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International Journal of Humanoid Robotics c World Scientific Publishing Company

ANALYTICAL AND AUTOMATIC MODELING OF DIGITAL HUMANOIDS

FABRICE GRAVEZ* OLIVIER BRUNEAU** FETHI BEN OUEZDOU* *Laboratoire d’Instrumentation et de Relations Individu Syst` eme Universit´ e de Versailles Saint-Quentin 10-12 Avenue de l’Europe 78140 Velizy Email: [email protected] and [email protected] ** Laboratoire Vision et Robotique Ecole Nationale Sup´ erieure d’Ing´ enieurs de Bourges 10 Boulevard Lahitolle 18020 Bourges Cedex Email: [email protected] Received (Day Month Year) Revised (Day Month Year) Accepted (Day Month Year) The aim of this article is to achieve a parametric modeling of kinematics, geometrical and inertial properties of the various joints and links which constitute an anthropomorphic biped. The awaited result is the automatic creation of virtual models of humanoid bipeds while respecting intrinsically the inertial and geometrical distribution of each link, according only to two parameters : the total mass and the total height of the system. Future developments are to use the analytical parameters of masses and inertia in analytical dynamic models in order to control humanoids having different masses and heights. Keywords: analytical modeling; anthropomorphic bipeds; automatic parameterization.

1. Introduction The modeling of the human body was the subject of many studies primarily in the field of medicine and biomechanics. The emergence of means of powerful calculations on computers allowed the creation of digital modeling of human being as well as the simulation of its dynamic behavior during a given task. The work described in this paper was carried out within the framework of the digital modeling of humanoid robots for which the inertial and geometrical properties are close to those of the human being. However this study can find application fields such as : biomechanics, medicine or ergonomics. The first objective of this parametrization is to allow automatic extraction of virtual anthropomorphic structures having different morphologies with only two input parameters which are the mass and the height. In 1

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further developments, this approach aims to integrate all the analytical forms of the parameters (namely lengths, masses and inertia) in analytical dynamic models in order to control bipeds having different masses and heights. However, the complexity of the human body, the number of degrees of freedom and the number of muscles (more than 350 pairs which are mono-articular or multi-articular, synergistic or antagonistic) make a realistic modeling to be a difficult task. And this will be more difficult if it is wished to be systematic, analytical and completely parameterized modeling. Furthermore, to analyze the dynamic behavior of such systems which interact with the environment, it is necessary to model, not only the geometry (dimensions of the bodies), the kinematics of the joint axes (directions of the axes and distances between joint axes) but also the inertial distribution (masses, positions of the centers of mass, moments and products of inertia of each body). Even if one manages to model a particular anthropomorphic system, it is not efficient to build a complete new numerical model for which some parameters such as the height and the total mass of the biped or dimensions and masses of some of its limbs change. To answer these problems, the work presented in this paper, is based on two main ideas : -the use of primitives described analytically to carry out the completely parameterized modeling of an anthropomorphic biped with respect to the inertial and geometrical distribution of a human being. - a parametrization for each primitive respecting the center of mass position, the mass, the density and dimensions of each member for a standard human with possibility of modifying the dimensions of each limb which are analytical functions of two parameters only : the height and total mass of the biped. Based on these two ideas, it is possible automatically build digital models of anthropomorphic systems of various weights and sizes by giving only these two inputs : height and mass. This allows saving time and reducing errors in the development of multiple digital models. The work presented in this article is organized in the following way. Section 2 gives the kinematic modeling of the structure and the procedure needed to determine the masses of limbs function of the total mass. Section 3 focuses on the distance calculations. These distances, function of the total height, concern the limb lengths and the relative positions of two consecutive joint axes. Section 4 deals with the primitives used to model each limb of the digital humanoid. This includes the primitives kind and the analytical expression of the main dimensions. Some results showing the ability of the proposed approach are given in section 5. Finally, conclusions and further developments of this work are detailed in the concluding section.

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2. Kinematic modeling and mass distribution 2.1. Kinematic modeling of the system A preceding work carried out by Olivier Bruneau 2 has established optimal kinematics and number of degrees of freedom necessary to an anthropomorphic biped structure to carry out dynamic walking and running gaits. Based on qualitative analysis, an optimal structure was obtained by simplifying variation of human kinematics with the use of criteria based on geometrical, mass and joint clearances considerations. The adopted kinematics for the final model is made up of 25 active d.o.f. (Figure 1). Some passive joints are also included in the foot. The degrees of freedom are distributed as shown in Figure 1 : - 3 d.o.f. for the neck, - 2 × 3 d.o.f. for the two arms (2 d.o.f. for the shoulder, 1 d.o.f. for the elbow, 0 d.o.f. for the wrist), - 4 d.o.f. for the trunk, - 2 × 6 d.o.f. for the two legs (3 d.o.f. for the hip, 1 d.o.f. for the knee, 2 d.o.f. for the ankle).

