Construction and Animation of Anatomically Based Human Hand Models

Eurographics/SIGGRAPH Symposium on Computer Animation (2003)D. Breen, M. Lin (Editors)

Construction and Animation of Anatomically BasedHuman Hand Models

Irene Albrecht, Jörg Haber, and Hans-Peter Seidel

MPI Informatik, Saarbrücken, Germany

a) b) c) d)

Construction and animation of the reference hand model: a) plaster cast of a human hand ready for 3D scanning; b) assembly ofskin mesh and individual bone meshes; c) and d) skin tissue deformation during animation.

AbstractThe human hand is a masterpiece of mechanical complexity, able to perform fine motor manipulations and pow-erful work alike. Designing an animatable human hand model that features the abilities of the archetype createdby Nature requires a great deal of anatomical detail to be modeled. In this paper, we present a human hand modelwith underlying anatomical structure. Animation of the hand model is controlled by muscle contraction values.We employ a physically based hybrid muscle model to convert these contraction values into movement of skin andbones. Pseudo muscles directly control the rotation of bones based on anatomical data and mechanical laws, whilegeometric muscles deform the skin tissue using a mass-spring system. Thus, resulting animations automaticallyexhibit anatomically and physically correct finger movements and skin deformations. In addition, we present adeformation technique to create individual hand models from photographs. A radial basis warping function is setup from the correspondence of feature points and applied to the complete structure of the reference hand model,making the deformed hand model instantly animatable.

Categories and Subject Descriptors(according to ACM CCS): I.3.5 [Computer Graphics]: Computational Geom-etry and Object Modelinghierarchy and geometric transformations, physically based modeling; I.3.7 [ComputerGraphics]: Three-Dimensional Graphics and Realismanimation.

1. Introduction

Our hands play a vital role in every aspect of our daily lives.We need them for eating, playing, writing, working, commu-nicating, in a nutshell: for everything. Most people take theeffectiveness and dexterity of their hands for granted withoutbeing aware of their complicated structure and the high level

of optimization. However, there is more than the mechanicalperfection to our hands:

Often the hands will solve a mystery thatthe intellect has struggled with in vain.

CARL GUSTAV JUNG

c© The Eurographics Association 2003.

I. Albrecht, J. Haber, and H.-P. Seidel / Construction and Animation of Anatomically Based Human Hand Models

Stretching from spiritual significance (e.g. blessing, palmreading), over idiomatic expressions (e.g. “to put one’s lifein someone’s hands”), to the act of shaking hands, not onlyfor greeting but also for expressing feelings like gratefulnessor sympathy, the central importance of hands is mirrored ina broad spectrum of symbolism.

In spite of the ubiquity of hands in daily life, but probablydue to their immense complexity, hands have not receivedmuch attention in computer graphics. Although the numberof possible applications is large, sophisticated hand modelshave not yet been developed. Virtual hand models can beused for teaching and practicing sign language, and for visu-alizing translations from speech or text into sign language.They come in handy for teaching other manual skills as well,for instance operating machines, and for giving online usageor assembly instructions. In immersive environments, handmodels are required in the simulation of the haptic dimen-sion: for manipulating a virtual object, visual feedback ishelpful. Close-ups in CG movies and games ask for naturalmodels with a lot of detail and convincing movements. Highdemands arise also from the medical field. In systems forhand surgery planning, a maximum of functionality of thehand must be provided to aid the surgeon in his decisions.

1.1. Contributions

In this paper, we present the following main contributions:

• a human hand model with anatomical structure, suitablefor real-time animation using physics-based simulation ofmuscles and elastic skin properties (Section 3);• a hybrid muscle model that comprises pseudo muscles and

geometric muscles. Pseudo muscles directly control therotation of bones based on anatomical data and mechani-cal laws, while geometric muscles deform the skin tissueusing a mass-spring system (Section 4);• a deformation technique based on feature points to warp

the complete structure of a reference hand model to an in-dividual hand model taken from a photograph (Section 5).

Our motivation to choose a physics-based approach con-trolled through muscle contraction values is given by themain advantage of such an approach: animations are anatom-ically and physically correct by default. The user does nothave to take care of anatomical or physical limitations whenpositioning the fingers or setting up dynamics of an anima-tion.

2. Related Work

2.1. Anatomy, Biomechanics, and Anthropometry

Research in anatomy and biomechanics has shown that thehuman hand is a very intricate and elegant mechanical de-vice, where many dedicated parts cooperate in an highly op-timized interplay to form a powerful union. Information on

the anatomical building blocks of the hand can be found in il-lustrated anatomy books35 or in more detail in11. The bookby Brand and Hollister8 is inclined more towards biome-chanics: meant as a textbook for hand surgeons, for instancewhen planning a tendon transfer operation, it provides a thor-ough description of the functioning of the hand.

Landsmeer24 developed a physics-based model for deter-mining tendon excursion from joint angle, depending on theway the tendon crosses the joint. Starting from this model,he develops criteria of how muscles must be arranged in ajoint system to be able to move the joints in any given way.In 2, tendon excursions of the index finger muscles have beenmeasured and the corresponding moment arms have beencomputed using Landsmeer’s tendon models.

A kinematic model for flexion and extension of the fingershas been developed by Lee and Kroemer25. Their model isbased on the assumption that the moment arms of the ten-dons at the joints are constant. Considering external forcesaffecting the joints, they compute the finger strength for thegiven joint configuration.

