tio. In other words, if the size of an individual is larger, theindividual will take longer or require considerably increasedASP to perform a given action.

Thus, the three variables, Arm length (or stature), meanmovement speed, and movement time, are not independent.Increase in individual size results in either an increase inmovement time or a decrease in mean movement speed, and itmust be presumed that this will limit the tendency to increasein size, as outlined with regard to size and power required forlower limb motion (Wang and Crompton 2003). However, itis not safe to derive definitive conclusions since biologicalsolutions to complex selective pressures or constraints couldbe impressive and unexpected.

Under the assumption of movement extent proportionalto Arm length, rof ∝ L, for a selected trajectory in thedimensionless Arm workspace, see Fig. 3A, Eqs. 29 and 32read χ ∝ L2, for fixed movement time, and χ ∝ L−1, forfixed mean movement speed, respectively. To determine howchange in L will affect change in χ , the effect of Arm lengthincrease on ASP can be expressed as:

dχ

dL= χo

Lo

· 2L

Lo

; (45)

dχ

dL= χo

Lo

· (−1)

(L

Lo

)2

; (46)

for fixed movement time and fixed movement speed, respec-tively, where χo and Lo are related to present-day age, andthen are known.

The derivative of ASP over Arm length gives the rate ofchange of χ with respect to L, see Fig. 7.A positive or negativesign indicates that ASP is enlarged or reduced, respectively,while Arm length is increased. However, the absolute val-ues of the derivative will increase or decrease, respectively,with increase in Arm length. In other words, with increase inArm length, the rate of change in ASP will rise or decline,respectively.

The former case offers no further interest, as ASP changerate is linearly proportional to Arm length, implying fasterand faster increase in ASP as Arm length increases. In the lat-ter case, whenArm length is small, the effect ofASP decreasewill be substantial; but when Arm length is larger, the effectwill be weak. An inspection to Fig. 7 shows that, after a rea-sonable ASP decrease is achieved, Arm length and presum-ably human stature, will increase very slowly, as outlinedwith regard to size and power required for lower limb motion(Wang and Crompton 2003).

5 Appendix A

5.1 Dimensionless coordinates and formulation of ASP

In a dimensionless formulation, Eulero-Poisson equations areuniversal, in the sense that they no longer depend on the Armlength, L, but only on the arm-to-forearm length ratio, h.Accordingly, an effective Arm workspace plotted in dimen-sionless spatial coordinates — as done in the current paper —may be thought of as universal for assigned h. Then it could beuseful a dimensionless formulation of Eulero-Poisson equa-tions and kinematic equations deduced from the minimumjerk principle.

To this aim, let us define the dimensionless spatial coor-dinates:

ξi = xi

L; λi = Li

L; i = 1, 2; (47)

L = L1 + L2; (48)

where L is theArm length, when arm and forearm are aligned.Accordingly, Eqs. 3, 4, and 5, translate into:{

ξ1 = λ1 cos (ϑ1) + λ2 cos (ϑ1 + ϑ2) ;ξ2 = λ1 sin (ϑ1) + λ2 sin (ϑ1 + ϑ2) ; (49)

ϑ1 = arctan(

χ2

χ1

)− arctan

(k2k1

);

ϑ2 = arctan(

s2c2

);

(50)

k1L

= λ1 + λ2 cos ϑ2;k2L

= λ2 sin ϑ2;c2 = ξ 2

1 +ξ 22 −λ2

1−λ22

2λ1λ2;

s2 = ±√

1 − c22;

(51)