Fig. 1. Selected kinematics of the biped

2.2. Mass distribution Biomechanical studies, express, most of the time, the mass of the limbs as a function of the total mass of the subject. A study related to the determination of the parameters of the body segments and their effects on calculation of the position of the center of pressure on the ground 9 shows that the Hanavan model 5 is particularly relevant compared to other studies carried out by Braune and Fischer 4 or by Dempster 3 . Thus, Hanavan technique is chosen to calculate the set of n (number of model elements) masses mi of each segment according to body mass M . This leads to a linear function : m i = ai M + b i

for

i = 1, 2, ..., n

(1)

For the adopted model, n is equal to 7. Hence, analytical expressions of these functions are the following :

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• • • • • • •

head + trunk : 0.47 × M + 5.44; the two arms : 0.08 × M − 1.31; the two forearms : 0.04 × M − 0.23; the two hands : 0.01 × M + 0.32; the two thighs : 0.18 × M + 1.45; the two tibiae : 0.11 × M − 0.86; the two feet : 0.02 × M + 0.68.

In addition, the selected model has to be coherent in term of distribution of limbs masses. this requires the following condition to be checked : n X

n  X

mi =

i=1

i=1

 ai M + b i = M

(2)

However, for the Hanavan model, because the fluidic and tissue losses are not taken into account, the sum of the limbs masses is not equal to the total mass M , but to 0.91 × M + 5.49. This leads to : n X

i=1 n X

ai = 0.91

(3)

bi = 5.49

(4)

i=1

the model of Hanavan is thus strictly valid only for M=61 kg. A modified model checking structurally the following condition for the coefficients αi and βi is established : n  n  X X mi = αi M + βi = M (5) i=1

i=1

the Equation (5) is checked for all total mass M values if : n X

i=1 n X

αi = 1

(6)

βi = 0

(7)

i=1

with i = 1, 2, ..., n, where n is the number of elements forming the model. To check structurally the Equation (5) and to preserve the ratio between two distinct αi values, which has to be equal to the ratio between two distinct ai values of the Equation (1) ( aaji = ααji ), we normalize the αi by taking : αi =

ai n X ai i=1

for

i = 1, 2, ..., n

(8)

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In this case, the Equation (6) is well checked. In addition, in order to obtain realistic limbs masses, the ith segment mass can be chosen to be equal to mref the value obtained by one of the studies developed i by Harless 6 , Braune & Fisher 4 , Dempster 3 7 , Clauser 8 , Winter 11 or Seward 10 . In addition, if one wishes results from the statistics, the ith segment mass mref is i chosen to be equal to the average of the values given by N reference studies :

= mref i

N X

mki

k=1

(9)

N

where : - mref is the final mass of a body element i and such as i N X

=M mref i

(10)

i=1

- N is the number of models, - mki is the mass of the body element i of the k th model. In this case, the αi (i =1,2,...,n) being already known and given by the Equation (8), the βi , checking linear relation mref = αi M + βi are easily determined : i − αi M βi = mref i

(11)

Using Equations (8) and (9) and relation (11), the latter becomes :

βi =

N X

mki

k=1

N

−

ai M n X ai

for

i = 1, 2, ..., n

(12)

i=1

mref i

Considering that βi = − αi M , using the Equations (6) and (10), the Equation (7) is well checked. Finallly, the values of the n αi and the values of the n βi (i =1,2,...,n) are respectively given by Equations (8) and (12). A modified model of Hanavan in the form mi = αi M + βi is then obtained. Its numerical values are as follows : • • • • • • •

head and trunk : 0.516 × M + 2.9 the two forearms : 0.088 × M − 1.8 the two arms : 0.044 × M − 0.5 the two hands : 0.011 × M + 0.375 the two thighs : 0.198 × M + 0.15 the two tibiae : 0.121 × M − 1.675 the two feet : 0.022 × M + 0.55

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Some body segments are not dissociated in the Hanavan model such as the head and the trunk. We use the other models referenced above to carry out a deeper decomposition of the set { head + trunk } into four elements : the head, the chest, the high pelvis and the low pelvis. The biomechanical studies, Winter 11 and Seward 10 , show that the head and the trunk represent respectively 13 % and 87 % of the mass of the set { head + trunk }. Hence the trunk can be divided into two parts : the chest and the pelvis representing respectively 34% × 87% and 66% × 87% of the set {head + trunk }. Moreover, in order to improve the model, we make the choice of dividing the pelvis into two equal shares : the high pelvis and the low one. Then each one represents 33% × 87% of the set {head + trunk }. This allows us to obtain the masses of the head, the chest, the high and the low pelvis. Thus, the limb masses mi are governed by the equation mi = αi M + βi with M the total mass of the system and the coefficients αi and βi given by Table 1.