In 4, the authors discuss a biomechanical model of the en-tire hand encompassing all principal muscles and degreesof freedom. Muscles are modeled by weightless expandablethreads. Weightless non-expandable loops surrounding thejoints describe the “line-of-action” of muscles. The authorsfound only the muscles at the wrist to possess some redun-dancy, i.e. the same wrist position can be obtained by severalmuscle combinations. To overcome this redundancy, muscleeffort is minimized.

For evaluation of the prehensile capabilities of the hu-man hand, Buchholz and Armstrong9 proposed a kinematicmodel based on collision detection between ellipsoids repre-senting the skin surface of the hand segments. Joint flexionangles and skin deformation for power grasp of ellipsoidalobjects are predicted and rendered as vector graphics.

The anatomical computer-generated hand model de-scribed in39 consists of bones, tendons, and soft tissue. Thelatter is modeled by an ellipsoid-shaped mass-spring net-work at every phalanx, and as an appropriately shaped mass-spring system at the palm. The outer surface of these net-works constitutes the skin. Tissue deformation during fingermovement is determined using a predictor-corrector method,which also takes into account incompressibility and collisionconstraints. Tendons are present via their mechanical effects,not geometrically. Their feedback action is modeled throughsprings opposing joint motion. The fingers are positioned au-tomatically by energy minimization. Although this modelingapproach seems to be somewhat similar to ours, there areseveral distinctions: the muscle force model we present ismore comprehensive, we model muscles additionally as ge-ometric objects with impact on the shape of the skin, and thetriangle mesh we use as skin has been obtained from a rangescan of a human hand.



Brandet al. 7 performed measurements of hand and fore-arm muscles to obtain potential excursion and relative ten-sion of the muscles. Potential excursion is the difference be-tween maximal stretch and maximal contraction of a muscle,i.e. the distance through which a muscle is able to contractactively. They found the potential excursion to be equal tothe resting length of the fibers of the muscle. Relative tensiondenotes the proportional tension of a muscle w.r.t. the over-all amount of possible tension of all studied muscles. Thesenumbers differ far less among individuals and within eachindividual over time than the absolute strength of a muscle.

Anthropometrical measurements have been carried out byWagner43, who extensively measured size and joint mobilityof the hands of pianists. He compared his results to studiesabout other musicians and non-musicians and found that ingeneral piano players have greater mobility in their handsthan the average.

2.2. Hand Models in Computer Graphics

In computer graphics, hand models have been developed forseveral typical applications. The most prominent applicationareas are model-based tracking (see for instance45 for anoverview), interactive grasping, and simulation systems usedfor e.g. surgery planning.

In 33, a simple volume-based animatable hand model con-structed from geometric primitives has been employed fortracking. The model includes anthropometrical and biome-chanical constraints: the size of the palm is correlated to thelength of the fingers and phalanges. Biomechanical laws de-termine the valid range and interdependencies of joint mo-tion, thereby reducing the number of degrees of freedom ofthe model. Heap and Hogg15 have built a statistical handshape model from simplex meshes fitted to MRI data fortheir tracking system. For model-based finger motion cap-turing, Lin et al. 28 employ a learning approach for the handconfiguration space to generate natural movement.

A parametric hand model has been designed for the semi-automatic grasping approach in29. In this model, skinningis based on joint-dependent local deformations, taking intoaccount rounding at joints and bulging. Another approachto grasping is described in14. The system uses finite elementsimulation of the skin and the grasped object in order to sim-ulate both skin and object deformations due to contact. In36,a simple hand model is described that likewise incorporatesconstraints on the movement range of joints. It was devel-oped for the animation of semi-automatic knowledge-basedgrasping, where objects are approximated by primitives withindividual grasping approach parameters. Another heuristicgrasping system has been introduced in37. Objects are storedtogether with primitives associated with the graspable partsof the object. The final position of the hand is determined byinverse kinematics and collision detection. Huanget al. 16

extended the previous model. A multi-sensor approach for

collision detection has been added, where the sensors areconstituted by spheres attached to the joint. Collision detec-tion between hand and object is performed with these sen-sors to naturally place the hand around the object.

In 21, artificial intelligence is used to position hand andwrist of a virtual violinist. Finger positions are determinedby best-first search, while wrist position and orientation aredecided by a neural network. Muleroet al. 32 present ananthropomorphic finger model with a tendon transmissionsystem based on pulleys and a position controller. The con-troller is modeled by a neural network and transforms tendonpull into joint motion. The system can work in an agonist-antagonist fashion. A model of the hand and arms based onmanifold mappings has been proposed by Kuniiet al. 23.They also consider inter-joint dependencies. Moccozetetal. 31 use Dirichlet free-form deformations (DFFDs) to sim-ulate the tissue and muscle layer between skin and bones.Muscles are not considered directly, but the use of DFFDsallows the authors to model wrinkles at joints and bulging ofsegments dependent on the angle of rotation of the respec-tive proximal joint. Ipet al. 18 have built an anatomy-basedhand model with muscles based on the work presented in4.The hand is modeled as a collection of hand segments con-nected by joints, where muscles are weightless expandablethreads. Soft tissue, tendons, and ligaments are not modeledexplicitly. Given the initial and final hand posture, the sys-tem is able to generate the in-between states. For describingthe hand postures, the authors use the Hand Action Cod-ing System17, a collection of muscle-based Hand ActionUnits that encode hand positions. Thompsonet al. 42 pre-sented a hand model capable of calculating relative musclelength, distance between pulley point / point of origin andtransformed insertion point, moment arm, and moment po-tential for hand muscles during motion. A wireframe skele-ton model is rendered together with the tendons, while thesingle parameters are displayed by bar graphs. Since the sys-tem was designed to aid medical doctors in planning ten-don transfers, replacement of one tendon/muscle unit by an-other can be simulated. In30, the joint movements of a handmodel composed of rigid bodies are constrained by biome-chanical laws. The model was designed for use in animatingAmerican Sign Language. An approach for skinning a handskeleton using eigendisplacements has been proposed in22.The resulting hand model can be animated in real-time usinggraphics hardware.