300 Emanuele Lindo Secco et al.

0.5

1.5

0.511.520

10

20

30

t = constv = constASP = const

L1/L

2

tf1

/ tf2

2

1

χχ

Fig. 6 A surface within a 3-D space represents relationships among the ratios of three parameters: ASP, χ , Arm length, L, and movement time,tf . A locus of constant movement time is marked by squares, showing that a taller individual therein needs to expend more ASP than does ashorter one to complete a given motion in dimensionless Arm workspace. A locus of constant Averaged Specific Power is roughly marked bycircles, indicating that when consuming an equal amount of ASP for assigned trajectory in dimensionless Arm workspace, a taller individualneeds a longer movement time than does a shorter one. A locus of constant mean movement speed is marked by triangles, implying that whentwo individuals move the hand at equal velocity along an assigned trajectory in dimensionless Arm workspace, a taller one spends less ASP butlonger movement time than does a shorter one

0.5 0.75 1-4

-2

0

2

a

b

0L/L

( )( )0

0

L/Ld

/d χχ

Fig. 7 The derivative of ASP over Arm length (normalized to present-day epoch) under the validity of two selective optimization criteria: a ASPmaximization at constant movement time, or b ASP minimization at constant mean movement speed. Curve a exhibits linear proportionality,implying more and more increasing ASP with increasing Arm length. Curve b shows that when Arm length is increased within the range of lowervalues, ASP will decrease substantially, but when Arm length is increased over a range of higher values, ASP decreases slowly. The last resultimplies that in the future, Arm length (and presumably human stature) may not increase quickly

Optimization of two-joint arm movements: a model technique or a result of natural selection? 301

and the combination of Eqs. 49 with their counterparts relatedto the initial position, ξi = ξio, ϑi = ϑio, i = 1, 2, yields:

ξ1 − ξ10 = λ1 [cos (ϑ1) − cos (ϑ10)]+λ2 [cos (ϑ1 + ϑ2) − cos (ϑ10 + ϑ20)] ;

ξ2 − ξ20 = λ1 [sin (ϑ1) − sin (ϑ10)]+λ2 [sin (ϑ1 + ϑ2) − sin (ϑ10 + ϑ20)] ;

(52)

where, in particular, the final position reads:

ξ1f − ξ10 = λ1[cos

(ϑ1f

)− cos (ϑ10)]

+λ2[cos

(ϑ1f + ϑ2f

)− cos (ϑ10 + ϑ20)] ;

ξ2f − ξ20 = λ1[sin(ϑ1f

)− sin (ϑ10)]

+λ2[sin(ϑ1f + ϑ2f

)− sin (ϑ10 + ϑ20)] ;

(53)

it is apparent that, in the (Oξ1ξ2) plane, Eqs. 52, depend onthe length ratio, h = L1/L2 = λ1/λ2, but are independentof the Arm length, L. The same is true for any result, whichis deduced hence from.

In terms of dimensionless coordinates, ξi , defined by Eqs.49, the kinematic equations deduced from the minimum jerkprinciple, Eqs. 8, read:ξi = ξio + (

ξif − ξio

)f (τ) ; i = 1, 2; (54)

which, being independent of the length ratio, h=L1 /L2 =λ1/λ2, may be thought of as universal, with regard to dimension-less distances, ξi = xi/L, and dimensionless time, τ = t/tf .The same holds for any result, which is deduced hencefrom.

By using Eqs. (47), the dimensionless velocity, υ(ξ1, ξ2,ξ3, τ ), and the dimensionless density, ς(ξ1, ξ2, ξ3, τ ), relatedto an infinitesimal Arm dimensionless volume element, d3σ= dx1 dx2 dx3/SArm, at the dimensionless time, τ , are:

υArm(ξ1, ξ2, ξ3, τ ) =[

3∑i=1

(dξi

dτ

)2]1/2

= tf

LvArm(x1, x2, x3, t); (55)

ςArm(ξ1, ξ2, ξ3, τ ) = SArm(t)

mArmρArm(x1, x2, x3, t)

= ρArm(x1, x2, x3, t)

ρArm(t); (56)

where SArm and ρArm = mArm/SArm represent the Arm vol-ume and mean density, respectively.