Table 1. Limbs masses according to the total mass and coefficients αi and βi

Body

Limb mass mi

αi

βi

Chest 0.34 ∗ 0.87 ∗ (0.516M + 2.9) α1 = 0.153 β1 = 0.857 High Pelvis 0.33 ∗ 0.87 ∗ (0.516M + 2.9) α2 = 0.148 β2 = 0.833 Low Pelvis 0.33 ∗ 0.87 ∗ (0.516M + 2.9) α3 = 0.148 β3 = 0.833 Head 0.13 ∗ (0.516M + 2.9) α4 = 0.067 β4 = 0.377 Hand (0.011M + 0.375)/2 α5 = 0.011/2 β5 = 0.375/2 Arm (0.088M − 1.8)/2 α6 = 0.088/2 β6 = −1.8/2 Forearm (0.044M − 0.5)/2 α7 = 0.044/2 β7 = −0.5/2 Thigh (0.198M + 0.15)/2 α8 = 0.198/2 β8 = 0.15/2 Tibia (0.121M − 1.675)/2 α9 = 0.121/2 β9 = −1.675/2 Foot (0.022M + 0.55)/2 α10 = 0.022/2 β10 = 0.55/2 10 4 10 10 4 4 X X X X X X mi mi + 2 αi = 1 βi = 0 Total M= αi + 2 βi + 2 i=1

i=5

i=1

i=5

i=1

i=5

This approach allows us to determine the masses of each part of the humanoid with a good approximation which is equivalent to the one in Hanavan’s model. However, it is possible to change locally the choice of mi of one limb to take into account a prosthetic device for the model. Furthermore, to create digital mockups automatically, the following required constraint is always respected in our approach : the sum of all the limbs is equal to the total mass of the digital humanoid.

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3. Limbs dimensions calculation This section aims, initially, to express analytically the whole limbs lengths according to the total height of the digital humanoid. Then, in order to completely define dimensions of each segment following the three directions of space, the relationship between transverse dimensions of each segment is defined. 3.1. Length of main segments The objective is to express here the limbs lengths according only to one parameter : the height H of the system. Among all the existing models 3 4 5 6 7 8 10 11 the models of Winter 11 and Seward 10 seem to be the complete ones. The Table 2 gives the ratios used between the limbs lengths and the height H of a human being along the vertical direction. It should be noted that the length concerning the foot corresponds to the distance between the heel and the end of the toes. The distances between the joint axes, are deduced directly from the length of the

Table 2. Segments lengths according to the height along the vertical direction

Body

ratio

limbs length

Chest High Pelvis Low Pelvis Head + Neck Hand Arm Forearm Thigh Tibia Foot

p1 = 0.103 p2 = 0.118 p3 = 0.118 p4 = 0.182 p5 = 0.083 p6 = 0.183 p7 = 0.156 p8 = 0.245 p9 = 0.245 p10 = 0.152

p1 × H p2 × H p3 × H p4 × H p5 × H p6 × H p7 × H p8 × H p9 × H p10 × H

segment concerned except for five particular distances extracted from the models of Winter and Seward : • the distance, along the vertical direction, between the ankle joint axis and the lower part of the foot when this one is flat on the ground, is equal to 0.039 × H • the distance between the heel and the ankle joint along the axis of the foot is equal to 0.045 × H • the distance between the bottom of the low pelvis and the hip joint along the vertical direction is equal to 0.05 × H • the distance between the two hip joints along the horizontal direction is

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equal to twice the large radius of the thigh. Furthermore the circumferences of the higher ends of the two thighs in initial configuration are tangential. • the distance between the shoulder joint axis and the bottom of the chest along the vertical direction is equal to half the height h of the chest : 0.5×h, or, by using Table 2 : 0.5 × p1 × H (see Figure 2)

Fig. 2. Position of the shoulder joint

3.2. Ratios between dimensions in the transverse plan In addition to the limb lengths, it is necessary to determine dimensions in the transverse plan (horizontal), and this, independently of the height H and mass M of the system to be modeled. Accordingly, we carry out the following assumption. First of all, when the ponderal mass of a human being increases, the additional mass is distributed in a quasi-isotropic way in the transverse plan. This leads to assume that the relationship between two dimensions of the transverse plan of a body element is invariant independently of the morphology of the human being and its increase of weight. Thus, the ratio between dimensions along the axes i and k are assumed to be constant (see Figure (1)). The ratios λi of the dimensions along these two axes (i/k) for all body segments are given in Table 3. For the foot, this is the inverse proportion of dimensions which is given (k/i). To conclude the two preceding sections, we defined on the one hand all the limbs masses according to a single parameter (the total mass M of the system) on the other hand all the limbs lengths and distances between joints according to a single parameter (the total height H of the system). In addition we gave the ratio of limbs transverse dimensions by assuming that this ratio remained constant for each segment independently of M and H. The aim of the following section is to completely determine the modeling of the various segments, namely their geometrical shapes and dimensions respecting the inertial characteristics in order to obtain a nice approximation of the geometrical and mass distribution of a human being. a For

the foot, this ratio is written λi =

ti Li

with ti the width of the foot and Li its length

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Table 3. Ratio between two dimensions of the transverse plan