In addition to the literature on human hand models, sev-eral approaches for anatomical modeling and physics-basedanimation of human faces and bodies have been presented.In particular, the mass-spring system approaches in41, 26, 27

and the muscle models proposed in38, 44, 19 are of interestwithin the scope of this paper.

Concerning the use of feature points for model deforma-tion, the work presented in40 should be mentioned: anatom-ical models of articulated creatures equipped with feature



points can be morphed to obtain new models of similarshape. Based on input measurements, the structure hierar-chy, bones, and muscles of the original model are deformed.From these deformed components, a new set of featurepoints is generated to deform the skin mesh using a localinterpolation approach based on radial basis functions.

3. The Reference Hand Model

The central component of our system is a prototype handmodel with anatomical structure, which is denoted as ourreference hand modelin the following. The building blocksof our reference hand model are:

• theskin surface, which is represented by a triangle meshconsisting of 3000 triangles;• theskeletonof the hand, composed of 29 triangle meshes

corresponding to the individual bones of the human handand forearm (cf. Figure 1);• a set ofvirtual muscles, which are embedded in between

the skin surface and the skeleton;• a mass-spring system, interlinking the skin, skeleton, and

muscles;• a joint hierarchy, which matches the structure of the skele-

ton, with an individually oriented coordinate system ateach joint center defining valid axes of joint rotation.

The skin mesh of our reference hand model has been ob-tained by scanning a plaster cast of a human hand, see alsothe figure on the first page. The resulting triangle mesh hasbeen reduced to a size of 3000 triangles to allow for real-timesimulation of skin deformations. The triangle meshes of theindividual bones have been taken from a publicly availableskeleton model1 and scaled to match the proportions of theskin mesh.

Using the hierarchy of coordinate systems, we can modelthe degrees of freedom (DOFs) for each joint easily. Theonly joints we do ignore are the joints between the individ-ual wristbones (cf. Figure 1). This is justified, since theircontribution to the overall movement is negligible. The PIPand DIP joints of the fingers and the IP joint of the thumbhave one DOF each for flexion / extension, while the MCPjoints of the fingers have a second DOF for adduction (to-wards the middle finger) and abduction (away from the mid-dle finger). In addition, depending on the current amount offlexion or extension, the finger MCP joints exhibit some ro-tation around their long symmetry axis. Likewise, the CMCjoint of the thumb is sometimes said to have three degrees ofmotion 11. The impression of rotation around a third axis isevoked, because the two real axes are not completely perpen-dicular to each other (see8, p. 41). To overcome the restric-tion of two orthogonal DOFs, we model the MCP joints ofthe fingers and the thumb CMC joint as having three DOFs.The muscles must be designed to accommodate the depen-dencies between the flexion / extension and rotation axes: ifa muscle flexes or extends the joint, it must also rotate it to

Figure 1: Bones of the human hand and forearm. Individ-ual bone names are underlined. Themetacarpal, proximalphalanx, anddistal phalanxbones exist in each finger of thehuman hand, while themiddle phalanxbones exist in all fin-gers but the thumb. Image taken from35.

some small degree. The CMC joints of the index and middlefinger are fixed, while the ring and little finger CMC jointshave two DOFs each with a very small range of motion.

Since muscles usually have greater strength and possibleexcursion than is required to move the limbs, it is also im-portant to constrain the range of each DOF of the joints toavoid movement which is in reality prohibited by the form ofthe joints, by the joint capsules, and by ligaments. For eachDOF, we set an upper and a lower limit according to28.

3.1. Animation

Our reference hand model is animated exclusively throughmuscle contraction values given over time. These contrac-tion values are specified in key frames with an arbitrary tem-poral distribution. During simulation, the contraction valuesare interpolated using a smooth spline function, which isevaluated at discrete points in time according to the desiredrendering frame rate. At each point in time, the deformationof all muscles and the position of each bone is computedfrom the current contraction values. In turn, muscle and bonemovements are used to update the positions of those nodesof the mass-spring system that attach to muscles and bones,respectively. In the final step of each simulation cycle, the



Lagrangian equations of motion are integrated through timefor all nodes of the mass-spring system employing a Verletleapfrog integration method. The resulting displacements ofthe nodes attached to the skin mesh represent the deforma-tion of the skin surface. Details about the geometric musclemodel, the mass-spring system, and the integration methodcan be found in19.

3.2. Rendering

Rendering is currently performed using plain OpenGL func-tionality. Conceptually, it would make no difference to out-put key frames for a more sophisticated rendering engine.However, we found the possibility to instantly view anima-tions running in real-time worthwhile enough to accept thesomewhat degraded rendering quality. Since the focus of thispaper is on the geometry of our hand model and its deforma-tion during animation, we have omitted textures that distractfrom the actual underlying geometry.

4. A Hybrid Muscle Model

Muscle mechanics of the human hand have evolved to a de-gree of complexity that is unique among mammals. Thisevolutionary process took place in order to allow us to per-form fine motor manipulations and powerful manual workalike. Modeling and simulating all the subtle anatomical de-tails of the muscles of the human hand is an impractical ap-proach. In this section, we present ahybrid muscle model,which is flexible enough to cover the rich variety of musclemechanics in the human hand and yet is easy to use.