The combination of Eqs. 17, 18, 55, and 56, yields thefollowing expression of Arm mean square velocity:

v2Arm(τ ) = L2

t2f

×∫ ∫ ∫

σArm

ςArm(ξ1, ξ2, ξ3, τ )υ2Arm(ξ1, ξ2, ξ3, τ )d3σ ;

(57)where σArm = 1 is the dimensionless Arm volume.

The substitution of Eqs. 17, 19, and 57 into 20, with noexternal mass attached to the hand and no external work per-formed, makes the ASP reduce to:

χ = v2of

tfImax; (58)

Imax = I (τmax) = v2Arm(τmax)

v2of

=∫ ∫ ∫

σArm

ςArm(ξ1, ξ2, ξ3, τmax)υ2Arm

(ξ1, ξ2, ξ3, τmax)d3σ ; (59)

which, for assigned arm-to-forearm length ratio and Arm tra-jectory in dimensionless coordinates, depends only on move-ment time and mean movement speed.

Let us define the effective Arm mass as:

meff(τ ) = 2TArm(τ )

v2hand(τ )

= mArmv2Arm(τ )

v2hand(τ )

; (60)

the combination of Eqs. 10, 12, 15, 16, 59, and 60 yields:

Imax = 900τ 4max(1 − τmax)

4 meff(τmax)

mArm; (61)

and Eq. 58 takes the equivalent form:

χ = 900τ 4max(1 − τmax)

4 meff(τmax)

mArm

v2of

tf; (62)

which depends on the ratio of effective to actual Arm massat the dimensionless time, τmax, related to the maximum Armkinetic energy during the movement.

Using Eqs. 16, 58, and 59, a dimensionless ASP may bedefined as:

κ = L

v3of

χ = L

rof

Imax

= 900τ 4max(1 − τmax)

4 meff(τmax)

mArm

L

rof

; (63)

which is constant for assigned arm-to-forearm length ratioand Arm trajectory in dimensionless coordinates.

The validity of Eqs. 57–63 maintains even if some exter-nal mass is attached to the hand during the Arm movement,and/or external work is performed. More specifically, theabove procedure may be generalized provided the system,Arm plus attached mass, is considered instead of the soleArm, where the velocity of the attached mass equals thehand velocity, expressed by Eqs. 10 and 12. Then the ratio,meff(τmax)/mArm, has to be replaced with the ratio, [meff(τmax)+ mO]/[mArm + mO], being mO the attached external mass.

With regard to an assigned Arm movement, let W(τ ) bethe external work performed up to the dimensionless time,τ , against friction and/or active forces, such as Earth gravi-tational field. At the end of the Arm movement, the externalwork may be expressed as:

W = W(1) = −mOv2of Imax; (64)

where mO is a fictitious attached mass, and Imax is related tothe selected Arm movement. Owing to Eqs. 19, 20, and 64,theASP remains unchanged in the case under discussion, andEq. 58 holds.

In the limiting situation where the Arm mass is concen-trated on the free extremity, v2

Arm(τ ) = v2hand(τ ) via Eq. 57,

302 Emanuele Lindo Secco et al.

meff (τ ) = mArm via Eq. 60, and τmax = 1/2 owing to Eqs. 10and 12. Accordingly, Eqs. 61, 62, and 63 reduce to:

Imax = 225

64; (65)

χ = 225

64

v3of

rof

= 225

64

r2of

t3f

; (66)

κ = 225

64

L

rof

; (67)

in addition, the instantaneous specific power, may be ex-pressed using Eqs. 10, 11,15,16, and 17, as:

χ (τ) = d

dt

T (t)

m= 1

2

r2of

t3f

df

dτ

d2f

dτ 2= 1

2

v3of

rof

df

dτ

d2f

dτ 2; (68)

which, owing to Eqs. 12, 13, takes the explicit form:

χ (τ) = 1800r2

0f

t3f

τ 3 (1 − τ)3 (1 − 2τ)

= 1800v3

of

rof

τ 3 (1 − τ)3 (1 − 2τ) ; (69)

and is represented in Fig. 8.