Body

Ratio (l/L)a

Chest High Pelvis Low Pelvis Head Hand Arm Forearm Thigh Tibia Foot

λ1 = 0.5 λ2 = 0.7 λ3 = 0.7 λ4 = 1 λ5 = 1 λ6 = 1 λ7 = 1 λ8 = 1 λ9 = 1 λ10 = 0, 33

4. Inertias of the segments In this section, a complete model for all segments needs to take into account : - the constraints given in the preceding section : distribution of the masses (according to M), distribution of the limbs lengths (according to H) and ratios λi of transverse dimensions, - the inertial constraints such as the positions of the centers of mass and the density of all the links. There are two ways to satisfy these constraints. The first one is to respect exactly the curves of the whole of the human segments and the nonuniform mass distribution. In this case, modeling can be extremely precise but can neither be analytical, nor be parameterized. The second one is to use geometrical primitives which can be analytically described with their dimensions and for which the tensors of inertia can be analytically calculated. The aim of this second method is the definition of a generic approximation of the human being. This is the second approach which was chosen. Thus, a full parameterized and analytical modeling of a system of the humanoid type according only to the parameters M and H is finally obtained. 4.1. Inertial constraints and choice of the primitives At first, the inertial constraints such as the positions of the centers of mass and the density of the various solids that must satisfy the modeled systems humanoids will be given. The positions of the limbs centers of mass significantly influence the dynamic behavior of the system. Based on the studies of Winter 11 and Hanavan 5 , they are given in Table 4 and expressed as a percentage of segments lengths starting from the proximal point of each segment. These ratios are represented by the variables γi .

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For the foot, it should be noted that the position of the center of mass is expressed as a percentage of the overall length of the foot starting from the heel.In addition, each segment has its own density. The volume and consequently the dimensions of the primitives are determined using the known masses. The densities of the various segments of the human body are given in Table 4. The objective is now to carry

Table 4. Positions of the centers of gravity expressed as a percentage of the limbs length starting from the proximal point of each segment and density

Body

Cog Position

Density kg/m3

Chest High Pelvis Low Pelvis Head + Neck Hand Arm Forearm Thigh Tibia Foot

γ1 = 50% γ2 = 50% γ3 = 50% γ4 = 50% γ5 = 50% γ6 = 43% γ7 = 43% γ8 = 43% γ9 = 43% γ10 = 43%

ρ1 = 1060 ρ2 = 1060 ρ3 = 1060 ρ4 = 860 ρ5 = 1160 ρ6 = 1080 ρ7 = 1130 ρ8 = 1060 ρ9 = 1080 ρ10 = 1100

out the choice to model the segments in the most realistic way. The approach of Hanavan 5 consists in making the assumption that the solids can be approached by simple geometrical primitives. This method is chosen in our modeling. However, different choices of primitives for the modeling of some segments in order to lay out a better biomimetic approximation were carried out : • the foot is modeled using a set made up of n right-angled parallelepipeds of volumes decreasing starting from the heel to the end of the toes. These primitives are connected through rotary joints provided with springs of torsion in order to finely model the interaction of flexible feet with the ground 1 . • the trunk is also modeled using several right-angled parallelepipeds articulated between them in order to lay out of a finer modeling than that made up with only one block. • the hand is not modeled any more with a sphere but with an ellipsoid of revolution in order to carry out a better biomimetic approximation. The Table 5 specifies the geometrical primitives used by Hanavan and by our approach for modeling the segments and the Figure 3 gives an outline of the system modeled using these primitives.

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Table 5. Geometrical primitives used by Hanavan and by our approach

Body

Hanavan

Head + Neck

Ellipsoid of revolution

Trunk

Elliptic cylinder

Arm Forearm Hand Thigh Tibia Foot

Truncated cone Truncated cone Sphere Truncated cone Truncated cone Truncated cone

Primitives used Ellipsoid of revolution Chest → Parallelepiped High Pelvis → Parallelepiped Low Pelvis → Parallelepiped Truncated cone Truncated cone Ellipsoid of revolution Truncated cone Truncated cone Set of parallelepipeds

Fig. 3. Modeling of the body segments using the primitives

4.2. Analytical determination of segments dimensions To summarize what was developed in the preceding sections, the constraints which must respect dimensions of the geometrical primitives are for each segment i as