Our hybrid muscle model comprisespseudo musclesandgeometric muscles. Both of these muscle types are animatedexclusively through muscle contraction values within therange[0,1], where 0 means no contraction at all, and 1 meansfull contraction. Pseudo muscles directly control the rotationof the bones of the hand, while geometric muscles accountfor skin tissue deformation through physics-based simula-tion employing a mass-spring system that connects muscles,skin, and bones. Though each of these two muscle types canbe used individually, we typically use a combination of apseudo muscle and a geometric muscle to represent the ef-fects of an anatomical muscle in the human hand. For in-stance, theopponens pollicisis implemented by a pseudomuscle that rotates theproximal phalanxof the thumb and bya geometric muscle (muscle (2) in Figure 2 left) that bulgesthe skin. Table 1 lists the pseudo muscles of our system to-gether with their specific parameters (taken from8, 2).

For each frame during animation, all pseudo muscles areevaluated to update the position of the bones. The segmentsof geometric muscles that are attached to bones are trans-formed correspondingly. Next, the geometric muscles’ de-formation due to contraction is computed. Finally, the mass-spring system is updated to evaluate the resulting skin defor-mation.

4.1. Pseudo Muscles

Pseudo muscles are virtual muscles that convert a given con-traction valuec∈ [0,1] into rotation anglesϕk for each DOFof each jointJk they affect. Our model for this conversionis based on anatomical data and mechanical laws. However,our implementation is only valid under two assumptions:

1. The bones that are rotated are long bones, which are rep-resented as solid cylinders in our mathematical model.This is true for all bones of the human hand with the ex-ception of the wristbones.

2. When rotating a hierarchy of bones, the number of lev-els in that hierarchy has to be less or equal to three. Inour hand model, this is true for the fingers starting at theknuckles and for the thumb starting at thetrapezium.

The second restriction is solely due to computational effi-ciency. In Section 4.1.2 we describe a technique to efficientlycompute the rotation of chains of bones up to length three,which imposes the restriction above. To avoid this limitation,we are currently investigating the possibility to apply simi-larity transforms and the parallel axis (Steiner) theorem3

for transforming inertia tensors from one coordinate frameto another. Our preliminary results show that this approachremoves the limitation of the hierarchy depth at the cost ofmore expensive computations. In addition, the moment ofinertia needs to be stored as a tensor to allow for the ap-plication of the parallel-axis theorem for non-parallel rota-tion axes. In the approach described below, we simplify ourmodel by taking into account only the magnitude of torqueand moment of inertia. Treating these variables as vector-valued tensors would render the computational costs of eval-uating our model too high for real-time simulation.

Each pseudo muscle represents an anatomical musclewith a given maximum contraction force~Fmax. Relative val-ues of~Fmax for all relevant hand muscles are listed in7, thedirection of~Fmax has to be estimated from the layout and theattachment point of the muscle/tendon (see, e. g.,35). Thecontraction force of a muscle is not constant, but depends onthe current fiber length̀ of the muscle: a muscle that is ei-ther (passively) stretched or (actively) contracted has a lowercontraction force than a muscle at its fiber resting length`0. The nonlinear relationship between the contraction force~Fcontr and` is depicted in8. We fitted a quadratic curve tothe diagrams shown there and obtained the relationship:

~Fcontr(`) =[1− 4 · (`/`0−1.1)2

]·~Fmax.

In addition to the contraction force, each anatomical mus-cle exhibits a stretch force: a muscle that is (passively)stretched counteracts the stretch with a force~Fstretch, whichdepends on the muscle’s current fiber length`. Obviously,the stretch force is equal to zero if` < `0. Again, we fitted acurve to the diagrams shown in8 and obtained:

~Fstretch(`) =

{2.77· (`/`0−1)2 ·~Fmax , `≥ `0

0 , ` < `0.



According to 8, the inequation 0.6`0 ≤ `≤ 1.6`0 musthold, i.e. a muscle cannot become arbitrarily short or elon-gated. These upper and lower limits for` are — both in re-ality and in our model — usually never reached, since thecorresponding joints are constrained in their rotations (cf.Section 3). The current fiber length̀of a pseudo muscleis initialized to the resting length̀0 and updated by the arclength of the rotation, see below.

4.1.1. Rotation of a Single Bone

To see how our conversion model works, let us assume fornow that there is exactly one (cylindrical) bone that is rotatedabout the joint’s axis of rotation due to the contraction ofone pseudo muscle. Given a contraction valuec∈ [0,1], theresulting force the muscle exerts on the bone is:

~F = c ·~Fcontr(`) + ~Fstretch(`) . (1)

Let~r denote the lever arm of the force working point. Theamount of torque is computed as follows:

T = sgn(〈~a,~r×~F〉) · ‖~r×~F‖ , (2)

where〈~a,~r×~F〉 denotes the dot product of the rotation axisand the vector-valued torque. In addition, the following rela-tionship between torqueT, angular velocityω, and momentof inertiaJ holds:

T = J · dωdt

. (3)

Since the angular velocityω equals the first temporal deriva-tive of the rotation angleϕ:

ω =dϕdt

, (4)

we can discretize time and compute:

∆ω(3)= ∆t ·J−1 ·T ,

ωnew = ωold + ∆ω ,

∆ϕ(4)= ∆t ·ωnew,

` ← ` − ∆ϕ · ‖~r‖ .