6 Appendix B

6.1 Different arm approximations and evaluation of ASP

A simple evaluation of the ASP is possible only in the specialcase, where the Arm mass is supposed to be concentrated onthe hand, as shown in Appendix A. More realistic approxi-mations imply harder evaluations. For this reason, we aim tosee if it is possible to establish some correlations between thesimplest approximations and more realistic ones, with regardto the averaged square velocity, Eq. 57, when the Arm kineticenergy attains a maximum. The above mentioned correla-tions, provided they exist, allow simple evaluations of ASPto an acceptable extent. To this aim, the following approxi-mations shall be exploited:

1. Arm, forearm, and hand mass concentrated on the hand;2. Arm and forearm mass concentrated on the shoulder, hand

mass on the hand;3. Forearm and hand mass concentrated on the hand, arm

mass on the elbow;4. Forearm mass concentrated on the elbow, arm mass on

the shoulder, hand mass on the hand;5. Arm mass and forearm plus hand mass uniformly distrib-

uted within the first and the second link, respectively;

and the related averaged square velocities v2max= 2Tmax/ mArm, shall be calculated at the Arm kinetic en-

ergy maximum, τ = τmax.Masses of the arm, forearm, and hand, are estimated from

the total body weight, assuming mbody = 78 kg, through

anthropometric relationships (Winter 1990): marm = 0.028mbody, mforearm = 0.016 mbody, mhand = 0.006 mbody, whereasdistances of the centers of mass of arm and forearm-handfrom the centers of rotation of the proximal joint are esteemedas LG1 = 0.436L1 and LG2 = 0.682L2, while moment ofinertia with respect to the center of mass are estimated respec-tively as IG1 = marm (0.322 L1)2 and IG2 = (mforearm + mhand)(0.468 L2)2.

With regard to approximations (1) and (2), the relatedASP has been evaluated in Appendix A.

Concerning approximations (3) and (4), the Arm kineticenergy may be expressed as (e.g., Landau and Lifchitz 1966;Secco 2001):

TArm = 1

2melbowL2

1φ21 + 1

2mhand

[L2

1φ21

+L22φ

22 + 2L1L2φ1φ2 cos(φ1 − φ2)

] ; (70)

φ1 = ϑ1 − π

2; φ2 = ϑ1 + ϑ2 − π

2; (71)

where melbow and mhand represent the total mass concentratedin elbow and hand, respectively. Then the ASP may be eval-uated, as shown in Appendix A.

With regard to approximation (5), the Arm kinetic energymay be expressed as:

TArm =(

1

2I1q

21

)

+(

1

2IG2q

22 + 1

2(mforearm + mhand)v

2G2

); (72)

where vG2 is the velocity of the barycentre of the secondjoint. The Inertias of each element are computed appealingto Huygens-Steiner Theorem:

I1 = IG1 + m1L2G1; (73)

while vG2 comes from:{xG1 = L1

2 cos q1;yG1 = L1

2 sin q1; (74)

and deriving it yields:

v2G2 = x2

G2 + y2G2 = L2

1q21 + L2

2

4q2

2

+L1L2q1q2 cos (q1 − q2) ; (75)

q1 = ϑ1; q2 = ϑ1 + ϑ2; (76)

Then the ASP may be evaluated, as shown in Appendix A.A viable expression of the ASP would be possible, pro-

vided a correlation exists between maximum kinetic energiesduring the Arm movement, related to the simplest approxi-mation 1, 2 on one hand, and more complex but more real-istic approximations 3, 5 on the other hand. To this aim,the decimal logarithm of the mean square velocity, v2

max =v2

Arm(τmax), related to approximations 3, 5, has been plottedversus its counterpart related to approximation (1), with re-gard to the whole Arm workspace made of allowed final hand

Optimization of two-joint arm movements: a model technique or a result of natural selection? 303