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follows : a) the position of the center of mass of each segment i with regard to the proximal point expressed by γi , percentage of the height hi or of the length Li of each segment (see Table 4) b) the height hi or the length Li of each segment i (expressed by pi , see Table 2) c) λi the ratio of dimensions of the primitives in the transverse plan (given in Table 3) d) the mass mi function of αi and βi coefficients (see Table 1) e) the density ρi (given by Table 4) Condition a) is structurally checked for the chest, the high pelvis and the low one (modeled with right-angled parallelepipeds) and for the head and the hand (modeled with ellipsoids of revolution). Indeed, γi = 50 % ∀ i = 1, 2, 3, 4, 5. Condition a) is not automatically checked for the arms, the forearms, the thighs and the tibiae which are modeled with truncated cones and for the feet which are modeled with a set of right-angled parallelepipeds. For i = 6, 7, 8, 9, 10, condition a) will be thus written : hGi = γi Li

(13)

Condition b) gives the height hi of each parallelepiped, the height 2Li of each ellipsoid of revolution, the height Li of each truncated cone, and the length Li of each foot with the following equations : hi = p i H

for

2Li = pi H L i = pi H

i = 1, 2, 3

(14)

i = 4, 5

(15)

i = 6, 7, 8, 9, 10

(16)

for

for

Condition c) is structurally checked for the head and the hands (modeled with ellipsoids of revolution), and for the arms, the forearms, the thighs and the tibiae (modeled with truncated cones) since λi = 1 ∀i = 4, 5, 6, 7, 8, 9. For the chest, the high pelvis and the low pelvis which are modeled with right-angled parallelepipeds, condition c) on the ratio of transverse dimensions gives : λi =

li Li

for

i = 1, 2, 3

(17)

Condition c) on the ratio between depth of the foot ti and length of the foot Li is given by the following equation : λi =

ti Li

for

i = 10

(18)

It should be noted, in addition, that the conditions d) and e) concerning the mass mi and the density ρi of each segment are taken into account in the calculation of

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geometrical dimensions of the primitives using the calculation of the volume Vi of the segment concerned i : αi M + βi mi = for i = 1, 2, ..., 10 (19) Vi = ρi ρi The height or the length hi , Li and 2Li of each segment i being already known (function of H), the objective is then to analytically calculate for each primitive the two other dimensions located in the transverse plan. In what follows, we successively give analytical forms of the primitives dimensions. 4.2.1. Right-angled parallelepiped (chest, high and low pelvis)

Fig. 4. Right-angled parallelepiped

hi being known according to H (Eq. (14)), it remains to determine Li and li (see Figure 4). The volume of a right-angled parallelepiped is given by : Vi = L2i λi hi

(20)

By equalizing the Equations (19) and (20), we obtain : s αi M + βi Li = ρi λi hi Then, using (17), the second transverse dimension is obtained : s (αi M + βi )λi li = ρi h i

(21)

(22)

By injecting now the Equation (14) in (21) and (22), we explicitly obtain finally the two dimensions in the transverse plan according to fully known parameters : s αi M + βi for i = 1, 2, 3 (23) Li = ρi λi pi H li =

s

(αi M + βi )λi ρi p i H

for

i = 1, 2, 3

(24)

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4.2.2. Ellipsoid of revolution (head and hands)

Fig. 5. Ellipsoid of revolution

2Li being known according to H (see Eq. (15)) , it remains to determine Ri according to H (see Figure 5). The volume of an ellipsoid of revolution is the following : Vi =

4 2 πR Li 3 i

(25)

Ri =

r

(26)

From (25), we obtain : 3Vi 4πLi

From the Equation (15) and Equation (19) injected into the Equation (26), we explicitly obtain finally the transverse parameter of the ellipsoid of revolution according to completely known parameters : s 3(αi M + βi ) for i = 4, 5 (27) Ri = 2πpi Hρi 4.2.3. Truncated cone (arms, forearms, thighs and tibiae)

Fig. 6. Truncated cone

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The constraint given by Equation (13) is not automatically checked for an unspecified choice of truncated cone. Thus we have to calculate dimensions of the truncated cone in order to satisfy this constraint of center of mass position. Knowing Li according to H (see Equation 16), the objective is thus to determine the large radius Ri and the small radius ri of each cone of length Li (see Figure 6) in order to satisfy : - the condition Ri ≥ ri - the selected center of mass position hGi given by the Equation (13) where the values γi for i = 6, 7, 8, 9 are given by the able 4) - the selected volume Vi of each truncated cone, corresponding to the constraint on the mass and the density of each modeled body given by the Equation (19) First of all, the volume of a truncated cone is given by : Vi =

1 πLi (Ri2 + Ri ri + ri2 ) 3

(28)

In addition, the center of mass position is given by :

hG i = or by using the ratio γi =

1 (Ri2 + 2Ri ri + 3ri2 ) Li 4 (Ri2 + Ri ri + ri2 )

(29)

hG i : Li

γi =

1 (Ri2 + 2Ri ri + 3ri2 ) 4 (Ri2 + Ri ri + ri2 )

(30)