Using this approach, we can compute the increment∆ϕof the rotation angle from a contraction valuec. The onlyunknown variable is the moment of inertiaJ. AlthoughJ isquite expensive to compute for an arbitrarily shaped body, itcan be easily computed for a solid cylinder of lengthl andmassm that is rotated about an axis orthogonal to its lengthaxis and passing through one of its ends13:

J =13·m· l2 . (5)

In our case, the lengthl is the length of the bone rotatedabout the joint’s axis. The massm, however, is the mass ofthe bone plus the mass of the tissue surrounding the bone.Values for this bone-plus-tissue mass can be found in4.

The above formulas do not consider friction yet. This

α

β

Ji

Ji−1

Ji−2

JiJi−1

Ji−2

Figure 3: Moments of inertia for a chain of bones.

means that a rotation, once it has started due to muscle con-traction, will not stop again. To take into account friction,we have to modify Equation (2) by subtracting the torque offriction:

T = sgn(〈~a,~r×~F〉) ·[‖~r×~F‖ − µ· |ωold|

], (2′)

whereµ is the coefficient of friction. In accordance to medi-cal literature, we useµ= 0.015.

Finally, we extend our mathematical model to allow for anarbitrary numbern of pseudo muscles that affect the rotationof the bone. Each pseudo musclei (i = 1, . . . ,n) exerts theforce:

~Fi = ci ·~Fcontr,i(`i) + ~Fstretch,i(`i) . (1′)

Since each muscle has its own lever arm~r i , the total amountof torque is given by:

T = sgn(〈~a, ~T〉) ·[‖~T ‖ − µ· |ωold|

], (2′′)

with~T =

n

∑i=1

~r i ×~Fi .

4.1.2. Rotation of Chains of Bones

For the derivation of the conversion formulas in the previousparagraph we assumed that the rotated bone is an end seg-ment, i.e. one of thedistal phalanges. If, however, we wantto rotate a chain of bones, for instance the threephalangesof a finger, the moment of inertiaJ depends on the positionof all bones in that chain.

Figure 3 depicts this situation: when rotating about jointJi , the moment of inertiaJi of the rotated bone is constantand can be computed according to Equation (5). The totalmoment of inertia for a rotation about jointJi−1 is com-posed ofJi−1 and the moment of inertia of the end segment.The latter, however, is not simplyJi in this case: the axis ofrotation does not pass through the end of the rotated end seg-ment as required for Equation (5). Thus the position of theend segment has to be transformed into the coordinate sys-tem ofJi−1 and the moment of inertiaJ∗i is computed bysumming up the squared distances of the transformed bonemesh vertices to the rotation axis multiplied by the mass of



Figure 2: Geometric muscles of our hand model. Left:adductor pollicis(1),opponens pollicis(2),1st dorsal interosseus(3),1stpalmar interosseus(4), 2nd palmar interosseus(5), 3rd palmar interosseus(6), opponens digiti minimi(7), flexor digiti minimibrevis(8), andabductor digiti minimi(9). Middle: the complex shape of muscles can be observed in this close-up view of (1)and (2) with all other muscles removed. Right: different muscle layers are set up automatically. The vertical muscles (7) and (8)slide freely below the horizontal muscle (9). See also color plate 8.

the bone. This computation becomes more and more costlywhen longer chains of bones are rotated.

Fortunately, the moment of inertiaJ∗i of the transformedbone depends only on the rotation angleα. Thus we pre-computeJ∗i (α) for a discrete set of angles (typically in stepsof five degrees) and store the arrayJ∗i [α] in the jointJi forfurther look-up. The total moment of inertiaJ for a rota-tion aboutJi−1 can thus be simply computed asJ = Ji−1 +J∗i [α]. Similarly, the total moment of inertia for a rotationaboutJi−2 is given by the sumJi−2 +J∗i−1[β]+J∗i [α][β].

Precomputing the moments of inertia for chains of boneswith more than three segments would require storing arraysof dimension three and more in the joints. To avoid this ex-haustive memory consumption, we restrict the computationof the moments of inertia to hierarchies with at most threelevels. For the rotation of the complete hand about the wristwe assume a constant moment of inertia of the hand.

4.2. Geometric Muscles

In our system, geometric muscles are embedded in betweenthe skin surface and the underlying bone structure. Geomet-ric muscles have an actual geometric shape assigned to them,which deforms and bulges during contraction. Springs areused to connect the surface of the muscle’s geometry to skinand bones. We have adopted the approach presented in19 forthe embedding of muscles into a mass-spring system. How-ever, some modifications of that approach were necessary toallow for a more complex muscle layout. In particular, wehave introduced the following changes:

• Each individual muscle has its own minimum and maxi-mum thickness. Rather thick muscles, e.g. theopponenspollicis (2), can thus be created as well as thin sheet mus-cles, e.g. theadductor pollicis(1), see Figure 2 (middle).• The distance between the skin surface and the surface of

the muscle can be set individually for each muscle to al-

Figure 4: Geometric muscles are created automatically froma muscle grid (shown in white) painted onto the skin surface.The bright dots mark the vertices of the skin mesh (shown asa wireframe), which are influenced by the muscle. See alsocolor plate 9.