0 0.5 1-4

-2

0

2

4

( )2of

3f

r1800

t

⋅

τχ

τ

( )2of

3f

r1800

t

⋅

τχ

Fig. 8 The dimensionless, instantaneous specific power of the Arm,t3f χ(τ)

1800·r20f

as a function of the dimensionless time, τ = t/tf , in the limiting

situation where the Arm mass is concentrated on the free extremity . The portion of the curve below zero corresponds to an ideal situation, wherethe energy loss related to deceleration of the hand comes back to the ‘body’. The mirrored image (with respect to the horizontal axis) of the curvecorresponds to a real situation, where the energy loss under discussion is dissipated into heat, according to the second law of the thermodynamics.

The abscissa of the extremum points is τ = (1/2)(1 ± 1/√

7). The horizontal, dashed line corresponds to the dimensionless, ASP,t3f χ(τ )

1800·r20f

= 1512

positions (86 cases in total), with equal movement time. Thecorrelation translates into a linear dependence to an accept-able extent, provided movement extents longer than aboutone half the Arm length, are considered. From this point on,our attention shall be restricted to the most realistic (amongthose listed above) approximation (5). The related trend isshown in Fig. 9, where the dimensionless movement extents,rof /L, are divided in ten different classes, ordered in length.Classes 4–10, implying rof /L > 0.45, exhibit a linear depen-dence to an acceptable extent, and the application of a leastsquare (LS) method5 yields:

log(v2

max)1

u2v

= a log(v2

max)5

u2v

+ b; (77)

a = 0.4830 ± 0.0177; b = 0.3469 ± 0.0114; (78)

rof

L> 0.45; (79)

5 Curve Fitting Toolbox 1.1 — Copyright 2001–2002 The Math-Works, Inc — finds coefficients of a polynomial p(x) of degree n = 1fitting x-data in a LS sense (details in Draper and Smith 1998; Bevingtonand Robinson 1992)

being uv the velocity unit, and the goodness of fit readsSSE: 0.0884; R-square: 0.9798; adjusted R-square: 0.9794;RMSE: 0.03776. In addition, Eq. 77 is consistent witha = 0.500, b = 0.3424226 = log(11/5).

Two parallel straight lines in Fig. 9 correspond to a =0.500 and b = 0.250, 0.430 (±0.001), respectively, andinclude between them all the points belonging to classes4–10. It can also be seen that two movements of class 3 liewithin the above mentioned parallel straight lines, three fallimmediately outside, and the remaining five well outside. Inconclusion, 69 movements on a total of 86 (about 80%) areconsistent with a linear trend, expressed by Eq. 77, partic-ularized to a = 0.500 and b = 0.3424226 = log(11/5).The movements under consideration are expected to be themost efficient, in the sense that they are related to longer handpaths, and for this reason our attention shall be restricted tothe validity of the relation:

log(v2

max)1

u2v

= 1

2log

(v2max)5

u2v

+ log11

5; (80)

rof

L> 0.45; (81)

which may be assumed to an acceptable extent.

304 Emanuele Lindo Secco et al.

0 0.5 10

4

t [s]

2*T

/m=

v2 rms

[m2 /s

2 ]

0 0.5 10

4

tau [s/s]

mef

f[k

g]

-1 0 1

0

1

x

x [m]

Configurationn 27 (of 225) x

init= 0.745; y

init= 0

xfinal

= 0.05; yfinal

= 0.55rof

=0.8863Total admissibleconfigurations: 86

y[m

]