The variable Vi being related to known parameters (see Equation (19)) and γi being known (values γi for i = 6, 7, 8, 9 given by the Table 4), it is simple to calculate Ri and ri according to Vi and γi and checking the Equations (28) and (30) to obtain the two radius according to known parameters. The resolution of the nonlinear system composed of the two Equations (28) and (30) with two unknown factors Ri and ri , finally gives : r KVi (31) ri = πLi Ri =

r

Vi (3(4γi − 1) − 2K) πLi K

(32)

q 2(6γi − 6γi2 − 1)

(33)

with : K = 6γi − 1 −

1 1 < γi < , which is checked 4 2 for the arms, forearm, thighs and tibiae according to Table 4). It should be noted that this result is valid only for

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By introducing Equation(19) in (31) and (32), we obtain finally the two transverse parameters of the truncated cone explicitly according to fully known parameters :

ri =

Ri =

s

s

K(αi M + βi ) πLi ρi

for

(αi M + βi ) (3(4γi − 1) − 2K) ρi πLi K

i = 6, 7, 8, 9

for

i = 6, 7, 8, 9

(34)

(35)

with : K = 6γi − 1 −

q 2(6γi − 6γi2 − 1)

(36)

4.2.4. Set of right-angled Parallelepipeds

Fig. 7. Set of right-angled parallelepipeds

The objective is to calculate the dimensions lk , hk and tk of the set of the parallelepipeds composing the foot according to known parameters (see Figure 7). The foot is composed of n right-angled parallelepipeds. We note hk , lk , OGk and Vk respectively the height, the length, the center of mass position and the volume of each right-angled parallelepiped. Knowing the overall length of the foot Li , the length lk of each right-angled parallelepiped (see Figure 7) is given by : l k = δk L i

(37)

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   0 ≤ δk < 1     δ0 = 0   δk > δk−1 with  δk = nk for the case of regular cutting     foot with k ∈ [0; n] and n the number    of right-angled parallelepipeds b

By injecting (16) in (37), we finally obtain lk function only of known parameters : l k = δk p i H

for

i = 10

(38)

In addition, by noting hsup the height of the first parallelepiped (hsup = h1 ) and hinf the height of the n + 1th virtual parallelepiped (hinf = hn+1 ), the height hk of each right-angled parallelepiped (see Figure 7) is given by : hk = (hinf − hsup )δk−1 + hsup with δ0 = 0

(39)

Knowing the total volume of the foot Vi and the center of mass position of the foot hGi , the objective is to find hk and lk of each right-angled parallelepiped according to H, Li and ρi so that : n X Vi = Vk (40) k=1

with Vk the volume of each right-angled parallelepiped k and n X OGk mk k=1

(41) mi with mi the mass of the foot and mk the mass of each right-angled parallelepiped k. Knowing the depth of the foot ti = tk , the volume Vk of each parallelepiped k is written then : hG i =

Vk = hk (lk − lk−1 )tk with l0 = 0

(42)

By introducing the expressions (37) and (39) in (42) then by integrating the result into (40), we obtain, by noticing that tk = ti = λi Li : Vi (43) Ahinf + Bhsup = 2 Li λi  n X    A = δk−1 (δk − δk−1 )   k=1 with n X    (1 − δk−1 )(δk − δk−1 ) B =   k=1

b the coefficients δ can be chosen differently with non-regular cutting foot if it is needed by the k user of the software.

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In addition, taking into account that the density is the same one for each parallelepiped, the Equation (41) becomes :

hG i =

n X

OGk Vk

k=1

(44)

Vi

Furthermore the position of the center of mass of a parallelepiped k along the axis x is given by : OGk =

lk + lk−1 with l0 = 0 2

(45)

Thus, the expressions (37) and (39) and the relation tk = ti = λi Li introduced in Equation (42) gives a new equation. Then, by introducing this new equation and the Equations (45) and (13) in Equation (44), we have : Chinf + Dhsup =

with

2γi Vi L2i λi

(46)

 n X    C = δk−1 (δk − δk−1 ) (δk + δk−1 )   k=1 n

X    D = (1 − δk−1 )(δk − δk−1 ) (δk + δk−1 )   k=1

It is then simple to solve the system of two Equations (43) and (46) with two unknown factors where these unknown factors are hsup and hinf . We obtain then : Vi 2γi B − D L2i λi CB − DA Vi C − 2γi A = 2 Li λi CB − DA

(47)

hinf = hsup

By introducing (16) and (19) in (47) we obtain hinf and hsup according to known parameters : αi M + βi 2γi B − D ρi p2i H 2 λi CB − DA αi M + βi C − 2γi A = ρi p2i H 2 λi CB − DA

hinf =

for

i = 10

hsup

for

i = 10

It should be noted that the determinant CB − DA is never equal to zero because of the conditions on the δk : δ0 = 0 δk < δk+1 δn = 1