Figure 5: Bulging of geometric muscles. Left: pseudo mus-cles are used to move the bones. Right: combining pseudoand geometric muscles results in additional skin deforma-tion due to bulging. See also color plate 10.

low for several layers of muscles (e.g. superficial and deeplayer) to be created automatically, see Figure 2 (right).• Muscles are allowed to attach to bones on both muscle

ends. Such types of muscles do not exist among the facialmuscles (with the exception of themasseter, which wasnot present in19), but are prevalent in the human hand.



Figure 6: Deformation of the reference hand model. Left: photograph of an individual’s hand including a ruler. Middle: positionof the 26 source feature points on the reference hand model. Right: resulting hand model after applying the warping function.See also color plate 11.

• Muscles may be assigned to several individual bones.Thus, individual segments of large or long muscles movewith the bones they are assigned to. Theabductor digitiminimi (9), for instance, is assigned to thecarpal bones(wristbone), and themetacarpalandproximal phalanxofthe little finger (cf. Figures 1 and 2 (left)).

Geometric muscles are created by interactively paintingmuscle gridsonto the skin surface (see Figure 4). From theshape of a muscle grid, the corresponding muscle is createdautomatically to fit in between skin and bone surfaces, tak-ing into account the individual muscle thickness and skindistance. This fitting process is similar to the one describedin 19, with the obvious extensions for individual muscle pa-rameters.

During contraction of a geometric muscle, the assignedgeometric shape deforms and bulges, see Figure 5. On themuscle surface, the attachment points of the springs con-necting the muscle to skin and bones move accordingly. Thedisplacement of these nodes exerts force on the mass-springmesh, which is updated to compute the corresponding skindeformation (cf. Section 3.1).

5. New Hand Models from Photographs

We employ a deformation technique based on feature pointsto warp the complete reference hand model to an individualhand model. Our approach is similar in spirit to the tech-nique proposed by Kähleret al. 20 for deformation of humanhead models. However, we do not require a 3D target handmodel to be obtained in a time-consuming scanning process.Instead, we use a simple photograph of the individual handto be modeled. The photograph merely needs to show a sim-ple ruler as depicted in Figure 6 (left). Since there are noother prerequisites for the photograph, low-cost consumercameras can be used for the acquisition.

First, we identify a small set of feature points in the in-put photograph. Our reference hand model is already taggedwith the same feature points by default. Section 5.1 presents

details about this step. Next, the complete structure of thereference hand model is deformed to match the shape ofthe individual hand from the photograph. The warp functionis set up using correspondence of feature points, see Sec-tion 5.2 for the details.

5.1. Feature Points

We use a small set of feature points on both the 3D referencehand model and in the 2D photograph. These feature pointscan be easily identified without anatomical knowledge. Inthe following, the feature points on the reference hand modelare denoted as thesource feature points, whereas those in theinput photograph are calledtarget feature points.

The reference hand model is equipped with 26 source fea-ture points by default (T = thumb, I = index finger, M = mid-dle f., R = ring f., L = little f.; cf. Figures 1 and 6 (middle)):

# location

5 tip of {T,I,M,R,L}4 distal interphalangeal joint(DIP) of {I,M,R,L}1 interphalangeal joint(IP) of T4 proximal interphalangeal joint (PIP) of

{I,M,R,L}5 metacarpophalangeal joint (MCP) of

{T,I,M,R,L}4 interdigital skin (between each pair of adjacent

fingers)2 radial and ulnar wrist (inner and outer side of

the wrist)1 head of ulna

Upon loading the input photograph into our system, theuser is asked to identify as many target feature points fromthe above set as possible in the photograph. Feature pointswhose positions are not clearly visible in the photograph canbe omitted. In our simulations, we obtained reasonable re-sults using a subset of only 16 feature points.



In addition to selecting the target feature points, a calibra-tion process is carried out to measure the size of the handin the input photograph. To this end, the user performs twomouse clicks at a known distance on the division scale ofthe ruler shown in the photograph, for instance at the points“0 cm” and “20 cm”. From their real world distance, the sys-tem automatically computes the scale of the photograph anduses this information to transform the positions of the tar-get feature points into the coordinate system of the referencehand model.

5.2. Warping the Reference Hand Model

Given two sets ofN corresponding source and target featurepoints, we are looking for a functionf that maps the sourcefeature pointssi to the target feature pointst i , (i = 1, . . . ,N).A natural solution to this interpolation problem is to employa radial basis function (RBF), see for instance10 for mathe-matical details. We use biharmonic basis functionsΦi(x) :=‖x−si‖2, which minimize bending energy12. This choice isin consonance with Bookstein’s suggestion to use thin-platesplines for the deformation of biological tissues5, 6.

Before setting up the functionf, we need to transform thetarget feature pointst i into the coordinate system of thesi .This is necessary to make the transformed hand model ap-pear in approximately the same place as the initial referencehand model. We transform the 2D pointst i into the fittingplane of the 3D pointssi using a rigid body transformation.The (uniform) scaling factor of this transformation is takenfrom the calibration step described in the previous section.Within the fitting plane of thesi , the target feature points arerotated to align the largest diameter of their set to the largestdiameter of the set{si}. Next, we assign to each target fea-ture point the height of its corresponding source feature pointabove/below the fitting plane. These heights are additionallyscaled by a (uniform) scaling factor obtained from the ra-tio of the largest diameters of the source and target featurepoints, respectively. Converting the 2D target feature pointsinto 3D points is essential to ensure that the transformedhand model will not be flattened but possesses a thicknessproportional to its overall size. Finally, we can set up ourradial basis warping function as described in standard litera-ture34, 10.

After setting up the warping functionf, we transform thecomplete structure of our reference hand model as follows:

1. Skin and bone meshes are transformed by applying thefunctionf to each vertex of the meshes. The connectivityof the meshes is not changed.