1234

-2 -1 0 1

-2

-1

0

1

log (vREALmax2 )

log

(v1m

ax2

)

ymax

= 0.500 x + 0.430

ymin

= 0.500 x + 0.250

yLS

= 0.483 x + 0.347

LS = least square estimation

Class 5

Class 4

Class 3

Class 2

Class 1

Class 6

Class 7

Class 8

Class 9

Class 10

a

b

Fig. 9 a The decimal logarithm of the mean square velocity, v2max = v2

Arm(τmax), related to approximation (5), versus its counterpart relatedto approximation (1), with regard to the whole Arm workspace made of allowed final hand position (86 cases in total), with equal movementtime. The dimensionless movement extents, rof /L, are divided in ten different classes, ordered in length. (class 1: 0.00 ≤ rof/L ≤ 0.15 −class 2: 0.15 ≤ rof/L ≤ 0.30 − class 3: 0.30 ≤ rof/L ≤ 0.45 − class 4: 0.45 ≤ rof/L ≤ 0.60 − class 5: 0.60 ≤ rof/L ≤ 0.75 −class 6: 0.75 − class 6: 0.75 ≤ rof/L ≤ 0.90 − class 7: 0.90 ≤ rof/L ≤ 1.05 − class 8: 1.05 ≤ rof/L ≤ 1.20 − class 9: 1.20 ≤ rof/L ≤1.35 − class 10 : 1.35 ≤ rof/L ≤ 1.50). The dashed line represents the best LS fit to classes 4–10 (64 points), implying rof /L > 0.45,y = ax + b, a = 0.4830 ∓ 0.0177, b = 0.3469 ∓ 0.0114, and the goodness of fit reads SSE: 0.0884; R- square: 0.9798; adjusted R-square:0.9794; RMSE: 0.03776. All the points interested to the above mentioned fit lie between two parallel straight (full) lines, defined by a = 0.500and b = 0.250, 0.430, (±0.001), respectively. In addition, two points belonging to class 3 lie within the above mentioned parallel straight lines,three fall immediately outside, and the remaining five well outside. In conclusion, 69 movements on a total of 86 (about 80%) are consistent with alinear trend, defined by a = 0.500, b = 0.3424226 = log(11/5). b Comparison of the kinetic energy approximations and the effective Arm mass— as defined in Appendix B — for movement 27 (of 225) of the Present-day man from (0,0.745) — stretch configuration — to (0.05,0.55) [m]

Optimization of two-joint arm movements: a model technique or a result of natural selection? 305

In dealing with approximations (3) and (4) a similar trendis found, where the values of the parameters are different andthe number of movements consistent with a linear trend isincreased.

In the special case, (vmax)5 = (vmax)1 = (vmax)∗1, Eq. 77

reduces to:

(1 − a) log(v2

max)∗1

u2v

= b; (82)

and the related, general expression, may be cast under theequivalent form:

(v2max)5

(v2max)1

=[

(v2max)1

(v2max)

∗1

]1/a−1

=[

v2of

(v2of )∗

]1/a−1

; (83)

where (vmax)1 = (15/8)vof , (vmax)∗1 = (15/8)v∗

of , via Eqs.59 and 65, according to approximation (1). Owing to Eqs. 16and 47, and keeping in mind that the movement time remainsunchanged for different hand displacements in the case underdiscussion, Eq. 83 translates into:

(v2max)5

(v2max)1

=[

ξ 2of

(ξ 2of )∗

]1/a−1

; (84)

finally, the explicit expression of the square velocity ratio,Imax, results from the combination of Eqs. 59, 65, and 84, as:

Imax = (v2max)5

v2of

= 225

64I ′

max; (85)

I ′max = (v2

max)5

(v2max)1

=(

ξof

ξ ∗of

)2(1/a−1)

; (86)

where the ratio, ξof /ξ ∗of , for an assigned Arm movement,

depends only on the Arm workspace in dimensionless coor-dinates

The particularization of Eq. 86 to the case under consid-eration, a = 1/2, reads:

I ′max =

(ξof

ξ ∗of

)2

; (87)

which provides a formal demonstration to Eq. 24.

Acknowledgements Thanks to prof. Giancarlo Alciati, Maria AntoniaCapitanio and Andrea Drusini for bibliographic indication and discus-sion. Francesca De Conti is acknowledged for careful reading of themanuscript and helpful suggestions. This work was funded by grantsfrom: MIUR

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