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Knowing hsup and hinf , all the hk are completely determined by the Equation (39). Dimensions lk , hk and tk of the set of the parallelepipeds composing the foot are function of known parameters. 5. Results We calculated in the preceding sections various dimensions of the primitives used for the modeling of anthropomorphic systems according to known structural parameters (density, ratio of transverse lengths, position of the centers of gravity) and of the two parameters giving the total height H and the total mass M of the system. However it is possible to change locally the mass mi , the height hi of each parallelepiped, the height 2Li of each ellipsoid of revolution, the height Li of each truncated cone or the transverse dimensions of these primitives if it is needed by the user of the software for instance for modeling of integrated prosthesis. We summarize here, in Table 6, the set of the mathematical formulas allowing to define geometrical dimensions of each body given by Figure 8 according to the biomechanical data.

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(a) Right-angled parallelepiped

(b) Ellipsoid of revolution

(c) Truncated cone

(d) Set of right-angled parallelepipeds

Fig. 8. Primitives used for the body segments

All the parameters defined in the preceding sections allow us to build bipeds of different masses and heights only by changing two parameters : the mass and the height of the system. Some bipeds are shown by Figure 9.

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Table 6. Segments modeling .Primitives used Body elements

Dimensions of bodies

.Known parameters

Right-angled parallelepiped chest (i = 1) high pelvis (i = 2) low pelvis (i = 3)

Figure 8(a) H, M αi , βi , ρi , pi λi

Ellipsoid of revolution head (i = 4) hands (i = 5)

hi = pi H v uα M +β u i i Li = t ρi λi pi H v u u (αi M + βi )λi li = t ρi pi H

Li =

Figure 8(b) H, M αi , βi , ρi , pi

Ri =

pi H 2 v u 3(α M + β ) u i i t 2πpi Hρi

K = 6γi − 1 − Truncated cone arms (i = 6) forearms (i = 7) thighs (i = 8) tibiae (i = 9)

Figure 8(c) H, M αi , βi , ρi , pi γi

q

2(6γi − 6γ 2 − 1) i

Li = pi H v u u K(αi M + βi ) ri = t πLi ρi v u u (αi M + βi )
3(4γi − 1) − 2K Ri = t ρi πLi K A =

n X

δk−1 (δk − δk−1 )

k=1 B =

Set of right-angled parallelepipeds feet (i = 10)

Figure 8(d) H, M αi , βi , ρi , pi λi ,γi , n

C =

n X (1 − δk−1 )(δk − δk−1 ) k=1 n X

k=1 D =

  δk−1 (δk − δk−1 ) δk + δk−1

n   X (1 − δk−1 )(δk − δk−1 ) δk + δk−1 k=1

hinf =

hsup =

αi M + βi 2γi B − D ρi p2 H 2 λi CB − DA i αi M + βi C − 2γi A ρi p2 H 2 λi CB − DA i

hk = (hinf − hsup )δk−1 + hsup lk = δk p i H

21

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Fabrice GRAVEZ, Olivier BRUNEAU, Fethi BEN OUEZDOU

(a) 1.40 m;50 kg

(b) 1.40 m;75 kg

(c) 1.40 m;100 kg

(d) 1.78 m;50 kg

(e) 1.78 m;75 kg

(f) 1.78 m;100 kg

(g) 2.10 m;50 kg

(h) 2.10 m;75 kg

(i) 2.10 m;100 kg

Fig. 9. Evolution of the biped according to the size and the mass

6. Conclusion In this paper, we have obtained a completely parameterized and analytical modeling of a system of the humanoid type according only to the parameters M and