2. Joint positions and positions of feature points are trans-formed in the same way by direct application of the warp-ing function.

3. Geometric muscles are defined by muscle grids (cf. Sec-tion 4.2). To warp the muscles, only the control pointsof the muscle grids have to be transformed. The shape of

Figure 7: A typical computer graphics scientist’s hand pose.Side-by-side comparison of our hand model (top) with a pho-tograph of the hand that was scanned to build the model(bottom). To allow for a better comparison of the actual ge-ometries, textures and sophisticated shaders have been omit-ted from the hand model.

the warped muscles is computed automatically to fit inbetween the transformed skin and bone meshes.

4. Pseudo muscles do not have any spatial parameters thatneed to be transformed.

6. Results

We created several animation sequences to verify the effi-ciency and correctness of our hybrid muscle model. In spiteof some simplifications, we found our pseudo muscle modelto work very well and yield plausible results. Figure 7 showsa side-by-side comparison of our hand model with a photo-graph of the (female) hand that has been scanned to build thereference hand model. Clearly, our hybrid muscle model isable to correctly reproduce the pose of the hand. When cre-ating a new animation, however, estimating the right musclecontraction values (i.e. those that result in the desired move-ment of the fingers) is not always a straightforward process.We are currently working on an optimization process thatcomputes minimal energy muscle contractions for a giventarget position of the fingers to eliminate the process of spec-ifying individual muscle contraction values. Creating the ge-



ometric muscles is simple and fast: the complete set of geo-metric muscles shown in Figure 2 (left) has been created inless than an hour.

Using our hybrid muscle model, animations were runningat interactive frame rates of 5–10 fps on a 1.7 GHz Pen-tium Xeon machine with a GeForce3 graphics board. Themain bottleneck is the integration of the equations of mo-tion for the mass-spring network, which comprises approx-imately 1500 nodes and 4500 springs. Yet our integrationtechnique runs stable for a reasonable choice of stiffness pa-rameters for the skin model. In all our tests, the skin meshnever lost integrity after mass-spring simulation. In particu-lar, our method does not break the skin while moving frompose to pose, since the connectivity of the skin mesh is neverchanged.

Our deformation technique works reliably and is easy touse. We warped our reference hand model, which has beenbuilt from scan data of a female, to match the size and pro-portions of a male’s and a child’s hand. The interactive spec-ification of the target feature points in the photograph takesabout a minute. Since all components of the hand modelare transformed, the resulting hand model is instantly ani-matable. However, the animation parameters (i.e. the musclecontraction values) have to be adapted to the warped handmodel: different proportions of reference and target handmodel result in different torques, moments of inertia, andfinally different rotation angles.

7. Conclusion and Future Work

We presented an approach for the construction and ani-mation of human hand models with underlying anatomicalstructure. Our system is built around a reference hand model,which is animated using muscle contraction values. We in-troduced a hybrid muscle model that comprises pseudo mus-cles and geometric muscles. While pseudo muscles controlthe rotation of bones based on anatomical data and mechan-ical laws, the deformation of geometric muscles causes real-istic bulging of the skin tissue. As a result, the created anima-tions automatically exhibit anatomically and physically cor-rect behavior. In addition, we proposed a deformation tech-nique based on feature points to create individual hand mod-els from photographs. Warping the complete structure of thereference hand model results in deformed hand models thatare instantly animatable.

Although our system is working reliably and rather effi-ciently, there are further ways of improvement. Ideally, ge-ometric muscles should move the bones. This, however, in-volves modeling tendons as well as setting up a mass-springsystem where rigid objects (such as the bones) can be moveddue to spring forces. In addition, gravity should be includedinto our hybrid muscle model. While the effect of gravity isprobably negligible for the deformation of skin tissue in thehuman hand, it plays an important role for the computationof bone positions from given muscle contraction values.

To facilitate the creation of animation sequences, we areworking on an optimization process that computes minimalenergy muscle contractions for a given target position of thefingers. Moreover, it would be helpful to include collisiondetection among the parts of the hand. Given these two add-ons (optimization process and collision detection), graspingof external objects would be rather easy to implement.

Finally, we would like to automatically generate texturesfrom the photographs that are used to create individual handmodels. However, the parameterization of the skin mesh isnot trivial, if the texture is to be used for OpenGL rendering.

Acknowledgments

The authors would like to thank Kolja Kähler for his valu-able comments regarding the geometric muscle model andthe mass-spring system.

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anatomical name `0 joint(s) / DOF ||~r||

flexor carpi radialis 52 wrist flex. 17.5wrist abd. 10.5

palmaris longus 50 wrist flex. 21wrist abd. 1.5

flexor digitorum superf. index 72 wrist flex. 15wrist add. 3MCP index flex. 11.9MCP index add. 3PIP index flex. 6.2

flexor digitorum superf. middle 70 wrist flex. 15(analogous: wrist add. 3.0flexor digitorum superf. ring MCP middle flex. 11.9flexor digitorum superf. pinky) MCP middle add. 1.7

PIP middle flex. 6.2flexor carpi ulnaris 42 wrist flex. 18.5

wrist add. 15flexor digitorum prof. index 66 wrist flex. 6(analogous: wrist add. 13flexor digitorum prof. middle MCP index flex. 11.1flexor digitorum prof. ring MCP index add. 6flexor digitorum prof. pinky) PIP index flex. 7.9

DIP index flex. 4.1flexor pollicis longus 59 wrist flex. 5

wrist abd. 13CMC thumb add. 10CMC thumb opp. 10MCP thumb flex. 7.5IP thumb flex. 5.5

extensor carpi radialis longus 93 wrist ext. 10wrist abd. 21

extensor carpi radialis brevis 61 wrist ext. 13wrist abd. 24

extensor digitorum index 55 wrist ext. 13(analogous: wrist add. 7.5extensor digitorum middle MCP index ext. 8.6extensor digitorum ring MCP index abd. 0.2extensor digitorum pinky) PIP index ext. 2.8

DIP index ext. 2.2extensor digiti minimi 59 wrist ext. 13

wrist add. 7.5MCP pinky ext. 8.6PIP pinky ext. 2.6DIP pinky ext. 1.9

extensor carpi ulnaris 45 wrist ext. 6wrist add. 25

extensor pollicis longus 57 wrist ext. 9wrist abd. 10.5CMC thumb ext. 5CMC thumb add. 10MCP thumb ext. 2.5IP thumb ext. 2

extensor indicis 55 wrist flex. 1.4wrist abd. 0.4MCP index ext. 9MCP index add. 1.3PIP index ext. 2.6DIP index ext. 1.9

abductor pollicis longus 46 wrist flex. 7.4wrist abd. 24CMC thumb ext. 0.5

extensor pollicis brevis 43 wrist flex. 3.2wrist abd. 23CMC thumb ext. 4.5CMC thumb abd. 3MCP thumb ext. 3

anatomical name `0 joint(s) / DOF ||~r||

abductor digiti minimi 40 CMC pinky opp. 6MCP pinky abd. 4PIP pinky ext. 2.5DIP pinky ext. 2

flexor digiti minimi brevis 34 CMC pinky opp. 6MCP index flex. 4MCP index abd. 4

opponens digiti minimi 34 CMC pinky opp. 6abductor pollicis brevis 37 CMC thumb opp. 3.5

CMC thumb abd. 7.5MCP thumb flex. 1

flexor pollicis brevis 36 CMC thumb opp. 9CMC thumb add. 1MCP thumb flex. 7

opponens pollicis 24 CMC thumb opp. 4CMC thumb add. 8.5

adductor pollicis 36 CMC thumb opp. 4.5CMC thumb add. 9MCP thumb flex. 7

lumbrical I 55 MCP index flex. 9.3MCP index radial abd. 4.8PIP index ext. 1.8DIP index ext. 0.7

lumbrical II 66 MCP middle flex. 5MCP middle radial abd. 4.8PIP middle ext. 1.8DIP middle ext. 0.7

lumbrical III 60 MCP ring flex. 5MCP ring radial abd. 4.8PIP ring ext. 1.8DIP ring ext. 0.7

lumbrical IV 49 MCP pinky flex. 5MCP pinky radial abd. 4.8PIP pinky ext. 1.8DIP pinky ext. 0.7

palmar interosseus I 15 MCP index flex. 6.6MCP index add. 5.8DIP index ext. 2.6PIP index ext. 1.6

palmar interosseus II 15 MCP ring flex. 6.6MCP ring add. 5.8DIP ring ext. 2.6PIP ring ext. 1.6

palmar interosseus III 15 MCP pinky flex. 6.6MCP pinky add. 5.8DIP pinky ext. 2.6PIP pinky ext. 1.6

dorsal interosseus I 25 MCP index flex. 3.7MCP index abd. 6.1PIP index ext. 2.6DIP index ext. 1.6

dorsal interosseus II 25 MCP middle flex. 3.7MCP middle radial add. 6.1PIP middle ext. 2.6DIP middle ext. 1.6

dorsal interosseus III 25 MCP middle flex. 3.7MCP middle ulnar add. 6.1PIP middle ext. 2.6DIP middle ext. 1.6

dorsal interosseus VI 25 MCP pinky flex. 3.7MCP pinky abd. 6.1PIP pinky ext. 2.6DIP pinky ext. 1.6

Table 1: List of the pseudo muscles of our system with their fibre resting length`0 [mm], the affected joints, and the lever arms||~r|| [mm]. Abbreviations used in this table:superf.= superficialis, prof. = profundus, flex. = flexion, ext.= extension, abd.=abduction, add.= adduction, opp.= opposition.



Figure 8: Geometric muscles of our hand model. Left:adductor pollicis(1),opponens pollicis(2),1st dorsal interosseus(3),1stpalmar interosseus(4), 2nd palmar interosseus(5), 3rd palmar interosseus(6), opponens digiti minimi(7), flexor digiti minimibrevis(8), andabductor digiti minimi(9). Middle: the complex shape of muscles can be observed in this close-up view of (1)and (2) with all other muscles removed. Right: different muscle layers are set up automatically. The vertical muscles (7) and (8)slide freely below the horizontal muscle (9).

Figure 9: Geometric muscles are created automat-ically from a muscle grid (shown in blue) paintedonto the skin surface. The green dots mark thevertices of the skin mesh (shown as a wireframe),which are influenced by the muscle.

Figure 10: Bulging of geometric muscles. Left: pseudo muscles areused to move the bones. Right: combining pseudo and geometric mus-cles results in additional skin deformation due to bulging.

Figure 11: Deformation of the reference hand model. Left: photograph of an individual’s hand including a ruler. Middle:position of the 26 source feature points on the reference hand model (cf. Section 5.1). Right: resulting hand model after applyingthe warping function.


Construction and Animation of Anatomically Based Human Hand Models

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