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H. The direct application of this idea is the possibility of automatically building digital models of anthropomorphic systems of various weight and sizes by giving only these two inputs : height and mass, thus avoiding long and tiresome work and source of errors of construction segment by segment. Then, it will be possible to simulate the dynamic behavior of a great variety of anthropomorphic bipeds for different tasks such as walking, running, standing, sitting and sliding. In further developments, this approach will be also used to integrate all the analytical calculated parameters (namely lengths, masses and inertia) in analytical dynamic models in order to control bipeds having different masses and heights. This study can however find applications in others fields such as : biomechanics to improve comprehension of the transmission of the efforts and movements in the human body during various human activities by simulating the dynamic behavior of the human being; medicine for the automatic control of pathologies of the locomotor apparatus; ergonomics through the optimization of working stations; the synthesis of creatures moving in virtual worlds by simulating realistic dynamic behaviors. References 1. O. Bruneau, F. B. Ouezdou and J.-G. Fontaine, Dynamic Walk of a Bipedal Robot Having Flexible Feet, in Proc. IEEE International Conf. on Intelligent Robots and Systems (IROS), (Maui, USA, 2001),pp. 512-517. 2. O. Bruneau, F. B. Ouezdou and J.C. Guinot, Dynamic Simulation Tool for Biped Robots CISM - IFToMM - Symposium on Theory & Practice of Robots & Manipulators (RoManSy), (Paris, France, 1998), pp 361-368. 3. W. T. Dempster, Space requirements of the seated operator Aerospace Medical Research Laboratories, WADC-55-159, AD-087-892, (Wright-Patterson Air Force Base, Dayton, Ohio, USA, 1955). 4. O. Fischer and W. Braune, Determination of the Moments of Inertia of the Human Body and Its Limbs Springer-Verlag New York, Incorporated, (1988). 5. E. P. Hanavan, A Mathematical Model of the Human Body Aerospace Medical Research Laboratories, AMRL-TR-64-102, AD-608-463, (Wright-Patterson Air Force Base, Dayton, Ohio, USA, 1964). 6. E. Harless, he Static Moments of the Component masses of the Human Body Aerospace Medical Research Laboratories, English Translation NTIS AD 279 649, (WrightPatterson Air Force Base, Dayton, Ohio, USA, 1860), Transactions of the MathPhysica, Royal Bavarian Academy Of Science, March, (1962). 7. W. T. Dempster, The anthropometry of body action Annals New York Academy of Sciences, (1956),pp. 559-585. 8. C. E. Clauser and J.T. McConville and J.W. Young, Weight, volume and center of mass of segments of the human body Aerospace Medical Research Laboratories, AMRL-TR69-70, AD-710-622, (Wright-Patterson Air Force Base, Dayton, Ohio, USA, 1969). 9. S. Koozekanami and J. Duerk, Determination of Body Segment Parameters and their Effect in Calculation of the Position of Center of Pressure During Postural Sway, IEEE - Transactions on Biomedical Engineering,(1985). 10. D. W. Seward and A. Bradshaw and F. Margrave, The Anatomy of a Humanoid Robot, Robotica, volume 14, Cambridge University Press, 437-443 (1996). 11. D. A. Winter, Biomechanics of Human Movement, (John Wiley & Sons, New York, 1979).

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Fabrice GRAVEZ, Olivier BRUNEAU, Fethi BEN OUEZDOU

Fabrice Gravez received his M.S. and Ph.D. degrees from the University Pierre et Marie Curie (Paris 6, France), in 1998 and 2003, respectively. Since 2003, he is Researcher at the laboratory LIRIS (Laboratoire d’Instrumentation et de Relations Individu Syst`eme), and is working on the ROBIAN biped project. FabriceGravez is the author of over 7 technical Publications and proceedings. His research interests include Humanoid robots, Human locomotion system and dynamic simulation.

Olivier Bruneau received his M.S. and Ph.D. degrees from the University Pierre et Marie Curie, Paris 6, in 1994 and 1998, respectively. From 1998 to 1999, he was Researcher at the Laboratoire de Robotique de Paris in the field of dynamic bipedal robots. Since 1999 he is Associate Professor in robotics and theoretical mechanics at the E.N.S.I. de Bourges. Olivier Bruneau is the author of over 20 technical Publications concerning the dynamic walking systems. His research interests include dynamics of locomotion, humanoid robots and learning methods of walking for the bipedal robots. Olivier Bruneau currently works at the laboratoire Vision et Robotique of Bourges. He is an active member of the research group working on the control of the biped robot called ”RABBIT” for walking and running within the framework of a ROBEA CNRS project. He works on the anthropomorphic biped robot called ROBIAN in collaboration with the the laboratory LIRIS (Laboratoire d’Instrumentation et de Relations Individu Syst`eme) of Versailles. He was a coorganizer of the 12th International Symposium on Measurement and Control in Robotics, towards Advanced Robot Systems and Virtual Reality, taking place in Bourges from the 20 to June 22, 2002.

Fethi Ben Ouezdou received his M.S. from the Ecole Nationale Sup´erieure des Arts & M´etiers and Ph.D. degrees from the University Pierre et Marie Curie (Paris 6), in 1986 and 1990, respectively. From 1990 to 1991, he was Researcher at the University Paris 6, and worked on the project of quadruped locomotion. From 1991, he was Associate Professor, at the University of Versailles. Now, he holds Position at the Departement of Mechanical Enginnering of the University of Versailles. He is also the head of the Departement. He participated to the creation of LIRIS laboratory (Laboratoire d’Instrumentation et Relation Individu Syst`eme) which belongs to the CNRS and the University of Versailles. He becomes Professor at the University of Verailles in February 2005. He is the leader of ”Biomechanics and Simulation” group in the

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LIRIS Laboratory. F.B. Ouezdou is the author of over 55 technical publications, proceedings, editorials and books. His research interests include humanoid robots, biomechanics of locomotion systems and biologically inspired design and control system. He is the coordinator of the collaborative research network SHARMES (Simulation and Modeling) at the University of Versailles, France. He is an active member of the History of Mechanisms commission of IFTOMM, and was the coordinator of the ROBIAN Project (Biped Robot) funded by the Ministry of Research France.